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1,314,259,996,663 | arxiv | \section{Introduction}
\label{s1}
Hierarchical clustering of the large-scale structure of the universe induces supersonic flow motions of baryonic matter, which result in the formation of weak shocks with sonic Mach numbers $M_{\rm s} \lesssim 4$ in the hot intracluster medium (ICM) \citep[e.g.,][]{ryu2003,vazza2009,ha2018a}.
In particular, shocks associated with mergers of subcluster clumps have been observed in X-ray and radio \citep[e.g.,][]{brunetti2014, vanweeren19}. These ICM shocks are thought to accelerate cosmic ray (CR) protons and electrons via diffusive shock acceleration (DSA) \citep{bell1978,drury1983}.
Although the acceleration of relativistic electrons can be inferred from the so-called giant radio relics \citep[e.g.,][]{vanweeren19}, the presence of the CR protons produced by ICM shocks has yet to be established \citep[e.g.,][]{pfrommer2004,pinzke2010,zandanel2014,vazza2016,kang2018}.
The inelastic collisions of CR protons with thermal protons followed by the decay of $\pi^0$ produce diffuse $\gamma$-ray emission, which has not been detected so far \citep{ackermann2016}.
Previous studies using cosmological hydrodynamic simulations with some prescriptions for CR proton acceleration
suggested that the non-detection of $\gamma$-ray emission from galaxy clusters would constrain the acceleration efficiency
$\eta \lesssim 10^{-3}$ for ICM shocks with $2\lesssim M_{\rm s}\lesssim 5$ \citep[e.g.,][]{vazza2016};
the acceleration efficiency is defined in terms of the shock kinetic energy flux, as
$\eta\equiv E_{\rm CR,2} u_2/(0.5\rho_1 u_{\rm sh}^3)$ \citep{ryu2003}.
Hereafter, the subscripts $1$ and $2$ denote the preshock and postshock states, respectively.
And $\rho$ is the density, $u$ is the flow speed in the shock-rest frame, $u_{\rm sh}$ is the shock speed, and $E_{\rm CR,2}$ is the postshock CR proton energy density.
Proton injection is one of the key processes that govern the DSA acceleration efficiency.
In the so-called thermal leakage model, suprathermal particles in the tail of the postshock thermal distribution were thought to re-cross the shock from downstream to upstream and participate in the Fermi I process \citep[e.g.][]{malkov97, kang2002}.
Through hybrid simulations, however, \citet[][CS14a, hereafter]{caprioli2014a} showed that
in quasi-parallel ($Q_\parallel$, hereafter, with $\theta_{\rm Bn}\lesssim 45\degree$) shocks,
protons are injected through specular reflection off the shock potential barrier,
gaining energy via shock drift acceleration (SDA), and that the self-excitation of upstream turbulent waves is essential for
multiple cycles of reflection and SDA.
Here, $\theta_{\rm Bn}$ is the obliquity angle between the shock normal and the background magnetic field direction.
They considered relatively strong ($M_{\rm s}\gtrsim 6.5$) $Q_\parallel$-shocks in plasmas with $\beta \sim 1$,
where $\beta = P_{\rm gas}/P_{\rm B}$ is the ratio of the gas to magnetic pressures.
As CRs are accelerated to higher energies, the CR energy density increases in time
before the acceleration saturates at $E_{\rm CR,2}/(E_{\rm CR,2}+ E_{\rm th,2})\approx 0.06-0.13$ for $M_{\rm s}\approx 6.3-63$ (see Figure 3 of CS14a)
with the injection fraction, $\xi \sim 10^{-4}-10^{-3}$ (see Equation [\ref{inj}] below).
Here, $E_{\rm th,2}$ is the energy density of postshock thermal protons.
As a result, the postshock thermal distribution gradually shifts to lower temperatures as the CR power-law tail increases its extent (see Figure 1 of CS14a).
Moreover, CS14a found that in the immediate postshock region, the proton momentum distribution can be represented by three components: the Maxwellian distribution of thermal particles,
$f_{\rm th}(p)$, the CR power-law spectrum, $f_{\rm CR}(p)$, and the suprathermal `bridge' connecting smoothly $f_{\rm th}$ and $f_{\rm CR}$ (see Figure 2 of CS14a).
This suprathermal bridge gradually disappears as the plasma moves further downstream away from the shock, because the electromagnetic turbulence
and ensuing kinetic processes responsible for the generation of suprathermal particles decrease in the downstream region.
Far downstream from the shock, the transition from the Maxwellian to CR distributions occurs
rather sharply at the so-called injection momentum, which can be parameterized as
$p_{\rm inj} \approx Q_{\rm i}\ p_{\rm th,p}$, where $p_{\rm th,p} = \sqrt{2m_p k_B T_2}$ is the postshock thermal proton momentum and $Q_{\rm i}\sim 3-3.5$ is the injection parameter.
Here, $T_2$ is the temperature of postshock thermal ions, $m_p$ is the proton mass, and $k_B$ is the Boltzmann constant.
They suggested that the CR energy spectrum can be modeled by the DSA power-law attached to the postshock Maxwellian at $p_{\rm inj}$,
although their hybrid simulations revealed a picture that is quite different from the thermal leakage injection model.
Later, \citet[][CPS15, hereafter]{caprioli2015} presented a minimal model for proton injection that accounts for quasi-periodic shock reformation
and multicycles of reflection/SDA energization, and predicted the CR spectrum consistent with the hybrid simulations of CS14a.
Recently, \citet[][HRKM18, hereafter]{ha2018b} studied, through Particle-in-Cell (PIC) simulations, the early acceleration of CR protons in weak ($M_{\rm s} \approx 2 - 4$) $Q_\parallel$-shocks in hot ICM plasmas where $\beta \sim 100$ \citep[e.g.,][]{ryu2008}.
In the paper, they argued that only supercritical $Q_\parallel$-shocks with $M_{\rm s}\gtrsim2.25$ develop overshoot/undershoot oscillations in their structures, resulting in a significant amount of incoming protons being reflected at the shock and injected into the DSA process.
Subcritical $Q_\parallel$-shocks with $M_{\rm s}\lesssim2.25$, on the other hand, have relatively smooth structures,
and hence the preacceleration and injection of protons into DSA are negligible.
Thus, it was suggested that ICM $Q_\parallel$-shocks may accelerate CR protons only if $M_{\rm s}\gtrsim2.25$.
{
Although the simulations followed only to the very early stage of DSA where the maximum ion momentum reaches up to $p_{\rm max}/m_ic \sim 0.5$ ($m_i$ is the reduced ion mass\footnote{Throughout the paper, we differentiate $m_i$ from $m_p$, because the effects of the reduced mass ratio, $m_i/m_e$ ($m_e$ is the electron mass), in PIC simulations remain to be fully understood. In the simulations of HRKM18, for example, $m_i=100-800\ m_e$ was used.}),
HRKM18 attempted to quantify proton acceleration at ICM $Q_\parallel$-shocks.
The simulated CR spectrum indicated the injection parameter of $Q_{\rm i}\approx 2.7$, which led to a rather high injection fraction, $\xi \approx 2\times 10^{-3} - 10^{-2}$, for shocks with $M_{\rm s} = 2.25 - 4$.
If we simply extrapolate this injection fraction to the relativistic regime of $p_{\rm max}/m_ic\gg 1$,
the ensuing DSA efficiency would be rather high, $\eta > 0.01$,
which is in strong disagreement with the existing observations of $\gamma$-rays from galaxies clusters.}
{
In a `fluid-version' of numerical studies of DSA, on the other hand, the time-dependent diffusion-convection equation for the isotropic part of the momentum distribution function, $f_{\rm CR}(p)$, is solved, adopting
a Bohm-type spatial diffusion coefficient ($\kappa\propto p$) and a `macroscopic' prescription for thermal leakage injection ($\tau_{\rm esp}$) \citep[e.g.,][]{kang2002}.
Previous studies using this approach managed to follow the
evolution of CR proton spectrum into the relativistic energies of up to $p_{\rm max}/m_pc \sim 50$ for shocks with a wide range of sonic Mach numbers \citep[e.g.,][]{kang2005}.
They showed that, as the CR pressure increases in time, the subshock weakens and $T_2$ decreases accordingly,
resulting in the gradual reduction of the injection rate and $f_{\rm CR}(p_{\rm inj})$ [see Figure 5 of \citet{kang2005}].
This leads to the decrease of the injection fraction $\xi(t)$ with time,
although the postshock CR pressure reaches an approximate time-asymptotic value [see Figure 6 of \citet{kang2002}].
These results are consistent with those of the hybrid simulations described above.}
{
Previously, \citet{kang2010} considered an analytic model for $f_{\rm CR}(p)$ in the test-particle regime of DSA for weak ICM shocks.
They suggested that the test-particle solution of $f_{\rm CR}(p)$ could be valid only if $Q_i\gtrsim 3.8$, which results in
the injection fraction $\xi \lesssim 10^{-3}$ and the CR pressure $P_{\rm CR,2}/\rho_1 u_{\rm sh}^2 < 0.1$.
In that study, however, the changes of $T_2(t)$ and $\xi(t)$ with the increase of $p_{\rm max}$ were not included self-consistently,
because $Q_{\rm i}$, although a free parameter, has a fixed value, and $T_2$ was estimated simply from the Rankine-Hugoniot relation, relying on the test-particle assumption.
Hence, the model failed to incorporate the full aspect of DSA observed in the previous simulations.}
{
Based on the earlier studies of DSA using hybrid, PIC, and fluid simulations, we here propose an improved analytic model
that is designed to approximately emulate the CR proton spectrum of DSA for given shock parameters.
The basic formulation is still based on the test-particle solution with a thermal leakage injection recipe with a free parameter, $Q_{\rm i}$,
as in \citet{kang2010}.
The main improvement is, however, the inclusion of the reduction of the postshock thermal energy density due to the transfer of the shock energy
to the CR population in a self-consistent manner; also the model considers a more realistic range of $Q_{\rm i} \approx 3.0 - 3.5$ that reflects the results of the hybrid simulations of
CS14a and CPS15.
In the next section, we first review what has been learned about proton injection and acceleration at $Q_\parallel$-shocks from recent plasma simulations.}
In Section \ref{s3}, we describe our analytic DSA model for the CR proton spectrum produced at weak $Q_\parallel$-shocks,
along with the injection fraction and acceleration efficiency that characterize the DSA of CR protons.
A brief summary follows in Section \ref{s4}.
\begin{figure*}[t]
\vskip -0.4 cm
\hskip -0.6 cm
\centerline{\includegraphics[width=1.2\textwidth]{f1.pdf}}
\vskip -0.1 cm
\caption{\label{f1}
(a) Postshock energy spectrum, $dN/d\gamma$, of ions with $m_i=100m_e$, taken from PIC simulations
for the ICM shock of $M_s=3.2$ with $\theta_{\rm Bn}=13^{\circ}$, $\beta=100$, and $T_1=8.6$ keV ($10^8$ K).
For $\Omega_{\rm ci}t \approx 94$~(red), the simulation data reported in HRKM18 are adopted,
while for $\Omega_{\rm ci}t \approx 240$ (blue), those from the new extended simulation described in Section \ref{s2.2} are used.
The red and blue dashed lines show the fits for the respective spectra (solid lines) to Maxwellian and test-particle power-law forms.
The vertical dotted magenta line marks the injection energy, $\gamma_{\rm inj}$, where the two fitting forms cross each other.
(b) Time evolution of the injection fraction $\xi(t)$, calculated with the postshock energy spectra for the shock model shown in panel (a). The red and blue arrows denote the points for $\Omega_{\rm ci}t \approx 94$, and 240, respectively.}
\end{figure*}
{
\section{Implications from Plasma Simulations}
\label{s2}
Although the structure and time variation of collisionless shocks are primarily governed by the dynamics of reflected protons and the waves excited by them in the foreshock region,
the roles of electron kinetic processes in proton injection to DSA has not yet been fully explored \citep[e.g.,][]{balogh13}.
Only PIC simulations can follow from first principles various microinstabilities and wave-particle interactions due to ion and electron kinetic processes.
Owing to greatly disparate time and length scales of ion and electron processes, however, the runs of PIC simulations are limited to only several $\times10^2\ \Omega_{\rm ci}^{-1}$, depending on $m_i/m_e$, $\beta$, and the dimension of simulations.
Here, $\Omega_{\rm ci}^{-1} = {m_i c}/{eB_0}$, is the ion cyclotron period where $c$ is the speed of light, $e$ is the electron charge,
and $B_0$ is the background magnetic field strength.
Typically, the injection and early acceleration of protons can be followed up to the maximum momentum of $p_{\rm max} /p_{\rm th,i}\sim30$ ($p_{\rm th,i} = \sqrt{2m_i k_B T_2}$) in PIC simulations \cite[e.g.,][HRKM18]{park2015}.
Hybrid simulations, in which electrons are modeled as a charge-neutralizing fluid,
can be run to several $\times10^2-10^3\ \Omega_{\rm cp}^{-1}$ (where $\Omega_{\rm cp}^{-1} = {m_p c}/{eB_0}$), neglecting details of electron kinetic processes.
Yet they can follow proton acceleration only up to $p_{\rm max}/m_pu_{\rm sh}\sim30$ or so (e.g., CP14a).
With currently available computational resources, both PIC and hybrid simulations can only study the early development of suprathermal and nonthermal protons.
Thus, it would be a rather challenging task to extrapolate what we have learned about DSA from existing plasma simulations to the relativistic regime of $p_{\rm max}/m_pc\gg 1$.
\subsection{Hybrid Simulations}
\label{s2.1}
As discussed in the introduction, the injection and acceleration of protons at $\beta\approx 1$ $Q_\parallel$-shocks with $M_{\rm s}\gtrsim 6.3$
were studied extensively through 2D hybrid simulations (CS14a and CPS15).
A small fraction of incoming protons can be injected to DSA after undergoing two to three cycles of SDA, followed by reflection off the shock potential drop.
In addition, at low-$\beta$ ($\beta\lesssim1$) shocks, the proton reflection can be facilitated by the magnetic mirror force due to the compression of
locally perpendicular magnetic fields in upstream MHD turbulence,
which are self-excited by back-streaming protons \citep[e.g.,][]{sundberg2016}.
The efficiency of proton injection could be quantitatively different at weak ICM shocks with $\beta\sim100$,
because the shock potential drop is smaller at lower $M_{\rm s}$ shocks and the magnetic mirror force is weaker in higher $\beta$ plasmas.
\citet[][CS14b, hereafter]{caprioli2014b}, on the other hand, showed that the magnetic field amplification due to resonant and
non-resonant streaming instabilities increases with the Alfv\'en Mach number, $M_{\rm A}\approx \beta^{1/2} M_{\rm s}$.
Hence, the level of upstream turbulence is expected to be higher for higher $\beta$ shocks at a given $M_{\rm s}$.
Therefore, higher $\beta$ could have two opposite effects on the efficiency of proton injection, i.e., weaker magnetic mirror but stronger turbulence in the foreshock.
Unfortunately, so far hybrid simulations for high-$\beta$ ($\beta\gg1$) shocks have not been published in the literature yet.
CPS15 suggested that the proton injection at weak shocks may be different from their findings for strong shocks in the following senses:
(1) the overshoot in the shock potential is smaller at weaker shocks, leading to a smaller reflection fraction at each confrontation with the shock,
(2) the fractional energy gain at each SDA cycle is smaller, so more SDA cycles are required for injection,
(3) the levels of turbulence and magnetic field amplification are weaker.
As a result, the proton injection and acceleration efficiencies should be smaller at weaker shocks.
According to Figure 3 of CS14a, for the $M_{\rm s}\approx 6.3$ shock ($M=5$ in their definition), the DSA efficiency is $\eta \approx 0.036$,
so a smaller $\eta$ is expected for ICM shocks with $M_{\rm s}\lesssim 4$.
Moreover, CS14b showed in their Figure 9 that the normalization (amplitude) of postshock $f_{\rm CR}$
decreases as $p_{\rm max}(t)$ increases with time.
We interpret that this trend is caused by the increase in the number of SDA cycles required for injection to DSA,
because the subshock weakens gradually due to the CR feedback, and so the energy gain per SDA cycle is reduced.
Considering that the ratio of $p_{\rm max}/p_{\rm th,p}$ reaches only to $\sim 30$ in these hybrid simulations,
the normalization of $f_{\rm CR}$ may continue to decrease as the CR spectrum extends to the relativistic region with $p_{\rm max}/m_p c\gg 1$.
\subsection{Particle-in-cell Simulations}
\label{s2.2}
HRKM18 explored for the first time the {\it criticality} of high-$\beta$ $Q_\parallel$ shocks and showed that
protons can be injected to DSA and accelerated to become CRs only at supercritical shocks with $M_{\rm s} \gtrsim 2.25$.
Figure 7 of HRKM18 showed that the shock criticality does not sensitively depend on $m_i/m_e$ and numerical resolution,
but the acceleration rate depends slightly on $\beta$.
As mentioned before, turbulence is excited more strongly for higher $\beta$ cases due to higher $M_{\rm A}$.
But the reflection fraction is smaller for higher $\beta$ due to weaker magnetic mirror forces, leading to lower reflection fraction and
lower amplitude of $f_{\rm CR}$ near $p_{\rm inj}$.
In order to get a glimpse of the long-term evolution of the CR proton spectrum,
we extend the 1D PIC simulation reported in HRKM18 from $\Omega_{\rm ci} t_{\rm end}= 90$ to 270
for the model of $M_{\rm s}=3.2$, $\theta_{\rm Bn}=13^{\circ}$, $m_i/m_e=100$, $\beta=100$, and $T_1=8.6$ keV ($10^8$ K).
Details of numerical and model setups can be found in HRKM18 (see their Table 1).
The main change is that a different computation domain, $[L_x, L_y] = [3\times 10^4,1]~(c/w_{\rm pe})^2$, is adopted here
in order to accommodate the longer simulation time.
Because of severe computational requirements, in practice, it is difficult to extend this kind of PIC simulations to a much larger box for a much longer duration.
In this simulation the average velocity of ions is $\sqrt{18.36}$ times higher than that of real protons for the given temperature.
Figure \ref{f1} shows the time evolution of the postshock energy spectra of ions, $dN/d\gamma$ (where $\gamma$ is the Lorentz factor),
and the injection fraction, $\xi(t)$ [see Eq. (11) of HRKM18].
We adopt the simulation data of HRKM18 for $\Omega_{\rm ci}t \approx 94$~(red), while the data from the new extended simulation is used for
$\Omega_{\rm ci}t \approx 240$~(blue).
The region of $(1.5 - 2.5) r_{L,i}$ behind the shock is included, where $r_{L,i}$ is the ion Larmor radius defined with the incoming flow speed.
Note that the spectrum near the energy cutoff might not be correctly reproduced due to the limited size of the simulation domain.
We notice the following features in Figure \ref{f1}(a) :
(1) the postshock temperature decreases slightly with time,
(2) the injection parameter, $Q_{\rm i}=p_{\rm inj}/p_{\rm th,i}$, increases from $\sim2.7$ to $\sim3.0$
as the time increases from $\Omega_{\rm ci} t \approx 90$ to 240, and
(3) the amplitude of $dN/d\gamma (\gamma_{\rm inj})$ decreases gradually.
Figure \ref{f1}(b) shows the resulting gradual decrease of $\xi(t)$, which may continue further in time.
As in HRKM18, a somewhat arbitrary value of $p_{\rm min}=\sqrt{2}p_{\rm inj}$ is adopted (see the next section for a further discussion).
We interpret the bump in the evolution of $\xi(t)$ near $\Omega_{\rm ci} t \approx 210$ as a consequence of shock reformation.
\begin{figure*}[t]
\vskip -0.8cm
\hskip 0 cm
\centerline{\includegraphics[width=1\textwidth]{f2.pdf}}
\vskip -6.2 cm
\caption{\label{f2}
Proton distribution function, $f(p)p^4$, calculated with Equations (\ref{finj})-(\ref{fN}).
Panel (a): $f(p)p^4$ in a $M_{\rm s}=3.2$ shock with $Q_{\rm i,0}=$ 3.0 (blue line), 3.3 (black line), and 3.5 (red line), when the maximum momentum is $p_{\rm max} \gg p_{\rm inj}$.
The vertical dashed line shows the injection momentum, $p_{\rm inj}$, with $Q_{\rm i,0}=3.3$.
Panels (b)-(d): Change of $f(p)p^4$ in $M_{\rm s}=2.5$, 3.2, and 4.0 shocks with $Q_{\rm i,0}=3.5$, as $p_{\rm max}$ increases. Here, $T_1=10^8$ K.
{\color{red}The DSA test-particle slope, $q_{\rm tp}$, is given in each panel.}
Due to the energy transfer to the CR component, the temperature reduction factor, $R_{\rm T}$, decreases.
Hence, while $p_{\rm inj}$ is fixed, the injection parameter, $Q_{\rm i}=Q_{\rm i,0}/\sqrt{R_{\rm T}}$, increases, leading to the reduction of the normalization factor, $f_{\rm N}$. }
\end{figure*}
\section{Analytic Model for CR Proton Spectrum}
\label{s3}
\begin{figure*}[t]
\vskip -0.8 cm
\hskip -0.2 cm
\centerline{\includegraphics[width=1.05\textwidth]{f3.pdf}}
\vskip -6.5 cm
\caption{\label{f3}
Change of the injection fraction, $\xi$, the temperature reduction factor, $R_{\rm T}$,
the postshock CR energy fraction, $E_{\rm CR,2}/E_{\rm sh}$, and the CR acceleration efficiency, $\eta$,
as $p_{\rm max}$ increases.
Here, $T_1=10^8$ K, $p_{\rm min}=p_{\rm inj}$, $Q_{\rm i,0}=3.3$ (dashed lines) and 3.5 (solid lines) are adopted.
As $R_{\rm T}$ decreases, the injection parameter increases as $Q_{\rm i} =Q_{\rm i,0}/\sqrt{R_{\rm T}}$,
which results in the reduction of $f_{\rm N}$ as in Equation (\ref{fN}).}
\end{figure*}
{
The analytic model presented here inherits the test-particle DSA model with a thermal leakage injection recipe, which was suggested by \citet{kang2010}.
It describes the downstream CR proton spectrum, $f_{\rm CR}(p)$, for weak shocks.}
For the preshock gas with the density, $n_1$, and the temperature, $T_1$,
the postshock vales, $n_2$, and $T_{2,0}$\footnote{Here, $T_{2,0}$ denotes the temperature of the thermal gas when the postshock CR energy density, $E_{\rm CR,2}$, is negligible, reserving $T_2$ for the cases of non-negligible $E_{\rm CR,2}$.}, can be calculated from the Rankine–Hugoniot jump condition.
For example, the shock compression ratio is given as $r = n_2/n_1 = (\gamma_{\rm g} + 1)/(\gamma_{\rm g} - 1 + 2/M^2_{s})$, where $\gamma_{\rm g}=5/3$ is the gas adiabatic index.
{
Following \citet{kang2010}}, we parameterize the model as follows:
(1) The CR proton spectrum follows the test-particle DSA power-law, as $f_{\rm CR}(p)\propto p^{-q}$, where $q=3r/(r-1)$.
(2) The transition from the postshock thermal to CR spectra occurs at the injection momentum
\begin{equation}
p_{\rm inj}= Q_{\rm i} \cdot p_{\rm th,p},
\label{IP}
\end{equation}
where $Q_{\rm i}$ is the injection parameter.
{The main improvement here is that
the postshock temperature, $T_2$, decreases slightly from $T_{2,0}$, and hence $p_{\rm th,p}$ does too,
as the fraction of the shock energy transferred to CRs increases.}
Our model leads to the following form of the CR proton spectrum,
\begin{equation}
f_{\rm CR}(p) \approx \psi \cdot f_{\rm N} \left({p \over p_{\rm inj}}\right)^{-q} \exp \left[-\left({p \over p_{\rm max}}\right)^2\right].
\label{finj}
\end{equation}
Here, the maximum momentum of CR protons, $p_{\rm max}$, increases with the shock age \citep[e.g.,][]{kang2010}.
The normalization factor can be approximated as
\begin{equation}
f_{\rm N} = {n_2 \over \pi^{1.5}} p_{\rm th,p}^{-3} \exp(-Q_{\rm i}^2),
\label{fN}
\end{equation}
assuming the CR power-law spectrum is hinged to the postshock Maxwell distribution at $p_{\rm inj}$.
Therefore, in our model, $Q_{\rm i}$ is the key parameter that controls $f_{\rm N}$.
In addition, we introduce an additional parameter, $\psi\sim 1$, to accommodate any uncertainties in determining the value of $Q_{\rm i}$ and the resulting amplitude, $f_{\rm N}$.
Throughout this paper, however, $\psi= 1$ is used.
Figure \ref{f2}(a) shows the model spectrum, $f_{\rm CR}(p)$, calculated with Equations (\ref{finj})-(\ref{fN}), which illustrates the transition from the thermal to nonthermal CR spectra at $p_{\rm inj}$.
{
The PIC simulation described in section \ref{s2.2} indicates $Q_{\rm i} \approx 3$ when $p_{\rm max}/m_i c \approx 0.5$, but $Q_{\rm i}$ may further increase for $p_{\rm max}/m_i c \gg 1$, as noted above.
On the other hand, hybrid simulations for strong shocks of $\beta\approx 1$ (CS14a, CPS15) showed that it is expected to range as $Q_{\rm i} \approx 3.0-3.5$.
As discussed in section \ref{s2.1}, higher $\beta$ could have two opposite effects on proton reflection, weaker magnetic mirror but stronger upstream turbulence.
So it is difficult to make quantitative predictions on the long-term evolution of $Q_i$ in high-$\beta$ shocks without performing plasma simulations of very long duration.
Here, we will consider the range of $Q_{\rm i}=3.3-3.5$ as an educated guess from the previous plasma simulation.
Moreover, in our analytic model, $Q_{\rm i} < 3.3$ would give the DSA efficiency of $\eta \gtrsim 0.01$ for $3\lesssim M_{\rm s} \lesssim 5$ (see Figure \ref{f4} below),
which would be incompatible with the non-detection of $\gamma$-ray emission from galaxy clusters \citep{vazza2016}.}
\begin{figure*}[t]
\vskip -0.8 cm
\hskip -0.2 cm
\centerline{\includegraphics[width=1.05\textwidth]{f4.pdf}}
\vskip -8.4 cm
\caption{\label{f4}
The injection fraction, $\xi$, the temperature reduction factor, $R_{\rm T}$, the postshock CR energy fraction,
$E_{\rm CR,2}/E_{\rm sh}$, and the CR acceleration efficiency, $\eta$, as a function of $M_{\rm s}$,
for $p_{\rm min}=p_{\rm inj}$ and $p_{\rm max}=10^5 m_p c$. Here, $T_1=10^8$ K.
The black and red filled circles connected with solid lines are the results for $Q_{\rm i,0}=3.3$ and $3.5$, respectively.
The two points for $M_{\rm s}=1.5$ and $2.0$ are connected with the dotted lines,
because subcritical shocks with $M_{\rm s}<2.25$ may not preaccelerate
and inject thermal protons to the full DSA process according HRKM18.
The open triangles represent the values calculated with $p_{\rm min}=780~{\rm MeV}/c$.
}
\end{figure*}
From the model $f_{\rm CR}(p)$ in Equation (\ref{finj}), we calculate the injection fraction of CR protons by
\begin{equation}
\xi \equiv \frac{4\pi}{n_2}\int_{p_{\rm min}}^{p_{\rm max}} f_{\rm CR}(p) p^2 dp,
\label{inj}
\end{equation}
as in HRKM18.
The postshock CR energy density is estimated by
\begin{equation}
E_{\rm CR,2} = 4 \pi c \int_{p_{\rm min}}^{p_{\rm max}} (\sqrt{p^2+ (m_pc)^2}-m_pc) f_{\rm CR}(p) p^2 dp.
\label{ECR}
\end{equation}
In the case of very weak shocks, where the CR spectrum is dominated by low energy particles,
both $\xi$ and $E_{\rm CR,2}$ depend sensitively on the lower bound of the integrals, $p_{\rm min}$ \citep[e.g.,][]{pfrommer2004}.
We here adopt $p_{\rm min}\approx p_{\rm inj}$ for fiducial models, while $p_{\rm min}=780\ {\rm MeV}/c$, the
threshold energy of $\pi$-production reaction, will be considered as well for comparison.
As mentioned above, $E_{\rm CR,2}$ may increase, as $f_{\rm CR}(p)$ extends to higher $p_{\rm max}$, resulting in the decrease of the postshock gas temperature from $T_{2,0}$ to $T_2$
{(see Figure 5 of \citet{kang2005} and Figure 1 of CS14a).}
Thus, we introduce the temperature reduction factor,
\begin{equation}
R_{\rm T}= {{E_{\rm th}(T_2,0) - E_{\rm CR,2}}\over {E_{\rm th}(T_{2,0})} }.
\label{RN}
\end{equation}
Then, $T_2= R_{\rm T} T_{2,0}$ is the reduced postshock temperature.\footnote{The fraction of thermal particles that becomes CR protons is assumed to be small, i.e., $\xi \ll 1$.}
{CPS15 suggested that when the postshock CR energy density approaches to
$E_{\rm CR,2} \approx 0.1 E_{\rm sh}=0.1 (\rho_1 u_{\rm sh}^2/2)$, the subshock weakens substantially,
which suppresses the proton reflection and injection.
Hence, the normalization of $f_{\rm CR}$ is expected to decrease as $p_{\rm max}$ increases.}
Our model is designed to mimic such a behavior by finding the self-consistent postshock thermal distribution with a lower temperature,
while $p_{\rm inj}$ is assumed to be fixed.
Then, the injection parameter increases as $Q_{\rm i} = Q_{\rm i,0}/\sqrt{R_{\rm T}}$, where $Q_{\rm i,0}$ is the initial value, leading to smaller values of $f_{\rm N}$.
Note that $p_{\rm inj}$ at shocks with different parameters ($M_s$, $\theta_{\rm Bn}$, and $\beta$) is controlled by a number of complex kinetic process, and hence should be studied through long-term plasma simulations, beyond the current computational capacity.
Considering that the proton injection into DSA is yet to be fully understood,
fixing $p_{\rm inj}$ while slightly increasing $Q_{\rm i}$ in our model should be regarded as a reasonable assumption.
Figure \ref{f2}(a) shows the model spectrum, including that of the self-consistent thermal distribution, in a $M_{\rm s}=3.2$ shock for $Q_{\rm i,0}=3.0-3.5$;
the spectrum depends on the adopted value of $Q_{\rm i,0}$.
Panels (b)-(d) illustrate the change of the model spectrum as $p_{\rm max}$ increases in shocks with $M_{\rm s} = 2.5,$ 3.2, and 4.0, respectively.
As $p_{\rm max}$ and also $E_{\rm CR,2}$ increase,
the Maxwellian part shifts to slightly lower $T_2$, and $R_{\rm T}$ decreases accordingly.
Because $p_{\rm inj}$ is assumed to be fixed, $Q_{\rm i}$ increases and thus the normalization factor $f_{\rm N}$ decreases in our model.
Figure \ref{f3} shows the change of $\xi$, $R_{\rm T}$, $E_{\rm CR,2}/E_{\rm sh}$, and $\eta$, calculated with Equations (\ref{finj})-(\ref{RN}), as $p_{\rm max}$ increases for {$Q_{\rm i,0}=3.3$ (dashed lines) and 3.5 (solid lines) in shocks with $M_{\rm s}=2.25-4.0$.}
The CR acceleration efficiency is related to the postshock CR energy density, as $\eta = E_{\rm CR,2}/r E_{\rm sh}$.
Figure \ref{f3}(b) plots how $R_T$ decreases, as $p_{\rm max}$ increases.
{
The injection fraction, $\xi$, increases with increasing $p_{\rm max}$ during the early acceleration phase, but decreases for $p_{\rm max}/m_pc \gg 1$.
The latter behavior results from the gradual reduction of $f_{\rm CR}(p_{\rm inj})$, which is caused by the self-adjustment of the shock structure, that is, the
cooling of the postshock thermal protons, the growing of the precursor, and the weakening of the subshock due to the dynamical feedback of the CR pressure.}
$E_{\rm CR,2}/E_{\rm sh}$ and $\eta$, on the other hand, monotonically increase and approach to asymptotic values for $p_{\rm max}/m_pc \gtrsim 10^2$.
Figure \ref{f4} shows the asymptotic values of those quantities as a function of $M_{\rm s}$ (filled circles and lines)
for $Q_{\rm i,0}=3.3$ (black) and 3.5 (red), which would cover the most realistic range for ICM shocks (CS14a).
As mentioned in the introduction, HRKM18 showed that ICM $Q_{\parallel}$-shocks with
$M_{\rm s}<2.25$ may not inject protons into the DSA process, resulting in inefficient CR proton acceleration.
We here include the $M_{\rm s}= 1.5$ and $2.0$ cases (connected with dotted lines) for illustrative purposes, showing the values estimated with our model.
{Note that the asymptotic value of $\xi(M_{\rm s})$ decreases with increasing $M_{\rm s}$ for supercritical shocks with $M_{\rm s}\ge 2.25$.
This behavior is opposite to the relation, $\xi \propto M_{\rm s}^{1.5}$, during the very early acceleration stage of the PIC simulations reported in HRKM18.
In those PIC simulations, $p_{\rm max}/m_pc \lesssim 0.5$, so the CR feedback effect is not very significant.
However, our analytic model is designed to take account for the dynamic feedback of the CR pressure to the shock structure when $p_{\rm max}/m_pc \gg 1$,
so $\xi$ could be smaller at higher $M_{\rm s}$.
With the adopted value of $Q_{\rm i,0}=3.3-3.5$, $E_{\rm CR,2}/E_{\rm sh}< 0.1$, so the test-particle assumption should be valid.
The acceleration efficiency increases with $M_{\rm s}$ and is close to $\eta\approx0.01-0.02$ in the range of $M_{\rm s}=3-5$.
Obviously, if the injection parameter is larger than what the hybrid simulations of CS14a indicated, that is, $Q_{\rm i,0}>3.5$, then DSA would be even less efficient.}
In the studies of $\gamma$-ray emission from simulated galaxy clusters, the lower bound of $f_{\rm CR}$
is often taken as $p_{\rm min}=780~{\rm MeV}/c$, as noted above.
The open triangles in Figure \ref{f4} show $E_{\rm CR,2}/E_{\rm sh}$ and $\eta$ calculated with this $p_{\rm min}$,
otherwise adopting the same analytic spectrum given in Equations (\ref{finj})-(\ref{fN}).
For $M_{\rm s}=2.25$, the acceleration efficiency with $p_{\rm min}=780~{\rm MeV}/c$ is smaller by a factor of 3.3
than that with $p_{\rm min}=p_{\rm inj}$.
But the two estimations are similar for $M_{\rm s}\gtrsim 4$.
The efficiency with $p_{\rm min}=780~{\rm MeV}/c$ is $\eta\sim0.01$ in the range of $M_{\rm s}=3-5$, while $\eta\sim 10^{-3}$ for $M_{\rm s}=2.25$.
If this result is extended to the case of $M_{\rm s} \sim 6$, $\eta$ would be still close to 0.01, which is about three times smaller than the efficiency reported by CS14a
(i.e., $\eta \approx 0.036$ at $M_{\rm s} \sim 6.3$).
Note that this estimate is somewhat larger than the upper limit of $\eta\lesssim 10^{-3}$, quoted to be consistent with the non-detection of $\gamma$-ray emission from galaxy clusters by \citet{vazza2016}.
On the other hand, as HRKM18 shown, $\eta$ may be very small and negligible for shocks with $M_{\rm s}<2.25$, for which the fraction of the total shock dissipation in the ICM was shown to be substantial \citep[e.g.,][]{ryu2003}.
Hence, the consistency of our model for proton acceleration with the non-detection of cluster $\gamma$-rays
should be further examined
by considering the details of the characteristics of shocks in simulated galaxy clusters.
\section{Summary}
\label{s4}
{
The DSA efficiency for CR protons at low $M_{\rm s}$ $Q_\parallel$-shocks in the high-$\beta$ plasmas of the ICM has yet to be investigated through kinetic plasma simulations.
HRKM18 studied the injection and the early acceleration of protons up to $p_{\rm max}/m_ic \approx 0.5$ at such shocks through 1D PIC simulations, adopting reduced mass ratios of $m_i/m_e$.
On the other hand, CS14a, CS14b, and CPS15 carried out hybrid simulations to study the DSA of protons, but considered only high $M_{\rm s}$ shocks in $\beta\approx 1$ plasmas.
Here, we revisited the test-particle DSA model for low $M_{\rm s}$ shocks with a thermal leakage injection recipe that was previously presented in \citet{kang2010}.
Reflecting new findings of recent plasma simulations, we improved the analytic DSA model by accounting for the
transfer of the postshock thermal energy to the CR energy and the weakening of the subshock due to the dynamical feedback of the CR pressure to the shock structure.}
We first set up an approximate analytic solution, $f_{\rm CR}(p)$, for CR protons in weak $Q_\parallel$-shocks.
We then calculated the injection fraction, $\xi$, the postshock CR energy fraction, $E_{\rm CR,2}/E_{\rm sh}$, and the acceleration efficiency,
$\eta$, of CR protons.
The main aspects of our model and the main results are summarized as follows.
1. In weak shocks with $M_{\rm s} \lesssim 5$, above the injection momentum, $ p_{\rm inj}=Q_i\ p_{\rm th,p}$, $f_{\rm CR}(p)$ follows the test-particle DSA power-law, whose slope is determined by the shock compression ratio.
2. According to plasma simulations such as CS14a, CPS15, and HRKM18, as CR protons are accelerated to higher energies, the postshock gas temperature $T_2$ and the normalization of $f_{\rm CR}$ decreases (see Figure \ref{f2}).
Thus, in our model, while the injection momentum, $p_{\rm inj}$, is assumed to be fixed, the injection parameter increases as $Q_i= Q_{i,0}/\sqrt{R_{\rm T}}$, where $R_{\rm T}$ is the reduction factor of the postshock temperature. Then $Q_i$ determines the CR spectrum according to Equations (\ref{finj})-(\ref{RN}).
We adopt $Q_{i,0}\approx 3.3-3.5$, extrapolating the results of previous hybrid simulations.
{
3. In our model, as $f_{\rm CR}(p)$ extends to higher $p_{\rm max}/m_pc\gg 1$, $\xi$ first increases and then decreases due to the reduction of $T_2$ and the increase of $Q_{\rm i}$, although $\eta$ monotonically increases and approaches a time-asymptotic value.
Such a behavior was previously seen in fluid DSA simulations \citep[e.g.,][]{kang2002}.
4. Both $\xi$ and $E_{\rm CR,2}/E_{\rm sh}$ depend on $Q_{i,0}$ and also the lower bound of the integrals, $p_{\rm min}$, especially in the case of very weak shocks (see Figure \ref{f4}).
For $p_{\rm min}\approx p_{\rm inj}$ and $Q_{i,0}=3.5$, the CR acceleration efficiency ranges as $\eta\approx 3.5\times 10^{-3} - 0.01$ for $2.25\lesssim M_{\rm s}\lesssim 5.0$.
If $p_{\rm min}\approx780~{\rm MeV}/c$ is adopted, it decreases to $\eta\approx 1.1 \times 10^{-3}-0.01$ for the same Mach number range.
If $Q_{i,0}=3.3$ is adopted, $\eta$ becomes larger by a factor of $1.5-2$, compared to the case with $Q_{i,0}=3.5$.}
5. In subcritical shocks with $M_{\rm s}<2.25$, protons may not be efficiently injected into DSA, so we expect that $\eta$ would be negligible at these very weak shocks (HRKM18).
{
In a parallel paper \citep{ha2019}, we will investigate the $\gamma$-ray emission as well as the neutrino emission from simulated galaxy clusters due to the inelastic collisions of CR protons and ICM thermal protons,
based on the analytic CR proton spectrum proposed in this paper.
In particular, we will check whether the prediction for $\gamma$-ray emission complies with the upper limits imposed by Fermi LAT observations.}
\acknowledgments
{
We thank the anonymous referee for critical comments that help us improve this paper from its initial form.}
D.R. and J.-H. H. were supported by the National Research Foundation of Korea (NRF) through grants 2016R1A5A1013277 and 2017R1A2A1A05071429.
H.K. was supported by the Basic Science Research Program of the NRF through grant 2017R1D1A1A09000567.
|
1,314,259,996,664 | arxiv | \section{Introduction}
\label{sect:intro}
A large number of cold dark matter (CDM) N-body simulations agree that the
haloes formed have, on average, a universal broken power law density
profile. While there is some debate over the logarithmic slope of the profile
within the break, or scale radius $r_{\rm s}$, and there is some scatter between
haloes, it seems that the original profile of \cite{NFW97} (NFW) still
provides a reasonable fit to the simulation data. This profile seems to
be a generic feature of haloes formed in the hierarchical model of
structure formation.
A number of authors have made significant progress in understanding non-linear
effects in structure formation using halo models, where the number density,
correlation function and mass density profile of CDM haloes are fitted to the
numerical simulations and then used in a simplified model of large scale
structure \cite{M+W96,CHM00,T+J03}. The NFW profile has played a
prominent role in this enterprise. One application of the halo model is in the
investigation of the halo occupation distribution, that characterises the
substructure within a larger halo. This has long been one of the more
controversial topics in CDM theory, with predictions and observations often at
odds. In order to build up an accurate picture of a hierarchical mass
distribution, the stripping of the sub-haloes by tidal gravitational
forces must be modelled \cite{T+B04,O+L04}; measurements of halo
stripping form an important test of the detailed predictions of the
CDM simulations \cite{Hay++03,T+B05}.
Moreover, it is now clear that the effect of baryons on the shapes and profiles
of total mass distributions cannot be ignored. In galaxies, the stellar
component of the mass dominates at small radii giving rise to a peakier
observed total density profile than seen in pure dark matter simulations
\cite{Koo++06}. The surface brightness profiles of massive galaxies seem to
be consistently well-fitted by a Sersic profile of index $3-4$
\cite{Tru++04}; a logarithmic slope of $-2$ in total density
in the inner regions appears ubiquitous \cite{Tre++06}. Moreover, the
dark matter profile itself is expected to steepen during the formation
of the galaxy, by the process of adiabatic contraction
\cite{Zel++80,Blu++86}. Typically this leads to more centrally
concentrated, rounder haloes \cite{Gne++04,Kaz++04}. The details of the
mass distributions of real galaxies are therefore a probe of the galaxy
formation physics claimed by the simulations.
Gravitational lensing allows us to probe the mass distributions of galaxies,
groups and clusters in a unique way. Insensitive to the dynamical state of the
lens system, both weak and strong lensing effects depend only on the projected
(and scaled) gravitational potential. Gravitational lensing has already been
used to investigate the density profile in galaxy clusters
\cite{Kne++03,Gav++03,San++04,Bro++05}. Substructure studies
have also been undertaken, making use of the galaxy-galaxy lensing
effect in clusters \cite{N+S04}. The galaxy scale halo mass profiles
have also been measured, using both strong lensing \cite{RKK03,D+W05,
Koo++06} and, in a more statistical fashion, galaxy-galaxy weak lensing
\cite{She++04, HYG04}.
Lensing studies provide direct tests of the CDM simulations, and typically
involve (at some point) fitting the parameters of an NFW-like model to the data.
However, this model is also well-suited to a more general analysis, building up
a data model from a linear combination of NFW-like potentials \cite{Mar06}.
This approach has applications in substructure characterisation, and
also template-based cluster finding. Characterising stripped substructure
both require an accurate treatment of the outer regions of haloes. However, in
order to measure accurately density profile slopes and concentrations, the
baryonic mass component must be included.
In the perpetually applicable thin lens approximation it is the projected
Newtonian gravitational potential that gives rise to the gravitational lensing
observables. Making the simplifying assumption that projected stellar mass
density is proportional to optical surface brightness leads us to seek the
potential that corresponds to the Sersic density profile. Likewise, for the
dark matter component we require a lens potential that corresponds to the
universal profile seen in simulations, but that also includes the effects of
tidal stripping.
Analytic lens potentials are convenient to work with: they, and their
derivatives that are needed for lensing data modelling, can be computed quickly
and accurately; the introduction of ellipticity to the halo can be done very
straightforwardly; more complex potentials can be constructed by simple linear
combination of analytic functions. This last feature allows concave isodensity
contours to be avoided in the case of high ellipticity. It also allows total
density distributions to be constructed from mixtures of dark and luminous
matter.
In this work we present analytic forms for a smoothly truncated universal CDM
gravitational potential, and also for the potentials corresponding to a subset
of the Sersic profiles. If the underlying potential is analytic, so are all the
derivatives needed in gravitational lens studies. An outline of the paper is
as follows. In Sections~\ref{sect:NFW} and~\ref{sect:sersic} we present our
suggested analytic potential models, for both dark and baryonic matter
components, and outline the derivation of the quantities relevant to
gravitational lensing. We leave the full formulae to an appendix, but in
Section~\ref{sect:obs} we plot the predicted observables, and compare them to
those from an unstripped baryon-free NFW form. In Section~\ref{sect:discuss} we
briefly discuss our results, and point the reader towards some publically
available computer code.
\section{Smoothly truncated dark matter haloes}
\label{sect:NFW}
The NFW profile for the CDM density $\rho$ of a halo is
\begin{equation}
\rho(r)=\frac{4\delta_{\rm c}
\rho_{\rm crit}}{\left(\frac{r}{r_{\rm s}}\right)\left(1+\frac{r}{r_{\rm s}}\right)^2}=
\frac{M_0}{4\pi r(r+r_{\rm s})^2}.
\label{eq:originalNFW}
\end{equation}
The characteristic overdensity $\delta_{\rm c}$ is the density at the scale radius
$r_{\rm s}$, in units of the critical density $\rho_{\rm crit}$. Alternatively, we can express
this as $\rho(r_{\rm s})=\delta_{\rm c}\rho_{\rm crit}=M_0/(16\pir_{\rm s}^3)$. The NFW profile is
analytically integrable along the line of sight; the most frequently used
formulae for the weak lensing shear \cite{W+B00} and strong lensing image
positions \cite{Bar96} were derived assuming the integral to extend over all
space. Given that the NFW profile has divergent total mass, \cite{T+J03}
suggest using a modified form that is sharply truncated at the virial
radius. The projection integral is then more realistic, with only mass
physically associated with a finite-sized halo being modelled.
We might expect real CDM haloes to be truncated due to tidal effects; a
step-function density cutoff may not offer a very physical picture of the edges
of haloes. With the \cite{T+J03} mass distribution, the lensing deflection
angle and shear are tractable (if somewhat less simple), and the actual
potential is worse still, involving (at least) polylogarithms. Also, the
convergence and shear are not differentiable at the truncation radius. A
power-law cutoff in the potential is more attractive in this regard. We should
insist that the truncated profile match that of NFW as closely as possible
within the tidal radius, which is introduced as the third parameter of the
profile. This is important for the results from previous work on fitting the
outputs from N-body simulations pertaining not only to the density profile but
also the mass function \cite{Jen++01}.
With these desiderata in mind, we suggest the following functional form for a
smoothly truncated universal 3-d mass density profile:
\begin{equation}
\rho(r)=\frac{4\delta_{\rm c} \rho_{\rm crit}}
{ \left(\frac{r}{r_{\rm s}}\right)
\left(1+\frac{r}{r_{\rm s}}\right)^2
\left(1+\left(\frac{r}{r_{\rm t}}\right)^2\right)^n}=
\frac{M_0}{4\pi r(r+r_{\rm s})^2}\,\left(\frac{r_{\rm t}^2}{r^2+r_{\rm t}^2}\right)^n.
\label{eq:newdensity}
\end{equation}
Here, $r_{\rm t}$ is a new parameter which should correspond to the tidal
radius for tidally truncated halos \cite{BT87}. The parameter $n$
controls the sharpness of the truncation; we will investigate the
cases $n=1,2$. For relatively isolated haloes, we expect the tidal radius to be
much larger than the scale radius. We define $\tau=r_{\rm t}/r_{\rm s}$, expecting
$\tau>>1$. Note that $\tau$ is not necessarily the ``concentration''
parameter, defined as the ratio of a ``virial'' radius to $r_{\rm s}$. In the
left-hand panel of Figure~\ref{fig:density}, we plot this profile in the usual
way (with logarithmic axes), and compare with the original NFW profile. We show
the effect of the tidal radius in providing a smooth edge to the halo. In the
right-hand panel of Figure~\ref{fig:density} we show the integrated projected
mass profile, for the same set of density profiles. In this panel radius is
projected radius, and the logarithmic divergence of the original NFW profile
can be clearly seen. Projected mass within some appropriate radius is
(approximately) the quantity that is best constrained by gravitational lensing
-- in Section~\ref{sect:obs} we show the predicted observables of gravitational
lensing in more detail.
\begin{figure}
\begin{minipage}[t]{0.48\linewidth}
\centering\epsfig{file=density.ps,width=0.95\textwidth}
\end{minipage} \hfill
\begin{minipage}[t]{0.48\linewidth}
\centering\epsfig{file=totalmass.ps,width=0.95\textwidth}
\end{minipage}
\caption{Density (left) and integrated projected mass (right) profiles
for NFW haloes with various truncation schemes. The solid lines indicate
the original NFW halo, with and without a hard cutoff at $\tau =
r_{\rm t} / r_{\rm s} = 10$. Dotted lines show the $n=1$ cutoff prescription,
with $\tau=10,20$. Dashed lines show the $n=2$ cutoff
prescription, again with $\tau=10,20$. For smoothly truncated
models, ratios of masses outside the virial radius to the virial
mass, $[M_{\rm tot}-M(<10)]/M(<10)$, are 17\% for ($n$,
$\tau$)=(1, 10), 4.6\% for (2, 10), 36\% for (1, 20), and 17\% for
(2,20). Note that this ratio is infinity for the original NFW halo
because it has divergent total mass.}
\label{fig:density}
\end{figure}
After some experimentation we found that the form recommended above is indeed
the simplest one that gives an analytic potential while ensuring a
non-diverging total mass for all values of the tidal radius. In the case where
the tidal radius~$r_{\rm t}$ is outside the scale radius~$r_{\rm s}$, the $n=1$ profile
falls off as $r^{-5}$, steep enough to mimic a sharp cutoff. If the tidal
radius were to lie inside the scale radius, then the density would decrease as
$r^{-3}$ initially before turning over to $r^{-5}$ outside~$r_{\rm s}$. Since this
would imply some memory of the original halo after the presumably violent act
of tidal stripping, we suggest that if $r_{\rm t}<r_{\rm s}$, the $n=2$ version of the
density profile be used. For $r_{\rm t} < r_{\rm s}$, this profile turns over to $r^{-5}$
at $r_{\rm t}$, which is effectively a sharp cutoff. The further turnover to
$r^{-7}$ at $r_{\rm s} > r_{\rm t}$ has little effect.
The close agreement of the unstripped halo with the original NFW profile is
comforting. For example, for a halo with a concentration of 10, if we set
$r_{\rm t}$ to twice the virial radius $(\tau=20)$, the masses contained within the
virial radius are the same to within $6\%$. Of course the total mass of the
truncated halo is 50\% larger than the virial mass, while the total mass of
the untruncated halo (formally) diverges. We will thus take $r_{\rm t}=2r_{200}$ as
our fiducial tidal radius for an unstripped halo. We note that this
choice of the truncation radius is simply a working assumption in this
paper, and the more appropriate value should eventually be obtained in
$N$-body simulations and observations.
\section{Lensing by stellar mass in galaxies}
\label{sect:sersic}
The Sersic profile, found to fit well the optical surface brightness~$I$ of
undisturbed galaxies \cite{Ser68}, is
\begin{equation}
I(r)= I_{\rm e} \exp\left[\kappa_n \left(1 - \frac{r}{r_{\rm e}}\right)^{1/n}
\right],
\label{eq:sersic}
\end{equation}
where the effective radius $r_{\rm e}$ is the radius within which half
the flux is contained, and $n$ is the Sersic index. For elliptical
galaxies, an index of around 4 is often seen \cite{deV48}, while the
characteristic exponential profile of galaxy disks corresponds to a
Sersic index of~1. In fact, a broad range of Sersic index values have
been seen in fits to observed galaxy light profiles \cite{Bla05}.
In addition, there has been some arguments that density profiles of CDM haloes
can also be fitted well by the Sersic profile with an index of $2-3$.
\cite{Mer05,T+G05}.
Assuming that stellar mass follows light, we can substitute surface mass
density $\Sigma$ for surface brightness in equation~\ref{eq:sersic}. In the
appendix, we show that the lens potential sourced by this mass distribution
\cite{Car04} is analytically tractable for integer and half-integer $n$.
\section{Predicted observables}
\label{sect:obs}
As we show in the appendix, both density profiles introduced above (truncated
NFW and Sersic) have analytic lens potentials. (The NFW profile has an analytic
three-dimensional potential, which can itself be projected analytically.) The
expressions for the lensing potential, while somewhat lengthy, are rapidly
calculated and differentiable to all orders. In this section we plot some of
these derivatives, pointing out their application in gravitational lens data
modelling. We note that in the limit of radii beyond the tidal radius the
lensing properties of our model haloes do indeed approach those of a point mass,
as required.
\subsection{Weak lensing}
We first address the issue of not truncating the NFW profile on the weak
lensing shear and convergence (see \cite{GL/Sch06} for a good
introduction to these quantities). The lefthand panel of
Figure~\ref{fig:shear} shows the convergence (projected mass density)
profile for the set of haloes first introduced in Figure~\ref{fig:density}.
We see that using a truncated profile with the same virial mass somewhat
reduces the predicted lensing effect. The corresponding shear profiles are
plotted in the right-hand panel of Figure~\ref{fig:shear}. Taking the central
density profiles to be equal (mimicking a well-constrained
central strong lensing region, for example), for $\tau=20$ we note
that the virial mass
(mass within 10$r_{\rm s}$) is 6\% less for the $n=1$ truncated halo. The shear for
the two profiles only differs by 3\%, however. The total {\em projected} mass
within the projected virial radius is some 12\% lower than that of
the untruncated profile. Lastly, the surface
density of the truncated halo is 30\% lower at 10$r_{\rm s}$.
The difference in reduced shear $\gamma/|1-\kappa|$ thus depends on
the absolute value of the convergence, relative to the critical
surface density, but can be significant. Very roughly, from
Figure~\ref{fig:shear} we expect that different truncations examined
in this paper can yield $\sim 10\%$ difference in $\gamma$ at around
the virial radius. Although this is smaller than the accuracy of shear
measurements for most massive clusters of galaxies (e.g., \cite{Bro08}),
the accuracy can be reachable in the weak lensing analysis of
stacked cluster samples (e.g., \cite{Joh07}).
\begin{figure}
\begin{minipage}[t]{0.48\linewidth}
\centering\epsfig{file=convergence.ps,width=0.95\textwidth}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\linewidth}
\centering\epsfig{file=shear.ps,width=0.95\textwidth}
\end{minipage}
\caption{Convergence (left) and shear (right) for NFW haloes. The curve
line styles are the same as in Figure~\ref{fig:density}. Projected mass
density $\Sigma$ (directly proportional to convergence $\kappa$) is plotted
on the left, showing that a hard cutoff in density results in a softer, but
still non-differentiable, cutoff in the convergence. Actually plotted on the
right is $\langle\Sigma\rangle-\Sigma$, which is directly proportional to the
shear $\gamma$ for axisymmetric haloes. Notice that a hard cutoff in density
means the shear is finite, but not differentiable at the cutoff radius.}
\label{fig:shear}
\end{figure}
\subsection{Strong and intermediate lensing}
Figure~\ref{fig:strong} shows the amplitude of the deflection angle for
a strongly lensed source. Here we see that using a truncated profile has
very little effect on the deflection angle in the regime where it is measurable
as a multiple-image separation ($r < r_{\rm s}$). This
is unsurprising given that the strong lensing is
dominated by the central part of the profile which is, by design,
little changed in our new model.
\begin{figure}
\begin{minipage}[t]{0.48\linewidth}
\epsfig{file=deflection.ps,width=0.95\textwidth}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\linewidth}
\epsfig{file=flexion.ps,width=0.95\textwidth}
\end{minipage}
\caption{Deflection angle (left) and flexion (right) for NFW haloes.
Curves are the same as in
Figure~\ref{fig:density}. Notice that the hard cutoff in density causes the
flexion to diverge at the cutoff radius. As the flexion approaches
$-\infty$, so it changes sign as well. This can be seen in
Figure~\ref{fig:shear}, where the shear actually starts to increase as the
cutoff radius is approached.}
\label{fig:strong}
\end{figure}
In contrast, the so-called ``flexion'' may be more strongly affected
by the truncation because it is essentially the higher-order
derivative of the lens potential. In addition it is measurable over a
wide range of scales from the Einstein radius to (in a statistical
sense) the virial radius. In Figure~\ref{fig:strong} we plot the third
derivative of the lens potential: the most interesting component of
flexion for circular symmetry is in fact just this, the radial
gradient of the shear. Marked differences in signal strength arise
when the haloes are truncated.
A plausible model for an elliptical galaxy lens consists of two parts: the
stellar component and the dark halo. Modelling the stellar component as an
$n=4$ Sersic profile and the dark halo as a truncated NFW profile with
concentration 10 and $\tau=20$, a reasonable fit to lensing data can
be made \cite{Gav07}. The salient feature is that the total mass profile is
approximately isothermal. This can be arranged with the following
prescription (for $\tau=20$ and $n=4$):
\begin{equation}
\frac{M_{\rm NFW}}{M_{\rm deV}}\approx 4.75\,\frac{r_s}{r_e}.
\end{equation}
The NFW mass is the total mass. The virial mass (mass within 10 scale radii in
this case) is 0.66 of the total. The broad isothermal region obtained by this
prescription is illustrated in Figure~\ref{fig:sersic}. We note that
adiabatic contraction \cite{Zel++80,Blu++86,Gne++04} modifies the NFW
profile, leading to more centrally concentrated profile of dark matter.
However, the effect of adiabatic contraction is most pronounced at the
very center of the halo where the baryonic (Sersic) component is
dominated, and thus its effect on the total mass profile is not
substantial.
\begin{figure}
\begin{minipage}[t]{0.48\linewidth}
\centering\epsfig{file=nfw_sersic_sigma.ps,width=0.95\textwidth}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\linewidth}
\centering\epsfig{file=nfw_sersic.ps,width=0.95\textwidth}
\end{minipage}
\caption{Sersic (n=4) profile combined with NFW profile. We plot the
convergence (left) and circular velocity squared $M(x)/x$ (right)
associated with each component (there are two curves for the NFW
profile: one untruncated and one with $\tau=20$),
along with the total (dashed line). The truncated NFW profile is 70 times
more massive than the Sersic profile (the virial mass is 50 times larger than
the stellar mass), and the scale radius is 15 times larger than the half
light radius. With these reasonable parameters, it is clear that the total
profile is nearly isothermal (logarithmic slope of -1 in convergence) around
the half light radius. In fact, we find that (for $\tau=20$), the relation
$M_{\rm NFW}/M_{\rm deV}=4.75 \,r_s/r_e$ gives a flat region in velocity
dispersion.}
\label{fig:sersic}
\end{figure}
Finally we move to two dimensions and illustrate the construction of
elliptically symmetric isophotes in the convergence distribution. It has been
noted \cite{K+K93} that the isodensity contours of an
elliptical lens potential can become dumbbell-shaped at low values of the axis
ratio. In the appendix we show how isodensity contours that are elliptical to
third order in the axis ratio can be constructed following a simple recipe. In
Figure~\ref{fig:ell} we illustrate this procedure, showing the constituent
concentric elliptical potentials and the resulting convergence contours. It is
found that our new model not only avoids the unphysical concave isodensity but
also gives much better fit to the elliptical isodensity than the elliptical
lens potential. For large axis ratios (3:1 is as large as is feasible by our
technique), the isodensity contours are slightly disky. We note that this
recipe preserves the radial profile of the (self-similar) constituent
potentials. Although the procedure is derived assuming a pure
power-law, the nearly isothermal distribution of the composite model
illustrated in Figure~\ref{fig:sersic} suggests that our prescription
is useful for such composite model, at least as long as the
ellipticities of Sersic and NFW components are similar.
\begin{figure}
\begin{minipage}[t]{0.48\linewidth}
\epsfig{file=ellipses_e06.ps,width=0.95\textwidth}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\linewidth}
\epsfig{file=ellipses_e08.ps,width=0.95\textwidth}
\end{minipage}
\caption{Convergence (isodensity) contours for three stacked elliptical
potentials. True ellipses are shown as solid curves, three stacked
potentials are shown as dashed curves, and single elliptical potentials are
shown as dotted curves. The left panel illustrates the case $\epsilon_{\rm
iso}=0.6$ (axis ratio 2:1), while the right panel illustrates
$\epsilon_{\rm iso}=0.8$ (axis ratio 3:1). The slight diskiness of the
isodensity contours can be seen especially for the $\epsilon_{\rm iso}=0.8$
case. The fitting procedure is described in Appendix C.}
\label{fig:ell}
\end{figure}
\section{Discussion}
\label{sect:discuss}
We have introduced a simple smoothly-truncated extension of the NFW density
profile. To date the majority of cluster and galaxy lens modelling that has been
performed using the NFW profile has used the untruncated profile. We find that,
if haloes are indeed tidally-moulded leading to the kinds of smooth truncation
that we propose, then the masses of the haloes may have been overestimated by
some 10\% or so during a weak lensing analysis.
This number pertains to the situation where the inner profile is inferred to be
the same for each model profile, as might be the case when good strong lensing
data are available. We find that the smooth truncation of a halo does not
significantly affect the deflection angles at the image positions (which lie
typically well within the scale radius). If strong lensing data are not
available then the degeneracy between the truncation radius and the halo mass
will give rise to a broader inferred marginal probability distribution for the
halo mass, with the mean shifting to lower values than for the untruncated
profile.
The truncation of galaxy haloes in field galaxy-galaxy lensing is always likely
to be masked by the effects of large scale structure on the outer parts of the
mass profile (the ``two-halo'' term). However, in clusters a 10\% systematic
error is comparable to that introduced in other parts of a current weak (plus
strong) lensing analysis. The uncertain background galaxy redshift distribution,
additional mass along the line of sight, cluster member galaxy contamination,
projection effects, and shear calibration errors can easily be of order 10\% in
the halo mass. However, as survey sizes increase, and the goals of
cosmological cluster-counting experiments become loftier, uncertainties such as
that introduced by halo truncation may become important.
Sharp truncation \cite{T+J03} introduces discontinuities in the shear
that are unlikely to cause problems in data modelling; however, the same may
not be true about flexion, where a singularity appears at the truncation
radius. Smooth truncation (or indeed, no truncation) avoids this problem.
Galaxy-galaxy weak lensing studies within clusters of galaxies have already
succeeded in producing a measurement of a truncation radius \cite{NKS02}. The
combined Sersic plus NFW model currently popular in field galaxy lens modelling
could profitably be applied in a cluster galaxy-galaxy lensing. The photometry
provides extra constraints on the stellar mass part of the density profile
\cite{N+K97}, allowing the dark matter structure of galaxies in clusters to be
probed. The model suggested here would straightforwardly allow strong and
intermediate lensing effects to be incorporated; we are not far from possessing
a useful sample of strong gravitational lenses lying in clusters.
The question of how best to model tidal-truncation of dark matter haloes has been
approached here in a phenomenological and pragmatic way: we wanted an analytic
form for the lens potential. We believe the forms presented here would provide
good fits to the haloes seen in numerical simulations, based on the
successes of others with very similar profiles \cite{T+B04,D+W05}.
We have shown that plausible models of gravitational lenses can be constructed
from the superposition of simple analytic profiles, including the generation of
elliptically-symmetric isodensity contours. An interesting extension of this
would be to attempt to build up still more complex mass distributions, from
misaligned and offset building blocks \cite{Mar06}. Again, whether the
haloes and sub-haloes observed in numerical simulations can be well-enough
approximated by such a model remains to be seen. At present it seems that the
signal-to-noise in Einstein rings is sufficiently high to constrain such a more
complex model \cite{Koo05,Koo++06}.
Finally, we discuss the importance of truncation, and a smooth one at that, in
when simulating lensing effects in large surveys. Gravitational lensing, weak,
intermediate and strong, may be expected to be an important component of
multiple-pronged dark energy investigation. Simulations of large fields will
play a vital role in improving our understanding of the astrophysical
systematic
errors present in cosmic shear, and cluster mass function, measurements. Halo
models are a cheap and efficient way of doing this, capturing the pertinent
physical effects without the need for further CPU-expensive N-body simulations.
However, ray-tracing through halo models does present some technical challenges
\cite{T+J03,O06}. The smooth analytic truncation proposed here allows the mass
budget to be balanced, while allowing all gravitational lensing effects to be
calculated rapidly to machine precision. In fact, it has been shown
that the shear angular correlation function becomes $\sim 20$\%
smaller if the truncation at around the virial radius is included, and
that the calculation with the truncation shows better agreement with
$N$-body simulations \cite{T+J03}.
A by-product of the future large optical and radio imaging surveys
will be an interesting sample of strong lenses showing higher-order
catastrophes beyond the usual cusps and folds. These systems provide
very high magnifications, and are very sensitive to the mass structure
in the lens, and as such promise to be interesting
laboratories. However, modelling them will require an accurate
multi-scale approach; we leave the development of this project to
further work.
The code used in this work is plain ISO C99 and can be freely downloaded from
\vspace{\baselineskip}
\begin{center}
\tt http://kipac.stanford.edu/collab/research/lensing/ample/
\end{center}
\section*{Acknowledgments}
We thank James Taylor, Peter Schneider, Stelios Kazantzidis, and
Masahiro Takada for useful discussions and encouragement. We also
thank an anonymous referee for many suggestions. This work was
supported in part by the U.S. Department of Energy under contract number
DE-AC02-76SF00515. PJM acknowledges support from the TABASGO foundation
in the form of a research fellowship.
\bibliographystyle{unsrt}
|
1,314,259,996,665 | arxiv | \section{Introduction}
Dynamic Mode Decomposition (DMD) is a data-driven and model-free technique to decompose complex flows into fundamental spectral components.
These components correspond to spatio-temporal features that characterize periodicity, damping, (temporal) segmentation, and long-time behavior of the flow.
Basically, the algorithm results in three components: DMD modes, amplitudes, and eigenvalues.
Whereas the modes represent spatial contribution to the flow and the amplitudes specify their impact, each associated eigenvalue characterizes the temporal development.
The objective of this paper is to gain a better understanding of these components such that a more insightful analysis is achieved.
Moreover, as the visualization community has not paid much attention to DMD so far, we want to make DMD more accessible for both its users and the visualization research community.
DMD is supposed to identify spatial patterns associated with frequencies and growth rates that determine the behavior of a system.
So far, the investigation via DMD has been performed by the study of individual DMD components.
In addition, spatial and temporal properties of components are treated independently.
Since DMD is based on the interplay of spatio-temporal components, this traditional analysis process is insufficient.
It has several negative implications:
First, the relevance of the components to the entire system is not clearly specified.
Thus, an appropriate selection of components (for the analysis process) is not possible.
Second, the existing DMD visualizations could be misleading as the mutual dependencies of the components are not taken into account.
To address these problems, we focus on the representation of the components and their visualization.
Our approach is guided by the needs of unsteady flow but could be extended to general time-dependent grid-based data.
Figure~\ref{pic_overview} illustrates the analysis process following our approach.
Our contributions can be summarized as follows:
\begin{itemize}
\item Conceptual contribution: We clarify drawbacks of the traditional DMD components and provide a new perspective on DMD based on a comparison with the discrete Fourier transform (DFT).
\item New visualizations: We improve DMD components and their representations using novel visualizations that respect the spatio-temporal character of DMD.
\item Data analysis contribution: We introduce two clustering approaches to aggregate components that segment the flow into physically relevant sections and can therefore be used for the selection of DMD components.
\end{itemize}
We also discuss the mathematical foundation of DMD by providing a derivation and a specific formulation of DMD.
Moreover, with artificial and simulated examples, we show that our approach is able to identify characteristic features of unsteady flow fields.
\section{Related Work}
The visualization and analysis of unsteady flow are a challenging research topic.
A variety of decomposition techniques has been proposed to extract different kind of features from a flow.
The characteristics of a feature strongly depend on the method and are often difficult to define.
In the context of unsteady flow, we distinguish between two types of decomposition techniques.
The first type directly operates on the vector field, such as the Helmholtz Hodge decomposition (HHD) \cite{6365629} or the Morse decomposition \cite{Zhang2007}.
The HHD decomposes the vector field into two spatial components that are divergence- and curl-free, respectively.
Recent work~\cite{676} deals with an extension of the HHD based on Fourier transformation.
Wiebel et al.~\cite{4293009} propose a similar decomposition into a potential flow from the boundary and a localized flow to extract features.
Rojo and G\"unter's~\cite{Rojo:2020VFTopoUnsteady} splitting method decomposes the flow into a steady and ambient part, enabling the description of the motion of topological elements and feature curves.
However, these decomposition techniques do not encode temporal patterns like DMD.
The Morse decomposition divides a vector field into disjoint invariant sets, called Morse sets.
The connection of those Morse sets is illustrated by a directed graph giving an overview of the topological skeleton of the flow field.
While this approach highlights only spatial relations of the decomposition, we address spatio-temporal patterns.
Bujack et al.'s state-of-the-art report \cite{Bujack:2020:PhysByMath} interprets physical features of several decomposition techniques of that type in terms of mathematical properties.
We follow a similar approach, however, DMD is of another type and therefore extracts different features.
The second type of decompositions makes use of temporal coherence by performing the decomposition on the full time series, instead of considering each step individually.
Besides DMD, Principal Component Analysis (PCA) \cite{lumley67, berkooz:1993:pod_flows}, also kwown as Proper Orthogonal Decomposition (POD), is a technique of this type.
PCA hierarchically decomposes the data into an orthogonal basis of spatially correlated modes, called principal components (or POD modes), modulated by appropriate random time-coefficients.
Therefore, the dynamic information is often neglected and particular emphasis is put on the spatial components.
Pobitzer et al's work~\cite{Pobitzer:2011:PODplus} deals with an extension of POD based on feature detectors.
The visualization of POD components and the additional use of feature detectors do not take spatio-temporal properties of the decomposition into account, which is again the main difference to our DMD approach.
DMD was introduced by Schmid and Sesterhenn \cite{schmid:2008:APS}.
Schmid~\cite{schmid:2010:dmd_numerical_data} improved the DMD algorithm by using a reduced singular value decomposition (SVD).
Tu et al.~\cite{tu:2014:on_dmd_theorey_and_app} formulated the latest version of DMD, called exact DMD.
As described in the introduction, the components of DMD are used separately and important spatio-temporal relations are not taken into account, especially not for the visualization.
In this way, DMD has been applied on diverse flow setups, e.g., the analysis of wake flows \cite{tu:2014:on_dmd_theorey_and_app, bagheri:2013:KoopamnCylinderWake, zhang:2014:idenfication_of_cs_using_pod_and_dmd, sampath:2014:pod_and_dmd_of_time_resolved_piv}, cavity flows \cite{seena:2011:dmd_of_turbulent_cavity_flows,lusseyran:2011:flow_coherent_structures}, mixing layer flow \cite{sayadi:2012:dmd_of_h_type, sayadi:2013:dmd_of_controlled_h_and_k_type}, and jet flows \cite{rowley:2009:spectral_nonlinear_flow, schmid:2011:applications_of_dmd}.
In the context of Lagrangian coherent structures, the interaction of DMD and Finite Time Lyapunov Exponent (FTLE) was considered to provide a feature-based description of the entire flow field \cite{ali:2017:turbulent_boundary_layer_lcs_pod_dmd, weheliye:2018:dmd_pod_lcs, Nair:2017:lcs_dmd}.
Kou and Zhang \cite{KOU:2017:ModeCriterion} propose a new criterion for the selection of dominant modes.
However, this approach uses traditional DMD components and is based on the energy over the full time.
Regarding visualization and computer graphics, DMD was used for background/foreground video separation \cite{kutz:2017:video_separation}, background modeling \cite{pendergrass:2017:dmd_for_background_modeling, erichson:2016:compress_dmd_for_background_modeling}, edge detection \cite{bi:2017:dmd_edge_detection}, and the visualization of large-scale power systems \cite{mohapatra:2016:dmd_large_scale_power_systems}.
However, all above mentioned publications use the traditional DMD components.
With our novel visualizations of the improved components, we show that our techniques overcome drawbacks of the conventional DMD approach for visual flow analysis.
\begin{algorithm}[t]
\caption{Exact Dynamic Mode Decomposition}\label{Algorithm_DMD}
\begin{algorithmic}[1]
\Function{DMD}{$x_0,\dots,x_m$}
\State $X = \begin{bmatrix} x_0 & \dots & x_{m-1} \end{bmatrix}, Y = \begin{bmatrix} x_1 & \dots & x_{m} \end{bmatrix}$
\State Calculate the reduced SVD $X = U \Sigma V^*$ with $\textnormal{rank}(X) = r$.
\State Calculate $S = U^* Y V \Sigma^{-1}$.
\State Calculate $\lambda_1,\dots,\lambda_r$ and $v_1,\dots,v_r$ of $S$.
\For {$\lambda_i \neq 0$}
\State $\vartheta_i = \frac{1}{\lambda_i} Y V \Sigma^{-1} v_i$
\EndFor
\State $\Lambda = \text{diag}(\lambda_1,\lambda_2,\dots,\lambda_{r_0})$ with $\lambda_1, \lambda_2, \dots, \lambda_{r_0} \neq 0$
\State $\Theta = \begin{bmatrix} \vartheta_1 & \vartheta_2 & \dots & \vartheta_{r_0} \end{bmatrix}$
\State Calculate $a = \Lambda^{-1} \Theta^+ x_1$ with $a = (a_1,\dots,a_{r_0})$
\State Calculate if exist $c_0 = -\sum_{j=1}^{r_0} \frac{1}{\lambda_j} \prod_{k \neq j} \frac{1}{\lambda_j - \lambda_k}$
\State\Return $\lambda_j, a_j, \vartheta_j, c_0$
\EndFunction
\end{algorithmic}
\end{algorithm}
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{pictures/photoshop_comps/dft_compare/dft_compare_comp.pdf}
\caption{
A comparison of DMD (left) with DFT (right).
Both methods lead to a structurally similar decomposition.
However, DMD computes its own complex frequencies (depending on a frequency and growth rate) by the eigenvalues of a least-squares fit matrix, whereas DFT uses fixed real frequencies given by the (complex-valued) roots of unity.
The components of DMD can therefore converge or diverge over time, which creates new opportunities for investigating data as the simple complex frequency domain highlights.
}
\label{pic_DMD_DFT}
\end{figure*}
\section{Mathematical Foundation}
We concentrate on time-dependent flow fields defined on a grid and sampled uniformly in time.
More precisely, for a 2D velocity field on a grid with $N$ points at time $t_k$, the data is characterized by $u_1(t_k), v_1(t_k), \dots, u_N(t_k), v_N(t_k) \in \mathbb{R}$, where $u$ and $v$ represent the velocity components.
For DMD, the snapshots $x_0,\dots,x_m \in \mathbb{R}^{2N}$ are given by
\begin{equation*}
\begin{bmatrix}
| & | & & | \\
x_0 & x_1 & \dots & x_{m} \\
| & | & & | \\
\end{bmatrix}
=
\begin{bmatrix}
u_1(t_0) & u_1(t_1) & & u_1(t_m) \\
v_1(t_0) & v_1(t_1) & \dots & v_1(t_m) \\
\vdots & \vdots & \ddots & \vdots \\
u_N(t_0) & u_N(t_1) & & u_N(t_m) \\
v_N(t_0) & v_N(t_1) & \dots & v_N(t_m) \\
\end{bmatrix}.
\end{equation*}
A 3D flow field is defined analogously with three instead of two components.
Before formulating the algorithm of DMD, we summarize the concepts of DMD \cite{kutz:2016:DMD_Book,krake:2019:DMD} and compare them with DFT.
These aspects help understand the visualization techniques and their principles.
\subsection{Derivation of DMD}
DMD performs a spectral decomposition on arbitrary data.
Therefore, we generally consider complex-valued snapshots $x_0, x_1, \dots, x_m \in \mathbb{C}^{n}$, which are usually uniformly sampled in time.
The size of a snapshot is typically substantially greater than the number of snapshots, i.e., $n \gg m$.
For the snapshots, we consider the following minimization problem:
\begin{equation} \label{eq:basic_minimization_problem}
\min_{A \in \mathbb{C}^{n \times n}} \sum_{j=0}^{m-1} \lVert A x_j - x_{j+1} \rVert_2^2~,
\end{equation}
where $\lVert \cdot \rVert_2$ denotes the Euclidean norm.
In other words, we search a matrix $A$ that optimally connects the subsequent snapshots in a least squares sense.
The idea of DMD is to calculate a low-dimensional representation of $A$ and to perform an eigenvalue decomposition on it to detect frequency patterns in the data.
To this end, we insert the snapshots as column vectors into the following two matrices:
\begin{equation} \label{eq:data_matrix}
X = \begin{bmatrix}
| & & | \\
x_0 & \dots & x_{m-1} \\
| & & |
\end{bmatrix},
\quad
Y = \begin{bmatrix}
| & & | \\
x_1 & \dots & x_{m} \\
| & & |
\end{bmatrix}.
\end{equation}
The optimization problem in Equation~\ref{eq:basic_minimization_problem} can now be rewritten as $\min_{A \in \mathbb{C}^{n \times n}} \lVert A X - Y \rVert_F^2~$, where $\lVert M \rVert_F \coloneqq (\sum_{i = 1}^n \sum_{j = 1}^m \lvert m_{ij} \rvert)^{1/2}$ denotes the Frobenius norm.
Consequently, a best-fit matrix is explicitly given by
\begin{equation}
A = YX^+ \in \mathbb{C}^{n \times n}~,
\end{equation}
where $X^+$ is the Moore-Penrose pseudoinverse, a generalized inverse \cite{moore:1920:moore_penrose} of the non-square matrix $X$.
Note that $A$ is a (large) matrix of size ${n \times n}$.
We assume that $A$ is diagonalizable (which is almost always the case, if $n \gg m$).
This means that there exists an invertible matrix $V = \begin{bmatrix} v_1 & v_2 & \dots v_n \end{bmatrix} \in \mathbb{C}^{n \times n}$ consisting of eigenvectors, as well as a diagonal matrix $\Lambda = \text{diag}(\lambda_1,\lambda_2,\dots,\lambda_n) \in \mathbb{C}^{n \times n}$ with corresponding eigenvalues, such that $V^{-1} A V = \Lambda$.
Therefore, we can approximate the $k$th snapshot by
\begin{equation} \label{eq:basic_decomposition}
x_k \approx A^k x_0
= V \Lambda^k V^{-1} x_0
= \sum_{j=1}^n b_j \lambda_j^k v_j~,
\end{equation}
where $b = \begin{pmatrix} b_1 & b_2 & \dots & b_n \end{pmatrix}^T = V^{-1} x_0$.
Note that entries of $b$ are coefficients of the linear combination of $x_0$ in the eigenvector basis.
Given that $n \geq m$, the rank of $A$ cannot be higher than $m$ due to the dimension restriction and, hence, the number of non-zero eigenvalues is at most $m$.
Consequently, the dynamic behavior will be captured by $m$ summands, consisting of the triple $(b_j,\lambda_j,v_j) \in \mathbb{C} \times \mathbb{C} \times \mathbb{C}^n$ for $j=1,2,\dots,m$.
DMD calculates exactly these (non-zero) eigenvalues and eigenvectors referred to as DMD eigenvalues and DMD modes, respectively.
For simplicity, we denote them as eigenvalues and modes throughout the rest of the paper.
The coefficients $b_1,\dots,b_m$ in the decomposition of Equation~\ref{eq:basic_decomposition} need to be computed by DMD in a different way.
We call them DMD amplitudes.
However, as the modes usually do not form a basis, an error will occur in the reconstruction of the first snapshot $x_0$.
\subsection{DMD Algorithm} \label{ssec:DMD_algorithm}
Algorithm~\ref{Algorithm_DMD} shows the DMD method derived from the previous section.
However, it differs in the calculation of amplitudes from the standard literature.
The new definition of amplitudes, which uses the second snapshot instead of the first one for the reconstruction, was introduced by Krake et al. \cite{krake:2019:DMD}.
We make use of this new formulation by improving the representation of DMD components and integrating them to arrive at novel and more adequate DMD visualizations.
In Algorithm~\ref{Algorithm_DMD}, a triple $(a_j,\lambda_j,\vartheta_j)$ similar to the one in Equation~\ref{eq:basic_decomposition} is determined without explicitly computing $A$.
This is achieved by using a reduced SVD applied to the data-matrix $X$, which results in two unitary matrices $U \in \mathbb{C}^{n \times r}$ and $V \in \mathbb{C}^{m \times r}$ (which are real-valued for real-valued data) as well as a real-valued diagonal matrix $\Sigma \in \mathbb{R}^{r \times r}$ with $r = \text{rank}(X)$ (lines 2--3).
The low-dimensional representation of $A$ is given by (line 4)
\begin{equation} \label{eq:matrixS}
S = U^* A U = U^* Y V \Sigma^{-1}~.
\end{equation}
Next, we compute the eigenvalues $\lambda_j$ and eigenvectors $v_j$ of $S$ for $j = 1,\dots,r$ (line 5).
Then, we transform the eigenvectors with non-zero eigenvalues into modes $\vartheta_j$ (lines 6--8).
Finally, the amplitudes $a_j$ and the error scaling $c_0$ are calculated (lines 9--12).
By this choice of amplitudes, Krake et al. \cite{krake:2019:DMD} proved the following reconstruction property:
\begin{equation} \label{eq:theroem}
x_0 = \sum_{j=1}^m {a_j \vartheta_j} + c_0 \cdot q~,
\qquad
x_k = \sum_{j=1}^m {\lambda_j^k a_j \vartheta_j}~,
\end{equation}
for $k = 1,\dotso,m$ and $q = x_m - XX^+ x_m$.
This property holds when both $x_0,\dotso,x_{m-1}$ and $x_1,\dots,x_m$ are linearly independent and $\lambda_1,\dots,\lambda_m$ are distinct (in this case $r_0 = r = m$).
The assertion states that DMD inherits the property of Equation~\ref{eq:basic_decomposition} with appropriate coefficients.
This version of DMD captures the entire system by providing a structured spectral decomposition into temporal and spatial aspects.
Thus, we are able to clarify the impact of (aggregated) components precisely.
Moreover, the interplay of the eigenvalues, modes, and amplitudes can be interpreted in a new and clearer way.
Based on these insights, we create novel visualizations that respect the spatio-temporal character of DMD.
\subsection{DMD and DFT} \label{ssec:DMD_DFT}
Discrete Fourier transform (DFT) is a well-understood tool to analyze time-dependent data.
It is able to extract frequency-based features like periodicity.
Since DFT and DMD lead to a structurally similar decomposition of data, it is possible to translate properties and procedures from DFT to DMD.
More precisely, DFT yields
\begin{equation} \label{eq:DFT}
x_k = \sum_{j = 0}^m \mu_j^k \hat{x}_j~,
\end{equation}
where $\mu_j = e^{\frac{2 \pi j}{m+1}}$ are the roots of unity and $\hat{x}$ is the Fourier transform.
Thus, the DFT converts 1D snapshots $x_k$ into complex numbers $\hat{x}_j$ (for vector-valued data $x_k$ into complex vectors $\hat{x}_j$) each depending on a root of unity $\mu_j$.
Using the exponential form, i.e., $\mu_j = e^{i \varphi_j}$, we observe that the components $\mu_j$ have magnitude $1$.
Thus, they are only determined by a real frequency $f_j = \frac{\varphi_j}{2\pi}$.
This leads to a real frequency domain representation as shown in Figure~\ref{pic_DMD_DFT} on the right side, where the magnitude of a summand $\mu_j^k \hat{x}_j$ does not change over time.
In contrast, DMD computes the triplets: eigenvalues $\lambda_j$, modes $\vartheta_j$, and amplitudes $a_j$.
A direct comparison of the components is difficult since the interplay of DMD components is more complex than for DFT.
Nonetheless, we can bring DMD and DFT in accordance using Equations \ref{eq:theroem} and \ref{eq:DFT}.
In the decompositions, the temporal components are given by $\lambda_j$ and $\mu_j$, respectively.
Each set of temporal components characterize a decomposition entirely as the spatial contributions, i.e., the Fourier transformed vectors $\hat{x}_j$ or the scaled modes $a_j \vartheta_j$, are simply fitted to those in a unique way.
Therefore, the decompositions only differ in the choice of temporal components.
In fact, the eigenvalues $\lambda_j$ computed by DMD are variable, whereas the DFT uses roots of unity $\mu_j$.
Hence, the eigenvalues $\lambda_j = r_j e^{i \varphi_j}$ are characterized by both a frequency $f_j = \frac{\varphi_j}{2\pi}$ and a magnitude $r_j$, or, in other words, by a complex frequency.
Thus, DMD produces a complex frequency representation of the data as illustrated in Figure~\ref{pic_DMD_DFT} on the left side.
The DMD procedure can be summarized as a two-stage method:
First, DMD computes appropriate complex frequencies based on the data.
Then, the data is transformed into complex numbers (for vector-valued data into complex vectors) that depend on those complex frequencies.
From a certain point of view, DMD can thus be seen as an extension of DFT.
This opens up new possibilities to extract complex frequency-based features, like periodicity, damping, and temporal segmentation.
In Figure~\ref{pic_DMD_DFT}, we observe that DMD needs three (non-vanishing) complex frequencies to reconstruct the signal, whereas DFT a number of real frequencies.
In a more complex scenario, the challenge is to find relevant DMD components.
This task is more complicated than in the case of DFT, since the interplay of eigenvalues $\lambda_j$, modes $\vartheta_j$, and amplitudes $a_j$ needs to be taken into account in order to respect the spatio-temporal character of DMD.
Starting with the eigenvalues $\lambda_j$ (complex frequencies), these should be used to highlight the impact of modes over time.
Previously, the modes $\vartheta$ should be adjusted by their amplitudes $a_j$ according to DFT such that the scaled modes $a_j \vartheta_j$ (analogously to the Fourier-transformed vectors) characterize the spatial contribution and the impact.
Based on these observations, we provide improved components and new visualizations clarifying the interplay of components.
Moreover, a cluster method is developed on the principles of DFT (aggregation of harmonics) that is used for the selection of relevant DMD components.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{pictures/karman/pic_karman_DMDMode.png}
\caption{
Traditional visualization of a 2D mode $\vartheta_j$ with arrow glyphs and color-coded velocity magnitude (blue = low, red = high).
}
\label{pic_karman_modes_old}
\end{figure}
\section{Investigation Approach} \label{sec:4:VisTech}
In this section, we first describe the conventional DMD approach with the traditional DMD components and visualizations by highlighting their benefits and drawbacks.
Then, we show how these components can be improved and visualized in a novel way to resolve these issues.
On this basis, we present two clustering methods that segment the flow into physically relevant sections.
Finally, an approach for the selection of DMD components is proposed.
\begin{figure}[t]
\centering
{\includegraphics[width=0.63\columnwidth]{pictures/old_new_plots/old_dom.png}}
\hfill
{\includegraphics[width=0.36\columnwidth]{pictures/old_new_plots/old_ew.png}}
\caption{
The traditional visualization of the amplitudes (left) and eigenvalues (right) for the 2D Karman vortex street.
The absolute values of amplitudes $\lvert a_j \rvert$ are visualized in a bar diagram that is sorted by the frequency of their corresponding eigenvalues.
The visualization neither takes temporal aspects nor connections to the eigenvalue visualization into account.
The eigenvalues are represented in the complex plane encoded by their corresponding amplitudes, where light gray indicates insignificant components.
For real-valued data (which is the case for flow data), the eigenvalues occur in complex conjugate pairs.
Hence, redundant information is visualized.
Moreover, the radial representation has several drawbacks for the identification of patterns.
}
\label{pic_karman_old_ew_dom}
\end{figure}
\subsection{Traditional DMD Components}
In the following, we point out the pros and cons of the conventional DMD approach.
DMD computes the following triplets: modes $\vartheta_j$, amplitudes $a_j$, and eigenvalues $\lambda_j$.
Traditionally, the three groups of components are mainly considered and visualized separately.
\paragraph{\textbf{DMD Modes}}
The modes represent the spatial contribution to the flow field and are supposed to display local and global features, like symmetry, mixing, transient response, and long-time behavior.
Each entry of a vector-valued mode $\vartheta_j$ corresponds to a spatial location of the velocity field as shown in Figure~\ref{pic_karman_modes_old}.
For the investigation with DMD, traditionally, the modes are considered individually.
However, as DMD is based on the superposition principle, modes that are inspected must be selected carefully.
Otherwise, the validity of spatial features is not ensured.
Moreover, the modes are complex-valued and therefore mostly visualized by their real and imaginary part such that an interpretation is even more complicated.
A noteworthy mode is the one with corresponding eigenvalue $\lambda_j = 1$.
If it exists, it represents the constant flow, often called time-averaged flow, which does not change over time.
\paragraph{\textbf{DMD Amplitudes}}
The amplitude $a_j$ determines the influence of the mode $\vartheta_j$ to the flow.
As mentioned before, the amplitudes are traditionally computed by $a=\Theta^+x_0$ instead of $a=\Theta^+x_1$ (compare Algorithm~\ref{Algorithm_DMD}) such that a reconstruction like in Equation~\ref{eq:theroem} does not hold.
These complex numbers are then visualized by their absolute value, i.e., $\lvert a_j \rvert$, in a bar diagram sorted by the frequency of their corresponding eigenvalues as illustrated in Figure~\ref{pic_karman_old_ew_dom} on the left.
The visualization gives an overview of the distribution of modes' influence.
On that basis, the selection of relevant modes is performed.
More precisely, a choice of the $k$-th most dominant modes is made by the $k$ highest absolute values of amplitudes.
This approach does not take temporal patterns into account, since the amplitudes only precisely reflect the influence of modes at time $t_0$.
Additionally, the influence is not precisely exact for the traditional amplitudes and the corresponding modes are not necessarily normalized.
Therefore, a misleading choice of modes could be made that do not represent any features of the flow, especially the ones that evolve over time.
Moreover, redundant components are highlighted in Figure~\ref{pic_karman_old_ew_dom} (as we will prove later) that may lead to unnecessary selection and visualization of modes.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{pictures/old_new_plots/new_ew_dom_small.png}
\caption{Our proposed visualizations of the eigenvalues (bottom) and the dominance structure (top) for the 2D Karman vortex street.
The non-redundant eigenvalues are represented by their frequencies (arguments) and growth rates (magnitudes) and are grayscaled by the norms of their associated scaled modes, where light gray indicates insignificant components.
The dominance structure visualizes the influence of non-redundant scaled modes.
For this, the norms of non-redundant scaled modes are illustrated in a bar diagram that is sorted by the frequencies of their corresponding eigenvalues.
In addition, the temporal development of each scaled mode at specific time steps is integrated into the representation.
}
\label{pic_karman_new_ew_dom}
\end{figure}
\paragraph{\textbf{DMD Eigenvalues}}
An eigenvalue of a corresponding mode describes its temporal behavior.
The position within the complex plane provides information about frequency and growth rate. These quantities are given by their arguments and magnitudes, respectively.
Usually, the eigenvalues are (gray) scaled by the absolute value of their corresponding amplitudes, as illustrated in Figure~\ref{pic_karman_old_ew_dom} on the right.
For real-valued data, the eigenvalues occur in complex conjugate pairs, i.e., if $\lambda = r \; e^{i\varphi}$ is an eigenvalue, then $\overline{\lambda} = r \; e^{-i\varphi}$ is an eigenvalue, too.
Therefore, the traditional visualization in the complex plane, as shown in Figure~\ref{pic_karman_old_ew_dom} on the right, displays redundant information.
Using the exponential form $\lambda = r \; e^{i\varphi}$ with frequency $f = \frac{\varphi}{2\pi}$ and magnitude $r$ as well as Equation~\ref{eq:theroem}, the eigenvalues can be categorized in the following way:
\begin{itemize}
\item $r_j<1$: These components describe transient responses, because the potentiation of the eigenvalues will cause the component to vanish.
\item $r_j>1$: Such components show contrary behavior: due to the potentiation of the eigenvalue, the components will grow and diverge.
\item $r_j=1$: In this case, potentiation of the eigenvalue will cause a rotation on the unit circle with a specific frequency.
These components characterize the steady state.
\end{itemize}
Even though the categorization into these three cases is possible and the interpretation of eigenvalue potentiation is more accessible (see Figure~\ref{pic_karman_old_ew_dom}), the detection of recurring and salient patterns suffer from the radial representation.
\subsection{Improved DMD Components}
In this subsection, we present an improvement to the conventional DMD components and visualizations that resolves the issues explained in the previous subsection.
\paragraph{\textbf{DMD Eigenvalues}}
For real-valued data (as given in many applications), the eigenvalues occur in complex conjugate pairs (compare Figure~\ref{pic_karman_old_ew_dom} on the left).
This well known fact follows from a real-valued SVD resulting in the real-valued matrix $S$ (see Equation~\ref{eq:matrixS} or appendix).
Therefore, the representation of the eigenvalues can be restricted to the upper half plane (complex numbers with non-negative imaginary part), which hides redundant components and leads to a clearer view.
Furthermore, the visualization in the complex plane has some disadvantages as well.
It is difficult to detect recurring and salient patterns due to the radial representation of the eigenvalues.
Therefore, we propose a characterization with more emphasis on the frequency and growth rate.
To this end, we remove the redundant eigenvalues and plot the remaining ones in a coordinate system with axes set by argument and magnitude, as visualized in Figure~\ref{pic_karman_new_ew_dom} at the bottom.
This representation enhances the identification of repetitive patterns significantly, which is the basis for the cluster methods later on.
In addition, the eigenvalues are grayscaled by the norms of the so-called scaled DMD modes that will be presented in the following.
In short, the norms represent the correct influence of components.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{pictures/karman/karman_Mode_EW1.png}
\hfill
\includegraphics[width=\columnwidth]{pictures/karman/karman_Mode_EW1_s.png}
\caption{
The traditional DMD mode to the eigenvalue $\lambda = 1$ is compared to our scaled DMD mode.
Both are taken from the 2D Karman vortex street data set.
The velocity vectors of the traditional DMD mode point in the wrong direction, as the inflow is on the left side.
Scaling the DMD mode resolves this issue.
}
\label{pic_dmd_mode_EW1_compare}
\end{figure}
\paragraph{\textbf{Scaled DMD Modes}}
The DFT can be linked to DMD because DMD can be seen as a two-stage method where eigenvalues computed first and the modes and amplitudes are fitted subsequently to those.
The Fourier-transformed vectors are the counterpart to the modes $\vartheta_j$ multiplied with their amplitudes $a_j$ (see Section~\ref{ssec:DMD_DFT}).
Thus, we propose scaling the modes for the analysis and denoting these new objects $a_j \vartheta_j$ as scaled DMD modes.
Due to this combination, the new representation contains both the spatial contribution to the decomposition and the influence to the system.
Before we discuss these two aspects, some advantageous mathematical properties of the representation are presented.
One important property of the scaled modes is that they occur in complex conjugate pairs, i.e., the scaled modes of a complex conjugate pair of eigenvalues $\lambda, \overline{\lambda}$ are given by $a \vartheta, \overline{a \vartheta}$.
A detailed mathematical proof of this fact can be found in the appendix.
Hence, the superposition of these two scaled modes can be expressed by twice the real part, since $a \vartheta + \overline{a \vartheta} = 2 \Re(a \vartheta)$.
Furthermore, the spatio-temporal development is given by
\begin{equation} \label{eq:combining_counterparts}
\lambda^k a \vartheta + \overline{\lambda^k a \vartheta}
= 2 \Re(\lambda^k a \vartheta)~, \quad k \in \mathbb{N}~.
\end{equation}
As a result, we can combine these complex conjugate pairs, which facilitates the analysis approach.
In particular, the number of eigenvalues $\lambda_j$ and scaled modes $a_j \vartheta_j$ (with an eigenvalue having a non-zero imaginary part) is reduced by a factor of two.
This aspect is crucial for all following steps.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{pictures/karman/karman_reconstruction_thin_cluster_stuff.pdf}
\caption{
The reconstruction error over time (left) of different subsets of components (I, II, and III) from the 2D Karman vortex street data set.
The eigenvalues of the respective clusters are shown on the right.
Whereas the subset III consists of all components, leading to an error-free reconstruction of the flow, the subsets I and II are thinned out and used for the cluster approaches that lead to a selection of modes.
Different thresholds facilitate the identification of patterns and support the clustering of components (red and blue on the right).
}
\label{pic_karman_reconst_and_cluster}
\end{figure*}
\textit{(a) Spatial Properties:}
The multiplication of a mode $\vartheta_j$ by its (complex) amplitude $a_j$ corrects its orientation.
To demonstrate this, the traditional mode to the eigenvalue $\lambda = 1$ is depicted in Figure~\ref{pic_dmd_mode_EW1_compare} (top), often referred to as time-averaged flow.
We observe that the velocity vectors point in the wrong direction, as the flow moves from the left to the right.
Since this mode is real-valued and the only one that has impact on the boundary regions, the flow direction is truly incorrect and therefore the mode does not display the physical phenomena correctly (the time-averaged flow).
This is due to the fact that the modes are eigenvectors, which can be scaled by any complex factor.
To represent real physical properties, a mode $\vartheta_j$ needs to be linked to the data by multiplying it with its amplitude $a_j$.
In Figure~\ref{pic_dmd_mode_EW1_compare} (bottom), the scaled mode to the eigenvalue $\lambda = 1$ is visualized.
Now, the velocities point in the correct direction.
This simple example demonstrates that the combination of amplitudes and modes is crucial for the interpretation of spatial features and improves the fundamental (numerical) representation.
For the visualization of scaled modes, we make use of Equation~\ref{eq:combining_counterparts}.
In the analysis of real-valued data, the scaled modes and, in particular, their temporal development can thus be restricted to the real part, since the imaginary part will vanish in a superposition, as Equation~\ref{eq:combining_counterparts} shows.
\textit{(b) Influence Properties:}
The traditional amplitudes (typically used for the selection of modes) suffer from an imprecise computation as Equation~\ref{eq:theroem} does not hold exactly.
As a result, the influence of modes is not reflected correctly.
Using the proposed improved representation, i.e., the scaled mode $a_j \vartheta_j$ computed by Algorithm~\ref{Algorithm_DMD}, the influence is given by its norm $\lVert a \vartheta \rVert$.
This formula is in accordance to the influence of Fourier-transformed vectors and is more precise.
Even though a correct choice of the most dominant components is now more likely, the norm only represents the influence at time $t_0$.
A selection of components only based on the norm of scaled modes (or simply on the absolute value of amplitudes) is insufficient as a temporal encoding is not included (compare Figure~\ref{pic_karman_old_ew_dom} on the left).
To integrate this, we propose a visual representation, referred to as dominance structure, that uses both the scaled modes and the eigenvalues.
This representation is illustrated in Figure~\ref{pic_karman_new_ew_dom} (top).
Basically, the norms of the scaled DMD modes $\lVert a \vartheta \rVert$ are visualized in a bar diagram that is sorted by the frequency of their corresponding eigenvalues.
However, it also takes the following aspects into account:
\begin{itemize}
\item Since the scaled modes occur in complex conjugate pairs (like the eigenvalues), it is sufficient to display one of them, as they have the exact same impact.
\item Due to the sorting according to the frequency (argument), it is possible to differentiate between the impact of low and high frequencies, which is in the spirit of DFT (see Figure~\ref{pic_DMD_DFT}) and the eigenvalue visualization (see Figure~\ref{pic_karman_new_ew_dom}).
Therefore, the two visualizations can be linked as both rely on the same quantities.
More precisely, the frequency is represented in an ordinal way for the dominance structure (i.e., sequence of numbers) and a quantitative way for the eigenvalues (i.e., exact positions).
\item The temporal development of a scaled mode $a_j \vartheta_j$ (according to the norm) is integrated into the representation.
This development is given by the values $\lVert \lambda_j^k a_j \vartheta_j \rVert$ at specific time steps $k$, which are visualized by color-coded bars.
The corresponding time steps $k$ are illustrated by a color map on the right.
For the sake of visibility, a color-coded bar should be always either placed in the foreground (if $\lvert \lambda_j \rvert<1$) or background (if $\lvert \lambda_j \rvert>1$).
\end{itemize}
Since the time step $k=0$ (here, highlighted by dark blue) is considered, the visualization is an extension and an improvement to the traditional one shown in Figure~\ref{pic_karman_old_ew_dom}.
It allows for new insights into the influence of scaled modes and supports the understanding of the interplay.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{pictures/quadgyre/quadgyre_comp_v2.png}
\caption{
The procedure of the harmonic cluster approach applied to the superposed quadgyre data set:
On the left, selected snapshots from the two unsteady flow fields (Flow I and II) are shown that characterize the superposed quadgyre.
These data sets consist in each case of four vortices moving periodically from left to right (Flow I) or from top right to the bottom right in a crescent-shaped move (Flow II).
The eigenvalue representation in the middle reveals two different frequency patterns that can be captured by the harmonic clustering approach.
For each aggregation, the three most dominant modes are represented by their temporal development.
It can be observed that each aggregation characterizes one respective base flow, which verifies the usefulness of the cluster method.
}
\label{pic_superposed_quadgyre}
\end{figure*}
\subsection{Aggregation and Selection of Components}
DMD decouples time-dependent flow into spatial and temporal components.
For an appropriate selection of components, however, the spatio-temporal character of DMD needs to be taken into account.
Mathematically, a selection of components can be formalized by a subset $\mathcal{C} \subseteq \{1,\dots,r_0\}$ of all components.
Having defined a subset $\mathcal{C}$, we denote the temporal development of it at step $k$ as:
\begin{equation} \label{eq:cluster}
\lambda^k_\mathcal{C}a_\mathcal{C}\vartheta_\mathcal{C} \coloneqq \sum_{i\in\mathcal{C}}{\lambda^k_i a_i \vartheta_i}~.
\end{equation}
If the traditional dominance-based approach for the selection of components is consulted, then the components are chosen according to the norms of scaled modes (or actually to the absolute values of the amplitudes).
Figure~\ref{pic_karman_reconst_and_cluster} (left) shows the reconstruction error over time (i.e., $\lVert x_k - \lambda^k_\mathcal{C}a_\mathcal{C}\vartheta_\mathcal{C} \rVert$ plotted for $k=0,\dots,m$) for three examples of subsets $\mathcal{C} \in \{\mathcal{C}_I,\mathcal{C}_{II},\mathcal{C}_{III}\}$.
The non-redundant eigenvalues belonging to the three subsets are illustrated in Figure~\ref{pic_karman_reconst_and_cluster} on the right (compare Figure~\ref{pic_karman_new_ew_dom}).
The subset $\mathcal{C}_{III}$ contains all components and is the maximally achievable order of accuracy
(the reconstruction is exact, since the conditions of Equation~\ref{eq:theroem} are satisfied).
The other two subsets $\mathcal{C}_I$ and $\mathcal{C}_{II}$ contain the components with the $k$ highest influence, where different thresholds have been chosen.
However, this traditional approach for the selection of components neither clarifies the interplay of the chosen components nor classifies them appropriately.
In addition, due to the superposition principle, an incoherently selection may lead to components that eliminate each other.
In sum, incoherently selected (scaled) components may not describe any relevant spatial and temporal features of the flow.
Another approach for the selection of components can be performed on the basis of the improved components and visualizations.
Since temporal aspects are encoded in the visualization of the dominance structure, it probably provides a better tool for the selection of components.
In general, it can be conducted in the following way (where we always select only non-redundant components):
First, the components with a high influence at a certain time step $k_0$ are selected.
This is achieved by selecting every component whose value $\lVert \lambda_j^{k_0} a_j \vartheta_j \rVert$ is higher than a chosen fixed threshold.
If $k_0 = 0$, then the procedure is equal to the traditional one.
In a second step, undesirable components are sorted out manually.
Using the new dominance structure, non-selected components that represent important steady state parts can be added.
In contrast, selected components that vanish extremely fast can be excluded.
With the traditional visualization (Figure~\ref{pic_karman_old_ew_dom} on the left), this is not possible, even if the eigenvalue visualization (Figure~\ref{pic_karman_old_ew_dom} on the right) is consulted additionally.
To assist this procedure, we keep attention to the reconstruction error over time (see Figure~\ref{pic_karman_reconst_and_cluster}) using the selected ones as well as all modes as a reference.
This shows the precision of the chosen components and helps substantiate the selection of components.
The proposed approach based on our visualizations is useful for a first impression or an explorative analysis.
Our experiences have shown that it is expedient for simpler data sets.
For complex systems that exhibit several different time-dependent phenomena, the interplay, interpretation, and classification of selected components still may remain unclear.
So far, the selected (scaled) modes were classified by the location of their corresponding eigenvalues.
This can lead to the fact that components eliminate each other for certain sections of the flow.
Therefore, an appropriate criterion for the selection of components is to classify those which represent a certain section of the flow accurately.
Mathematically, a discrete optimization problem can be formulated: For a certain time section $0 \leq k_1 < \dots, k_2 \leq m$, we look for a subset $\mathcal{C}_M$ with $0<M<r_0$ elements that satisfies
\begin{equation} \label{eq:measure}
\min_{\mathcal{C}_M} \sum_{k = k_1}^{k_2} \lVert x_k - \lambda^k_\mathcal{C}a_\mathcal{C}\vartheta_\mathcal{C} \rVert_2^2~.
\end{equation}
This technique can be obviously used for the selection of components, since each computed subset may characterize a section of the flow and the components its features.
For (almost) periodic data sets, the segmentation of the flow into sections is irrelevant, however, a classification into different base frequencies is of importance.
The discrete optimization problem (Equation~\ref{eq:measure}) can practically not be solved as the computational complexity grows with $\binom{r_0}{M}$ (for non-rank-deficient data $\binom{m}{M}$) for the selection of $M$ components.
Instead of using this time-consuming (PCA-like energy sorted) selection technique, we propose two fast clustering approaches for aggregating components that will detect the same components (as we will show later).
As shown above, the aggregated components reveal relevant physical features classifying the flow into segments such as the transient response or the steady state.
Subsequently, the components can be investigated individually with regard to the specific detected feature.
The following two clustering approaches operate on the eigenvalues and their new representation, i.e., we aggregate on the basis of temporal patterns.
Therefore, every aggregation include the complex conjugate counterpart of a components such that the temporal development should be evaluated using the real part as in Equation~\ref{eq:combining_counterparts}.
\paragraph{\textbf{Distance-Based Clustering}}
Several flow phenomena like damping processes are characterized by modes that exhibit very similar frequencies and growth rates.
Therefore, we propose aggregating components with closely located eigenvalues.
For this procedure, we first thin out the potential eigenvalues such that the flow can be reconstructed adequately by them.
Then, a distance-based clustering method is applied to those eigenvalues.
The methodology is highlighted in Figure~\ref{pic_karman_reconst_and_cluster} by (I).
It often leads to multiple clusters $C_{11} ,C_{12}, ...$ that may reveal features.
However, the eigenvalue $\lambda =1$ is always added manually to each cluster.
To check the relevance of a cluster, the temporal development is consulted.
For instance, the reconstruction error over time of cluster $C_{11}$ is shown in Figure~\ref{pic_overview} on the right.
\paragraph{\textbf{Harmonic Clustering}}
For the identification of temporal patterns, DFT uses harmonics, i.e., multiples of frequencies.
We adapt this concept and aggregate components that exhibit patterns of harmonics.
The clustering approach uses again an appropriately thinned out set of eigenvalues whose components reconstruct the flow adequately.
Then, we search for multiples in the eigenvalue representation, either manually or algorithmically.
The choice depends on the distribution and complexity of eigenvalues.
The process is demonstrated in Figure~\ref{pic_karman_reconst_and_cluster} by (II) (or Figure~\ref{pic_superposed_quadgyre}).
If there is more than one cluster, the eigenvalue $\lambda = 1$ should not be included in the cluster as constant parts are always mixed and not separable.
However, the dynamic behavior is separated by this approach, which is the key point.
For instance, the reconstruction error over time of cluster $C_{21}$ is shown in Figure~\ref{pic_overview} on the right.
In sum, we propose using the two clustering approaches for the selection of components.
A selection consists of united clusters (found from the clustering approaches), each reveals features of the flow by inspecting the respective individual components.
This is due to the fact that the aggregations segment the flow into relevant sections and therefore classify the components.
In Figure~\ref{pic_overview}, the two clustering approaches are illustrated representing different phenomena of the flow.
\section{Results}
To demonstrate the usefulness of our proposed techniques, we apply them to unsteady flow fields to identify different frequency-based features.
The first example is a generated synthetic flow, called superposed quadgyre.
It is an overlay of two artificial 2D flow fields, called quadgyre, which are extensions of the double gyre~\cite{SHADDEN:2005:LCS} flow field
This scenario demonstrates the application of DMD to periodic flows and illustrates the correctnesss of our aggregation approaches.
The next example is a simulated von Karman vortex street that is a more complex unsteady flow resulting in an equilibrium state.
Our improved techniques allows us to identify the relevant components that are associated to the transient response and steady state.
Finally, we consider a 3D von Karman vortex street to show how the approach carries over to 3D.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{pictures/quadgyre/quadgyre_dominance_new.png}
\caption{
The dominance structure of the superposed quadgyre dataset.
Besides of decreasing behavior for higher frequencies, we immediately observe no converging and diverging parts.
We conclude the periodicity of the flow.
Moreover, the representation indicates the existence of two different decay patterns.
}
\label{pic_quadgyre_new_dom}
\end{figure}
\subsection{Example 1: Superposed Quadgyre}
When applying DMD to overlapping periodic phenomena, such as the superposed quadgyre data set, we obtain characteristic features with DMD.
Our improved techniques identify these and the clusters represent each individual periodic phenomena as we will demonstrate in the following:
The superposed quadgyre data set is a superposition of two unsteady flow fields with different base movements and frequencies.
A full period of each flow is illustrated in Figure~\ref{pic_superposed_quadgyre} (left).
The analytical formula of both flows is given by
\begin{align*}
u(x,y,t) &= -\pi A \sin{(\pi f(x,t))} \cos{(\pi g(y,t))} \frac{dg}{dy}(y,t)~, \\
v(x,y,t) &= \pi A \cos{(\pi f(x,t))} \sin{(\pi g(y,t))} \frac{df}{dx}(x,t)~,
\end{align*}
where $f(x,t) = \varepsilon \sin{(\omega_f t + s_f)}x^2 + x-2 \varepsilon \sin{(\omega_f t + s_f})x$ and $g(y,t) = \varepsilon\sin{(\frac{\omega_g}{2} t + s_g)}y^2 + y -2 \varepsilon \sin{(\frac{\omega_g}{2} t + s_g)}y$.
Both data sets consist of four vortices that move periodically.
The vortices in the first dataset move right to left and back with parameters $A = 1$, $\varepsilon = 1$, $\omega_f = \frac{8}{27}\pi$, $\omega_g = 0$, $s_f = \frac{1}{5}\pi$, and $s_g = 0$.
The vortices from Flow II move from top right to bottom right in a crescent-shaped motion with parameters $A = 1$, $\varepsilon = 1$, $\omega_f = \omega_g = \frac{1}{2}\pi$, and $s_f = s_g = \frac{1}{3}\pi$.
The flow fields are sampled at a resolution of 201 $\times$ 201 cells (resulting in a snapshot dimension of $n = 80802$) and have 50 time steps.
To analyze the flow decomposition, we first inspect the eigenvalue plot (Figure~\ref{pic_superposed_quadgyre}, center).
We observe that all eigenvalues have an absolute value of one, i.e., the flow is periodic and there are no converging or diverging phenomena.
This is confirmed by the dominance structure (Figure~\ref{pic_quadgyre_new_dom}) as all bars are colored red.
The next step is to select relevant components.
As mentioned before, for periodic systems, it is more of importance to classify the data into different base frequencies.
Hence, instead of using the traditional dominance-based approach for the selection of modes, we use the proposed harmonic clustering approach.
Two clusters $C_I$ and $C_{II}$ are detected accurately as demonstrated in Figure~\ref{pic_superposed_quadgyre}.
To verify the relevance and correctness of the two found aggregations, where each belong to a different base frequency phenomena, we compare each of them with the suitable analytical base flow.
More precisely, we consider the error between the temporal development of an aggregation (without the constant part) and the respective base flow, where we subtracted the mean.
The error plot is depicted in Figure~\ref{pic_quadgyre_reconst_2datasets}.
It can be observed that the relative error is approximately 4\% for both clusters.
Hence, each found cluster characterizes one of the base flows (which form the superposed quadgyre by superposition).
This shows that our clustering approach can extract and classify overlapping phenomena with different frequency patterns.
Therefore, the components of the two aggregations can now be analyzed separately from each other.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{pictures/quadgyre/quadgyre_reconst_2datasets2.pdf}
\caption{
The plot shows the error between the temporal development of the two aggregations found by the harmonic clustering approach applied to the superposed quadgyre data set and the respective original flows, which are mean subtracted in order to eliminate constant parts (compare Figure~\ref{pic_superposed_quadgyre}).
Since the error is very small, each cluster captures the dynamic behavior of one original flow accurately.
}
\label{pic_quadgyre_reconst_2datasets}
\end{figure}
In Figure~\ref{pic_superposed_quadgyre} (right), the three most important modes of each cluster are depicted as well as their temporal development.
The upper aggregation belongs to the standard quadgyre.
A similar flow was investigated by Brunton et al. \cite{Brunton:2015:CompressedDMD}, however, our approach suppresses the negligible complex conjugate scaled modes.
The upper three modes show symmetric vanishing and recurring vortices indicating a periodic flow in the horizontal direction.
The lower three modes indicate a crescent shape movement of vortices.
The temporal development of the first mode shows the two vortices in the top and bottom right as well as the emerging vortex on the left.
This vortex can be found in the other two modes as well.
Whereas the second one mainly describes the crescent-shaped movement (since the frequency is twice as the base frequency), the last one highlights the vortices in the left upper and lower corners.
The temporal evolution of the modes (Figure~\ref{pic_superposed_quadgyre} right) shows that the symmetry properties of both systems are conserved.
If the DFT is consulted for such a frequency-based investigation, a classification like this would not work as the frequencies cannot be identified directly.
As the DFT uses uniformly distributed frequencies depending on the total number of snapshots, the mixed frequencies of the superposed quadgyre will not be determined.
Therefore, the frequency detection with DFT is blurred and none of the original flows is precisely detected.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{pictures/karman/pic_karman_sDMDMode_1.png} \includegraphics[width=\columnwidth]{pictures/karman/pic_karman_sDMDMode_2.png} \includegraphics[width=\columnwidth]{pictures/karman/pic_karman_sDMDMode_3.png} \includegraphics[width=\columnwidth]{pictures/karman/pic_karman_sDMDMode_4.png}
\caption{
Visualization of scaled modes from the 2D Karman vortex street.
These are selected from an aggregation that represent the steady state.
The aggregation was found by the harmonic clustering approach.
}
\label{pic_karman_dmd_modes}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{pictures/old_new_plots/new_ew_dom_karman3D.png}
\caption{
Our visualizations of the dominance structure and eigenvalues for the 3D Karman vortex street.
We observe the identical patterns as in the 2D counterpart.
}
\label{pic_karman3d_ew_dom}
\end{figure}
\subsection{Example 2: 2D Karman Vortex Street}
To compare our improved components and visualizations with the traditional ones, a flow past a circular object forming a von Karman vortex street is investigated (see Figure~\ref{pic_overview} on top).
This typical analysis example for DMD was simulated on a grid with 881 $\times $ 166 points.
Applying DMD to the 251 snapshots results in 250 DMD components.
In Figure~\ref{pic_karman_new_ew_dom}, the dominance structure and eigenvalues are visualized with our new approach.
In this example, we are faced with a rather complicated dominance structure.
It can be observed that a lot of scaled modes have a large influence on the entire system.
However, we can also immediately observe that nearly all components characterize damping phenomena, in particular those with a high impact at the beginning.
After about 50 time steps, many components have vanished and parts of the steady state (highlighted by pure red bars) will become more dominant.
Hence, by the temporal encoding in the dominance visualization, we get a precise understanding of the component's impact.
In addition, as the two axes of the dominance and eigenvalue visualization are linked, we observe a similar wave-shaped contour.
The traditional visualizations in Figure~\ref{pic_karman_old_ew_dom} do not detect or highlight any of these aspects.
Based on the observations of our improved visualizations, it may be possible to make an adequate selection of components, however, we want to demonstrate the usefulness of our cluster approaches for the selection of components.
In Figure~\ref{pic_karman_reconst_and_cluster}, the reconstruction error over time for different subsets of components is shown where the corresponding eigenvalues are represented on the right.
For the aggregation of components, we use both clustering approaches on these different representations that facilitate the detection of patterns.
The whole procedure is demonstrated in Figure~\ref{pic_karman_reconst_and_cluster} and the two chosen clusters $C_{1,1}$ and $C_{2,1}$ segment the flow into physically relevant parts.
Figure~\ref{pic_overview} shows the errors over time of the components belonging to the two aggregations C1,1 and C2,1.
For the first aggregation, we observe a small error at the beginning of the simulation.
Therefore, the transient response can be described by the individual components of C1,1, whereas aggregation C2,1 characterizes the steady state as the error decreases over time.
Thus, we have detected two important phenomena that classify the components.
A comprehensive time-continuous and -discrete analysis of wake flows with DMD was conducted by Bagheri \cite{bagheri:2013:KoopamnCylinderWake}.
In this context, a classification was achieved that equals the result of our cluster approaches.
To validate the cluster approaches once again, we computed the discrete minimization problem in Equation~\ref{eq:measure}.
We applied it both to a section in the beginning and in the end, where we search for a subset $\mathcal{C}_M$ with $M=4$ components.
The resulting components correspond to the ones found by the cluster approaches.
Consequently, these components are the best selection for the representation of the transient response in the beginning and the steady state in the end.
Tu et al.\cite{tu:2014:on_dmd_theorey_and_app} combine multiple experiments to detect harmonics of a von Karman vortex street.
They showed that in a simple experiment the patterns are harder to identify.
However, using our improved components and visualizations as well as the harmonic clustering approach, the patterns can be detected easily.
For the further analysis of these phenomena, we restrict the investigation to the components within these two aggregations.
The scaled mode as well as the traditional mode corresponding to the eigenvalue $\lambda=1$ (which is contained in both clusters) are compared in Figure~\ref{pic_dmd_mode_EW1_compare}.
As mentioned before, the scaled mode represents the flow accurately, unlike the traditional mode.
Figure~\ref{pic_karman_dmd_modes} shows the four scaled modes from cluster C2,1.
These components represent the characteristic Karman vortices by superposition.
In accordance with the sorting, the scaled modes reveal more and more fine scale patterns.
This is due to the fact that fine scale patterns are represented by higher frequencies.
Moreover, the four scaled modes are absolutely symmetrical (with regard to the y-axis), whereas the traditional modes do not exhibit these structures \cite{tu:2014:on_dmd_theorey_and_app}.
\subsection{Example 3: 3D Karman Vortex Street}
DMD is independent of the dimensionality of the spatial domain.
For an analysis with DMD, we just need to adjust the visualization of the scaled modes to the dimensionality of the spatial domain.
Therefore, our investigation approach with DMD can be equally applied to 3D unsteady flow, except for the computational time.
To demonstrate this, we analyze a 3D von Karman vortex street representing an extension to the 2D counterpart.
The grid has a resolution of $228 \times 44 \times 44$ and 381 snapshots were considered.
The flow field starts analogously as the 2D scenario in an emerging phase of the vortex street.
In Figure~\ref{pic_karman3d_ew_dom}, the dominance structure and eigenvalue visualization are depicted.
In both visualizations, we observe the same structure as in the 2D counterpart in Figure~\ref{pic_karman_new_ew_dom}.
An evaluation can be indeed conducted analogously.
For the investigation of spatial properties, we exemplarily visualize a mode in Figure~\ref{pic_karman3d_dmd_mode}.
Our visualization for the 3D scaled mode does not aim to highlight features, it is used to show similarities with the 2D counterpart.
We use a volume based representation as well as a cross section that shows the interaction inside.
The cross section exhibits similar structures as the modes from the 2D counterpart (Figure~\ref{pic_karman_dmd_modes}).
For 3D data sets, the computation time is affected moderately as only the SVD of the matrix $X$ and the computation of the amplitudes scales linearly with the additional dimension.
Furthermore, our visualizations and clustering approaches are only based on multiple calculations with the DMD eigenvalues.
Therefore, no overhead is produced.
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{pictures/karman_3D/karman3D_mode.png}
\caption{
Visualization of a scaled mode for the 3D von Karman vortex street.
A volume-based representation with a cross section is used to show the connection to the 2D case.
The frequency of the second 2D mode in Figure~\ref{pic_karman_dmd_modes} is approximately equal and, therefore, the same spatial behavior can be observed in the cross section.
}
\label{pic_karman3d_dmd_mode}
\end{figure}
\section{Conclusion}
In this paper, we have thoroughly recapped the mathematical foundation of DMD that allows us to combine and improve the DMD components such that the underlying physics is represented more adequately, e.g., by the construction of scaled modes.
Moreover, the interplay of the components is clarified and a new view on DMD is given by comparing it to DFT.
These new insights are used to design appropriate visualizations such that the spatio-temporal character of DMD is respected and redundant parts are hidden.
Therefore, a more adequate selection of components can be made and the identification of specific patterns and features is facilitated.
The two novel clustering approaches should be additionally consulted for the selection of components as these segment the flow into physically relevant sections.
In sum, a deeper understanding of the DMD components is gained that make DMD more accessible for users.
These new techniques may also be combined with other visualization methods that highlight further features in the components.
For instance, we could apply methods like FTLE to the temporal development of scaled modes.
In this case, FTLE could be seen as a post-processing step to gain another view on the spatial components.
Another interesting research direction might be the application of our techniques to other data.
So far, we only used data represented by a grid that consists of velocity components.
It would be interesting to choose a different data representation, like particle-data, and evaluate different quantities, e.g., pressure or vorticity.
We plan to apply our proposed techniques to other (real-valued) data and evaluate the effectiveness for different applications again.
For example, the clustering approaches are only useful, if the data exhibits similar frequency patterns such as fluid flow.
\acknowledgments{
This work is partly supported by “Kooperatives Promotionskolleg Digital Media” at Hochschule der Medien and the University of Stuttgart.}
\bibliographystyle{abbrv-doi}
|
1,314,259,996,666 | arxiv | \section{Introduction}
User identification technology is an indispensable element for building smart indoor applications, such as elderly and child care, personalized custom service, and
surveillance in sensitive zones. As one example, a smart home recognizing its elderly host (also his/her walking direction) may predict his/her intentions and switch on home appliances (e.g., lights or TVs) accordingly. Taking supermarket as another example, identifying customer identities and hence retrieving the relevant shopping histories may enable a salesman (or simply an advertising system) to make more appropriate recommendations and thus more likely to cut a deal. Existing solutions for user identification, ranging from conventional computer vision~\cite{SphereFace,ArcFace,CenterLoss,FaceNet} and biometrics~\cite{BiometricSensing1,BiometricSensing2,BiometricSensing3} to the emerging WiFi sensing~\cite{WiFiSensing1,WiFiSensing2,WiWho,WiFiID}, have shown promising results for several important applications.
However, the inherent limitations of existing solutions have prevented them from being widely applicable.
In particular, computer vision techniques require a good lighting condition and line-of-sight (LoS), while they may raise critical privacy concerns.
Biometrics often demand wearable instruments or users' active involvements, hence causing potential discomfort and inconvenience. WiFi sensing approaches leveraging gait profile or breathing pattern appear to be viable, but they often fail to work in practice due to the severe interference from WiFi's main function of communications and other co-spectrum devices.
Recent developments on passive behavioral biometrics~\cite{Footstep2, Footstep3} (e.g., footstep-enabled identification~\cite{FootprintID,Footstep1})
can be promising alternatives to the aforementioned solutions as they incur no privacy issues and have less interference due to their limited sensing range. However,
these approaches often require special sensing hardware~\cite{Footstep1,Footstep2,Footstep3} or multiple long-time footstep measurements~\cite{FootprintID}. Besides, they can only handle one user each time, rendering them less appealing for practical applications where multiple users may appear at the same time.
In this paper, we revisit the footstep-enabled passive behavioral biometrics, and we propose PURE\ (Passive mUlti-peRson idEntification) to achieve a lightweight user identification, exploiting both commodity sensing devices and up-to-date deep learning techniques.
PURE\ is built on a key observation that footsteps carry ``footprints'' unique to individuals and thus can be leveraged for effective user identification. These footprints can be passively captured by commodity acoustic sensing hardware, totally removing the need for active user involvements. PURE\ aims to extract such footprint information from as few as a single step, thus enabling a much faster identification than existing gait-oriented systems. In addition, PURE\ targets simultaneous multi-user identification, robustness against various background interference, as well as immunity to replay attack.
Implementing the above ideas into a practical system entails several technical challenges.
First of all, background noise and interferences (especially voices) may overwhelm footsteps,
significantly affecting the system performance. Both detecting and extracting footsteps under such strong interference can be a formidable task.
Second, footstep variations, caused by factors such as different walking paces, can lead to distinctive features hence affect the identification performance. In addition,
footstep can carry environment-dependent information (e.g., floor material). Such domain conditions are largely irrelevant to individuals' unique footprints, and can thus seriously degrade the identification performance if not sufficiently removed.
Last but not least, replay attack that records someone's footstep and cheats an identification system later by playing the recorded sounds could be a major hurdle for practical adoption in certain applications.
\begin{figure*}[b]
\centering
\includegraphics[width=0.98\textwidth]{Figs/stft1.pdf}
\caption{Mel-Frequency Cepstral (MFC) of footsteps generated by five persons clearly show distinctive profiles.}
\label{fig:mfcc of different users}
\end{figure*}
In~PURE, we tackle these challenges via a series of delicate designs. In the presence of continuous voice signals, we explore the rhythmic patterns in the time-frequency (TF) representation to detect footsteps and employ a blind source separation algorithm with the \textit{a priori} knowledge to extract footsteps.
To exclude feature variations caused by environment-dependent information and walking pace discrepancy for user identification, we train the user identifier (a predictor) via an adversarial domain adaptation scheme to improve its generalizability.
Finally, we leverage the following fact to thwart replay attack: the replayed sounds exhibit static spatial characters (e.g., Angle-of-Arrival) and reveal inconsistency between walking speed and step frequency.
To summarize, this paper makes the following contributions:
\begin{itemize}
\item We propose PURE, an acoustic passive multi-person identification system with little infrastructure cost.
%
%
\item We innovatively employ an adversarial learning scheme to combat feature variations introduced by environment-independent information or heterogeneous walking paces, thus improving the system generalizability and identification performance.
%
\item We leverage the dynamic and smoothly changing spatial characters extracted from both structure-borne and air-borne footsteps to thwart the challenging replay attack.
%
\item We implement PURE\ prototype using commodity hardware and extensively evaluate its performance under various practical settings; the results demonstrate
a cross-domain identification accuracy up to 90\%.
\end{itemize}
The rest of this paper is organized as followed: Sec.~\ref{sec:background} discusses footstep basics. In Sec.~\ref{sec:system design}, we elaborate on the details of system design. Sec.~\ref{sec:performance evaluation} reports extensive performance evaluation results.
We present a literature review in Sec.~\ref{sec:related work}, and finally conclude the paper in Sec.~\ref{sec:conclusion}.
\section{Background and Motivation}
\label{sec:background}
We first explain the basic theories about footstep, then we further provide rationales for our later designs via a few preliminary measurement studies.
\subsection{Basic Acoustics of Footstep}
\label{ssec:basics}
When a foot touches the ground, it causes minor vibrations at the impact point and radiates energy via both the air and solid medium behind the surface. The acoustic pressure radiated from this impact point can be characterized by Rayleigh's surface integral~\cite{AcousticPhysics}:
\begin{small}
\begin{align}\label{eq:air-borne property}
p\left(r, \theta, t\right) &= \frac{-2\rho}{\pi m d}\int_R \int_{\phi} R \sum _j \sum _k \left(-1\right)^{\frac{j-1}{2}}\sin \left[ j\pi \left( \frac{1}{2} + \frac{R}{a}\right) \right] \notag\\
&\times \left[ \frac{\text{d}F}{\text{d}t} \left( t - d/c \right) * \cos \left( \omega_{jk} \left( t -d/c \right) \right) \right] \mathrm{d}R\mathrm{d}\phi,
\end{align}
\end{small}
\noindent where $(r, \theta)$ describes the position relative to the impact point with respect to the floor plane, $\rho$ is the mass density that characterizes the medium properties, $c$ is the speed of acoustic signals in a certain medium, $\omega$ is the frequency, $a$ is a constant, $d$ is the length of the leg, $F$ is the impact force being zero at all time except during the impact period, and $*$ denotes convolution. The acoustic pressure, generated by the impact event, mostly radiates through two common mediums. The one propagating through piston-like and non-dispersive air channel (we refer to as \textit{air-borne} hereafter) has a constant speed (e.g., a speed of 340 m/s at a temperature of 25$^\circ$C~\cite{AcousticPhysics}) and the corresponding waveform remains identical along the propagating path as far as there are no multipath reverberations. The other one traversing in solid medium, known as blending wave (we refer to as \textit{structure-borne} hereafter), exhibits a dispersive phenomenon where the propagation speed $c_f$ (ranging from 2000~\!m/s to 3000~\!m/s) of a specific signal component is a function of its frequency $f$:
\begin{equation}\label{eq:structure-borne property}
{c_f} = \sqrt[4]{Eh{f^2} \left[12\rho (1 - v_p^2) \right]^{-1}},
\end{equation}
where $E, \rho, h$ are constants that characterize the property of a medium: $E$ quantifies the elastic property, $\rho$ is the mass density that indicates the stiffness, $h$ is the thickness, and $v_p$ is the phase velocity. Eqn.~\eqref{eq:structure-borne property} implies that when detecting structure-borne footsteps from different distances relative to the impact point, the corresponding waveform can be different due to the frequency-dependent propagation speeds.
\begin{figure}
\centering
\subfigure[Feature embeddings of different users under different environments.]{
\label{fig:user and environment impacts}
\includegraphics[width=0.42\columnwidth]{Figs/impact_of_user_env.pdf}
}
\hspace{0.01\textwidth}
\subfigure[Feature embeddings under walking pace variations. ]{
\label{fig:walking pace variation}
\includegraphics[width=0.42\columnwidth]{Figs/impact_of_pace.pdf}
}
\caption
MFC features embeddings (a) under different environments, and (b) under different walking speeds: s1 $=$ 0.2~\!m/s, s2 $=$ 0.5~\!m/s, s3 $=$ 1~\!m/s, ``other'' denotes other users at 0.2~\!m/s.}
\label{fig:measurements}
\end{figure}
\subsection{Richness of Footstep Acoustic Profile}
Eqn.~\eqref{eq:air-borne property} and~\eqref{eq:structure-borne property} together show that footstep contains both air-borne and structure-borne signals and they involve a rich set of frequency components, carrying unique identity information. For example, both a person's weight (impact force) and walking style (duration of impact) are closely related to the generated acoustic pressures.
To verify the above intuition, we record footsteps from five persons under the same circumstance and inspect the corresponding Mel-Frequency Cepstrals (MFC)~\cite{MFCC}. The MFC features depicted in Fig.~\ref{fig:mfcc of different users} clearly show distinctive visual clues and thus imply the possibility of using footsteps for person identification. We then carry out measurements under different environments and Fig.~\ref{fig:user and environment impacts} plots the embedded features. One may observe that the low-dimensional features among different users show clear boundaries, demonstrating the possibility of effective identification using footsteps. We also conduct a simple user study by playing several recording of footsteps produced by different persons; all audiences participating in our user study indeed claim that they can tell perceivable differences among these recordings. The distinctiveness of acoustic features in footsteps from different persons lays the foundation for our footstep enabled user identification system. However, the footstep features of the same person can be affected by
\textit{domain conditions} such as varying walking paces as demonstrated by Fig.~\ref{fig:walking pace variation}. These feature distinctions, introduced by domain conditions (including, e.g., environment heterogeneity and walking pace variations), can be particularly detrimental to the identification accuracy and thus should be excluded.
\subsection{Differences between Structure- and Air-borne Footsteps}
The structure-borne and air-borne components of footsteps have a sharp difference in their propagation speeds, as already discussed in Sec.~\ref{ssec:basics};
this offers us an opportunity to physically separate them. For instance, when the sampling rate is 192~\!kHz and we detect the footsteps at a distance of 2~\!m, a clean set of structure-borne signals (due to propagation difference) lasts for $\frac{1}{340} - \frac{1}{3000} \approx 5.4$~\!ms, an equivalent of 1000 samples.
\begin{figure}[htp]
\centering
\includegraphics[width=.58\columnwidth]{Figs/footstep_waveform1.pdf}
\caption{A footstep waveform contains both structure- and air-borne components that are temporally separated.}\label{fig:footstep waveform}
\end{figure}
As clear shown by the measurements illustrated in Fig.~\ref{fig:footstep waveform}, the structure-borne component indeed arrives ahead of the air-borne one.
Separating these two components allows us to exam their respective natures. As the air-borne component appears to have more complicated waveforms, we conjecture that it may be more suitable for the identification purpose than its structure-borne counterpart.
To verify the above hypothesis, we have conducted measurements on the five persons in several environments using commodity microphones.
We then utilize a Gaussian Mixture Model (GMM)~\cite{GMM} to identify those persons based on either structure-borne or air-borne components. The identification accuracy comparison under different Signal-to-Noise-Ratio (SNR) is shown in Fig.~\ref{fig:identification comparison}, which confirms the advantage of adopting the air-borne component for achieving a higher identification accuracy. Also, the experiments reveal that using traditional GMM method for identification is practically not feasible since the required SNR is over 60~\!dB.
\begin{figure}[t]
\centering
\subfigure[Identification accuracy comparison.]{
\label{fig:identification comparison}
\includegraphics[width=0.42\columnwidth]{Figs/identification_accuracy_gmm2.pdf}
}
\hspace{0.001\textwidth}
\subfigure[Feature embeddings from different locations.]{
\label{fig:structure-borne measurements}
\includegraphics[width=0.42\columnwidth]{Figs/impact_of_distance.pdf}
}
\label{fig:structure-borne measurements identification and ranging}
\caption{Examining the properties of structure-borne and air-borne components of footsteps. (a) GMM applied on air- and structure-borne signals for identification. (b) Structure-borne component carries distance information.}
\end{figure}
\subsection{The Edge of Structure-borne Footstep}
Although structure-borne footstep is less representative than its air-borne counterpart for identifying persons, it has a unique property, namely acoustic dispersion, as shown by Eqn.~\eqref{eq:structure-borne property}. The dispersion phenomenon indicates that the structure-borne waveform is modulated by distance, suggesting the possibility of using it for ranging. Our measurements in Fig.~\ref{fig:structure-borne measurements}, showcasing clear boundaries between feature embeddings of structure-borne footsteps from different locations, further demonstrate the feasibility of using structure-borne footsteps for ranging. This range information, together with the angle of signal arrival, may give us an edge over attackers who replay footstep recordings, because the smoothly changing spatial characters of human footsteps cannot be fully imitated.
\begin{figure}[t]
\centering
\includegraphics[width=0.92\columnwidth]{Figs/system_architecture-ICDCS5.pdf}
\caption{The system architecture of PURE.}
\label{fig:system structure}
\end{figure}
\section{System Design}
\label{sec:system design}
PURE consists of a pipeline of signal processing and deep learning modules as shown in Fig.~\ref{fig:system structure}. It employs a microphone array~\cite{MicrophoneArray} to capture raw audio streams and then denoise them via a background spectral subtraction, removing most stationary noises. Then footstep detection is performed by inspecting the energy change, as well as by adopting an audio classifier. During the above process elaborated in Sec.~\ref{ssec:background suppression},
if there exist continuous interferences such as voices, a source separation algorithm kicks in to decouple footsteps from the sound mixture, as detailed in Sec.~\ref{ssec:footstep extraction under continous interference}.
Finally, PURE applies a neural network to identify the user behind each footstep (Sec.~\ref{ssec:domain adapted identification}), and it also extracts spatial information to counter replay attack (Sec.~\ref{ssec:prevementing replay attack}).
\subsection{Background Noise Suppression and Footstep Detection}
\label{ssec:background suppression}
Indoor places often contain common color noises~\cite{BackgroundSubtraction} that can affect the SNR of captured footsteps and thus should be removed. From the measurements in Fig.~\ref{fig:measurements}, we can see that the spectra of footsteps almost spread over the entire available bandwidth. Therefore, using common bandwidth oriented filter to trim the out-of-band noise is infeasible. To this end, we utilize a multi-band spectral subtraction method~\cite{BackgroundSubtraction} to suppress background noise and obtain about 3~\!dB gain in SNR.
\begin{figure}[b]
\centering
\includegraphics[width=0.6\columnwidth]{Figs/gmm_clusters1.pdf}
\caption{Low-dimensional outputs of GMM classifier show clear boundaries between different acoustic noises.}
\label{fig:gmm clusters}
\end{figure}
After that, we apply abrupt energy detection, characterized by Root Mean Square (RMS)~\cite{RMS}, for candidate footstep detection. The RMS of a sequence $\mathbf{x} = \{ x_1, x_2, \cdots, x_L\}$ is defined by $E_\mathrm{RMS}(\mathbf{x}) = \sum _{i=1}^{L} \sqrt {\frac{x^2_1 + x^2_2 + ... + x^2_L}{L}}$. If the detected energy $E_{RMS}$ is above a certain threshold, a footstep may be captured. Since many transient noises can also exhibit high energy, a simple energy-based detection method would not be sufficient.
To this end, we further utilize a Gaussian Mixture Model (GMM)~\cite{GMM} based audio classifier to recognize footsteps. We train this GMM classifier against common background noises and it almost achieves an oracle performance: as shown in Fig.~\ref{fig:gmm clusters}, different acoustic signals can be clearly identified.
\subsection{Footstep Extraction under Continuous Interference }
\label{ssec:footstep extraction under continous interference}
To extract footstep overwhelmed by continuous strong interference such as voice signals, we further utilize a Flexible Audio Source Separation Toolbox (FASST)~\cite{FASST}. This procedure is often suspended for the sake of efficiency; it is only invoked when footsteps are heavily interfered. To leverage FASST, we use the rhythmic pattern of footsteps in frequency domain to detect the ongoing walking paces. More specifically, we utilize auto-correlation of STFT magnitude to detect the presence of footsteps. The reason for not directly using auto-correlation in time-domain is that the repetitive features in time-domain waveform are likely to be under noise floor due to their relatively low volume.
Suppose $\mathbf{V}, \mathbf{V} \in \mathbb{R}^{P \times Q}$ denotes the STFT power spectrum where $P, Q$ are the dimensions of time and frequency, respectively, we first calculate the auto-correlation of $\mathbf{V}$ along the time frames to obtain $\mathbf{B}$, the operation of which can enhance the rhythmic pattern of footsteps:
\begin{equation}\label{eq:detection in frequency domain}
\mathbf{B}(i, j) = \frac{1}{P-j+1} \sum _{k=1}^{P-j+1} \mathbf{V}(i, k) \mathbf{V}(i, k+j-1).
\end{equation}
We then take the average of $\mathbf{B}$ over the frequency dimension and normalize the result with its first term, $b(j) = \frac{1}{Q}\sum _{i=1}^{Q}\mathbf{B}(i, j), b(j) = \frac{b(j)}{b(1)}$. The walking rhythmic features can then be inspected in $\mathbf{b}=[b(j)]$, which we denote as the Averaged Spectrogram Auto-Correlation Coefficient (ASACC). We further check whether the periodicity exhibited in $\mathbf{b}$ lies in a reasonable range as the frequency of normal walking pace is usually within $[0.8, 2]$Hz~\cite{Zee}. Ideally, we can utilize $\mathbf{b}$ to obtain a soft binary mask for $\mathbf{V}$ and then extract the footsteps in the same vein as the proposal in~\cite{REPET}. However, as explain by~\cite{FASST}, the mask-based approaches, even with ground truth labels, are inferior to the local instantaneous Gaussian mixture models in source separation. Therefore, we still resort to FASST for better separation performance. Our measurements in Fig.~\ref{fig:correlation demonstration} further verify the above intuition as correlation in time domain shown in Fig.~\ref{fig:waveform correlation content} exhibit no rhythmic features while the rhythm is evident in frequency domain as shown in Fig.~\ref{fig:spectrogram correlation demonstration}. To further remove the residual noises in the separated signals, we apply a trained time-domain denoising network in~\cite{DEMUS} for footstep enhancement.
\begin{figure}[h]
\centering
\subfigure[Time domain correlation.]{
\label{fig:waveform correlation content}
\includegraphics[width=0.44\columnwidth]{Figs/waveform_autocorrelation.pdf}
}
\hspace{0.001\textwidth}
\subfigure[Spectrum domain correlation.]{
\label{fig:spectrogram correlation demonstration}
\includegraphics[width=0.45\columnwidth]{Figs/spectrogram_autocorrelation.pdf}
}
\caption{Time domain correlation of a mixture of footsteps and voice fails to reveal rhythmic features (a), while in frequency domain, such features can be clearly observed (b). }
\label{fig:correlation demonstration}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.93\columnwidth]{Figs/DAT1.pdf}
\caption{Identification network (ID-Net) based on domain adversarial adaptation.}
\label{fig:DAT}
\end{figure}
\subsection{Domain Adapted Identification}
\label{ssec:domain adapted identification}
As we have mentioned in Sec.~\ref{ssec:basics}, the domain conditions, including walking speed variations and environment dynamics, can be particularly detrimental to our identification system. To preclude these domain information hence improve identification accuracy, we adopt domain adversarial adaptation.
We formulate the identification problem as a \textit{classification problem}: given a specific footstep $\mathbf{x} \in {X}$, we aim to learn a maximum likelihood estimator $\mathcal{G}: {X} \rightarrow {I}$, where $X$ and $I$ represent the spaces of input (STFT of footstep waveform) and user identity, respectively.
In reality, an footstep $\mathbf{x}$ is sampled from a joint distribution $P(\mathbf{x}, u, d, v, e)$, where $u \in I$ that characterizes footstep diversity, $d \in D$ denotes distance information, $v \in V$ represents walking pace, and $e \in E$ describe specific properties incurred by environment dynamics. We denote $d, v, e$ potentially harmful to our identification purpose as \textit{domain conditions}. Apparently, only features characterizing the joint distribution $P(\mathbf{x}, u)$ are desirable for our identification purpose but features induced by domain conditions $d, v, e$ should be eliminated. To this end, we use a deep neural network $G(\mathbf{x})$ to approximate $\mathcal{G}(\mathbf{x})$ and adopt an adversarial learning~\cite{DomainAdv} to train $G$ so as to preclude the impact from $d, v, e$.
Our ID-Net is made up of three parts, namely feature extractor, identity predictor, and domain discriminator, as shown in Fig.~\ref{fig:DAT}. The feature extractor $\mathbf{f} = G_f(\mathbf{x}, {\boldsymbol\theta}_f)$, parameterized by ${\boldsymbol\theta}_f$, compresses an STFF of footstep waveform $\mathbf{x}$ into $\mathbf{f} \in \mathbb{R}^Q$, a lower-dimensional feature vector. With $\mathbf{f}$, the identity predictor aims to recognize the user behind the footstep,
while the goal of domain discriminator is to identify different domains. The input to the domain discriminator is $\mathbf{f}$ weighted by a latent vector extracted from the identity predictor. The goal of ID-Net is to extract a \textit{domain independent} representation $\mathbf{f}$ so as to i) achieve high-accuracy user identification and ii) deceive the domain discriminator to misidentify domains. It is this domain independent $\mathbf{f}$ (involving only identity-specific features) that allows a cross-domain generalization of ID-Net.
The identity predictor $\hat P_u = G_u(\mathbf{f}, {\boldsymbol\theta}_u)$ outputs a probability matrix, whose element $\hat p^{(i, j)}_{u}$ represents the probability of the $i$-th footstep belonging to the $j$-th user. The loss function for training the parameter ${\boldsymbol\theta}_u$ is categorical cross-entropy:
\begin{equation}
\mathcal{L}_u = -|I|^{-1}\textstyle{\sum _{i=1}^{|I|}\sum_{j=1}^{N_u}} \log \left(\hat p^{(i, j)}_{u} \right),
\end{equation}
where $N_u$ denotes the number of users. To further force the identity predictor to learn features sufficiently discriminative and generalizable to identify unseen users, we introduce a center loss~\cite{CenterLoss} in training the identity predictor:
\begin{equation}
\mathcal{L}_C = 0.5~\textstyle{\sum_{i = 1}^{|I|}} \left \| \mathbf{f}_i - \mathbf{c}_{z_i} \right \|^2_2,
\end{equation}
where $\mathbf{c}_{z_i} \in \mathbb{R}^Q$ is the center for the $z_i$-th class deep features, $\mathbf{f}_i \in \mathbb{R}^Q$ is the $i$-th deep feature, and the summation is performed over the input set $I$. We update $\mathbf{c}_{z_i}$ in a mini-batch (with size $m$) manner where the gradient of $\mathcal{L}_C$ with respect to $\mathbf{f}_i$ is calculated as $\mathbf{f}_i - \mathbf{c}_{z_i}$ with
$
\mathbf{c}_j = \mathbf{c}_j + \frac{\sum^{|I|}_{i=1} \mathcal{I} (z_i = j)(\mathbf{c}_j - \mathbf{f}_i) }{1 + \sum ^{|I|}_{i=1}\mathcal{I} (z_i = j)}
$,
$\mathcal{I}(z_i = j)$ is an \textit{indicator function} whose value is 1 if $z_i = j$; otherwise 0. We combine the center loss $\mathcal{L}_c$ with the categorical cross-entropy loss $\mathcal{L}_u$ to train the identity predictor as
$
\mathcal{L}_\eta = \mathcal{L}_u + \lambda \mathcal{L}_c
$,
where $\lambda$ is a scalar that balances the two losses. This loss function enables ID-Net's identity predictor to maximize inter-class margins and minimize intra-class distances, thereby improving its generalizability.
The loss for training the domain discriminator $G_d(\mathbf{f}, {\boldsymbol\theta}_d)$ is also categorical cross-entropy:
\begin{equation}
\mathcal{L}_\delta = -|I|^{-1} \textstyle{\sum _{i=1}^{|I|} \sum _{j=1}^{N_d}} \log(\hat {\delta} ^{(i,j)}),
\end{equation}
where $\hat\delta ^{(i,j)}$ is the output probability for the $i$-th footstep originating from the $j$-th domain and $N_d$ denotes the number of domains.
The overall training process aims to minimize both $\mathcal{L}_\eta$ and $\mathcal{L}_{\delta}$ by tuning their respective network parameters $\mathbf{\theta}_u$ and $\mathbf{\theta}_d$. In the meantime, $\mathbf{\theta}_f$ is tuned to minimize $\mathcal{L}_\eta$ but maximize $\mathcal{L}_{\delta}$ (via gradient reversal).
This procedure forces $\mathbf{f}$ to retain only user-specific properties but discard those induced by domains, allowing ID-Net to handle footstep samples taken from unseen domains.
\subsection{Thwarting Replay Attack via Spatial Clues}
\label{ssec:prevementing replay attack}
As a user identification system, PURE\ may be adopted in applications where security is a concern (e.g., assisting authentication and surveillance). Under these circumstances, PURE\ needs to be resilient to potential attacks, among which replay attack is the most lethal one. Similar to all other acoustic-related identification methods (e.g., speaker recognition~\cite{podder2018speaker}), footsteps can be overheard (and recorded) by a microphone and then be replayed to attack the system in the sense of faking a certain user. Fortunately, footsteps contain dynamic and smoothly changing spatial clues induced by user movements. On the contrary, a recording clip of footsteps often exhibit only static spatial characters or may otherwise suggest abnormal trajectories, so it should be readily recognizable upon fully extracting the spatial clues contained in footsteps.
To thwart replay attacks, we propose to leverage two pieces of spatial information, namely Time-of-Arrival (ToA) and Angle-of-Arrival (AoA), to filter out replayed footsteps. In particular, ToA is extracted from the structure-borne footstep and AoA is obtained from the air-borne counterpart.
The key rationale is that footsteps from a lively walking person exhibit two traits: i) the moving speed suggested by ToA is bounded, as one should not move too fast indoors (it is also suspicious for a real person to move too fast in a security-concerned area)
and ii) the moving trajectory implied by ToA should be naturally irregular, i.e., no person may move along a straight line.
On the contrary, an replayed footstep clip always violate at least one of these two criteria, as explained next, regardless of whether the attack is aware of our countermeasure or not.
We formulate the defense against replay attack as a Hypothesis Test~\cite{Hypothesis}, in which we define significance thresholds to judge whether a footstep can be accepted as an authentic one or rejected as a replayed one.
In our implementation, two thresholds are defined: i) we use Spearman coefficient~\cite{spearman} $\pi$ to quantify the correlation between walking speeds $\mathbf{v}$ and step frequency $\mathbf{f}$, as replayed footsteps often exhibit rather low correlation while those of authentic footsteps are high. ii) we exploit the maximum difference between any detected AoA $\gamma_\mathrm{diff}$ to check the motion states:
the AoAs should exhibit variance instead of being static. To be more specific, if $\pi \ge \bar\pi$ and $\gamma_\mathrm{diff} \ge \bar\gamma_\mathrm{diff}$ are both satisfied, we accept the footsteps; otherwise, we reject them, where $\bar\pi = 0.8$ and $\bar\gamma_\mathrm{diff} = 10^\circ$ are empirically set based on measurements.
We apply beamforming techniques~\cite{AudioBeamforming} to extract AoAs from footstep, but obtaining ToA, or equivalently range, is non-trivial. Intuitively, a range-sensitive acoustic fingerprinting strategy based on Sec.~\ref{ssec:basics} can be used,
but such fingerprints can be interfered by domain conditions.
Therefore, we again resort to adversarial learning to preclude domain conditions with respect to ranging.
Essentially, we train a neural network, R-Net, to infer range from a footstep $\mathbf{x}$.
This R-Net follows the same design as ID-Net except that the identity predictor is replaced by a range estimator (for regression) and the domain conditions are modified accordingly. We omit the training details given their similarity to those presented in Sec.~\ref{ssec:domain adapted identification}.
We emulate a case to show how replayed footsteps exhibit abnormal properties and can thus be detected.
We let a user (hence his/her footsteps) move from $[1, 0]$~\!m to $[1, 3.4]$~\!m, with a speed of 0.7~\!m/s (0.1~\!m/s variance) and a step frequency of 1~\!Hz (0.05~\!Hz variance). In the meantime, the microphone of PURE\ is located at $[1.5, 2]$~\!m, while an attacker hides at the origin to record the footsteps. After a while, the attacker replays the recorded footsteps, trying to impersonate the legitimate user. As shown in Fig.~\ref{fig:replay attack}, $\mathbf{v}_\mathrm{replay}$ has a significantly lower correlation with $\mathbf{f}$, compare with that of $\mathbf{v}_\mathrm{live}$: $\pi_\mathrm{replay} = 0.32 < \bar{\pi} = 0.8 < \pi_\mathrm{live} = 0.87$.
Meanwhile, Fig.~\ref{fig:spatial parameters} shows a fixed $\gamma_\mathrm{diff} = 0$
for replayed footsteps, as opposed to the meaningful one for the live ones.
\begin{figure}[h]
\centering
\subfigure[Correlation between $\mathbf{v}$ and $\mathbf{f}$.]{
\label{fig:replay attack}
\includegraphics[width=0.46\columnwidth]{Figs/coefficient4.pdf}
}
\subfigure[AoA vs. range changes.]{
\label{fig:spatial parameters}
\includegraphics[width=0.46\columnwidth]{Figs/spatial_parameters3.pdf}
}
\caption{Emulated scenario to compare (a) walking speed $\mathbf{v}$ vs. step frequency $\mathbf{f}$ (representing $\pi$) and (b) AoA vs. range changes (suggesting $\gamma_\mathrm{diff}$) between live and replayed footsteps.}
\label{fig:replay attacks}
\end{figure}
\section{Implementation and Performance Evaluation}
\label{sec:performance evaluation}
\begin{figure}[t]
\centering
\subfigure[A microphone array.]{
\label{fig:microphone array}
\includegraphics[width=0.44\columnwidth]{Figs/micarray2.JPG}
}
\hspace{0.001\textwidth}
\subfigure[Experiment setting.]{
\label{fig:experiment setting}
\includegraphics[width=0.43\columnwidth]{Figs/testbed.jpg}
}
\caption{Implementation with a microphone array (a) and corresponding experiment setting for evaluations (b).}
\label{fig:microphone array and setting}
\end{figure}
\begin{figure*}[!t]
\centering
\subfigure[Clean footsteps.]{
\label{fig:clean footstep autocorr}
\includegraphics[width=0.6\columnwidth]{Figs/foot_results.pdf}
}
\hspace{0.003\textwidth}
\subfigure[Voice.]{
\label{fig:voice autocorr}
\includegraphics[width=0.6\columnwidth]{Figs/voice_results.pdf}
}
\hspace{0.003\textwidth}
\subfigure[Mixture.]{
\label{fig:mixture autocorr}
\includegraphics[width=0.6\columnwidth]{Figs/mixed_results.pdf}
}
\caption{ASACC calculated from (a) clean footsteps, (b) voice, and (c) a mixture of clean footsteps and voice. }
\label{fig:ASACC illustration}
\end{figure*}
\subsection{Implementation}
We implement a PURE\ prototype using a circular array backed by a Raspberry Pi4, as shown in Fig.~\ref{fig:microphone array}. We configure the sampling rate as 192~\!kHz, the highest configuration on this platform, to capture as much structure-borne signals as possible due to their short duration. However, we downsample the air-borne footsteps to 16~\!kHz for identification purpose, in order to achieve computational efficiency. The signal processing modules (including footstep detection, GMM based classifier, and FASST audio source separation) are implemented using C++. The deep learning modules, namely the denoising network, ID-Net, and R-Net,
are implemented using Tensorflow~\cite{TF}. For the denoising network, we follow the routines in~\cite{DEMUS}, which allows PURE\ to achieve a salient denoising performance and also a low runtime complexity.
ID-Net takes STFT magnitude of dimension $32\times16$ as its input, representing a footstep that lasts around $30$~\!ms.
The feature extractor has two convolution layers each followed by a batch normalization layer and ReLU activation layer. The first one has 32 filters with a $5\times 3$ kernel and the second one has 64 filters with a $3\times2$ kernel. After then, we flatten the output of the last convolution layer, add a dropout layer with drop probability of 0.65, and project it into a 16-dimensional feature vector. The identity predictor and domain discriminator both have only one fully connected layer with 16 neurons and adopt a sigmoid activation function. Their respective outputs depend on the number of users and domains involved in the training set.
For the R-Net, the input is a footstep waveform with 500 samples. The feature extractor in R-Net has a 1D convolution layer, followed by a pooling layer, and a fully-connected layer. The corresponding filter size is 64, 16, and 16, respectively. A dropout layer with a probability of 0.5 is inserted before the fully-connected layer. The range estimator, together with the domain discriminator has only one hidden layer, and both has a filter size of 16. To gather training data for R-Net, we first deploy a centimeter-level localization system using Decwave UWB-based sensors~\cite{Decwave}. We tie one sensor on a user's foot when he/she walks so as to obtain ground truth locations, i.e., ToA and AoA labels.
Also, we attach an IMU sensor on the user's leg to help triggering the microphone array, so that it may correctly capture a footstep.
We synthesize a training data set for the denosing network with clean footsteps from~\cite{Footstepsound1, Footstepsound2} and speech from TED talks~\cite{TEDtalks}. The noise is extracted from Diverse Environments Multichannel Acoustic Noise Database (DEMAND)~\cite{DEMAND}. We also collect footsteps under common floors (wood, stone) and circumstances (hall, indoor office, home appliance). We configure the signal SNR in a range of 5~\!dB, 10~\!dB, 20~\!dB, and 30~\!dB during training. We also utilize the footsteps from source separation to train the network in gaining the ability of minimizing residual interference signals.
\subsection{Performance Evaluation}
We present extensive experiments in this section. We start with evaluating the source separation algorithm, followed by the denoising network. Then we seriously verify the identification accuracy. Finally, we report performance of defending against the replay attack. The experimental statistics, unless otherwise noted, are all obtained by repeating the same experiment 1,000 times.
\subsubsection{Source Separation Performance}
As we mentioned in Sec.~\ref{ssec:footstep extraction under continous interference}, the source separation module is activated if we detect rhythmic features in STFT spectrogram. Therefore, before source separation, we first evaluate the performance of this detection algorithm. Recall that we utilize the ASACC $\mathbf{b}$, calculated by Eqn.~\eqref{eq:detection in frequency domain} for footstep detection, as the strongest energy of footsteps lies in the low frequency range, we thus only use the first three bins. As shown in Fig.~\ref{fig:ASACC illustration}, only footsteps that exhibit rhythmic features can generate periodic peaks in ASACC of Fig.~\ref{fig:clean footstep autocorr}, whereas voice signals hold no such properties in Fig.~\ref{fig:voice autocorr}. And when footsteps are mixed with voice, using ASACC can still identify this rhythmic feature as shown in Fig.~\ref{fig:mixture autocorr}. The detailed steps of this detection algorithm proceed as follows.
After obtaining $\mathbf{b}$, we estimate the beating frequency $k$ in $\mathbf{b}$ via Discrete Fourier Transform. If the magnitude $m_{k}$ of bin $k$ in spectrum goes beyond the average of its subsequent 20 bins by a certain threshold (10 in our case), namely $m_{k} > {\frac{1}{20}\sum ^{k+20}_{i=k} m_i} + 10$, we accept that current audio signals contain footsteps; otherwise, no footstep is detected and source separation is deactivated.
This detection method allows us to achieve a 100\% footstep detection even when the magnitude of voice is higher than footstep.
To inspect the source separation performance, we first use clean footsteps blended with voice signals under different configurations as inputs and then check the quality of separated signals. Specifically, we synthesize mixtures of footstep and voice under different Source to Interference Ratio (SIR)~\cite{EvaluationMatric} as inputs and evaluate the performance using SIR and Source to Distortion Ratio (SDR)~\cite{EvaluationMatric}. These two evaluation matrices, namely SIR and SDR, are widely used to quantify source separation performance where $\mathrm{SIR} = 10\log\frac{\| \mathbf{s}_\mathrm{tgt} \|^2_2}{\| \mathbf{e}_\mathrm{itf} \|^2_2}$, $\mathrm{SDR} = 10\log\frac{\| \mathbf{s}_\mathrm{tgt} \|^2_2}{\| \mathbf{e}_\mathrm{itf} + \mathbf{e}_\mathrm{nie} + \mathbf{e}_\mathrm{atf}\|^2_2}$, with $\mathbf{s}_\mathrm{tgt}, \mathbf{e}_\mathrm{itf}, \mathbf{e}_\mathrm{nie}$, and $\mathbf{e}_\mathrm{atf}$ being respectively the target signal, interference signal, noise signal, and signal artifacts. The higher the value of these matrices, the better the target signal quality is. Both SIR and SDR in this experiment are calculated given footsteps as the primary signals, as opposed to common speech enhancement tasks where voices are the major concerns.
\begin{figure}[b]
\centering
\subfigure[SIR before separation vs SDR after separation.]{
\label{fig:sir-sdr}
\includegraphics[width=0.43\columnwidth]{Figs/sir_sdr1.pdf}
}
\hspace{0.01\textwidth}
\subfigure[SIR before separation vs SIR after separation.]{
\label{fig:sir-sir}
\includegraphics[width=0.43\columnwidth]{Figs/sir_sir1.pdf}
}
\hspace{0.001\textwidth}
\caption{Source separation performance. SDR (a) remains almost constant and SIR (b) is enhanced by source separation.}
\label{fig:separation visualization}
\end{figure}
The results are shown in Fig.~\ref{fig:separation visualization}.
From Fig.~\ref{fig:sir-sdr}, we can see that the SDR after source separation remains almost constant under different SIRs. This simply implies that the source separation algorithm introduces little distortion to the original footsteps, which is notably important for our later identification performance. As a matter of fact, we can barely perceive any distortion when playing the separated footsteps, except some residual voices.
It is observable that after source separation in Fig.~\ref{fig:sir-sir}, SIR is significantly boosted, indicating a success removal of voice interference.
\begin{figure}[t]
\centering
\subfigure[Time domain waveform.]{
\label{fig:separation performance in time domain}
\includegraphics[width=0.43\columnwidth]{Figs/separation_strong_interference.pdf}
}
\hspace{0.01\textwidth}
\subfigure[STFT spectrogram.]{
\label{fig:separation performance in frequency domain}
\includegraphics[width=0.43\columnwidth]{Figs/separation_strong_interference_stft3.pdf}
}
\caption{Time domain waveform (a) and STFT spectrogram (b) of mixed, original, and separated footsteps. }
\label{fig:separation performance}
\end{figure}
We finally showcase waveforms and STFT spectrogram of footsteps after source separation, compared with the mixed and original recorded ones, in Fig.~\ref{fig:separation performance in time domain} and Fig.~\ref{fig:separation performance in frequency domain}, respectively. In this experiments, the maximum voice magnitude is identical to that of footsteps, under which case, a user mostly notice voice but ignores footstep hence interference is strong. Though under severe interference, the separated footstep waveform is already rather clean as we can see from both Fig.~\ref{fig:separation performance in time domain} and Fig.~\ref{fig:separation performance in frequency domain}: minor distortions and residuals may exist, but none of them introduce perceivable artifacts.
\subsubsection{Denoising Performance}
The denoising network is used to filter out interference from the background subtraction and get rid of residual signals from the source separation. To evaluate the denoising performance, we synthesize mixtures of real-life recorded footsteps and voices under different SNRs and SDRs, generating a total of 1400 sound clips whose duration is within 20~\!s. Then we check the respective SNR and SDR after denosing.
Our measurements in Fig.~\ref{fig:snr gain} reveal a maximum SNR gain around 30~\!dB (depending on the background noise type). The SNR gain can be noticeable when the SNR of input noisy signals is relatively low, e.g., below 20~\!dB. However, the network introduces little distortion to its inputs when SNR is high. The same goes
for SDR as shown in Fig.~\ref{fig:sdr gain}. But this little distortion introduces no perceivable difference to the inputs. Meanwhile, we can deactivate the denoising module when SNR or SDR is high so as to prevent the possible distortion since the SNR or SDR can be roughly calculated. We finally showcase our denoising network in residual removal in Fig.~\ref{fig:denoising for residual removal}.
Both time domain waveform (Fig.~\ref{fig:denoising at time domain}) and Frequency domain spectrogram (Fig.~\ref{fig:denoising at frequency domain}) indicate the success of residual noise removal. The noise removal effect can be visualized by the less signal magnitude variations in time domain waveform and the less ``blurred image'' in spectrogram.
\begin{figure}[t]
\centering
\subfigure[SNR.]{
\label{fig:snr gain}
\includegraphics[width=0.43\columnwidth]{Figs/SNR_gain.pdf}
}
\hspace{0.01\textwidth}
\subfigure[SDR.]{
\label{fig:sdr gain}
\includegraphics[width=0.43\columnwidth]{Figs/SDR_gain.pdf}
}
\caption{SNR (a) and SDR (b) before and after denoising.}
\label{fig:snr and sdr after denoising}
\end{figure}
\begin{figure}[b]
\centering
\subfigure[Time domain waveform.]{
\label{fig:denoising at time domain}
\includegraphics[width=0.43\columnwidth]{Figs/sir_10_timedomain1.pdf}
}
\hspace{0.01\textwidth}
\subfigure[STFT spectrogram.]{
\label{fig:denoising at frequency domain}
\includegraphics[width=0.43\columnwidth]{Figs/sir_10_stft.pdf}
}
\caption{Denoising performance for residual removal. }
\label{fig:denoising for residual removal}
\end{figure}
\subsubsection{Identification Performance}
\begin{figure}[t]
\centering
\subfigure[Accuracy over SNR.]{
\label{fig:accuracy over snr}
\includegraphics[width=0.43\columnwidth]{Figs/accuracy_snr1.pdf}
}
\hspace{0.01\textwidth}
\subfigure[Accuracy over SIR.]{
\label{fig:accuracy over sir}
\includegraphics[width=0.43\columnwidth]{Figs/accuracy_interference5.pdf}
}
\caption{Identification accuracy at various SNRs and SIRs.}
\label{fig:identification performance}
\end{figure}
We conduct extensive measurements to evaluate the identification performance utilizing real-life recorded footsteps from six users.
In our first study, we utilize data samples from the same domain (the same environment and walking speed) but only vary the user identity. In this study, we deactive center loss and apply no adversarial learning for ID-Net.
The results shown in Fig.~\ref{fig:accuracy over snr} reveal that even under a SNR of -12.5~\!dB,
ID-Net~still achieves 87\% accuracy, demonstrating the feasibility of applying footstep for user identification.
\begin{figure}[b]
\centering
\subfigure[Identification accuracy after different processing methods.]{
\label{fig:run once}
\includegraphics[width=0.43\columnwidth]{Figs/process.pdf}
}
\hspace{0.01\textwidth}
\subfigure[Impact of sampling rate.]{
\label{fig:impact of sampling rate}
\includegraphics[width=0.43\columnwidth]{Figs/impact_of_samplerate1.pdf}
}
\caption{(a) Identification accuracy improved after each processing step. (b) Improving the sampling rate can boost the identification accuracy at low SNR.}
\label{fig:impacts}
\end{figure}
It should be noted that we only utilize one step for identification, if we incorporate multiple footsteps and use a majority voting strategy, the accuracy can be boosted to $1 - C^2_3\times \left(1 - 0.90 \right)^2\times 0.9 - C_3^3(1-0.9)^3 = 97.2$\%.
We then check the identification accuracy under different levels of voice interference (SIR).
The results in Fig.~\ref{fig:accuracy over sir} show that even under severe interference (SIR = 0), ID-Net can still achieve an accuracy up to 87.13\%. And if adding noise and thereby reducing the SNR for footsteps, the accuracy would drop to 60\%, indicating ID-Net's vulnerability to strong background interference. This also emphasizes the need for source separation and denoising, as the former can deliver SIR gain and the latter provides SNR gain, thereby promoting the identification accuracy.
We next extensively explore the identification accuracy after source separation and denoising. The results shown in Fig.~\ref{fig:run once} reveals that source separation can achieve a maximum accuracy gain of 59.9\% while denoising network can boost the performance by an average of 5\%.
We then explore the impact of sampling rate on the final identification performance and Fig.~\ref{fig:impact of sampling rate} shows the results. It can be observed that improving the sampling rate can contribute to better identification performance. The performance gain is marginal when SNR is sufficiently high ($>20$~\!dB), if the sampling rate hits 48~\!kHz. Since a higher sampling rate requires more computational power but achieves little performance improvement, we therefore adopt 16~\!kHz in our system.
\begin{figure}[t]
\centering
\subfigure[Low-dimensional features without center loss.]{
\label{fig:without center loss}
\includegraphics[width=0.43\columnwidth]{Figs/without_centerloss.pdf}
}
\hspace{0.01\textwidth}
\subfigure[Low-dimensional features with center loss.]{
\label{fig:with center loss}
\includegraphics[width=0.43\columnwidth]{Figs/with_centerloss.pdf}
}
\caption{Low-dimensional feature visualization without (a) and with (b) center loss. It clearly shows the power of center loss in maximizing inter-class boundaries while minimizing intra-class distances.}
\label{fig:power of center loss}
\end{figure}
We next conduct an ablation study on the effectiveness of center loss, the impact of which on the classified features can be visualized in Fig.~\ref{fig:power of center loss}. It is observable that center loss can effectively maximize the inter-class boundaries and minimize intra-class distances. This ability could not only enhance the identification performance but also improve the generalizability of ID-Net. Applying center loss sometimes can push the identification accuracy to almost 100\%.
\begin{table}[htp]
\centering
\small
\caption{Accuracy without domain adaptation.}
\label{tab:accuracy under different domains}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Accuracy (\%)} & \textbf{Distance} & \textbf{Speed} & \textbf{Environment} \\ \hline
Distance & 76.4 & 57.2 & 9.11 \\ \hline
Speed & 57.2 & 62.01 & 14.31 \\ \hline
Environment & 9.11 & 14.31 & 12.04 \\ \hline
\end{tabular}
\end{table}
We test the identification performance under different domains including speed, distance, and environment variations in the following experiments.
Specifically, our footsteps are captured under:
1) three levels of walking speed, namely 0.2~\!m/s, 0.5~\!m/s, and 1~\!m/s,
2) different distances ranging from $0$ to $3$~\!m,
3) heterogeneous environments including common indoor office, home appliance, hall, corridor, etc that exhibit different ground materials and background interference.
We first show the impact of domains on the identification performance when we deactivate center loss and domain predictor. The results are displayed in Table~\ref{tab:accuracy under different domains} and they tells us that domain conditions can have a notable impact on the identification accuracy. To read the statistics in Table~\ref{tab:accuracy under different domains}, each row and column indicate the number of domains involved in the training data. For instance, $ (\text{row}, \text{column}) = \left( \text{Speed}, \text{Distance} \right) = 57.2\%$ means when the training data involves speed and distance variations, the identification accuracy is 57.2\%.
According to Eqn.~\eqref{eq:air-borne property}, distance should not impose any negative impacts on the final results. But when data only contains distance variations, the identification accuracy is only 76.42\%. We believe that this is caused by 1) SNR degradation due to propagation loss and 2) structural differences from place to place that cause heterogeneous features. Speed variations, equivalently leading to different impact forces, sabotage the identification accuracy to only 62.01\%. And environment dynamics, introducing different medium properties, undermine the identification accuracy most.
\begin{table}[b]
\centering
\small
\caption{Accuracy with domain adaptation.}
\label{tab:accuracy with domain adaptation}
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Accuracy (\%)} & \textbf{0 Domain} & \textbf{2 D.} & \textbf{2 D.} & \textbf{3 D.}\\ \hline
One footstep & 1 & 88.6 & 84.92 & 81.75\\ \hline
2/3 & 1 & 95.92 & 90.33 & 88.16\\ \hline
3/5 & 1 & 96.53 & 94.47 & 90.73\\ \hline
\end{tabular}
\end{table}
We next explore the identification accuracy under domain adaptation. Particularly, we evaluate the identification accuracy under different number of domains and number of footsteps. The average results from 100 trials are shown in Table~\ref{tab:accuracy with domain adaptation} and ``2/3'' means we incorporate three steps to identify each user and we accept the result if the same identity appears twice. It can be observable that domain adaptation can significantly improve identification accuracy, pushing the resulting accuracy from a minimal of 9\% (without domain adaptation) to 81.75\%.
And if incorporating multiple footsteps, the accuracy can be further improved to 90.73\%.
\begin{figure}[t]
\centering
\subfigure[Identification accuracy under one footstep interference.]{
\label{fig:identification accuracy under two footstep overlaped}
\includegraphics[width=0.435\columnwidth]{Figs/overlap3.pdf}
}
\hspace{0.001\textwidth}
\subfigure[Identification accuracy under multiple footsteps.]{
\label{fig:identification accuracy under multiple footstep}
\includegraphics[width=0.435\columnwidth]{Figs/multiple_users1.pdf}
}
\caption{Identification accuracy when one footstep is mixed with partial another footstep components (a). Such a collision can reduce the identification accuracy. Identification accuracy drops when the number of users increases (b). As collision rate increases when more users are involved, the accuracy drops. }
\label{fig:multiple user identification accuracy}
\end{figure}
We finally explore the identification accuracy under multiple user scenarios.
We first explore the identification accuracy under the case when each footstep is interfered by only one another footstep. We check the identification accuracy when the footsteps are overlapped at different percentages, the results of which are displayed in Fig.~\ref{fig:identification accuracy under two footstep overlaped}. As the figure tells, the identification accuracy drops monotonically if the percentage of overlapped region increases. However, the accuracy is still around 50\% if two steps are totally overlapped. This simply implies that ID-Net can still recognize these two footsteps but is unable to distinguish them. We then evaluate the performance when multiple person randomly walk in an indoor meeting room where we place the microphone array in the center. The results in Fig.~\ref{fig:identification accuracy under multiple footstep} show that an identification accuracy around 80\% can still be achieved even when there are three users. As the number of user increases, the collision between different footsteps happens more frequently hence worse performance.
\subsubsection{Defend Against Replay Attack}
PURE~leverages R-Net to extract spatial information, including range and AoA, from multiple consecutive footsteps to defend against replay attack. In this section, we first present the performance of R-Net in spatial information extraction and then inspect the defending performance based on these signatures. We run over 1000 trials in an indoor office $6.8\times4.2$~\!m$^2$ where we place the microphone array in the center.
Fig.~\ref{fig:spatial clues} shows the ranging and AoA estimation performance. In Fig.~\ref{fig:ranging performance}, We verify the ranging performance of R-Net under three cases to demonstrate its capability in domain adaptation. First, we utilize training data (70\% of all data) from all the identities and domains to train R-Net, and we then verify the ranging performance using test data (the remaining 30\%), which we refer to as Test with Domain Adaptation (Test w/DA). Second, we randomly remove one identity from the training data and after training, we apply inference on this particular identity, referred as Test new samples with Domain Adaptation (Test new samples w/DA). Third, we cut the domain predictor from R-Net and apply inference using the same setting with the second case, denoted as Test new samples without Domain adaptation (Test new samples w/o DA). The results in Fig.~\ref{fig:ranging performance} exhibit an median error of around 0.3~\!m, even testing on samples that come from other domains and never participant in the training process. And if without domain adaptation, the median ranging errors would reach up to 1~\!m. The comparison of afore-mentioned results clearly demonstrates the salient performance of R-Net in domain adaptation, as well as in ranging. Fig.~\ref{fig:AoA performance} shows AoA estimation errors are below $10^\circ$. These salient performance lays the foundation of our defending mechanism against replay attack.
\begin{figure}[t]
\centering
\subfigure[Ranging performance.]{
\label{fig:ranging performance}
\includegraphics[width=0.44\columnwidth]{Figs/toa.pdf}
}
\hspace{0.001\textwidth}
\subfigure[AoA estimation performance.]{
\label{fig:AoA performance}
\includegraphics[width=0.45\columnwidth]{Figs/AoA.pdf}
}
\caption{Performance in spatial clues extraction. }
\label{fig:spatial clues}
\end{figure}
We have checked our defending mechanism under several replay attack scenarios including different walking trajectories and hacking positions. Our measurements reveal that if the trajectories contain complex shapes such as ``L'' or circle shape segments, the detected $\pi$ could easily violates the threshold $\bar{\pi}$ so that we achieve 100\% success in defending these attacks. When there involves only straight line trajectories,
the variance of detected AoA revealed by replayed attacks never exceeds the preset $10^\circ$ threshold while live footsteps reveal a minimum variation around 32.8$^\circ$ due to location swing caused by the alternation between left and right legs.
In conclusion, PURE\ can successfully defend against replay attack.
\section{Related Work}
\label{sec:related work}
In this section, we survey the literature on user identification.
Whereas common identification techniques have a broad categories, ranging from traditional computer vision, fingerprint sensing, and iris scan, they are rather irrelevant to our proposal. Therefore, we shall not review these common techniques but focus only on solutions that leverage emergent sensing techniques and adopt behavioral biometrics for user identification.
The proposals of~\cite{WiFiID,WiWho,WiFiSensing1,WiFiSensing2} exploit the gait information to identify users during their walking. The basic idea behind these systems is that the particular walking cycle of each user can be sampled by WiFi signals. However, they may not be able to adapt to environment dynamics, and even walking direction variations can severely affect the identification accuracy. Meanwhile, they often fail to work in practice due to the severe interference from WiFi’s main function of communications and other co-spectrum devices.
FootprintID~\cite{FootprintID} is a structural vibration based identification system. It employs Gephone~\cite{Gephone} to sense the structural vibration caused by a footstep. For identification, it again relies on the gait patterns extracted from multiple structural vibration measurements; the reported identification accuracy for 10 people may reach up to 96\%. However, this promising solution still leaves many open issues, including sensor location variation, multiple pedestrians interference, footwear variation. Other similar behavioral biometrics enabled identification system can be found in~\cite{GaitAccelerometer1,GaitAccelerometer2,GaitAccelerometer3}, a well as~\cite{Footstep3} where accelerometer and camera are used together as sensors.
To summarize, existing technologies driven by behavioral biometrics often require multiple measurements hence long latency identification experience. On the contrary, PURE\ solves this problem elegantly by requiring as few as only one step. While PURE\ can be deemed as a type of behavioral biometrics, it is actually quite related to voiceprint recognition~\cite{VoiceRecognition,VoiceRecognition1}; it can be deemed as a ``footstep-print'' enabled identification system. PURE\ is similar to those acoustic fingerprint based systems~\cite{VoiceRecognition,VoiceRecognition1} but PURE~is totally passive and thus can provide better user experience.
The most similar work to PURE\ is the one from~\cite{Footstep1} where footstep patterns rather than gaits are used for identification. But this proposal requires an excessive number of piezoelectric sensors to capture footstep signals while PURE\ utilizes only commodity microphone, significantly reducing the deployment cost and rendering itself widely applicable for indoor scenarios.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we have explored the possibility of exploiting footsteps for passive user identification. We have proposed PURE\ as a multi-person identification system driven by a pipeline of signal processing and deep learning techniques. PURE\ demands as few as a single footstep to enable user identification and is immune to replay attacks. PURE\ is even feasible to work under continuous voice interference, thanks to a novel source separation and denoising network. To have PURE\ working across different domains, we have exploited domain adversarial adaptation scheme with a center loss to further enhance its generalization ability across different domains. We have implemented a prototype for PURE\ and extensively evaluated its performance; the results confirm that PURE\ achieves a cross-domain identification accuracy up to 90\%. Since PURE\ outperforms existing passive identification system in both deployment cost and identification latency, we have the reason to believe that PURE\ has the potential for a wide adoption.
\bibliographystyle{IEEEtran}
|
1,314,259,996,667 | arxiv | \section{Introduction: Euler characteristic reciprocity}
\label{sec:intro}
For posets $P$ and $Q$, the set of increasing maps from $P$ to $Q$,
denoted by $\Hom^{<}(P, Q)$, is defined as
\begin{equation}
\Hom^{<}(P, Q)=\{\eta:P\longrightarrow Q\mid
p_1<p_2\Longrightarrow \eta(p_1)<\eta(p_2)\}.
\end{equation}
The set of weakly increasing maps $\Hom^{\leq}(P, Q)$ is similarly defined.
For finite posets $P$ and $Q$, the cardinality
$\left|\Hom^{<(\leq)}(P, Q)\right|$
is an important object of study in enumerative combinatorics and
theory of polytopes (\cite{sta-ec}).
In particular, the following result by Stanley
is one of the early results which leads
recent active research
on combinatorial reciprocities (\cite{bec-san}).
\begin{theorem}
\label{thm:sta}
\cite{sta-chr, sta-ord}
Let $P$ be a finite poset and
$[n]$ denote the totally ordered set $\{1<2<\dots <n\}$. Then,
\begin{itemize}
\item[(i)] (Order polynomials) there exist polynomials
$\ord^<(P, t), \ord^\leq(P, t)\in\Q[t]$ that satisfy
\begin{eqnarray}
\ord^\leq(P, n)&=&\left|\Hom^\leq(P, [n])\right|,\\
\ord^<(P, n)&=&\left|\Hom^<(P, [n])\right|,
\end{eqnarray}
for $n\geq 1$.
\item[(ii)] (Reciprocity)
\begin{equation}
\label{recip}
\ord^<(P, t)=(-1)^{|P|}\cdot\ord^\leq(P, -t).
\end{equation}
\end{itemize}
\end{theorem}
Let $t=n$ in formula (\ref{recip}). The left-hand side
makes sense in terms of the cardinality of $\Hom^<(P, [n])$.
However, the right-hand side, the cardinality of $\Hom^{\leq}(P, [-n])$,
is meaningless as it is.
For this reason, it is a natural question to give a definition of
``$-Q$'' for the poset $Q$ and give meaning to the formula of
the form
\begin{equation}
\label{expect}
\text{`` }
\#\Hom^<(P, Q)=(-1)^{|P|}\cdot\#\Hom^\leq(P, -Q).
\text{ ''}
\end{equation}
The cardinality of a finite set is a non-negative integer,
however for our purposes we need an extension of ``finite sets''
such that it takes whole integers (including negative integers)
as ``cardinality''. Such a problem has been discussed in \cite{sch-neg},
and one natural answer is topological spaces
(in particular, semialgebraic sets) and their Euler characteristics.
In fact, number of generalizations of combinatorial results have been
obtained using the Euler characteristic \cite{eas-hug, str-eul}.
In \cite{hmy}, the definition ``$-Q:=Q\times\R$ with lexicographic order'' was
proposed for this purpose.
Then, based on this definition, the above formula (\ref{expect})
can be formulated as an identity for the Euler characteristics.
In order to state the main result of \cite{hmy},
let us recall the Euler characteristic of a semialgebraic set \cite{bpr}.
Let $X\subset\R^N$ be a semialgebraic set. Then, there exists a
finite partition $X=\bigsqcup_{\lambda\in\Lambda}X_\lambda$ into semialgebraic
sets $X_\lambda$ which is semialgebraically homeomorphic to
the open simplex $\sigma_{d_\lambda}$, where
$\sigma_d=\{0<x_1<x_2<\dots x_d<1\}\subset\R^d$ is the $d$-dimensional
open simplex (note that $\sigma_0$ is the point).
Then the Euler characteristic $e(X)$ of $X$ is defined as
$e(X):=\sum_{\lambda\in\Lambda}(-1)^{d_\lambda}$.
Note that
if $X$ is compact, then $e(X)$ coincides with the usual Euler characteristic.
More generally, if $X$ is locally compact, then $e(X)$ is coincides with
the Euler characteristic of the Borel-Moore homology group \cite{bcr}.
A poset is called a \emph{semialgebraic poset} if
its ground set is a semialgebraic set and
order structure is semialgebraically defined.
Finite posets and the real line $\R$ are semialgebraic posets.
The Euler characteristic of a semialgebraic poset is a natural
generalization of the cardinality of a finite poset.
For example, for a finite poset $P$ ,
$e(P)$ is equal to the cardinality $|P|$.
Furthermore, due to the multiplicativity of the Euler characteristic
and $e(\R) = -1$, for a semialgebraic poset $Q$, we have
\[
e(-Q) = -e(Q).
\]
\begin{theorem}
\label{thm:hmy}
\cite{hmy} Let $P$ be a finite poset and $Q$ be a semialgebraic poset.
Then $\Hom^<(P, Q)$ and $\Hom^{\leq}(P, Q)$ are semialgebraic sets.
Furthermore,
\begin{itemize}
\item[(i)] the Euler characteristics of these spaces satisfy
\begin{eqnarray}
\label{eq:euler01}
e(\Hom^<(P, Q))=(-1)^{|P|}\cdot e(\Hom^{\leq}(P, -Q)), \\
\label{eq:euler02}
e(\Hom^<(P, -Q))=(-1)^{|P|}\cdot e(\Hom^{\leq}(P, Q)).
\end{eqnarray}
\item[(ii)]
If furthermore $Q$ is totally ordered, then the
Euler characteristics of these spaces
can be expressed using ordered polynomials as follows.
\[
\begin{split}
e(\Hom^<(P, Q))&=\ord^<(P, e(Q)), \\
e(\Hom^\leq(P, Q))&=\ord^\leq(P, e(Q)).
\end{split}
\]
\end{itemize}
\end{theorem}
Note that Stanley's reciprocity Theorem \ref{thm:sta}
can be obtained by considering the totally ordered set $Q=[n]$.
Moreover,
Theorem \ref{thm:hmy} asserts that the reciprocity
holds for any finite poset $Q$,
not necessarily for the poset of the form $Q=[n]$.
\begin{remark}
Note that, in Theorem \ref{thm:hmy} (i),
since $-(-Q)\neq Q$, the two formulas
(\ref{eq:euler01}) and (\ref{eq:euler02}) are not equivalent.
\end{remark}
This paper is organized as follows. In \S \ref{sec:euler},
we discuss the refinement of Theorem \ref{thm:hmy} (i),
i.e.,
whether the claim of the Theorem follows from the homeomorphism of spaces.
In \S \ref{sec:main} we formulate the main result.
In \S \ref{sec:semi} we summarize the properties of upper
semicontinuous functions needed for the proof, and
in \S \ref{sec:proof} we give the proof of the main result.
\section{A refinement of Euler characteristic reciprocity}
\label{sec:euler}
Theorem \ref{thm:hmy} (i) asserts that the Euler characteristics
of two spaces are equal up to sign factor.
Let us reformulate these formulas: noting that
$e(\R^{|P|})=(-1)^{|P|}$,
the two formulas of Theorem \ref{thm:hmy} (i)
can be rewritten as:
\begin{eqnarray}
\label{eq:nonhomeo}
e(\Hom^{\leq}(P, Q\times\R))=e(\Hom^<(P, Q)\times\R^{|P|}), \\
\label{eq:homeo}
e(\Hom^<(P, Q\times\R))=e(\Hom^{\leq}(P, Q)\times \R^{|P|}).
\end{eqnarray}
It is a natural question to ask whether the equality between
these Euler characteristics can be refined.
More precisely, are the spaces in the left-hand sides and
the right-hand sides homeomorphic?
The main result of this paper is to prove that the second equality
(\ref{eq:homeo})
holds at the level of space, that is, there exists a homeomorphism
\begin{equation}
\label{eq:homeo02}
\Hom^<(P, Q\times\R)\simeq\Hom^{\leq}(P, Q)\times \R^{|P|}.
\end{equation}
(See Theorem \ref{thm:main} and Corollary \ref{cor:semialg}
for the precise statement).
\begin{remark}
For the first equality (\ref{eq:nonhomeo}),
the spaces
$\Hom^{\leq}(P, Q\times\R)$ and $\Hom^<(P, Q)\times\R^{|P|}$
are not homeomorphic in general.
For example, when $P=[2]$ and $Q=[1]$,
$\Hom^<(P, Q)=\emptyset$, therefore,
the space in the right-hand side of (\ref{eq:nonhomeo})
is empty, while, the left-hand side is non-empty.
As another example, let us consider the case $P=Q=[2]$.
Then, $\Hom^<(P, Q)$ consists of a point and
$\Hom^<(P, Q)\times\R^{|P|}$ is a connected space.
On the other hand,
$\Hom^{\leq}(P, Q\times\R)$ looks like
Figure \ref{fig:example}, which has three connected components.
(The figure is drawn using the identification $\R$ with the open interval $(0,1)$.)
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1.2]
\draw[->] (-0.5,0)--(4.5,0);
\draw[->] (0,-0.5)--(0,4.5);
\draw[very thin] (-0.5,-0.5)--(4.5,4.5);
\filldraw[fill=gray!20!white, draw=black, dashed, very thin]
(1,3)--(2,3)--(2,4)--(1,4)--cycle;
\filldraw[fill=white, draw=black] (1,3) circle (2pt) ;
\filldraw[fill=white, draw=black] (1,4) circle (2pt) ;
\filldraw[fill=white, draw=black] (2,3) circle (2pt) ;
\filldraw[fill=white, draw=black] (2,4) circle (2pt) ;
\filldraw[fill=gray!20!white, draw=black, dashed, very thin]
(1,1)--(2,2)--(1,2)--cycle;
\draw[very thick] (1,1)--(2,2);
\filldraw[fill=white, draw=black] (1,1) circle (2pt) ;
\filldraw[fill=white, draw=black] (1,2) circle (2pt) ;
\filldraw[fill=white, draw=black] (2,2) circle (2pt) ;
\filldraw[fill=gray!20!white, draw=black, dashed, very thin]
(3,3)--(4,4)--(3,4)--cycle;
\draw[very thick] (3,3)--(4,4);
\filldraw[fill=white, draw=black] (3,3) circle (2pt) ;
\filldraw[fill=white, draw=black] (4,4) circle (2pt) ;
\filldraw[fill=white, draw=black] (3,4) circle (2pt) ;
\end{tikzpicture}
\caption{$\Hom^{\leq}([2], [2]\times\R)$.}
\label{fig:example}
\end{figure}
It is a natural problem
to explore the reasons that lead to the equality (\ref{eq:nonhomeo})
of the Euler characteristics even though the spaces are not
homeomorphic.
\end{remark}
\section{Metrizable posets and main result}
\label{sec:main}
Let $P$ and $Q$ be posets.
From the definition of lexicographic order,
a pair $(\eta, \theta)$ of maps
$\eta:P\longrightarrow Q$ and $\theta:P\longrightarrow\R$
is contained in $\Hom^<(P, Q\times\R)$ if and only if for every
$p_1, p_2\in P$ with $p_1<p_2$,
either ``$\eta(p_1)<\eta(p_2)$'' or
``$\eta(p_1)=\eta(p_2)$ and $\theta(p_1)<\theta(p_2)$'' holds.
It follows that $\eta\in\Hom^\leq(P, Q)$.
Thus, we obtain the natural projection
$\pi:\Hom^<(P, Q\times\R)\longrightarrow\Hom^{\leq}(P, Q)$
(also similarly
$\pi:\Hom^\leq(P, Q\times\R)\longrightarrow\Hom^{\leq}(P, Q)$).
\begin{definition}
A poset $Q$ is a \emph{metrizable poset} if its ground set is equipped with
metrizable topology.
\end{definition}
The main result of this paper is as follows.
\begin{theorem}
\label{thm:main}
Let $P$ be a finite poset and $Q$ be a metrizable poset. Then there exists
a homeomorphism $\varphi:\Hom^{<}(P,Q\times \R)\stackrel{\simeq}{\longrightarrow}\Hom^{\leq}(P,Q)\times\R^{|P|}$ which makes the following
diagram commutative:
\begin{equation}
\label{diag:main}
\xymatrix{
\Hom^{<}(P,Q\times \mathbb{R}) \ar[r]^{\varphi} \ar[d]_{\pi} & \Hom^{\leq}(P,Q)\times\mathbb{R}^{|P|} \ar[d]_{\pi} \\
\Hom^{\leq}(P,Q) \ar[r]^{id} & \Hom^{\leq}(P,Q).
}
\end{equation}
\end{theorem}
Before giving the proof, let us discuss special cases of this result.
\begin{example}
If $Q=[1]$ is the poset with one element, then the result gives a homomorphism
$\varphi:\Hom^{<}(P, \R)\stackrel{\simeq}{\longrightarrow}\R^{|P|}$.
Let $P=\{p_1, \dots, p_n\}$. Then $\Hom^{<}(P, \R)$ is expressed as follows.
\[
\Hom^{<}(P, \R)=\{(t_1, \dots, t_n)\in\R^n\mid t_i<t_j \mbox{ if $p_i<p_j$ in $P$}\},
\]
which is a convex open subset of $\R^n$. Hence it is homeomorphic to $\R^n$.
\end{example}
\begin{example}
Suppose $P=[2]$ and $Q=\R$. Then we have
\[
\Hom^{<}(P, Q\times\R)=\{((q_1, t_1), (q_2, t_2))\in(\R\times\R)^2\mid q_1<q_2
\mbox{, or $q_1=q_2$ and $t_1<t_2$}\}.
\]
Restricting diagram (\ref{diag:main}) to $(q_1, t_1)=(0,0)$,
Theorem \ref{thm:main} asserts that
\[
\{(q_2, t_2)\in\R^2\mid q_2>0\mbox{, or $q_2=0, t_2>0$}\}
\]
is homeomorphic to $[0, \infty)\times\R$ (Figure \ref{fig:homeo}).
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1]
\fill[fill=gray!20!white]
(0,0)--(0,4)--(4,4)--(4,0)--cycle;
\draw[very thick] (0,2)--(0,4);
\draw[very thin, dashed] (0,2)--(0,0);
\filldraw[fill=white, draw=black] (0,2) circle (2pt) ;
\fill[fill=gray!20!white]
(6,0)--(6,4)--(10,4)--(10,0)--cycle;
\draw[very thick] (6,0)--(6,4);
\end{tikzpicture}
\caption{$\{(q_2, t_2)\in\R^2\mid q_2>0\mbox{, or $q_2=0, t_2>0$}\}$
and $[0, \infty)\times\R$.}
\label{fig:homeo}
\end{figure}
\end{example}
This example shows that considerations of the upper
semicontinuous functions are key to the proof of Theorem \ref{thm:main}.
\section{Upper semicontinuous functions on metrizable spaces}
\label{sec:semi}
Recall that a function $f:X\longrightarrow \R\cup\{-\infty\}$
on a topological space $X$ is said to be \emph{upper semicontinuous} if
for every $\alpha\in\R$, $f^{-1}([-\infty, \alpha))\subset X$ is open.
\begin{lemma}
\label{lem:upper}
For a function $f:X\longrightarrow \R\cup\{-\infty\}$, let
$X_f:=\{(x, t)\in X\times\R\mid t>f(x)\}$. If $X$ is metrizable and
$f$ is upper semicontinuous, then there exists a homeomorphism
$\varphi: X_f\longrightarrow X\times\R$
that makes the following diagram commutative
\begin{equation}
\xymatrix{
X_f \ar[r]^{\varphi} \ar[d]_{\pi} & X\times\R \ar[d]_{\pi} \\
X \ar[r]^{id} & X.
}
\end{equation}
\end{lemma}
\begin{proof}
It is classically known (\cite[Chapter 9, \S 2]{bourbaki}) that
there exists a sequence $f_n:X\longrightarrow \R\cup\{-\infty\}, (n\geq 1)$
of continuous functions such that
\begin{itemize}
\item for each $x\in X$, $f_1(x)>f_2(x)>\dots>f_n(x)>\dots$, and
\item $\lim_{n\to\infty}f_n(x)=f(x)$.
\end{itemize}
Then, define $\varphi:X_f\longrightarrow X\times \R$ as
\[
\varphi ( x , t ) =
\begin{cases}
(x, t - f_0(x)) & (t\geq f_0(x) ), \\
(x,-(i+1) + \dfrac{t - f_{i+1}(x)}{f_i(x) - f_{i+1}(x)} & (f_i(x)\geq t \geq f_{i+1}(x)).
\end{cases}
\]
This $\varphi$ gives a desired homeomorphism.
\end{proof}
\section{Proof of the main result}
\label{sec:proof}
We give the proof of Theorem \ref{thm:main} in this section.
We fix a numbering $P=\{p_1, \dots, p_n\}$
in such a way that $1\leq i<j\leq n$ implies $p_j\not\leq p_i$.
Such a numbering can be obtained, for example, by letting
$p_1$ be a minimal element of $P$ and
$p_i$ be a minimal element of $P\smallsetminus\{p_1, \dots, p_{i-1}\}$
for $i>1$.
For $1\leq k\leq n$, let us define the subset
$X_k\subset\Hom^{\leq}(P, Q)\times\R^n$ as follows.
\[
\begin{split}
X_k:=
\{
(q_1, \dots, q_n, t_1, \dots, t_n)\in\Hom^{\leq}(P, Q)\times\R^n\mid
&\mbox{For $1\leq \forall i<\forall j\leq k$}, \\
&\mbox{if $p_i< p_j$ and $q_i=q_j$, then $t_i<t_j$}
\},
\end{split}
\]
where $(q_1, \dots, q_n)=(\eta(p_1), \dots, \eta(p_n))$
for $\eta\in\Hom^{\leq}(P, Q)$ and
$t_i\in\R$.
Note that $X_1=\Hom^{\leq}(P, Q)\times\R^n$ and
$X_n=\Hom^{<}(P, Q\times\R)$.
Let $1\leq k\leq n-1$. Define the map
$\pi_k:X_k\longrightarrow\Hom^{\leq}(P, Q)\times\R^{n-1}$ by
$\pi_k(q_1, \dots, q_n, t_1, \dots, t_n)\longmapsto
(q_1, \dots, q_n, t_1, \dots, t_k, t_{k+2}, \dots, t_n)$, and
$Y_k:=\pi_k(X_k)$. It follows from the definition that $X_k=Y_k\times\R$.
Next, for $1\leq j\leq k\leq n-1$, define the function
$f_{jk}:Y_k\longmapsto\R\cup\{-\infty\}$ as follows.
\[
f_{jk}(q_1,\ldots,q_n,t_1,\ldots,t_k,t_{k+2},\ldots,t_n) =
\begin{cases}
-\infty
(p_j \nleq p_{k+1} \mbox{ or } q_j < q_{k+1}), \\
t_j
(p_j \leq p_{k+1} \mbox{ and } q_j = q_{k+1}).
\end{cases}
\]
Then $f_{jk}$ is an upper semicontinuous function.
In fact, when $p_j \nleq p_{k+1}$, $f_{jk}$ is upper semicontinuous
because it is a constant function.
When $p_j \nleq p_{k+1}$, we need to verify $f_{jk}^{-1}([-\infty, \alpha))$
is open for $\forall\alpha\in\R$. Indeed, we have
\[
\begin{split}
f_{jk}^{-1}([-\infty, \alpha))=
&
\{(q_1, \dots, q_n, t_1, \dots, t_k, t_{k+1}, \dots, t_n)\in Y_k\mid
q_j\neq q_{k+1}\}\\
&
\cup
\{(q_1, \dots, q_n, t_1, \dots, t_k, t_{k+1}, \dots, t_n)\in Y_k\mid t_j<\alpha\}.
\end{split}
\]
The first set is open because $Q$ is Hausdorff. The second set is
clearly open.
Now we consider the function $f_k:=\max\{f_{1k}, \dots, f_{kk}\}$ on $Y_k$.
Since the maximum of finitely many upper semicontinuous functions is
upper semicontinuous, $f_k$ is upper semicontinuous.
By Lemma \ref{lem:upper}, there exists a homeomorphism $\varphi_i$
that makes the following diagram commutative.
\[
\xymatrix{
Y_k\times \mathbb{R} \ar[r]^{\varphi_k} \ar[d]_{\pi} &
{Y_k}_{f_k} \ar[d]_{\pi} \\
Y_k \ar[r]^{id} & Y_k,
}
\]
where ${Y_k}_{f_k}=\{(y, t_{k+1})\in Y_k\times\R\mid t_{k+1}>f_k(y)\}$.
Furthermore, by definition, we have ${Y_{k}}_{f_k}=X_{k+1}$.
Hence there exists a homeomorphism $X_k\simeq X_{k+1}$ which commutes
with the projection to $\Hom^{\leq}(P, Q)$. In particular, we have
$\Hom^{\leq}(P, Q)\times\R^n=X_1\simeq\dots\simeq X_n=\Hom^<(P, -Q)$.
This completes the proof of Thoerem \ref{thm:main}.
Since a semialgebraic set is metrizable, we have the following.
\begin{corollary}
\label{cor:semialg}
Let $P$ be a finite poset and $Q$ be a semialgebraic poset. Then
$\Hom^{\leq}(P, Q)\times\R^{|P|}$ and $\Hom^<(P, -Q)$ are homeomorphism.
\end{corollary}
\begin{remark}
\label{rem:homeo}
It is known that the Euler characteristics of homeomorphic
semialgebraic sets coincide (\cite{beke}). Therefore,
the equality of Euler characteristics
(\ref{eq:euler02}) in Theorem \ref{thm:hmy}
can be obtained from Corollary \ref{cor:semialg}.
However, it is not clear whether
$\Hom^{\leq}(P, Q)\times\R^{|P|}$ and $\Hom^<(P, -Q)$ are
semialgebraically homeomorphic or not, because in Lemma \ref{lem:upper},
we use maps that are not semialgebraic.
Note that there exist two semialgebraic sets that are homeomorphic,
but not semialgebraically homeomorphic (\cite{sy-tri}).
\end{remark}
\medskip
\noindent
{\bf Acknowledgements.}
Masahiko Yoshinaga
was partially supported by JSPS KAKENHI
Grant Numbers JP19K21826, JP18H01115.
|
1,314,259,996,668 | arxiv | \section{Introduction}
In 1880 Andrei A. Markov, a 24-year old student from St Petersburg, discovered in his master's thesis \cite{Markov} a remarkable connection between Diophantine analysis and the following Diophantine equation
\beq{equa}
x^2+y^2+z^2=3xyz,
\eeq
known nowadays as the {\it Markov equation.} The solutions of this celebrated equation are known as {\it Markov triples} and can be found from the obvious one $(1,1,1)$ by compositions of Vieta involutions
\beq{invo}
(x,y,z) \rightarrow (x,y, 3xy-z)
\eeq
and permutations of $x,y,z.$
The numbers, which appear in Markov triples, are called {\it Markov numbers}, the set of which we denote by $\mathcal{M}$. Their arithmetic was studied by Frobenius \cite{Frobenius}, see recent development in \cite{BGS}. For more history and details we refer to the very nicely written book \cite{Aigner} by Aigner.
The growth of Markov numbers
$$
m=1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325,\dots
$$
was studied by Don Zagier \cite{Zagier}, who proved that asymptotically
$$
m_n\approx\frac{1}{3}e^{C\sqrt{n}}
$$
with some constant $C$ (see also McShane and Rivin \cite{McShane}).
However, Markov triples naturally grow on a binary tree (see e.g. \cite{Bombieri}).
In our paper we study the growth of the numbers along each path {\it as the function of the path} on the Markov tree.
More precisely, we will be using the tree representation with Markov numbers living in the connected components of the {\it complement} to a planar binary tree, using the graphical representation of Vieta involution shown in Fig. 1.
\begin{figure}[h]
\begin{center}
\includegraphics[trim = 0mm 50mm 140mm 50mm, clip, height=25mm]{x-y-z-eq}
\caption{\small Graphical representation of Vieta involution}
\end{center}
\end{figure}
The corresponding {\it Markov tree} is shown in Fig. 2 next to the {\it Farey tree}, for which at each vertex we have fractions $\frac{a}{b}$, $\frac{c}{d}$ and their {\it Farey mediant} $$\frac{a}{b}*\frac{c}{d}=\frac{a+c}{b+d}.$$
\begin{figure}[h]
\begin{center}
\includegraphics[height=48mm]{MarkovTree2} \hspace{8pt} \includegraphics[height=48mm]{FareyTree21}
\caption{\small Correspondence between Markov numbers and Farey fractions}
\end{center}
\end{figure}
This defines the {\it Farey parametrisation} of the Markov numbers $m=m(\frac{p}{q})$ by the fractions $\frac{p}{q} \in [0,\frac{1}{2}]$, which goes back to Frobenius \cite{Frobenius} and will be crucial for us.
Using the Farey tree we can assign to every infinite path $\gamma$ on a rooted planar binary tree a point $x \in [0,\frac{1}{2}]$ by considering the limit of the Farey fractions along the path (see Fig. 3).
\begin{figure}[h]
\begin{center}
\includegraphics[width=32mm]{Fib_Path_Farey} \hspace{8pt} \includegraphics[width=32mm]{Fib_Path_Markov} \hspace{8pt}\includegraphics[height=32mm]{Fib_Path_Euclid}
\caption{\small Farey, Markov and Euclid trees with the ``golden" path}
\end{center}
\end{figure}
Let $m_n(x)$ be the $n$-th Markov number along the path $\gamma(x)$ and define the corresponding Lyapunov exponent
$\Lambda(x)$ as
\beq{defL}
\Lambda(x)=\limsup_{n\to\infty}\frac{\ln(\ln m_n(x))}{n}.
\eeq
Equivalently, following \cite{Cohn_1979, Zagier} one can consider the ``tropical version" of the Markov tree: the Euclid tree describing the Euclidean algorithm with integer triples $(u,v,w)$ satisfying the relation
\beq{euclid}
u+v=w
\eeq
and define the {\it Lyapunov exponent} as
\beq{defeucl}
\Lambda(x)=\limsup_{n\to\infty}\frac{\ln w_n(x)}{n},
\eeq
where $w_n(x)$ is the last (largest) number in the $n$-th triple along path $\gamma(x).$
To see the equivalence of these definitions one can consider (following Mordell \cite{Mordell}) a modification of the Markov equation given by
\beq{mod}
x^2+y^2+z^2=3xyz+\frac{4}{9},
\eeq
related to (\ref{euclid}) by the simple change
\beq{modch}
x=\frac{2}{3}\cosh u, \, y=\frac{2}{3}\cosh v, \, z=\frac{2}{3}\cosh w,
\eeq
which explains the double logarithm in the definition (\ref{defL}). Alternatively, one can use the simple arguments from Zagier \cite{Zagier}.
We prove that the Lyapunov exponent exists for all paths and can be naturally extended to the function $\Lambda(x), \, x \in \mathbb{R}P^1$, which is $GL_2(\mathbb Z)$-{\it invariant}:
\beq{gl2}
\Lambda\left(\frac{ax+b}{cx+d}\right)=\Lambda(x), \quad x \in \mathbb{R}P^1,\, \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix} \in GL_2(\mathbb Z).
\eeq
and almost everywhere vanishing (see next section). This interesting function is the main object of our study.
The set $Spec_\Lambda=\{\Lambda(x), \,\, x \in \mathbb{R}P^1\}$ of all possible values of $\Lambda(x)$ is called the {\it Lyapunov spectrum} of Markov and Euclid trees.
\begin{Theorem}
The Lyapunov spectrum of Markov and Euclid trees is
\beq{specl}
Spec_\Lambda=[0, \ln \varphi],
\eeq
where $\varphi=\frac{1+\sqrt 5}{2}$ is the golden ratio.
\end{Theorem}
In particular, for all $x \in \mathbb{R}P^1$
$$
\Lambda(x) \leq \Lambda(\varphi)= \ln \varphi,
$$
so the ``golden path" has the maximal Lyapunov exponent.
To state our second main result we need to introduce the following set of the ``most irrational numbers" $\mathbb{X} \subset \mathbb R.$
Recall that Hurwitz \cite{Hurwitz} proved that the golden ratio and its equivalents have the maximal possible Markov constant, which can be considered a measure of irrationality (see \cite{Burger} and section 4 below).
The celebrated Markov theorem claims that Markov constants $\mu$ larger than $1/3$ have the form
\beq{mspec}
\mu=\frac{m}{\sqrt{9m^2-4}},
\eeq
where $m$ is a Markov number (see details in Delone \cite{Delone} and Bombieri \cite{Bombieri}).
Corresponding equivalence classes of these most irrational numbers are naturally labeled by the Markov numbers $m \in \mathcal{M}$. They have special representatives $x_m$ (which we call Markov-Hurwitz numbers) with pure periodic continued fractions with period consisting of $1$ and $2$:
$$
x_1=[\overline{1}]=\frac{\sqrt 5-1}{2}, \, x_2=[\overline{2}]=\sqrt 2 -1, \, x_5=[\overline{2,2,1,1}]=\frac{\sqrt {221}-9}{14}, \, \dots,
$$
where
$$[a_1, a_2, \dots]:=\frac{1}{a_1 +\frac{1}{a_2+ \dots}}$$
(see Section 4). Note that we use a version of continued fractions with $a_0=0$, which will allow us to avoid zeros in continued fractions (cf. \cite{Khin}).
The set $\mathbb{X}$ of all Markov-Hurwitz numbers is countable and has only one isolated point: $x_1=\frac{\sqrt 5-1}{2}\approx 0.6180$, which is also the maximal number in $\mathbb{X}$. The minimal number is $x_2=\sqrt 2 -1\approx 0.4142,$ and the maximal limiting point of $\mathbb{X}$ is
$$x_*=[2,2,\overline{1}]=\frac{7+\sqrt{5}}{22}\approx 0.4198.$$
Using the Farey parametrization of Markov numbers $m=m(\frac{p}{q})$ we can denote the corresponding number $x_m$ as $x(\frac{p}{q}).$
\begin{Theorem}
The restriction $\Lambda_\mathbb{X}$ of the Lyapunov function on the set of Markov-Hurwitz numbers is monotonically increasing from $$\Lambda(x_2)=\frac{1}{2}\ln(1+\sqrt 2) \quad {\text to} \quad \Lambda(x_1)=\ln \left(\frac{1+\sqrt 5}{2}\right).$$
In the Farey parametrization, $\Lambda(x(\frac{p}{q}))$ is convex as a function of $\frac{p}{q}.$
\end{Theorem}
The proof is based on the interpretation of Markov numbers as geodesics on the punctured torus with hyperbolic metric, which was found by Gorshkov \cite{Gorshkov} in his thesis in 1953 and, independently, by Cohn \cite{Cohn}.
Since then this relation has been very much in use, see in particular, Goldman \cite{Goldman}, Bowditch \cite{Bowditch} and a nice exposition by Series \cite{Series}.
Our general approach is close to Chekhov and Penner \cite{Chekhov}, who discussed similar questions in quantum theory of Teichm\"uller spaces.
The key result for us is due to V. Fock \cite{Fock}, who proved using Thurston's laminations that a certain function defined in terms of Markov numbers can be extended to a convex function on a real interval.
We present also a generalisation of these results to the countable sets $\mathbb{X}_a$ of quadratic irrationals depending on a natural number $a$. They are related to the solutions of the Diophantine equation
$$
X^2+Y^2+Z^2=XYZ+4-4a^6, \, a \in \mathbb N,
$$
studied by Mordell \cite{Mordell}, and geometrically to the geodesics on the one-hole hyperbolic tori.
For $a=1$ we have the scaled Markov equation and Markov-Hurwitz set $\mathbb{X}_1=\mathbb{X}.$
\section{Farey tree, monoid $SL_2(\mathbb N)$ and Lyapunov exponent}
Let $\mathcal T$ be a binary (= $3$-valent) tree. It is well-known (see e.g. nicely written notes by Hatcher \cite{Hatcher}) that $\mathcal{T}$ can be embedded in the hyperbolic plane $\mathbb{H}$ as the dual graph to the Farey tessellation of $\mathbb{H}$ into ideal triangles (see left hand side of Fig. 4, which we have borrowed from \cite{Hatcher} with author's permission).
\begin{figure}[h]
\begin{center}
\includegraphics[height=38mm]{FareyPlusDual} \hspace{8pt} \includegraphics[height=38mm]{FareyTreeLarge}
\caption{\small Dual tree for Farey tessellation and positive Farey tree}
\end{center}
\end{figure}
It will be enough for us to consider only the upper half of the tree, which can be considered as the Farey tree $T_F$ of all positive fractions (see Fig. 4).
The Farey tree shown in Fig. 2 is the branch of this tree corresponding to the fractions lying between 0 and $\frac{1}{2}$.
Let $SL_2(\mathbb N) \subset SL_2(\mathbb Z)$ be the set of matrices with non-negative entries. Such matrices are closed under multiplication and contain the identity, and thus form a monoid.
The positive Farey tree gives a nice parametrisation of this monoid. Indeed, for every (naturally oriented) edge $E$ of $T_F$ we have two fractions $\frac{a}{c}$, $\frac{b}{d}$ adjacent to it, so we can consider the matrix
$$
A_E=\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix},
$$
which belongs to $SL_2(\mathbb N)$. This is in a good agreement with Frobenius \cite{Frobenius}, who considered the pairs of coprime numbers $(p,q)$ rather than fractions $\frac{p}{q}.$ One can easily show that every matrix $A \in SL_2(\mathbb N)$ appears in this way exactly once.
Recall now that the {\it spectral radius} $\rho(A)$ of a matrix $A$ is defined as the maximum of the modulus of its eigenvalues. For a non-triangular (hyperbolic) matrix $A$ from $SL_2(\mathbb N)$
the eigenvalues are $\lambda, \lambda^{-1}$, where $\lambda=\lambda(A)>1$ and
$$
\rho(A)=\lambda(A)
$$
(for triangular (parabolic) matrices $\rho(A)=1$).
Consider now the path $\gamma(x)$ in the Farey tree.
\begin{prop}
\label{spect}
The Lyapunov exponent can be equivalently defined as
\beq{defeucl}
\Lambda(x)=\limsup_{n\to\infty}\frac{\ln \rho(A_n(x))}{n},
\eeq
where $A_n(x) \in SL_2(\mathbb N)$ is attached to $n$-th edge along path $\gamma(x)$
and $\rho(A)$ is the spectral radius of matrix $A$.
\end{prop}
\begin{proof}
Let us assume for convenience that $x \in [0,1]$, which corresponds to right half of the positive Farey tree shown in Fig. 4.
The left half of the tree is related by $x \rightarrow 1/x$ and the change of matrices
$$
A=\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
\rightarrow
\begin{pmatrix}
d & c \\
b & a \\
\end{pmatrix}
=S^{-1}AS, \quad
S=\begin{pmatrix}
0 & 1 \\
1 & 0\\
\end{pmatrix}.
$$
\begin{figure}[h]
\begin{center}
\includegraphics[height=30mm]{FareyTreeBold_Large} \hspace{8pt} \includegraphics[trim = 0mm 25mm 0mm 30mm, clip, height=30mm]{EuclidTreeLarge11_Bold}
\caption{\small Farey and Euclid rooted trees with a path}
\end{center}
\end{figure}
We see that the denominators of the fractions form the Euclid tree shown on the right of the Figure 5.
Let
$$
A_n(x)=\begin{pmatrix}
a_n & b_n \\
c_n & d_n \\
\end{pmatrix}
$$
be the matrix assigned to $n$-th edge of $\gamma(x)$. Then $$w_n(x)=\max(c_n(x),d_n(x))$$ is the corresponding sequence from the Euclid tree.
Let $\lambda_n$ be the maximal eigenvalue of $A_n(x)$. Since $\lambda_n + \lambda_n^{-1}=a_n+d_n$ with $\lambda_n^{-1}\leq 1$ we have
$$
\lambda_n \leq a_n+d_n <c_n+d_n \leq 2\max(c_n, d_n)=2 w_n,
$$
where we have used that $a_n<c_n$, which is valid on this half of the tree.
To have the estimate of $\lambda_n$ from below we need to consider the cases $x=0$ and $x>0$ separately.
For $x=0$
$$
A_n(0)=\begin{pmatrix}
1 & 0 \\
n & 1 \\
\end{pmatrix}
$$
with $\lambda_n=1$ and $w_n=n$, so
$$
\limsup_{n\to\infty}\frac{\ln \lambda_n}{n}=0=\limsup_{n\to\infty}\frac{\ln w_n}{n}=\limsup_{n\to\infty}\frac{\ln n}{n}
$$
in this case.
If $x>0$ since $a_n/c_n \to x$ as $n \to \infty$ we have for large $n$ the inequality
$a_n>\frac{x}{2}c_n$. This means that
$$
\lambda_n\geq\frac{1}{2}(a_n+d_n)>\frac{1}{2}\left(\frac{x}{2}c_n+d_n\right)>\frac{x}{4}\max(c_n,d_n)=\frac{x}{4}w_n.
$$
Thus we have for $x>0$ and large $n$ that
$
\frac{x}{4}w_n(x) < \lambda_n(x)<2w_n(x),
$
which implies that
\beq{lims}
\limsup_{n\to\infty}\frac{\ln \lambda_n(x)}{n}=\limsup_{n\to\infty}\frac{\ln w_n(x)}{n}
\eeq
provided any of these limits exists, which we show next.
\end{proof}
\begin{Theorem}
The Lyapunov exponent $\Lambda(x)$ exists for all real $x\geq 0$ and satisfies
\beq{ineq}
0\leq \Lambda(x)\leq \ln \varphi,
\eeq
where $\varphi= \frac{1+\sqrt 5}{2}$ is the golden ratio,
and every value in $[0, \ln \varphi]$ is attained.
\end{Theorem}
\begin{proof}
Recall that the {\it norm} of a matrix $A$ acting on a Euclidean space is defined as
$$
||A||=max_{|x|=1}|Ax|.
$$
The norm is related to the spectral radius by the formula
$$
||A||^2=\rho(A^*A)
$$
and satisfies the inequalities (see e.g. \cite{Lax})
$$
\rho(A)\leq ||A||
$$
and
$$
||AB||\leq ||A||\cdot||B||.
$$
Now note that the matrices $A_n(x)$ along a path $\gamma(x)$ have the product form
$$
A_n(x)=X_1\dots X_n,
$$
where $X_i$ are either $L$ or $R$ defined as
$$
L=\begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}, \quad
R=\begin{pmatrix}
1 & 0 \\
1 & 1 \\
\end{pmatrix},
$$
depending on whether we turn left or right on the tree.
Since
$$
RL=\begin{pmatrix}
1 & 1 \\
1 & 2 \\
\end{pmatrix}
$$
has maximal eigenvalue $$\lambda(RL)=\frac{3+\sqrt 5}{2}=\left(\frac{1+\sqrt 5}{2}\right)^2,$$
the norms
$$
||L||=||R||=\frac{1+\sqrt 5}{2}=||X_i||.
$$
Therefore
$$
\rho(A_n)\leq ||A_n|| \leq ||X_1\dots X_n|| \leq ||X_1|| \dots ||X_n|| =\left(\frac{1+\sqrt 5}{2}\right)^n,
$$
which implies that the sequence
$$\frac{\ln \rho(A_n)}{n} \leq \ln \frac{1+\sqrt 5}{2}$$ is bounded.
In particular,
$$
\Lambda(x) = \limsup_{n \to \infty} \frac{\ln \rho(A_n(x))}{n}
$$
exists and satisfies the inequality
$$
\Lambda(x) \leq \ln \frac{1+\sqrt 5}{2}.
$$
The equality is attained at $x=\frac{\sqrt 5-1}{2}$ since the corresponding
$$A_{2n}=(RL)^n=\begin{pmatrix}
1 & 1 \\
1 & 2 \\
\end{pmatrix}^n.
$$
Similarly, for the very right path $\gamma_0$ we have $$A_n=R^n=\begin{pmatrix}
1 & 0 \\
1 & 1 \\
\end{pmatrix}^n =\begin{pmatrix}
1 & 0 \\
n & 1 \\
\end{pmatrix},
$$
and thus
$\Lambda(0)=0.$
To show that every value $\Lambda_0 \in(0, \ln \varphi)$ is attained we will use the following lemma.
Let $X_n, \, n\in \mathbb N$ be a sequence of matrices, which are equal to either $L$ or $R$
and $B$ is any matrix from $SL_2(\mathbb N).$
Consider the products
$$
A_n=X_1\dots X_n, \quad B_n=BX_1\dots X_n, \, n \in \mathbb N.
$$
\begin{Lemma}
For every matrix $B \in SL_2(\mathbb N)$
\beq{limits0}
\limsup_{n\to\infty}\frac{\ln \rho(B_n)}{n}=\limsup_{n\to\infty}\frac{\ln \rho(A_n)}{n}.
\eeq
In particular, for every such $B$
\beq{limits}
\lim_{n \to \infty} \frac{\ln \rho(BR^n)}{n}=0, \quad \lim_{n \to \infty} \frac{\ln \rho(B(RL)^n)}{2n}=\ln \varphi.
\eeq
\end{Lemma}
Indeed, let $$B=\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}, \quad A_n=\begin{pmatrix}
a_n & b_n \\
c_n & d_n \\
\end{pmatrix},$$
and assume again for convenience that $a_n<c_n, \, b_n < d_n.$
Then
$$B_n=\begin{pmatrix}
\hat a_n & \hat b_n \\
\hat c_n & \hat d_n \\
\end{pmatrix},$$
where $\hat c_n=c a_n+d c_n,\, \hat d_n=c b_n+d d_n,$ so
$$
\hat w_n=\max (c a_n+d c_n, c b_n+d d_n) \geq \max(d c_n, d d_n)=d w_n.
$$
On the other hand, since $a_n<c_n, \, b_n <d_n$
$$
\max (c a_n+d c_n, c b_n+d d_n)<\max (c c_n+d c_n, c d_n+d d_n)=(c+d)\max(c_n,d_n).
$$
Thus we have
$$
d w_n <\hat w_n<(c+d) w_n
$$
which implies that
$$
\limsup_{n\to\infty}\frac{\ln \hat w_n}{n}=\limsup_{n\to\infty}\frac{\ln w_n}{n}
$$
and the claim follows from Proposition \ref{spect}.
The second part follows from the equalities
$$
\lim_{n\to\infty}\frac{\ln \rho(R^n)}{n}=0, \quad \lim_{n\to\infty}\frac{\ln \rho((RL)^n)}{2n}=\ln \varphi,
$$
which are easy to check.
Now the strategy is the following: we can start with any matrix $A_0 \in SL_2(\mathbb N)$
with $\rho(A_0)<e^{\Lambda_0}$ and apply multiplication by $RL$ from the right several times until
we get to the matrix with spectral radius larger than $e^{\Lambda_0}.$ As soon as this happens we start multiplying from the right by matrix $R$ until we have matrix with spectral radius less than $e^{\Lambda_0},$ and then repeat all this.
It is easy to check that this process leads to a sequence of matrices $A_n$ such that
$$
\lim_{n \to \infty} \frac{\ln \rho(A_n)}{n}=\Lambda_0.
$$
\end{proof}
Let us extend $\Lambda$ to negative $x$ by $\Lambda(-x)=\Lambda(x)$ and define $\Lambda(\infty)=0.$
\begin{Corollary}
The function $\Lambda(x), \,\, x \in \mathbb RP^1$ is $GL_2(\mathbb Z)$-invariant:
$$
\Lambda\left(\frac{ax+b}{cx+d}\right)=\Lambda(x), \quad x \in \mathbb{R}P^1
$$
for all integer $a,b,c,d$, satisfying $ad-bc=\pm 1.$
\end{Corollary}
Indeed, it is well-known that two irrational numbers $x, y \in \mathbb R$ are $GL_2(\mathbb Z)$-equivalent, which means that
$$
y=\frac{ax+b}{cx+d}, \quad \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix} \in GL_2(\mathbb Z),
$$
if and only if $x$ and $y$ have continued fraction expansions which eventually coincide (see e.g. \cite{LeVeque}).
This implies that the corresponding paths $\gamma_x$ and $\gamma_y$ have eventually the same sequence of left and right turns (see sections 5 and 6 below), and by Lemma 1 have the same Lyapunov exponents.
In particular, $\Lambda(x+1)=\Lambda(x)$ is periodic, so one can consider it only at the segment $[0,1]$.
\begin{Theorem}
The Lyapunov exponent $\Lambda(x)=0$ for almost every $x \in [0,1].$ In particular, for almost every $x$ the limsup in the definition of $\Lambda(x)$ can be replaced by the usual limit.
\end{Theorem}
\begin{proof}
For rational $x$ we have $\Lambda(x)=0$, so assume that $x$ is irrational. Let $x = [a_1,a_2,\ldots]$ be its expansion as a continued fraction, $\frac{p_n(x)}{q_n(x)}=[a_1,a_2,\ldots,a_n]$ be the $n$-th convergent and $s_n(x)=a_1+\dots+a_n.$
Then by definition (\ref{defeucl}) we have
$$
\Lambda(x)=\limsup_{n\to\infty}\frac{\ln q_n(x)}{s_n(x)}=\limsup_{n\to\infty}\frac{\ln q_n(x)}{n} \frac{n}{s_n(x)}.
$$
But by the classical result of Paul L\`evy \cite{Levy} for almost all $x$
$$
\lim_{n\to\infty}\frac{\ln q_n(x)}{n} =\frac{\pi^2}{12 \ln 2}.
$$
Now the result follows from the known fact (see \cite{Corn}, Th. 4 in Ch. 7, Section 4) that for almost every $x$ $$\lim_{n\to\infty}\frac{s_n(x)}{n}=\infty.$$
\end{proof}
An interesting question is to study more the set
$$supp(\Lambda)=\{x \in \mathbb R: \Lambda(x)\neq 0\},$$
and, in particular, to find its Hausdorff dimension (cf. e.g. \cite{Hensley}).
For the quadratic irrationals the values of $\Lambda$ can be described explicitly.
Let $x=[a_1,\dots, a_k, \overline{b_1, b_2, \dots, b_{2n}}]$ be the continued fraction expansion of a quadratic irrational $x$, which is known
after Lagrange to be periodic. We assume that the length of the period is even by doubling it if necessary.
Define the matrix $B(x) \in SL_2(\mathbb N)$ as the product
$$
B=R^{b_1}L^{b_2}\dots R^{b_{2n-1}}L^{b_{2n}}=\begin{pmatrix}
1 & 0 \\
b_1 & 1 \\
\end{pmatrix}
\begin{pmatrix}
1 & b_2 \\
0 & 1 \\
\end{pmatrix}
\dots
\begin{pmatrix}
1 & 0 \\
b_{2n-1} & 1 \\
\end{pmatrix}
\begin{pmatrix}
1 & b_{2n} \\
0 & 1 \\
\end{pmatrix}.
$$
Let $\tau=tr\, B(x)$ be the trace and $$\lambda(x)=\frac{\tau+\sqrt{\tau^2-4}}{2}$$ be the largest eigenvalue (or spectral radius) of $B(x).$
\begin{prop}
\label{spect}
The Lyapunov exponent of the quadratic irrational $$x=[a_1,\dots, a_k, \overline{b_1, b_2, \dots, b_{2n}}]$$ can be described explicitly as
\beq{defeucl}
\Lambda(x)=\frac{\ln \lambda(x)}{s(x)},
\eeq
where $s(x)=b_1+\dots+b_{2n}.$
\end{prop}
The proof follows easily from the results of this section.
In particular, we have for $x=\sqrt{2}, \sqrt{3}, \sqrt{5}$ the periods $\overline{2,2}$, $\overline{1,2}$, $\overline{4,4}$ respectively, so
$$
\Lambda(\sqrt{2})=\frac{1}{4} \ln (3+2\sqrt{2}), \,\, \Lambda(\sqrt{3})=\frac{1}{3} \ln (2+\sqrt{3}), \,\, \Lambda(\sqrt{5})=\frac{1}{8} \ln (9+4\sqrt{5}).
$$
\section{Markov forms and the Cohn tree}
Before we proceed to the most irrational numbers let us introduce the notion of the Markov binary quadratic form \cite{Markov}.
Let $(k,l,m)$ be a Markov triple:
$$
k^2+l^2+m^2=3klm
$$
with $m$ being the largest number.
The {\it Markov form} $f_m(x,y)$ associated to this Markov triple has the form
\beq{mform}
f_m(x,y)=mx^2+(3m-2p)xy+(q-3p)y^2,
\eeq
where
\beq{pq}
p:= \min \{x : lx \equiv \pm k \text{ }(\text{mod } m) \}, \quad
q:= \frac{1}{m} (p^2 + 1).
\eeq
This is an indefinite binary quadratic form with the discriminant
$$\Delta(f_m)=9m^2-4$$
and with
$
m(f_m)=m,
$
where by definition
$$
m(f):=\min_{(x,y) \in \mathbb Z^2\setminus (0,0)} |f(x,y)|.
$$
Markov studied the possible values of the ratio
$$
M_f=\frac{m(f)}{\sqrt{\Delta(f)}}
$$
for the indefinite integral binary forms and showed that all possible values $M=M_f$ larger than $1/3$ are given by
$$
M=\frac{m}{\sqrt{9m^2-4}},
$$
where $m$ is a Markov number, and realised by the Markov forms (see \cite{Delone}).
The corresponding positive roots $x=\alpha_m$ of $f_m(x,1)=0$ give the most irrational numbers, which we will discuss in the next section.
They have the continued fraction expansion
$$
\alpha_m = [\overline{a_1, \ldots, a_{2n}}]
$$
with the following properties (Markov \cite{Markov}, Frobenius \cite{Frobenius}; see also Cusick-Flahive \cite{Cuhive}, Ch. 2 Th.3 ):
$$
m=K(a_1, \ldots, a_{2n-1}), \\
p=K(a_2, \ldots, a_{2n-1}), \\
q=K(a_2, \ldots, a_{2n-3}),
$$
where $K(s_1, \ldots, s_n)$ is the continuant, which is the numerator of the continued fraction $[s_1, \ldots, s_n]$. We also have
$$
a_1=a_{2n}=2, \,\, a_{2n-2}=a_{2n-1}=1,
$$
and the sequence $a_2,\ldots,a_{2n-3}$ is palindromic.
We would like to explain now the connection of the Markov forms with the following ``quantum version" of the Euclid tree, known as the {\it Cohn tree.}
Cohn \cite{Cohn} proposed to replace the addition of integer numbers in $u+v=w$ by multiplication of matrices in $SL_2(\mathbb N)$, so the triples on Cohn tree are $(A,B,C)$ with $C=AB$ with initial matrices
\begin{equation}
\label{initial}
A=\left(\begin{array}{cc} 1 & 1 \\ 1 & 2
\end{array}\right), \quad B=\left(\begin{array}{cc} 3 & 4 \\ 2 & 3
\end{array}\right)
\end{equation}
(see Fig. 6). The relation between Cohn and Markov trees are given simply by the ``trace map"
$$
C \rightarrow m = \frac{1}{3}tr \, C.
$$
\begin{figure}[h]
\label{fig:CFTree}
\begin{center}
\includegraphics[trim = 0mm 30mm 0mm 22mm, clip, height=38mm]{QETree2} \hspace{8pt} \quad \includegraphics[trim = 0mm 30mm 0mm 22mm, clip, height=38mm]{MarkovTree2}
\caption{\small Cohn and Markov trees related by trace map}
\end{center}
\end{figure}
To state the relation with Markov forms we need to recall a standard relation between matrices from $SL_2(\mathbb Z)$ and integral binary quadratic forms (see e.g. \cite{LeVeque}).
Let $A=\left(\begin{array}{cc} a & b \\ c & d
\end{array}\right) \in SL(2,{\mathbb Z})$ be a hyperbolic matrix from $SL_2(\mathbb Z)$. Consider $A$ as the automorphism of the lattice
${\mathcal L} = {\mathbb Z} \oplus {\mathbb Z} \subset {\mathbb R}^2$ by choosing
some basis $e_1, e_2$ in this lattice. Then we can define the following integral binary
quadratic form $Q_A$ by the formula
\begin{equation}
\label{QA}
{\bf v}\wedge A{\bf v} = Q_A ({\bf v}) e_1 \wedge e_2,
\end{equation}
where ${\bf v}$ is a vector from ${\mathbb R}^2.$ Explicitly if ${\bf
v} = x e_1 + y e_2$ then
\begin{equation}
\label{formQ}
Q_A(x,y) =
\det\left(\begin{array}{cc}
x & a x+b y
\\ y & c x + d y \end{array}\right) = c x^2 + (d-a) xy - b y^2.
\end{equation}
The main property of this form (easily seen from the definition) is that this form is invariant
under the action of $A$:
$$
Q_A(A{\bf v}) = Q_A({\bf v}).
$$
Note that the discriminant of $Q_A$ is
$$
D = (d-a)^2 +
4bc = (a+d)^2 - 4(ad-bc)=
(a+d)^2 -4,
$$
which is exactly the discriminant of the
characteristic equation of $A$:
$$
\lambda^2 - (a+d)
\lambda + 1 = 0.
$$
In particular, since $A$ is hyperbolic the form
$Q_A$ is indefinite.
The following theorem (which seems to be new) gives a direct link between Cohn matrices and Markov forms.
\begin{Theorem}
Let $A_m$ be the matrix from Cohn tree corresponding to Markov number $m.$
Then Markov form $f_m(x,y)$ can be written as
\beq{cohnform}
f_m(x,y)=Q_{m}(x+y,y)
\eeq
where $Q_m=Q_{A_m}$ is the binary form (\ref{formQ}) corresponding to $A_m$.
\end{Theorem}
\begin{proof}
We use the results of Aigner \cite{Aigner}, who showed that Cohn matrix $A_m$ has the form
\beq{aigner}
A_m=\begin{pmatrix}
m+p & 2m+p-q \\
m & 2m-p\\
\end{pmatrix},
\eeq
where $p$ and $q$ are the same as in the definition of Markov form (see Thm. 4.13 in \cite{Aigner}, bearing in mind that Aigner's version of Cohn matrices is transposed to ours).
Using (\ref{formQ}) we have
$$
Q_{m}(x,y)=mx^2+(m-2p)xy -(2m+p-q)y^2,
$$
$$
Q_{m}(x+y,y)=m(x+y)^2+(m-2p)(x+y)y -(2m+p-q)y^2
$$
$$=mx^2+(3m-2p)xy+(q-3p)y^2,
$$
which is exactly the Markov form (\ref{mform}).
\end{proof}
Note that there is also a very deep relation of the Cohn tree with combinatorial group theory and with the automorphisms of free group $\mathbb F_2$, for which we refer to Chapter 6 in Aigner \cite{Aigner}. This is based on a well-known fact that the mapping class groups of a torus and a punctured torus are both isomorphic to $GL_2(\mathbb Z).$
\section{Markov-Hurwitz most irrational numbers}
We start with the definition of the Markov constant, which is considered (after Markov and Hurwitz) as the measure of the irrationality of a number.
The {\it Markov constant} $\mu(\alpha)$ of an irrational number $\alpha$ is defined as the minimal number $c$ such that the inequality
\begin{equation} \label{eq:markov1}
\left|\alpha-\frac{p}{q}\right|\leq\frac{c}{q^2}
\end{equation}
holds for infinitely many $\frac{p}{q}$.
One can show for $\alpha$ given as an infinite continued fraction $\alpha = [a_1,a_2,\ldots]$ the Markov constant can be computed as
\beq{formula}
\mu(\alpha) = \liminf_{N\to\infty}([0,a_{N+1}, a_{N+2} \ldots]+[a_N,a_{N-1},\ldots,a_1])^{-1}
\eeq
(see e.g. \cite{Burger}).
A well-known result of Hurwitz \cite{Hurwitz} claims that for all irrational $\alpha$ we have
\begin{equation*}
\mu(\alpha)\leq\frac{1}{\sqrt{5}},
\end{equation*}
and $\mu(\alpha)=\frac{1}{\sqrt{5}}$ if and only if $\alpha$ is equivalent to $\frac{1+\sqrt{5}}{2}$.
In other words, the golden ratio (and its equivalents) are the most irrational numbers.
One can ask the natural question of what happens if we exclude values of $\alpha$ equivalent to $\frac{1 + \sqrt{5}}{2}$ from the consideration.
The answer is that for the remaining numbers
$\mu(\alpha) \leq 1/\sqrt 8$
(see e.g. \cite{Burger}) and $\mu(\alpha)=1/\sqrt 8$ if and only if $\alpha$ is equivalent to $1+\sqrt{2}=[\overline{2}]$ (the ``silver ratio").
One can continue this to derive the ``bronze ratio"
$$
\alpha=[\overline{2,2,1,1}]=\frac{9+\sqrt{221}}{10}, \quad \mu(\alpha)=\frac{5}{\sqrt{221}},
$$
which one might find already puzzling.
A remarkable theorem of Markov explains the situation with the top irrational numbers, linking this question with Markov equation.
\begin{Theorem}[Markov \cite{Markov}]
All Markov constants $\mu(\alpha)>\frac{1}{3}$ have the form
$$
\mu = \frac{m}{\sqrt{9m^2 -4}},
$$
where $m \in \mathcal M$ is Markov number.
\end{Theorem}
The original Markov result was stated in terms of binary quadratic forms, considered in the previous section. For modern proofs we refer to Bombieri \cite{Bombieri} and Cusick and Flahive \cite{Cuhive}.
To describe the corresponding most irrational numbers we need the following version of the Markov tree.
It is well-known that the most irrational numbers have periodic continued fractions with even periods consisting of 1's and 2's only (see e.g. \cite{Burger}).
Let us define the {\it conjunction} operation of two periods as
\begin{equation}
\label{eq:HurwitzEqn}
[\overline{s_1, \ldots,s_n}] \odot [\overline{t_1, \ldots,t_m}] = [\overline{s_1, \ldots, s_n, t_1, \ldots, t_m}]
\end{equation}
and construct the new tree using this operation and starting with $A=2_2$ and $B=1_2$,
where by $k_n$ we mean the sequence $k,\dots, k$ of numbers $k$ taken $n$ times.
As a result we have the following {\it Markov-Hurwitz tree} (see Fig.7).
\begin{figure}[h]
\label{fig:CFTree}
\begin{center}
\includegraphics[trim = 0mm 30mm 0mm 22mm, clip, height=38mm]{MarkovTree2} \hspace{8pt} \includegraphics[trim = 0mm 30mm 0mm 22mm, clip, height=38mm]{ContFracTree2}
\caption{\small Markov and Markov-Hurwitz trees}
\end{center}
\end{figure}
Let $y_m$ be the number on Markov-Hurwitz tree corresponding to Markov number $m$.
The following result can be extracted from Cusick and Flahive \cite{Cuhive} (see Lemma 4 in Chapter 2 of \cite{Cuhive}), who made the detailed analysis of the roots $\alpha_m$ of $f_m(x,1)=0$ for Markov forms $f_m(x,y).$
\begin{Theorem} [\cite{Cuhive}] The Markov constant
$$
\mu(y_m) = \frac{m}{\sqrt{9m^2 -4}},
$$
so $y_m$ are representatives of the most irrational numbers.
\end{Theorem}
{\bf Remark.} It follows from the results of \cite{Cuhive} that for $m>1$
\begin{equation}
\label{cfy}
y_m=\mu_m+1=\frac{5c-2d+\sqrt{9c^2-4}}{2c},
\end{equation}
where
$v_m=(\mu_m,1)$ is the eigenvector with the largest eigenvalue of the corresponding matrix
$$
A_m=\left(\begin{array}{cc} a & b \\ c & d
\end{array}\right)
$$
from the Cohn tree.
\section{Paths in Farey tree and Minkowski's $?(x)$ Function}
To describe the most irrational paths in Farey tree we will need the following {\it question mark function} introduced by Minkowski \cite{Mink} and denoted by $?(x)$. It was studied later by Denjoy and by Salem (see more history and references in \cite{ViaParadis}) and can be uniquely defined by the following properties:
\begin{itemize}
\item $?(0)=0, \,\, ?(1)=1.$
\item If $\frac{a}{b}$ and $\frac{c}{d}$ are neighbours in a Farey sequence (which means that $|ad-bc|=1$), then the value of question mark function on their mediant is the arithmetic mean of corresponding values:
\beq{deffar}
?\left(\frac{a+c}{b+d}\right)=\frac{1}{2}\left(?\left(\frac{a}{b}\right)+?\left(\frac{c}{d}\right)\right)
\eeq
\item $?$-function is continuous on $[0,1].$
\end{itemize}
One can show that that it has also the following properties (see e.g.\cite{ViaParadis}):
\begin{itemize}
\item $x$ is rational iff $?(x)$ has finite binary representation (dyadic rational)
\item $x$ is a quadratic irrational iff $?(x)$ is rational, but not dyadic rational
\item $?(x)$ is strictly increasing and defines a homeomorphism of $[0,1]$ to itself
\item $?'(x) = 0$ almost everywhere
\end{itemize}
Salem \cite{Salem} gave a very convenient definition of $?(x)$ in terms of continued fractions.
Namely, if $x$ is given as a continued fraction
\begin{equation*}
x = [a_1, a_2, \dots, a_n, \dots],
\end{equation*}
then
\begin{equation}
\label{eq:Salemfrac}
?(x) = \frac{1}{2^{a_1 - 1}} - \frac{1}{2^{a_1 + a_2 - 1}} + \frac{1}{2^{a_1 + a_2 + a_3 - 1}} - \dots .
\end{equation}
We claim that Minkowski's function $?(x)$ encodes the path $\gamma$ leading to $x$ on the Farey tree (see Fig.4).
More precisely, let $\gamma_x$ be such a path for $x \in [0,1]$ and define the {\it path function} using binary representation as
\begin{equation}
\label{eq:BinMap}
\pi(x)=\pi(\gamma_x): = [0. \epsilon_{1} \epsilon_{2} \dots \epsilon_{j} \dots]_2,
\end{equation}
where
\begin{equation}
\label{eq:binmapdef}
\epsilon_{j} = \left\{
\begin{array}{l l}
0 & \quad \text{if the $j$th step of $\gamma_x$ is a right-turn;}\\
1 & \quad \text{if the $j$th step of $\gamma_x$ is a left-turn.}
\end{array} \right.
\end{equation}
For example, for the path $\gamma$ in Fig. 4 we have
$\pi(\gamma)=[0.1010\dots]_2.$
\begin{Theorem}The path function $\pi(x)$ is nothing other than Minkowski's question mark function.
\end{Theorem}
\begin{proof}
We simply check that $\pi(x)$ satisfies the defining properties of Minkowski's function.
First, we have by definition that
\begin{equation*}
\pi(0) = [0.0000 \dots]_{2} = 0, \quad \pi(1) = [0.1111 \dots]_{2} = 1,
\end{equation*}
and
\begin{equation*}
\pi \left( \frac{1}{2} \right) = [0.1000 \dots]_{2} = \frac{1}{2} = \frac{0 + 1}{2} = \frac{ \pi (0) + \pi (1) }{2}.
\end{equation*}
Now let's check that $\pi(x)$ satisfies the main property (\ref{deffar}):
$$
\pi\left(\frac{a+c}{b+d}\right)=\frac{1}{2}\left(\pi\left(\frac{a}{b}\right)+\pi\left(\frac{c}{d}\right)\right).
$$
Let $\frac{a}{c}$ and $\frac{b}{d}$ be two Farey neighbours assuming that $\frac{a}{c} < \frac{b}{d}$. At every point on the Farey tree apart from $x = \frac{1}{2}$, either $\frac{a}{c}$ or $\frac{b}{d}$ is `higher up' the Farey tree: the binary expansions will be of different lengths. There are two cases to consider.
Case 1: Assume that $\frac{b}{d}$ is `higher up' the Farey tree than $\frac{a}{c}$. Since $\frac{a}{c}$ and $\frac{b}{d}$ are neighbours, we know that
$$
\pi \left( \frac{a}{c} \right) = [0.b_1 b_2 \dots b_n 1]_2, \quad
\pi \left( \frac{b}{d} \right) = [0.b_1 b_2 \dots b_n \beta_1 \beta_2 \dots \beta_k 1]_2,
$$
where $\beta_1 \beta_2 \dots \beta_k = [10 \dots 0]$. Then from the definition \eqref{eq:binmapdef} of $\pi$ we have
$$
\pi \left( \frac{a+b}{c+d} \right) = [0.b_1 b_2 \dots b_n \beta_1 \beta_2 \dots \beta_k 01]_2 = \frac{b_1}{2} + \frac{b_2}{2^2} + \dots + \frac{b_n}{2^n} + \frac{1}{2^{n+1}} + \frac{1}{2^{n+k+2}}$$
$$
= \frac{1}{2} \left[\frac{b_1}{2} + \frac{b_2}{2^2} + \dots + \frac{b_n}{2^n} + \frac{1}{2^{n+1}} \right] + \frac{1}{2} \left[\frac{b_1}{2} + \frac{b_2}{2^2} + \dots + \frac{b_n}{2^n} + \frac{1}{2^{n+1}} + \frac{1}{2^{n+k+1}} \right]$$
$$
= \frac{1}{2}\left(\pi \left( \frac{a}{c} \right) + \pi \left( \frac{b}{d} \right)\right).
$$
Case 2: Now assume that $\frac{a}{c}$ is `higher', so that
$$
\pi \left( \frac{a}{c} \right) = [0.b_1 b_2 \dots b_n \beta_1 \beta_2 \dots \beta_k 1]_2, \quad
\pi \left( \frac{b}{d} \right) = [0.b_1 b_2 \dots b_n 1]_2,
$$
where $\beta_1 \beta_2 \dots \beta_k = [01 \dots 1]$. Again from \eqref{eq:binmapdef} we have
$$
\pi \left( \frac{a+b}{c+d} \right) = [0.b_1 b_2 \dots b_n \beta_1 \beta_2 \dots \beta_k 11]_2 $$
$$= \frac{b_1}{2} + \frac{b_2}{2^2} + \dots + \frac{b_n}{2^n} + \frac{1}{2^{n+2}} + \frac{1}{2^{n+3}} + \dots+ \frac{1}{2^{n+k}} + \frac{1}{2^{n+k+1}} + \frac{1}{2^{n+k+2}}$$
$$ = \frac{1}{2} \left[\frac{b_1}{2} + \frac{b_2}{2^2} + \dots
+ \frac{b_n}{2^n} + \frac{1}{2^{n+2}} + \frac{1}{2^{n+3}} + \dots + \frac{1}{2^{n+k}} + \frac{1}{2^{n+k+1}} \right]
\quad$$
$$ + \frac{1}{2} \left[\frac{b_1}{2} + \frac{b_2}{2^2} + \dots + \frac{b_n}{2^n} + \frac{1}{2^{n+1}} \right]
= \frac{1}{2}\left(\pi \left( \frac{a}{c} \right) + \pi \left( \frac{b}{d} \right)\right).$$
So in either case, we have
\begin{equation*}
\pi \left( \frac{a+b}{c+d} \right) = \left(\pi \left( \frac{a}{c} \right) + \pi \left( \frac{b}{d} \right)\right),
\end{equation*}
which means that $\pi$ coincides with Minkowski function on all rational numbers.
Since $\pi(x)$ is monotonic it must coincide with $?(x)$ for all $x \in [0,1].$
\end{proof}
\section{Most irrational paths and Minkowski tree}
Let now $x=\alpha$ be quadratic irrational and assume that $\alpha$ has a pure periodic continued fraction expansion
$
\alpha=[\overline{a}]
$
with even period $a=a_1, \ldots, a_{2n}$ (if the period is odd we will double it to make even).
It follows from Salem's formula (\ref{eq:Salemfrac}) that the value of Minkowski's function
$?(\alpha)$ has a pure periodic binary representation
$$
?(\alpha)=[0.\overline{A}]_2
$$
with period $A$ of length $a_1+\dots +a_{2n}$ consisting of $a_1-1$ 0's followed by $a_2$ 1's, then followed by $a_3$ 0's etc until we have $a_{2n}$ 1's followed by one final $0.$
For convenience we will drop the initial zero and write simply $[\overline{A}]_2$ instead of $[0.\overline{A}]_2.$
In particular, we have
$$
?([\overline{1,1}]) = [\overline{10}]_2, \, \,\,
?([\overline{2,2}]) = [\overline{0110}]_2, \,\,\,
?([\overline{2,2,1,1}]) = [\overline{011010}]_2.
$$
Using Salem's representation we can prove the following conjunction property of Minkowski's function.
\begin{prop}
Let
$
[\overline{a}]=[\overline{a_1, \ldots, a_{2n}}], \,\, [\overline{b}]=[\overline{b_1, \ldots, b_{2m}}]
$
be two continued fractions of even periods and
$$
?([\overline{a}])=[\overline{A}]_2, \quad ?([\overline{b}])=[\overline{B}]_2.
$$
Then
\begin{equation}
?([\overline{ab}])=[\overline{AB}]_2.
\end{equation}
\end{prop}
\begin{proof}
Observe that
$$
?([\overline{ab}]) = \frac{1}{2^{a_1 - 1}} - \frac{1}{2^{a_1 + a_2 - 1}} + \dots - \frac{1}{2^{a_1 + \ldots + a_{2n} - 1}} $$
$$
+ \frac{1}{2^{a_1 + \ldots + a_{2n} + b_1 - 1}} - \frac{1}{2^{a_1 + \ldots + a_{2n} + b_1 + b_2 - 1}} + \ldots - \frac{1}{2^{a_1 + \ldots + a_{2n} + b_1 + \ldots + b_{2m} - 1}} $$
$$
+ \frac{1}{2^{a_1 + \ldots + a_{2n} + b_1 + \ldots + b_{2m} + a_1 - 1}} - \frac{1}{2^{a_1 + \ldots + a_{2n} + b_1 + \ldots + b_{2m} + a_1 + a_2 - 1}} + \ldots $$
$$
=[A]_2 +[\underbrace{0 \ldots 0}_{\sum{a_{i}}}B]_2 +[\underbrace{0 \ldots 0}_{\sum{a_i + b_i}}A]_2 + \ldots =[\overline{AB}]_2.
$$
\end{proof}
Thus Minkowski's $?(x)$ function maps the most irrational numbers to particular binary expansions, specifically those which mirror the continued fraction expansion of the most irrational numbers with ``1,1'' replaced by ``10'' and ``2,2'' replaced by ``0110''.
Applying Minkowski's function to the Markov-Hurwitz tree we have the {\it Minkowski tree}, encoding the paths to the most irrational numbers (see Fig.8, where $i_k$ means $i$ repeated $k$ times).
\begin{figure}[h]
\label{fig:CFTree}
\begin{center}
\includegraphics[trim = 0mm 30mm 0mm 22mm, clip, height=38mm]{ContFracTree2} \hspace{12pt} \includegraphics[trim = 0mm 30mm 0mm 30mm, clip, height=38mm]{Minkowski-tree-sub-3}
\caption{\small Markov-Hurwitz and Minkowski trees related by $?$-function}
\end{center}
\end{figure}
\section{Lyapunov exponents of the most irrational paths.}
Let $m(\frac{p}{q})\in \mathcal M$ be the Markov number corresponding to the Farey fraction $\frac{p}{q}\in \frac{1}{2}$ (see Fig.2), and
$x(\frac{p}{q})$ be a representative of the corresponding class of the most irrational numbers.
It would be convenient for us to choose such representative as the inverse of the corresponding number $y_m$ from Markov-Hurwitz tree:
$x_m=y_m^{-1} \in [0,1].$ We call these representatives
{\it Markov-Hurwitz numbers} and denote by $\mathbb{X}$ the set of all these numbers
$$
\mathbb{X}=\{x_m=y_m^{-1}: m \in \mathcal M\}.
$$
\begin{Theorem}
The function $\Lambda(x(\frac{p}{q}))$ is convex as function of $\frac{p}{q}\in \mathbb Q.$
The restriction $\Lambda_\mathbb{X}$ of $\Lambda(x)$ on the set of Markov-Hurwitz numbers $\mathbb{X}$ is monotonically increasing from $$\Lambda(x_2)=\frac{1}{2}\ln(1+\sqrt 2) \quad {\text to} \quad \Lambda(x_1)=\ln \left(\frac{1+\sqrt 5}{2}\right).$$
\end{Theorem}
\begin{proof}
Following Fock \cite{Fock} consider the following function $\psi(\xi), \, \xi \in [0,\frac{1}{2}].$
First, define it for rational $\xi = \frac{p}{q} \in [0,\frac{1}{2}]\cap \mathbb Q$ as follows
\begin{equation}\label{fock}
\psi \left(\frac{p}{q}\right) = \frac{1}{q} \arcosh \left(\frac{3}{2} m \left(\frac{p}{q}\right)\right).
\end{equation}
Fock proved the following, crucial for us, result (see item 6 in Section 7.3 of \cite{Fock}).
\begin{Theorem} [V. Fock \cite{Fock}]
The function $\psi$ can be extended to a continuous convex function of all $\xi \in \mathbb R$ with the property
\begin{equation}
\label{fock2}
\psi(1-\xi)=\psi(\xi).
\end{equation}
\end{Theorem}
For readers' convenience we present here a version of Fock's proof following \cite{SorVes}.
\begin{proof}
We use the following remarkable relation of Markov numbers to the lengths of closed geodesic on a punctured torus (see \cite{Cohn, Gorshkov, Series}).
Consider the one punctured equianharmonic torus $T_*$ with hyperbolic metric. The corresponding Fuchsian group is generated by Cohn matrices (\ref{initial}) and coincides with the commutator subgroup of $SL_2(\mathbb Z)$ (see e.g. \cite{Aigner}). Then Markov numbers can be interpreted as
$$
m(\frac{p}{q})=\frac{1}{3} tr \, A_m=\frac{2}{3} \cosh L(p,q),
$$
where $L(p,q)$ is the length of the closed geodesic (known to be unique) in primitive homology class $(p,q) \in H_1(T, \mathbb Z),$ and $A_m$ is the matrix from Cohn tree corresponding to $m=m(\frac{p}{q})$.
The length function $L(p,q)$ obviously satisfies the inequality
$$
L(p_1,q_1)+L(p_2,q_2) \le L(p_1+p_2, q_1+q_2).
$$
This allows to us extend this function by homogeneity and continuity to the norm on the real homology $L(x,y), \, (x,y) \in H_1(T_*, \mathbb R)\cong \mathbb R^2$, known as the {\it stable norm} \cite{GLP}.
Its restriction to the real line $x=\xi, y=1$ coincides with Fock's function $\psi$ at the rational points $\xi=p/q.$ Indeed, $L(p/q,1)=\frac{1}{q}L(p,q)$ by homogeneity. Now Fock's claim follows from the general fact that any norm restricted to a line is a continuous convex function.
The property (\ref{fock2}) follows from the symmetry $L(p,q)=L(q-p,q).$
\end{proof}
{\it Remark.} Combining this with Theorem 2.1 from McShane and Rivin \cite{McShane} we can deduce that Fock's function is differentiable at every irrational and non-differential at every rational point (see \cite{SorVes}).
We claim now that our function
$$\Lambda(x(\frac{p}{q}))=\frac{1}{2}\psi(\frac{p}{q})$$
is simply half of Fock's function.
Indeed, let $x_m$ be a Markov-Hurwitz number and
$?(x_m)=[\overline{a}]_2$, with $a=\epsilon_1, \dots, \epsilon_{2q}$, be its image under Minkowski's function. It is easy to see that the length of the period $2q$ is exactly twice the denominator of the Farey fraction $\frac{p}{q}$ corresponding to $m$ (see Fig.8).
Now we should use the path defined by $a$ to climb up the Farey tree.
The second key observation is that we will come to the matrix $A_m \in SL_2(\mathbb N)$, which is nothing other than the Cohn matrix corresponding to $m.$
Indeed, for $x_1=\frac{\sqrt 5-1}{2}=[\overline{11}]$ we have $?(x_1)=[\overline{10}]_2$ and the corresponding matrix
$$A_1=\begin{pmatrix}
1 & 0 \\
1 & 1 \\
\end{pmatrix} \begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}=\begin{pmatrix}
1 & 1 \\
1 & 2 \\
\end{pmatrix}.
$$
Similarly, for $x_2=\sqrt 2 -1=[\overline{22}]$ we have $?(x_2)=[\overline{0110}]_2$ and
$$A_2=\begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}\begin{pmatrix}
1 & 0 \\
1 & 1 \\
\end{pmatrix} \begin{pmatrix}
1 & 0 \\
1 & 1 \\
\end{pmatrix} \begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}=\begin{pmatrix}
3 & 4 \\
2 & 3 \\
\end{pmatrix}.
$$
The general case follows from the conjunction rule for the Minkowski tree and the product rule for the Cohn tree.
This means that the Lyapunov exponent
$\Lambda(x_m)=\frac{\ln \lambda(m)}{2q},$
where $\lambda(m)$ is the largest eigenvalue of $A_m$.
But we know that the Cohn matrix $A_m$ has the trace $3m$ and thus the characteristic equation
$$\lambda^2-3m\lambda+1=0.$$
Thus the Lyapunov exponent is
\begin{equation}
\Lambda(x_m)=\frac{1}{2q}\ln{\left(\frac{3m+\sqrt{9m^2-4}}{2}\right)} = \frac{1}{2q}\arcosh \left( \frac{3m}{2} \right),
\end{equation}
which is exactly half of the Fock function. This proves the convexity of $\Lambda(x(\frac{p}{q}))$.
To prove the monotonicity we note first that the function $x(\frac{p}{q})$ is monotonically decreasing, which follows from the conjunction construction of Markov-Hurwitz tree.
Since Fock's function $\psi$ is convex and satisfies (\ref{fock2}) it has the minimum at $\xi=\frac{1}{2}.$ This means that $\Lambda(x(\frac{p}{q}))$ is monotonically decreasing when $\frac{p}{q} \in [0,\frac{1}{2}]$, and thus $\Lambda(x)$ is strictly increasing on $\mathbb X.$
\end{proof}
\section{Generalised Markov-Hurwitz sets}
Part of theorem 8 can be generalised to the following sets.
Let $a\in \mathbb Z_{>0}$ be an integer parameter and
consider a version of the Hurwitz tree starting with the continued fractions $[\overline{2a_2}]=[\overline{2a,2a}]$, $[\overline{a_2}]=[\overline{a,a}]$
and the corresponding version of the Minkowski tree growing from $[\overline {0_{2a-1}1_{2a} 0}]_2=[\overline{\underbrace{0 \ldots 0}_{2a-1}\underbrace{1 \ldots 1}_{2a}0}]_2$ and $[ \overline {0_{a-1}1_{a} 0}]_2=[\overline{\underbrace{0 \ldots 0}_{a-1}\underbrace{1 \ldots 1}_{a}0}]_2$, where we continue to drop the initial zero as before (see Fig.9).
\begin{figure}[h]
\label{fig:CFTree-a}
\centering{
\includegraphics[trim = 0mm 30mm 0mm 30mm, clip, height=42mm]{MH-a-tree1}
\includegraphics[trim = 0mm 30mm 0mm 30mm, clip, height=42mm]{Minkowski-a-tree-sub1}
\caption{\small Generalised Markov-Hurwitz and Minkowski trees}}
\end{figure}
Let us denote by $\mathbb{X}_{a}$ the set of the inverses of the corresponding quadratic irrationals from this version of the Hurwitz tree.
When $a=1$ we have the set $\mathbb{X}_1=\mathbb{X}$ considered before.
The corresponding version of Cohn tree starts with the generalisation of Cohn matrices (\ref{initial})
\begin{equation}
\label{cohna}
M_a = \begin{pmatrix}
1-a+a^2 & a^2 \\
a & a+1 \\
\end{pmatrix}, \quad M_{2a} =
\begin{pmatrix}
1-2a+4a^2 & 4a^2 \\
2a & 2a+1 \\
\end{pmatrix}.
\end{equation}
Indeed, it is easy to check that
$$
\begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}^{2a-1}\begin{pmatrix}
1 & 0 \\
1 & 1 \\
\end{pmatrix}^{2a}\begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}=\begin{pmatrix}
1 & 2a-1 \\
0 & 1 \\
\end{pmatrix}\begin{pmatrix}
1 & 0 \\
2a & 1 \\
\end{pmatrix}\begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}=M_a.
$$
The trace map $A \rightarrow tr \, A$ produces the $a$-generalisation of Markov tree shown in Fig.10.
\begin{figure}[h]
\centering
\includegraphics[trim = 0mm 30mm 0mm 30mm, clip, height=38mm]{Markov-a-tree1} \hspace{8pt} \quad \includegraphics[trim = 0mm 30mm 0mm 30mm, clip, height=38mm]{FareyTree21}
\label{fig:M-a-tree}
\caption{The $a$-generalisation of Markov tree and corresponding Farey fractions.}
\end{figure}
The corresponding triples are the integer solutions of the following version of Markov equation studied by Mordell \cite{Mordell}
\begin{equation}
\label{xyza}
X^2 + Y^2 + Z^2=XYZ+4-4a^6.
\end{equation}
Note that when $a=1$ we have the equation
\begin{equation}
\label{xyz}
X^2 + Y^2 + Z^2=XYZ,
\end{equation}
which is a simply rescaled version of Markov equation (\ref{equa}) and has integer solutions being Markov triples multiplied by 3: $$X=3x, \,Y=3y,\, Z=3z.$$
The modified equation (\ref{xyza}) has no fully symmetric solutions, but has a solution with $X=Y$:
$$X=Y=a^2+2, \, Z=4a^2+2.$$
Applying to this solution Vieta involution
$
(X,Y,Z) \rightarrow (X,Y, XY-Z)
$
and permutations we have the generalised Markov tree above.
Let $A(a,\frac{p}{q})$ be the matrix from the $a$-Cohn tree corresponding to the fraction $\frac{p}{q}$ from the Farey tree, $m(a,\frac{p}{q})=tr\, A(a,\frac{p}{q})$ be the corresponding $a$-Markov number:
$$
m(a,\frac{0}{1})=a^2+2, \, m(a,\frac{1}{2})=4a^2+2,\, m(a,\frac{1}{3})=4a^4+9a^2+2,\dots,
$$
$y(a,\frac{p}{q})$ be the corresponding quadratic irrational from the $a$-version of Markov-Hurwitz tree, $x(a,\frac{p}{q})=y(a,\frac{p}{q})^{-1}$.
Note that, as in the previous case (see Remark at the end of Section 4), we have
$$y(a,\frac{p}{q})=\mu(a,\frac{p}{q})+1,$$
where
$v=(\mu(a,\frac{p}{q}),1)^T$ is the eigenvector with the largest eigenvalue of the matrix $A(a,\frac{p}{q}).$
The key observation is that on our set $\mathbb{X}_a$ the values of the Lyapunov function have the form
\begin{equation}
\label{gena}
\Lambda(x(a,\frac{p}{q}))= \frac{\ln \lambda(a,\frac{p}{q})}{2aq},
\end{equation}
where
$$\lambda(a,\frac{p}{q})=\frac{m+\sqrt{m^2-4}}{2}, \quad m=m(a,\frac{p}{q})$$ is the largest eigenvalue of the Cohn matrix $A(a,\frac{p}{q}).$
The proof is a straightforward generalisation of the arguments from the previous section.
Geometrically the equation (\ref{xyza}) describes the lengths of the closed geodesics on the equianharmonic hyperbolic torus with a hole (see e.g. \cite{Cohn, Fock}) of length
$$l=\arcosh (2a^6-1).$$
This follows from the {\it Fricke identities} \cite{Fricke}: for any $A,B \in SL_2(\mathbb R), C=AB$ we have
$$
tr\, AB + tr\, AB^{-1}=tr\, A\, tr\, B,
$$
\begin{equation}
\label{fricke}
(tr\, A)^2+(tr\, B)^2+(tr\, C)^2=tr\, A \, tr\, B\, tr\,C +tr\, (ABA^{-1}B^{-1})+2.
\end{equation}
This means that $X=tr\, A,\, Y=tr\, B,\, Z=tr\, C$ satisfy (\ref{xyza}) with $$tr\, ABA^{-1}B^{-1}=2-4a^6.$$
The matrices $A=M_a$ and $B=M_{2a}$ generate the Fuchsian subgroup $G_a$ of $SL_2(\mathbb R)$, which is free. The corresponding quotient of the upper half-plane is a hyperbolic torus with a hole. The length of the hole satisfies $$2 \cosh l=|tr\,ABA^{-1}B^{-1}|= 4a^6-2$$ giving $l=\arcosh (2a^6-1).$
When $a=1$ we have the punctured torus with $l=0$ and the scaled version of the Markov equation.
Repeating the proof of Fock's theorem for the stable norm of one-hole torus we have the following result.
\begin{Theorem}
The function $\Lambda(x(a,\frac{p}{q}))$ is convex as function of $\frac{p}{q}$ for all $a \in \mathbb N$. The restriction of $\Lambda$ to the set $\mathbb{X}_{a}$ is monotonically increasing.
\end{Theorem}
\section{Concluding remarks}
Our results can be applied to study the topological entropy of the modular group dynamics on the corresponding affine cubic surfaces
\beq{cantat}
x^2+y^2+z^2=3xyz+D, \quad x,y,z \in \mathbb C.
\eeq
Indeed, Cantat and Loray \cite{Cantat} showed that the topological entropy of the dynamics generated by the action of $A \in SL_2(\mathbb Z)$ is equal to the logarithm of the spectral radius of $A$ (see also Iwasaki and Uehara \cite{Iwasaki}).
Thus our function $\Lambda(x)$ can be interpreted as the average topological entropy along the path $\gamma_x$ on binary tree.
One can view our work as part of the theory of $SL_2(\mathbb Z)$ dynamics, or more generally, of braid group $B_3$ actions \cite{Veselov}.
The examples of such dynamical systems naturally come from the theory of Yang-Baxter maps \cite{V2007}.
A more interesting example, due to Dubrovin \cite{D1}, comes from the theory of Painlev\'e-VI equation, where the algebraic solutions correspond to the finite orbits of the braid group $B_3$, which are classified in \cite{LT}. It is remarkable that the Markov orbit corresponds to a very special, non-algebraic, solution of Painlev\'e-VI, which is related to the enumerative geometry and quantum cohomology of $\mathbb CP^2$ (Kontsevich and Manin, Dubrovin). Another remarkable appearance of the Markov numbers in algebraic geometry is related to the notion of the exceptional vector bundles, see \cite{Rud}.
The most intriguing question about the Lyapunov function $\Lambda$ is whether it is already known in some parts of mathematics.
The invariance under modular group suggests that $\Lambda(x)$ might be interpreted as the limit values of some modular function on the real boundary of the hyperbolic plane (see e.g. \cite{Reyna} for Riemann's approach to this problem).
\section{Acknowledgements}
We are very grateful to Martin Aigner, Wael Bahsoun, Alexey Bolsinov, Leonid Chekhov, Nikolai P. Dolbilin, Dmitri Orlov, Alfonso Sorrentino and Boris Springborn for very helpful discussions, to Andy Hone for telling us about Minkowski function, to Vladimir Fock for explaining the proof of his results from \cite{Fock} and to Peter Sarnak for encouragement.
Special thanks go to the referee for very constructive criticism, which helped us to substantially improve the paper.
The work of K.S. was supported by the EPSRC as part of PhD study at Loughborough.
|
1,314,259,996,669 | arxiv | \section{Introduction}
In the best understood examples of the AdS/CFT correspondence, the gravity description is accomplished in terms of strings and D-branes in Anti de Sitter spaces with a constant dilaton, reflecting the conformal symmetry of the quantum field theory description. Considering heavy objects on the gravity side naturally leads to backreaction in which case the isometries of AdS are only preserved asymptotically and the dilaton is no longer constant. In the quantum field theory description this situation corresponds, typically, to the computation of expectation values, not in the vacuum of the theory, but in some states related to operators with large quantum numbers. This setup deviates from conformal invariance and in this manuscript we explore one of its explicit still controlled instances.
When deviating from strict AdS spaces there are not as many exact results as in conformal situations where one can explore the scenario described in the previous paragraph by comparing string theory with gauge theory results explicitly. One rare example of such exact results in non-conformal situations is the computation of the partition function, Wilson loops expectation values and correlators in ${\cal N}=2^{*}$ super Yang-Mills and its holographic dual \cite{Buchel:2013id,Bobev:2013cja,Chen-Lin:2015dfa,Chen-Lin:2015xlh,Liu:2017fiq}.
A different setup to study the AdS/CFT correspondence in non-conformal situations, which we intend to explore in this article, arises with the computation of Wilson loop correlators in cases where one of them is taken in a large rank representation. On the gravity side, such large rank representation Wilson loops are described in terms of $\tfrac{1}{2}$-BPS backreacted spaces, with isometry group $SO(2, 1)\times SO(3)\times SO(5)$ and which present a running dilaton and fluxes turned on. The construction of these bubbling geometries (see \cite{Lin:2004nb} for bubbling geometries associated to the insertion of chiral fields) took various steps \cite{Lunin:2006xr,Yamaguchi:2006te} before culminating in \cite{D'Hoker:2007fq},
where these type IIB supergravity solutions were found in terms of two harmonic functions on a Riemann surface $\Sigma$ on whose boundary the dual Wilson loop representation data is encoded. These supergravity solutions are highly involved and arguably represent the state-of-the-art as a far as supergravity solutions are concerned. Strings and minimal area surfaces in this kind of bubbling geometries have been studied in \cite{Benichou:2011aa,Gentle:2014lva}, in order to compute gravitational potential between open strings and to account for entanglement entropies holographically.
The expectation value of $\tfrac{1}{2}$-BPS circular Wilson loops for arbitrary representations can be computed with a Gaussian matrix model. This was first conjectured by Erickson, Semenoff and Zarembo in \cite{Erickson:2000af} and Drukker and Gross in \cite{Drukker:2000rr}, and it was finally proven by Pestun using supersymmetric localization \cite{Pestun:2007rz}. Remarkably, if the Wilson loop is taken in the fundamental representation, the matrix model solution leads to an explicit expression via orthogonal polynomials which is exact in the 't Hooft coupling $\lambda$ as well as in the rank of the gauge group, $N$, \cite{Drukker:2000rr}. For higher rank representations the holographic dictionary was established in \cite{Gomis:2006sb,Gomis:2006im}, however, with few exceptions \cite{Fiol:2013hna}, exact expressions for generic $\lambda$ and $N$ seem currently out of reach. Nevertheless, for totaly symmetric and antisymmetric representations, it is possible to obtain expressions that hold in the planar and large $\lambda$ limit \cite{Hartnoll:2006is}, that successfully match the associated D-branes on-shell actions \cite{Drukker:2005kx,Yamaguchi:2006tq}, as predicted by the AdS/CFT correspondence.
Later on, localization techniques were used for other kinds of Wilson loops of arbitrary shapes, preserving less supersymmetry \cite{Drukker:2007dw,Drukker:2007yx,Drukker:2007qr} or to account for correlators of supersymmetric Wilson loops \cite{Giombi:2009ms,Bassetto:2009rt,Giombi:2009ds,Bassetto:2009ms,Giombi:2012ep}, but most of the explicit results have been found for the fundamental representation.
When the Wilson loop representation is even larger, for instance, when the associated Young tableau possesses a number of order $N^2$ boxes, the dual description involves a large number of D-branes that back-react on the geometry. The corresponding matrix model can be solved with a saddle point approximation in the large-$N$ limit provided the sizes the of Young tableau edges $\{n_i,k_i\}$ are taken to be of order $N$ \cite{Okuda:2007kh}. The eigenvalue distribution can be determined in terms of geometric data on the spectral curve which, moreover, is identified with the hyperelliptic surface characterizing the bubbling geometry as beautifully demonstrated in \cite{Okuda:2008px}.
The main purpose of this paper is to compute correlators $\langle W_\textbf{R} W_\textbf{r} \rangle $, between Wilson loops in large representations $\textbf{R}$, whose Young tableau edges $\{n_i,k_i\}$ are of order $N$, and Wilson loops in a ``small'' representation, let us say, fundamental, completely symmetric and completely anti-symmetric. We will consider in particular the case in which both Wilson loops are defined over the coincident circle and coupled to the same scalar, so that both are invariant under the same set of symmetries and supersymmetry transformations. This allows to compute the correlator directly in the field theory using the matrix model that is obtained by supersymmetric localization. The gist of our matrix model calculation is that the ``small'' Wilson loop does not back-react on the eigenvalue distribution of the large representation Wilson loop. Thus, the correlator is eventually given by an expectation value in the eigenvalue distribution of the large representation Wilson loop.
According to the AdS/CFT correspondence, the correlator of Wilson loops of the form $\langle W_{\mathbf{R}} W_{\text{fund}}\rangle$ can be computed, in the large 't Hooft coupling $\lambda$ limit, as the on-shell action of certain strings in the bubbling geometries found in \cite{D'Hoker:2007fq}. Among the many strings that can propagate in the bubbling geometries, the ones that can be related to the particular correlator given are those invariant under the same symmetries and supersymmetries of the background. We demonstrate in this manuscript that there is precise agreement between the two sides of the correspondence.
The paper is organized as follows. In section \ref{reviewbubbling} we review the bubbling geometries dual to large representation Wilson loops and the relation between their charges and the Young tableau parameters. In section \ref{strings} we present the minimal area string configurations in generic bubbling geometries. We consider in detail the case of strings in genus one bubbling geometries, dual to a Wilson loop in a rectangular Young tableau representation, and give explicit expressions for the on-shell actions that will be later compared with matrix model results. At the end of this section, we extend our results to general genus $g$ backgrounds. In section \ref{matrixmodel} we turn to the matrix model description of the correlator of Wilson loops. We first focus on the correlator of a Wilson loop in the fundamental representation and one in a representation given by a rectangular Young tableau, but we later consider more generic cases. We finally conclude and comment our results in section \ref{conclu}. We also include various appendices for the readers interested in further details on the results presented in the main text.
\section{ Review of bubbling geometries dual to ${1\over 2}$-BPS Wilson loops}
\label{reviewbubbling}
The general bubbling geometry background corresponds to solutions of type IIB supergravity that preserves a $SO(2,1)\times SO(3)\times SO(5)$ isometry group and 1/2 of the total supersymmetry \cite{D'Hoker:2007fq}. The resulting metric is the one associated with an $\mathbb{H}_2$, $S^2$ and $S^4$ fibration over a 2-dimensional complex Riemann surface $\Sigma$. The metric in the Einstein frame can be written as
\begin{equation}
ds^2 = G^E_{MN} \,dx^M\, dx^N=f_1^2 ds^2_{\mathbb{H}_2}+f_2^2ds^2_{S^2} +f_4^2ds^2_{S^4} +d\Sigma^2\,.
\label{metric}
\end{equation}
A quite remarkable fact about these solutions is that all the geometric functions and fluxes are completely determined by two holomorphic functions $\mathcal{A}$ and $\mathcal{B}$ defined on the Riemann surface $\Sigma$. Equivalently, the geometry can be specified in terms of four real harmonic functions defined as
\begin{align}
h_1 &=\mathcal{A}+\bar{\mathcal{A}}\,, \quad & \widetilde{h}_1 =\,{\rm i}\,\left(\mathcal{A}-\bar{\mathcal{A}}\right)\,, \nonumber\\
h_2 &=\mathcal{B}+\bar{\mathcal{B}}\,, \quad & \widetilde{h}_2 =\,{\rm i}\,\left(\mathcal{B}-\bar{\mathcal{B}}\right)\,.
\label{h}
\end{align}
There are various ways of describing functions on a Riemann surface \cite{farkas2012riemann}.
For example, as functions in the upper half-plane with $g+1$ branch cuts satisfying appropriate boundary conditions. This formulation usually provides a clearer scheme for describing general properties of the geometry. Alternatively functions $h_1$ and $h_2$ can be represented in terms of hyperelliptic functions of the $2g$-periodic variables $(z,\bar{z})$ on a genus $g$ Riemann surface without boundaries. Along this article we will alternate between both descriptions and refer to the background with metric (\ref{metric}) generically as the genus $g$ solution.
Consider $\Sigma$ to be the half plane described by coordinates $(u,\bar{u})$. The main properties of an arbitrary genus $g$ solution are encoded in the boundary conditions satisfied by the harmonic functions over the real axis. More precisely, the $h_2$ function satisfies Dirichlet boundary conditions all along the boundary of $\Sigma$, whereas $h_1$ satisfies alternating Dirichlet and Neumann boundary conditions. The points where the boundary condition changes are denoted by $\tilde{e}_a$ and determine the position of the branch cuts. A genus $g$ solution is obtained for a Riemann surface $\Sigma$ with $2g+2$ branch points on its boundary. It is customary to use conformal symmetry to bring a branch point, let us say $\tilde{e}_{2g+2}$, to minus infinity and consider the ordering $\tilde{e}_{2g+2}<\ldots<\tilde{e}_2<\tilde{e}_1$. Additionally, the remaining branch points are subjected to the constraint $\sum_{a=1}^{2g+1} \tilde{e}_a=0$.
The general form of these functions satisfy the following equations
\begin{equation}
\partial_u h_1(u)= \frac{{\rm i}\, P(u)}{(u-u_0)^2\, s(u)}\,, \qquad \partial_u h_2 (u)=-\frac{\rm i}{(u-u_0)^2}\,,
\label{udef}
\end{equation}
where $u_0$ is a singular point where the geometry is asymptotically $AdS_5\times S^5$, $P(u)$ is a polynomial of degree $g+1$ with real coefficients and
\begin{equation}
s(u)^ 2=(u-\tilde{e}_1) \prod_{i=1}^{g} (u-\tilde{e}_{2i})(u-\tilde{e}_{2i+1})\,.
\end{equation}
Alternatively, making a conformal transformation one can get rid of the pole at the singular point. We will denote these coordinates as $(v,\bar{v})$, for which a direct relation with the matrix model resolvent $w(x)$ can be established \cite{D'Hoker:2007fq,Okuda:2008px}.
\begin{equation}
{\cal A}(v)= { {\rm i}\, \alpha' \over 8\, g_s} \, \left[ 2 \, v-w(v) \right] \,, \qquad {\cal B}(v)=\frac{ {\rm i}\, \alpha'\, v}{4} \,.
\label{AB}
\end{equation}
In order to follow the same conventions as in \cite{Okuda:2008px}, we use $e_a$ to denote the branch point locations in $(v,\bar{v})$ coordinates.
Clearly, the use of $u$ or $v$-coordinates is a matter of taste with no significant difference in the physical picture. Turning to the $(z,\bar{z})$ formulation, we can write
\begin{equation}
d\Sigma^2=4\sigma^2 dzd\bar{z},
\label{rieman1}
\end{equation}
where the radius $\sigma$ is a real function of $(z,\bar{z})$. The warping functions $f_1$, $f_2$, $f_4$, $\sigma$ and dilaton $\Phi$ are given by\footnote{Note that conventions in \cite{D'Hoker:2007fq,Benichou:2011aa} is $\phi=\Phi/2$. }
\begin{eqnarray}
f_1^4= -4e^\Phi h_1^4\frac{W}{N_1}\,, \quad f_2^4= 4e^{-\Phi} h_2^4\frac{W}{N_2}\,, \quad f_4^4= 4e^{-\Phi} \frac{N_2}{W}\,, \quad \sigma^8 = -\frac{WN_1N_2}{h_1^4 h_2^4}\,, \quad
e^{2\Phi} = -\frac{N_2}{N_1}\,,
\label{Eq:warps}
\end{eqnarray}
where
\begin{align}
N_1 &= 2\, h_1 \, h_2 | \partial h_1|^2 -h_1^2\, W\,,
\!\!\!\!\! && W = \partial h_1 \,\bar \partial h_2 + \partial h_2\, \bar \partial h_1 \,, \nonumber\\
N_2 &= 2\,h_1 \,h_2\, |\partial h_2|^2 -h_2^2 \, W \, ,
\!\!\!\!\!&& \ V= \partial h_1 \bar \partial h_2 - \partial h_2 \bar \partial h_1\,.
\end{align}
and $\partial=\partial_z$, $\bar\partial=\partial_{\bar z}$.
Also the NS and RR fluxes can be written in the following way
\begin{equation}
H_{3}=dB_{2}\,, \quad F_{3}=d C_{2}\, , \quad F_5=dC_4+\frac18
\left(B_2\wedge F_3-C_2\wedge H_3\right),
\end{equation}
and the corresponding potentials are
\begin{equation}
B_2= b_1\, \hat{e}_{\mathbb{H}_2}\,, \qquad C_2= b_2\, \hat{e}_{S^2}\,, \qquad C_4=-4\,j_1\,\hat{e}_{\mathbb{H}_2} \wedge \hat{e}_{S^2}+4\, j_2\, \hat{e}_{S^4}\,,
\label{flux}
\end{equation}
where $\hat{e}_{\mathbb{H}_2}$, $\hat{e}_{S^2}$ and $\hat{e}_{S^4}$ are the unit volume elements of $\mathbb{H}_2$, $S^2$ and $S^4$, respectively and
\begin{align}
b_1 &=-2 \, {\rm i}\, \frac{h_1^2\, h_2\, V}{N_1}-2\widetilde{h}_2-b_1^0\,,\nonumber\\
b_2 &=-2\, {\rm i}\frac{h_1\,h_2^2\, V}{N_2}+2\,\widetilde{h}_1-b_2^0\,, \nonumber\\
j_2 &= {\rm i} \,h_1\,h_2\, \frac{V}{W}-\frac32 \left(\widetilde{h}_1\,h_2-h_1\widetilde{h}_2\right)+3\,{\rm i}\left(\mathcal{C-\bar{C}}\right)\,. \label{b}
\end{align}
with $d \mathcal{C}=\mathcal{B}\partial\mathcal{A}-\mathcal{A}\partial\mathcal{B}$.
The integration constants $b_1^0$, $b_2^0$ are gauge redundancies that will be fixed later by requiring that the two-form fluxes precisely vanishes at the $AdS_5$ singular point, {\it i.e.} $b_1(z_0)=b_2(z_0)=0$.
The function $j_1$ can be computed by using the self-duality of the RR 5-form obtaining
\begin{align}
\partial j_1 &= -{\rm i} \frac{f_1^2\,f_2^2}{f_4^4}\partial j_2 +\frac18 \left(b_1 \, \partial b_2-b_2 \, \partial b_1\right)\,.
\end{align}
\subsection{Charges and representation parameters}
To complete the description of the solution we find it convenient to go back to the $(u,\bar{u})$ formulation. The harmonic function $h_1$ satisfies Dirichlet boundary conditions on the intervals $ (\tilde{e}_{2i+1}, \tilde{e}_{2i})$ and Neumann boundary conditions on the intervals $(\tilde{e}_{2i},\tilde{e}_{2i-1})$ for $i,j=1,\ldots, g+1$. Moreover, the $S^2$ and $S^4$ spheres shrink to zero size along Neumann and Dirichlet intervals respectively, as can be seen from the relation between the warping factors $f_i$ and the functions $h_i$ in Eq. (\ref{Eq:warps}).
The free parameters of the solutions, {\it i.e.} the positions and lengths of branch cuts can be related to the lengths of the rows and columns of the Young Tableau associated to the representation of the dual Wilson loop. However, the precise relation is in general very involved and can be established through flux integrals over the non-trivial cycles of the geometry. We shall present here some general aspects for arbitrary genus and leave a more detailed discussion of this relation for the genus one example described in section \ref{strings}. A fairly complete treatment of this subject can be found in \cite{D'Hoker:2007fq,Benichou:2011aa} and we will mainly follow the ideas presented there.
The geometric structure described so far allows to define a series of non-trivial 3- and 5-cycles encircling either Dirichlet or Neumann type intervals along the boundary of $\Sigma$\footnote{There are additional non-trivial 7-cycles given by $S^2\times \gamma_i$ and $S^4\times \tilde{\gamma}_j$ warped products which measure the fundamental string charges of the D-brane configuration\cite{Benichou:2011aa}. These charges are in turn related to the number of boxes contained in each sub-diagram of the Young tableau associated to the dual Wilson loop. }.
Such 3- and 5-cycles have topology $S^3$ and $S^5$ respectively hence being charged under either 3- or 5-form RR fluxes. More precisely, we define the 5-cycle $\gamma_i$ as the fibration of an $S^4$ over the contour surrounding the Neumann interval $(\tilde{e}_{2i}, \tilde{e}_{2i-1})$. Analogously, the 3-cycle $\tilde{\gamma}_j$ corresponds to an $S^2$ fibration over the contour around the Dirichlet interval $(\tilde{e}_{2i+1},\tilde{e}_{2i})$. The corresponding charges can be computed by the following integrals
\begin{align}
Q^{i}_{\text{D3}}&=\oint_{\gamma_i}d C_4 \, ,\\
Q^{j}_{\text{D5}}&=\oint_{\tilde{\gamma}_j} F_3
\end{align}
Using the Cauchy theorem and expanding the fluxes near the boundary, the integrals above can be deformed to the following integrals over the branch cuts \cite{Benichou:2011aa}:
\begin{align}
Q^{i}_{\text{D3}}&=12 {\rm i}\, \text{Vol}(S^4)\int_{\tilde{e}_{2i}}^{\tilde{e}_{2i-1}}d\mathcal{C} + \text{c.c.}\label{QD3}\, ,\\
Q^{j}_{\text{D5}}&=2 {\rm i}\, \text{Vol}(S^2)\int_{\tilde{e}_{2j+1}}^{\tilde{e}_{2j}}d\mathcal{A}+\text{c.c.}\label{QD5}\, ,
\end{align}
where
\begin{equation}
d \mathcal{C}=\mathcal{B}\partial\mathcal{A}-\mathcal{A}\partial\mathcal{B}.
\end{equation}
These integrals giving the D5 and D3 RR charges are naturally associated with the Wilson loop representation parameters (see Fig. \ref{youngbranch}) in the following way
\begin{equation}
Q_{\text{D3}}^i=(4\pi^2\alpha')^2 n_i\, , \quad Q_{\text{D5}}^j=-(4\pi^2\alpha') k_j\, \label{nkd3d5}
\end{equation}
\begin{figure}[h]
\centering
\def12cm{10cm}
\input{youngbranch2.pdf_tex}
\caption{Branch cuts and generic Young tableau assigned to the dual Wilson loop. Representation parameters $\{k_j,n_i\}$ are linked to geometric parameters through flux integrals over non-trivial 3- and 5-cycles $\tilde{\gamma}_j$ and $\gamma_i$. }
\label{youngbranch}
\end{figure}
\section{Strings in bubbling geometries}
\label{strings}
Let us introduce a fundamental string in the bubbling geometry background just presented in the previous section and search for minimal area solutions. Our interest in these configurations is kindled by the fact that the corresponding on-shell action can be related to the correlator of two Wilson loops, one in the fundamental representation whose dual is the fundamental string and the other in some large rank representation whose holographic dual is the background bubbling geometry itself. More precisely, in the large 't Hooft coupling limit
\begin{equation}
\langle W_{\text{fund}}\rangle_{\mathbf{R}}=\frac{\langle W_{\textbf{R}} W_{\rm fund} \rangle}{\langle W_{\textbf{R}} \rangle} \simeq \sum_{ \{ z^* \} } e^{-S_{\text{on-shell}}(z^*)}\,,
\label{correstring}
\end{equation}
relating the correlator between the Wilson loops in the large 't Hooft coupling limit to the
gravity partition function evaluated at the points $\{ z^* \}$ of minimum action for the
fundamental string in the bubbling background. In general there will be many different classical string embeddings in a genus $g$ background, which should correspond to different specifications of the fundamental Wilson loop $W_{\rm fund}$, namely different curves and orientations in the internal space.
Since we would like to eventually compare string theory with matrix model results, we shall focus on string configurations corresponding to fundamental Wilson loops preserving the same $SO(2,1)\times SO(3)\times SO(5)$ symmetry as the large rank representation one. This is necessary for the two Wilson loop operators to preserve the same set of supercharges. The restriction on the symmetries implies that both Wilson loops should be taken on coincident circles (with one orientation or the other)
and with same or opposite internal space orientations. Therefore, we will in turn restrict our attention to very specific dual classical string configurations.
To explicitly compare with matrix model results, we will find particular examples of these configurations and evaluate their on-shell actions. To build up our intuition we first present the general set up for the calculation and then turn to explicit examples for genus zero and one.
\subsection{General set up}
Our aim is to solve the equations of motion derived from the Nambu-Goto action
\begin{equation}
S=\frac{1}{2\pi \alpha'} \int d^2\sigma \sqrt{ {\rm det} (G^{(S)}_{MN} \partial_\alpha X^M \partial_\beta X^N) } +\frac{1}{2\pi \alpha'} \int P\left[B_2\right]\label{nambugoto}\,,
\end{equation}
with $G_{MN}^{(\text{S})}$ the metric in the string frame related to that one in the Einstein frame via $G^{(\text{S})}=e^{\frac{\Phi}{2}}G^{(\text{E})}$.
$P[B_2]$ is the pull-back of the NS 2-form flux over the worldsheet\footnote{Being metric independent, the coupling of the string to the $B$-field in the action remains unchanged in the new frame.}.
We consider string world sheets extended all along the $\mathbb{H}_2$ factor parameterized by global coordinates $(\rho,\phi)$ such that $ds^2_{\mathbb{H}_2}=d\rho^2+\sinh^2\rho \,d\phi^2$ and sitting at an arbitrary point on both the $S^2$ and $S^4$. Notice that, given this parametrization for the $\mathbb{H}_2$ factor, the corresponding string describes a circular contour on the $AdS$ boundary\footnote{Recall that, in global coordinates, the regularized $\mathbb{H}_2$ volume is finite and equals to $-2\pi$. Should we have taken the $\mathbb{H}_2$ factor in Poincar\'e coordinates, then the regularized volume would be zero. This last parametrization is associated to a single straight Wilson line, which has trivial vacuum expectation value $\langle W\rangle=1$. }. Furthermore, we work in the formulation where $\Sigma$ is a genus $g$ Riemann surface described by coordinates $(z,\bar{z})$ which we further assume can only depend on the worldsheet coordinate $\rho$. Plugging this {\it ansatz} into Eq. \eqref{nambugoto} and using the explicit form for both the metric and the antisymmetric tensor given in Eqs. \eqref{metric}, \eqref{flux} and \eqref{b} yields
\begin{equation}
S=\frac{1}{2\pi \alpha'} \int d\phi\, d\rho\, \sinh\rho \, e^{\frac{\Phi}{2}}{f_1^2}\,\sqrt{1+\frac{4\,\sigma^2 }{ f_1^2} |z'|^2} + \frac{1}{2\pi \alpha'} \int d\phi \,d\rho \, \sinh\rho \, b_1\, ,
\end{equation}
with $z'={dz}/{d\rho}$.
The Euler-Lagrange equation becomes
\begin{align}
\partial_z \left( e^{\Phi\over 2} f_1^2 \right)\sqrt{1+\frac{4\sigma^2 |z'|^2 }{ f_1^2}}
+ e^{\Phi\over 2} f_1^2 \partial_z \sqrt{1+\frac{4\sigma^2 |z'|^2 }{ f_1^2}} +\partial_z b_1 =\frac{1}{\sinh\rho}\frac{d}{d\rho}\left(\frac{2e^{\frac{\Phi}{2}}\sigma^2 \bar{z}'}{\sqrt{1+\frac{4\sigma^2|z'|^2 }{ f_1^2}}}\right)\,.
\label{Eq:motion}
\end{align}
Although finding a general solution to the above equation looks like a daunting task in the general case, there is a particularly simple solution. Indeed, if there is a point $z=z^*$ in the Riemann surface such that
\begin{equation}
\partial_z \left( e^{\Phi\over 2} \,f_1^2 \right)=\partial_z b_1=0 \label{saddle}\,,
\end{equation}
then keeping $z=z^*$ constant, {\it i.e.} $z'=0$, gives a solution of the equations of motion. Fortunately, solutions with the aforementioned symmetry restrictions will be found within this class. For these solutions the on-shell action reads
\begin{equation}
S_{\text{on-shell}}=\left.\frac{{\rm vol}({AdS_2})}{2\pi \alpha'} \left( e^{\Phi\over 2} \, f^2_1+b_1\right)\right|_{z=z^*}
=\left.-\frac{1}{\alpha'} \left( e^{\Phi\over 2} \, f^2_1+b_1\right)\right|_{z=z^*}\,,
\label{stringaction}
\end{equation}
where we used the regularized volume $ {\rm vol}({AdS_2})=-2\pi$.
At this point we would like to come back to the issue of fixing the gauge ambiguity of the background fluxes. In particular, a gauge transformation of the $B$-field changes the string action by a boundary term, thus leaving
the classical configurations unaffected because the equations of motion remain invariant. However, the gauge choice does affect the evaluation of the on-shell action. As already mentioned, we fix the gauge redundancy of the $B$-field by requiring that $b_1(z_0) = 0$. This means that the $B$-field vanishes at the singular point where the background is asymptotically $AdS_5\times S^5$, thus being identified with the dual CFT vacuum. Otherwise, if $b_1(z_0)$ were non-vanishing,
a non-trivial source should be turned on at the boundary CFT that would take us away from the vacuum.
In the following subsections we will find classical solutions to the Euler-Lagrange equations and evaluate the on-shell action for strings in genus zero and genus one supergravity backgrounds.
\subsection{Strings in genus zero background}
To familiarize the reader with the details of the presentation of the solution we review the computation of a minimal string area on $AdS_5\times S^5$, which corresponds to the genus zero background geometry. Despite being a well known result, a reformulation of this problem in the geometrical language just presented in the previous section would introduce some hints about the manipulations that we will perform in the genus one case.
The $AdS_5\times S^5$ solution in the $(v,\bar{v})$ formulation is obtained by taking
\begin{equation}
{\cal A} =- \frac{ \alpha' }{4 \, g_s }\, \sqrt{\lambda- v^2 } \, \qquad~~~~~~~~~
{\cal B} = {\rm i} \frac{ \alpha'\, v }{4} .
\label{Eq:h-genus0}
\end{equation}
with $\alpha'$, $\lambda$ and $g_s$ related to the radiue $L$, the RR flux $N$ and the dilaton $\Phi_0$ of the $AdS_5\times S^5$ solution via\footnote{ $L^4$ is proportional to $N$ in the Einstein frame and to $\lambda$ in the string frame.}
\begin{equation}
L^4=4\pi N \alpha'^2\quad , \quad e^{\Phi_0}=g_s\quad , \quad \lambda=4\, \pi\, g_s\, N
\end{equation}
More precisely, plugging (\ref{Eq:h-genus0}) one finds the dilaton and warping factors
\begin{eqnarray}
f_1^2-f_2^2=L^2,\quad\quad~~~~~~~ \sigma^2={ L^2 \over 4 |1- { v^2 \over \lambda} | }, \quad\quad~~~~~~ e^{\Phi}=e^{\Phi_0}, \label{39}
\end{eqnarray}
The gauge fixed $B$ field is vanishing.
Note that $h_1={\cal A}+{\cal \bar A}$ satisfies Neumann boundary conditions along the real segment $(-\sqrt{\lambda},\sqrt{\lambda})$ and Dirichlet along the remaining segments of the real axis. Moreover, given (\ref{39}), we note that $f_1$ becomes constant wherever $f_2$ vanishes, namely for $v^*\in[-\sqrt{\lambda},\sqrt{\lambda}]$. Therefore, any point lying on this segment corresponds to a solution of the equations of motion. Furthermore, all these solutions lead to the same on-shell action
\begin{equation}
S_{\text{on-shell}}=-\frac{ e^{\Phi_0 \over 2} \, f^2_1(v^*)}{\alpha'}=-\frac{ e^{\Phi_0 \over 2} \, L^2 }{\alpha'}=-\sqrt{\lambda}\,,
\end{equation}
From the foliation of the solution it should be clear that the Riemann surface provides the radial coordinate for $AdS_5$ to be written as a foliation of $AdS_2 \times S^2$ and the angular coordinate to write $S^5$ as a foliation of $S^4$. This becomes evident if we perform the following change of variables
\begin{equation}
v = \sqrt{\lambda}\, \cosh(\eta-{\rm i \,} \theta)\, , \quad 0\leq \eta<\infty\, , \quad 0\leq\theta\leq\pi,
\end{equation}
under which the metric takes the familiar form
\begin{equation}
ds^2= L^2\,\left(d\eta^2 + \cosh^2\eta\, ds^2_{\mathbb{H}_2}+\sinh^2\eta\, ds^2_{S^2}+d\theta^2+\sin^2\theta\, ds^2_{S^4}\right)\,.
\end{equation}
On the other hand, the solution segment $v^*\in [-\sqrt{\lambda},\sqrt{\lambda} ]$ gets mapped to the segment $\eta=0\,,\, 0\leq\theta\leq\pi$ thus making manifest that different choices of $v^*$ correspond to different polar angles on the $S^5$. In particular the branch points $v^*=\pm \sqrt{\lambda} $ corresponds to the north and south poles of $S^5$ and solutions placed at these points will be dual to configurations associated to Wilson loops coupled with
opposite orientation in the six-dimensional internal space.
\subsection{Strings in genus one backgrounds}
In this section we will consider genus one backgrounds since they can be explicitly realized in terms of Weierstrass elliptic functions \cite{D'Hoker:2007fq}. These geometries arise due to the backreaction of a Wilson loop in a representaion given by a rectangular Young tableau with $n_1=n$ rows and $k_1=k$ columns, see Fig.\ref{fig:RecRep}. In this case, the most convenient approach corresponds to taking $\Sigma$ as a torus described by coordinates $(z,\bar{z})$ with periods $2 \omega_1$ and $2 \omega_3$. The Weierstrass elliptic functions provide the mapping between the torus and the half complex plane. In particular, taking $z_0=1$, the holomorphic functions take the form
\begin{align}
{\cal A}&= {\rm i}\, \kappa_1 \left(\zeta(z-1)+\zeta(z+1)-2\frac{\zeta(\omega_3)}{\omega_3}z \right)\,,
\nonumber
\\
{\cal B} &= {\rm i}\, \kappa_2 \left(\zeta(z-1)-\zeta(z+1) \right)\,,
\label{h2}
\end{align}
where $\zeta$ denotes the Weierstrass $\zeta$-function, a primitive of the Weierstrass $\wp$-function
\begin{equation}
\wp(z)=-\zeta'(z)\,,
\end{equation}
satisfying the condition $\lim_{z\to 0} (\zeta(z)-1/z)=0$. The functions $\zeta(z)$ and $\wp(z)$ depend implicitly on two numbers $g_2,g_3$ (or equivalently $\tilde{e}_1,\tilde{e}_2$ ) specifying the periods of the torus. More precisely, $\wp(z)$ can be defined as the solution of the differential equation
\begin{equation}
\left[ \wp'(z) \right]^2=4 \left[ \wp(z) \right]^3-g_2\, \wp(z) -g_3=4 \left[ \wp(z)-\tilde{e}_1 \right] \left[ \wp(z)-\tilde{e}_2 \right] \left[ \wp(z)-\tilde{e}_3 \right] \label{diff}\,,
\end{equation}
with $\tilde{e}_1+\tilde{e}_2+\tilde{e}_3=0$ and
\begin{equation}
g_2=2\left(\tilde{e}_1^2+\tilde{e}_2^2+\tilde{e}_3^2\right)\,, \qquad g_3= 4 \tilde{e}_1 \tilde{e}_2 \tilde{e}_3\,.
\end{equation}
At the half periods, $\omega_i$, one finds $\wp(\omega_i)=e_i$ and $\wp '(\omega_i)=0$, so Eq. (\ref{diff}) is verified. Given the branch points $\tilde{e}_1,\tilde{e}_2$ one can compute the periods $2\omega_1$ and $2\omega_3$ using the standard elliptic formulas
\begin{equation}
\omega_1= \frac{ K\left(\tfrac{\tilde{e}_2-\tilde{e}_3}{\tilde{e}_1-\tilde{e}_3}\right)}{\sqrt{\tilde{e}_1-\tilde{e}_3}} \,, \qquad
\omega_3= {\rm i}\,\frac{K\left(\tfrac{\tilde{e}_1-\tilde{e}_2}{\tilde{e}_1-\tilde{e}_3}\right)}{\sqrt{\tilde{e}_1-\tilde{e}_3}} \,,\qquad \omega_2=\omega_1+\omega_3\,,
\label{halfperiods}
\end{equation}
where $K$ is the complete elliptic integral of the first kind. Finally $\kappa_1$ and $\kappa_2$ are determined by requiring that the geometry reduces asymptotically to $AdS_5\times S^5$ when $z\to z_0=1$.
Near this point one finds
\begin{align}
{\cal A}& \underset{z\to 1}{\approx} {\rm i}\, \kappa_1 \left[ {1\over (z-1)} +\zeta(2)-2\frac{\zeta(\omega_3)}{\omega_3} - \left(\wp(2)+2 \, {\zeta(\omega_3)\over \omega_3} \right)(z-1) -{ \wp '(2) \over 2} (z-1)^2+\ldots \right]\,,
\nonumber
\\
{\cal B} & \underset{z\to 1}{\approx} {\rm i}\, \kappa_2 \left( {1\over (z-1)} -\zeta(2) +\wp(2)(z-1) +{ \wp '(2) \over 2} (z-1)^2 +\ldots \right)\,.
\label{nearz1}
\end{align}
Comparing with Eq. (\ref{Eq:h-genus0}), one finds that the match requires
\begin{align}
\kappa_1&= \frac{L^2}{8} e^{-\frac{\Phi_0}{2}}
\left(\wp(2)+\frac{\zeta(\omega_3)}{\omega_3}\right)^{-\frac12}, \\
\kappa_2&=\frac{L^2}{8}e^{\frac{\Phi_0}{2}}
\left(\wp(2)+\frac{\zeta(\omega_3)}{\omega_3}\right)^{-\frac12}\,.
\end{align}
Moreover, requiring that $b_1=0$ at $z=1$ one finds
\begin{equation}
b_1^0 = 2\kappa_2\left(\frac{\wp'(2)}{\wp(2)+\frac{\zeta(\omega_3)}{\omega_3}}-2\zeta(2)\right)\,.
\label{gaugefix}
\end{equation}
\begin{figure}[h!]
\beginpgfgraphicnamed{Rec-Rep}
\begin{center}
\begin{tikzpicture}[>=stealth]
\draw[thick] (0,0) grid [xstep=.2,ystep=.2] (5,3);
\draw[densely dotted,<->,red] (0,3) .. controls (2,3.8) and (3,3.8) .. (5,3) node[above,pos=.5,black]{$k$};
\draw[densely dotted,<->,red] (0,3) .. controls (-0.6,2.3) and (-0.6,0.7) .. (0,0) node[left,pos=.5,black]{$n$};
\end{tikzpicture}
\end{center}
\endpgfgraphicnamed
\caption{Number of rows and columns in the tableau are related to the charges $Q^1_{\text{D3}}$ and $Q^1_{\text{D5}}$.}
\label{fig:RecRep}
\end{figure}
\vspace{0.5cm}
The number of rows and columns in a rectangular Young tableau are directly related to the charges $Q^1_{\text{D3}}$ and $Q^1_{\text{D5}}$ of the supergravity solution, given by the expressions \eqref{QD3} and \eqref{QD5} respectively, while the rank $N$ of the gauge group is related to $Q^0_{\text{D3}}=Q^2_{\text{D3}}+Q_{\text{D3}}^1$. Indeed, for the genus one case there are two non-trivial 5-cycles $\gamma_1$ and $\gamma_2$ and one non-trivial 3-cycle $\tilde{\gamma}_1$ (see Figure \ref{torus}), these charges have been computed explicitly \cite{Benichou:2011aa} obtaining\footnote{Here we used formula D.8
of \cite{Benichou:2011aa} and the identity $ \wp(2) = \tfrac{1}{4}\left(\frac{\wp ''(1)}{\wp '(1)}\right)^2 -2 \wp(1) $. }
\begin{eqnarray}
N -n\!&=&\! \frac{Q^2_{\text{D3}}}{(4\pi^2 {\alpha'})^2}\,, \nonumber\\
n \!&=&\! \frac{Q^1_{\text{D3}}}{(4\pi^2 {\alpha'})^2} = \frac{N \omega_3}{2\pi \, {\rm i}\,}
\left(4\left(\zeta(1)-\tfrac{\zeta(\omega_3)}{\omega_3}\right)+\frac{\left(\wp(1)+\tfrac{\zeta(\omega_3)}{\omega_3}\right)\wp''(1)-\wp'(1)^2}
{\left(\wp(2)+\tfrac{\zeta(\omega_3)}{\omega_3}\right)\wp'(1)}\right)\,,
\nonumber\\
k \!&=&\! -\frac{Q^1_{\text{D5}}}{4\pi^2 {\alpha'}} = \frac{\sqrt{\pi}\, {\rm i}\,}{\omega_3}\sqrt{\frac{N}{g_s}} \left(\wp(2)+\tfrac{\zeta(\omega_3)}{\omega_3}\right)^{-1/2}\,.
\label{N5}
\end{eqnarray}
\begin{figure}[h]
\centering
\def12cm{12cm}
\input{torus.pdf_tex}
\caption{Mapping from the torus to the half-plane. The boundary of the fundamental domain of the Weierstrass elliptic functions delimited by $\{0,\omega_1,\omega_2,\omega_3\}$ gets mapped to the boundary at the real axis ($\wp(\omega_i)=\tilde{e}_i$).}
\label{torus}
\end{figure}
In what follows let us find the solutions $z=z^*$ of Eq. (\ref{Eq:motion}) for this particular case. Recall that we are interested in string configurations preserving the same $SO(2,1)\times SO(3)\times SO(5)$ symmetry as the background. It turns out that the only points on the Riemann surface consistent with this condition are those where both the $S^2$ and the $S^4$ shrink to zero size, which corresponds precisely to the branch points where the warping factors $f_2$ and $f_4$ vanish.
In order to show that they actually satisfy Eq. (\ref{saddle}) we consider the expansions of the holomorphic functions ${\cal A}$ and ${\cal B}$ around the four branch points located at $z=\omega_a$, $a=0,1,2,3$, with $\omega_0 = 0$. Given the periodic property of the elliptic functions $\zeta(z+2\omega_i)=\zeta(z)+2\zeta(\omega_i)$, formulas (\ref{h2}) drastically simplify to
\begin{eqnarray}
{\cal A} (z) & \underset{z\to \omega_a}{\approx} & c^A_0(\omega_a)+c_1(\omega_a) (z- \omega_a) +c_3(\omega_a) (z-\omega_a)^3 +\ldots \nonumber\\
{\cal B} (z) & \underset{z\to \omega_a}{\approx} & c^B_0(\omega_a)+c_2(\omega_a) (z- \omega_a)^2 +c_4(\omega_a) (z-\omega_a)^4 +\ldots \label{nearbranch}
\end{eqnarray}
with\footnote{We recall that $\wp''(z)=6\, \wp(z)^2-g_2/2$.}
\begin{eqnarray}
c_1(\omega_a) &=& 2 \, {\rm i} \, \kappa_1 {\zeta(\omega_3) \over \omega_3 }\,, \qquad ~~~~c_3(\omega_a) = - \, \frac{ {\rm i} \, \kappa_1 }{ 3} \wp ''(1+\omega_a)\,, \nonumber\\
c_2(\omega_a) &=& {\rm i} \, \kappa_2 \, \wp '(1+\omega_a)\,, \qquad c_4(\omega_a) = {\rm i} \, \kappa_2 \, \wp(1+\omega_a) \wp ' (1+\omega_a)\,, \nonumber\\
c_0^A(\omega_a) &=& -2 \, {\rm i} \, \kappa_1 \left( {\zeta(\omega_3) \over \omega_3 }\, \omega_a +\zeta(1-\omega_a)- \zeta(1+\omega_a) \right)\,, \nonumber\\
c_0^B &=& - \, {\rm i} \, \kappa_2\,
\left( \zeta(1+\omega_a) +\zeta(1-\omega_a) \right)\,.
\end{eqnarray}
Plugging the expansions (\ref{nearbranch}) into the background fields (\ref{Eq:warps}) and (\ref{b}) we find
\begin{eqnarray}
e^{\Phi\over 2} \, f_1^2 (z) & \underset{z\to \omega_a}{\approx} & \left|{ 2 \, {\rm i}\, c_1\,c_2^2 \over 2 \,c_2\, c_3-c_1\, c_4}\right| + {\mathcal O}\left[ (z-\omega_a)^2 \right],\nonumber\\
b_1(z) & \underset{z\to \omega_a}{\approx} & { 2 \, {\rm i}\, c_1\,c_2^2 \over 2 \,c_2\, c_3-c_1\, c_4} -b_1^0-2\, {\rm i}\, c_0^{B} + {\cal O}\left[ (z-\omega_a)^2 \right]\ ,
\end{eqnarray}
showing that $z=\omega_a$ solves Eq. (\ref{saddle}). Moreover, the on-shell action is\footnote{One may use
$ \zeta\left(1+\omega_i\right)= \zeta
\left(1-\omega_i\right)+2 \zeta(\omega_i)$. }
\begin{align}
& S_{\rm on-shell}(\omega_a)
= -\frac{1}{ \alpha'} \,\frac{L^2 \sqrt{g_s} } {
4 \sqrt{\wp(2)+\frac{\zeta \left(\omega_3\right)}{\omega_3}}} \left( 2 \zeta(2) - 2\left[ \zeta\left(1+\omega_a\right)+ \zeta
\left(1-\omega_a\right) \right] -\frac{ \wp '(2)}{\wp(2) +{ \zeta \left(\omega_3\right)\over \omega_3} } \right. \nonumber\\
&
+\left| \frac{3\, \wp '\left(1+\omega_a\right)\, \left( \wp\left(1+\omega_a\right)+{\zeta
\left(\omega_3\right) \over \omega_3} \right) }{ \wp ''\left(1+\omega_a\right)-3 \wp\left(1+\omega_a\right) \left( \wp\left(1+\omega_a\right) + \zeta
{\left(\omega_3\right)\over \omega_3} \right) }\right|
- \left.\frac{3\, \wp '\left(1+\omega_a\right)\, \left( \wp\left(1+\omega_a\right)+{\zeta
\left(\omega_3\right) \over \omega_3} \right) }{ \wp ''\left(1+\omega_a\right)-3 \wp\left(1+\omega_a\right) \left( \wp\left(1+\omega_a\right) + \zeta
{\left(\omega_3\right)\over \omega_3} \right) } \right) \label{action}
\end{align}
The string configurations we have found for the genus one case, and eventually their on-shell actions \eqref{action} are written as functions of the branch point positions $\tilde{e}_i$ . In order to make a comparison with the gauge field theory results it is necessary to express them in terms of the numbers of rows and columns $n$ and $k$ of the corresponding Young tableau. To do this we have to invert (\ref{N5}) to give the branch points $\tilde{e}_i$ and the half-periods $\omega_i$ in terms of $n$ and $k$. Although, the relation between the two sets of variables is pretty involved for generic values of $n$ and $k$, here, we are interested in the precise regime, for which $n$ is order $N$ and $k$ is order $N$ or larger.
Accessing the regime of interest requires to take $\omega_3\to 0$ and $\omega_1$ to approach 2. In order to implement this limit, it is convenient to introduce
\begin{equation}
\omega_1 = 2 -\frac{x}{\Lambda}\,,\qquad \omega_3 = \frac{\, {\rm i}\,\pi}{2\Lambda}\,,
\end{equation}
and consider that $\Lambda$ is large and $x$ finite. Inverting the formulas for the periods in the limit, one finds
\begin{align}
\tilde{e}_1 &= {\Lambda^2 \over 3} \left(1+24 \, e^{2x-\Lambda} +24 \, e^{4x-2\Lambda}+ {\cal O}(e^{6x-3\Lambda}) \right)\,, \nonumber\\
\tilde{e}_2 &= {\Lambda^2 \over 3} \left(1-24 \, e^{2x-\Lambda} +24 \, e^{4x-2\Lambda}+ {\cal O}(e^{6x-3\Lambda}) \right)\,,
\end{align}
while the Weierstrass elliptic zeta function can be expressed as\footnote{Following sub-leading orders would not influence the on-shell evaluation of the action in the regime considered.}
\begin{align}
\zeta(z)\simeq &
-\frac{\Lambda^2 z}{3}\left(1-\frac{3}{\Lambda z} \coth(\Lambda z)\right)
+8 \,\Lambda^2\, z\, e^{4 \,x-8\Lambda}\left(1 -\frac{ \sinh\left(2\Lambda z\right)}{2\, \Lambda \, z } \right)+{\cal O}(e^{6x-3\Lambda})\,,
\label{zetalim}
\end{align}
and $\wp(z)=-\zeta '(z)$. In this limit the charges (\ref{N5}) adopt the form
\begin{align}
n = &\,\frac{e^{4x}}{1+e^{4x}} N \,,
\label{nlim}
\\
k = &\, \frac{2 e^{2\Lambda}}{\sqrt{\lambda}\sqrt{1+e^{4x}}} N\,.
\label{klim}
\end{align}
Similarly, if we use the expansions (\ref{zetalim}), for the on-shell actions (\ref{action}) we find
\begin{align}
S_{\rm on-shell}(0)=S_{\rm on-shell}(\omega_3)= &\, -\frac{\sqrt{\lambda}}{\sqrt{1+e^{4x}}} + \frac{\sqrt{\lambda} e^{2\Lambda} e^{4x}}{2(1+e^{4x})^{3/2}}\,,
\nonumber\\
S_{\rm on-shell}(\omega_1)= S_{\rm on-shell}(\omega_2)=& \, - \frac{\sqrt{\lambda}e^{2x}}{\sqrt{1+e^{4x}}}-\frac{\sqrt{\lambda} e^{2\Lambda}}{2(1+e^{4x})^{3/2}}\,,
\end{align}
which can be put in terms of the number of rows and columns using (\ref{nlim}) and (\ref{klim})
\begin{align}
S_{\rm on-shell}(0)=S_{\rm on-shell}(\omega_3) = & -\sqrt{\lambda\left(1-\tfrac{n}{N}\right)}+\frac{k n\lambda}{4N^2}\label{sw03}\,, \\
S_{\rm on-shell}(\omega_1)= S_{\rm on-shell}(\omega_2)=& -\sqrt{\lambda \tfrac{n}{N}}-\frac{k(N-n)\lambda}{4N^2}\label{sw14}\,.
\end{align}
We notice that the pair of solutions with $z^*=0,\omega_3$ or $z^*=\omega_1,\omega_2$ share the same on-shell action. They can be distinguished by the position of the fundamental string on $\Sigma$ and we would like to identify which correlators of Wilson loops can be related with each of them, according to the AdS/CFT correspondence. Because fundamental strings at any of the four branch points correspond to $SO(2,1)\times SO(3)\times SO(5)$ symmetric configurations, they should correspond to correlators of Wilson loops on the same circle with either the same or the opposite internal space orientations. In the remaining of this section we will argue that the contributions of the saddle points $z^*=0,\omega_1$ has to be taken into account altogether for a given orientation of the fundamental string, and $z^*= \omega_2,\omega_3$ for the opposite one.
By considering an $AdS_5\times S^5$ limit of the bubbling geometry, it is possible to argue that strings at $z^*=0$ and $z^*=\omega_3$ are the
dual description of correlators in which the fundamental Wilson loops have opposite internal space orientations. More precisely, we consider
the large $\omega_1$ limit, which corresponds to the collapse of one of the branch cuts (namely $\tilde{e}_2\to \tilde{e}_1$). In this limit, when the usual $AdS_5\times S^5$ background is restored (see Appendix \ref{AppProbe}), $z^*=0$ and $z^*=\omega_3$ become the antipodal points on the $S^5$, and strings located there
correspond to fundamental Wilson loops which couple to the scalars with opposite orientation in the internal space. Therefore, for the correlator of a back-reacting Wilson loop with a fundamental one with the same internal space orientation, either $z^*=0$ or $z^*=\omega_3$ has to be considered but not both.
The existence of four saddle point solutions is a non-trivial consequence of the genus one geometry. We will argue that for the dual one type of correlator (same or opposite internal space orientation) $z^*=\omega_1$ has to be taken into account altogether with $z^*=0$, while $z^*=\omega_3$ has to be taken into account altogether with $z^*=\omega_2$. This is related to the non-trivial topology of the target space. In particular, the definition domain of the generating functions is two-sheeted and then we need a two-fold boundary condition in order to have a well defined variational problem. Evidence that $z^*=0$ and $z^*=\omega_1$ corresponds to the same correlator in the dual CFT comes from the fact that $z^*=0$ and $z^*=\omega_1$ configurations are related by a large gauge transformation. If we consider for instance the transformation $z\to z+\omega_i$, the holomorphic functions $\mathcal{A}$ and $\mathcal{B}$ change as
\begin{align}
\mathcal{A}(z,z_0) \to &\ \mathcal{A}(z,z_0+\omega_i)+ \alpha_i \,, \qquad \alpha_1=\alpha_2=\, {\rm i}\, \frac{\pi\kappa_1}{|\omega_3|}\,, \qquad \alpha_3=0\,,\nonumber
\\
\mathcal{B}(z,z_0)\to &\ \mathcal{B}(z,z_0+\omega_i)+ \beta_i \,, \qquad \beta_i =\, {\rm i}\, 2\kappa_2\zeta(\omega_i)\,,
\end{align}
where we slightly changed the notation to make the position of the singular point manifest. The singular point can be shifted by a conformal transformation of the target space and, since $\zeta(\omega_1)$ is real, the configurations at $\omega_0=0$ and $\omega_1$ are related by an imaginary shift of the holomorphic functions. Imaginary shifts on the holomorphic functions are related to large gauge transformations of the background fluxes which induce redefinitions of the charges, since they are fluxes integrals over non-trivial cycles. The relation of these gauge transformations to the Hanany-Witten effect is discussed in \cite{Benichou:2011aa}. Since invariance under this kind of gauge transformations is expected, both configurations $z^*=0$ and $z^*=\omega_1$ should contribute to the saddle point dual to a given Wilson loops correlator. An analogous relation is found for $\omega_2$ and $\omega_3$.
This gauge transformation of the background can be associated to a symmetry already present in the dual gauge theory. For a generic Wilson loop representation $\mathbf{R}$, this symmetry is the invariance under the change of $\mathbf{R}$ by its complex conjugate $\bar{\mathbf{R}}$. The conjugate representation is obtained by inverting the Maya diagram assigned to a given tableau \cite{Yamaguchi:2006te,Okuda:2007kh} (see Figure \ref{maya}). Black segments in the Maya diagram are a direct representation of the cuts of the density of eigenvalues $\rho(x)$ in the associated matrix model that will be encountered in next section.
\begin{figure}[H]
\centering
\def12cm{13.8cm}
\input{Maya.pdf_tex}
\caption{ Young tableaux for $\mathbf{R}$ and $\bar{\mathbf{R}}$ and associated Maya diagrams.}
\label{maya}
\end{figure}
\vspace{0.5cm}
In the gravity description, this conjugation symmetry can be interpreted as viewing the geometry from either one or the other Riemann sheet (see fig. \ref{riemann}) and the roles played by branch point $\tilde{e}_4=-\infty$ ($z=0$) and $\tilde{e}_1$ ($z=\omega_1$) are exchanged; the same occurs with the roles played by $\tilde{e}_2$ ($z=\omega_2$) and $\tilde{e}_3$ ($z=\omega_3$). Additionally, the non-trivial cycles get interchanged, giving rise to the usual $n\to N-n$ transformation.
\begin{figure}[H]
\centering
\def12cm{15cm}
\input{riemman2.pdf_tex}
\caption{Left: Red lines denote the branch cuts and dotted blue lines indicate that cycles are closing on the second sheet of the Riemann surface. Right: Branch points and cycles interchange roles when viewing from one sheet or the other.}
\label{riemann}
\end{figure}
Collecting the two contributions together and defining $\nu=\frac{n}{N}$ we can write the final AdS/CFT result for the correlator
\begin{equation}
\label{eq:HolCorr}
\langle W_{\rm fund} \rangle_{\mathbf{R}} \approx e^{\sqrt{\lambda(1-\nu)}-\frac{k\nu \lambda}{4N}} + e^{\sqrt{\lambda\, \nu}+\frac{k(1-\nu)\lambda}{4N}}
\end{equation}
As a final remark, we notice that the result is invariant under $n\to N-n$ when also taking $k\to-k$, suggesting that the conjugation of the representation is related
to a different choice of orientations of the brane system.
So far, as it has been stressed before, the string configurations we have found are the dual description of correlators between two Wilson loops defined along the same circular contour with either the same or the opposite orientations in the internal space. However, this does not exhaust all the possible configurations consistent with the symmetry $SO(2,1)\times SO(3)\times SO(5)$. Indeed, we should allow for the possibility of correlators between two Wilson loops defined along circular contours with opposite space-time orientations with either the same or the opposite internal space orientations.
The dynamics of a string dual to a Wilson loop with opposite space-time orientations is governed by a similar Nambu-Goto action, but with a sign changed in front of the $B$-field term. Interestingly, the configurations at the points $z=\omega_a$ also satisfy the Euler-Lagrange of this alternative problem. The on-shell actions for these strings with opposite space-time orientations are
\begin{align}
\tilde S_{\rm on-shell}(0) = \tilde S_{\rm on-shell}(\omega_3) = & -\sqrt{\lambda\left(1-\tfrac{n}{N}\right)}-\frac{k n\lambda}{4N^2}\label{stw03}\,, \\
\tilde S_{\rm on-shell}(\omega_1)= \tilde S_{\rm on-shell}(\omega_2)=& -\sqrt{\lambda \tfrac{n}{N}}+\frac{k(N-n)\lambda}{4N^2}\label{stw14}\,.
\end{align}
Reasoning as before, one can conclude that $z^*=0$ and $z^*=\omega_1$ or $z^*=\omega_2$ and $z^*=\omega_3$ contribute to this other type of correlators,
depending on the relative internal space orientation. Thus, the AdS/CFT result for this other type of correlators is
\begin{equation}
\label{eq:HolCorr2}
\langle \, \widetilde W_{\rm fund} \, \rangle_{\mathbf{R}}\approx e^{\sqrt{\lambda(1-\nu)}+\frac{k\nu \lambda}{4N}} + e^{\sqrt{\lambda\,\nu}-\frac{k(1-\nu)\lambda}{4N}}.
\end{equation}
We will find in the next section that the matrix model computation matches the result above, giving an indirect support to our interpretation. In appendix \ref{susywl} we study the supersymmetric properties of this configuration of Wilson loops from the field theory side.
\subsection{Strings in genus $g$ backgrounds}
Finally, we consider a fundamental string in a general genus $g$ background. We work in the half-plane formulation, where the supergravity solution is specified by a single holomorphic function $w(v)$ in the upper half-plane with $g+1$ cuts along the real line. This function can be identified with the resolvent of the dual matrix model description \cite{Okuda:2008px}. We will first prove that, given a genus $g$ background geometry, fundamental strings sitting at any of the $2g+2$ branch points $e_{a}$ give rise to solutions of the Euler-Lagrange equations and then we will evaluate the action of the fundamental string at these points.
In the $v$-plane, the functions ${\cal A}$ and ${\cal B}$ are given by
\begin{equation}
{\cal A}(v)= { {\rm i}\, \alpha' \over 8\, g_s} \, \left[ 2 \, v-w(v) \right] \,, \qquad {\cal B}(v)=\frac{ {\rm i}\, \alpha'\, v}{4} \,.
\label{generalg}
\end{equation}
In these coordinates the $AdS_5\times S^5$ asymptotic region is approached as $v\to\infty$. The asymptotic behavior
of the holomorphic function $w(v)$ is given by
\begin{equation}
w(v)=\frac{\lambda}{v}+\frac{\lambda w_1}{v^2}+\mathcal{O}(v^{-3})\,.
\label{decay}
\end{equation}
Plugging \eqref{generalg} and \eqref{decay} into the gravity solution one finds that the potential $b_1$ vanishes for $v\to\infty$ provided
$b_1^0 = \alpha' w_1$.
Let us now consider the string action in the vicinity of the branch points $e_a$. Expansions of $h_1$ and $h_2$ near the real line have been performed in \cite{D'Hoker:2007fq}. If we write $v=x+{\rm i}\,y$ and expand all functions near the boundary $y\approx 0$, we get
\begin{align}
h_1 &={\cal A}+{\cal \bar A}= a_0(x) +a_1(x)y + a_2(x)y^2 +a_3(x)y^3 + \mathcal{O}(y^4)\,,\nonumber\\
h_2 &={\cal B}+{\cal \bar B}= -\alpha' \,{y \over 2} \label{hdir20}\,.
\end{align}
The coefficient $a_{2k}$ and $a_{2k+1}$ are completely determined in terms of $a_0$ and $a_1$ respectively by means of the harmonic equation $(\partial_x^2+\partial_y^2)h_1=0$. In particular
\begin{equation}
a_2(x)=-\frac12 a_0''(x)\, , \quad a_3(x)=-\frac16 a_1''(x)\,,
\label{harmonicsol0}
\end{equation}
and so on. Moreover, along the real line, $h_1$ satisfies either Neumann or Dirichlet boundary conditions and therefore either $a_0(x)$ or $a_1(x)$ vanish along the real line. So one can write
\begin{equation}
h_1 (x+{\rm i} y) =\left\{
\begin{array}{ccc}\label{hnd}
a_0(x) + a_2(x)y^2 + \ldots & N:& x\in (e_{2i},e_{2i-1}) \\
a_1(x)y +a_3(x)y^3 +\ldots & D: & x\in (e_{2j+1},e_{2j}) \\
\end{array}
\right.
\end{equation}
For example, approaching the real line along an interval with Neumann boundary conditions, using (\ref{hdir20}-\ref{hnd}), we obtain the expansions
\begin{align}
W&=\alpha' \,{a_0''(x)\, y\over 4} +\mathcal{O}(y^3)\,, \quad~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ V=-\alpha' {i a_0'(x)\over 4}+\mathcal{O}(y^2)\,,\\
N_1 &=-\alpha' \, { a_0(x)\, \over 4} \left[ a_0'(x)^2+a_0(x)\,a_0''(x)\right]\, y + \mathcal{O}(y^3)\,,\quad \quad
N_2 = -\frac{(\alpha' )^3}{16}a_0(x) y + \mathcal{O}(y^3)\,.
\end{align}
leading to
\begin{align}
e^{\Phi\over 2} \, f_1^2 =&\ \alpha' \left| { \sqrt{ a_0(x)^3\, a_0''(x) } \over a_0'(x)^2+a_0(x)\, a_0''(x) } \right| +\mathcal{O}(y^2)\,,
\nonumber\\
b_1 =&\ \alpha' x-\alpha' { a_0(x)\, a_0'(x) \over a_0'(x)^2+a_0(x)\, a_0''(x) } - b_1^0 +\mathcal{O}(y^2)\,,
\label{actionbranch}
\end{align}
At the branch points, $h_1$ satisfies both Neumann and Dirichlet boundary conditions and therefore we impose $a_0$ to vanish there. Moreover, ${\cal A}$ has to develop a branch cut discontinuity at those points. Taking
\begin{equation}
a_0(x) = (x-e_a)^{1\over 2} \left[ C_{a,0}+C_{a,1}\,(x-e_a)+C_{a,2}\,(x-e_a)^2+ {\cal O}(x-e_a)^3 \right]\,,
\label{a0exp}
\end{equation}
where $C_{a,i}$ are numerical coefficients.
Expanding (\ref{actionbranch}) around these points we find
\begin{align}
e^{\Phi\over 2} \, f_1^2 = &\ \alpha' \left| {C_{a,0}\over 4\, C_{a,1}}- {3\over 8 \, C_{a,1}^2} ( C_{a,1}^2+2\,C_{a,0} \, C_{a,2})(x-e_a)\right|+ \mathcal{O}(x-e_a)^2+ \mathcal{O}(y)^2\,,
\nonumber\\
b_1 = & \ \alpha' \left[ e_1\, - { C_{a,0} \over 4 \, C_{a,1}} + {3\over 8 \, C_{a,1}^2} ( C_{a,1}^2+2\,C_{a,0} \, C_{a,2})(x-e_a)\right] -
b_1^0
+ \mathcal{O}(x-e_a)^2+ \mathcal{O}(y)^2\,,
\label{stringactiongen}
\end{align}
We therefore see that branch points are minima of the action if the expansion coefficients satisfy the relation
\begin{equation}
C_{a,1}^2+2\,C_{a,0} \, C_{a,2}=0\,.
\label{relation}
\end{equation}
We will verify in a particular regime that this relation is satisfied. The corresponding on-shell action becomes
\begin{equation}
S_{\text{on-shell}}(e_a)=-\frac{1}{\alpha'} \left.\left( e^{\Phi\over 2} \, f^2_1+b_1\right)\right|_{v = e_a}
=- e_a+{ C_{a,0} \over 4 \, C_{a,1}}-\left|{C_{a,0}\over 4\, C_{a,1}}\right|+ \frac{b_1^0}{\alpha'}\,.
\label{stringactiong}
\end{equation}
The general results above can be made more precise in a special limit of the underlying genus $g$ surface where the physics becomes more transparent and a concrete expression for $w(v)$ can be proposed. In particular we consider the limit where intervals with Neumann boundary conditions or branch cuts are sufficiently far away from each other. Thus, in the surroundings of a particular branch cut, the information about the other cuts can be dismissed and $h_1$ behaves essentially as in the genus zero case. In the dual matrix model description some analogous implication will be observed for the dual resolvent function $w(v)$ in the limit where the dual Young tableau is made of large blocks.
Let us denote the $g+1$ branch cuts by $L_i$ and consider they are centered at $c_i$ and with lengths $2\mu_i$. In other words, the $2g+2$ branch points are located at $e_{2i}=c_i-\mu_i$ and $e_{2i-1}=c_i+\mu_i$. Then we propose the following expressions for $w$ over the real axis, valid for cuts well separated, {\it i.e.} $|c_i-c_j| \gg 1$. For $ x\in L_i$ or $c_i-\mu_i<x<c_i+\mu_i$
\begin{align}
w(x)= & \, 2(x-c_i)-2{\rm i}\sqrt{\mu_i^2-(x-c_i)^2} + 2 \sum_{k=1}^{i-1} \left(x-c_k+\sqrt{(x-c_k)^2-\mu_k^2}\right) \nonumber
\\
& + 2\sum_{k=i+1}^{g+1}\left(x-c_k-\sqrt{(x-c_k)^2-\mu_k^2}\right)\,.
\label{win}
\end{align}
While for $x$ between two cuts, {\it i.e.} $c_{i+1}+\mu_{i+1}<x<c_i-\mu_i$
\begin{equation}
w(x)= 2 \sum_{k=1}^{i} \left(x-c_k+\sqrt{(x-c_k)^2-\mu_k^2}\right)
+ 2\sum_{k=i+1}^{g+1}\left(x-c_k-\sqrt{(x-c_k)^2-\mu_k^2}\right)\,.
\label{wout}
\end{equation}
Therefore, in the vicinity of the branch cut $L_i$, and provided that $|c_i-c_j| \gg 1$, we have
\begin{equation}
w(x)\approx\left\{\begin{array}{lcr}
2(x-c_i)+2\sqrt{(x-c_i)^2-\mu_i^2}\, & \quad & x<c_i-\mu_i\\
2(x-c_i)-2{\rm i}\sqrt{\mu_i^2-(x-c_i)^2}\, & \quad & c_i-\mu_i<x<c_i+\mu_i\\
2(x-c_i)-2\sqrt{(x-c_i)^2-\mu_i^2}\, & \quad & x> c_i+\mu_i \\
\end{array}\right.
\label{wnearcut}
\end{equation}
where $\approx$ means that we are discarding terms of order $\mathcal{O}\left(\frac{1}{c_i-c_j}\right)$.
Note moreover that, when taking $x\to \infty$, we have
\begin{equation}
w(x) = \frac{1}{x} \sum_{i=1}^{g+1}{\mu_i^2} +\frac{1}{x^2}\sum_{i=1}^{g+1}{c_i \mu_i^2}+ \mathcal{O}(x^{-3})\,,
\end{equation}
thus, our requirement that $b_1$ has to vanish in the region asymptotically $AdS_5\times S^5$ implies that
\begin{equation}
b_1^0 = \alpha'\frac{\sum_{i=1}^{g+1} c_i \mu_i^2 }{\sum_{i=1}^{g+1}\mu_i^2}\,.
\end{equation}
At this point we should express the branch point parameters $\{c_i,\mu_i\}$ in terms of the brane fluxes, which are directly related to the integers
$\{n_i,k_j\}$ specifying the representation of the dual Wilson loop. These relations can be obtained from (\ref{QD3}-\ref{nkd3d5}), which gives
\begin{align}
(4\pi^2\alpha')^2 n_i\, , =&\ 32\, \pi^2\, {\rm i}\, \int_{e_{2i}}^{e_{2i-1}}d\mathcal{C} + \text{c.c.} = {4 \, \pi^2 \, (\alpha')^2\over g_s} \, {\rm i} \int_{e_{2i}}^{e_{2i-1}} w(x) dx + \text{c.c.} \, , \label{intws3} \\
(4\pi^2\alpha') k_j =&\ -8\, \pi\, {\rm i}\, \int_{e_{2j+1}}^{e_{2j}}d \mathcal{A}(x)+ \text{c.c.} =-{ \pi\,\alpha'\over g_s} \int_{e_{2j+1}}^{e_{2j}}d \left[ w(x)-2x \right] + \text{c.c.} \, \, , \label{intws5}
\end{align}
where in the first line we integrated by parts and used the fact that $x w(x)$ is real once evaluated at the branch points.
If we now use (\ref{wnearcut}) and since the integral is defined slightly above the real axis, we obtain
\begin{align}
n_i \approx &\ \frac{1}{2\, \pi^2 g_s} \int_{c_i-\mu_i}^{c_i+\mu_i} \sqrt{\mu_i^2-(x-c_i
)^2}= \frac{N}{\lambda} \mu_i^2\,,
\\
k_j \approx &\ -{ 1\over 4 \pi g_s} \int_{e_{2j+1}}^{e_{2j}}d \left[ w(x)-2x \right] + \text{c.c.} =
{4 N\over \lambda}(c_j-c_{j+1})\,.
\end{align}
We now define $\nu_i = \frac{n_i}{N}$ and $K_j = \sum_{i=j}^g k_i$, so that we can write $k_j = K_j-K_{j+1}$ and conclude that
$\mu_i = \sqrt{\lambda \nu_i}$ and $c_i = \frac{\lambda K_i}{4N}+c_0$. Since $\sum_{i=1}^{g+1}\nu_i = 1$ the gauge
fixing constant becomes
\begin{equation}
b_1^0 = \alpha'\sum_{i=1}^{g+1} c_i \nu_i\,.
\end{equation}
In order to obtain an explicit evaluation of (\ref{stringaction}) we need the coefficients $C_{a,n}$ of the expansion of
$a_0(x)$. For the proposal (\ref{wnearcut}) and for $x\in L_i$ we have
\begin{align}
a_0(x) = \frac{\alpha'}{2g_s}\sqrt{\mu_i^2-(x-c_i)^2}\,,
\label{ghexpneu}
\end{align}
Moreover, expanding around the right endpoint of the cut $x\approx e_{2i-1}=c_i+\mu_i$ we obtain an expansion of the form (\ref{a0exp}) with
\begin{equation}
C_{2i-1,0}={{\rm i} \,\alpha'\over g_s} \sqrt{\mu_i \over 2} \qquad , \qquad C_{2i-1,1}={C_{2i-1,0}\over 4\mu_i} \qquad , \qquad
C_{2i-1,2}=-{C_{2i-1,0}\over 32}\,.
\end{equation}
We notice that these coefficients satisfy the relation (\ref{relation}) and the on-shell string action (\ref{stringactiongen}) at the branch point reduces to
\begin{align}
S_{\rm on-shell}(e_{2i-1}) =&\ -e_{2i-1} + \frac{b_1^0}{\alpha'} = -c_i-\mu_i +\frac{1}{\lambda}\sum_{j=1}^{g+1} c_i \mu_i^2
\nonumber\\
=&\ -\sqrt{\lambda \nu_i} -\frac{\lambda}{4N} \left(K_i - \sum_{j=1}^{g}K_j \nu_j \right)\,.
\end{align}
Notice that going from the first to the second line, the dependence on the arbitrary constant $c_0$ cancels out, thus implying that the on-shell action is invariant under rigid translations of the branch cuts.
On the other hand, the coefficients for the expansion around the left endpoint of the cut $x\approx e_{2i}=c_i-\mu_i$ are
\begin{equation}
C_{2i,0}= {\alpha'\over g_s} \sqrt{\mu_i \over 2} \qquad , \qquad C_{2i,1} = -{C_{2i,0}\over 4\mu_i} \qquad , \qquad
C_{2i,2}= -{C_{2i,0}\over 32}\,.
\end{equation}
They also satisfy the relation (\ref{relation}), but the on-shell string action (\ref{stringactiongen}) is in this case
\begin{equation}
S_{\rm on-shell}(e_{2i})=-c_i+\mu_i -\left| \frac{C_{2i,0}}{2C_{2i,1}}\right| + \frac{b_1^0}{\alpha'} = -c_i-\mu_i + \frac{b_1^0}{\alpha'} = -e_{2i-1} + \frac{b_1^0}{\alpha'} \,.
\end{equation}
Similar results are obtained using the expansion along the interval with Dirichlet boundary conditions. In analogy with the genus one case, configurations at the endpoints of the same brunch cut have identical on-shell actions but only $g+1$ configurations will contribute to the saddle point approximation that computes the dual correlator of Wilson loops,
\begin{equation}
\boxed{
\langle W_{\rm fund} \rangle_{\mathbf{R}}\approx \sum_{i=1}^{g+1}e^{-S_{\rm on-shell}(e^*_i)} =
\sum_{i=1}^{g+1}e^{\sqrt{\lambda \nu_i} +\frac{\lambda}{4N} \left(K_i - \sum_{j=1}^{g}K_j \nu_j \right)} ,
}
\label{correstringgenric}
\end{equation}
where $\{e^*_i\}$ is the subset of branch points corresponding to the compatible string embeddings. For the genus one case we have seen that $\{e^*_i\}=\{e_1,e_4\}$.
As discussed above, for the correlator of Wilson loops with opposite orientations we have to change the sign in the $b_1$ contribution to the on-shell action.
Repeating the same analysis as before we obtain
\begin{equation}
\langle \widetilde W_{\rm fund} \rangle_{\mathbf{R}}\approx \sum_{i=1}^{g+1}e^{-S_{\rm on-shell}(e^*_i)} =
\sum_{i=1}^{g+1}e^{\sqrt{\lambda \nu_i} -\frac{\lambda}{4N} \left(K_i - \sum_{j=1}^{g}K_j \nu_j \right)} .
\label{correstringgenric2}
\end{equation}
\\
\section{Correlator of $\frac{1}{2}$-BPS Wilson Loops in $\mathcal{N}=4$ SYM}
\label{matrixmodel}
We now turn to the dual field theory description of the object we have been considering, {\it i.e.}, the correlator of $\frac{1}{2}$-BPS Wilson Loops in $\mathcal{N}=4$ super Yang-Mills. Specifically, we will consider the correlator of two Wilson loops
\begin{equation}
\label{eq:HolCorr0}
\langle \, W_\mathbf{r} \, \rangle_\mathbf{R} = \frac{\langle \, W_{\mathbf{R}} \, W_\mathbf{r} \,\rangle}{\langle \, W_\mathbf{R} \, \rangle}\,,
\end{equation}
with the Wilson loops defined as
\begin{equation}
W_{\mathbf{R}} = {\rm tr}_{\mathbf{R}} P \exp\left[\oint_{\cal C}ds\left({\rm i} A_\mu \dot x^\mu + \vec n \cdot \vec \Phi |\dot x| \right) \right]\,.
\label{wilsonloop}
\end{equation}
The two Wilson loops in the correlator will be taken over the same circle, {\it i.e.} one on top of each other sharing the orientation in the internal space, namely be $\vec n(\tau) = \vec n_0$ with $\vec{n}_0$ a constant unitary vector in the six-dimensional internal space. By $\mathbf{R}$ and $\mathbf{r}$ we mean large and small rank representations respectively. As small representations we will successively consider the fundamental, the totally symmetric and totally anti-symmetric. We notice that the correlator $\langle \, W_\mathbf{r} \, \rangle_\mathbf{R}$ is dimensionless, and there are no other scales besides the radius of the loop, so the result should be a radius-independent function of the coupling constant.
A remarkable fact is that the expectation value of operators (\ref{wilsonloop}) is given in terms of expectation values in a Gaussian matrix model obtained through localization \cite{Pestun:2007rz}. When the rank of the representation $\mathbf{R}$ is very large, the insertion of this Wilson loop competes with the quadratic terms of the matrix model. This backreaction in the eigenvalue distribution is the field theory counterpart of the gravitational backreaction, as the dual geometry is no longer $AdS_5\times S^5$ \cite{D'Hoker:2007fq,Okuda:2008px}. This suggests
$\langle \, W_{\rm fund} \, \rangle_\mathbf{R}$ should be compared with the string theory result
(\ref{correstringgenric}).
To be more specific, we are interested in computing the correlator between a Wilson loop that backreacts on the geometry and another which does not. We are going to use the intuition of \cite{Okuda:2008px}, to first consider the correlator between backreacting Wilson loop in a representation given by a large rectangular Young tableau and a Wilson loop in the fundamental. Finally we will consider the case where the light Wilson loops is in the totally symmetric or totally antisymmetric representations by generalizing the approach of \cite{Hartnoll:2006is}. We further extend all results to the case in which the backreacting Wilson loop is in an arbitrary large representation of the gauge group.
\subsection{The back-reacting Wilson loop}
In this section we review the computation of a Wilson loop in an arbitrary representation ${\bf R}$ of the gauge group \cite{Okuda:2008px}.
First, we consider the result for representations of $U(N)$ and then comment on how to obtain the result for $SU(N)$. The expectation value of a circular Wilson loop in ${\cal N}=4$ is computed by the localization formula
\begin{equation}
\langle \, W_{\bf R} \,\rangle =\frac{1}{Z}\int d a \, \Delta(a) \,e^{-{2N\over \lambda} \sum_r a_r^2 }\, \text{tr}_{\bf R} e^{ a }\,, \label{wra}
\end{equation}
with
\begin{equation}
Z=\int d a \, \Delta(a) \,e^{- {2N\over \lambda} \sum_r a_r^2 }\, ,
\end{equation}
and $da=\prod_{r=1}^N da_r$, $\Delta(a)=\prod_{r<s} (a_r-a_s)^2$ is the Vandermonde determinant and $a_r$ the eigenvalues of the matrix $a$ in the fundamental representation. A representation ${\bf R}$ of $U(N)$ is specified by the Dynking labels $\lambda=(\lambda_1,\lambda_2,\ldots \lambda_{N-1})$, or equivalently by a Young tableau with rows of length $\ell_r$ given by
\begin{equation}
\ell_r=1+\sum_{s=r}^{N-1} \lambda_s \quad\quad r=1,\ldots N \,.
\end{equation}
It is convenient to associate to any representation a Young tableau with an extra column of length $N$. We introduce the orthonormal basis $\{ e_r \}$ with
$e_r \in \mathbb{R}^N$ and write the $U(N)$ simple roots as $\alpha_r=e_r-e_{r+1}$ for $r=1,\ldots N-1$. The character of a representation is given by the Weyl formula
\begin{equation}
\text{tr}_{\bf R}\, e^{ a }=\sum_{\alpha \in {\bf R} } e^{ a\cdot \alpha }= { {\rm det}_{r,s} e^{ a_r (\ell_s +N-s ) } \over {\rm det}_{r,s} e^{ a_r (N-s ) } }\,,
\label{trace}
\end{equation}
with the sum running over the set of weights $\{ \alpha \}$ defining the representation ${\bf R}$. The determinant in the numerator can be written as
\begin{equation}
{\rm det}_{r,s}\, e^{ a_r (\ell_s +N-s ) } = \sum_{\sigma\in S_N} (-1)^{\sigma}\, \prod_{r=1}^N e^{ a_{\sigma(r)} (\ell_{r} +N-r ) }\,,
\end{equation}
while the one in the denominator can be explicitly written in the form
\begin{equation}
{\rm det}_{r,s} \, e^{ a_r (N-s ) } =\prod_{r<s} \left( e^{ a_r } - e^{ a_s } \right) \,. \label{denominator}
\end{equation}
Alternatively the denominator can be written as
\begin{equation}
\prod_{r<s} \left( e^{ a_r } - e^{ a_s }\right) =(-1)^\sigma\prod_{r<s} \left( e^{ a_{\sigma(r)} } - e^{ a_{\sigma(s)} }\right)\,.
\end{equation}
with $\sigma \in S_N$ an arbitrary permutation. Eq. \eqref{trace} can then be rewritten as
\begin{equation}
\text{tr}_{\bf R}\, e^{ a }=\sum_{\sigma\in S_N} \, \frac{\prod_{r=1}^N e^{ a_{\sigma(r)} (\ell_r +N-r ) }}{\prod_{r<s} \left( e^{ a_{\sigma(r)} } - e^{ a_{\sigma(s)} }\right)}\,. \label{trrea}
\end{equation}
Plugging (\ref{trrea}) into (\ref{wra}) and renaming the dummy variables $a_{\sigma(r)} \to a_r$ one finds that any element in the sum over $\sigma$ gives the same result.
Discarding the ${\bf R}$-independent $N!$ factor we obtain
\begin{eqnarray}
\langle \, W_{\bf R} \,\rangle &=& \frac{1}{Z}\int d a \, \Delta(a) \,e^{- {2N\over \lambda} \sum_r a_r^2 }\,
{ \prod_{r=1}^N e^{ a_r (\ell_r +N-r ) } \over \prod_{r<s} \left( e^{ a_r } - e^{ a_s } \right) } \nonumber\\
&=&
\frac{1}{Z}\int d a \, \Delta(a) \,e^{ \sum_r \left( -{ Na_r^2 \over 2\lambda} + a_r \, \ell_r \right) }\,
\prod_{r<s} \left( 1 - e^{ a_s -a_r } \right)^{-1}\,.
\end{eqnarray}
In the limit where the t'Hooft coupling $\lambda$ is large, the main contributions come from $a_r$ large, so assuming $a_r>a_s$ for $r<s$ the exponential terms can be dropped leading to
\begin{equation}
\langle \, W_{\bf R} \,\rangle = {1\over Z}\int d a \, \Delta(a) \,e^{ \sum_r \left( - {2\,N\over \lambda} a_r^2 + a_r \, \ell_r \right) }\, \label{wlr1} .
\end{equation}
Taking the Wilson loops made of blocks of $n_i $ rows of length $K_i$ and exponentiating the Vandermonde determinant one finds
\begin{equation}
\langle \, W_{\bf R} \,\rangle = {1\over Z}\int d a \,\exp\left({ - {2\,N\over \lambda} \sum_{r} a_r^2 +\sum_{r<s} \log(a_r-a_s)^2 + \sum_{i=1}^{g+1} K_i \, \sum_{r\in {\cal I}_i} a_r
}\right)\, \label{wlr},
\end{equation}
where we have split the range of $r\in \left[ 1,N\right]$ into segments ${\cal I}_i$,
of length $n_i$, ${\cal I}_1=\left[ 1,n_1\right]$, ${\cal I}_2=\left[ n_1+1,n_1+n_2\right]$ and so on. Notice that $n_{g+1}=N-(n_1+n_2+\ldots n_g)$ and $K_{g+1}=0$.
We display the generic Young tableau in Fig. \ref{fig:GenRep}.
\begin{figure}[H]
\beginpgfgraphicnamed{Gen-Rep}
\begin{center}
\begin{tikzpicture}[>=stealth]
\draw[thick] (0,0) grid [xstep=.25,ystep=.25] (5,1);
\draw[font=\huge] (2.25,0) node {$\mathbf{R}$};
\draw[font=\huge] (9.5,0) node {$\mathbf{R}$};
\draw[thick] (7,0)rectangle (12,1);
\draw[densely dotted,<->,red] (12,0) .. controls (10.5,0.5) and (8.5,0.5) .. (7,0) node[above,pos=.5,black]{$K_1$};
\draw[densely dotted,<->,red] (5,0) .. controls (5.3,0.3) and (5.3,0.7) .. (5,1) node[right,pos=.5,black]{$n_1$};
\draw[thick] (7,-1)rectangle (11,0);
\draw[densely dotted,<->,red] (11,-1) .. controls (9.5,-0.5) and (8.5,-0.5) .. (7,-1) node[above,pos=.5,black]{$K_2$};
\draw[densely dotted,<->,red] (4,0) .. controls (4.3,-0.3) and (4.7,-0.3) .. (5,0) node[below,pos=.5,black]{$k_1$};
\draw[thick] (0,-1) grid [xstep=.25,ystep=.25] (4,0);
\draw[densely dotted,red] (2,-1.2)--(2,-1.8);
\draw[densely dotted,red] (9,-1.2)--(9,-1.8);
\draw[thick] (0,-3) grid [xstep=.25,ystep=.25] (3,-2);
\draw[thick] (7,-3)rectangle (10,-2);
\draw[thick] (0,-4) grid [xstep=.25,ystep=.25] (2,-3);
\draw[thick] (0,-5) grid [xstep=.25,ystep=.25] (1,-4);
\draw[densely dotted,<->,red] (1,-4) .. controls (1.3,-4.3) and (1.3,-4.7) .. (1,-5) node[right,pos=.5,black]{$n_g$};
\draw[thick](0,-4)--(0,-6);
\draw[densely dotted,<->,red] (1,-5) .. controls (0.7,-5.3) and (0.3,-5.3) .. (0,-5) node[below right,pos=.6,black]{$k_g$};
\draw[thick] (7,-4)rectangle (9,-3);
\draw[densely dotted,<->,red] (9,-4) .. controls (8.6,-3.6) and (7.6,-3.6) .. (7,-4) node[above,pos=.5,black]{$K_{g-1}$};
\draw[thick] (7,-5)rectangle (8,-4);
\draw[densely dotted,<->,red] (8,-5) .. controls (7.8,-4.6) and (7.2,-4.6) .. (7,-5) node[above,pos=.5,black]{$K_{g}$};
\draw[densely dotted,<->,red] (0,-5) .. controls (0.3,-5.3) and (0.3,-5.7) .. (0,-6) node[below right,pos=.7,black]{$n_{g+1}$};
\draw[densely dotted,<->,red] (0,1).. controls (-1,-2) and (-1,-4)..(0,-6) node[left,pos=0.4,black]{$N$};
\end{tikzpicture}
\end{center}
\endpgfgraphicnamed
\caption{A general representation $\mathbf{R}$ with steps given $n_i$ and $k_i$, in the right a decomposition of the representation in $g$ rectangles of edges $n_i$, $K_i=\sum_{j=i}^g k_j$, all of order $N$.}
\label{fig:GenRep}
\end{figure}
Completing the squares in (\ref{wlr}), one can write the expectation value of the Wilson loop as
\begin{equation}
\langle \, W_{\bf R} \,\rangle = {v_{\mathbf{R}} \over Z}\int d a \,
\exp\left({ - {2\,N\over \lambda} \sum_i \sum_{r\in {\cal I}_i} \left( a_r - c_i \right)^2 +\sum_{r<s} \log(a_r-a_s)^2
}\right)\, , \label{wlr2}
\end{equation}
with\footnote{Note that the centers $c_i$ of the matrix model branch cuts are intimately related to the centers of the branch cuts of the supergravity solution introduced in section \ref{strings} up to an arbitrary constant $c_0$ which in the matrix model is completely fixed. }
\begin{equation}
c_i={K_i\, \lambda\over 4\, N} \quad , \quad v_{\mathbf{R}}=\exp\left({\sum_i {n_i \, K_i^2\, \lambda\over 8 \, N}}\right)
\end{equation}
We are interested in the limit of large $N$ with $K_i, n_i\approx N$. In this limit all contributions in the sum are of order $N^2 $ and cannot be dropped when using the saddle point approximation. The saddle point equations then read
\begin{eqnarray}
\label{eq:saddleq}
&&-\frac{4 N}{\lambda} (a_r-c_i)+2\sum_{s\neq r} \frac{1}{ a_r-a_s} =0\, , \qquad r\in {\cal I}_i\,,
\end{eqnarray}
or in its continuous version\footnote{Here $\rho(x)={1\over N} \sum_r \delta(x-a_r)$. }
\begin{equation}
-\frac{4 N}{\lambda} (x-c_i) +2 N \int dy \, {\rho(y) \over x-y } =0\,, \qquad c_i-\mu_i < x < c_i+\mu_i\,, \label{eqsaddle}
\end{equation}
with $\mu_i>0$ some real numbers. These equations are solved \cite{Okuda:2008px} by taking the matrix model resolvent $w(x)$
\begin{equation}
w(z)=\lambda \int_{-\infty}^\infty {\rho(y) \over z-y}\,,
\end{equation}
to be given by the integral
\begin{equation}
w(z)=\int_{\infty}^z \alpha\,,
\label{doble v}
\end{equation}
of a meromorphic one form
\begin{equation}
\alpha(z) =2\left( 1-{a_{g+1}(z)\over \sqrt{H_{2g+2}(z)} } \right)\,,
\label{alfa}
\end{equation}
defined on the hyperelliptic curve $y^2=H_{2g+2}(z) $ with $H_{2g+2}(z)$ and $a_{g+1}(z)$ polynomials of order $2g+2$ and $g+1$ respectively. The parameters specifying these polynomials are uniquely given in terms of $K_i$ and $n_i$. By considering integrals of \eqref{doble v} and \eqref{alfa} over non-trivial cycles on the hyperelliptic surface, one finds constraints analogous to the expressions \eqref{intws3} and \eqref{intws5} giving the supergravity charges of the dual bubbling geometry. Then, it is natural to identify the matrix model resolvent with the holomorphic function introduced in \eqref{AB} as proposed in \cite{Okuda:2008px}.
%
\subsubsection{ Multi-cut Wigner semicircle distribution}
To make an explicit comparison with string theory results, here we focus on the case where the distances between the cuts are large. First, we observe that for a single cut, (\ref{eqsaddle}) is solved by taking $ \rho(y)={2\over \pi \, \mu }\, \sqrt{ \mu^2-y^2}$.
In the limit where the interactions between the eigenvalues within different intervals can be neglected, the solution to (\ref{eqsaddle}) can be found as\footnote{ Note this eigenvalue distribution is in complete agreement with the proposed gravity solution in terms of the $w$ function \eqref{win},\eqref{wout}, if we further identify this function with the resolvent of the matrix model, namely
\begin{equation}
w(z)=\lambda\int \frac{\rho(y)}{z-y}\approx \frac{2}{\pi}\sum_{i=1}^{g+1}\int_{c_i-\mu_i}^{c_i+\mu_i}\frac{\sqrt{\mu_i^2-(y-c_i)^2}}{z-y}
\end{equation} }
\begin{equation}
\rho(x)= \Bigg \{ \begin{array}{ccc}
{2\over \pi \, \lambda }\, \sqrt{ \mu_i^2-(x-c_i)^2} & , & c_i - \mu_i < x < c_i + \mu_i, \\
0 & , & {\rm otherwise}\, ,
\label{twocutdensity}
\end{array}
\end{equation}
with centers and half-lengths given by
\begin{eqnarray}
c_i &=&{K_i\, \lambda\over 4\, N} \qquad , \qquad \mu_i=\sqrt{\lambda \nu_i } \qquad {\rm for} \qquad i=1,\ldots g+1 \nonumber\\
K_{g+1} &=& 0 \qquad , \qquad \nu_{g+1}= 1-\sum_{i=1}^g \nu_i \,,
\end{eqnarray}
where we have defined $\nu_i=\frac{n_i}{N}$ and normalised the eigenvalues distributions as
\begin{equation}
\int_{c_i-\mu_i}^{c_i+\mu_i} \rho(x) \,dx=\nu_i
\end{equation}
Finally, the expectation value \eqref{wlr2} evaluated in the multi-cut eigenvalue distribution reduces to
\begin{equation}
\left\langle W^{U(N)}_{\mathbf{R}} \right\rangle \approx \exp\left(\frac{\lambda}{8N} \sum_{i=1}^g n_i\, K_i^2\right)\,,
\label{matrix2}
\end{equation}
where $\approx$ here implies we are discarding subleading contributions of order $N^2\log\lambda$.
In the case of $SU(N)$ there is an additional factor of $(\det(e^M))^{-\frac{|\mathbf{R}|}{N}}$ in the matrix model integral with
$ |\mathbf{R}|= N \sum_{i=1}^g K_i\, \nu_i $. This insertion results simply into
a rigid shift of all centers by $-\frac{|\mathbf{R}|\, \lambda}{4N^2} $ or equivalently
\begin{equation}
K_i \to K_i- \sum_{j=1}^g K_j\, \nu_j\,.
\label{eq:shift}
\end{equation}
For the expectation value of the Wilson loop one finds
\begin{equation}
\left\langle\, W^{SU(N)}_{\mathbf{R}} \, \right\rangle \approx \exp\left({\frac{\lambda}{8N} \sum_{i=1}^g n_i\, \left( K_i-\sum_{j=1}^g K_j\, \nu_j \right)^2}\right)\,.
\label{matrix2SU}
\end{equation}
After having reviewed the distribution of eigenvalues found in \cite{Okuda:2008px}, we proceed to compute correlators with other Wilson loops, by evaluating expectation values of appropriate insertions. We will first consider the correlator with a fundamental Wilson loop and then move to the cases of correlators with totally symmetric and anti-symmetric Wilson loops.
\subsection{Adding a fundamental Wilson loop}
Computing the correlator between a large Wilson loop and a Wilson loop in the fundamental representation translates in the matrix model to evaluating the expectation value of the operator $\sum_{r=1}^N e^{a_r}$ in the matrix model integral \eqref{wlr2}
\begin{eqnarray}
\langle \, W_{\bf R} \,W_{\rm fund} \,\rangle &=& \frac{1}{Z}\int d a \, \Delta(a) \,e^{-{2N\over \lambda} \sum_r a_r^2 }\, \text{tr}_{\bf R}\, e^{ a }\, \text{tr}_{\rm fund} \,e^{ a }\,, \nonumber\\
&=& {v_{\bf R} \over Z}\int d a \sum_{i=1}^{g+1} \sum_{r\in {\cal I}_i} \,
e^{-S_r}
\end{eqnarray}
with
\begin{equation}
S_r= {2\,N\over \lambda} \sum_{i=1}^{g+1} \sum_{s\in {\cal I}_i} \left( a_s - c_i \right)^2 -\sum_{s<t} \log(a_{s}-a_{t})^2 -a_r\,,
\end{equation}
This insertion is not back-reacting in the sense that it does not modify the $\rho$-distribution discussed in the previous subsection. Taking the ratio with
$\langle \, W_{\bf R} \,\rangle $, the factor $v_{\bf R}$ cancels between numerator and denominator, and after the large $N$ limit one finds
\begin{equation}
\langle \, \,W_{\rm fund} \,\rangle_{\bf R} = \int_{-\infty}^\infty \, dx\, \rho(x)\, e^x \approx {2\over \pi \, \lambda }\sum_{i=1}^{g+1} \int_{c_i-\mu_i}^{c_i+\mu_i} \, dx\, \, \sqrt{ \mu_i^2-(x-c_i)^2}\, e^x\,,
\label{intrho}
\end{equation}
where $\approx$ denotes the approximation where centers are far away from each other, {\it i.e.} $K_i -K_j \gg N$ and the interactions between the regions ${\cal I}_i$ have been neglected. By doing the integrals we get the typical Bessel functions,
\begin{equation}
\label{eq:WrectaWf}
\langle \, \,W_{\rm fund} \,\rangle_{\bf R} \approx \sum_{i=1}^{g+1} \frac{2 \mu_i}{ \lambda}\, e^{c_i}\, I_1(\mu_i)\approx\sum_{i=1}^{g+1} e^{c_i+\mu_i}\,.
\end{equation}
For comparison with the string theory results in the context of the AdS/CFT correspondence, we should focus on the $SU(N)$ matrix model. In that case
\begin{equation}
\boxed{\langle \, \,W_{\rm fund}^{SU(N)} \,\rangle_{\bf R}
\approx \sum_{i=1}^{g+1} e^{\sqrt{\lambda\nu_i}+\frac{\lambda}{4N}\left(K_i-\sum_{j}^{g}K_j \nu_j\right)}}\,.
\label{eq:WrectaWfsun}
\end{equation}
that matches precisely the AdS/CFT prediction \eqref{correstringgenric}.
For instance, in the case of a representation given by a rectangular Young tableau, the position of the centers are
\begin{eqnarray}
c^{SU(N)}_1 &=& { k\,\lambda\over 4\, N} (1-\nu)\,, \qquad \qquad c^{SU(N)}_2=- { k\,\nu\,\lambda\over 4\, N}\,,
\end{eqnarray}
and (\ref{eq:WrectaWfsun}) yields
\begin{equation}
\langle \, \,W_{\rm fund}^{SU(N)} \,\rangle_{\bf R} \approx e^{\sqrt{\nu\,\lambda}+\frac{k(1-\nu) \lambda}{4N}} + e^{\sqrt{\lambda(1-\nu)}- { k\,\nu\,\lambda\over 4\, N} } \, ,
\end{equation}
that matches the result \eqref{eq:HolCorr}.
Before moving to correlators in more general representations, let us consider the correlator with another fundamental Wilson loop that can also be computed with the matrix model. At the end of section \ref{strings} we considered the possibility of a correlator of two loops with opposite spatial orientations. It turns out, as shown in appendix \ref{susywl}, that if the internal orientation is also opposite, the two loops are invariant under the same set of supersymmetries and therefore their correlator can be accounted for by an expectation value in the Gaussian matrix model. Since the internal space orientation is opposite, the matrix model computation is in this case
\begin{equation}
\langle \, \,\widetilde{ W}_{\rm fund} \,\rangle_{\bf R} \approx \int_{-\infty}^\infty \, dx\, \rho(x)\, e^{-x}\,.
\label{intrho2}
\end{equation}
For the case of the $SU(N)$ matrix model, we get now
\begin{align}
\langle \, \,\widetilde{W}_{\rm fund}^{SU(N)} \,\rangle_{\bf R}
&\approx \sum_{i=1}^{g+1} \frac{2 \mu_i}{ \lambda}\, e^{-c_i}\, I_1(\mu_i)\approx\sum_{i=1}^{g+1} e^{-c_i+\mu_i} \nonumber\\
&\approx \sum_{i=1}^{g+1} e^{\sqrt{\lambda\nu_i}-\frac{\lambda}{4N}\left(K_i-\sum_{j}^{g}K_j \nu_j\right)}\,.
\label{eq:WrectaWfsun2}
\end{align}
Once again this is in agreement with the AdS/CFT prediction \eqref{correstringgenric2}.
If we restrict ourselves to the case of a representation given by a rectangular Young tableau, the result becomes
\begin{equation}
\langle \, \,\widetilde{W}_{\rm fund}^{SU(N)} \,\rangle_{\bf R} \approx e^{\sqrt{\nu\,\lambda}-\frac{k(1-\nu) \lambda}{4N}} + e^{\sqrt{\lambda(1-\nu)}+ { k\,\nu\,\lambda\over 4\, N} }\,,
\end{equation}
thus matching the explicit result \eqref{eq:HolCorr2}.
So far we have computed correlators of Wilson loops defined over coincident circular contours. This amounted to compute the expectation value of the product ${\rm tr}_{\bf R}e^M {\rm tr}_{\bf r}e^M$. However, there is an alternative and interesting point of view, which arises from the ring structure of the characters of the gauge group representations, namely
\begin{eqnarray}
\text{tr}_{\mathbf{R}} e^{M}\text{tr}_{\mathbf{r}} e^{M}=\text{tr}_{\mathbf{R}\otimes\mathbf{r} } e^{M}=\sum_{\mathbf{R}_i\in \text{irreps}}C_{\mathbf{R}\mathbf{r}\mathbf{R}_i} \, \text{tr}_{\mathbf{R}_i}e^{M},
\end{eqnarray}
where $C_{\mathbf{R}\mathbf{r}\mathbf{R}_i}$ are the multiplicities and ``irreps'' denote the irreducible components of ${\bf R}\otimes {\bf r}$. For the products we have considered in this section, $\mathbf{R}$ is a `large' back-reacting representation associated to a Young diagram made of $g$ blocks and $\mathbf{r}$ is the fundamental one. In this case, the decomposition is rather simple, leading to a sum of $g+1$ irreps all of them with multiplicities $C_{\mathbf{R}\mathbf{r}\mathbf{R}_i}$ equal to 1, as schematically depicted in Figure \ref{fig:ProdYT}.
\begin{figure}[H]
\beginpgfgraphicnamed{Tensor-Prod}
\begin{center}
\begin{tikzpicture}[>=stealth]
\path[draw] (4.2,-0.8)-- (4.2,1)--(6.2,1)--(6.2,0.6)--(5.8,0.6)--(5.8,0.2)--(5.5,0.2)--(5.5,-0.2)--(5.1,-0.2)--(5.1,-0.6);
\draw[font=\tiny] (6.5,0) node {$\otimes$};
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\draw[font=\tiny] (7.2,-0.02)node{$=$}(7.2,-0.02);
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\path[draw,red] (9.5,1)--(9.65,1)--(9.65,.85)--(9.5,.85)--(9.5,1);
\draw[font=\tiny] (9.9,0.0) node {$\oplus$};
\path[draw] (10.4,-0.8)-- (10.4,1)--(12.4,1)--(12.4,0.6)--(12.0,0.6)--(12.0,0.2)--(11.6,0.2)--(11.6,-0.2)--(11.2,-0.2)--(11.2,-0.6);
\path[draw,red] (12.0,0.6)--(12.15,0.6)--(12.15,0.45)--(12.0,.45)--(12.0,0.6);
\draw[font=\tiny] (12.6,0.0) node {$\oplus$};
\path[draw] (13.0,-0.8)-- (13.0,1)--(15.0,1)--(15.0,0.6)--(14.6,0.6)--(14.6,0.2)--(14.2,0.2)--(14.2,-0.2)--(13.8,-0.2)--(13.8,-0.6);
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\draw[font=\tiny] (15.6,0.0) node {$\oplus$\,\,\dots};
\end{tikzpicture}
\end{center}
\endpgfgraphicnamed
\caption{Tensor product between a `large' representation and a fundamental one. }
\label{fig:ProdYT}
\end{figure}
Note that this exactly coincides with the number of saddles points we considered in our string theory computation, and also with the number of contributions that appeared in the matrix model computation. This field theory remark also suggests and interpretation for each saddle point contribution in string theory, as coming from a $g+1$ bubbling solution where one of the branch cuts is collapsing (see figure \ref{fig:GenYTplusbox}).
\begin{figure}[H]
\beginpgfgraphicnamed{Gen-Rep}
\begin{center}
\begin{tikzpicture}[>=stealth]
\node[yscale=-1,inner sep=0,outer sep=0,rotate=-45] at (-0.5,2.0) {\huge $\mathbf{R}$};
\path[draw](-6.8,0)--(6.8,0);
\coordinate [label=below left: {\tiny $\tilde{e}_{2g+1}$}] (2g+1) at (-5,0);
\coordinate [] (2g) at (-4.5,0);
\coordinate [label=below right: {\tiny $\tilde{e}_{2g-1}$}] (2g-1) at (-4,0);
\coordinate [] (2g-2) at (-3.5,0);
\coordinate [label=below: {\tiny $\tilde{e}_{7}$}] (7) at (0.5,0);
\coordinate [label=below: {\tiny $\tilde{e}_{6}$}] (6) at (1,0);
\coordinate [label=below: {\tiny $\tilde{e}_{5}$}] (5) at (1.65,0);
\coordinate [label=below: {\tiny $\tilde{e}_{4}$}] (4) at (2.5,0);
\coordinate [label=below: {\tiny $\tilde{e}_{3}$}] (3) at (3.5,0);
\coordinate [label=below: {\tiny $\tilde{e}_{2}$}] (2) at (4.5,0);
\coordinate [label=below: {\tiny $\tilde{e}_{1}$}] (1) at (5,0);
\draw[red,very thick] (2g+1)--(2g);
\draw[red,very thick] (2g-1)--(2g-2);
\draw[red,very thick] (7)--(6);
\draw[red,very thick] (5)--(4);
\draw[red,very thick] (3)--(2);
\draw[red,very thick] (1)--(6.8,0);
\draw[red,very thick] (1.35,0)--(1.5,0);
\fill[red] (-5,0) circle (0.05cm);
\fill[red] (-4.5,0) circle (0.05cm);
\fill[red] (-4,0) circle (0.05cm);
\fill[red] (-3.5,0) circle (0.05cm);
\fill[red] (1,0) circle (0.05cm);
\fill[red] (1.65,0) circle (0.05cm);
\fill[red] (2.5,0) circle (0.05cm);
\fill[red] (3.5,0) circle (0.05cm);
\fill[red] (4.5,0) circle (0.05cm);
\fill[red] (5,0) circle (0.05cm);
\fill[red] (1.35,0) circle (0.05cm);
\fill[red] (0.5,0) circle (0.05cm);
\fill[red] (1.5,0) circle (0.05cm);
\path[draw](-2.5,4.5)--(-3.5,5.5)--(-4.0,5.0)--(-4.5,5.5)--(-5,5)--(0,0)--(5,5)--(4.5,5.5)--(3.5,4.5)--(2.5,5.5)--(1.5,4.5)--(1.0,5)--(0.5,4.5)--(0.0,5);
\draw[dotted](-5,5)--(-5,0);
\draw[dotted](-4.5,5.5)--(-4.5,0);
\draw[dotted](-4.0,5.0)--(-4,0);
\draw[dotted](-3.5,5.5)--(-3.5,0);
\draw[dotted](-2,4.0)--(0,4.0);
\draw[dotted](5.0,5.0)--(5,0);
\draw[dotted](4.5,5.5)--(4.5,0);
\draw[dotted](3.5,4.5)--(3.5,0);
\draw[dotted](2.5,5.5)--(2.5,0);
\draw[dotted](1.65,4.65)--(1.65,0);
\draw[dotted,red](1.5,4.5)--(1.5,0);
\draw[dotted](1.0,5)--(1,0);
\draw[dotted](0.5,4.5)--(0.5,0);
\path[draw,red,fill] (1.5,4.5)--(1.65,4.65)--(1.5,4.8)--(1.35,4.65)--(1.5,4.5) ;
\draw[dotted,red](1.35,4.65)--(1.35,0);
\end{tikzpicture}
\end{center}
\endpgfgraphicnamed
\caption{One of the diagrams depicted in figure \ref{fig:ProdYT} of a general bubbling geometry with an additional box in red. From the gravity side, the additional red box corresponds to the collapse of one branch cut in a genus $g+1$ geometry. This is pictorially interpreted as the additional red cut that collapses and approaches $\tilde{e}_5$ in the figure.}
\label{fig:GenYTplusbox}
\end{figure}
\subsection{Small loops in symmetric or antisymmetric representations}
In this section we consider other examples of correlators of a backreacting rectangular Young tableau representation Wilson loop with non backreacting Wilson loops in the totally symmetric and totally antisymmetric representation. We write
\begin{equation}
\langle W_{\mathbf{r}} \, \rangle_{\mathbf{R}} = \frac{\langle\, W_{\mathbf{R}}\, W_{\mathbf{r}} \, \rangle}{\langle \,W_{\mathbf{R}}\, \rangle}
=\int_{-\infty}^\infty \rho(x) \Omega_{\mathbf{r}}(x)\,,
\label{correlatorO}
\end{equation}
where $\Omega_{\mathbf{r}}(x)$ is some function corresponding to the insertion $W_{\mathbf{r}}$ in the continuous large $N$ limit and the eigenvalue distribution, $\rho(x)$, is given by the two-cuts case of (\ref{twocutdensity}). We stress, again, that this distribution is reliable in the limit where both semicircles are sufficiently far away from each other, that is, when $\frac{k\lambda}{4N}$ is sufficiently large.
The normalized correlator with (anti)-symmetric Wilson loops can be written compactly using the generating function of characteristic polynomials as in \cite{Hartnoll:2006is}:
\begin{equation}
\langle W_{S_l,A_l}\rangle_{\mathbf{R}}
=
\frac{1}{{\rm dim}_{S_l,A_l}}
\oint_\Gamma \frac{dt}{2\pi\, {\rm i}} \frac{1}{t^{l+1}} \exp\Big(\mp N \int_{-\infty}^\infty dx \rho(x)\log(1\mp\, t\, e^x)\Big)\,,
\label{generating}
\end{equation}
where we take the $-$ sign for the totally symmetric representation, $S_l$, and the $+$ sign for the totally anti-symmetric representation, $A_l$.
The contour $\Gamma$ encloses the pole at $t=0$. We want to evaluate the integral (\ref{generating}) for large $N$, and for a general (anti)-symmetric representation even when $l$ is large, but not as large that can possibly back-react on the eigenvalue distribution.
\subsubsection{Correlator with a totally symmetric Wilson loop}
\label{sec:Symm}
We start by considering the totally symmetric case. We have to evaluate the integral \eqref{generating} for the two-cut density distribution (\ref{twocutdensity}).
It is convenient to change variables $x \to c_i - a_i x$ along each cut ${\cal I}_i$ in such a way as to bring the $x$-integrals to the intervals $\left[ -1,1\right]$
\begin{equation}
\int_{-\infty}^\infty dx \rho(x)\log(1-\, t\, e^x) = \sum_{i=1}^2 \mu_i \, \int_{-1}^1 \sqrt{1-x^2} \log(1-e^{-\mu_i x+c_i} t)\,.
\end{equation}
It is also convenient to change the $t$ variable, $t=e^{z}$, which yields
\begin{align}
\oint_{\tilde{\Gamma}}dz
\exp\!& \left[ -N \left( \sum_{i=1}^2 \tfrac{2\, \mu_i^2}{\pi\lambda}\!\int\limits_{-1}^{1}\!dx \sqrt{1-x^2}\log(1-e^{-\mu_1 x+c_i+ z} ) +f\, z\right) \right]\,,
\end{align}
where $f=\frac{l}{N}$. The integral above has two branch cuts in $z$ due to the $\log$. They are given by
\begin{eqnarray}
\label{eq:zcuts1}
{-\mu_i-c_i}\leq z\leq {\mu_i-c_i}\quad \text{with }\quad i=1,2\,.
\end{eqnarray}
The contour $\tilde{\Gamma}$ is picking now the pole at infinity, so it can be deformed to pass just above and below the cuts. Using Jordan Lemma the contour integral reduces to the discontinuity across the cuts of the integral:
{\small
\begin{align}
\label{eq:ImWL}
& \langle W_{S_l} \rangle_{\mathbf{R}} \approx \frac{1}{\pi} {\rm Im}
\Bigg\{ \sum_{j=1}^2
\int \limits_{-c_j-\mu_j}^{-c_j+\mu_j} \! \!dz
\exp\Bigg[ -{N \over \lambda} \Big( \sum_{i=1}^2
\frac{2 \, \mu^2_i}{\pi } \int\limits_{-1}^{1}\!dx\sqrt{1-x^2}
\log(1 -e^{-\mu_i x+c_i + z }) +f\, \lambda\, z\Big) \Bigg] \Bigg\}\\
&= {\rm Im}
\Bigg\{ \sum_{j=1}^2 \, \frac{\mu_j}{\pi} \,
\int \limits_{-1 }^{1 } \! \!dz
\exp\Bigg[ -{N \over \lambda} \Big( \sum_{i=1}^2
\frac{2 \, \mu^2_i}{\pi} \int\limits_{-1}^{1}\!dx\sqrt{1-x^2}
\log(1 -e^{-\mu_i x+c_i-c_j+ \mu_j z}) +f \, \lambda\, (\mu_j\,z-c_j ) \Big) \Bigg] \Bigg\}\,,\nonumber
\end{align}}
where in the second line we made the change of variables $ z\to \mu_j z-c_j $. The $x$-integrals here are formal because the integrand has branch cuts along the integration region. A way to cure this is to give $z$ a small imaginary part ${\rm i}\, \epsilon$, so we are passing through a line slightly above the real axis. The integrals for $j=1,2$ can be evaluated separately using the large-$N$ saddle point method.
The $z$-integral is dominated by the region $z\approx z^*$ extremizing the exponential term. Let us consider the case $j=1$ and take $c_1-c_2\gg 1$. In this limit, only the $i=1$ term in the sum contributes.
To compute the saddle equations
it is convenient to break the $x$-integral into pieces such that the argument of the log is always positive. We write
\begin{eqnarray}
\int\limits_{-1}^{1}\!dx\sqrt{1-x^2}
\log(1 -e^{\mu_1 (z-x) }) &=& \int\limits_{-1}^{z}\!dx\sqrt{1-x^2}
\log(e^{\mu_1 \, z} -e^{\mu_1 x }) + \int\limits_{z}^{1}\!dx\sqrt{1-x^2}
\log(e^{\mu_1 \, x} -e^{\mu_1 z }) \nonumber\\
&& + {\rm i} \, \pi \, \int\limits_{-1}^{z}\!dx\sqrt{1-x^2}\,. \label{decomp}
\end{eqnarray}
We are going to look for solutions when $\text{Re}(z)<-1$ in this case the saddle equation becomes\footnote{For $\text{Re}(z)>1$ there are no solutions to the saddle equation.}
\begin{equation}
{ 2\, \mu_1\, \over \pi} \int\limits_{-1}^{1}\!dx
\frac{\sqrt{1-x^2}}{ 1- e^{\mu_1 (x-z) }}
+4\, {\rm i}\, \mu_1 \sqrt{1-z^2} + \lambda \, f =0. \label{sad0}
\end{equation}
In this domain,
the integral term in Eq. (\ref{sad0}) can be discarded when $\mu_1$ is large and the saddle point equation reduces to
\begin{equation}
4\, {\rm i}\, \mu_1 \sqrt{1-z^2} + \lambda \, f =0 \label{sad00}\,,
\end{equation}
with solution
\begin{equation}
z^{*}=-\sqrt{1+\kappa_1^2}\,, \qquad\text{with}\qquad\kappa_1=\frac{f\lambda}{2 \mu_1}= {l \over 2N} \sqrt{\lambda\, \over \nu }\,.
\end{equation}
Evaluating (\ref{decomp}) at the saddle point $z^*$ and discarding $e^{\mu_1 z}$-terms inside of the logs one finds\footnote{The integral is computed using
\begin{equation}
{\rm i} \int_{-1}^z \sqrt{1-x^2} dx=-\int_0^{{\rm arccosh}\, z} \sinh^2 y \, dy=\frac{1}{2} (y-\sinh y \, \cosh y )\Big|_0^{{\rm arccosh \, z } } =
\frac{1}{2} \left({\rm arccosh \, z }-z\sqrt{z^2-1} \right)
\end{equation}
}
\begin{eqnarray}
\label{eq:I11}
\int\limits_{-1}^{1}\!dx\sqrt{1-x^2}
\log(1 -e^{\mu_j (z^*-x) }) & \underset{ \mu_i \to \infty}{\approx} & 2{\rm i} \, \pi \, \int\limits_{-1}^{z^*}\!dx\sqrt{1-x^2} \nonumber\\
&=& \pi \left({\rm arccosh \, z^* }-z^*\sqrt{(z^*)^2-1} \right)
\end{eqnarray}
To get the contribution from this saddle point we need to evaluate the exponential in \eqref{eq:ImWL} at $z^*$. Strictly speaking this quantity is not well defined due to the branch cuts of the exponent and for that we have added an small imaginary part to $z$, so, we will do the same for $z^*$, indeed, the well defined quantity is the imaginary part \eqref{eq:ImWL}, we essentially need to evaluate the right hand side of \eqref{eq:I11} taking into account this imaginary shift, and evaluate the full answer with this small deformation.
Taking $z^*=-\sqrt{1+\kappa_1^2}+{\rm i} \epsilon$ one finds
\begin{equation}
\label{eq:I1}
{\rm i} \,\int\limits_{-1}^{z^*}\!dx\sqrt{1-x^2} = \frac{1}{2} \left(\kappa_1 \sqrt{1+\kappa_1 ^2}-{\rm arcsinh}\, \kappa_1 \right)+\text{imaginary part}\,.
\end{equation}
Plugging the solution into (\ref{eq:ImWL}) one finally finds for the contribution of the first saddle point
\begin{equation}
\langle W_{S_l} \rangle_{\mathbf{R}}^{(1)} \approx
\exp\Bigg[
\frac{2\, N\, \mu_1^2}{\lambda } \left(\kappa_1\sqrt{1+\kappa_1^2} + \text{arcsinh}\, \kappa_1\right)+ N\, f\, c_1
\Bigg]\,,
\end{equation}
where we have discarded a large $N$ phase in the result above. For $j=2$, one follows the same steps but now we have an extra contribution coming from the term with $i \neq j$ given by
\begin{equation}
\frac{2 \, \mu^2_1}{\pi} \int\limits_{-1}^{1}\!dx\sqrt{1-x^2}
\log(1 -e^{-\mu_1 x+c_1-c_2+ \mu_2 z}) \approx \mu^2_1\, ( c_1-c_2+ \mu_2 z)\,.
\end{equation}
The saddle point equation now becomes
\begin{equation}
4\, {\rm i}\, \mu_2 \sqrt{1-z^2} + \lambda \, f + \mu^2_1 =0\,.
\end{equation}
The solution is now given by
\begin{eqnarray}
z^*=-\sqrt{1+\kappa_2^2}, \quad \text{with} \quad\kappa_2=\frac{\lambda f}{4 \mu_2}+\frac{\mu_1^2}{4\, \mu_2}=\frac{f\sqrt{\lambda}}{4\sqrt{1-\nu}}+\frac{\sqrt{\lambda}}{4}\frac{\nu}{\sqrt{1-\nu}}\,.
\end{eqnarray}
Plugging this into (\ref{eq:ImWL})
\begin{eqnarray}
\langle W_{S_l}\rangle_{\mathbf{R}}^{(2)} \approx \exp\Bigg[\frac{2N \mu_2^2}{\lambda } \left(\kappa_2\sqrt{1+\kappa_2 ^2} + \text{arcsinh}\, \kappa_2\right)+ N(1+f)c_2 \Bigg]\,,
\end{eqnarray}
where the $N c_2$ term comes from the extra term $-N\, \mu_1^2(c_1-c_2)/\lambda$. Finally, the total contribution to the correlator with the $S_l$ representation adds up to,
\begin{align}
\langle W_{S_l}\rangle_{\mathbf{R}} \approx &\exp\Bigg[2N(1- \nu) \left(\kappa_2\sqrt{1+\kappa_2 ^2} + \text{arcsinh}\, \kappa_2-\frac{1+f}{1-\nu}\frac{k\lambda}{8N}\nu\right) \Bigg]\nonumber\\&
+\exp\Bigg[2N \nu \left(\kappa_1\sqrt{1+\kappa_1 ^2} + \text{arcsinh}\, \kappa_1+f\frac{1-\nu}{\nu}\frac{k\lambda}{8N}\right)\Bigg].
\label{symfinal}
\end{align}
A comment is in order, in \cite{Hartnoll:2006is} there was an additional solution to the saddle point equations which in the large $\lambda$ regime and $\kappa_i$ fixed or $\frac{l}{N}$ fixed, was sub-leading with respect to the contribution of the saddle point considered here. We report these contributions in the Appendix \ref{sec:2ndsad}.
\subsubsection{Correlator with a totally anti-symmetric Wilson loop}
Let us now turn our attention to the correlator with a Wilson loop in a totally anti-symmetric representation which is given by \eqref{generating}
with the two-cut distribution given in \eqref{twocutdensity}.
Performing the transformation $t=e^{\mu_2z-c_2}$ and defining $f=\frac{l}{N}$ the integral above can be rewritten as
\begin{align}
\langle W_{A_l}\rangle_{\mathbf{R}}\approx\int_{\tilde{\Gamma}} dz\exp\Big[& \frac{2N}{\lambda \pi}\Big( \mu_2^2 \int\limits_{-1}^{1} dx \sqrt{1-x^2}\log\left(1+ e^{-\mu_2(x-z)}\right)\nonumber\\
+&\mu_1^2 \int\limits_{-1}^{1} dx \sqrt{1-x^2}\log\left(1+ e^{-\mu_1x+\mu_2z+(c_1-c_2)}\right)-\frac{\pi\lambda }{2}(\mu_2z-c_2)f\Big)\Big].
\label{antisym}
\end{align}
Note that the branch cuts of the integrand are now along the horizontal segments $[-1+{\rm i}\,\pi,1+{\rm i}\,\pi]$ and $[-\frac{1}{\mu_2}(c_1-c_2)-\frac{\mu_1}{\mu_2}+{\rm i}\,\pi,-\frac{1}{\mu_2}(c_1-c_2)+\frac{\mu_1}{\mu_2}+{\rm i}\,\pi]$, together with the images obtained by shifting the imaginary part by multiples of $2\pi$. As in the symmetric case, we deform the contour $\tilde{\Gamma}$ to lay along the real axis and approximate the integral by its large $N$ saddle point. Unlike the previous case, the saddle point value is not located over any branch cut, making the evaluation much more straightforward. The saddle point equation reads
\begin{equation}
\mu_2^2\int_{-1}^{1} dx \frac{\sqrt{1-x^2}}{1+ e^{\mu_2(x-z)}}+\mu_1^2\int_{-1}^{1} dx \frac{\sqrt{1-x^2}}{1+ e^{\mu_1x-\mu_2z-(c_1-c_2)}}-\frac{\pi\lambda }{2}f=0\,.
\label{antisymeom}
\end{equation}
Now we search for solutions in the large $\mu_i$ regime. It turns out that the solutions can only be placed along the segments $[-1,1]$ and $[-\frac{1}{\mu_2}(c_1-c_2)-\frac{\mu_1}{\mu_2},-\frac{1}{\mu_2}(c_1-c_2)+\frac{\mu_1}{\mu_2}]$. Otherwise, the integrals in \eqref{antisymeom} become $z$-independent thus not having any solution there.
Let us first consider the region $-1<z<1$. Taking into account that $c_1-c_2=\frac{k\lambda}{4N}\gg 1$, equation \eqref{antisymeom} reduces to
\begin{equation}
\mu_2^2\int_{-1}^z dx\sqrt{1-x^2}+\frac{\pi \mu_1^2}{2}-\frac{\pi\lambda }{2}f=0\,,
\end{equation}
which yields
\begin{equation}
\text{arccos}(z)-z\sqrt{1-z^2}=\pi\left(1+\frac{\mu_1^2}{\mu_2^2}-\frac{\lambda}{\mu_2^2}f\right)\,.
\end{equation}
The solution is $z=\cos \theta_{2}$ with $\theta_{2}$ such that
\begin{align}
\theta_{2}-\sin \theta_{2}\cos\theta_{2}&=\pi\left(1+\frac{\mu_1^2}{\mu_2^2}-\frac{\lambda}{\mu_2^2}f\right) = \pi \left(1+\frac{\nu}{1-\nu}-\frac{l}{N(1-\nu)}\right)\,,
\label{theta2}
\end{align}
and then the integral \eqref{antisym} results in
\begin{align}
\langle W_{A_l}\rangle_{\mathbf{R}}^{(2)}&\approx\exp\Big[\frac{2N}{\lambda\pi}\left(\mu_2^3\int_{-1}^{\cos\theta_2} dx\, x\sqrt{1-x^2}+\frac{\pi \mu_1^2}{2}(c_1-c_2)+\frac{\pi\lambda}{2}f c_2\right)\Big]\,,\\
&=\exp\Big[N\left(\frac{2\sqrt{\lambda}}{3\pi}\left(\sqrt{1-\nu}\sin\theta_{2}\right)^3+(1-f)\frac{k\nu\lambda}{4N}\right)\Big].
\label{antiaction1}
\end{align}
There is an additional saddle point sitting on the interval $[-\frac{1}{\mu_2}(c_1-c_2)-\frac{\mu_1}{\mu_2},-\frac{1}{\mu_2}(c_1-c_2)+\frac{\mu_1}{\mu_2}]$. In this case, the first integral on equation \eqref{antisymeom} vanishes, whereas the second one only receives contributions from $0<x<\tilde{z}$ with
\begin{equation}
\tilde{z}=\frac{1}{\mu_1}\left(\mu_2z+c_1-c_2\right)\,, \quad -1<\tilde{z}<1\, ,
\end{equation}
thus obtaining the following equation
\begin{equation}
\text{arccos}(\tilde{z})-\tilde{z}\sqrt{1-\tilde{z}^2}=\pi\left(1-\frac{\lambda}{\mu_1^2}f\right)\, ,
\end{equation}
which is solved in this other case by $\tilde{z}=\cos\theta_{1}$ such that
\begin{align}
\theta_{1}-\sin\theta_{1}\cos\theta_{1}&=\pi\left(1-\frac{\lambda}{\mu_1^2}f\right)
=\pi\left(1-\frac{l}{N \nu}\right)\,.\label{theta1}
\end{align}
The integral \eqref{antisym} evaluated at this saddle contributes as
\begin{align}
\langle W_{A_l}\rangle_{\mathbf{R}}^{(1)}&\approx\exp\big[\frac{2N}{\pi\lambda}\left(\mu_1^3\int_{-1}^{\cos\theta_1} dx\, x\sqrt{1-x^2}(-\mu_1x+c_1-c_2)+\frac{\pi\lambda}{2}f c_2\right)\Big]\,,\\
&=\exp\Big[N\left(\frac{2\sqrt{\lambda}}{3\pi}\left(\sqrt{\nu} \sin\theta_{1}\right)^3+f \frac{k(1-\nu)\lambda}{4N}\right)\Big]\,.
\end{align}
Hence, the result for the correlator from both saddle points is
\begin{align}
\langle W_{A_l}\rangle_{\mathbf{R}} \approx & \exp\left[{N\left(\frac{2\sqrt{\lambda}}{3\pi}\left(\sqrt{\nu} \sin\theta_{1}\right)^3+f \frac{k(1-\nu)\lambda}{4N}\right)}\right]\nonumber \\
&+\exp\left[{N\left(\frac{2\sqrt{\lambda}}{3\pi}\left(\sqrt{1-\nu}\sin\theta_{2}\right)^3+(1-f)\frac{k\nu\lambda}{4N}\right)}\right]\,.
\label{antisymfinal}
\end{align}
It is worth noting that implementing the following conjugation, $\nu\to 1-\nu$ and $l\to N-l$ in \eqref{theta2} and \eqref{theta1} we find that, $\theta_1\to \pi-\theta_2$ and $\theta_2\to \pi-\theta_1$ thus leaving \eqref{antisymfinal} invariant.
\subsubsection{Back-Reacting Wilson loops in general representations}
\label{sec:GenR}
We can go further and generalize our results \eqref{symfinal} and \eqref{antisymfinal} for correlators of Wilson loops in symmetric and anti-symmetric representations with a general large representation ${\bf R}$ dual to a genus $g$ bubbling geometry. In order to do so we have to make use of the general multi-cut eigenvalue distribution \eqref{twocutdensity} proposed previously, together with the definitions of the $\mu_i$ and $c_i$ given there.
Let us consider first the symmetric case. We deform the contour of the $z$ variable to lay over the $g+1$ branch cuts of the integrand, thus obtaining the natural generalization of integral \eqref{eq:ImWL}
\begin{align}
\langle W_{S_l}\rangle_{\mathbf{R}}\approx\text{Im}\sum_i^{g+1}\frac{\mu_i}{\pi}\int_{c_i-\mu_i}^{c_i+\mu_i}\exp\Bigg[&- \frac{2N }{\pi \lambda}\Big(\mu_i^2\int_{-1}^1\sqrt{1-x^2}\log\left(1-e^{-\mu_i(x-z)}\right) + \frac{\pi\lambda}{2}f(\mu_i z-c_i)\nonumber\\
& +\sum_{j\neq i}\mu_j^2\int_{-1}^1\sqrt{1-x^2}\log\left(1-e^{-\mu_jx + \mu_iz +c_j-c_i)}\right)\Big) \Bigg]\,.
\end{align}
For the $i$-th term, the saddle point is located at the left of the $i$-th branch cut, but still to the right of the $(i+1)$-th one\footnote{Provided the cuts are far away from each other, this is guaranteed.}. Thus, from the sum in the second line, only the terms with center $c_j>c_i$ contribute. In our notation, this implies $j<i$, and the saddle point equations are solved by
\begin{eqnarray}
z_i^*=-\sqrt{1+\kappa_i^2}\,, \quad \text{with} \quad\kappa_i=\frac{\lambda f}{4 \mu_i}+\frac{1}{4\mu_i}\sum_{j<i}\mu_j^2\,.
\end{eqnarray}
The integral evaluated at these saddle points result
\begin{equation}
\exp\Bigg[\frac{2N\mu_i^2}{\lambda } \left(\kappa_i\sqrt{1+\kappa_i ^2} + \text{arcsinh}\, \kappa_i\right)+ 4N\mu_i\kappa_ic_i-\frac{N}{\lambda}\sum_{j<i}\mu_j^2 c_j + i \phi_i \Bigg]\,,
\end{equation}
where $\phi_i$ denotes an irrelevant phase. Taking the imaginary part and collecting all together we obtain
\begin{equation}
\langle W_{S_l}\rangle_{\mathbf{R}}\approx\sum_i^{g+1}\exp\Bigg[\frac{2N\mu_i^2}{\lambda } \left(\kappa_i\sqrt{1+\kappa_i ^2} + \text{arcsinh}\, \kappa_i\right)+ 4N\mu_i\kappa_ic_i-\frac{N}{\lambda}\sum_{j<i}\mu_j^2 c_j \Bigg]\,.
\end{equation}
Finally, let us now turn to the antisymmetric case. Making the change of variable $t=e^{c_{g+1}-\mu_{_{g+1}} z}$, expression \eqref{generating} can be taken to the form
\begin{align}
\langle W_{A_l}\rangle_{\mathbf{R}}\approx\int_{\tilde{\Gamma}} dz\exp\Big[& \frac{2N}{\lambda \pi}\Big( \mu_{g+1}^2 \int\limits_{-1}^{1} dx \sqrt{1-x^2}\log\left(1+ e^{-\mu_{g+1}(x-z)}\right)
\label{antigeneral}
\\
+&\sum_{i}^{g}\mu_i^2 \int\limits_{-1}^{1} dx \sqrt{1-x^2}\log\left(1+ e^{-\mu_ix+\mu_{_{g+1}}z+(c_i-c_{g+1})}\right)-\frac{\pi\lambda }{2}(\mu_{g+1}z-c_{g+1})f\Big)\Big]\,.\nonumber
\end{align}
As for the genus one case, the contour can be deformed to run along the real axis and the integral can be approximated by evaluating the integrand at the $g+1$ saddle points sitting at
\begin{equation}
z^*_i=\Big[\frac{1}{\mu_{g+1}}(c_{g+1}-c_i-\mu_i),\frac{1}{\mu_{g+1}}(c_{g+1}-c_i+\mu_i)\Big]\,, \quad i=1,\ldots, g+1\,.
\end{equation}
Defining $\tilde{z}^*_i=\frac{1}{\mu_i}(\mu_{g+1}z^*_i+c_i-c_{g+1})$, the solution to the saddle point equations can be written as $\tilde{z}^*_i=\cos\theta_{i}$ with $\theta_{i}$ such that
\begin{equation}
\theta_{i}-\sin\theta_{i}\cos\theta_{i}=\pi\left(1+\sum_{j<i}\frac{\mu_j^2}{\mu_i^2}-\frac{\lambda}{\mu_i^2}f\right),
\label{generaltheta}
\end{equation}
hence the result for the correlator can be written as
\begin{equation}
\langle W_{A_l}\rangle_{\mathbf{R}}\approx \sum_{i}^{g+1}\exp\Big[N\left(\frac{2}{3\pi \lambda}\left(\mu_i \sin\theta_{i}\right)^3+fc_i+\sum_{j<i}\frac{\mu_j^2}{\lambda}(c_j-c_i)\right)\Big].
\label{antigeneralfinal}
\end{equation}
Furthermore, it can be seen that the last expression is manifestly invariant under conjugation of the representation ${\bf R}$. Indeed, under conjugation $\nu_i\to\nu_{g+2-i}$ and $k_i\to k_{g+1-i}$ together with $f\to 1-f$, so from \eqref{generaltheta} it can be shown that
\begin{equation}
\theta_{i}\to \pi - \theta_{g+2-i}\,,
\end{equation}
and from the definition of the centers, it can be shown that $c_i\to -c_{g+2-i}$. This together with the property $c_i+\sum_{j>i}\nu_j(c_j-c_i)=-\sum_{j<i}\nu_j(c_j-c_i)$ shows that \eqref{antigeneralfinal} is invariant under conjugation.
\section{Conclusions}
\label{conclu}
We have found classical fundamental string solutions in the background of bubbling geometries dual to Wilson loops in large rank representations.
For a general genus $g$ background we have shown that minimal area configurations are found at the points $z=z^*$ of the Riemann surface $\Sigma$ that minimize both the area (given by the product of the dilaton and the warping factor $e^{\Phi\over 2} f_1^2$) and the $B$-field component $b_1$. We have also found that the critical points, in the upper half-plane coordinates, are precisely located at the branch points $e_a$.
Furthermore, we have argued that $g+1$ out of the $2g+2$ solutions correspond to string configurations preserving the same symmetries and supersymmetries as the bubbling geometries. Thus, only the former have to be taken into account in the saddle point approximation that is related to the strong coupling limit of the correlator between a large representation Wilson loop and a fundamental Wilson loop.
In order to write down the explicit expressions for the corresponding on-shell actions, we have considered in great detail the case of strings in genus one backgrounds. In this case the on-shell actions display quite a non-trivial structure, since two classical configurations contribute to the saddle point approximation.
In the case of genus one background, the dual large representation Wilson loop is characterized by a rectangular Young tableau. The matrix model computation we performed for its correlator with a fundamental Wilson loop is valid in the large-$N$ limit and requires $\frac{k\lambda}{4N}\gg 1$ as well. Remarkably, the large $\lambda$ limit of this correlator, given in terms of a combination of two Bessel functions, was shown to be in perfect agreement with the two contributions to the string theory saddle point approximation.
In addition, the correlator of a fundamental and a generic Young tableau representation Wilson loop was similarly solved in the large-$N$ limit, provided the edges of the tableau are all size of order $N$. The resulting expression for the correlator is again given by a combination of $g+1$ Bessel functions. Finally, we went on to compute correlators of more general configurations including, for instance, a large rectangular representation with totally symmetric and totally anti-symmetric representations.
Let us close with some comments about open problems that could be interesting complements of the results presented in this article.
Our computation for correlators between rectangular and totally symmetric/anti-symmetric representation Wilson loop provides a prediction for
probes D3 and D5 branes in the bubbling geometry background. Thus, it would be interesting to find those D-brane configurations and evaluate their on-shell actions.
Alternatively, it would be interesting to consider the gravity picture suggested by the product of characters formula in the field theory side, and check that each saddle point in the on-shell string action indeed coincides with a bubbling geometry of one genus higher, in a limit where one branch cut collapses.
Our work, together with the very interesting results of \cite{Gomis:2008qa} where correlators of large Wilson loops with local operators were discussed, creates a platform for the computation of more general correlators. Following some of the development in \cite{Alday:2011pf}, it seems now feasible to tackle more complicated insertions, for example, two Wilson loops and a local operator. Clearly, one of our driving motivations has been a concrete exploration of non-conformal gauge/gravity pairs. However, we secretly hope that some thread of the beautiful integrability techniques that have been so successful in understanding the structure of three-point correlators \cite{Zarembo:2010rr,Gromov:2012vu} might still be extracted from our explicit computations.
Finally, and certainly more ambitiously, there is the question of sub-leading corrections on both sides of the correspondence. On the field theory side, there are well established techniques to go beyond the large-$N$ limit and they have been applied to the computation of Wilson loops in the context of the Gaussian matrix model \cite{Faraggi:2014tna,Liu:2017fiq,Gordon:2017dvy}; there are also techniques to explore the large $\lambda$ expansion in some cases \cite{Horikoshi:2016hds,Chen-Lin:2016kkk}. It will be instructive to extend these computations to correlators of Wilson loops. The holographic computation, although conceptually clear \cite{Faraggi:2011bb,Faraggi:2011ge,Buchbinder:2014nia}, seems more daunting at the moment.
\section*{Acknowledgments}
We are thankful to X. Chen-Lin, J. Liu, G. Silva, D. Trancanelli and S. Zhou for various discussions on closely related topics. LAPZ is partially supported by the US Department of Energy under Grant No. \ DE-SC0017808 -- {\it Topics in the AdS/CFT Correspondence: Precision tests with Wilson loops, quantum black holes and dualities}.
DHC and JAD are supported by CONICET and grants PICT 2012-0417 and PIP 0681 and PI {\it B\'usqueda de nueva F\'\i sica}. V.I. Giraldo-Rivera is grateful to the string theory group at IFLP for hospitality during the completion of this work. He is happy to thank the members of String theory group at ICTS-TIFR, for the support and encouragement. He also acknowledges support from Simons Foundation. The work of FF and JFM is partially supported by the MIUR PRIN Contract 2015MP2CX4-{\it Non-perturbative Aspects Of Gauge Theories And Strings}.
|
1,314,259,996,670 | arxiv | \section{Introduction} \label{sec:intro}
Over several decades, much research interest in dynamical systems has been devoted to the study of transport phenomena.
One studied notion is that of \emph{coherence}.
Unfortunately, there is no generally-agreed-upon-definition of \emph{coherent structure}, other than the necessity for a structure to show some persistence over a longer time horizon.
One way to address the question in fluid mechanics is to consider tracer particles in the flow and study local and global geometric deformations of fluid elements attached to their trajectories under advection. This gives rise to a \emph{Lagrangian} analysis of coherent structures, see~\citet{romkedar_etal_1990, wiggins_92, haller1998finite, aref_02, jones2002invariant, wiggins2005dynamical, shadden2005definition, froyland_padberg_09, Thi12, FrPa14, KaKe20, KoRe18, HaKaKo18} for geometric and probabilistic approaches to this.
For this study, we focus on the notion of (finite-time) \emph{coherent sets} as introduced by \citet{FrSaMo10} and \cite{froyland2013analytic}, which are defined as sets of particles that are minimally dispersive, or hard for the tracer particles to escape from during a fixed finite time interval.
The methods detecting coherent sets, or even any coherent structures, are relying on dynamical information in the form of trajectories of individual tracer particles or quantities from which these can be obtained, e.g., velocity fields; see~\citet{hadjighasem2017critical} for an overview.
In this work, we consider the case, where the underlying flow map is not accessible and merely densities of a passive (advected) scalar are given at some initial and final time instance.
In a space-discrete version, this amounts to unordered sets of particle positions at the two time instances.
We can think of this situation as forgetting the ``identity'' or ``label'' information of individual particles, i.e., losing almost all information of the full dynamics.
Due to this, anything we can hope to identify are large and robust dynamical structures, like coherent sets. Previous works addressing the task of finding coherent structures from this kind of data are based on optical flow (more precisely, Advection Corrected Correlation Image Velocimetry) and Lagrangian coherent structure analysis~\citep{hadjighasem2016geodesic}, motion segmentation~\citep{almomani2018go}, and ensemble Kalman filter~\citep{santitissadeekorn2019iterative} for image sequences.
Typical applications include cases where the dynamics are only indirectly observed through radar or satellite images of precipitation intensity, see \citet{dwd} for an example, or other quantities.
The fields of computer vision and image processing have long been concerned with the problem of recovering dynamical information (motion) from sequences of images, for an overview see \citet{BPS2014}.
Optical flow and motion segmentation are just two of them, which can be used to build a coherence analysis upon. Our main goal---and contribution---is to show that the theories (and state of the art computation) of \emph{optimal transport} (OT) and of coherent sets are naturally connected.
Fig.~\ref{fig:double_gyre_clustering} shows a segmentation example based on our findings, see Subsection~\ref{sec:Ex4}.
Throughout this paper, we
choose intentionally a level of detail on which this connection can be emphasized in a largely self-contained manner.
\renewcommand\curfolder{img/double_gyre}
\renewcommand\curwidth{0.35\textwidth}
\begin{figure}[tbp]
\centering
\subfloat[Initial time step.]
{\includegraphics[width=\curwidth]{\curfolder/kmeans_0.pdf}}
\hspace*{1cm}
\subfloat[Final time step.]
{\includegraphics[width=\curwidth]{\curfolder/kmeans_500.pdf}}
\caption{Initial and end configuration of a dynamical system consisting of two counter-rotating groups of unlabeled particles.
The whole movement of the system can be seen in the video in the supplementary material.
The color scheme encodes a partition into ``coherent'' sets obtained with our method in Subsection~\ref{sec:Ex4}.}
\label{fig:double_gyre_clustering}
\end{figure}
In a nutshell, we construct Frobenius--Perron operators from transport plans of (unbalanced) regularized optimal transport and use them to find coherent sets in evolving densities or particle ensembles.
Such transport plans can be interpreted as small random perturbations of deterministic maps, naturally introducing a small amount of ``noise'' that is used for the definition of coherence~\citep{froyland2013analytic}.
Moreover, regularized optimal transport is an optimal choice in the sense that it yields a kind of most likely transport plan, given all that we know of the dynamics is how it maps a single distribution from an initial to a final time and that particles move according to Brownian motion.
The outline of this paper is as follows:
In Section~\ref{sec:basics}, we recall basic notions from measure theory, and in Section~\ref{sec:OT}, we introduce properties of OT, regularized OT
and unbalanced regularized OT.
The segmentation model under consideration and the corresponding optimization problem
as well as its relaxation are presented in Section~\ref{sec:segm}.
Section~\ref{sec:frobenius_perron} deals with Frobenius--Perron transfer operators and the relation to OT. In particular, we construct appropriate kernels
of transfer operators in two different ways, namely by i) smoothing of OT transport plans,
and ii) by using regularized OT plans, where the smoothing is already inherent.
Both in i) and ii) the kernels converge to the OT plan if the smoothing parameter
goes to zero.
As we point out in Section~\ref{sec:schroedinger}, the usage of regularized OT plans can be motivated from a statistical physics perspective.
To this end, we give accessible insights, namely just for two time steps, for the Schr\"odinger question.
We outline a discrete numerical approach in Section~\ref{sec:discrete}.
Various proof-of-concept examples are presented in Section~\ref{sec:numerics}.
Finally, in Section~\ref{sec:conclusions}, conclusions are drawn.
\section{Preliminaries} \label{sec:basics}
Throughout this paper, let $\mathbb{X},\mathbb{Y} \subset \mathbb R^d$ be compact sets equipped with the Borel $\sigma$-algebras $\mathcal{B}(\mathbb{X})$, $\mathcal{B}(\mathbb{Y})$ induced by the subspace topology, respectively.
Assume that the boundaries have Lebesgue measure (denoted by $\lambda$) zero and that the sets fulfill $\min_{x \in \mathbb{X}}\lambda(B_\varepsilon(x)), \min_{y \in \mathbb{Y}}\lambda(B_\varepsilon(y)) > 0$ for every $\varepsilon > 0$ and
\begin{equation}\label{eq:boundary}
\sup_{\varepsilon > 0} \frac{\max_{x \in \mathbb{X}}\lambda(B_\varepsilon(x))}{\min_{x \in \mathbb{X}}\lambda(B_\varepsilon(x))} < C_\mathbb{X},
\qquad
\sup_{\varepsilon > 0} \frac{\max_{y \in \mathbb{Y}}\lambda(B_\varepsilon(y))}{\min_{y \in \mathbb{Y}}\lambda(B_\varepsilon(y))} < C_\mathbb{Y},
\end{equation}
for some $C_\mathbb{X}, C_\mathbb{Y} >0$,
with balls $B_\varepsilon(x) \coloneqq \{x' \in \mathbb{X}: \|x'-x\|_2 < \varepsilon \}$ in the appropriate spaces.
This condition holds for domains satisfying the uniform cone condition, see \citet{AdamsFournier03}, which is fulfilled if $\mathbb{X}$ and $\mathbb{Y}$ have Lipschitz boundaries.
By $\mathcal M(\mathbb{X})$ we denote the linear space of all finite signed Borel measures on $\mathbb{X}$,
by $\mathcal M^+(\mathbb{X})$ the subset of non-negative measures,
and by $\mathcal{P}(\mathbb{X})$ the set of Borel probability measures on $\mathbb{X}$.
The closed set $\textnormal{supp}(\mu) \coloneqq \{ x \in \mathbb{X}: B \subset \mathbb{X} \text{ open, }x \in B \implies \mu(B) >0\}$ is called the \emph{support} of a measure $\mu$.
Further, the \emph{total variation} measure of $\mu \in \mathcal M(\mathbb{X})$ is defined by
\[
|\mu|(B) \coloneqq \sup \Bigl\{ \sum_{k=1}^\infty |\mu(B_k)|:
\bigcup\limits_{k=1}^\infty B_k = B, \, B_k \; \mbox{pairwise disjoint}\Bigr\}.
\]
Equipped with the norm $\| \mu\|_{\mathcal M} = |\mu|(\mathbb{X})$ the space $\mathcal M(\mathbb{X})$ becomes a Banach space.
By $C(\mathbb{X})$ we denote the Banach space of continuous, real-valued functions on
$\mathbb{X}$ with norm $\| \varphi\|_{C(\mathbb{X})} \coloneqq \max_{x \in \mathbb{X}} |\varphi(x)|$.
The space $\mathcal M(\mathbb{X})$ can be identified via Riesz' representation theorem with the dual space of $C(\mathbb{X})$
and
the weak-$\ast$ topology on $\mathcal M(\mathbb{X})$ gives rise to the \emph{weak convergence of measures}, i.e., a sequence $\{\mu_k\}_{k \in \ensuremath{\mathbb{N}}} \subset \mathcal M(\mathbb{X})$ converges \emph{weakly} to $\mu$ and we write $\mu_k \weakly \mu$, if
\begin{equation}
\lim_{k \to \infty} \int_{\mathbb{X}} f \,\mathrm{d} \mu_k = \int_{\mathbb{X}} f \,\mathrm{d} \mu \qquad \text{for all } f \in C(\mathbb{X}).
\end{equation}
Note that the set $\mathcal{P}(\mathbb{X})$ is weakly compact.
For a non-negative, finite measure $\mu$ and $p \in [1,\infty)$, let $L_p(\mathbb{X},\mu)$
be the Banach space (of equivalence classes) of complex-valued functions with norm
\[\|f\|_{L_p(\mathbb{X},\mu)} = \Bigl( \int_\mathbb{X} |f|^p \,\mathrm{d} \mu \Bigr)^\frac1p < \infty.\]
For the Hilbert space $L_2(\mathbb{X},\mu)$ we use the notation
$\langle f,g \rangle_\mu \coloneqq \int_\mathbb{X} f g \,\mathrm{d} \mu$.
By $1_A$ we denote the characteristic function of a (measurable) set $A$, defined by
\[
1_A(x)\coloneqq
\begin{cases}
1 & x\in A \\
0 & x\not\in A.
\end{cases}
\]
Let $\mathcal A \subseteq \mathcal B(\mathbb{X})$ be a sub-$\sigma$-algebra.
A mapping $g\colon \mathbb{X} \to \ensuremath{\mathbb{R}}$ is called \emph{conditional expectation}
of $f\in L_1(\mathbb{X}, \mu)$ if $g$ is $\mathcal A$-measurable and for all $A \in \mathcal A$ it holds
\begin{equation} \label{cond_exp}
\int_A g \,\mathrm{d} \mu = E(1_A g) = E(1_A f) = \int_A f \,\mathrm{d} \mu.
\end{equation}
In this case, we write $g = E(f|\mathcal A)$.
A measure $\nu \in \mathcal M(\mathbb{X})$ is \emph{absolutely continuous} with respect to $\mu \in \mathcal M(\mathbb{X})$,
abbreviated by $\nu \ll \mu$,
if for every $A \in \mathcal B(\mathbb{X})$ with $\mu(A) = 0$ we have $\nu(A) = 0$.
If $\mu, \nu \in \mathcal M^+(\mathbb{X})$ satisfy $\nu \ll \mu$, then the \emph{Radon--Nikodym derivative} $\tfrac{\,\mathrm{d} \nu}{\,\mathrm{d} \mu}= \sigma_\nu \in L_1(\mathbb{X},\nu)$ exists and $\nu = \sigma_\nu \mu$.
Let $T\colon \mathbb{X} \to \mathbb{Y}$ be a measurable function, i.e., $T^{-1}(A) \in \mathcal{B}(\mathbb{X})$ for all $A \in \mathcal{B}(\mathbb{Y})$.
Then, the \emph{push-forward} measure of $\mu$ by $T$ is defined as $T_\# \mu \coloneqq \mu \circ T^{-1}$.
A measurable function $f$ on $\mathbb{Y}$ is integrable with respect to $\nu \coloneqq T_\# \mu$
if and only if the composition $f \circ T$ is integrable with respect to the measure $\mu$.
In this case, the integrals coincide, i.e., it holds
\begin{align}\label{eq:push_f}
\int_\mathbb{Y} f \,\mathrm{d} \nu &= \int_{\mathbb{X}} f\circ T \,\mathrm{d} \mu.
\end{align}
For the Lebesgue measure $\lambda$, we abbreviate $\,\mathrm{d} x$ instead of $\,\mathrm{d} \lambda(x)$
throughout the paper.
\section{Optimal Transport and its Regularization} \label{sec:OT}
In this section, we collect results on OT and its regularized version by the Kullback--Leibler divergence, which we couple with so-called Frobenius--Perron operators in order to segment images
in the subsequent sections.
Moreover, we describe unbalanced OT, which appears to be useful in our numerical examples.
The following discussion is based on \citet{CP2019} and \cite{S2015}.
\paragraph{Optimal Transport}
For a non-negative, symmetric and Lipschitz continuous cost function $c \in C(\mathbb{X} \times \mathbb{Y})$
and given measures $\mu\in \mathcal P(\mathbb{X}) ,\nu \in \mathcal P(\mathbb{Y})$, the \emph{Monge problem of optimal transport} consists
in finding a measurable function $\hat T\colon \mathbb{X} \to \mathbb{Y}$, called \emph{optimal transport map}, that realizes
\begin{equation}
\inf_{T} \Bigl\{ \int_{\mathbb{X}} c\bigl(x,T(x)\bigr) \,\mathrm{d} \mu(x) \colon \; T_{\#}\mu = \nu\Bigr\}.
\end{equation}
If a map $T\colon \mathbb{X} \to \mathbb{Y}$ solely fulfills $T_{\#}\mu = \nu$, we call it a \emph{transport map}
between $\mu$ and $\nu$.
In contrast to Monge's problem, \emph{Kantorovich's relaxation} allows the mass to be split, i.e., it aims to find a minimizer of
\begin{equation}\label{Monge_Kantorovich_problem}
\OT(\mu,\nu) \coloneqq \inf_{\pi \in\Pi(\mu,\nu)} \int_{\mathbb{X} \times \mathbb{Y}} c(x,y) \,\mathrm{d} \pi(x,y),
\end{equation}
where $\Pi(\mu,\nu)$
denotes the set of all joint probability measures $\pi$ on $\mathbb{X} \times \mathbb{Y}$ with marginals $\mu$ and $\nu$.
We refer to the measures of $\Pi(\mu,\nu)$ as \emph{transport plans}
between $\mu$ and $\nu$.
In our setting, the OT functional $\pi \mapsto \int_{\mathbb{X} \times \mathbb{Y}} c \,\mathrm{d} \pi$ is continuous and
\eqref{Monge_Kantorovich_problem} has a solution $\hat \pi$, called \emph{optimal transport plan}.
Every transport map $T\colon \mathbb{X} \to \mathbb{Y}$ between $\mu$ and $\nu$ induces a transport plan
$\pi = (\id_\mathbb{X},T)_{\#}\mu \in \Pi(\mu,\nu)$, i.e.,
$$
\int_{\mathbb{X} \times \mathbb{Y}} h(x,y) \,\mathrm{d} \pi(x,y) = \int_\mathbb{X} h\bigl(x,T(x)\bigr) \,\mathrm{d} \mu(x) \quad \text{for all} \, h \in C(\mathbb{X} \times \mathbb{Y}).
$$
Further, the \emph{$c$-transform} $\varphi^{c} \in C(\mathbb{Y})$ of $\varphi \in C(\mathbb{X})$ is defined as $\varphi^{c}(y) = \min_{x\in\mathbb{X}}\{c(x,y)-\varphi(x)\}$
and a function $\varphi^{c} \in C(\mathbb{Y})$ is called \emph{$c$-concave} if it is the $c$-transform of some function $\varphi \in C(\mathbb{X})$.
The dual formulation of the OT problem \eqref{Monge_Kantorovich_problem} reads
\begin{equation}\label{Wdual}
\OT(\mu,\nu) = \max_{ \substack{(\varphi,\psi)\in C(\mathbb{X})\times C(\mathbb{Y})\\
\varphi(x)+\psi(y)\leq c(x,y)}}
\int_{\mathbb{X}}\varphi \,\mathrm{d} \mu + \int_{\mathbb{Y}} \psi \,\mathrm{d} \nu.
\end{equation}
Maximizing pairs are of the form $(\varphi,\psi) = (\hat \varphi, \hat \varphi^c)$ for a $c$-concave function
$\hat \varphi$ and fulfill $\hat \varphi(x) + \hat \varphi^c(y) = c(x,y)$ in $\textnormal{supp}(\hat \pi)$, where $\hat \pi$ is any optimal transport plan.
The function $\hat \varphi$ is called
(Kantorovich) \emph{potential} for the couple $(\mu,\nu)$.
In our applications, we want to focus on settings, where an optimal transport map exists. The following theorem can be found, e.g., in \citet[Thm.~1.17]{S2015}.
\begin{theorem}\label{thm:tmap}
Let $\mu,\nu\in \mathcal P (\mathbb{X})$, where $\mu$ is absolutely continuous
with respect to the Lebesgue measure and let $c(x,y)=h(x-y)$ with a strictly convex function $h$.
Then, there exists a unique optimal transport plan $\hat \pi\in \Pi(\mu,\nu)$ that is induced by the optimal transport map $\hat T$.
Moreover, there exists a Kantorovich potential $\hat \phi$ which is linked to $\hat T$ via
$\hat T(x) =x-(\nabla h)^{-1}(\nabla \hat \phi(x))$.
\end{theorem}
For $c(x,y) \coloneqq \|x-y\|_p^p$, $p \in [1,\infty)$, the optimal transport cost induces the \emph{$p$-Wasserstein distance}
\begin{align} \label{eq:OTprimal}
W_p(\mu,\nu) \coloneqq \OT(\mu,\nu)^\frac{1}{p}
= \biggl( \min_{\pi \in \Pi(\mu,\nu)}
\int_{\mathbb{X}^2} \| x- y\|_p^p \mathrm{d} \pi(x,y) \biggr)^\frac{1}{p},
\end{align}
which metrizes the weak topology on $\mathcal P(\mathbb{X})$.
Indeed, due to boundedness of $\mathbb{X}$, we have that $\mu_k \weakly \mu$
if and only if $\lim_{k \to \infty} W_p (\mu_k, \mu) = 0$.
In our numerical examples, we use the Wasserstein-2 distance.
By Theorem~\ref{thm:tmap}, we have in particular for
$c(x,y) \coloneqq \tfrac{1}{2}\norm{x-y}_{2}^{2}$
that
$\hat T(x)=x-\nabla \hat \phi(x)$.
\paragraph{Regularized Optimal Transport}
We recall that the \emph{Kullback--Leibler divergence} $\mathrm{KL}\colon{\mathcal M^+}(\mathbb{X}) \times {\mathcal M^+}(\mathbb{X}) \rightarrow \mathbb [0, +\infty]$ is defined for $\mu \ll \nu$ by
\begin{equation} \label{KLdef}
\mathrm{KL} (\mu,\nu) \coloneqq
\int_{\mathbb{X}} \log\Bigl(\frac{\,\mathrm{d} \mu}{\,\mathrm{d} \nu} \Bigr) \, \,\mathrm{d} \mu + \nu(\mathbb{X}) - \mu(\mathbb{X}) ,
\end{equation}
and by $\mathrm{KL} (\mu,\nu) \coloneqq + \infty$ otherwise.
For $\mu,\nu \in \mathcal{P}(\mathbb{X})$ the last two summands in \eqref{KLdef} cancel each other.
The \emph{Kullback--Leibler divergence} is strictly convex and weakly lower-semicontinuous with respect to the first variable.
For $ \mu \in {\mathcal P}(\mathbb{X})$, $ \nu \in \mathcal P(\mathbb{Y})$ and $\varepsilon > 0$, the \emph{regularized OT} problem is defined as
\begin{align}
\OT_{\varepsilon}(\mu,\nu)
\coloneqq&
\min_{\pi \in \Pi(\mu,\nu)}
\,
\Big\{
\int_{\mathbb{X} \times \mathbb{Y}} c\, \mathrm{d} \pi
+ \varepsilon \mathrm{KL} (\pi,\mu \otimes \nu) \Big\}\label{sinkhorn}\\
=&\; \varepsilon
\min_{\pi \in \Pi(\mu,\nu)}
\, \mathrm{KL} (\pi, \exp(-c/\varepsilon) \mu \otimes \nu) - \varepsilon \int_{\mathbb{X} \times \mathbb{Y}} \exp(-c/\varepsilon) -1 \,\mathrm{d} (\mu \otimes \nu).\label{sinkhorn_kernel}
\end{align}
A dual formulation is given in the next theorem, see \citet{CP2019,neumayer2020optimal}.
\begin{theorem}\label{prop:dual}
The (pre-)dual problem of $\OT_\varepsilon$ is given by
\begin{align}
\max_{(\varphi,\psi)\in C(\mathbb{X}) \times C(\mathbb{Y})}
\Big\{ \int_{\mathbb{X}}\varphi \,\mathrm{d} \mu + \int_{\mathbb{Y}} \psi \,\mathrm{d} \nu
- \varepsilon \int_{\mathbb{X} \times \mathbb{Y}} \exp\Bigl(\frac{\varphi(x) + \psi(y) - c(x,y)}{\varepsilon}\Bigr) -1
\,\mathrm{d} (\mu \otimes \nu) \Big\}.\label{pre-dual}
\end{align}
The optimal potentials $\hat \varphi_\varepsilon \in C(\mathbb{X})$, $\hat \psi_\varepsilon \in C(\mathbb{Y})$
exist and are unique on $\textnormal{supp}(\mu)$ and $\textnormal{supp}(\nu)$, respectively (up to an additive constant).
They are related to the optimal transport plan $\hat \pi_\varepsilon$ by
\begin{equation}\label{eq:PDrelation}
\hat\pi_\varepsilon = \exp\Bigl(\frac{\hat \varphi_\varepsilon(x) + \hat \psi_\varepsilon(y) - c(x,y)}{\varepsilon}\Bigr) \mu \otimes \nu.
\end{equation}
\end{theorem}
Under certain assumptions, see Proposition \ref{prop:conv-reg}, the regularized OT plans $\hat \pi_\varepsilon$ converge weakly to the OT plan $\hat \pi$ as $\varepsilon$ goes to zero.
The corresponding discrete optimization problem can be solved efficiently using Sinkhorn's algorithm, see \citet{C2013,FSVATP2019} for more details.
\paragraph{Unbalanced Optimal Transport}
In practical applications, we often have to deal with noisy data.
In this case, the computation of a (regularized) optimal transport plan is often unreasonable. e.g., if $\mu, \nu$ are positive measures such that $\mu(\mathbb{X})\neq \nu(\mathbb{Y})$.
To resolve this issue, we can use unbalanced regularized optimal transport, which allows small deviations between the input measures and the marginals of the associated transport plan.
This approach corresponds to the optimization problem
\begin{equation}\label{eq:unbalanced_entropic_ot}
\min_{\pi\in \mathcal{M}^+(\mathbb{X} \times \mathbb{Y})} \left\{ \int_{\mathbb{X}\times \mathbb{Y}} c\,\mathrm{d} \pi + \varepsilon \text{KL}(\pi\vert \mu\otimes\nu)
+ \kappa \bigl(\text{KL}(\pi_0 \vert \mu) + \text{KL}(\pi_1\vert \nu)\bigr)\right\},
\end{equation}
where $\pi_0$ and $\pi_1$ denote the marginals with respect to the first and second component, respectively, see \citet{CPSV18}.
Similarly as in the balanced case, we can rewrite the problem in dual form
\begin{align}
\max_{(\varphi,\psi)\in C(\mathbb{X}) \times C(\mathbb{Y})} \left\{\vphantom{\int_\mathbb{X}}\right.&\kappa \biggl(\int_\mathbb{X} 1 - \exp\bigl(-\tfrac{\varphi}{\kappa}\bigr) \,\mathrm{d} \mu +\int_\mathbb{Y} 1- \exp\bigl(-\tfrac{\psi}{\kappa}\bigr)\,\mathrm{d}\nu\biggr)\\ &- \left.\varepsilon\int_{\mathbb{X}\times\mathbb{Y}} \exp\bigl(\tfrac{\varphi(x) + \psi(y) -c(x, y)}{\varepsilon}\bigr) - 1 \,\mathrm{d} (\mu \otimes \nu)\right\}
\end{align}
and the optimal solution $\hat \pi_{\varepsilon,\kappa}$ of the primal problem is related to the optimal solutions $\hat \varphi_{\varepsilon,\kappa}, \hat \psi_{\varepsilon,\kappa}$ of the dual problem via
\begin{equation} \label{eq:ot-unbalanced}
\hat \pi_{\varepsilon,\kappa} = \exp\bigl(\tfrac{\hat \varphi_{\varepsilon,\kappa}(x) + \hat \psi_{\varepsilon,\kappa}(y) -c(x,y)}{\varepsilon}\bigr)\mu \otimes \nu.
\end{equation}
Further, the formal limit $\kappa \to \infty$ results in the original balanced problem.
Note that the OT plan $\hat \pi_{\varepsilon,\kappa}$ does not have marginals $\mu$ and $\nu$.
However, if $\kappa$ is large, the marginals $\mu_\kappa$ and $\nu_\kappa$ of $\hat \pi_{\varepsilon,\kappa}$
are close to the original input measures and in some sense we can interpret them as smoothed versions of those.
Similarly as for regularized OT, we can use a variant of the Sinkhorn algorithm to efficiently solve the corresponding discrete problem, see \citet{SFVTP19}.
\section{Segmentation Model}\label{sec:segm}
Following the lines of \citet{froyland2013analytic}, we introduce the segmentation model that we want to apply for coherent structure detection.
More precisely, we use the concept of \emph{transfer operators}, which are rigorously introduced in Section~\ref{sec:frobenius_perron}.
It is sufficient to note that the transfer operator $L$ below is the adjoint of the pullback operator $f\mapsto f\circ T$ as an operator between the respective $L^p$ spaces.
Further, it is a functional extension of the push-forward operator $T_{\#}$ from Section~\ref{sec:basics}.
Assume that we are given full information about some dynamical system by means of its linear, bounded transfer operator
$L\colon L_2(\mathbb{X},\mu) \to L_2(\mathbb{Y},\nu)$. Let $\dot\cup$ denote the disjoint union of sets.
We aim to find measurable partitions $\mathbb{X}= X_1\dot\cup X_2$ and $\mathbb{Y}= Y_1\dot\cup Y_2$ so that it holds for $k=1,2$ that
\begin{itemize}
\item $L 1_{X_k} = 1_{Y_k}$ (\emph{coherence}) and
\item $\mu(X_k) = \nu(Y_k)$ (\emph{mass conservation}).
\end{itemize}
Note that the first condition readily implies $L 1_{\mathbb{X}} = 1_{\mathbb{Y}}$.
Based on these conditions, a natural model in order to find the partitions would be
\begin{align}\label{eq:coh_set_problem}
\max_{\substack{X_1\dot\cup X_2=\mathbb{X},\, Y_1\dot\cup Y_2=\mathbb{Y}}} \biggl\{\frac{\langle L 1_{X_1}, 1_{Y_1} \rangle_\nu}{\mu(X_1)}
+ \frac{\langle L 1_{X_2}, 1_{Y_2}\rangle_\nu }{\mu(X_2)} \biggr\},
\end{align}
trying to optimize the alignment of $L 1_{X_k}$ and $1_{Y_k}$ in $\mathbb{Y}$, $k=1, 2$. The expression $\langle L \frac{1_{X_k}}{\mu(X_k)}, 1_{Y_1} \rangle_\nu$ quantifies the probability that a $\mu$-distributed random initial condition, being in $X_k$, is mapped by the dynamics into $Y_k$.
Using standard arguments, see \citet{froyland2013analytic} or also \citet{Luxburg2007} in connection with
graph cuts, problem~\eqref{eq:coh_set_problem} can be rewritten as
\begin{align}\label{eq:coh_set_problem_1}
1 + \max_{\substack{X_1\dot\cup X_2 = \mathbb{X},\, Y_1\dot\cup Y_2=\mathbb{Y}}} \langle L\psi_{X_1, X_2}, \psi_{Y_1, Y_2} \rangle_\nu,
\end{align}
where
$\psi_{X_1, X_2} \coloneqq \sqrt{\tfrac{\mu(X_2)}{\mu(X_1)}} 1_{X_1} - \sqrt{\tfrac{\mu(X_1)}{\mu(X_2)}} 1_{X_2}$ and
$\psi_{Y_1, Y_2} \coloneqq \sqrt{\tfrac{\nu(Y_2)}{\nu(Y_1)}} 1_{Y_1} - \sqrt{\tfrac{\nu(Y_1)}{\nu(Y_2)}} 1_{Y_2}$.
These functions fulfill
\[
\|\psi_{X_1, X_2}\|_{L_2(\mathbb{X},\mu)} = \mu (\mathbb{X})^\frac12, \quad
\|\psi_{Y_1, Y_2}\|_{L_2(\mathbb{Y},\nu)} = \nu (\mathbb{Y})^\frac12, \quad
\langle \psi_{X_1, X_2}, 1_\mathbb{X} \rangle_\mu = \langle \psi_{Y_1, Y_2}, 1_\mathbb{Y} \rangle_\nu = 0.
\]
Clearly, if we could find $\psi_{X_1, X_2}$ with corresponding $\psi_{Y_1, Y_2}$, the desired partition can be obtained by thresholding these functions at zero.
Unfortunately, problem \eqref{eq:coh_set_problem} or equivalently \eqref{eq:coh_set_problem_1} is hard to solve,
indeed NP-hard in its finite dimensional variant.
Instead, we relax the problem by replacing
$\psi_{X_1, X_2}$ with $f/\|f\|_{L_2(\mathbb{X},\mu)} \in L_2(\mathbb{X},\mu)$ and
$\psi_{Y_1, Y_2}$ with $g/\|g\|_{L_2(\mathbb{Y},\nu)} \in L_2(\mathbb{X},\nu)$.
Then, the relaxed problem
\begin{align} \label{problem}
\max_{
\substack{
(f,g) \in L_2(\mathbb{X},\mu) \times L_2(\mathbb{Y},\nu)
} }
\biggl\{
\frac{ \langle L f, g \rangle_\nu}{ \Vert f\Vert_{L_2(\mathbb{X},\mu)} \Vert g \Vert_{L_2(\mathbb{Y},\nu)} }:
\langle f, 1_\mathbb{X} \rangle_\mu = \langle g, 1_\mathbb{Y} \rangle_\nu = 0
\biggr\}
\end{align}
allows for fuzzy instead of hard assignments.
In the following, we set ${\mathcal L} \coloneqq L^* L$ and assume that
\begin{enumerate}[label=(A\arabic*)]
\item\label{enum:op_marg} $L 1_\mathbb{X} = 1_\mathbb{Y}$;
\item\label{enum:adjoint_marg} $\mathcal L 1_\mathbb{X} = \|L\|^2 1_\mathbb{X}$, which by \ref{enum:op_marg} is equivalent to $L^* 1_\mathbb{Y} = \|L\|^2 1_\mathbb{X}$, where we abbreviate
$\|L\| \coloneqq \|L\|_{L_2(\mathbb{X},\mu)\to L_2(\mathbb{Y},\nu)}$ for the operator norm;
\item\label{enum:operator_cpt_simple} $L$ is compact and the largest eigenvalue $\lambda_1 = \|L\|^2$ of ${\mathcal L} = L^* L$ is simple.
\end{enumerate}
Then, problem \eqref{problem} is equivalent to finding the second largest singular value of $L$
and the corresponding left and right singular vectors. This can be seen as follows:
The compact, self-adjoint, positive operator $\mathcal L$
has a non-negative spectrum, where the countable set of eigenvalues
$\lambda_1 > \lambda_2 \ge \ldots \ge 0$ of $\mathcal L$
fulfills the Courant's minimax principle \citep[Thm.~4, p.~212]{BS1986},
\[
\lambda_k = \min_{V: \mathrm{codim} V \le k-1} \max_{0\not = f \in V}
\frac{\langle \mathcal{L} f,f \rangle_\mu}{\langle f,f \rangle_\mu} =\min_{V: \mathrm{codim} V \le k-1} \max_{0\not = f \in V} \frac{ \| L f\|^2_{L_2(\mathbb{Y},\nu)} }{ \|f\|^2_{L_2(\mathbb{X},\mu)} }, \quad k \in \mathbb N.
\]
Using the definition of the norm, this can be rewritten as
\begin{equation}\label{courant}
\sigma_k \coloneqq \lambda_k^{\frac12}
=
\min_{V: \mathrm{codim} V \le k-1} \max_{ \mystack{0\not = f \in V}{0 \not = g \in L_2(\mathbb{Y},\nu)} }
\frac{ \langle L f,g \rangle_\nu }{ \|f\|_{L_2(\mathbb{X},\mu)} \|g\|_{L_2(\mathbb{Y},\nu)}}, \quad k \in \mathbb N.
\end{equation}
The values $\sigma_k$ are the non-zero singular values of $L$, and the maximizing functions $f$ and $g$ are so-called
left and right singular vectors of $L$ belonging to $\sigma_k$,
respectively. Note that the left singular vectors are the eigenvectors of $\mathcal L$ and
right singular vectors are given by $g = Lf$.
By assumptions~\ref{enum:adjoint_marg} and \ref{enum:operator_cpt_simple}, $f = 1_\mathbb{X}$ is an eigenvector of $\mathcal L$ belonging to the simple, largest eigenvalue $\lambda_1$
so that the eigenvectors belonging to smaller eigenvalues are perpendicular to~$1_\mathbb{X}$.
Thus,
\eqref{courant} becomes
\begin{equation}\label{courant_1}
\sigma_2
=
\max_{ \mystack{0\neq f \in L_2(\mathbb{X}, \mu)}{0 \neq g \in L_2(\mathbb{Y},\nu)} }
\biggl\{ \frac{ \langle L f,g \rangle_\nu }{ \|f\|_{L_2(\mathbb{X},\mu)} \|g\|_{L_2(\mathbb{Y},\nu)} }:
\langle f, 1_\mathbb{X} \rangle_\mu = \langle g, 1_\mathbb{Y} \rangle_\nu = 0 \biggr\},
\end{equation}
where the orthogonality condition on $g$ can be added since $f\perp 1_\mathbb{X}$ implies $Lf\perp 1_\mathbb{Y}$:
To summarize, if $L\colon L_2(\mathbb{X},\mu) \to L_2(\mathbb{Y},\nu)$
fulfills
the assumptions~\ref{enum:op_marg}--\ref{enum:operator_cpt_simple}, then model~\eqref{problem} can be solved by finding the left and right singular functions of $L$ belonging to the second largest singular value of $L$.
Later, we focus on compact operators arising from non-negative kernels $K\in L_2(\mathbb{X}\times \mathbb{Y},\mu\otimes\nu)$ via
$$
(L f)(y) \coloneqq \int_\mathbb{X} f(x) K(x, y) d\mu (x), \quad (L^* g)(x) \coloneqq \int_\mathbb{Y} g(y) K(x, y) d\nu (y).
$$
Then, \ref{enum:op_marg}--\ref{enum:operator_cpt_simple} are met if the following corresponding assumptions hold for $K$:
\begin{enumerate}[label=(K\arabic*)]
\item\label{enum:nu_marg} $\int_\mathbb{X} K(x, \cdot) d\mu(x) = 1_\mathbb{Y}\quad\nu$-a.e.,
\item \label{enum:mu_marg}$\int_\mathbb{Y} K(\cdot, y) d\nu(y) = \|L\|^2 \, 1_\mathbb{X}\quad \mu$-a.e.
\item \label{enum:cpt_simple} The largest eigenvalue of $L^*L$ is simple.
\end{enumerate}
\begin{remark}[Relation to graph cut segmentation]
Partitioning data using the eigenvector corresponding to the second largest eigenvalue of the so-called graph Laplacian operator, which is self-adjoint for undirected graphs,
is a frequently applied method, e.g., in image processing.
As illustrated by Fig.~\ref{fig:double_gyre_clustering} in the introduction, our approach is related to dynamical data sets,
and we aim to partition the sets simultaneously at different times using the singular vector pairs
of a not necessarily self-adjoint transfer operator, whose construction is addressed in the next sections.
\end{remark}
\section{Frobenius--Perron Operators}\label{sec:frobenius_perron}
In this section, we consider the so-called Frobenius--Perron operator $L=P_T$ of a measurable map $T\colon \mathbb{X} \to \mathbb{Y}$.
In Subsection \ref{sec:frobenius_perron_1}, we will see that this transfer operator $L$
fulfills our assumptions \ref{enum:op_marg} and \ref{enum:adjoint_marg} provided that the operator $T$
is a transport map between measures $\mu \in \mathcal P(\mathbb{X})$ and $\nu \in \mathcal P(\mathbb{Y})$.
Unfortunately, the resulting transfer operator will be useless for our segmentation task since it has only singular values $1$ and thus it violates \ref{enum:operator_cpt_simple}.
As a possible solution, we provide a kernel-based definition of the Frobenius--Perron operator and propose two approaches for smoothing the transport maps.
The first technique is suggested in Subsection \ref{sec:frobenius_perron_2}.
It adds a small random perturbation to the kernel of the operator, an idea that goes back to \citet{froyland2013analytic}.
The second approach, which we will favor in our numerical examples, uses regularized optimal transport plans.
\subsection{Frobenius--Perron Operators and Transport Maps}\label{sec:frobenius_perron_1}
For completeness, we introduce the Frobenius--Perron operator between two Lebesgue spaces $L^p(\mathbb{X},\mu)$ and~$L^p(\mathbb{Y},\nu)$. Analogous results for $\mathbb{X}=\mathbb{Y}$ and $\mu = \nu$ can be found, e.g., in~\citet[Chap.~4]{boyarsky1997laws} or~\citet{brin2002introduction,LasotaAndrzej1994CFaN}.
Assume that $T\colon \mathbb{X} \to \mathbb{Y}$ is \emph{non-singular} with respect to $\mu \in \mathcal{P}(\mathbb{X})$ and $\nu\in \mathcal{P}(\mathbb{Y})$,
namely that $\nu(A) = 0$ implies $\mu\bigl(T^{-1} (A)\bigr) = 0$ for all $A \in \mathcal{B}(\mathbb{Y})$.
It is immediately clear that a transport map $T$ is non-singular with respect to $\mu$ and $\nu$.
For a non-singular, measurable map $T\colon \mathbb{X} \to \mathbb{Y}$, the linear operator
$P_T\colon L_1(\mathbb{X},\mu) \to L_1(\mathbb{Y},\nu)$
is called \emph{Frobenius--Perron operator} of $T$ if it satisfies
\begin{equation}\label{eq:PerFro}
\int_A P_T \psi \,\mathrm{d} \nu = \int_{T^{-1} (A)} \psi \,\mathrm{d} \mu \quad \mathrm{for \; all} \quad A \in \mathcal{B}(\mathbb{Y}) .
\end{equation}
In other words, the function $P_T \psi \in L_1(\mathbb{Y}, \nu)$ is given for each $\psi \in L_1(\mathbb{X},\mu)$ as the density of $\smash{T_{\#}(\psi\mu) = \int_{T^{-1}(\cdot)}}\psi\,\mathrm{d} \mu$ with respect to $\nu$, which exists because
\[
\nu(A) = 0 \implies \mu(T^{-1}(A)) = 0 \implies (T_{\#}\psi\mu)(A) = 0 \quad \mathrm{for \; all} \quad A \in \mathcal{B}(\mathbb{Y}),
\]
i.e., $T_{\#}(\psi\mu) \ll \nu$.
In particular, integrals are preserved $\int_\mathbb{Y} P_T \psi \,\mathrm{d} \nu = \int_{\mathbb{X}} \psi \,\mathrm{d} \mu$.
Note that often the Frobenius--Perron operator is defined as an operator just mapping from $L_1(\mathbb{X},\mu)$ to itself, see \citet{LasotaAndrzej1994CFaN}.
As $T$ is non-singular, the linear operator
$U_T \colon L_\infty(\mathbb{Y},\nu) \to L_\infty(\mathbb{X}, \mu)$ given by
\begin{equation} \label{eq:koopmann}
U_T \psi = \psi \circ T
\end{equation}
is also well-defined.
It is known as \emph{Koopman operator} and the adjoint of $P_T$.
\begin{remark}
\citet{froyland2013analytic} defined the Frobenius--Perron operator $P\colon L_1(\mathbb{X}, \lambda) \to L_1(\mathbb{Y}, \lambda)$
for a map $T$ that is non-singular
with respect to the Lebesgue measure, i.e.,
\begin{equation}\label{eq:PerFro_leb}
\int_A (P \psi) (y) \,\mathrm{d} y
= \int_{T^{-1} (A)} \psi(x) \,\mathrm{d} x
\quad \mathrm{for \; all} \quad A \in \mathcal{B}(\mathbb{Y}).
\end{equation}
For $\mu$ and $\nu$ being absolutely continuous with respect to the Lebesgue measure
with densities $\sigma_\mu$ and $\sigma_\nu$, respectively, a transfer operator $L\colon L_1(\mathbb{X}, \mu) \to L_1(\mathbb{Y}, \nu)$ was determined by $L\psi \coloneqq P(\psi \sigma_\mu)/\sigma_\nu$.
This implies for all $A \in \mathcal{B}(\mathbb{Y})$ that
$$
\int_A L \psi (y) \,\mathrm{d} \nu(y) = \int_A P( \psi \sigma_\mu)(y) \,\mathrm{d} y
=
\int_{T^{-1}(A)} \psi(x) \sigma_\mu(x) \,\mathrm{d} x
= \int_{T^{-1}(A)} \psi(x) \,\mathrm{d} \mu(x).
$$
Comparing this with \eqref{eq:PerFro}, we see that for this special case our Frobenius--Perron operator coincides with the above transfer operator
$P_T = L$.
\end{remark}
If $T$ is a transport map between $\mu$ and $\nu$, the following proposition shows that $P_T$ has the properties \ref{enum:op_marg} and \ref{enum:adjoint_marg},
but unfortunately only singular values $1$, thus violating \ref{enum:operator_cpt_simple}.
\begin{proposition}\label{lem:id}
Let a measurable map $T\colon \mathbb{X} \to \mathbb{Y}$ fulfill $\nu = T_\# \mu$ and let $P_T$, $U_T$ be defined by \eqref{eq:PerFro} and \eqref{eq:koopmann}, respectively.
Then, the following holds true:
\begin{itemize}
\item[i)] For any $\psi \in L_1(\mathbb{X},\mu)$ we have $U_T \circ P_T (\psi) = E(\psi|T^{-1}(\mathcal B(\mathbb{Y})))$, which is the conditional expectation of $\psi$ with respect to the $\sigma$-algebra $T^{-1}(\mathcal B(\mathbb{Y}))$.
If $T\colon \mathbb{X} \to \mathbb{Y}$ maps Borel sets of $\mathbb{X}$ to those of $\mathbb{Y}$ and is also $\mu$-essentially injective, i.e., there exists a measurable set $B$ with $\mu(B) = 1$ on which $T$ is injective, this simplifies to $U_T \circ P_T = \text{Id}$.
\item[ii)] For any $p \in [1, \infty]$, the Frobenius--Perron operator, as well as its adjoint,
can be restricted to $P_T\colon L_p(\mathbb{X},\mu) \to L_p(\mathbb{Y},\nu)$ and $U_T\colon L_p(\mathbb{Y},\nu)\to L_p(\mathbb{X},\mu)$, respectively.
\item[iii)] The operators satisfy $P_T 1_\mathbb{X} = 1_\mathbb{Y}$ and $U_T 1_\mathbb{Y} = 1_\mathbb{X}$.
\end{itemize}
\end{proposition}
\begin{proof}
i) Recall the definition of conditional expectations in \eqref{cond_exp}.
We have for any $A \in \mathcal B(\mathbb{Y})$ that
\begin{align*}
\int_{T^{-1}(A)} \!\psi \,\mathrm{d} \mu {\stackrel{\eqref{eq:PerFro}}{=}} \int_A P_T \psi \,\mathrm{d} \nu
{\stackrel{\eqref{eq:push_f}}{=}} \int_{\mathbb{X}} (1_{A} P_T \psi)\circ T \,\mathrm{d} \mu= \int_{T^{-1}(A)} (P_T \psi)\circ T \,\mathrm{d} \mu
= \int_{T^{-1}(A)} \! U_T \circ P_T (\psi) \,\mathrm{d} \mu,
\end{align*}
which implies the first claim.
If the conditions for the second part are fulfilled, we can choose $A = T(A' \cap B)$ for any $A' \in \mathcal B(\mathbb{X})$.
Then, the second claim follows from
\[\int_{A'} \psi \,\mathrm{d} \mu = \int_{A' \cap B} \psi \,\mathrm{d} \mu = \int_{A'\cap B} U_T \circ P_T (\psi) \,\mathrm{d} \mu
= \int_{A'} U_T \circ P_T (\psi) \,\mathrm{d} \mu.\]
ii)
Using part i) and the Jensen inequality for conditional expectations, it holds for any $p \in [1, \infty)$ that
\begin{align}
\int_\mathbb{Y} \vert P_T\psi\vert^p \,\mathrm{d} \nu &{\stackrel{\eqref{eq:push_f}}{=}} \int_\mathbb{X} \vert U_T \circ P_T(\psi) \vert^p \,\mathrm{d} \mu = \int_\mathbb{X} \bigl\vert E\bigl(\psi|T^{-1}(\mathcal B(\mathbb{Y}))\bigr)\bigr\vert^p \,\mathrm{d} \mu\\
&\leq \int_\mathbb{X} E\bigl(\vert \psi \vert^p |T^{-1}(\mathcal B(\mathbb{Y}))\bigr) \,\mathrm{d} \mu = \int_\mathbb{X} \vert \psi\vert^p \,\mathrm{d} \mu < \infty
\end{align}
and
\begin{align}
\int_\mathbb{X} \vert U_T\psi\vert^p \,\mathrm{d} \mu &= \int_\mathbb{X} \vert \psi \circ T \vert^p \,\mathrm{d} \mu = \int_\mathbb{Y} \vert \psi\vert^p \,\mathrm{d} \nu < \infty.
\end{align}
For $p = \infty$, we only need to consider $P_T$ and estimate
\begin{align}
\Vert P_T\psi\Vert_{L_\infty(\mathbb{Y},\nu)} = \Vert (P_T\psi)\circ T\Vert_{L_\infty(\mathbb{X},\mu)} = \bigl\Vert E\bigl(\psi|T^{-1}(\mathcal B(\mathbb{Y}))\bigr) \bigr\Vert_{L_\infty(\mathbb{X},\mu)} \leq \Vert \psi \Vert_{L_\infty(\mathbb{X},\mu)} < \infty.
\end{align}
iii) First, we have $\int_A P_T 1_\mathbb{X} \,\mathrm{d} \nu = \mu\bigl(T^{-1}(A)\bigr) = \nu(A)$, which readily implies $P_T 1_\mathbb{X} = 1_\mathbb{Y}$.
Second, we obtain by definition that $U_T 1_\mathbb{Y} = 1_\mathbb{Y} \circ T = 1_\mathbb{X}$.
\end{proof}
By Proposition~\ref{lem:id} i), we conclude that $L=P_T$ is not suited for our segmentation problem: Since $U_T\circ P_T(1_A) = 1_A$ with $A\in T^{-1}(\mathcal B (\mathbb{Y}))$, the leading eigenvalue $1$ of $L^*L$ is in general not simple.
Additionally, this also implies that neither $P_T$ nor $U_T$ can be compact as otherwise the identity operator would be compact.
A possible remedy is to smooth the operator~$P_T$.
To this end, we need a kernel representation of $P_T$.
Using the family of probability measures
$(\pi_x)_{x \in \mathbb{X}}$ with $\pi_x\in \mathcal P(\mathbb{Y})$ given by $\pi_x = \delta_{T(x)}$
for all $x\in \mathbb{X}$, condition \eqref{eq:PerFro} can be rewritten as
\begin{equation}\label{kernel_def}
\int_A P_T \psi \,\mathrm{d} \nu = \int_{\mathbb{X}} \pi_x(A)\psi(x) \,\mathrm{d} \mu(x) \quad \mathrm{for \; all} \quad A \in \mathcal B(\mathbb{Y}).
\end{equation}
Note that $(\pi_x)_{x\in \mathbb{X}}$ are the disintegrations of the transport plan $\pi$ that is induced by the transport map $T$ with respect to $\mu$.
Here, the measure-valued function $x\mapsto \pi_x$ is usually called the \emph{transition function}.
This can be generalized to non-deterministic systems by choosing other families of probability measures.
Assuming that each probability measure $\pi_x$ has the density $K(x,\cdot)$ w.r.t.~$\nu$ with $K\in L_1(\mathbb{X} \times \mathbb{Y},\mu \otimes \nu)$,
we obtain for all $A \in \mathcal{B}(\mathbb{Y})$ by \eqref{kernel_def} and Fubini's theorem
\begin{align*}
\int_A (P_K \psi) (y) \,\mathrm{d} \nu(y) = \int_{\mathbb{X}} \int_A K(x,y) \psi(x) \,\mathrm{d} \nu(y) \,\mathrm{d} \mu(x) = \int_A \int_{\mathbb{X}} K(x,y) \psi(x) \,\mathrm{d} \mu(x) \,\mathrm{d} \nu(y).
\end{align*}
Hence, we conclude that
\begin{equation}
(P_K \psi) (y) = \int_{\mathbb{X}} K(x,y) \psi(x) \,\mathrm{d} \mu(x)
\end{equation}
holds $\nu$-a.e., which corresponds to the definition of the Frobenius--Perron operator for kernels, see also \citet{KNKW18}.
The corresponding adjoint operator $U_K \colon L_\infty(\mathbb{Y},\nu) \to L_\infty(\mathbb{X}, \mu)$ is given by
\[(U_K \psi) (x) = \int_{\mathbb{Y}} K(x,y) \psi(y) \,\mathrm{d} \nu(y).\]
In this framework, we call $K\in L_1(\mathbb{X} \times \mathbb{Y},\mu \otimes \nu)$ a \emph{transition kernel with reference measures $\mu$ and $\nu$}, to which $P_K$ as defined above is associated.
There is also a dynamical reason for the introduction of the transition density $K(x, \cdot)$.
In the following, we will view them as $T(x)$ with a small random perturbation.
As shown by \citet{froyland2013analytic}, segmentation of such perturbed dynamics yields partitions $\mathbb{X}=X_1\dot\cup X_2$ and $\mathbb{Y}=Y_1\dot\cup Y_2$ that are robust to small noise.
\subsection{Smoothed Transition Kernels}\label{sec:frobenius_perron_2}
In the previous subsection, we have seen that the operators $P_T$ arising from transport maps fulfill
only some of the required properties to serve as operators in our segmentation task.
Next, we are interested in blurred kernels that fulfill \ref{enum:nu_marg} and \ref{enum:mu_marg} and have a nontrivial spectrum.
Compared to the approach proposed by \citet{froyland2013analytic}, we prefer to avoid the domain padding
and adapt the weights in the kernels instead.
This leads to modifications of the corresponding proofs.
For $\varepsilon > 0$ and
a measurable map $T\colon \mathbb{X} \to \mathbb{Y}$ with $\nu = T_\# \mu$, we define $\eta_{\varepsilon,x} \coloneqq 1_{B_\varepsilon(x)}/\lambda(B_\varepsilon(x))$
and $k_\varepsilon\colon \mathbb{X} \times \mathbb{Y} \to \ensuremath{\mathbb{R}}$ by
\begin{align}
k_\varepsilon (x,y) &\coloneqq \int_\mathbb{X} \eta_{\varepsilon,x}(z) \eta_{\varepsilon,T(z)}(y) \,\mathrm{d} z
=
\frac{1}{\lambda(B_\varepsilon(x))} \int_{B_\varepsilon(x) \cap T^{-1} \left(B_\varepsilon(y) \right)}
\frac{1}{\lambda\left(B_\varepsilon \left(T(z) \right)\right)} \, \,\mathrm{d} z.
\end{align}
Due to \eqref{eq:boundary}, $k_\varepsilon$ is bounded from above by
$C_{\varepsilon} \coloneqq 1/\min_{y \in \mathbb{Y}} \lambda(B_\varepsilon(y)) = C_\mathbb{Y} / \max_{y \in \mathbb{Y}} \lambda(B_\varepsilon(y))$.
Hence, $\sigma_\varepsilon (y) \coloneqq \int_\mathbb{X} k_\varepsilon(x,y) \,\mathrm{d} \mu(x)$ is well-defined and bounded and we can introduce a new measure $\nu_\varepsilon$ via $\nu_\varepsilon \coloneqq \sigma_\varepsilon \lambda$.
Finally, our smoothed kernel $K_\varepsilon\colon \mathbb{X} \times \mathbb{Y} \to \ensuremath{\mathbb{R}}$
reads as
\begin{equation} \label{eq:kernel_smoothed}
K_{\varepsilon}(x,y)
\coloneqq
\frac{k_\varepsilon(x,y)}{\sigma_\varepsilon(y)}.
\end{equation}
The following proposition shows that the
operator $L_\varepsilon\colon L_2(\mathbb{X},\mu) \to L_2(\mathbb{Y},\nu_{\varepsilon})$ defined by
\begin{equation} \label{FP_kernel}
L_\varepsilon \psi (y) \coloneqq \int_\mathbb{X} K_\varepsilon (x,y) \psi(x) \,\mathrm{d} \mu(x)
\end{equation}
is suited for our segmentation model \eqref{problem}.
\begin{proposition} \label{lem:prop_kern}
The kernel $K_\varepsilon\colon \mathbb{X} \times \mathbb{Y} \to \ensuremath{\mathbb{R}}$ defined in \eqref{eq:kernel_smoothed} is double-stochastic, i.e., $K_\varepsilon \in L_2(\mathbb{X} \times \mathbb{Y}, \mu \otimes \nu_\varepsilon)$ is non-negative and
\begin{align}
\int\limits_\mathbb{X} K_{\varepsilon}(x,\cdot) \,\mathrm{d} \mu(x) = 1_\mathbb{Y} \,\,\,\nu_\varepsilon\text{-a.e.}, \quad
\int\limits_\mathbb{Y} K_{\varepsilon}(\cdot,y) \,\mathrm{d} \nu_\varepsilon(y) = 1_\mathbb{X} \,\,\,\mu\text{-a.e.}
\end{align}
Further, $L_\varepsilon$ fulfills the properties \ref{enum:nu_marg} and \ref{enum:mu_marg}.
\end{proposition}
\begin{proof}
The first marginal property follows directly by definition of $K_\varepsilon$.
Further, the second one can be verified by
\begin{align*}
\int\limits_\mathbb{Y} K_{\varepsilon} (x,y) \,\mathrm{d} \nu_\varepsilon(y)
=
\int_\mathbb{Y} \int_\mathbb{X} \eta_{\varepsilon,x} (z) \eta_{\varepsilon,T(z)} (y) \,\mathrm{d} z \,\mathrm{d} y =
\int_\mathbb{X} \eta_{\varepsilon,x} (z) \int_\mathbb{Y} \eta_{\varepsilon,T(z)} (y) \,\mathrm{d} y \,\mathrm{d} z
=
\int_\mathbb{X} \eta_{\varepsilon,x} (z) \,\mathrm{d} z = 1.
\end{align*}
Further, non-negativity of $K_\varepsilon$ follows directly by definition, and square-integrability can be checked as follows
\begin{align*}
\int_\mathbb{X} \int_\mathbb{Y} K_\varepsilon(x,y)^2 \,\mathrm{d} \nu_\varepsilon(y) \,\mathrm{d} \mu(x)
&=
\int_\mathbb{X} \int_\mathbb{Y} \frac{k_\varepsilon (x,y)^2}{\sigma_\varepsilon(y)^2} \sigma_\varepsilon(y) \,\mathrm{d} y \,\mathrm{d} \mu(x)=
\int_\mathbb{X} \int_\mathbb{Y} \frac{k_\varepsilon (x,y)^2}{\sigma_\varepsilon(y)} \,\mathrm{d} y \,\mathrm{d} \mu(x)\\
&=
\int_\mathbb{Y} \frac{\int_\mathbb{X} k_\varepsilon (x,y)^2 \,\mathrm{d} \mu(x)}{\int_\mathbb{X} k_\varepsilon (x,y) \,\mathrm{d} \mu(x)} \, \,\mathrm{d} y \le
\int_\mathbb{Y} C_\varepsilon \frac{\int_\mathbb{X} k_\varepsilon (x,y) \,\mathrm{d} \mu(x)}{\int_\mathbb{X} k_\varepsilon (x,y) \,\mathrm{d} \mu(x)}\, \,\mathrm{d} y = C_\varepsilon \lambda(\mathbb{Y}).
\end{align*}
Finally, \cite[Thm.~6.18]{Folland1984} implies $\Vert L_\varepsilon \Vert = 1$ and hence $L_\varepsilon$ fulfills the properties \ref{enum:nu_marg} and \ref{enum:mu_marg}.
\end{proof}
\begin{remark}\label{rem:simplicity}
Simplicity of the largest eigenvalue of $L_\varepsilon^*L_\varepsilon$ can be shown in a similar way as in \citet[Prop.~3]{froyland2013analytic}, e.g., for $\mathbb{X}=\mathbb{Y}$ being a connected domain and $T$ being a diffeomorphism with Jacobi determinant uniformly bounded from above and below.
We omit the proof here, but remark that due to compactness, the multiplicity $q$ of $\lambda_1$ is finite. In case $q>1$, the solution to \eqref{problem} is $\smash{ \sigma_1=\lambda_1^{1/2} }$ and the maximizing functions are in the eigenspaces corresponding to $\lambda_1$.
\end{remark}
In case that $T$ is a transport map, we have the following convergence behavior of $K_\varepsilon$
and $\nu_\varepsilon$ as $\varepsilon \rightarrow 0$, stating that the system associated with $K_{\varepsilon}$ is a \emph{small random perturbation} of $T$, see also~\citet{Kha63} and \citet{Kif86}.
\begin{proposition}\label{prop:conv}
Let $\mu$ and $\nu$ be absolutely continuous measures w.r.t.~the Lebesgue measure with densities
$\sigma_\mu$ and $\sigma_\nu$, respectively.
Assume that the densities are positive a.e.~and that $\sigma_\mu \in L_\infty(\mathbb{X})$ is continuous a.e.
Let $T\colon \mathbb{X} \to \mathbb{Y}$ be a measurable map with $\nu = T_{\#}\mu$
and set $\pi \coloneqq (\id_\mathbb{X},T)_{\#}\mu$.
Then, the kernel $K_{\varepsilon}$ and the measure $\nu_\varepsilon$ defined with respect to $T$ satisfy
$K_{\varepsilon}\, (\mu \otimes \nu_\varepsilon) \rightharpoonup \pi$ and $\nu_\varepsilon \rightharpoonup \nu$ as $\varepsilon \to 0$.
\end{proposition}
\begin{proof}
For the first claim, it suffices to show
$\lim_{\varepsilon \to 0} K_{\varepsilon}\, (\mu \otimes \nu_\varepsilon)(A \times B) = \pi(A \times B)$
for all $A \in \mathcal B(\mathbb{X})$ and $B \in \mathcal B(\mathbb{Y})$ with $\mu(\partial A) = 0$ and $\nu(\partial B)=0$,
see~\citet[Thm.~2.8(i)]{Billingsley99}.
Due to our assumptions on the densities, these sets also satisfy $\lambda(\partial A)=0$ and $\lambda(\partial B)=0$.
Using the definition of $\nu_\varepsilon$ and Fubini's theorem, we get
\begin{align*}
K_{\varepsilon} (\mu \otimes \nu_\varepsilon) (A \times B)
&=
\int_A \int_B k_\varepsilon (x,y) \,\mathrm{d} y \,\mathrm{d} \mu(x)
= \int_A \int_B \int_\mathbb{X} \eta_{\varepsilon,x} (z) \eta_{\varepsilon,T(z)} (y) \,\mathrm{d} z \,\mathrm{d} y \,\mathrm{d} \mu(x)\\
&= \int_\mathbb{X} \int_A \eta_{\varepsilon,x}(z) \,\mathrm{d} \mu(x) \, \int_B \eta_{\varepsilon, T(z)}(y)
\,\mathrm{d} y\, \,\mathrm{d} z.
\end{align*}
Next, we define
\begin{align}
f_{\varepsilon,A}(z) &\coloneqq \int_A \eta_{\varepsilon,x}(z) \,\mathrm{d} \mu(x) , \qquad
g_{\varepsilon,B}(z) \coloneqq \int_B \eta_{\varepsilon,T(z)}(y) \,\mathrm{d} y.
\end{align}
First, we show $f_{\varepsilon,A}(z) \to \sigma_\mu(z)1_A(z)$ for a.e.~$z \in \mathbb{X}$.
Fix $z \in \mathring{\mathbb{X}} \setminus \partial A$.
Then, it holds for $\varepsilon$ small enough that
$\eta_{\varepsilon, x}(z) = 1_{B_\varepsilon(z)}(x)/\lambda(B_\varepsilon(z))$.
Thus, we get for every such $z$ where $\sigma_\mu$ is continuous,
\[
\lim_{\varepsilon \to 0} f_{\varepsilon,A}(z)
= \lim_{\varepsilon \to 0} \int_A \frac{1_{B_\varepsilon(z)}(x)}{\lambda(B_\varepsilon(z))} \sigma_\mu(x) \,\mathrm{d} x
= \lim_{\varepsilon \to 0} \frac{1_A(z)}{\lambda(B_\varepsilon(z))} \int_{B_\varepsilon(z)} \sigma_\mu(x)\,\mathrm{d} x = \sigma_\mu(z)1_A(z)
\]
and hence convergence a.e.
Next, we show $g_{\varepsilon,B}(z) \to 1_{T^{-1}(B)}(z)$ a.e.
For $z \in T^{-1}(\mathring B)$ and $\varepsilon$ small enough, we conclude that $g_{\varepsilon,B}(z) = 1$.
If $z \in T^{-1}(\mathbb{Y} \setminus \text{cl}(B))$, we get for $\varepsilon$ small enough that $g_{\varepsilon,B}(z) = 0$.
As $T$ is non-singular, the set $T^{-1}(\partial B)$ has Lebesgue measure zero and the claim follows.
By definition, we realize that $g_{\varepsilon,B}(z) \leq 1$ and by \eqref{eq:boundary} that
$f_{\varepsilon,A}(z) \leq \Vert \sigma_\mu\Vert_{L_\infty(\mathbb{X})} C_\mathbb{X}$.
Now, the first assertion of the theorem follows by applying Lebesgue's dominated convergence theorem
\begin{align*}
\lim_{\varepsilon \to 0} K_{\varepsilon}\, (\mu \otimes \nu_\varepsilon) (A \times B)
&= \lim_{\varepsilon \to 0} \int_\mathbb{X} f_{\varepsilon,A}(z) g_{\varepsilon,B}(z) \,\mathrm{d} z = \int_\mathbb{X} 1_A(z) 1_{T^{-1}(B)}(z) \,\mathrm{d} \mu(z) = \pi(A \times B).
\end{align*}
In order to show $\nu_\varepsilon \rightharpoonup \nu$, it suffices to prove $\lim_{\varepsilon \to 0} \nu_\varepsilon(B) \to \nu(B)$ for all $B \in \mathcal B(\mathbb{Y})$ with $\nu(\partial B)=0$.
Repeating the same calculations as in the first part of the proof with $A = \mathbb{X}$, we obtain
\[\lim_{\varepsilon \to 0} \nu_\varepsilon(B)
= \lim_{\varepsilon \to 0} \int_B \int_\mathbb{X} k_\varepsilon(x,y) \,\mathrm{d} \mu(x) \,\mathrm{d} y
= \int_\mathbb{X} 1_{T^{-1}(B)}(z) \,\mathrm{d} \mu(z)
\stackrel{\eqref{eq:push_f}}{=} \int_\mathbb{Y} 1_B (y) \,\mathrm{d} \nu(y)
= \nu(B).\]
\end{proof}
By Proposition \ref{prop:conv} and Theorem \ref{thm:tmap} we could use the OT map $\hat T$ induced by $\hat \pi$ and create a smoothed kernel with respect to this map.
\subsection{Kernels from Regularized OT}\label{sec:frobenius_perron_3}
Having the OT results in mind, we propose to construct kernels based on regularized OT plans
$\hat \pi_\varepsilon$ in \eqref{eq:PDrelation}, respectively $\hat \pi_{\varepsilon, \kappa}$ in \eqref{eq:ot-unbalanced}
with the properties~\ref{enum:nu_marg}--\ref{enum:cpt_simple} and to use them as kernels
for transfer operators $L_\varepsilon$ in \eqref{FP_kernel}.
A motivation from statistical physics for this approach is given in Section~\ref{sec:schroedinger}.
Focusing on $\hat \pi_\varepsilon$, we use
\begin{equation} \label{sm_kern_ot}
K_\varepsilon (x,y)
\coloneqq
\frac{\,\mathrm{d} \hat \pi_\varepsilon(x,y)}{\,\mathrm{d}(\mu \otimes \nu)(x,y)}
=
\exp\Bigl(\frac{\hat \varphi_\varepsilon(x) + \hat \psi_\varepsilon(y) - c(x,y)}{\varepsilon}\Bigr) .
\end{equation}
Indeed, we can use this kernel for our segmentation problem \eqref{problem}, as the following proposition shows.
\begin{proposition}
For a Lipschitz cost function $c$, the kernel $K_\varepsilon$ and the associated operator $L_\varepsilon$ fulfill \ref{enum:nu_marg}--\ref{enum:cpt_simple}.
\end{proposition}
\begin{proof}
Using the marginal property of $\hat \pi_\varepsilon(x,y) = K_\varepsilon(x,y) \sigma_\mu(x) \sigma_\nu(y) \lambda \otimes \lambda$, we obtain by integration that $\int_\mathbb{X} K_\varepsilon (x,y) \sigma_\mu(x) \sigma_\nu(y) \,\mathrm{d} x = \sigma_\nu(y)$ and hence
\begin{align*}
\int_\mathbb{X} K_\varepsilon (x,y) \,\mathrm{d} \mu(x) = \int_\mathbb{X} K_\varepsilon (x,y) \sigma_\mu(x) \,\mathrm{d} x = 1 \quad \nu\text{-a.e.},
\end{align*}
which is \ref{enum:nu_marg} and similarly we show \ref{enum:mu_marg}.
Further, it is well known, that for Lipschitz $c$ the Kantorovich potentials are Lipschitz and hence bounded on compact domains, see \citet[Prop.~5.7]{neumayer2020optimal}. Therefore, square-integrability and non-negativity follow directly from \eqref{eq:PDrelation}.
Finally, the assumptions of Lemma $3$ in \citet{froyland2013analytic} are fulfilled as $K_\varepsilon > 0$ and
consequently the largest eigenvalue of $\mathcal L_\varepsilon = L_\varepsilon^*L_\varepsilon$ is simple, i.e, \ref{enum:cpt_simple} holds.
\end{proof}
A counterpart of Proposition \ref{prop:conv} is given below, see~\citet{CDPS17}.
Recall Theorem \ref{thm:tmap} for the uniqueness of the optimal transport plan
and its relation to the optimal transport map.
\begin{proposition}\label{prop:conv-reg}
Let $\mu,\nu\in \mathcal P (\mathbb{X})$, where $\mu$ is absolutely continuous
with respect to the Lebesgue measure and let $c(x,y)=h(x-y)$ with a strictly convex function $h$.
Then, it holds for the kernel $K_{\varepsilon}$ in \eqref{sm_kern_ot} that
$\hat \pi_\varepsilon = K_{\varepsilon}\, (\mu \otimes \nu) \rightharpoonup \hat \pi$,
where $\hat \pi$ is the minimizer of the original OT problem \eqref{Monge_Kantorovich_problem}.
\end{proposition}
\section{Motivation for Regularized OT Kernels from Statistical Physics} \label{sec:schroedinger}
In this section, we motivate that under certain assumptions our construction of $L_\varepsilon$ from the regularized OT plan is a reasonable choice.
For these purposes, the measures $\mu$ and $\nu$ are modeled as empirical measures corresponding to a large ensemble of $n$ indistinguishable particles with time-dependent random positions $(X^i_t)_t$, $i=1,\ldots,n$, $t\in \{0, 1\}$.
Then, any transition kernel choice for the transfer operator corresponds to a (joint) distribution of the particle ensemble.
In the absence of any other given dynamical information, it is unclear which joint distribution to pick.
Although this is unlikely to represent the true dynamics when conditioning on the given measures $\mu$ and $\nu$, one might therefore ask what happens if one makes the ``blind'' generic choice that each individual particle is simply following an independent Brownian path.
We show that this assumption interestingly leads precisely to the regularized optimal transport plan $\hat \pi_\varepsilon$ for the particle ensembles as a whole as the number of particles $n$ approaches infinity.
Although only the time-discrete case $t\in \{0, 1\}$ is considered, we note that the argumentation can be generalized to Wiener processes or even arbitrary positive path measures and state spaces.
Here, we refer to the survey of \citet{leonard2013survey} on Schr\"odinger's question, see \eqref{sch}, and to \citet{leonard2010entropic} for rigorous proofs.
Our self-contained approach may be more accessible than those for the general setting.
Let the particle positions be described by i.i.d.~$\ensuremath{\mathbb{R}}^d \times \ensuremath{\mathbb{R}}^d$-valued random vectors $(X_0^i,X_1^i)$, $i=1,\ldots,n$,
on a common probability space with conditional probability density
\begin{equation}\label{joint_density}
\sigma_{X_1 | X_0}^\varepsilon(x_0, x_1)
= \tfrac{1}{\sqrt{\pi \varepsilon}^d}\exp \left(-\Vert x_0 - x_1\Vert^2 / \varepsilon \right),
\quad \varepsilon > 0.
\end{equation}
Since we assume that the particles are indistinguishable,
the ensemble of particles at time $t \in \{0,1\}$ can be described using a $\mathcal P(\ensuremath{\mathbb{R}}^d)$-valued random variable,
in other words, a random probability measure,
\begin{equation} \label{Lnt}
Z_t^n \coloneqq \frac{1}{n} \sum_{i=1}^{n} \delta_{X_t^i}.
\end{equation}
Note that $\mathcal P(\ensuremath{\mathbb{R}}^d)$ shall again be equipped with the weak topology and we rely on the corresponding Borel $\sigma$-algebra.
The following proposition gives an intuition on the convergence behavior of $Z^n_1$ as $n\rightarrow \infty$
for such a process conditioned on starting points $X_0^i = x_0^i$ for which the corresponding empirical distribution converges weakly to $\mu$.
Note that we re-use $Z^n_0$ with a slight abuse of notation.
\begin{proposition} \label{rem:nice}
For any sequence of sampled initial points $(x_0^i)_{i \in \ensuremath{\mathbb{N}}}$ with
\begin{equation} \label{eq:weak_1}
Z_0^n = \frac{1}{n} \sum_{i=1}^{n} \delta_{x_0^i} \rightharpoonup \mu,
\end{equation}
the empirical random probability measure
\[\tilde Z^n_1 \coloneqq \frac1n \sum_{i=1}^n \delta_{X_1^i | X_0^i = x_0^i}\]
converges a.s.\ to the constant random measure $(\sigma^\varepsilon_{X_1 | X_0}(\cdot, 0) * \mu) \lambda$ as $n\to \infty$.
In other words, for a.e.~realization of
the random variables $(X_1^i| X_0^i = x_0^i)_{i\in \ensuremath{\mathbb{N}}}$
the associated empirical measure converges weakly to $(\sigma^\varepsilon_{X_1 | X_0}(\cdot, 0) * \mu) \lambda$ as $n\to \infty$.
\end{proposition}
\begin{proof}
For any $f \in C_b(\mathbb R^d)$,
the random variables $f(X_1^i| X_0^i = x_0^i)$, $i \in \mathbb N$, are independent with finite variances $V^i$ fulfilling
$\sum_i V^i/i^2 < \infty$, so that we obtain by Kolmogorov's strong law of large numbers \citep[p.~389]{Shiryaev1996} that
\begin{align}
\int_{\ensuremath{\mathbb{R}}^d} f(y) \,\mathrm{d} \tilde Z^n_1(y) = &\frac{1}{n}\sum_{i=1}^n f(X_1^i|X_0^i = x_0^i)
\xrightarrow{\text{a.s.}}
\lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^n E\bigr(f (X_1^i|X_0^i = x_0^i)\bigr).
\label{auch_noch}
\end{align}
By \eqref{joint_density}, we have
$$
\bigl(\sigma_{X_1 | X_0}^\varepsilon (\cdot,0) * Z_0^n\bigr)(y)
= \frac{1}{n}\sum_{i=1}^n \sigma_{X_1 | X_0}^\varepsilon (y-x_0^i, 0)
= \frac{1}{n}\sum_{i=1}^n \sigma_{X_1 | X_0}^\varepsilon (x_0^i, y) ,
$$
so that by \eqref{eq:weak_1} and weak continuity of convolutions and $Z_0^n\rightharpoonup \mu$, the limit in \eqref{auch_noch} becomes
\begin{align}
&\lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^n E\bigr(f (X_1^i|X_0^i = x_0^i)\bigr)
=
\lim_{n \to \infty} \int_{\ensuremath{\mathbb{R}}^d} f(y) \frac{1}{n}\sum_{i=1}^n \sigma_{X_1 | X_0}^\varepsilon (x^i_0, y) \,\mathrm{d} y\\
&= \lim_{n \to \infty} \int_{\ensuremath{\mathbb{R}}^d} f(y) \bigl(\sigma_{X_1 | X_0}^\varepsilon (\cdot,0) * Z_0^n\bigr)(y) \,\mathrm{d} y
= \int_{\ensuremath{\mathbb{R}}^d} f(y) \bigl(\sigma_{X_1 | X_0}^\varepsilon(\cdot, 0) * \mu\bigr)(y) \,\mathrm{d} y.
\end{align}
Thus, assuming \eqref{eq:weak_1}, we obtain the assertion.
\end{proof}
In contrast to the setting in the proposition, we assume now that the initial configuration at $t=0$ lies
in a small Wasserstein-2 ball
$B(\mu, r) \coloneqq \{\alpha\in \mathcal{P}(\R^d) : W_2(\alpha, \mu) \leq r \}$ with radius $r>0$
around $\mu$, and those of the observed particle distribution at $t=1$
in $\Bnuxi$.
The collective dynamical behavior of the ensemble is described by
the $\mathcal{P}(\ensuremath{\mathbb{R}}^d \times \ensuremath{\mathbb{R}}^d)$-valued random variable
\begin{equation} \label{Ln}
Z^n \coloneqq \frac{1}{n} \sum_{i=1}^{n} \delta_{(X_0^i, X_1^i)},
\end{equation}
which contains more information than the two random probability measures $Z^n_t$, $t=0,1$.
We are interested in estimating the conditional probability
\begin{align}\label{eq:cond_prob}
P\bigl(Z^n \in A \,\vert\, Z_0^n\in B(\mu, r), Z_1^n\in \Bnuxi\bigr)
= \frac{P\bigl(Z^n \in A, Z_0^n\in B(\mu, r), Z_1^n\in \Bnuxi\bigr)}{P\bigl(Z_0^n\in B(\mu, r), Z_1^n\in \Bnuxi\bigr)}
\end{align}
for large $n$ and all $A \in \mathcal{B}(\mathcal{P}(\ensuremath{\mathbb{R}}^d \times \ensuremath{\mathbb{R}}^d))$. For small $r$ and in the infinite particle limit, this is precisely
\emph{Schr\"odinger's question}, that can now be stated as follows: What is the limit
\begin{equation}\label{sch}
\lim_{r\to 0}\lim_{n\to \infty} P\bigl(Z^n \in A \,\vert\, Z_0^n\in B(\mu, r), Z_1^n\in \Bnuxi\bigr)?
\end{equation}
Interestingly, the limiting dynamical behavior of the particles is described approximately by the regularized optimal transport plan $\hat\pi_\varepsilon$; more precisely, the limit above and answer to Schr\"odinger's question is $\delta_{\hat\pi_\varepsilon}(A)$.
However, we have to be careful here.
The statement holds for the \emph{entropy regularized OT distance}
\begin{align}\label{eq:ot_ent}
\OT_{\varepsilon}(\mu,\nu)
&\coloneqq
\min_{\pi \in \Pi(\mu,\nu)}
\,
\int_{\mathbb{X} \times \mathbb{Y}} c\,\mathrm{d} \pi
+ \varepsilon \mathrm{KL} (\pi,\lambda \otimes \lambda)\\
&=
\varepsilon
\min_{\pi \in \Pi(\mu,\nu)}
\, \mathrm{KL} \bigl(\pi, \exp(-c/\varepsilon) \lambda \otimes \lambda\bigr) - \varepsilon \int_{\mathbb{X} \times \mathbb{Y}} \exp(-c/\varepsilon) -1 \,\mathrm{d} (\lambda \otimes \lambda).
\end{align}
The relation between the KL-regularized OT in~\eqref{sinkhorn} and the
entropy regularized OT in~\eqref{eq:ot_ent} is described in the following
remark. Roughly speaking, the KL-regularized OT is more general as minimizers always exist,
but there are many cases where both minimization problems coincide.
\begin{remark}\label{rem:entropy}
Let $\sigma_\mu$ and $\sigma_\nu$ denote the densities of $\mu$ and $\nu$, respectively.
For $\pi \ll \lambda \otimes \lambda$ with density $\sigma_\pi$ the entropy is defined by
$E(\pi) \coloneqq \mathrm{KL}(\pi,\lambda \otimes \lambda)$.
Note that
$ \pi \ll \mu \otimes \nu$ if and only if $\pi \ll \lambda \otimes \lambda$ for any $\pi \in \Pi(\mu,\nu)$.
If $\mathrm{KL} (\mu \otimes \nu, \lambda \otimes \lambda) < \infty$, we can show for any $\pi \ll \lambda \otimes \lambda$
with $\pi \in \Pi(\mu,\nu)$ that it holds
\begin{align}
\mathrm{KL} (\pi,\lambda \otimes \lambda) - \mathrm{KL} (\mu \otimes \nu, \lambda \otimes \lambda) \label{care} =\mathrm{KL} (\pi,\mu \otimes \nu).
\end{align}
Consequently, in this case the minimizer for KL-regularized OT~\eqref{sinkhorn} and entropy regularized OT~\eqref{eq:ot_ent} coincide.
The crux is the condition $\mathrm{KL} (\mu \otimes \nu, \lambda \otimes \lambda) < \infty$, which is equivalent to $\mu,\nu$ having finite entropy,
i.e., $\sigma_\mu, \sigma_\nu$ are in a so-called Orlicz space $L \log L$, see~\citet{CLMW19,NR2013}.
For more information we refer to \citet{neumayer2020optimal}.
\end{remark}
The following proposition establishes a relation between regularized optimal transport plans minimizing \eqref{eq:ot_ent} and Schr\"odinger's question.
As already mentioned, this fact is known in a more general setting.
For the sake of completeness, we add the proof for our setting.
\begin{proposition}\label{prop:schroedinger}
Let $(X_0^i,X_1^i)_{i\in \ensuremath{\mathbb{N}}}$ be i.i.d.~$\ensuremath{\mathbb{R}}^d \times \ensuremath{\mathbb{R}}^d$-valued random vectors
on a common probability space with conditional density \eqref{joint_density}, and let $Z^n_t$, $t = 0,1$, and $Z^n$ be given by \eqref{Lnt} and \eqref{Ln}, respectively.
Then, it holds for $\mu,\nu \in \mathcal P(\mathbb R^d)$ fulfilling $\mathrm{KL} (\mu \otimes \nu, \lambda \otimes \lambda) < \infty$ and every $A \in \mathcal B(\mathcal P(\ensuremath{\mathbb{R}}^d \times \ensuremath{\mathbb{R}}^d))$ with $\hat \pi_\varepsilon \notin \partial A$ that
\begin{equation}\label{schroedinger}
\lim_{r\to 0} \lim_{n\to \infty} P\bigl(Z^n \in A \,\vert\, Z_0^n \in B(\mu, r), Z_1^n \in \Bnuxi \bigr)
=
\delta_{\hat \pi_\varepsilon}(A),
\end{equation}
where $\hat \pi_\varepsilon$ is the regularized optimal transport plan minimizing \eqref{eq:ot_ent} with the Wasserstein-2 cost function.
\end{proposition}
\begin{proof}
For $r>0$,
set $C_r \coloneqq \{\pi \in \mathcal P(\ensuremath{\mathbb{R}}^d\times \ensuremath{\mathbb{R}}^d):\pi_0 \in B(\mu, r), \pi_1 \in \Bnuxi\}$
and consider the unique minimizers $\hat \pi_{r,\varepsilon} \coloneqq
\argmin_{\pi \in C_r} \mathrm{KL}(\pi ,\sigma^\varepsilon_{X_1 | X_0} \lambda \otimes \lambda)$, see after \eqref{KLdef}.
Clearly, we have by \eqref{sinkhorn_kernel} for any $r>0$ that
\begin{align} \label{soso}
\mathrm{KL}(\hat \pi_{r,\varepsilon} ,\sigma^\varepsilon_{X_1 | X_0}\lambda \otimes \lambda)
= \min_{\pi \in C_r} \mathrm{KL}(\pi ,\sigma^\varepsilon_{X_1 | X_0}\lambda \otimes \lambda) \le
\mathrm{KL}(\hat \pi_\varepsilon ,\sigma^\varepsilon_{X_1 | X_0}\lambda \otimes \lambda).
\end{align}
Choose a sequence $(r_j)_{j\in \ensuremath{\mathbb{N}}}$, $r_j>0$, with $r_j\to 0$ for $j\to \infty$.
Using the weak compactness of $\mathcal P(\ensuremath{\mathbb{R}}^d\times \ensuremath{\mathbb{R}}^d)$ and the weak closedness of any $C_r$,
we conclude that all accumulation points of
$(\hat\pi_{r_j,\varepsilon})_j$ are contained in $C_{r_j}$ for any fixed $j\in \ensuremath{\mathbb{N}}$.
Consequently, they are also contained in $\Pi(\mu,\nu) = \cap_{j\in\ensuremath{\mathbb{N}}} C_{r_j}$.
Take a weak accumulation point $\tilde\pi_\varepsilon$ using weak compactness,
and with abuse of notation, choose a subsequence $(r_j)_{j\in \ensuremath{\mathbb{N}}}$ such that $\hat\pi_{r_j,\varepsilon} \rightharpoonup \tilde \pi_\varepsilon \in \Pi(\mu,\nu)$ as $r_j\to 0$.
By weak lower-semicontinuity of $\mathrm{KL}(\cdot,\sigma_{X_1 | X_0}^\varepsilon \lambda \otimes \lambda)$,
we have
that
\[
\mathrm{KL}\bigl(\tilde \pi_\varepsilon ,\sigma^\varepsilon_{X_1 | X_0}\lambda \otimes \lambda \bigr)
\le
\lim_{j\to \infty} \mathrm{KL}\bigl(\hat \pi_{r_j,\varepsilon} ,\sigma^\varepsilon_{X_1 | X_0}\lambda \otimes \lambda\bigr)
\le \mathrm{KL} (\hat \pi_\varepsilon,\sigma^\varepsilon_{X_1 | X_0}\lambda \otimes \lambda \bigr).
\]
Since $\tilde \pi_\varepsilon$ is a feasible point of the regularized OT problem \eqref{sinkhorn},
this implies that the whole sequence
$(\hat \pi_{r_j,\varepsilon})_j$ converges weakly to $\tilde \pi_\varepsilon = \hat \pi_\varepsilon$ as $j\to\infty$.
Due to \eqref{eq:cond_prob}, we have
\begin{align}
\lim_{n\to \infty} \frac{1}{n} \log P\bigl(Z^n \in A \,\vert\, Z_0^n\in B(\mu, r), Z_1^n\in \Bnuxi\bigr) = \lim_{n\to \infty} \frac{1}{n}\log \frac{P(Z^n \in A \cap C_r)}{P(Z^n\in C_r)}.
\end{align}
Using logarithm laws and Sanov's Theorem~\ref{thm:sanov}, see~\citet[Thm.~6.2.10]{DZ2010}, for the respective summands, we obtain for any measurable $A$ satisfying $A = \text{cl} (\mathring A)$
and $\hat \pi_\varepsilon \notin \text{cl}(A)$--which by convergence of $(\hat \pi_{r_j,\varepsilon})_j$ also implies $\hat \pi_{r,\varepsilon} \notin \text{cl}(A)$ for $r>0$ small enough--that
\begin{align*}
\lim_{n\to \infty} \frac{1}{n}\log \frac{P(Z^n \in A \cap C_r)}{P(Z^n\in C_r)}
= \inf_{\pi \in C_r} \mathrm{KL}\bigl(\pi ,\sigma_{X_1 | X_0}^\varepsilon\lambda \otimes \lambda\bigr) -\inf_{\pi \in A \cap C_r}\mathrm{KL}\bigl(\pi ,\sigma_{X_1 | X_0}^\varepsilon\lambda \otimes \lambda\bigr) <0,
\end{align*}
where the last inequality follows from the strict convexity of $\mathrm{KL}$.
Hence, we obtain
\[\lim_{r\to 0}\lim_{n\to \infty} P\bigl(Z^n \in A \,\vert\, Z_0^n\in B(\mu, r), Z_1^n\in \Bnuxi\bigr) = 0.\]
Using complements, we conclude for any measurable $A$ satisfying $A = \text{cl} (\mathring A)$
and $\hat \pi_\varepsilon \in \mathring A$,
that
\[
\lim_{r\to 0}\lim_{n\to \infty} P\bigl(Z^n \in A \,\vert\, Z_0^n\in B(\mu, r), Z_1^n\in \Bnuxi\bigr) = 1,
\]
so that our claim follows from these two results using the monotonicity of measures.
\end{proof}
\begin{theorem}[Sanov]\label{thm:sanov}
Let $(X_i)_{i \in \ensuremath{\mathbb{N}}}$ be a sequence of i.i.d.~$\mathcal P(\mathbb{X})$-valued random variables
with distribution $m$ on some probability space, where $\mathcal P(\mathbb{X})$ is equipped with the weak topology.
Then, it holds for any measurable $A \in \mathcal B(\mathcal P(\mathbb{X}))$ with $A = \text{cl}(\mathring A)$ that
\[\lim_{n \to \infty} \frac1n \log P\biggl( \frac1n\sum_{i=1}^n \delta_{X_i} \in A \biggr) = - \inf_{\pi \in A} \mathrm{KL}(\pi, m).\]
\end{theorem}
To summarize, in the case that the particles make independent Gaussian jumps,
conditioning on initial and end configurations close to $\mu$ and $\nu$ means
having a dynamical behavior of the particles described approximately by the regularized optimal transport plan~$\hat \pi_\varepsilon$.
\section{Discretization}\label{sec:discrete}
In this section, we discuss the discrete settings for our numerical computations.
In Subsection~\ref{subsec:STK}, we consider the construction of smoothed transition kernels from OT plans and from entropic OT plans and illustrate their behaviour by a numerical example.
Since the second approach is much more efficient, we will choose it in the numerical part.
Further, we provide the corresponding segmentation algorithms in Subsection~\ref{subsec:entropic}.
\subsection{Discrete Kernels}\label{subsec:STK}
In view of numerical applications, we focus now on discrete OT for the kernel construction.
Let $(x_i)_{i \in I}$ with $I = \{i=1,\ldots,m\}\subset \ensuremath{\mathbb{Z}}$ and $(y_j)_{j \in J}$ with $J = \{j=1,\ldots,n\}\subset \ensuremath{\mathbb{Z}}$ be the support points of
$\mu = \sum_{i \in I} \mu_i \delta_{x_i}$ and $\nu = \sum_{j \in J} \nu_j \delta_{y_j}$, respectively.
Then, the OT plan is given by
$$
\hat \pi = \argmin_{\pi \in \Pi(\mu,\nu)} \sum_{i \in I} \sum_{j \in J} c(x_i,y_j) \pi(i,j),
$$
and similarly for the regularized OT plans.
Here, $\hat \pi$ can be interpreted as a mapping $\hat \pi\colon I \times J \to \ensuremath{\mathbb{R}}_{\ge 0}$.
For convenience,
we extend all vectors and mappings to the whole integers by setting them to zero outside of the index sets $I$ and $J$.
\paragraph{Smoothed Transition Kernels from OT}
The construction of smoothed transition kernels as proposed in Subsection~\ref{sec:frobenius_perron_2}
relies on the transport map $\hat T$ associated to the OT plan~$\hat \pi$.
Unfortunately, for the discrete transport problem,
this OT map does not necessarily exist.
Therefore, we replace the smoothed transition kernels described in Subsection~\ref{sec:frobenius_perron_2}
by a construction that uses transport plans instead of transport maps.
Let $\sigma_{\varepsilon}\colon \ensuremath{\mathbb{Z}} \to \ensuremath{\mathbb{R}}$ denote some positive, normalized smoothing kernel
centered around $i=0$ with finite width. For $i \in I$, we smooth $\hat \pi$ in $j$-direction to get
\[
K^1_{\hat \pi,\varepsilon}(i,j) \coloneqq \biggl(\frac{\hat \pi(i,\cdot)}{\mu_i\sum_{k \in J} \sigma_\varepsilon(k-\cdot)} \ast \sigma_\varepsilon\biggr) (j), \quad j \in J
\]
and set $K^1_{\hat\pi,\varepsilon}(i,j) \coloneqq 0$ for $j \not \in J$ or $i \not \in I$.
The rescaling of $\hat \pi$ ensures that mass is preserved, i.e.,
\begin{align}
\sum_{j \in J} K^1_{\hat \pi,\varepsilon}(i,j)
&= \sum_{j \in J} \sum_{r \in \mathbb Z} \frac{\hat \pi(i,r)}{\mu_i\sum_{k \in J} \sigma_\varepsilon(k-r)} \sigma_\varepsilon(j-r) = \sum_{r \in \mathbb Z} \frac{\hat \pi(i,r)}{\mu_i\sum_{k \in J} \sigma_\varepsilon(k-r)} \sum_{j \in J}\sigma_\varepsilon(j-r) = 1.
\end{align}
Next, we smooth in $i$-direction as
\[
K^2_{\hat \pi,\varepsilon}(i,j) \coloneqq \frac{K^1_{\hat \pi,\varepsilon}(\cdot,j) \ast \sigma_\varepsilon (i)}{\sum_{k \in I}\sigma_\varepsilon(i-k)}
\]
and set again $K^2_{\hat \pi,\varepsilon}(i,j)\coloneqq 0$ for $j \not \in J$ or $i \not \in I$. Here, the denominator ensures that the mass is only distributed between indices $i\in I$
and consequently, for any $i \in I$,
\begin{align*}
\sum_{j \in J} K^2_{\hat\pi,\varepsilon}(i,j)
=
\sum_{j \in J} \frac{\sum_{r \in \mathbb Z} K^1_{\hat \pi,\varepsilon}(r,j) \sigma_\varepsilon(i-r)}{\sum_{k \in I}\sigma_\varepsilon(i-k)} = \frac{\sum_{r \in I} \sigma_\varepsilon(i-r) }{\sum_{k \in I}\sigma_\varepsilon(i-k)} \sum_{j \in J} K^1_{\hat \pi,\varepsilon}(r,j)
= 1.
\end{align*}
Then, the final kernel is defined by
\[K_{\hat \pi,\varepsilon}(i,j) \coloneqq \frac{K^2_{\hat \pi,\varepsilon}(i,j)}{\sum_{i \in I} K^2_{\hat \pi,\varepsilon}(i,j) \mu_i}.\]
It is straightforward to check that this kernel fulfills \ref{enum:nu_marg},
and also \ref{enum:mu_marg}, with the smoothed marginal measure $\nu^\varepsilon_j = \sum_{i \in I} K^2_{\hat\pi,\varepsilon}(i,j) \mu_i$, i.e., $\sum_{i \in I} K_{\hat\pi,\varepsilon}(i,j) \mu_i = 1$ and $\sum_{j \in J} K_{\hat\pi,\varepsilon}(i,j) \nu^\varepsilon_j = 1$.
Property~\ref{enum:cpt_simple}, that is, simplicity of the largest singular value of $L_\varepsilon$ defined by \eqref{FP_kernel}, might fail in cases where $\hat\pi$ has a ``block-diagonal structure'' and the width of $\sigma_\varepsilon$ is small enough for the blurred kernel $K_{\hat \pi,\varepsilon}$ to retain this structure. In this case, either $\varepsilon$ needs to be increased, or one obtains ``perfectly'' coherent sets corresponding to the largest singular value, see Remark~\ref{rem:simplicity}.
\paragraph{Kernels from Regularized OT}
Having computed the OT plan $\hat \pi_\varepsilon > 0$ of the discrete regularized OT,
the corresponding kernel for our transfer operator is given by
\begin{equation} \label{sm_kern_ot_d}
K_\varepsilon (i,j)
\coloneqq
\frac{\hat \pi_\varepsilon(i,j)}{\mu(i)\nu(j)} , \quad i \in I, j \in J.
\end{equation}
For plans $\hat \pi_{\varepsilon,\kappa}$ arising from
unbalanced OT we have to choose
\begin{equation} \label{unbal_kern_ot_d}
K_\varepsilon (i,j)
\coloneqq
\frac{\hat \pi_{\varepsilon,\kappa} (i,j)}{\tilde \mu(i) \tilde \nu(j)} , \quad i \in I, j \in J,
\end{equation}
where
$\tilde \mu = \sum_j \hat \pi_{\varepsilon,\kappa} (\cdot,j)$
and
$\tilde \nu = \sum_i \hat \pi_{\varepsilon,\kappa} (i,\cdot)$
are the marginals of $\hat \pi_{\varepsilon,\kappa}$.
Then, the properties \ref{enum:nu_marg}--\ref{enum:cpt_simple} are again ensured with respect to $\tilde \mu$ and $\tilde \nu$.
\paragraph{Numerical Comparison}
The proposed methods for creating kernels are compared numerically for the cost $c(x_i,y_j) = (x_i-y_j)^2$.
\renewcommand\curfolder{img/kernel_comparison}
\begin{figure}[p!]
\centering
\includegraphics[width=0.9\textwidth]{\curfolder/kernel_comparison_data.pdf}
\caption{Probability distributions $\mu$ and $\nu$ for comparing different transition kernel constructions.}
\label{fig:kernel_comparison_data}
\vspace{.5cm}
\centering
\includegraphics[clip, trim=1.5cm 1.5cm .8cm 1.5cm, width=0.9\textwidth]{\curfolder/kernel_comparison.pdf}
\caption{Comparison of kernels from regularized OT (left) and smoothed kernels from OT with a Gaussian blur (middle) and a ball-averaging one (right). The vertical direction belongs to the first and the horizontal direction to the second component of the kernels under comparison.
The blur width $w = \sqrt{\varepsilon / 2}$ is adapted to the regularization parameter $\varepsilon$ of the regularized OT, such that all kernels have equal bandwidth.
}
\label{fig:kernel_comparison}
\end{figure}
For this purpose, we fix the two probability densities $\mu$ and $\nu$ on $[0,1]$
displayed in Fig.~\ref{fig:kernel_comparison_data},
which are sums of Gaussians.
The kernel proposed in \eqref{sm_kern_ot_d} for different standard deviations $w = \sqrt{\varepsilon/2}$ is shown in the left column of Fig.~\ref{fig:kernel_comparison}.
To get a clue about the smoothed kernel, we blur the OT plan $\hat \pi$ using two different functions, namely a Gaussian kernel $\sigma_w^1\colon \ensuremath{\mathbb{Z}} \to \ensuremath{\mathbb{R}}$
and an averaging blur $\sigma_w^2\colon \ensuremath{\mathbb{Z}} \to \ensuremath{\mathbb{R}}$ within a ball
of width $\lfloor w \rfloor$, both centered around $0$.
To ensure that $\sigma_w^1$ has finite width, we set the values below $10^{-4}$ to zero.
Both discrete functions are normalized, such that they sum up to one.
The resulting smoothed kernels are depicted in the middle and right columns in Fig.~\ref{fig:kernel_comparison}.
They look similar to the left ones, however, the averaging kernel appears to be artificially rough for large $w$.
As a consequence, it seems natural to use regularized OT plans instead of smoothed OT plans.
Note that the cost for computing $\hat \pi_\varepsilon$ with the Sinkhorn algorithm scales with $1/\varepsilon$,
i.e., $\varepsilon$ should not be to small, see~\citet{C2013}. This is not a real issue, since we need a certain amount of blur anyways in order to ensure that the leading singular value
in the corresponding transfer operator $L_\varepsilon$ is simple.
\subsection{Segmentation Algorithms}\label{subsec:entropic}
In our numerical examples in the next section, we apply the kernel \eqref{sm_kern_ot_d}. However, we will see that for the addressed applications
the unbalanced OT plan $\hat \pi_{\varepsilon,\kappa}$ with corresponding kernel \eqref{unbal_kern_ot_d} leads to more natural
results.
In the following, we use the matrix-vector notation
$$
\mu \coloneqq \left( \mu(i) \right)_{i=1}^m, \; \Sigma_\mu \coloneqq \mathrm{diag} \, \mu,\;
\nu \coloneqq \left( \nu(j) \right)_{j=1}^n, \Sigma_\nu \coloneqq \mathrm{diag} \, \nu,
$$
and $\|f\|_\mu \coloneqq \|\Sigma_\mu^{\scriptscriptstyle 1/2} f\|_2$, $\|g\|_\nu \coloneqq \|\Sigma_\nu^{\scriptscriptstyle 1/2} g\|_2$.
Further, let
$K_\varepsilon \coloneqq \left( K_\varepsilon(i,j) \right)_{i,j=1}^{m,n}$,
$\hat \pi_\varepsilon \coloneqq \left( \hat \pi_\varepsilon(i,j) \right)_{i,j=1}^{m,n}$,
and similarly for the unbalanced kernels.
Then \eqref{sm_kern_ot_d} becomes
$K_\varepsilon = \Sigma_\mu^{\scriptscriptstyle -1} \hat \pi_\varepsilon \Sigma_\nu^{\scriptscriptstyle -1}$.
To solve the segmentation model~\eqref{problem},
we have to find the second largest singular values of the discrete transfer operator
$L_\varepsilon \colon (\mathbb R^m, \|\cdot\|_\mu) \to (\mathbb R^n, \|\cdot\|_\nu)$
given by
\[
L_\varepsilon f = K_\varepsilon^\mathrm{T} \Sigma_\mu f = \Sigma_\nu^{-1} \hat \pi_\varepsilon^\mathrm{T} f.
\]
Then, the Rayleigh quotient in \eqref{problem} can be rewritten as
\begin{align}\label{svd}
\frac{ \langle L_\varepsilon f, g \rangle_\nu}{ \Vert f\Vert_{\mu} \Vert g \Vert_{\nu}}
&=
\frac{ f^\mathrm{T} \Sigma_\mu K_\varepsilon \Sigma_\nu g }{\|\Sigma_\mu^{\scriptscriptstyle 1/2} f\|_2 \, \|\Sigma_\nu^{\scriptscriptstyle 1/2} g\|_2}
=
\frac{ f^\mathrm{T} \hat \pi_\varepsilon g }{\|\Sigma_\mu^{\scriptscriptstyle 1/2} f\|_2 \, \|\Sigma_\nu^{\scriptscriptstyle 1/2} g\|_2}
=
\frac{ u^\mathrm{T} \Sigma^{\scriptscriptstyle -1/2}_\mu \hat \pi_\varepsilon \Sigma^{\scriptscriptstyle -1/2}_\nu v }{\|u\|_2 \, \|v\|_2},
\end{align}
where we substituted $u \coloneqq \Sigma_\mu^{\scriptscriptstyle 1/2} f$ and $v \coloneqq \Sigma_\nu^{\scriptscriptstyle 1/2} g$.
The constraint in \eqref{problem} becomes
$$0 = \langle f, 1_\mathbb{X}\rangle_\mu = f^\mathrm{T} \Sigma_\mu 1_\mathbb{X} = u^\mathrm{T} \Sigma_\mu^{\scriptscriptstyle 1/2} 1_\mathbb{X} = \langle u, \Sigma_\mu^{\scriptscriptstyle 1/2} 1_\mathbb{X} \rangle$$
and indeed $\Sigma_\mu^{\scriptscriptstyle 1/2} 1_\mathbb{X}$ is the left singular vector to the largest singular value of $\Sigma^{\scriptscriptstyle -1/2}_\mu \hat \pi_\varepsilon \Sigma^{\scriptscriptstyle -1/2}_\nu$.
This holds also accordingly for the dominant right singular vector.
To compute the singular vectors belonging to the largest singular values of this matrix, we apply a (truncated) singular value decomposition (SVD).
We summarize our steps in Algorithm~\ref{alg:r-ot}.
Readers familiar with \citet{FrSaMo10} will notice the evident analogies between our algorithm and \citet[Lem.~1]{FrSaMo10} in computing the segmentation (coherent sets).
The transition matrix $P$ therein connects to the objects used by us via~$\Sigma_{\nu}^{-1}P^{\mathrm{T}} = K_{\varepsilon}^{\mathrm{T}}$.
If the marginal requirement on $\mu$ and $\nu$ is not fulfilled, unbalanced OT may be preferable.
Similarly as above, the same considerations for unbalanced regularized OT lead to Algorithm~\ref{alg:u-r-ot}.
\begin{algorithm}[!tb]
\begin{algorithmic}
\State Input: Particle distributions $\mu$ and $\nu$ with $\mu^\mathrm{T} 1_m = \nu^\mathrm{T} 1_n$,
$\varepsilon >0$.
\State 1. Compute the regularized OT plan $\hat{\pi}_\varepsilon$ using Sinkhorn's algorithm.
\State 2. Compute the first non-trivial singular vectors $u_2, v_2$ of $\Sigma^{\scriptscriptstyle -1/2}_\mu \hat\pi_\varepsilon \Sigma^{\scriptscriptstyle -1/2}_\nu$ by truncated SVD.
\State 3. Set \smash{$\hat{f} \coloneqq \Sigma^{\scriptscriptstyle -1/2}_\mu u_2$} and \smash{$\hat{g} \coloneqq \Sigma^{\scriptscriptstyle -1/2}_\nu v_2$}.
\caption{Segmentation based on regularized OT}
\label{alg:r-ot}
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[!tb]
\caption{Segmentation based on unbalanced regularized OT}
\label{alg:u-r-ot}
\begin{algorithmic}
\State Input: Particle distributions $\mu$ and $\nu$, $\varepsilon >0$, $\kappa > 0$.
\State 1. Compute the unbalanced regularized OT plan $\hat{\pi}_{\varepsilon, \kappa}$ using Sinkhorn's algorithm.\\
\hspace*{0.3cm} Set $\tilde \mu \coloneqq \hat{\pi}_{\varepsilon, \kappa} 1_n$, $\tilde \nu \coloneqq \hat{\pi}_{\varepsilon, \kappa}^\mathrm{T} 1_m$.
\State 2. Compute the first non-trivial singular vectors $u_2, v_2$ of
\smash{$\Sigma^{\scriptscriptstyle -1/2}_{\tilde \mu} \hat{\pi}_{\varepsilon, \kappa} \Sigma^{\scriptscriptstyle -1/2}_{\tilde \nu}$} by truncated SVD.
\State 3. Set \smash{$\hat{f} \coloneqq \Sigma^{\scriptscriptstyle -1/2}_{\tilde \mu} u_2$} and \smash{$\hat{g} \coloneqq \Sigma^{\scriptscriptstyle -1/2}_{\tilde \nu} v_2$}.
\end{algorithmic}
\end{algorithm}
\begin{remark}[Multiphase Segmentation] \label{rem:multiphase}
In our numerical examples, we are also interested in partitions with more than
just two sets.
These can be obtained by using additional singular functions as it is also known in the extensive literature on graph cut partitions, see, e.g., \citet{Luxburg2007}.
To this end, we employ the maximizers of the problem
\begin{align} \label{third_eigenvector}
\max_{
\substack{
(f,g) \in L_2(\mathbb{X},\mu) \times L_2(\mathbb{Y},\nu)
} }
\biggl\{
\frac{ \langle L f, g \rangle_\nu}{ \Vert f\Vert_{L_2(\mathbb{X},\mu)} \Vert g \Vert_{L_2(\mathbb{Y},\nu)} }:
\langle f, 1_\mathbb{X} \rangle_\mu = \langle f, f_2 \rangle_\mu = \langle g, 1_\mathbb{Y} \rangle_\nu = \langle g, g_2\rangle_\nu = 0
\biggr\},
\end{align}
where $f_2, g_2$ are the solutions of \eqref{problem}, i.e., the second largest singular vector pair.
It is not hard to show that the solutions of \eqref{third_eigenvector} are given by the singular functions corresponding to the third largest singular value of $L$. Similarly, we can consider
further singular function pairs.
Based on these singular functions, we compute a multiphase partition using the $c$-means (or fuzzy $c$-means) algorithm.
\end{remark}
\section{Numerical Results} \label{sec:numerics}
In this section, we present various examples.
All involved transport plans are computed with respect to the squared Euclidean distance as cost function.
\subsection{Exploring Coherent Set Detection}\label{subsec:precipitation}
\renewcommand\curfolder{img/data_1}
\renewcommand\curwidth{0.235\textwidth}
\begin{figure}[tp]
\centering
\subfloat[15 Apr 2018, 08:35am.\label{subfig:rain_1_t0}]{\includegraphics[width=\curwidth]{\curfolder/data_mu.pdf}}
\,
\subfloat[15 Apr 2018, 08:55am.\label{subfig:rain_1_t1}]{\includegraphics[width=\curwidth]{\curfolder/data_nu.pdf}}
\renewcommand\curfolder{img/data_2}
\,
\subfloat[16 Apr 2018, 06:00am\label{subfig:rain_2_t0}]{\includegraphics[width=\curwidth]{\curfolder/data_mu.pdf}}
\,
\subfloat[16 Apr 2018, 06:30am.\label{subfig:rain_2_t1}]{\includegraphics[width=\curwidth]{\curfolder/data_nu.pdf}}
\caption{Two precipitation density pairs over Germany. Plotted in logarithmic scale for better visibility of low-precipitation-areas.}
\label{fig:data_clouds}
\end{figure}
First, we apply our method to a data set of precipitation densities over Germany, which was made freely available by \citet{dwd}.
We take two pairs of snapshots, see Fig.~\ref{fig:data_clouds}.
In order to cope with memory limitations, we have applied spatial averaging and masked out all indices $i, j$ with $\mu_i = 0$ or $\nu_j= 0$, resulting in the problem dimensions $\vert I\vert = 3930$ for Fig.~\ref{subfig:rain_1_t0}, $\vert J\vert = 3939$ for Fig.~\ref{subfig:rain_1_t1}, $\vert I\vert = 10725$ for Fig.~\ref{subfig:rain_2_t0} and $\vert J\vert = 10028$ for Fig.~\ref{subfig:rain_2_t1}.
In Figs.~\ref{subfig:rain_1_t0}--\ref{subfig:rain_1_t1}, we clearly see two main areas of precipitation: One in the north west, and another one in the south east.
Since these two areas do not move much, it should be easy to identify them as coherent sets.
In the second example, it is less obvious what the optimal partition should be.
Both discrete densities $\mu$ and $\nu$ are normalized such that $\sum_i \mu_i = \sum_j \nu_j = 1$.
Now, we can apply our proposed procedure from Section~\ref{sec:discrete}:
First, we compute the regularized optimal transport plan $\hat{\pi}_\varepsilon$ with regularization parameter $\varepsilon=0.01$ using the Python Optimal Transport package (POT)\footnote{Code available at \url{https://github.com/rflamary/POT} (accessed: 26.06.2020)}, where the distance between two neighboring pixels is $1$.
Then, we compute the singular vectors $u_2$ and $v_2$ of the matrix \smash{$\Sigma^{\scriptscriptstyle -1/2}_\mu \hat{\pi}_\varepsilon \Sigma^{\scriptscriptstyle -1/2}_\nu$} as well as the optimal partition vectors $\hat{f} = \Sigma^{\scriptscriptstyle -1/2}_\mu u_2$ and $\hat{g} = \Sigma^{\scriptscriptstyle -1/2}_\nu v_2$.
Numerically, we observe that choosing $\varepsilon$ small produces partition functions that are almost constant on the respective parts with a sharp transition between them, whereas very large $\varepsilon$ yields approximately affine partition functions, which is clearly not what we want.
In any case, $\varepsilon$ has to be chosen large enough such that the Sinkhorn algorithm converges in acceptable time.
\renewcommand\curfolder{img/balanced_1e-2_W2_1}
\renewcommand\curimgwidth{0.17\textwidth}
\renewcommand\curbarwidth{0.09\textwidth}
\renewcommand\curboxwidth{0.18\textwidth}
\begin{figure}[tbp]
\centering
\subfloat[$\hat{f} = \Sigma^{-1/2}_\mu u_2$.]{
\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/segm_t0.pdf}}}
\,
\subfloat[$\hat{g} = \Sigma^{-1/2}_\nu v_2$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/segm_t1.pdf}}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[Classifier $\hat{f} \lesseqgtr 0$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/hard_t0.pdf}}}
\,
\subfloat[Classifier $\hat{g} \lesseqgtr 0$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/hard_t1.pdf}}}
\caption{Computed coherent sets for the precipitation densities in Figs.~\ref{subfig:rain_1_t0}--\ref{subfig:rain_1_t1} using regularized OT.}
\label{fig:opt_partition_1}
\vspace{.5cm}
\renewcommand\curfolder{img/lambd1e-2_reg1_unnormalized_W2_1}
\centering
\subfloat[$\hat{f} = \Sigma^{-1/2}_\mu u_2$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/segm_t0.pdf}}}
\,
\subfloat[$\hat{g} = \Sigma^{-1/2}_\nu v_2$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/segm_t1.pdf}}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[Classifier $\hat{f} \lesseqgtr 0$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/hard_t0.pdf}}}
\,
\subfloat[Classifier $\hat{g} \lesseqgtr 0$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/hard_t1.pdf}}}
\caption{Computed coherent sets for the precipitation densities in Figs.~\ref{subfig:rain_1_t0}--\ref{subfig:rain_1_t1} using unbalanced regularized OT.}
\label{fig:unbalanced_1}
\vspace{.5cm}
\renewcommand\curfolder{img/lambd1e-2_reg1_unnormalized_W2_2}
\centering
\subfloat[$\hat{f} = \Sigma^{-1/2}_\mu u_2$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/segm_t0.pdf}}}
\,
\subfloat[$\hat{g} = \Sigma^{-1/2}_\nu v_2$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/segm_t1.pdf}}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[Classifier $\hat{f} \lesseqgtr 0$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/hard_t0.pdf}}}
\,
\subfloat[Classifier $\hat{g} \lesseqgtr 0$.]{\makebox[\curboxwidth][c]{\includegraphics[align=c, width=\curimgwidth]{\curfolder/hard_t1.pdf}}}
\caption{Computed coherent sets for the precipitation densities in Figs.~\ref{subfig:rain_2_t0}--\ref{subfig:rain_2_t1} using unbalanced regularized OT.}
\label{fig:unbalanced_2}
\end{figure}
Results are shown in Fig.~\ref{fig:opt_partition_1}.
Note that the dark-blue pixels in Fig.~\ref{fig:data_clouds} are only included to indicate the domain of $\mu$ and $\nu$ and not part of the computation.
Hence, they are discarded in Figs.~\ref{fig:opt_partition_1}--\ref{fig:unbalanced_2}.
We observe that the method works roughly as expected:
As shown in our theory part, the first singular value is one and belongs to the singular pair $(1_{\mathbb{X}}, 1_{\mathbb{Y}})$.
Further, the two precipitation areas are indicated by differing signs in the partition vectors corresponding to the second singular value.
However, taking a closer look, splitting the precipitation areas (or ``fuzzy classification'') does not work perfectly.
In Fig.~\ref{fig:opt_partition_1}, some parts of the bigger ``cloud'' (as we refer to precipitation areas from now on) have values greater than zero, although the rest has negative values.
Ceonsequently, some parts of the upper cloud are classified as ``red'', i.e., as part of the lower cloud.
This issue is even more pronounced for the included hard ``classification'' according to the criteria $\hat{f} \geq 0$ and $\hat{g} \geq 0$.
Note that simple thresholding was sufficient for our purposes, but more sophisticated approaches such as fuzzy $c$-means (which will be used in the next subsection) or sparsity-promoting clustering, see~\citet{froyland2019sparse}, are also applicable.
Next, we want to resolve the mentioned issue of wrong classification.
Of course, it is inaccurate to model precipitation densities as transported mass particles, since they can come down as rain and simply disappear.
Hence, the misclassification is not surprising as the OT plan needs to transport this disappearing mass somewhere else.
Indeed, an examination of the transport plan reveals that mass preservation does not hold in the two clouds, thus they cannot fulfill the coherence condition $L 1_{X_k} = 1_{Y_k}$ (see Section~\ref{sec:segm}) and the transport needs to shift some of the mass from the upper cloud to the lower one.
To compensate for this effect, we propose to use unbalanced regularized OT, relaxing the mass conservation condition on the marginals, see Section~\ref{sec:OT}.
Noteworthy, this modification does not introduce any major computational overhead.
Again, we use $\varepsilon=0.01$ and choose $\kappa=1$, see~\eqref{eq:unbalanced_entropic_ot}. As mentioned by \citet{SFVTP19}, the parameter $\kappa$ intuitively corresponds to a choice of radius for which, when exceeded, it is cheaper to produce or destroy mass instead of transporting it between the two clouds.
Hence, we should choose $\kappa$ small enough to have an effect compared to balanced OT, but large enough so that the marginals of the resulting transport plan are still close to the given $\mu$ and $\nu$.
The results for the data from Figs.~\ref{subfig:rain_1_t0}--\ref{subfig:rain_1_t1} and Figs.~\ref{subfig:rain_2_t0}--\ref{subfig:rain_2_t1}
are displayed in Figs.~\ref{fig:unbalanced_1} and~\ref{fig:unbalanced_2}, respectively.
As expected, the first singular pair is still given by $(1_{\mathbb{X}}, 1_{\mathbb{Y}})$.
Now, the area classification in Figs.~\ref{subfig:rain_1_t0}--\ref{subfig:rain_1_t1} is more or less perfect and hence also the second singular value is almost one.
This might not be surprising, as the areas are well separated and moving mass between them is quite expensive.
Compared to standard regularized transport, using unbalanced regularized transport for the data from Figs.~\ref{subfig:rain_2_t0}--\ref{subfig:rain_2_t1} does not change the partition much.
Overall, the results based on unbalanced regularized OT look very promising and hence we use such plans in all further experiments.
Note that a comparison with the coherent set methods mentioned in Section~\ref{sec:intro} is not possible, since these require a known dynamic.
\subsection{Particles Moving in a Potential} \label{subsec:wells}
In this subsection, we discuss the example of \citet[Sec.~6.2, p.~18]{koltai2018optimal} in a slightly modified form.
The dynamical system under consideration consists of particles moving according to standard Brownian motion with a drift term induced by a potential and a rotating force.
More precisely, our particle trajectories are solutions to the stochastic differential equation \[
\,\mathrm{d} x_t = (F_r + F_c)(x_t) \,\mathrm{d} t + \sqrt{2\beta^{-1}}\,\mathrm{d} w_t,
\]
where $w_t$ denotes standard Brownian motion, $F_r(x) = -\nabla W(x)$ is the force coming from the $3$-well potential
\[
W(x) = \cos(3\varphi) + 10(r-1)^2,\quad \text{where } x= \binom{r\cos(\varphi)}{r\sin(\varphi)}
\]
and $F_c$ is a circular driving force in clock-wise direction, given by
\[
F_c(x) \coloneqq \mathrm{e}^{-\beta W(x)}
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
x.
\]
In statistical physics, $\beta$ is called \emph{inverse temperature}, which is chosen as $2.0$ in this experiment. Note that $F_c$ is strongest in the valley of each well.
Overall, the particles tend to remain in a well for some time and occasionally hop to another one due to their diffusive motion and the circular force.
We expect the particle distribution to converge to an equilibrium with modes at the potential minima, slightly rotated in clock-wise direction.
First, we create $1000$ initial particle positions for the measure $\mu$ as follows:
We sample a single-particle-trajectory with $50000$ steps of time length~$0.01$ using an Euler--Maruyama-scheme starting from $(1,0)$, where the circular driving force $F_c$ is neglected.
Then, we draw $1000$ trajectory points uniformly at random from these points without replacement. The final discrete measure $\nu$ is simulated by
the numerical trajectories for each one of these $1000$ initial particle positions using $300$ Euler--Maruyama-steps of length~$0.01$, where $F_c$ is included now. This yields $1000$ pairs of starting and ending position $(x_0^i, x_1^i)$ displayed in Fig.~\ref{fig:data_wells}.
To visualize the trajectory of individual particles, we colored each particle in Fig.~\ref{subfig:wells_data_1} in the same color as in Fig.~\ref{subfig:wells_data_0}. Further, in Fig.~\ref{subfig:wells_lines}, we drew connecting lines between initial and terminal particle positions.
Next, we apply Algorithm \ref{alg:u-r-ot} to
\smash{$\mu = \frac{1}{n}\sum_{i=1}^n \delta_{x_0^i}$ and $\nu = \frac{1}{n}\sum_{i=1}^n \delta_{x_1^i}$} with $\kappa=1$.
The regularization parameter $\varepsilon$ is chosen so that the corresponding kernel $\exp(-c/\varepsilon)$ (see \eqref{sinkhorn_kernel}) has standard deviation one third of the particles mean distance at $t=0$.
We write $f_k = \Sigma^{\scriptscriptstyle -1/2}_\mu u_k$ and $g_k = \Sigma^{\scriptscriptstyle -1/2}_\nu v_k$ for the partition vectors belonging to the decreasingly ordered singular values $\sigma_k$, where $u_k$ and $v_k$ are the corresponding left and right singular vectors.
The obtained partition vector pairs for $k=2,3$ are shown in Fig.~\ref{fig:partition_wells}.
Using the information from $f_k, g_k$, $k=2, 3$, we can embed every point $x_t^i$ into $\ensuremath{\mathbb{R}}^2$ via $\Phi \colon x_0^i \mapsto ((f_2, f_3))_i$, $x_1^i \mapsto ((g_2, g_3))_i$.
Now, we partition the data using the fuzzy $c$-means algorithm\footnote{Implementation available at \url{https://github.com/omadson/fuzzy-c-means} (accessed: 26.06.2020)} as described in Sec.~\ref{sec:segm} on this embedded data for three clusters, see also \citet{SS2012}.
In contrast to applying fuzzy $c$-means directly on the individual snapshots at $t=0, 1$, we naturally obtain correspondences between the clusters at the different time steps, where OT serves as a proxy for the underlying dynamics.
The obtained hard clusters and fuzzy membership values are displayed in Fig.~\ref{fig:cmeans_wells}.
Note that applying the method in \citet{froyland2013analytic} under usage of the particle label information cannot yield coherent sets for such strong particle mixing.
Other methods such as the finite-time Lyapunov exponent \citep{shadden2005definition} also only work for short timespans.
On the other hand, comparing this with our method, it is clear from Fig.~\ref{fig:data_wells} that the OT assignment will be very different from the ground truth particle correspondences.
Thus, it aims for a partition based on the ``macroscopic'', ensemble density level rather than on the ``microscopic'' level of individual, distinguishable particles.
Indeed, we observe that the coherent sets are three denser blobs due to the energy landscape.
Since there was no circular force present during the construction of $\mu$, there is a slight offset between the initial and stationary distributions, which is captured by the coherent sets.
As expected, the likeliness of being in a cluster decreases if a point is close to the boundaries of the well.
Consequently, there is some uncertainty about the exact cluster boundaries.
\renewcommand\curfolder{img/wells}
\renewcommand\curwidth{0.27\textwidth}
\begin{figure}[htbp]
\centering
\subfloat[Data at time $t=0$.\label{subfig:wells_data_0}]{\includegraphics[align=c, width=\curwidth]{\curfolder/data_0.pdf}}
\,
\subfloat[Data at time $t=1$.\label{subfig:wells_data_1}]{\includegraphics[align=c, width=\curwidth]{\curfolder/data_1.pdf}}
\,
\subfloat[Movement of particles.\label{subfig:wells_lines}]{\includegraphics[align=c, width=\curwidth]{\curfolder/lines.pdf}}
\caption{Particles moving in a potential with circular driving force.
The colors in Figs.~(a)--(b) indicate the angular coordinates at time $t=0$, illustrating the particle mixing.}
\label{fig:data_wells}
\vspace{.5cm}
\renewcommand\curwidth{0.19\textwidth}
\renewcommand\curbarwidth{0.07\textwidth}
\subfloat[$f_2 = \Sigma^{-1/2}_\mu u_2$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_1_t0.pdf}}
\,
\subfloat[$g_2 = \Sigma^{-1/2}_\nu v_2$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_1_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_1_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[$f_3 = \Sigma^{-1/2}_\mu u_3$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_2_t0.pdf}}
\,
\subfloat[$g_3 = \Sigma^{-1/2}_\nu v_3$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_2_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_2_bar.pdf}}
\addtocounter{subfigure}{-1}
\caption{First and second partition vectors for data displayed in Fig.~\ref{fig:data_wells}.}
\label{fig:partition_wells}
\vspace{.5cm}
\renewcommand\curwidth{0.19\textwidth}
\renewcommand\curbarwidth{0.07\textwidth}
\subfloat[Hard, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/hard_t0.pdf}}
\,
\subfloat[Hard, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/hard_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/hard_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[Cluster $0$, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_0_t0.pdf}}
\,
\subfloat[Cluster $0$, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_0_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/cmeans_0_bar.pdf}}
\addtocounter{subfigure}{-1}
\par\smallskip
\subfloat[Cluster $1$, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_1_t0.pdf}}
\,
\subfloat[Cluster $1$, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_1_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/cmeans_1_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[Cluster $2$, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_2_t0.pdf}}
\,
\subfloat[Cluster $2$, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_2_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/cmeans_2_bar.pdf}}
\caption{Results of hard classification and fuzzy $c$-means.
The color scheme in (c)--(h) encodes the likeliness that a point belongs to the cluster $j$ with $j=0, 1, 2$.}
\label{fig:cmeans_wells}
\end{figure}
\subsection{Particles with Pairwise Lennart--Jones Potentials}\label{subsec:lennart-jones}
In the last example, we aim for a segmentation of particle groups with a slightly more realistic data set, consisting of $200$ particle trajectories created with the molecular dynamics simulation software LAMMPS \citep{lammps}.
In our simulation, the particles interact with each other in terms of a pairwise Lennart--Jones potential with cutoff, essentially repelling each other in close proximity but attracting each other otherwise, such that there is some optimal energy-minimizing pairwise distance \citep{rapaport2004art}.
Given some initial velocity for the particles, they start to stick to each other over time and slowly form groups, which in turn connect to larger groups and so on.
We take two snapshots of the simulation showing some group formation, see Fig.~\ref{fig:data_lj}.
As the domain is the two-dimensional torus, particles leaving the domain on one side come back in from the opposite side in the visualization.
\footnote{The script for generating the trajectories is available at the blog post under \url{http://nznano.blogspot.com/2017/11/molecular-dynamics-in-python.html\#Implementation-in-LAMMPS} (accessed: 26.06.2020)}
Then, we apply our computational scheme with the same parameter choices as in Subsection~\ref{subsec:wells} to obtain the clusters.
The computed segmentation vectors and the induced fuzzy clustering are depicted in Figs.~\ref{fig:partition_lj} and~\ref{fig:cmeans_lj}, respectively.
From a visual point of view, the clustering scheme ``correctly'' detects the stable group on the left, roughly three connecting groups in the middle and several smaller connecting groups on the right, independent of the mixing of several individual particles.
As we obtained the data by a molecular dynamics simulation, we can access the ``ground truth'' particle labels and track the position of every particle forward and backward in time.
Since the particle mixing in this example is not as strong as in Section~\ref{subsec:wells}, it is natural to ask whether the hard cluster labels of particle $i$ are the same at $t=0, 1$.
We observe that the cluster labels agree in $176$ of the total $200$ cases, that is, around $84\%$ of the particles retain their cluster label in time.
Furthermore, we may compare the hard cluster label of each particle $i$ at time $t=0$ with the one that particle $i$ is assigned at time $t=1$ in two scatter plots, where the position in the plot of each particle is fixed and the two different colorings indicate the hard cluster labels for $t=0$ and $t=1$ (that is, corresponding to the left or right singular vectors), respectively.
The results are shown in Fig.~\ref{fig:ground_truth_lj}.
\renewcommand\curfolder{img/lennart_jones}
\renewcommand\curwidth{0.27\textwidth}
\begin{figure}[htbp]
\centering
\subfloat[Data at time $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/data_0.pdf}}
\,
\subfloat[Data at time $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/data_1.pdf}}
\caption{Particles moving in a potential with circular driving force.
The colors indicate the horizontal coordinate at $t=0$, again illustrating the particle mixing.}
\label{fig:data_lj}
\vspace{.5cm}
\renewcommand\curwidth{0.19\textwidth}
\renewcommand\curbarwidth{0.07\textwidth}
\subfloat[$f_2 = \Sigma^{-1/2}_\mu u_2$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_1_t0.pdf}}
\,
\subfloat[$g_2 = \Sigma^{-1/2}_\nu v_2$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_1_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_1_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[$f_3 = \Sigma^{-1/2}_\mu u_3$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_2_t0.pdf}}
\,
\subfloat[$g_3 = \Sigma^{-1/2}_\nu v_3$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_2_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_2_bar.pdf}}
\addtocounter{subfigure}{-1}
\caption{First and second partition vectors for data displayed in Fig.~\ref{fig:data_lj}.}
\label{fig:partition_lj}
\vspace{.5cm}
\renewcommand\curwidth{0.19\textwidth}
\renewcommand\curbarwidth{0.07\textwidth}
\subfloat[Hard, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/hard_t0.pdf}}
\,
\subfloat[Hard, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/hard_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/hard_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[Cluster $0$, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_0_t0.pdf}}
\,
\subfloat[Cluster $0$, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_0_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/cmeans_0_bar.pdf}}
\addtocounter{subfigure}{-1}
\par
\subfloat[Cluster $1$, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_1_t0.pdf}}
\,
\subfloat[Cluster $1$, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_1_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/cmeans_1_bar.pdf}}
\addtocounter{subfigure}{-1}
\,
\subfloat[Cluster $2$, $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_2_t0.pdf}}
\,
\subfloat[Cluster $2$, $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/cmeans_2_t1.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/cmeans_2_bar.pdf}}
\caption{Results of hard classification and fuzzy $c$-means.
The color scheme in (c)--(h) encodes the likeliness that a point belongs to the corresponding cluster $j$ with $j=0, 1, 2$.}
\label{fig:cmeans_lj}
\end{figure}
\begin{figure}
\centering
\renewcommand\curwidth{0.23\textwidth}
\subfloat[Positions for $t=0$, cluster labels for $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/hard_t0.pdf}}
\,
\subfloat[Positions for $t=0$, cluster labels for $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/tracing_t0.pdf}}
\,
\subfloat[Positions for $t=1$, cluster labels for $t=1$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/hard_t1.pdf}}
\,
\subfloat[Positions for $t=1$, cluster labels for $t=0$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/tracing_t1.pdf}}
\caption{Comparison of hard cluster labels corresponding to left and right singular vectors, for particle positions at both $t=0$ and $t=1$, respectively. Here we have used the ground truth particle labels $i$.}
\label{fig:ground_truth_lj}
\end{figure}
\subsection{Particle Tracking with Concatenated Transfer Operators}\label{sec:Ex4}
Here, we discuss an example also analyzed by \citet{froyland2014almost} and \citet[Sec.~V.A]{banisch2017understanding}.
Consider the non-autonomous system
\begin{equation}\label{eq:ODE_Ex4}
\begin{aligned}
\frac{\,\mathrm{d} x}{\,\mathrm{d} t} &= -\pi A \sin\bigl(\pi f(t, x)\bigr)\cos(\pi y) \\
\frac{\,\mathrm{d} y}{\,\mathrm{d} t} &= \pi A \cos\bigl(\pi f(t, x)\bigr)\sin(\pi y) \frac{\,\mathrm{d} f}{\,\mathrm{d} x}(t, x)
\end{aligned}
\end{equation}
with $f(x, t) = \alpha \sin(\omega t)x^2 + (1-2\alpha \sin(\omega t))x$ and parameters $A=0.25$, $\alpha = 0.25$ and $\omega = 2\pi$.
This system describes two counter-rotating gyres, where the vertical boundary between them oscillates periodically.
Moreover, it preserves the Lebesgue measure on $\mathbb{X}=\mathbb{Y}=[0,1]\times[0,2]$.
We initialize $n = 450$ particles on an equispaced rectangular grid.
Then, we compute their trajectories by solving \eqref{eq:ODE_Ex4} with $500$ time steps of length $\Delta t=0.02$, i.e., on the time interval~$[0,10]$.
This yields $N=501$ measures $\mu_t=\frac 1 n \sum_{i=1}^n \delta x_t^i$, $t=0,\ldots,N$.
Note that $\Delta t$ is small enough to recover the ground truth particle correspondences in most cases.
Based on the entropy regularized optimal transport plans $\hat \pi_\varepsilon$ between $\mu_t$ and $\mu_{t+1}$ with regularization parameter $\varepsilon=10^{-3}$, we construct corresponding transfer operators $L_{\varepsilon, t}\colon L_2(\ensuremath{\mathbb{R}}^n)\to L_2(\ensuremath{\mathbb{R}}^n)$ as $L_{\varepsilon, t} = \Sigma^{\scriptscriptstyle -1/2}_{\mu_t} \hat \pi_\varepsilon^T \Sigma^{\scriptscriptstyle -1/2}_{\mu_{t+1}}$.
Again, the entropy regularization readily introduces the required diffusion.
Efficient implementations of the Sinkhorn algorithm can be achieved using, e.g., multiscale schemes \citep{schmitzer19} or parallelization on GPUs \citep{C2013}.
Next, we compute the concatenated transfer operator
$L_\varepsilon = \smash{\prod_{t=0}^{N-1}} L_{\varepsilon, t}$,
for which its matrix is shown in Fig.~\ref{fig:trans_mat}.
Its block-diagonal structure already indicates coherent sets.
Finally, the partitions corresponding to the second and third singular vectors $f_i$, $g_i$, $i=1,2$,
of its SVD are shown in Fig.~\ref{fig:partition_dg}.
The corresponding singular values are $\sigma_2 \approx 0.71$ and $\sigma_3 \approx 0.35$, respectively.
For a hard clustering using fuzzy $c$-means we refer to Fig.~\ref{fig:double_gyre_clustering} in the introduction.
Particle transitions between the left and the right half of the domain are very rare, as indicated by the optimal partition.
Further, the third singular vectors illustrate that the particles, which move in closed curves around the respective gyre cores, take a long time to transition from the gyre centers to their boundaries or vice versa.
In summary, this example illustrates how our method can be used to compute coherent sets for flows with unlabeled particles.
Here, OT is used to track them through observations of subsequent timesteps; see also Particle Image Velocimetry \citep{saumier2015piv}.
\renewcommand\curfolder{img/double_gyre}
\begin{figure}[htbp]
\centering
\renewcommand\curwidth{0.4\textwidth}
\includegraphics[align=c, width=\curwidth]{\curfolder/tf_op.pdf}
\caption{Matrix of concatenated transfer operator $L_\varepsilon$. The rows and columns are ordered according to the horizontal and then vertical coordinates of the corresponding particles in the initial configuration.}
\label{fig:trans_mat}
\vspace{.5cm}
\renewcommand\curwidth{0.35\textwidth}
\renewcommand\curbarwidth{0.07\textwidth}
\subfloat[$f_2 = \Sigma^{-1/2}_\mu u_2$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_1_t0.pdf}}
\,
\subfloat[$g_2 = \Sigma^{-1/2}_\nu v_2$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_1_t500.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_1_bar.pdf}}
\addtocounter{subfigure}{-1}
\vspace{.5cm}
\subfloat[$f_3 = \Sigma^{-1/2}_\mu u_3$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_2_t0.pdf}}
\,
\subfloat[$g_3 = \Sigma^{-1/2}_\nu v_3$.]{\includegraphics[align=c, width=\curwidth]{\curfolder/segm_2_t500.pdf}}
\,
\subfloat{\includegraphics[align=c, width=\curbarwidth]{\curfolder/segm_2_bar.pdf}}
\addtocounter{subfigure}{-1}
\caption{First and second partition vectors for double gyre time series.}
\label{fig:partition_dg}
\end{figure}
\section{Conclusions} \label{sec:conclusions}
This is the first paper that merges the theories of Frobenius--Perron operators and regularized optimal transport. We have elaborated how regularized (and possibly unbalanced) OT can be used to compute coherent sets if all we know about the dynamics of a moving particle system or a continuous quantity of moving mass is a pair of measures that constitute a preimage-image pair under the dynamical evolution.
We have also shown that the theory of (regularized/unbalanced) optimal transport is fitting well to the concept of coherent sets.
Moreover, it has natural dynamical interpretations as the regularization parameter $\varepsilon\to 0$ or the number of data points $n\to \infty$, see Proposition~\ref{prop:conv-reg} and Proposition~\ref{prop:schroedinger}, respectively.
In four numerical examples we have shown how the method performs.
These examples underline the initial suspicion that without further structural ``aid'' or dynamic information the knowledge of the one-step evolution of a single measure is not sufficient to identify coherent sets correctly.
It is necessary to incorporate additional dynamical information into the analysis.
Thus, a topic interesting to address is OT of multiple measure pairs (multiple steps of evolving one measure, or one step of evolving multiple measures), such as in segmentation of vector- and manifold-valued images,
see, e.g., \citet{CTA2018,FLS2016,FLS2017,KMDL2019,thorpe17}.
Further, working with discrete OT calls for a consistency result when approximating ground truth measures with atomic ones, as it has been given for the case of (static) spectral clustering in \citet{trillos18} using transportation distances between functions as provided in the aforementioned references.
Finally, the efficacy of unbalanced OT suggests to model certain scenarios as \emph{open dynamical systems}.
\section*{Acknowledgments}
Funding through the German Research Foundation (DFG) with\-in the project STE 571/16-1 is gratefully acknowledged by GS, through grant CRC 1114 ``Scaling Cascades in Complex Systems'', Project Number 235221301, projects A01 ``Coupling a multiscale stochastic precipitation model to large scale atmospheric flow dynamics'' by PK\ and B07 ``Selfsimilar structures in turbulent flows and the construction of LES closures'' by JvL.
The authors want to thank Henning Rust, Institute for Meteorology at FU Berlin, for his advice regarding the data of the example from Section~\ref{subsec:precipitation}.
|
1,314,259,996,671 | arxiv | \section{Introduction}
Coronal holes (CHs) are regions of predominantly unipolar coronal magnetic
fields with a significant component of the magnetic field open into the heliosphere.
They are visible in spectral lines emitting at coronal temperatures as dark areas when
compared to the quiet Sun, while in the chromospheric He~{\sc i}~10830~\AA\ line they
appear bright. For detailed introduction on coronal holes see \citet{2009arXiv0906.2556M} (hereafter paper I).
CHs are identified as the source of the fast solar wind
with velocities of up to $\approx$~800~\rm km~\;s$^{-1}$\ \citep{1973SoPh...29..505K}. In
contrast, the slow wind has velocities around 400~\rm km~\;s$^{-1}$\ and is more dense and
variable in nature when compared to the fast solar wind. \citet{1996ASPC..109..491V}
found from Ulysses satellite data that the elemental composition of the fast
wind is similar to the elemental composition of the photosphere. The slow solar
wind is enriched with low first ionization potential (FIP) elements by a factor
of 3--5 greater than in the photosphere (with respect to hydrogen) while higher
FIP elements were found at solar surface abundances. The FIP effect describes
the element abundance anomalies (the enhancement of elements with low FIP such
as Fe, Mg and Si over those with high FIP like Ne and Ar) in the upper solar
atmosphere and solar wind, and can give a clue on the origin of both the fast
and the slow solar winds. \citet{1996ASPC..109..491V} concluded that the fast and
slow solar winds not only differ in their kinetics but also in their composition
of elements.
\citet{2004ApJ...612.1171W} suggested that the release of trapped plasma in closed
loop structures by magnetic reconnection could play a significant role in the solar wind flow. Such reconnection between the open and closed magnetic field
lines presumably happens continuously at coronal hole boundaries. \citet{1998ApJ...498L.165W} investigated the ejection of plasma blobs from the streamer belt linked to
the slow wind and concluded that magnetic reconnection between the distended
streamer loops and the open magnetic field lines might be behind the plasma
ejection. They also suggested that this ejection cannot account for all the
slow solar wind and a major component should, therefore, originate outside the helmet
streamers, i.e. from inside the coronal holes. \citet{2004ApJ...603L..57M}
found non-Gaussian profiles along the boundaries in an equatorial extension
of a polar CH in the mid- and high-transition region lines N~{\sc iv}~765~\AA\
and Ne~{\sc viii}~770~\AA, respectively, recorded with the Solar Measurement of Emitted
Radiation (SUMER) spectrometer on-board the Solar and Heliospheric Observatory (SoHO).
The authors suggested that these profiles are the signature of magnetic
reconnection occurring between the closed magnetic field lines of the quiet Sun
and the open of the coronal hole. Similar activity was reported by
\citet{2006A&A...446..327D} along the boundary of a polar CH.
\begin{figure*}[!ht]
\centering
\vspace{16cm}
\special{psfile=fig1_11.eps hoffset=-20 voffset=240 hscale=65 vscale=65 angle=0}
\special{psfile=fig1_2.eps hoffset=220 voffset=240 hscale=65 vscale=65 angle=0}
\special{psfile=fig1_31.eps hoffset=-20 voffset=20 hscale=65 vscale=65 angle=0}
\special{psfile=fig1_4.eps hoffset=220 voffset=20 hscale=65 vscale=65 angle=0}
\vspace*{-0.7cm}
\caption{ Equatorial coronal hole (top left), polar CH (top right),
quiet Sun (bottom left) and quiet Sun with TCHs (bottom right)
with the positions of all the corresponding identified brightening
pixels over-plotted. The CH boundaries are outlined with a black line. The
over-drawn rectangles correspond to the field-of-views shown in
Figs.~\ref{fig5}, \ref{fig6} and \ref{fig7}. }
\label{fig1}
\end{figure*}
\begin{table*}
\begin{center}
\caption{Description of the XRT data used in the present study (ECH -- equatorial
coronal hole, PCH -- polar coronal hole, TCH -- transient coronal hole). }
\label{table1}
\begin{tabular} {ccccc}
\hline
Date & Remark & Observing period & Field-of-view & Exposure time \\
& & of time (UT) & (arcsec) & (sec) \\
\hline
09/11/07 & ECH & 06:35-14:59 & $366\times366$ & 16 \\
12/11/07 & ECH & 01:17-10:59 & $374\times374$ & 16 \\
14/11/07 & ECH & 00:12-11:11 & $374\times374$ & 16 \\
16/11/07 & ECH & 18:07-23:58 & $370\times370$ & 16 \\
16/12/08 & PCH & 10:02-17:52 & $362\times337$ & 23 \\
20/09/07 & PCH & 12:22-18:05 & $1018\times291$ & 16 \\
10/01/09 & QS & 11:30-17:27 & $370\times370$ & 23 \\
13/01/09 & QS& 11:22-17:41 & $370\times370$ & 23 \\
29/11/07 & QS with TCH& 18:17-23:59 & $506\times506$ & 23 \\
\hline
\end{tabular}
\end{center}
\end{table*}
In paper I we demonstrated that although isolated equatorial CH and
equatorial extension of polar CH maintain their general shape during several
solar rotations, a closer look at their day-by-day and even hour-by-hour
evolution demonstrates significant dynamics. We showed that small-scale
loops which are abundant along coronal hole boundaries contribute to the
small-scale evolution of coronal holes. We suggested that these dynamics
are triggered by continuous magnetic reconnection already proposed by
\citet{2004ApJ...603L..57M}. The next step of our research was to analyse
images taken with the X-ray Telescope (XRT) on-board Hinode.
Seen in XRT images, CHs are highly structured and dynamic at small scales.
High cadence XRT data reveal in great detail the fine structure of coronal
bright points (BPs) and X-ray jets associated with them. X-ray jets
are collimated transient ejection of coronal plasma, first reported with the
Solar X-ray telescope (SXT) onboard Yohkoh \citep{1992PASJ...44L.173S}. They
are believed to result from magnetic reconnection \citep{1994xspy.conf...29S} and represent
plasma outflows from the reconnection site. Recently, \citet{2008ApJ...673L.211M} presented
three--dimensional simulations of flux emergence in CH combined with spectroscopic and imager
observations from XRT and EIS/Hinode of an X-ray jet. The authors report that a jet resulting from
magnetic reconnection is expelled upward along the open
reconnected field lines with values of temperature, density, and velocity in agreement with the XRT and EIS
observations. \citet{1996PASJ...48..123S} reported 100 jets over 6 months in SXT
images from the Yohkoh while \citet{2007PASJ...59S.771S} upgraded
this number to an average of 60 jet events per day in polar coronal holes. The
authors concluded that jets preferably occur inside polar coronal holes (PCH).
We should note that although TRACE images which have higher spatial
resolution were used in paper I, the detailed structure of the dynamic
changes along CH boundaries was hard to distinguish. A reason for that is
the effect of stray light in the TRACE extreme-ultraviolet (EUV) telescope
reported recently by \citet{2009ApJ...690.1264D}. The authors found that 43\% of the light
which enters TRACE through the Fe~{\sc ix/x} 171~\AA\ filter is scattered
either through diffraction off the entrance filter grid or through other
non-specific effects. This creates a haze effect and especially effects
the visibility of small-scale bright structures.
\begin{figure}[!h]
\vspace{6cm}
\special{psfile= fig_app_5.eps hoffset=0 voffset=40 hscale=55 vscale=55 angle=0}
\vspace{-2.9cm}
\caption{Polar coronal hole observed by XRT on 2007 September 20 with the positions of all the identified brightening pixels over-plotted.}
\label{fig2}
\end{figure}
\begin{figure}[!h]
\vspace{7cm}
\special{psfile= fig_app_4.eps hoffset=-10 voffset=-10 hscale=65 vscale=65 angle=0}
\caption{Quiet Sun observed by XRT on 2009 January 13 with the positions
of all the identified brightening
pixels over-plotted.}
\label{fig3}
\end{figure}
Other transient structures seen in coronal holes are the so-called plumes
observed off-limb above the North and South polar coronal holes. They were
first observed in white light as ray like structures
\citep{1965PASJ...17....1S}. They are also observed at EUV
and soft X-ray temperature \citep{1978SoPh...58..323A} as coronal outflow
structures similar to coronal jets, but hazy in nature with no sharp boundaries unlike jets.
They represent denser and cooler outflows with respect to the surrounding media and are
observed to extend from coronal BPs. They can extend up to 30 R$_{\odot}$ from the solar disk center in a plane
image \citep{2001ApJ...546..569D} and are observed to be in a steady state
for at least 24 hours \citep{1997SoPh..175..393D}. The X-ray jets have been
identified as precursors for the plume formation \citep{2008ApJ...682L.137R}.
Recently, \citet{2008SoPh..249...17W} identified coronal plumes inside equatorial
coronal holes. They found that the plumes are analogous to polar coronal
plumes. On the disk they are seen as a diffuse structure with a bright core
and associated with EUV BPs.
The present study is a continuation of paper I and presents results from
the analysis of high-cadence/high-resolution images of coronal holes (equatorial,
polar and transient) and quiet Sun from XRT/Hinode. We aim to establish which type of
event generates the non-Gaussian profiles registered at CH boundaries by
\citet{2004ApJ...603L..57M} and how they are related to the small-scale BPs
evolution along coronal hole boundaries as reported in paper I. In
Sect.~\ref{data} we describe the data used for our study. Sect.~\ref{ip}
outlines an automatic brightening identification procedure. In
Sect.~\ref{results}, we give the obtained results and draw some
conclusions on the outcome of our study. Finally, in Sect.~\ref{conclusions}
we discuss the implication of our result to the understanding of the nature
of coronal hole boundaries evolution at small scale and the possible
contribution of these events to the formation of the slow solar wind.
\begin{figure*}[!ht]
\centering
\vspace{7cm}
\special{psfile=fig_app_1.eps hoffset=-40 voffset=25 hscale=50 vscale=50 angle=0}
\special{psfile=fig_app_2.eps hoffset=130 voffset=25 hscale=50 vscale=50 angle=0}
\special{psfile=fig_app_3.eps hoffset=300 voffset=20 hscale=50 vscale=50 angle=0}
\vspace*{-0.75cm}
\caption{ Equatorial coronal hole observed by XRT on 2007 November 9, 14 and 16 with the positions of all the identified brightening pixels over-plotted. The CH boundaries are over-plotted with a black solid line.}
\label{fig4}
\end{figure*}
\section{Data, reduction and preparation}
\label{data}
We used images from the X-ray Telescope \citep{2007SoPh..243...63G} on-board
Hinode taken during a dedicated observing run of an isolated equatorial
coronal hole (ECH), a Southern polar coronal hole and quiet Sun regions.
The ECH was tracked from the West to the East limb from 8 to 10 hours per day
for 4 days. The Southern polar CH was
observed for one day while the quiet Sun regions over 2 days. All data were
taken with an Al Poly filter which has a well pronounced temperature response
at logT$_{max}~\approx$~6.9~K. XRT images have an angular pixel size of
1$^{\prime\prime}$~$\times$~1$^{\prime\prime}$\ at full resolution. They were recorded with 16~{\rm s} and
23~{\rm s} exposure time and a cadence of about 40~{\rm s}. We also used
randomly selected quiet Sun data with transient coronal holes (TCHs) and a Northern
polar coronal hole observation. Further details on the data can be found
in Table~\ref{table1}.
The data were reduced using the standard procedures, which include flat-field
subtraction, dark current removal, despiking, normalisation to data number
per second to account for the variations in exposure time, satellite jitter
and orbital variation corrections. The images were then de-rotated to a
reference time to compensate for the solar rotation. A common field-of-view
(FOV) was selected from all the images for each day. We then prepared an
array with dimensions (nx, ny, nf), where nx is the number of Solar\_X pixels,
ny is the number of Solar\_Y pixels and nf is the number of images. Each image was
binned to 4~$\times$~4 pixels$^2$ in order to improve the signal-to-noise ratio and
reduce the data points (and subsequently
the computational time). The binned images were used to produce light
curves of nf points for each pixel. These light curves were the input for
an identification procedure which will be discussed in the next section.
\section{Brightening identification procedure}
\label{ip}
We developed an automatic identification procedure to distinguish small-scale
intensity enhancements in XRT images. While the visual identification of large
events such as jets from bright points give good results, it is difficult to
identify and track small-scale events, especially on the quiet Sun high
background emission or over pre-existing bright coronal loop structures
(e.g. bright points, active regions etc). We eliminated all light curves which show no activity or minimum activity comparable to the noise level.
\begin{table*}
\begin{center}
\caption{Number of events over 24 hours per 100 $\times$ 100 arcsec$^2$ identified inside the coronal holes (CH), in coronal hole boundary regions (CHBR) and in the quiet Sun (QS).}
\label{table2}
\begin{tabular} {lcccccc}
\hline \\
& \multicolumn{3}{c}{No of events identified} & \multicolumn{3}{c}{No of unresolved events } \\
Date & \multicolumn{3}{c}{with plasma outflows} & \multicolumn{3}{c}{identified with no plasma outflows} \\
\hline
& CH & CHBR & QS & CH & CHBR & QS \\
\hline
09/11/07 & 29 & 40 & 6 & 6 & 26 & 8 \\
12/11/07 & 16 & 57 & 6 & 2 & 17 & 7 \\
14/11/07 & 10 & 51 & 1 & 16 & 41 & 7 \\
16/11/07 & 56 & 32 & - & 44 & 33 & 12 \\
16/12/08 & 99 & 86 & 6 & 31 & 14 & 33 \\
20/09/07 & 57 & 72 & 4 & 5 & 7 & 9 \\
10/01/09 & - & - & 6 & - & - & 17 \\
13/01/09 & - & - & 9 & - & - & 14 \\
29/11/07 & - & 64 & 7 & - & 62 & 14 \\
\hline
\end{tabular}
\end{center}
\end{table*}
The first step of the identification procedure was to define the background
emission for each light curve. The light curves were smoothed over a window
of width 5 to remove the spikiness in the background. Due to a difference in the
background emission between the quiet Sun and the CHs, it was necessary to set two
different thresholds for further analysis. The threshold we used was
1.8 times the mean emission value for the CHs and 1.3 times the
mean emissivity for the QS. The comparatively higher threshold set for the
CH light curves helped to eliminate the high fluctuations of the low emission
background. Light curves with a maximum value less than these thresholds were
neglected. Any point in the light curve was considered as a peak if its
value was greater than the threshold and also greater than the average of the
two preceding points, and the average of two successive points. All the
values below the threshold were considered as local minima. Each identified
peak was traced back on either side to identify the minimum from the local
minima. The value of all the points between the two identified minima for each
peak were set to zero in the light curve and thereby from the average over
the rest of the light curve (I$_{av}$) we computed the standard deviation (SD).
The new background (BG) was obtained as BG = I$_{av}$ +1.1~$\times$~SD.
The next step was the actual identification of intensity enhancements. A
new threshold of 2~$\times$~BG for CHs and 1.3~ $\times$~BG for QS was set
using the above calculated background. Intensity increases above these
thresholds with corresponding minima less than BG and duration less than 45
minutes were identified. A pixel brightening was considered only if all the
above mentioned conditions were satisfied. The threshold was calculated with
a trial and error method.
The peaks having a duration of more than 45 minutes were examined separately.
The closest local minimum on either side of the peak were traced back. If
the difference between the peak and the minimum were greater than the BG for
the CHs and 0.3$\times$BG for the QS with the duration less than 45 minutes, then
they were considered. Also the peaks which have one minimum that was either
in the beginning or at the end of the light-curve were evaluated with the same
criteria, in order not to miss any real event. Any intensity enhancements
in the coronal holes, the quiet Sun or over pre-existing bright loop structures
which satisfy the above criteria could be
identified by our procedure.
\section{Results and discussion}
\label{results}
As it has been described in Sect.~2, we made a selection of data which comprised
observations of different features on the Sun: equatorial and polar coronal holes
as well as quiet Sun regions with and without transient coronal holes.
In Fig.~\ref{fig1}--\ref{fig4} we display examples of an X-ray image from each different region.
Our intention was to find out whether the changes we have seen so far along CHBs
(\citet{2004ApJ...603L..57M} and paper I) are unique for CH regions, i.e. regions
of open magnetic field lines. These data also permit to resolve the fine
structure of individual features and follow their dynamics at high cadence.
To each dataset we applied the identification procedure described above. This
procedure provided us with the following information: (i) light curves which
contain one or more radiance enhancement identified as brightenings following
the criteria given in Sect.~ 3; (ii) the start and end time of each radiance
enhancement; (iii) the brightening positions in pixel numbers. As we produced
light curves by binning over 4~$\times$~4 pixels$^2$, imprints of brightening
events with spatial scales larger than 4~$\times$~4 pixels$^2$ were observed
in more than one light curve. This made visual grouping of identified bright
pixels essential to distinguish each event. Grouping of the features into
individual events was done by playing the image sequence of each dataset with
the identified brightenings over-plotted at corresponding times (see the online
movies). Clusters of bright pixels identified next to each other with similar
lightcurves were grouped into events. The events showing plasma outflows (i.e.
plasma moving along quasi-straight trajectories) were classified as jets, while
events exhibiting plasma blobs moving along curved trajectories or just
brightening increase in a group of pixels were classified as unresolved
brightenings events. The so-called space-time plot was also used to
investigate the plasma motion in the form of a jet and to determine their proper
motion. A space-time plot was produced by averaging over a slice of 3 pixels wide and 100
pixels long from each image, cut along the jet, i.e. in the direction of plasma propagation and then plotting that in time
\citep[][and Subramianian, PhD thesis
2010]{2007PASJ...59S.745S} . We were able to group more
than $95\%$ of the identified bright pixels. The ungrouped pixels ($\leq5\%$)
comprise bright pixels identified at the edges of images and above bad
pixels. The pixels identified in the beginning of each dataset which
could not be classified due to the lack of coverage of the whole event and
the pixels identified with a time lapse over their lifetimes were also rejected
from counting.
The visual grouping of identified bright pixels into
events can be found in Table \ref{table2}. We defined a coronal hole boundary
region (CHBR) as the region $\pm$15$^{\prime\prime}$\ on both sides of the contour line
defining the CH boundary. Additionally, animated image sequences with over-plotted
identified brightenings at corresponding times can be seen online as
movie\_qs.mp4 (quiet Sun region on 2009 January 10), movie\_ech.mp4 (ECH on
2007 November 12,), movie\_pch.mp4 (PCH on 2008 December 16) and movie\_tch.mp4
(TCH on 2007 November 29).
The first and the most important result of this study is easily noticeable
from Figs.~\ref{fig1}--\ref{fig4} the boundaries of coronal holes are abundant with
brightening events which appear much larger than the same phenomena in the
quiet-Sun region. We separated the events visually into two groups, events
with plasma outflows or jet-like events and events without outflows or
simple brightenings. The equatorial coronal hole data, observed near the disk
center, shows twice as many jet-like and simple brightening events in
the CHBRs (as defined above) as compared to the CH regions. In contrast, polar
coronal hole data and ECH close to the West limb (2007 November 16) show a higher
number of events inside the CH as well as in the CHBRs suggesting that this
can be due to the line-of-sight effect. However, further investigation is
needed on a larger number of datasets.
If we assume that the magnetic reconnection responsible for the occurrence
of jet-like events takes place predominantly between closed and open
magnetic field lines, then the number of reconnection events producing
outflows will be always higher in the CH boundary region since open and
closed magnetic field lines are continuously pushed together by different
processes such as convection, differential rotation, meridian motions
etc. Inside coronal holes, where the number of bipolar systems and the
corresponding closed loop structures are limited, the number of jet-like
events will therefore be lower. For the transient coronal hole regions, the
separation of the coronal hole boundaries region (30$^{\prime\prime}$\ wide) from the coronal holes,
for estimating the number of events in each region, is more difficult due to the very small size of these coronal holes. Therefore, here
we consider that these CHs represent entirely a boundary region. The number of events found in these
TCHs is several times larger than in the quiet Sun (both with and without outflows).
The plasma ejected during the outflow events always originates from
pre-existing or newly emerging (at X-ray temperatures) bright points
both inside CHB regions and CHs. They typically start with a brightening
in just a few XRT pixels (4-6) somewhere in a pre-existing BP which
we believe to be the reconnection site. \cite{1996PASJ...48..123S}
reported from the statistical study of 100 jets in SXT/Yohkoh observations
that most of them were associated with micro-flares in the foot-points of
the jets. \citet{2007PASJ...59S.745S} resolved the fine structure of a
quiet Sun X-ray jet to be the expansion and eruption of loop structures colliding
with the ambient magnetic field, similar to the CH jet. Madjarska (2009) studied in great detail
one of the jet-like phenomena identified in the datasets analysed here. The author estimated that the
reconnection site reaches temperatures of up to 12 MK from observations
of Fe~{\sc xxiii}~263~\AA\ from EIS/Hinode, which confirms that some of these
events are very similar to large flares but on a much smaller spatial scale.
Seconds after the reconnection takes place, a cloud of plasma is blown
out from the BP. Madjarska (2009) reported that
although the event appears more like a jet (although some expanding loops
can also be distinguished) in X-ray images as observed in projection on
the solar disk, the two additional view points from STEREO/SECCHI reveal
that the phenomenon evolves as an expulsion of BP loops followed
by a collimated flow along the quasi-open field lines of the expanded
loops. The escaping plasma reaches temperatures of around 2~MK
\citep[][Madjarska 2009]{2007PASJ...59S.751C}.
\citet{2009SoPh..tmp..120N} found 5 out of 79 jets analysed from STEREO/SECCHI
observations exhibiting a three part structure typically of coronal mass
ejections (CMEs) - bright leading edge, a dark void and
bright trailing edge (e.g. the prominence material). The authors named
them micro-CMEs. The rest were called Eiffel tower-type jets, where the
reconnection appears to happen on top of the loops and lambda-type jets
with reconnection occurring in the jet foot-points. As most of the events
studied here were seen in projection on the solar disk, this visual
division into different groups is not possible. However, the visual examination
of the phenomena analysed here confirms that expulsion of BP loops
describes these features best, hence, we will further refer to them as EBPLs.
\begin{figure*}[!ht]
\vspace{4.6cm}
\special{psfile= fig2.eps hoffset=0 voffset=0 hscale=100 vscale=100 angle=0}
\caption{An example of a typical jet-like happening on 2007 November 12 at the
coronal hole boundaries. A refers to the jet while B refers to the BP.}
\label{fig5}
\end{figure*}
\begin{figure*}[!ht]
\vspace{3.2cm}
\special{psfile= fig4.eps hoffset=0 voffset=0 hscale=100 vscale=100 angle=0}
\caption{A bright point which produced several jet-like events at the boundaries of a transient coronal hole on 2007 November 29.}
\label{fig6}
\end{figure*}
\begin{figure*}[!ht]
\vspace{3.2cm}
\special{psfile= fig3.eps hoffset=0 voffset=0 hscale=100 vscale=100 angle=0}
\caption{An example of a brightening event identified in quiet Sun data obtained on 2009 January 10.}
\label{fig7}
\end{figure*}
Fig.~\ref{fig5} presents an example of a typical EBPL event
from a BP happening along an ECH boundary. The EBPL ends with the BP
vanishing at X-ray temperatures triggering the coronal hole to expand.
The plasma ejected from the BP seems to collide with structures on its way
(propagating towards the QS region seen in projection on the solar disk)
setting off a brightening in a pre-existing BP (denoted with B in
Fig.~\ref{fig5}) with no obvious plasma outflows.
We also found that the presence of a transient CH in the quiet Sun triggers
the occurrence of EBPL-like phenomena, similar to the equatorial and
polar CH ones. Yet again, the evolution of the BPs along the boundaries
changed the CHBs (Fig.~\ref{fig6}). Due to the smaller size of the TCH,
these changes lead to a large expansion or contraction of the TCH, in some
cases even a disappearance. In Fig.~\ref{fig7} we give a series of images
which exhibit one of the quiet Sun events. In comparison to CHs, the
events identified in the quiet Sun images rarely show outflows
(Table~\ref{table2}). Neither expanding loops nor collimated flows were
distinguishable in the XRT images (i.e. no EBPL).
This brings us to the numerical comparison of CHBRs and CHs with QS areas. We
find an average of 75 brightening events per 100 $\times$ 100 arcsec$^2$ per day
for the coronal holes and boundaries and 20 events in the quiet Sun. Approximately
70$\%$ of these brightenings in CHs and CHBRs showed plasma outflows while only
30$\%$ of the brightenings seen in the quiet Sun exhibit jet-like structures.
Previous works indicated fewer events per day either because of the poorer spatial resolution of the
instrument used or because of the visual identification methods.
The identified brightening events with no plasma outflows could either
be driven by two sided loop reconnection \citep{1994xspy.conf...29S} between
emerging fluxes and overlying coronal fields, in which the ejected plasma flow
along the closed loop structures, or reconnection driven brightenings with
plasma outflows at much lower temperatures. They can also represent flows in
loop structures, perhaps triggered by reconnection shocks from the
neighbourhood as seen in the example brightening event B in Fig.~\ref{fig5}.
\citet{2007PASJ...59S.745S} showed that a QS jet appears to be guided by the
closed magnetic field lines (loops), unlike the jets in the CHs and CHBRs which
are guided by the open magnetic field lines. This result immediately
raises the question whether the presence of open magnetic field lines is
crucial for the generation of outflow phenomena.
Comparison of the physical properties such as duration and size of the
jet-like events in ECHs and polar CHs show no difference. A good
correspondence was found between the duration and the size of the events,
irrespective of their position. The larger events have longer duration
of around 40 min and are mostly associated with pre-existing coronal
bright points or at least with features becoming visible in X-rays just
before the eruption. The smaller events have a shorter duration of around
20 min (mostly with no pre-existing features at coronal temperatures).
However, for most events the actual duration from the moment the
reconnection occurs (i.e. when the plasma is ejected) until the plasma
outflow is no longer evident is usually between 10 and 15~{\rm min}.
The amount of plasma ejected entirely depends on the magnetic
energy available before the reconnection and it will, therefore, differ
from event to event. Repetitive occurrence of jets in the same bright
points are more common in CHs than in the QS. No periodicity was found,
although a large number of BPs produced several jet-like events (from 2
to 5 times) during the course of the observations until all the stored magnetic
energy was exhausted and the bright point fully disappeared.
The proper motions of the outflows obtained from space-time plots are in
the range of 100$-$500~\rm km~\;s$^{-1}$ for most of the events. Because of the projection
effect of the jets with respect to the solar disk, the velocity we obtain gives the lower
boundary of the real velocity of the ejected plasma. Based
on their plasma velocity we divide X-ray jets into two groups: (i) jets
with pre-existing coronal structures (X-ray BPs) which have velocities
$\approx$350~\rm km~\;s$^{-1}$\ or greater (i.e. in the range of the Alfv\'en velocity
in the lower corona) and (ii) jets with no pre-existing structures at X-ray
temperatures, showing velocities of around 150~\rm km~\;s$^{-1}$\
(i.e. close to the Alfv\'en velocity in the transition region). Stereo EUV images
taken with the 171~\AA\ filter confirm the presence of a corresponding
reconnection jet-like structures at transition region temperatures.
\section{Conclusions}
\label{conclusions}
The present article confirms our findings from paper I that the evolution
of loop structures known as coronal bright points is associated with the
small-scale changes of CHBs. We were able to identify the true nature of these
changes which represent plasma outflows
associated with the expansion of the bright point loop structures. The plasma
trapped in the loop structures is consequently released along the ``quasi'' open
magnetic field lines. These
ejections appear to be triggered by magnetic reconnection, most probably
the so-called interchange reconnection \citep{2004ApJ...612.1196W} between
the closed magnetic field lines (BPs) and the open magnetic fields of coronal
holes. The ejected plasma is guided and accelerated further away from the
Sun by the open magnetic field lines with some jets reaching
several solar radii \citep{2009SoPh..tmp..120N}. Contrary, in the quiet Sun
the plasma ejected as a result of two or one sided loop reconnection, is
contained in the corona by the closed magnetic field lines.
Tall and extended coronal loops are very rare in coronal holes
\citep{2004SoPh..225..227W}, while closed (loop) magnetic structures of
varying physical properties are ubiquitous in the quiet Sun corona as seen
in EUV and X-ray observations. Coronal holes with predominant open magnetic
fields and minority closed loop structures (BPs) are encompassed by these loop
structures seen at transition region and coronal temperatures.
\citet{2004ApJ...612.1171W}, \citet{2001ApJ...555..426W} and
\citet{2005JGRA..11007109F} showed that the elemental abundance of
trapped plasma is proportional to the confinement time of the plasma in
loop structures. Newly emerged active region loop structures were found
to have initial photospheric abundances (FIP bias $\approx$1--2)
which increased with time, reaching $\approx$~5 in 1--3 days
\citep{2001ApJ...555..426W}. \citet{1998Natur.394..152S} concluded that
$\approx$1.5 days is the reconfiguration time-scale for the super-granulation
network magnetic fields where coronal BPs have their foot-points rooted.
This time period is in the range of the confinement time-scale needed for
the enhancement of the FIP bias.
Coronal BPs electron densities were derived from CDS/SoHO in the temperature
range 1.3--2$\times 10^6$~K by \citet{2005A&A...435.1169U}. The authors
concluded that the bright points plasma have properties which are more
similar to active region plasma rather than quiet Sun plasma, although BPs do
not show the increase of electron density at temperature over
Log$T_e$~$\sim$~6.2~K, observed in the core of active regions
\citep{2005A&A...435.1169U}. These results were later confirmed from data taken
with the Extreme-ultraviolet Imaging Spectrometer (EIS) for Log$T_e \sim$6.1
and 6.2~K \citep {2008A&A...492..575P}. The BP lifetime in EUV were found to be
on average 20~hrs in the EUV \citep{2001SoPh..198..347Z} and on average 8~hrs
in X-rays with some BPs lasting up to 40~hrs \citep{1974ApJ...189L..93G}. The
results of individually studied BPs by \citet{2004A&A...418..313U} give for two
BPs a lifetime of 38 and 51~hrs detected in Fe~{\sc xii}~195~\AA\ images from
Extreme-ultraviolet Imaging Telescope (EIT) on-board SoHO and by P\'erez-Suarez
(PhD thesis 2009) in five BPs: BP1 -- 48~hrs, BP2 more than 54~hrs, BP3 --
37~hrs, BP4 -- 45.2~hrs and BP5 -- 35 h (on the limb). The study by
\citet{1974ApJ...189L..93G} on BPs lifetime in X-rays has not been updated so
far using Hinode X-ray observations. The BPs properties given above strongly
suggest that their plasma can become enriched on low FIP elements.
\citet{2008ApJ...682L.137R} concluded that X-ray jets are the precursors
of polar plumes, and jets happening in pre-existing polar plumes enhance the
brightness of the plume haze. Polar plumes are observed even at several
solar radii \citep{1997SoPh..175..393D} and were found to contribute to the
solar wind stream. They have been reported to occur even at low latitudes
\citep{1995ApJ...446L..51W}. Jets, associated with BPs, were also recently
registered with the Large Angle and Spectrometric Coronagraph on-board SoHO
\citep{2008SoPh..249...17W} and SECCHI/STEREO \citep{2009SoPh..tmp..120N}.
Hence, the BP plasma cloud, which is ejected as a result of magnetic
reconnection, will therefore, escape from the Sun having the plasma
characteristics of the slow solar wind. We asked ourselves whether the plasma
ejections we observe can possibly be a source of the fast solar wind? This
possibility cannot be fully rejected, although it is a fact that these jets
happen sporadically rather than continuously, which is in contradiction with
the nature of the fast solar wind.
Our specially designed observing programs provided us with spectroscopic
co-observations from SUMER, CDS and EIS along with the XRT and SOT. In a
follow up paper we will derive the physical properties such as velocity,
density, temperature and others of a large number of events happening in the
FOV of the spectrometers.
\begin{acknowledgements} The authors thank ISSI, Bern for the support of
the team ``Small-scale transient phenomena and their contribution to coronal
heating''. Research at Armagh Observatory is grant-aided by the N.~Ireland
Department of Culture, Arts and Leisure. We also thank STFC for support via
grants ST/F001843/1 and PP/E002242/1. Hinode is a Japanese mission developed
and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC
(UK) as international partners. It is operated by these agencies in
co-operation with ESA and NSC (Norway). The STEREO/ SECCHI data used here are
produced by an international consortium of the Naval Research Laboratory (USA),
Lockheed Martin Solar and Astrophysics Lab (USA), NASA Goddard Space Flight
Center (USA), Rutherford Appleton Laboratory (UK), University of Birmingham
(UK), Max-Planck-Institut f\"{u}r Sonnensystemforschung (Germany), Centre Spatiale
de Li\`{e}ge (Belgium), Institut d'Optique Th\'{e}orique et Appliqu\'{e}e (France), and
Institute Astrophysique Spatiale (France).
\end{acknowledgements}
\bibliographystyle{aa}
\input CHB_madjarska.bbl
\end{document}
|
1,314,259,996,672 | arxiv |
\section{Introduction}\label{sec-intro}
One can distinguish between two main lines of research in low-regularity geometry. One approach is analytical, where one
lowers the differentiability assumptions on, for example, (pseudo-)Riemannian metrics below the level where curvature can be
classically defined. For example, one can study geometrical properties of (pseudo-)Riemannian metrics that have regularity
$C^0$, $C^{0, \alpha}$ or $C^{1, 1}$, etc., or so-called ``Geroch--Traschen'' metrics, for which the Christoffel symbols are
$L^2_{\mathrm{loc}}$, and the curvature is well-defined as a distribution~\cite{GT, LM:07, SV:09}. The other approach to
studying low-regularity geometries is by ``synthetic'' or metric space methods. Here, curvature bounds for Alexandrov spaces
and CAT$(k)$ spaces are defined in terms of comparison properties of geodesic triangles.
In the context of low-regularity Riemannian geometry, examples of a result of an analytical nature would be DeTurck and
Kazdan's study concerning harmonic coordinates~\cite{DK}, Taylor's results on regularity of isometries~\cite{T:06} and
Lytchak and Yaman's result~\cite{LY} that minimising curves for $C^{0, \alpha}$ Riemannian manifolds are $C^{1, \beta}$
curves, where $\beta = \frac{\alpha}{2-\alpha}$. Examples in this direction in the Lorentzian setting are the positive mass
theorem for distributional curvature \cite{LL:15, GT:14}, work on cone structures \cite{FS:12, Min:17, BS:18} and the recent
work of extending the classical singularity theorems to $\ensuremath{\mathcal{C}}^{1,1}$-regularity \cite{KSSV:15, KSV:15, GGKS:18}, which in
turn builds on previous results in low regularity Lorentzian geometry and causality \cite{CG:12, Min:15, KSS:14, KSSV:14,
Sae:16}.
In the synthetic direction, the theory of Alexandrov spaces with curvature bounded above and/or below is well-developed as an
appropriate generalisation of Riemannian geometry with sectional curvature bounds (see, for instance, \cite{BBI:01, BH:99,
Pap:14}), and the work of Lott--Villani--Sturm gives a generalisation of the notion of a Riemannian metric with lower bound
on the Ricci curvature to metric measure spaces~\cite{LV, S1, S2}.
In this paper, we will concentrate on a generalisation of Lorentzian geometry suitable for the low-regularity setting. More
precisely, we shall be interested in the problem of finding low-regularity extensions of spacetimes. Concerning this
question, approached from the analytical side, several fundamental contributions have appeared recently. Of particular
relevance to us, Sbierski has shown the $C^0$-inextendibility of the Schwarzschild solution~\cite{Sbi:18}. Building upon
Sbierski's work, Galloway, Ling and Sbierski established that global hyperbolicity combined with timelike geodesic
completeness implies $C^0$-inextendibility. Further developments in this direction are due to Galloway--Ling and Graf--Ling
(see below). In a related direction, Dafermos and Luk have recently shown
$C^0$-extendibility of the interior of the
Kerr solution~\cite{DL:17}.
In this paper, we will concentrate on the synthetic-geometrical approach to extendibility. In~\cite{KS:17}, the theory of
Lorentzian length spaces has been developed, which will form the framework of the present work. In this more axiomatic
approach, there is a notion of a geodesic (as a locally length maximising curve), which is not available in the
more analytical direction of research. Therefore, it is possible to mimic the classical proof that geodesic completeness implies
inextendibility (see, for example,~\cite[Prop.~6.16]{BEE:96}). Moreover, within this picture, it becomes clear precisely what
minimal geometric properties are underlying certain analytical extension results. In particular, for the first time, our
approach allows us to directly relate low-regularity inextendibility with (synthetic) curvature blow-up. Such a result does
not appear to be feasible in a purely analytical approach, due to the lack of a notion of a curvature for the extended
spacetime.
An additional advantage of our synthetic approach is that there is no requirement for the introduction of coordinate
systems, and regularity conditions (such as existence of smooth structures, or a certain level of differentiability) never
arise. In this regard, it should perhaps be noted that in the analytical work on low-regularity extensions, one has to carry
out standard geometrical constructions on the original manifold. As such, even though one works in a coordinate chart of the
extended manifold in which the metric is merely continuous, the metric on the intersection of the original manifold with the
coordinate chart must be $C^2$-regular.%
\footnote{I.e.\ one implicitly must assume that the metric is smooth on $\iota(M)$ in the coordinate chart on $\tilde{M}$ in
which the metric is just continuous.}
One could compare this situation with, for example, the fact that the Nash--Kuiper theorem~\cite{Nash, K:56}
implies that the flat metric on $T^2$ can be induced from a $C^1$ map $T^2 \to \mb{R}^3$.%
\footnote{See, for instance,~\cite{BJLT} for an illustration of this example.}
In the coordinate system in which the map is $C^1$, the induced metric will be merely $C^0$, even though we know that there exists a coordinate system in which the metric is smooth. As such, one could consider a more general notion of $C^0$ extensions of spacetimes, where one allows the regularity of the metric on the original manifold to drop. On the contrary, in our approach, such issues never arise. In fact, the extensions that we consider need not even be manifolds.
Our main references for Lorentzian geometry and causality theory are \cite{ONe:83,BEE:96,MS:08,Chr:11},
as well as \cite{CG:12} for the case of continuous Lorentzian metrics.
The plan of the paper is as follows: In Section \ref{sec:lls} we briefly recall some main concepts
and results on Lorentzian length spaces. Section \ref{sec:extensions} introduces extensions of
Lorentzian (pre-)length spaces, relates them to extensions of spacetimes and shows that the
future or past boundary of an extension is non-empty. In Section \ref{sec:geodesics} we
define geodesics in the synthetic setting and show that this notion reduces precisely to
that of pregeodesics for spacetimes. We also demonstrate that, as in the smooth case, extendibility as a geodesic
is equivalent to continuous extendibility. In Section \ref{sec:TC} we define an analogue of
timelike completeness: a Lorentzian pre-length space is said to have property (TC) if all
inextendible timelike geodesics have infinite length. This is the key property on which our main
inextendibility result (Theorem \ref{thm-inext-lls-lls}) rests. We then establish connections between inextendibility
and the occurrence of synthetic causal curvature singularities. Finally, in Section \ref{sec:relation}
we relate the results of the present work to the recent advances in the study of
the low regularity inextendibility
of spacetimes.
\section{A short introduction to Lorentzian length space s}\label{sec:lls}
Here we briefly recall some basic notions and results
from the theory of Lorentzian length spaces, following \cite{KS:17}, to which we
refer for further details and proofs.
A set $X$ endowed with a preorder $\leq$ and a transitive relation $\ll$
contained in $\leq$ is called a \emph{causal space}. We write $x<y$ if $x\leq y$ and $x\neq y$.
If $x\ll y$ respectively $x\le y$ we call $x$ and $y$ timelike respectively causally related.
Chronological and causal futures and pasts $I^\pm(x)$, $J^\pm(x)$ of a point $x$
are then defined in the usual manner based on these relations.
If $X$ is, in addition, equipped with a metric $d$
and a lower semicontinuous map
$\tau \colon X\times X \to [0, \infty]$ that satisfies the reverse triangle inequality
$\tau(x,z)\geq \tau(x,y) + \tau(y,z)$ (for all $x\leq y\leq z$), as well as $\tau(x,y)=0$ if $x\nleq y$ and $\tau(x,y)>0
\Leftrightarrow x\ll y$, then $(X,d,\ll,\leq,\tau)$ is called a \emph{Lorentzian pre-length space\/} and $\tau$ is called
the \emph{time separation function\/} of $X$. Note that lower semicontinuity of $\tau$ implies that $I^{\pm}(x)$ is open, for any $x \in X$.
A non-constant curve $\gamma \colon I\rightarrow X$ ($I$ an interval) is called
(future-directed) \emph{causal (timelike)} if $\gamma$ is locally Lipschitz continuous and if for all
$t_1,t_2\in I$ with $t_1<t_2$ we have $\gamma(t_1)\leq\gamma(t_2)$ ($\gamma(t_1)\ll\gamma(t_2)$). It
is called \emph{null\/} if, in addition to being causal, no two points on the curve are related with respect
to $\ll$. For strongly causal continuous Lorentzian metrics, this notion of causality
coincides with the usual one (\cite[Prop.\ 5.9]{KS:17}). In analogy to the theory
of metric length spaces, the length of a causal curve is defined via the time separation
function: For $\gamma \colon [a,b]\rightarrow X$ future-directed causal we set
\[
L_\tau(\gamma):=
\inf\Big\{\sum_{i=0}^{N-1} \tau(\gamma(t_i),\gamma(t_{i+1})): a=t_0<t_1<\ldots<t_N=b,\ N\in\mb{N}\Big\}.
\
\]
If the interval is (half-)open, say $I=[a,b)$, then the infimum is taken over all partitions
with $a=t_0<t_1<\ldots<t_N<b$, and similarly for the other cases.
For smooth and strongly causal spacetimes $(M,g)$ this notion of length coincides with the usual one:
$L_\tau(\gamma)=L_g(\gamma)$ (\cite[Prop.\ 2.32]{KS:17}).
A future-directed causal curve $\gamma \colon [a,b]\rightarrow X$ is
\emph{maximal\/} if it realizes the time separation, i.e., if $L_\tau(\gamma) = \tau(\gamma(a),\gamma(b))$.
Standard causality conditions (chronology, (strong) causality, global hyperbolicity, \dots) can also be imposed
on Lorentzian pre-length spaces, and substantial parts of the causal ladder (\cite{MS:08}) continue
to hold in this general setting. A Lorentzian pre-length space $X$ is called \emph{causally path connected\/} if for
all $x,y\in X$ with $x\ll y$ (respectively $x<y$) there is a future-directed
timelike (respectively causal) curve from $x$ to $y$. A neighbourhood $U$ of $x$ is called
\emph{causally closed\/} if the relation
$\leq$ is closed in $\bar{U}\times\bar{U}$, and $X$ itself is called
\emph{locally causally closed\/} if every point has a causally closed neighbourhood.
A key technical tool in smooth semi-Riemannian geometry is the existence of convex neighbourhoods,
in which the causality is particularly simple and where one has a complete description of length-maximising
curves. The analogue of this notion in the present context is the following:
A Lorentzian pre-length space $X$ is called \emph{localisable\/} if any $x\in X$ has an open, so-called \emph{localising\/} %
neighbourhood $\Omega_x$ such that:
\begin{enumerate}[label=(\roman*)]
\item \label{def-loc-LpLS-cau-comp} The $d$-length of all causal curves contained in $\Omega_x$
is uniformly bounded.
\item \label{def-loc-LpLS-om-con} There is a continuous map $\omega_x\colon \Omega_x \times \Omega_x\rightarrow
[0,\infty)$ such that
$(\Omega_x, d\rvert_{\Omega_x\times\Omega_x},$ $\ll\rvert_{\Omega_x\times \Omega_x},\leq\rvert_{\Omega_x\times\Omega_x},
\omega_x)$ is a Lorentzian pre-length space, and for every $y\in\Omega_x$ we have
$I^\pm(y)\cap\Omega_x\neq\emptyset$.
\item \label{def-loc-LpLS-max-cc} For all $p,q\in \Omega_x$ with $p<q$ there is a future-directed causal curve
$\gamma_{p,q}$ from $p$ to $q$ that is
maximal in $\Omega_x$ and satisfies $L_\tau(\gamma_{p,q}) = \omega_x(p,q) \leq \tau(p,q)$.
\end{enumerate}
If, in addition, the neighbourhoods $\Omega_x$ can be chosen such that
\begin{enumerate}
\item[(iv)]\label{def-loc-LpLS-4} Whenever $p,q\in\Omega_x$ satisfy $p\ll q$ then $\gamma_{p,q}$ is timelike
and strictly longer than any future-directed
causal curve in $\Omega_x$ from $p$ to $q$ that contains a null segment,
\end{enumerate}
then $(X,d,\ll,\leq,\tau)$ is called {\em regularly localisable}.
Lorentzian length spaces are close analogues of metric length spaces in the sense that the time separation function
can be calculated from the length of causal curves connecting causally related points. Precisely, a
locally causally closed, causally path connected and localisable Lorentzian pre-length space is called a \emph{Lorentzian length space\/} if
$\tau = \mathcal{T}$, where for any $x,y\in X$ we set
\begin{equation*}
\mathcal{T}(x,y):= \sup\{L_\tau(\gamma):\gamma \text{ future-directed causal from }x \text{ to } y\}\,,
\end{equation*}
if the set of future-directed causal curves from $x$ to $y$ is not empty. Otherwise let $\mathcal{T}(x,y):=0$.
If, in addition, $X$ is regularly
localisable, then it is called a regular Lorentzian length space.
Any smooth strongly causal spacetime is an example of a regular Lorentz\-ian length space (with metric $d = d^h$ induced by any Riemannian metric $h$ on the spacetime).
More generally,
any spacetime with a continuous, strongly causal and causally plain metric (see the remark preceding Corollary~\ref{cor-geo-compl-inext-lls} below) is a (strongly)
localisable Lorentzian length space. Further examples are provided by certain Lorentz-Finsler spaces
in the sense of \cite{Min:17} or, for the non-manifold setting, causal Fermion systems \cite{Fin:16,Fin:17}.
The final concept from the theory of Lorentzian length spaces we are going to require below is
that of synthetic curvature bounds, based on triangle comparison. We will confine ourselves to
causal triangle comparison here, as this is the only one we are going to employ. By an
\emph{admissible causal geodesic triangle\/} we mean a triple $(x,y,z)\in X^3$ with
$x\ll y \le z$ or $x \le y \ll z$ such that $\tau(x,z)<\infty$ and such that the sides (if non-trivial)
are realized by future-directed causal curves. Curvature bounds are formulated by comparing
such triangles with triangles of the same side lengths in one of the Lorentzian model spaces
$M_K$ of constant sectional curvature. Here,
\begin{equation}\label{eq:model_spaces}
M_K = \left\{ \begin{array}{ll}
\tilde S^2_1(r) & K=\frac{1}{r^2}\\
\mb{R}^2_1 & K=0\\
\tilde H^2_1(r) & K= -\frac{1}{r^2}.
\end{array}
\right.
\end{equation}
where $\tilde S^2_1(r)$ is the simply connected covering manifold of the two-dimensional Lorentzian pseudosphere
$S^2_1(r)$ (i.e., de-Sitter space), $\mb{R}^2_1$ is two-dimensional Minkowski space, and $\tilde H^2_1(r)$ is the simply connected covering manifold
of the two-dimensional Lorentzian pseudohyperbolic space (i.e., anti-de-Sitter space) . In order to guarantee the existence of comparison
triangles in one of the model spaces, one needs to impose size restrictions of the following kind:
Given $K\in \mb{R}$, let $(a,b,c)\in \mb{R}_+^3$ with $c\ge a+b$.
If $c=a+b$, then let $c<\frac{\pi}{\sqrt{K}}$. (Here, $\frac{\pi}{\sqrt{K}}:=\infty$
if $K\le 0$).
Otherwise, if $K<0$ then assume $c<\frac{\pi}{\sqrt{-K}}$. Then $(a,b,c)$ is said to {\em satisfy timelike size bounds} for $K$. These bounds ensure the existence of comparison triangles in the corresponding model space.
Using this terminology, a Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ is said to have causal curvature bounded below (above) by $K\in\mb{R}$
if every point in $X$ has a neighbourhood $U$ such that:
\begin{enumerate}[label=(\roman*)]
\item $\tau|_{U\times U}$ is finite and continuous.
\item Whenever $x,y \in U$ with $x < y$, there exists a causal curve $\alpha$ in $U$ with $L_\tau(\alpha) = \tau(x,y)$.
\item If $(x,y,z)$ is an admissible causal geodesic triangle in $U$, realized by maximal causal curves
(or a constant curve, respectively) $\alpha, \beta, \gamma$
whose side lengths satisfy timelike size bounds for $K$, and if $(\bar{x},\bar{y},\bar{z})$ is a comparison triangle of
$(x,y,z)$ in $M_K$ realized by causal geodesics (or a constant curve) $\bar\alpha$, $\bar{\beta}$, $\bar{\gamma}$,
then whenever $p$, $q$ are points on the timelike sides of $(x,y,z)$ and $\bar p$, $\bar q$ are corresponding
points of the timelike sides of $(\bar{x},\bar{y},\bar{z})$,
we have $\tau(p,q)\le \bar{\tau}(\bar p, \bar q)$ $($respectively $\tau(p,q)\ge \bar{\tau}(\bar p, \bar q))$.
\end{enumerate}
Such a neighbourhood $U$ is called a \emph{comparison neighbourhood with respect to $M_K$}.
\section{Extensions}\label{sec:extensions}
We start the main part of our work by defining the notion of an \emph{extension\/} of a Lorentzian pre-length space, requiring only conditions that are natural within our setting
This concept is fully compatible with the usual notion of extension for spacetimes, see Proposition~\ref{prop-ext-ext}.
\begin{defi}\label{def-ext}
Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space . A Lorentzian pre-length space $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ is called an \emph{extension} of $(X,d,\ll,\leq,\tau)$ if
\begin{enumerate}[label=(\roman*)]
\item \label{def-ext-conn} the metric space $(\tilde X, \tilde d)$ is connected,
\item there exists an isometry $\iota\colon (X,d)\rightarrow (\tilde X, \tilde d)$ of metric spaces,
\item the image $\iota(X)$ is a proper, open subset of $\tilde X$,
\item \label{def-ext-cr} $\iota$ preserves $\ll$ and $\leq$, i.e., $\forall x,y\in X$: if $x\leq y$ then $\iota(x)\
\tilde{\leq}\ \iota(y)$ and if $x\ll y$ then $\iota(x)\ \tilde\ll\ \iota(y)$, and
\item \label{def-ext-tau} a curve $\gamma \colon I \to X$ is timelike (respectively causal) if and only if
$\iota\circ \gamma$ is timelike (respectively causal) in $(\tilde X,\tilde d,\tilde \ll,\tilde \le,\tilde \tau)$.
Furthermore, $\iota$ preserves $\tau$-lengths, i.e., for any $\leq$-causal curve $\gamma\colon I\rightarrow
X$ we have
\begin{equation}\label{eq-def-ext-tau}
L_\tau(\gamma) = L_{\tilde\tau}(\iota\circ\gamma)\,.
\end{equation}
\end{enumerate}
In this case $(X,d,\ll,\leq,\tau)$ is called \emph{extendible}. If no extension exists, then $(X,d,\ll,\leq,\tau)$ is called \emph{inextendible} (as a
Lorentzian pre-length space).
\end{defi}
\begin{rem}
Of course, this definition also applies to Lorentzian length space s, i.e., a Lorentzian length space is \emph{extendible} if there is a Lorentzian length space $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ and
$\iota\colon (X,d)\rightarrow (\tilde X, \tilde d)$ with the above properties \ref{def-ext-conn}-\ref{def-ext-tau}. In
this case conditions \ref{def-ext-cr} and \ref{def-ext-tau} slightly simplify.
\end{rem}
\begin{lem}\label{lem-tau-ttau}
Let $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ be an extension of $(X,d,\ll,\leq,\tau)$, where both are Lorentzian length space s. Then $\tilde\tau \circ (\iota\times\iota) \geq \tau$.
\end{lem}
\begin{pr}
Let $p,q\in X$ with $\tau(p,q)>0$ (if $\tau(p,q)=0$ there is nothing to do). Let $\gamma$ be a future
directed $\leq$-causal curve from $p$ to $q$ (which exists due to $p\leq q$ and the causal path-connectedness of $X$).
Then $\iota\circ\gamma$ is $\tilde\leq$-causal and $L_\tau(\gamma)=L_{\tilde\tau}(\iota\circ\gamma)\leq \tilde\mathcal{T}(\iota(p),\iota(q))
= \tilde\tau(\iota(p),\iota(q))$. Taking the supremum over all future-directed $\leq$-causal curves from $p$ to $q$ we get $\mathcal{T}(p,q)\leq
\tilde\tau(\iota(p),\iota(q))$ and since $\mathcal{T}=\tau$ the claim follows.
\end{pr}
The following lemma shows that condition \ref{def-ext-tau} of Definition \ref{def-ext} required of an extension is in
fact not too strong. Moreover, it demonstrates that for smooth strongly causal spacetimes the time separation function
determines the metric completely.
\begin{lem}
Let $(M,g)$ and $(\tilde M,\tilde g)$ be smooth spacetimes (of the same dimension) with time separation functions $\tau$
and $\tilde\tau$, respectively. Let $(M,g)$ be strongly causal and let $\iota\colon M\rightarrow \tilde M$ be onto. Then
$\iota$ is an isometry if and only if $\iota$ preserves
causal curves and their lengths, i.e., a curve $\gamma$ is causal in $M$
if and only if $\iota\circ\gamma$ is causal in $\tilde M$ and for such
curves, $L_g(\gamma) =
L_{\tilde{g}}(\iota\circ\gamma)$.
\end{lem}
\begin{pr}
It is a classical result that goes back to Hawking, King and McCarthy \cite{HKM:76} (cf.\ \cite[Prop.\
3.34]{MS:08} or \cite[Thm.\ 4.17]{BEE:96}) that $\iota$ is an isometry if and only if it preserves $\tau$.
By definition of the time separation functions in spacetimes,
this latter condition is, in turn, implied by $\iota$ preserving the $g$-lengths
of causal curves.
\end{pr}
Furthermore, in the case of spacetimes the above result implies that there is no difference between an
extension in our sense, and in the usual sense of an isometric embedding (cf.\ \cite[Def.\ 2.15]{Sbi:18}.%
To be precise, we have the following result:
\begin{prop}\label{prop-ext-ext}
Let $(M,g)$ and $(\tilde M, \tilde g)$ be smooth, strongly causal spacetimes (of the same dimension) and let $\iota\colon
M\rightarrow \tilde M$ be a map such that $\iota(M)\subset \tilde M$. Then the induced Lorentzian length space of $(\tilde M,
\tilde g)$ extends the one coming from $(M,g)$ via $\iota$ if and only if $\iota$ is a (smooth) isometric embedding.
\end{prop}
\begin{pr}
We start with the following observation: Let $\tilde h$ be any Riemannian metric on $\tilde M$ with induced metric
$d^{\tilde h}$. This fixes the induced Lorentzian length space in the following sense: Any other Riemannian metric on $\tilde M$ also
induces the manifold topology and the notion of locally Lipschitz continuous curves is preserved (cf.\ \cite[Prop.\
2.3.1]{Chr:11}), thus fixing the spacetime $(\tilde M, \tilde g)$ and any Riemannian background metric determines
the resulting Lorentzian length space.
Assume that $(\tilde M,d^{\tilde h},\tilde\ll,\tilde\leq,\tilde \tau)$ extends $(M,d^h,\ll,\leq,\tau)$ via $\iota$. As
$\iota(M)$ is an open and connected subset of $\tilde M$ we consider the spacetime $(\hat M,\hat g):= (\iota(M),\tilde
g\rvert_{\iota(M)})$ with its time separation function $\hat \tau$. This means that
\begin{equation*}
\hat\tau(\tilde p,\tilde q) =
\sup\{L_{\tilde g}(\tilde \gamma): \tilde \gamma \text{ f.d.\ causal curve from } \tilde p\text{ to }\tilde q \text{ with }
\textrm{image}(\tilde\gamma)\subseteq\iota(M)\}\,.
\end{equation*}
By Definition \ref{def-ext},\ref{def-ext-tau} a curve $\gamma\colon I\rightarrow M$ is causal if and only if
$\iota\circ\gamma\colon I
\rightarrow \hat M$ is causal in $(\hat M,\hat g)$. This together with \eqref{eq-def-ext-tau} and \cite[Prop.\ 2.32]{KS:17}
implies that $\iota$ preserves $\hat\tau$, i.e.,
\begin{equation*}
\tau(p,q)=\hat\tau(\iota(p),\iota(q))\quad \forall p,q\in M\,.
\end{equation*}
Thus by \cite[Prop.\ 3.34]{MS:08} $\iota$ is an isometry $(M,g)\rightarrow (\hat M,\hat g)$.
\medskip
For the converse assume that $\iota$ is a smooth isometric embedding. Then we check points
\ref{def-ext-conn}-\ref{def-ext-tau} of Definition \ref{def-ext}. As $\tilde M$ is connected by
assumption,
the first point follows. Pulling back $\tilde h$ to $M$ gives a Riemannian metric $h:=\iota^*(\tilde
h\rvert_{\iota(M)})$. Denoting its induced metric by $d^h$ we obtain a metric isometry $\iota\colon (M,d^h)\rightarrow
(\tilde M, d^{\tilde h})$ and $\iota(M)$ is open and proper --- giving the second and third point. Let $p,q\in M$ with
$p<q$, i.e., there exists a future directed causal curve $\gamma$ from $p$ to $q$. As $\iota$ is an isometry of $(M,g)$ and
$(\tilde M, \tilde g)$, the curve $\iota\circ\gamma$ is future directed causal and connects $\iota(p)$ with $\iota(q)$. Thus
$\iota(p)\tilde <\iota(q)$. The case for $p\ll q$ is completely analogous, giving the fourth point. Finally, let
$\gamma\colon I\rightarrow M$ be a (locally Lipschitz continuous) curve. Then $\gamma$ is $g$-timelike/causal if and only if
$\iota\circ\gamma$ is $\tilde g$-timelike/causal by the isometric embedding property. Moreover, by \cite[Prop.\ 2.32]{KS:17}
we have
\begin{equation*}
L_\tau(\gamma) = L_g(\gamma) = L_{\tilde g}(\iota\circ\gamma) = L_{\tilde \tau}(\iota\circ\gamma)\,.
\end{equation*}
This gives the fifth point and finishes the proof.
\end{pr}
To illustrate that one can have extensions that are not manifolds we consider the following example, which is a Lorentzian
version of \cite[Ex.\ 4.2.5]{BBI:01}.
\begin{ex}
Let $\mb{R}^2_1$ be two-dimensional Minkowski space and embed it into $\mb{R}^3$ as a plane through the origin
orthogonal to the $z$-direction, i.e., $N:=\{(t,x,0): (t,x)\in\mb{R}^2\}$. We now add a half-ray to the origin and give the resulting
space the structure of a Lorentzian length space. Let $\Gamma:=\{(0,0,z):z\geq 0\}$ and set $\tilde M:= N \cup \Gamma$. On $N$ we use the
relations from Minkowski space and on $\Gamma$ we define $Z_1:=(0,0,z_1)\ll Z_2:=(0,0, z_2)$ if $z_1 < z_2$, and
$Z_1\leq Z_2$ if $Z_1\ll Z_2$ or $Z_1=Z_2$. For $p=(t,x,0)\in N$ and $Z\in \Gamma$ we define $p\ll Z$ if $(t,x)\ll 0$ in
$\mb{R}^2_1$ and analogously for the causal relation.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=72mm, height= 48mm]{extension.pdf}
\end{center}
\caption{Non-manifold extension}
\end{figure}
We define the time separation function $\tau$ as the time separation
function coming from Minkowski space on $N$, for points on $\Gamma$ we set $\tau((0,0,z_1),(0,0,z_2)):=z_2-z_1$ if $z_1\leq
z_2$ (zero otherwise) and for $p=(t,x,0)\in N$ and $Z=(0,0,z)$ we set $\tau(p,Z):=\sqrt{t^2-x^2} + z$ if $p\leq Z$ (and zero
otherwise). As $\tau$ is continuous this gives a Lorentzian pre-length space. In fact, this construction gives a Lorentzian length space as it is clearly
path-connected and locally causally closed.
Moreover, it is regularly localisable since maximal causal curves always exist
(they are the, possibly broken, straight lines) and the induced length agrees with the $\tau$-length by construction.
Furthermore, it is not hard to see that $\tilde M$ is strongly causal. In this space maximal curves branch: every maximal
curve from $J^-(0)$ to $J^+(0)$ has $0$ as a branching point, as the curve is allowed to continue into $N$ or $\Gamma$.
This implies via \cite[Cor.\ 4.13]{KS:17} that $\tilde M$ has timelike curvature unbounded below, i.e., a curvature
singularity in the sense of \cite[Def.\ 4.20]{KS:17}. Finally, $\tilde M$ extends $M\backslash\{(0,0)\}$, thereby providing an
example of a non-manifold extension. Note that $\tilde M$ does not extend $M$ since $M$ is not embedded into $\tilde M$ as an
open subset.
\end{ex}
At this point we can introduce the past and future boundary of Lorentzian pre-length spaces with respect to an extension in complete analogy to the
case of spacetime extensions, see \cite[Def.\ 2.1]{GL:17}.%
\begin{defi}
Let $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ be a Lorentzian pre-length space extending the Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ via the embedding $\iota$. The \emph{future/past boundary}
$\partial^+(X)$\,/\,$\partial^-(X)$ of $X$ is defined as the set of all points $\tilde p\in \partial\iota(X)$ that can be
reached by a future/past directed $\tilde\ll$-timelike curve $\gamma\colon[0,1]\rightarrow \tilde X$ such that
$\gamma([0,1))\subseteq \iota(X)$ and $\gamma(1)=\tilde p$.
\end{defi}
The following result establishes that for any extension of a Lorentzian length space the future or past boundary is non-empty.
It is a direct analogue of \cite[Lemma 2.17]{Sbi:18}).
\begin{lem}\label{lem:Sbi}
Let $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ be an extension of $(X,d,\ll,\leq,\tau)$, where both are Lorentzian length space s, and denote the corresponding isometry by
$\iota$. Then there is a $\tilde\ll$-timelike curve $\tilde\gamma\colon[0,1]\rightarrow \tilde X$ such
that $\tilde\gamma([0,1))\subseteq \iota(X)$ and $\tilde\gamma(1)\in\tilde X\backslash \iota(X)$, i.e.,
$\partial^+(X)\cup\partial^-(X)\neq\emptyset$.
\end{lem}
\begin{pr}
Since $\iota(X)$ is a proper and open subset of $\tilde X$ and $\tilde X$ is connected, we get that $\partial \iota(X)\neq
\emptyset$. Let $\tilde p\in \partial \iota(X)$ and let $\tilde \Omega$ be a localising neighbourhood of $\tilde p$ in $\tilde X$. Then,
$\tilde I^\pm(\tilde p)\cap\tilde\Omega\neq\emptyset$ and let $\tilde q\in \tilde I^-(\tilde p)\cap \tilde \Omega$. We now
consider two cases. First, if $\tilde q\in\iota(X)$, then since $\tilde q\tilde{\ll}\tilde p$ there is a
$\tilde{\ll}$-timelike curve $\tilde\gamma\colon[0,1]\rightarrow \tilde X$ such that $\tilde\gamma(0)=\tilde q$,
$\tilde\gamma(1)=\tilde p$. Set $s_0:=\sup\{s\in[0,1]: \tilde\gamma([0,s])\subseteq\iota(X)\}$, then since $\iota(X)$ is
open and $\tilde p\in\partial\iota(X)$ we have $\tilde\gamma(s_0)\in\tilde X\backslash\iota(X)$. Reparametrising
$\tilde\gamma\rvert_{[0,s_0]}$ to $[0,1]$ yields the result. The second case is when $\tilde q\in\tilde
X\backslash\iota(X)$. Now $\tilde{I}^+(\tilde q)\cap\tilde\Omega$ is a neighbourhood of $\tilde p\in\partial\iota(X)$, thus
$\iota(X)\cap (\tilde{I}^+(\tilde q)\cap\tilde\Omega)\neq \emptyset$. Let $\tilde r\in \iota(X)\cap (\tilde{I}^+(\tilde
q)\cap\tilde\Omega)$, then $\tilde q\tilde{\ll}\tilde r$ and the result follows as in the first case by arguing into the
past.
\end{pr}
\section{Geodesics}\label{sec:geodesics}
In this synthetic approach we have the tools at hand to define causal geodesics as locally length maximising curves.
Furthermore, we establish that for spacetimes the synthetic notion is compatible with the analytical one.
\begin{defi}\label{def:geo}
Let $(X,d,\ll,\leq,\tau)$ be a localising Lorentzian pre-length space and let $\gamma\colon I\rightarrow X$ be a future-directed causal curve. Then $\gamma$ is a
\emph{geodesic} if for every $t_0\in I$ there exists a localising neighbourhood $\Omega$ of $\gamma(t_0)$ and a
neighbourhood $J=[c,d]$ of $t_0$ in $I$ such that $\gamma\rvert_{J}$ is maximal in $\Omega$ from $\gamma(c)$ to $\gamma(d)$.
\end{defi}
\begin{rem}
Let $\gamma\colon I\rightarrow X$ be a geodesic and let $t_0\in I$, and $\Omega$ a localising neighbourhood of $\gamma(t_0)$ as
above. Then
\begin{equation*}
L_\tau(\gamma\rvert_{[c,d]}) = \omega_{\gamma(t_0)}(\gamma(c),\gamma(d))\,,
\end{equation*}
where $\omega_{\gamma(t_0)}$ is the local time separation function on $\Omega$, cf.\ \cite[Def.\ 3.16]{KS:17}.
\end{rem}
To show that for a smooth and strongly causal spacetime this notion is equivalent to the notion of causal pregeodesics
we need the following lemma stating a general property of strongly causal Lorentzian length spaces.
\begin{lem}\label{lem-str-cau-tau-om}
Let $(X,d,\ll,\leq,\tau)$ be a strongly causal Lorentzian length space. Then for all $x\in X$ and every localising neighbourhood $\Omega$ of $x$ with local
time separation function $\omega$ there is a neighbourhood $U$ of $x$, $U\subseteq \Omega$ such that $\omega\rvert_{U\times
U}$ is completely determined by $\tau$: $\forall p,q\in U:\ \omega(p,q)=\tau(p,q)$. In particular, $\tau$ is
continuous on a neighbourhood of the diagonal in $X\times X$.
\end{lem}
\begin{pr}
Let $x\in X$ and let $\Omega$ be a localising neighbourhood of $x$ with local time separation function $\omega$. By strong
causality and \cite[Lemma 2.38(iii)]{KS:17} there is a neighbourhood $U$ of $x$ with $U\subseteq \Omega$ such that all
causal curves with endpoints in $U$ are contained in $\Omega$. Let $p,q\in U$ with $p<q$, then by the properties of $\Omega$
(see Section \ref{sec:lls}) there is a causal curve $\gamma_{pq}$ that is maximal in $\Omega$ from $p$ to $q$ with
$L_\tau(\gamma_{pq}) = \omega(p,q)$. As $p,q\in U$, any causal curve connecting these points is contained in $\Omega$. Thus
$\gamma_{pq}$ is maximal even in $X$, and consequently we have $\tau(p,q) = \mathcal{T}(p,q) = L_\tau(\gamma_{pq}) = \omega(p,q)$.
The neighbourhood of the diagonal can be chosen to be the union of all such $U\times U$ as above.
\end{pr}
With the above lemma we can now establish the promised compatibility.
\begin{thm}\label{thm-pregeo}
Let $(M,g)$ be a smooth, strongly causal spacetime and let $(M,d^h,\ll,\leq,\tau)$ be the induced Lorentzian length space (\cite[Ex.\
3.24(i)]{KS:17}). Then a causal pregeodesic of $(M,g)$ is a geodesic in the sense of Definition \ref{def:geo} and vice versa.
\end{thm}
\begin{pr}
First, let $\gamma\colon I\rightarrow M$ be a causal pregeodesic of $(M,g)$, which we can assume without loss of
generality to be already parametrised as a geodesic. The localising neighbourhoods can be chosen to be (totally) normal
neighbourhoods. Let $t_0\in I$ and let $U$ be a totally normal neighbourhood of $\gamma(t_0)$. Let $J=[c,d]$ be a
neighbourhood of $t_0$ in $I$ such that
$\gamma(J)\subseteq U$ and set $x:=\gamma(c), y:=\gamma(d)$. Since $\gamma$ is a geodesic, it
has to be the radial geodesic from $x$ to $y$ in $U$. As such it is maximal in $U$ and because $L_g=L_\tau$ by \cite[Prop.\
2.32]{KS:17} we obtain%
\begin{equation*}
L_\tau(\gamma\rvert_{[c,d]}) = L_g(\gamma\rvert_{[c,d]}) =
\sqrt{-g_x(\exp_x^{-1}(y),\exp_x^{-1}(y))}=\omega(x,y)\,.
\end{equation*}
Conversely, let $\gamma\colon I\rightarrow M$ be a geodesic in the sense of Definition \ref{def:geo}. As this is a local
question, we can cover $\gamma(I)$ by open sets $U$, where $U\subseteq \Omega$ are as
in the proof of Lemma \ref{lem-str-cau-tau-om}, and show that the
segment of $\gamma$ in any such $U$ is a pregeodesic with respect to $g$. In fact, let $t_0\in I$ with $\gamma(t_0)\in
U_0\subseteq \Omega_0$ and let $J\subseteq I$ be an interval around $t_0$ such that $\gamma(J)\subseteq U_0$. Let $s_1, s_2
\in J$ with $s_1<s_2$, then we get from Lemma \ref{lem-str-cau-tau-om} that
\begin{equation*}
L_\tau(\gamma\rvert_{[s_1,s_2]}) = \omega(\gamma(s_1),\gamma(s_2)) = \tau(\gamma(s_1),\gamma(s_2))\,.
\end{equation*}
Therefore, again since $L_g=L_\tau$, $\gamma$ is maximal on $[s_1,s_2]$ and hence $\gamma$ is a pregeodesic (see e.g.\ \cite[Thm.\ 4.13]{BEE:96}).
\end{pr}
\begin{defi}
Let $(X,d,\ll,\leq,\tau)$ be a localising Lorentzian pre-length space and let $\gamma\colon [a,b)\rightarrow X$ be a future-directed geodesic. Then $\gamma$
is \emph{extendible as a geodesic} if there exists a (future-directed) geodesic $\bar{\gamma}\colon[a,b]\rightarrow X$ with
$\bar{\gamma}\rvert_{[a,b)}=\gamma$. Otherwise, $\gamma$ is called \emph{inextendible as a geodesic}.
\end{defi}
A well-known property of geodesics in smooth semi-Riemannian manifolds is the fact that extendibility
as a geodesic is equivalent to continuous extendibility. Its standard proof relies on the existence of
convex neighbourhoods. The following result is an analogue in the setting of Lorentzian pre-length spaces,
with localising neighbourhoods working as a substitute.
\begin{prop} Let $(X,d,\ll,\leq,\tau)$ be a strongly causal and localising Lorentzian pre-length space and let $\gamma\colon [a,b)\rightarrow X$
be a future-directed geodesic.
Then $\gamma$ is extendible as a geodesic if and only if it is extendible as a continuous curve to $[a,b]$.
\end{prop}
\begin{pr}
Only the `if' part requires a proof, so let us suppose that $\gamma\colon[a,b]\rightarrow X$ is continuous and that
$\gamma|_{[a,b)}$ is a geodesic. Let $\Omega$ be a localising neighbourhood of $\gamma(b)$ and choose $c\in (a,b)$ such that
$\gamma([c,b]) \subseteq \Omega$. Then for any $t\in (c,b)$ we have
\[
L_\tau(\gamma|_{[c,t]}) = \omega(\gamma(c),\gamma(t)),
\]
where $\omega\equiv\omega_{\gamma(b)}$ is the local time separation function on $\Omega$. As $t\nearrow b$,
the right hand side of this equation converges to $\omega(\gamma(c),\gamma(b))$. Concerning the left hand side,
for any $n\in \mb{N}$ with $\frac{1}{n}<b-c$ denote by $\gamma_n \colon [c,b]\to X$ a linear reparametrisation of
$\gamma|_{[c,b-\frac{1}{n}]}$. Then the $\gamma_n$ converge uniformly to $\gamma$ on $[c,b]$. Therefore,
\cite[Prop.\ 3.17]{KS:17} implies that
\[
L_\tau(\gamma|_{[c,b]}) \ge \limsup_n L_\tau(\gamma_n) = \limsup_n \omega(\gamma(c),\gamma(b-1/n)) = \omega(\gamma(c),\gamma(b)).
\]
As the converse of this inequality holds by the definition of localisability (cf.\ Section \ref{sec:lls}), the claim
follows.
\end{pr}
\section{Timelike completeness and inextendibility}\label{sec:TC}
As discussed in the introduction, our approach allows us to mimic the proof from the smooth case
that geodesic completeness implies inextendibility, i.e., \cite[Prop.\ 6.16]{BEE:96}. We first
introduce an appropriate notion of timelike geodesic completeness for Lorentzian pre-length space s.
\begin{defi}
Let $(X,d,\ll,\leq,\tau)$ be a localising Lorentzian pre-length space, then $X$ is said to have property $(TC)$ if all inextendible timelike geodesics have infinite
$\tau$-length.
\end{defi}
This notion is equivalent to timelike geodesic completeness in the case of smooth and strongly causal spacetimes:
\begin{lem}\label{lem-tc}
Let $(M,d^h,\ll,\leq,\tau)$ be the Lorentzian length space induced by a smooth and strongly causal spacetime $(M,g)$. Then $(M,g)$ is
timelike geodesically complete if and only if $(M,d^h,\ll,\leq,\tau)$ has property $(TC)$.
\end{lem}
\begin{pr}
First, let $(M,g)$ be not timelike geodesically complete, so that there exists an inextendible timelike geodesic
(without loss of generality inextendible to the future) $\gamma\colon[a,b)\rightarrow M$, with $b<\infty$, thus
$L_g(\gamma)<\infty$. Since $L_g=L_\tau$ by \cite[Prop.\ 2.32]{KS:17}, Theorem \ref{thm-pregeo} implies that property
(TC) cannot hold. Conversely, let $(M,g)$ be timelike geodesically complete and let $\gamma\colon[0,b)\rightarrow M$ be an inextendible timelike
geodesic (in the sense of Definition \ref{def:geo}). Then by Theorem \ref{thm-pregeo} $\gamma$ is a timelike pregeodesic of
$(M,g)$, hence by completeness $L_g(\gamma)=\infty$ (cf.\ \cite[p.\ 154]{ONe:83}). Since $L_g = L_\tau$, property $(TC)$
follows.
\end{pr}
Property (TC) does guarantee inextendibility, as the following result shows.
\begin{thm}\label{thm-inext-lls-lls}
Let $(X,d,\ll,\leq,\tau)$ be a strongly causal Lorentzian length space that has property $(TC)$. Then $(X,d,\ll,\leq,\tau)$ is inextendible as a regular Lorentzian length space.
\end{thm}
\begin{pr}
Assume, to the contrary, that there exists a regular Lorentzian length space $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ that extends $(X,d,\ll,\leq,\tau)$.
By Lemma \ref{lem:Sbi} there is a (without loss of generality) future-directed $\tilde{\ll}$-timelike curve\\
$\tilde\gamma\colon[0,1]\rightarrow \tilde X$ with $\tilde\gamma([0,1))\subseteq\iota(X)$ and $\tilde\gamma(1)=\tilde p\in
\tilde X\setminus\iota(X)$. Let $\tilde U$ be a localising neighbourhood of $\tilde p$ (with respect to $\tilde X$) and
$\tilde \omega$ its local time separation function. Let $t_0\in[0,1)$ be such that $\tilde\gamma([t_0,1])\subseteq \tilde U$.
Consequently, $q:=\tilde\gamma(t_0)\in \tilde U \cap\iota(X)$ and $q\tilde{\ll} \tilde p$. Thus there is an -- in $\tilde U$
-- $\tilde{\tau}$-maximal curve $\tilde\gamma_{q,\tilde p}\colon[0,1]\rightarrow \tilde U$ from $q$ to $\tilde p$, which is
$\tilde\ll$-timelike by regularity, see \cite[Thm.\ 3.18]{KS:17}. Since
$\iota(X)$ is open, $q\in\iota(X)$ and $\tilde p\notin\iota(X)$ there is a $t_*\in (0,1)$ such that
$\tilde\gamma_{q,\tilde p}([0,t_*))\subseteq\iota(X)$ and $\tilde r:=\tilde\gamma_{q,\tilde p}(t_*)\notin\iota(X)$. Then
$\tilde\gamma_{q,\tilde p}\rvert_{[0,t_*)}\colon[0,t_*)\rightarrow \tilde U\cap\iota(X)$ and we set
$\lambda:=\iota^{-1}\circ \tilde\gamma_{q,\tilde p}\rvert_{[0,t_*)}$. By Definition
\ref{def-ext},\ref{def-ext-tau}, $\lambda$ is $\ll$-timelike. We claim that $\lambda$ is a timelike $\tau$-geodesic. To
this end, recall that a maximal causal curve is maximal on any subinterval, see \cite[Prop.\ 2.34,(ii)]{KS:17}. Fix any
$0\leq s_0 < t_*$, and let $V$ be a neighbourhood of $\lambda(s_0)$ with $\iota(V)\subseteq \tilde U$.
As $X$ is strongly causal, there exists a neighbourhood $V'\subseteq V$ of $\lambda(s_0)$ such that
any causal curve that starts and ends in $V'$ is contained in $V$.
Now suppose that $s_1\le s_0<s_2$ are so close that $\lambda|_{[s_1,s_2]}$ is contained in $V'$.
Then in particular any future-directed $\leq$-causal curve connecting
$\lambda(s_1)$ to $\lambda(s_2)$ remains entirely in $V$. By Definition \ref{def-ext},\ref{def-ext-tau} we
therefore obtain
\begin{equation*}
\begin{split}
L_\tau(\lambda|_{[s_1,s_2]}) &= L_{\tilde \tau}(\iota\circ\lambda|_{[s_1,s_2]})\\
&=\max\{L_{\tilde \tau}(\tilde\alpha):\tilde\alpha \text{ f.d. } {\tilde\le\text{-causal from }} \iota\circ\lambda(s_1)
\text{ to }
\iota\circ\lambda(s_2) \text{ in } \tilde U \}\\
&\ge\max\{L_{\tilde\tau}(\iota\circ\alpha):\alpha \text{ f.d. } {\le\text{-causal from }} \lambda(s_1) \text{ to }
\lambda(s_2)
\text{ in } V \}\\
&=\max\{L_{\tau}(\alpha):\alpha \text{ f.d. } {\le\text{-causal from }} \lambda(s_1) \text{ to } \lambda(s_2) \text{ in } V
\}\\
&=\max\{L_{\tau}(\alpha):\alpha \text{ f.d. } {\le\text{-causal from }} \lambda(s_1) \text{ to } \lambda(s_2) \text{ in }
X \}\\
&= \mathcal{T}(\lambda(s_1),\lambda(s_2)))\ge L_\tau(\lambda|_{[s_1,s_2]})\,.
\end{split}
\end{equation*}
Thus $L_\tau(\lambda|_{[s_1,s_2]}) = \mathcal{T}(\lambda(s_1),\lambda(s_2))) = \tau(\lambda(s_1),\lambda(s_2)))$. By Lemma
\ref{lem-str-cau-tau-om}, any local time separation function is completely determined by $\tau$ on $V'$, hence the above shows
that $\lambda$ is a geodesic in $X$. Moreover, the length of $\lambda$ is given by
\begin{equation*}
L_\tau(\lambda) = L_{\tilde\tau}(\iota\circ\lambda) = \lim_{t\nearrow t_*}L_{\tilde\tau}(\tilde\gamma_{q,\tilde p}\rvert_{[0,t]}) =
\lim_{t\nearrow t_*}\tilde\omega (q,\tilde\gamma_{q,\tilde p}(t)) = \tilde\omega(q,r)<\infty\,,
\end{equation*}
as the local time separation function $\tilde\omega$ of $\tilde U$ (with respect to $\tilde X$) is continuous and finite.
Finally, $\lambda$ is inextendible as a geodesic in $X$ since it is not even extendible as a continuous curve ($\lim_{t\nearrow
t_*}\iota\circ\lambda(t) = \lim_{t\nearrow t_*}\gamma_{q,\tilde p}(t)= \tilde r\notin\iota(X)$) --- thus contradicting property
$(TC)$.
\end{pr}
We can now relate the low regularity inextendibility to a blow-up of curvature. More precisely, we have
the following result.
\begin{thm}\label{thm-tc-ext-cur-sing}
Let $(X,d,\ll,\leq,\tau)$ be a strongly causal Lorentzian length space that has property $(TC)$. If $X$ is extendible, the extension has a causal curvature
singularity (\cite[Def.\ 4.20]{KS:17}). Specifically, the extension cannot have bounded upper causal curvature.
\end{thm}
\begin{pr}
Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian length space that is strongly causal and has property $(TC)$. Assume that there exists a Lorentzian length space $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ extending $(X,d,\ll,\leq,\tau)$ and
having causal curvature bounded above. Then \cite[Rem.\ 4.16, Thm.\ 4.17 and Thm.\ 4.18]{KS:17} yield that $(\tilde X,\tilde d,\tilde \ll,\tilde \leq,\tilde \tau)$ is regular.
This contradicts the inextendibility result Theorem \ref{thm-inext-lls-lls} and yields that $X$ has a curvature singularity
in the sense of \cite[Def.\ 4.20]{KS:17}.
\end{pr}
We now specialise to the case where the object to be extended is a smooth spacetime. Firstly, recall that~\emph{causally plain\/} spacetimes are precisely those that do not exhibit the bubbling phenomenon. Roughly speaking, a metric is bubbling if it contains a point where the boundary of the future null cone has non-empty interior. (For a precise definition, see~\cite[Definition~1.16]{CG:12}; cf.~also the discussion preceding Lemma~5.6 in~\cite{KS:17}.) Spacetimes $(M, g)$ with $g$ a Lipschitz metric are causally plain~\cite[Corollary~1.17]{CG:12}.
The following result is now a direct corollary of Theorem \ref{thm-inext-lls-lls}.
\begin{cor}\label{cor-geo-compl-inext-lls}
Let $(M,g)$ be a smooth, strongly causal and timelike geodesically complete spacetime and let $(M,d^h,\ll,\leq,\tau)$ be its
induced Lorentzian length space. Then $(M,d^h,\ll,\leq,\tau)$ is inextendible as a regular Lorentzian length space, and hence also inextendible in the class of continuous,
strongly causal and causally plain spacetimes that are regular.
\end{cor}
\begin{pr}
By Lemma \ref{lem-tc} $(M,d^h,\ll,\leq,\tau)$ has property $(TC)$ and strong causality is the same notion for
spacetimes and the corresponding Lorentzian length space s by \cite[Lemma 2.21(i),(ii) and Lemma 2.38(iii)]{KS:17}.
Thus, Theorem \ref{thm-inext-lls-lls} applies, showing that
$(M,d^h,\ll,\leq,\tau)$ is inextendible as a regular Lorentzian length space. Furthermore, by \cite[Thm.\ 5.12]{KS:17} every continuous
strongly causal and causally plain spacetime $(\tilde M, \tilde g)$ gives rise to a Lorentzian length space.
\end{pr}
Also in this case of spacetimes we obtain the result that timelike geodesic completeness forces the extension to have a
curvature singularity, even though curvature cannot be defined in the usual sense via the Riemann
tensor.
\begin{cor}\label{cor-inext-st-cur-sing}
Let $(M,g)$ be a smooth, strongly causal and timelike geodesically complete spacetime and let $(M,d^h,\ll,\leq,\tau)$ be
its induced Lorentzian length space. If $(M,d^h,\ll,\leq,\tau)$ is extendible as a Lorentzian length space then the extension has a causal curvature singularity
(it cannot have causal curvature bounded above).
\end{cor}
\begin{pr}
This follows directly from Theorem \ref{thm-tc-ext-cur-sing}, similarly to the proof of Corollary \ref{cor-geo-compl-inext-lls}.
\end{pr}
\begin{rem}
In \cite{AB:08}, Alexander and Bishop introduced sectional curvature bounds for general semi-Riemannian manifolds. Moreover,
they characterized these curvature bounds via triangle comparison with small triangles in model spaces (i.e., the spaces
$M_K$ from \eqref{eq:model_spaces} in the Lorentzian setting), see \cite[Thm.\ 1.1]{AB:08}. As was shown in \cite[Ex.\ 4.9]{KS:17}, our definitions in
Section \ref{sec:lls} are compatible with these curvature bounds in this sense and in particular a curvature
singularity in our sense implies that there cannot be a corresponding sectional curvature
bound in the sense of \cite{AB:08}.
Corollary \ref{cor-inext-st-cur-sing} therefore implies that if the extension is assumed to be a smooth and strongly causal
spacetime itself, then its sectional curvature as defined in \cite{AB:08} must be unbounded above.
\end{rem}
To conclude this section we note that it is an interesting open question whether one can characterize completeness of
timelike geodesics in Lorentzian length space s via condition $(TC)$, analogous to the smooth case, cf.\ \cite[p.\ 154]{ONe:83}.
\section{Relation to other results on low regularity inextendibility}\label{sec:relation}
In this final section we relate our work to further current results on the low regularity inextendibility of spacetimes.
In \cite{GL:18} it was recently established that in a (locally) Lipschitz continuous spacetime maximal causal curves have a
causal character. This immediately gives that the induced Lorentzian length space $(M,d^h,\ll,\leq,\tau)$ of a strongly causal Lipschitz
spacetime $(M,g)$ is regular: By \cite[Cor.\ 1.17]{CG:12} and \cite[Thm.\ 5.12]{KS:17} $(M,d^h,\ll,\leq,\tau)$ is a Lorentzian length space and
by \cite[Thm.\ 1.1]{GL:18} it is regular (a fact that was already observed by Graf and Ling in \cite{GL:18}). From this they
deduce that a timelike geodesically complete smooth spacetime is inextendible in the class of Lipschitz spacetimes. Thus,
their result is slightly stronger than ours when restricted to spacetimes (compare Corollary \ref{cor-geo-compl-inext-lls})
as they do not need strong causality of the original spacetime. However, even when restricting to the case where the object
to be extended is a spacetime, our result is more general in the following sense:
\begin{itemize}
\item It allows the original spacetime to be of low regularity (continuous and causally plain) as well.
\item There might be continuous strongly causal, causally plain spacetimes inducing a regular Lorentzian length space where the metric is not locally
Lipschitz continuous.
\item It applies even to non-manifold extensions, and
\item it relates inextendibility with curvature blow-up (Theorem \ref{thm-tc-ext-cur-sing}).
\end{itemize}
In \cite{GLS:18} the authors show that a smooth, timelike geodesically complete and globally hyperbolic spacetime is
$\ensuremath{\mathcal{C}}^0$-inextendible, i.e., there is no spacetime with continuous metric extending the given spacetime. Again, as above,
their result is slightly stronger when restricting to spacetimes, since of course not all spacetimes with continuous metrics
give rise to a Lorentzian length space, as they need not be causally plain and strongly causal (see e.g.\ \cite[Ex.\ 1.11]{CG:12}). However,
our approach does not need the original spacetime to be globally hyperbolic and (as above) allows it to be of low regularity
as well. Moreover, as noted above our result also rules out non-manifold extensions (as long as they are regular Lorentzian length space s). A
closer inspection of the proof of Theorem \ref{thm-inext-lls-lls} reveals that one does not need that the entire extension is
regular. In fact, all that is needed is that a maximal causal curve $\gamma$ that is contained in the original space except for
its endpoint (which is on the boundary) is timelike whenever its starting point and endpoint are timelike related in the
extension. This is weaker than being regular, as it essentially only concerns points in the original space and its boundary.
Thus the main result of \cite{GLS:18} can be understood in this way: If the smooth spacetime is timelike geodesically complete
and globally hyperbolic, then maximal causal curves as above have a causal character. This then yields the inextendibility
result.
It should also be noted that in our framework one can define \emph{future/past one-connectedness} (\cite[Def.\
2.13]{Sbi:18}) and \emph{future/past divergence} (\cite[Def.\ 2.4(2)]{GL:17}) as for spacetimes. Since being extendible forces
the future or past boundary to be non-empty by Lemma \ref{lem:Sbi} a further line of study could be to see if, as for
spacetimes, future (past) one-connectedness together with future (past) divergence yield empty future (past) boundary (cf.\
\cite[Thm.\ 2.5]{GL:17}).
\bigskip
To summarize, we have developed a framework where we can show inextendibility of spaces that resemble timelike
geodesically complete spacetimes, in a similar spirit as the classical result (\cite[Prop.\ 6.16]{BEE:96}). Our approach provides a
new and unified perspective on the recent results \cite{GLS:18, GL:18}, see the discussion above.
Moreover, for the first time we can relate low regularity inextendibility with a (synthetic) curvature blow up --- a fact
that fits well with physical expectations. Finally, it shows that timelike geodesic completeness is a very robust property,
which carries over even to spaces that are not spacetimes or even manifolds.
\bigskip
\noindent
{\bf Acknowledgements.} This work was supported by research grants P26859 and P28770 of the Austrian Science Fund FWF. The
work of J.G.\ was partially supported by STFC Consolidated Grant ST/L000490/1.
|
1,314,259,996,673 | arxiv | \section{Introduction}
The silicon photomultiplier (SiPM) has evolved into an
established photodetector technology. They are used in high-energy physics
\cite{Ogawa2017,Moreau2010,Mannel2013}, astroparticle physics
\cite{Anderhub2013,Otte2015}, medical imaging \cite{Otte2005,Spanoudaki2007},
and LIDARs \cite{Perenzoni2017,Agishev2013}, to only name a few areas of
application. One key factor to the success of SiPMs is the continuing effort
made by manufacturers to reduce nuisance parameters like dark-count rate,
afterpulsing, and optical crosstalk.
Diagnostic tools are crucial in these efforts as they help to identify means
that further reduce nuisance parameters, which in turn, improves the performance
of SiPMs. One way to diagnose SiPMs is to measure their characteristics as
function of temperature and bias, model the data, and extract physical
meaningful quantities from the model parameters. We have taken that
approach in earlier work \cite{Otte2016a} and we use it again here.
In this paper we emphasize the use of the bias dependence of the breakdown
probability, which we already used in \cite{Otte2016a} to determine the origin
of optical crosstalk in a Hamamatsu device and extend it to a discussion of the
location of the avalanche region. The approach is not new, we first presented it
at \cite{Otte2015a} and it was used to characterize FBK devices
\cite{Zappala2016}. Compared to \cite{Zappala2016} we use a parameterization,
which is less dependent on the device specifics as we will discuss in detail.
\section{Devices used in this Study}
The Hamamatsu SiPM is a prototype named LVR2-6050-CN. The device has an
active area of $6\times6$\,mm$^{2}$ and is composed of $50\,\mu$m sized cells.
For better UV sensitivity the sensor is not covered with a protective layer. The
breakdown voltage at room temperature ($24^\circ$C) is 38.4\,V and the bias
voltage to achieve a 90\% breakdown probability for 400\,nm photons is about
42\,V (see later). That bias voltage is less than
the 56\,V required for the Hamamatsu LCT5 device we tested in \cite{Otte2016a}.
Whether the lower bias is due to a narrower high-field region in the present
device or due to other changes in the technology we do not know.
The second device is a KETEK PM3325 WB SiPM.\footnote{\url{https://www.ketek.net/store/category/sipm-standard-devices/wb-series/}} It has an active area of
$3\times3$\,mm$^{2}$ and $25\,\mu$m cells. The chip is protected with a
$400\,\mu$m thick glass window. The PM3325 does not feature trenches to suppress
optical crosstalk. The bias voltage to achieve a 90\% breakdown probability
when illuminated with 400\,nm photons is about 32\,V and the breakdown voltage
is 27.5\,V at room temperature.
\begin{figure}[!tb]
\centering
\includegraphics*[width=0.8\columnwidth]{CellSketch.pdf}
\caption{
Conceptual cross section of one cell of a \emph{p}-on-\emph{n} SiPM. Blue photons are absorbed mostly before
reaching the avalanche region and an electron (filled circle) drifts down into the
high-field region. Red photons are absorbed mostly after the avalanche region and a
hole (empty circle) drifts up into the high-field region. If the photon is absorbed
in the non-depleted bulk, the hole first has to diffuse into the depleted volume
before it can drift into the avalanche region.
}
\label{fig:sketch}
\end{figure}
\section{Probing the Avalanche Structure with Photon Detection Efficiency
Measurements}
The photon detection efficiency (PDE) is one example where the breakdown
probability plays a decisive role. Depending on the photon's absorption length
and the location and extension of the high-field region, a photon is either
absorbed before the high-field region (blue photons) or after it (red photons).
See Figure \ref{fig:sketch} for a conceptual sketch of one SiPM cell, which
illustrates the situation.
The photon absorption results in the generation of an electron and hole,
which - in case the absorption takes place in the active volume of the cell -
drift in opposite directions due to the electric field in the depleted volume.
If the photon is absorbed after the high-field region in a \emph{p}-on-\emph{n}
structure like the ones studied here, it is the hole that drifts into the high-field region, if the photon
absorbs before the high-field region, it is the electron that drifts down into the avalanche
region.
The probability to initiate a Geiger breakdown is smaller for holes than
for electrons (due to the lower mobility of holes in silicon, e.g.\ \cite{Oldham1972}).
If one could
measure the probability of a subsequent breakdown as a function of where the
electron/hole pair is released one would, therefore, reverse engineer the location and vertical extension of the
high-field region.
Such a mapping is indeed possible with bias dependent PDE
measurements as has been shown in \cite{Otte2015a,Zappala2016}.
For the Hamamatsu SiPM we measured the PDE at three wavelengths and for the
KETEK device at four wavelengths. A description of the setups and procedures
used for the PDE and all other measurements presented here is given in
\cite{Otte2016a}.
Like in our previous measurements we find that the PDE for
a given wavelength is well fit with the empirical model
\begin{equation}\label{eq:PDE}
PDE(U_{\mbox{\tiny rel}}) = PDE_{\mbox{\footnotesize max}}\left[1-e^{-
\mathcal{O}\cdot U_{\mbox{\tiny
rel}}} \right],
\end{equation}
where $U_{\mbox{\tiny rel}}= \left(U - U_{\mbox{\tiny BD}}\right)/{U_{\mbox{\tiny
BD}}}$ is the relative overvoltage above the breakdown
voltage $U_{\mbox{\tiny BD}}$. $PDE_{\mbox{\footnotesize max}}$ is the PDE in
saturation but is not necessarily the true saturation value because we cannot
measure the PDE at higher bias values. The term in square brackets is the breakdown
probability, which depends only on the product of the relative overvoltage and a
dimensionless parameter $\mathcal{O}$, which is mostly dependent on whether
an electron or a hole initiates a breakdown as we explain
later.\footnote{$\mathcal{O}$ was jocularly referred to as the \emph{Otte
number} at recent meetings.}
In that context it is interesting to remark that empirically all the bias
dependent physics of the breakdown is included in one single constant, or in a
linear function when larger relative overvoltages than measured here are taken
into account \cite{Zappala2016}. Because our data is well described with one
constant we do not need to consider the linear function, which would,
furthermore, not be sufficiently constrained by our data. The devices we tested
cannot be operated much beyond the measured voltage range.
While the overall fit function is the same as in \cite{Zappala2016} there are
two differences in its usage. Instead of plotting the breakdown probability as a
function of absolute bias voltage we use the relative overvoltage
$U_{\mbox{\tiny rel}}$. The second difference is that we characterize the
electron/hole initiation probability with $\mathcal{O}$ instead of the voltage
at which the PDE reaches 95\%. $\mathcal{O}$ and $U_{\mbox{\tiny rel}}$ are inherently less
dependent on the structure of the device and temperature than the absolute bias voltage as we
shall motivate in the following.
The avalanche and breakdown characteristics of a pn-junction are governed by the
ionization rates, which depend strongest on the electric field and much less on
device specifics like the doping profile, doping concentrations, or
temperature \cite{Sze2007}. By parametrizing the breakdown probability as a
function of the average electric field $\bar{E}$ in the high-field region and
not as a function of absolute voltage one arrives at a
parameterization that depends mostly on avalanche physics.
With such a parameterization it should then be possible to extract information
about the breakdown characteristics that can be compared with measurements from
other devices in a meaningful manner.
The bias $U = \bar{E}/w$ depends on the device specific
parameter $w$, \emph{i.e.}\ the \emph{effective} width of the high field
region and thus cannot fulfill the task of a device-independent characterization. The
relative overvoltage $U_{\mbox{\tiny rel}}$, on the other hand, is independent of $w$ and proportional
to $\bar{E}$.
\begin{equation}
U_{\mbox{\tiny rel}} = \frac{U - U_{\mbox{\tiny BD}}}{ U_{\mbox{\tiny BD}}} =
\frac{\bar{E}\cdot w - \bar{E}_{\mbox{\tiny BD}}\cdot w}{ \bar{E}_{\mbox{\tiny BD}}\cdot w} =
\frac{\bar{E} - \bar{E}_{\mbox{\tiny BD}}}{ \bar{E}_{\mbox{\tiny BD}}}
\end{equation}
where $\bar{E}_{\mbox{\tiny BD}}$ is the electrical field at breakdown averaged
across the high-field region.
We note that $w$ drops out if the width of the depleted region does not change
between breakdown and operating voltage. That assumption holds true for most
available SiPM including the tested devices where the gain as a function of bias
voltage is described by a linear function (see Figure \ref{fig:Gain}).
\begin{figure}[!tb]
\subfloat[Hamamatsu
LVR2]{\includegraphics*[width=\columnwidth]{HamamatsuGainvsBias.pdf}}
\subfloat[KETEK
PM3325]{\includegraphics*[width=\columnwidth]{KETEKGainvsBias.pdf}}
\caption{Gain as a function of absolute voltage for seven different
temperatures.
\label{fig:Gain}
}
\end{figure}
While, as mentioned above, most of the breakdown characteristics depend on the
electric field, other factors play a role too. $U_{\mbox{\tiny rel}}$ compensates
for some but admittedly, not all of the device and temperature dependencies by
normalizing to $\bar{E}_{\mbox{\tiny BD}}$.
We can show that at least the temperature dependencies of the breakdown
characteristics are properly taken care of. Optical
crosstalk measurements taken at 100\,K temperature difference fall on top of
each other when plotted as a function of $U_{\mbox{\tiny rel}}$ (see Figure
\ref{fig:PromptOC}), which would not be the case if plotting against $U_{\mbox{\tiny rel}}$ would
not compensate for temperature dependencies. The picture is very different when
optical crosstalk is plotted as a function of absolute voltage.
Using $U_{\mbox{\tiny rel}}$ in the argument of the exponential function of the
breakdown probability can be viewed as a Taylor series expansion about the
critical electric field $\bar{E}_{\mbox{\tiny BD}}$. The linear coefficient in
the expansion is $\mathcal{O}$, the constant term is obviously zero or so small
that it is not relevant, higher order terms can be relevant \cite{Zappala2016}. $\mathcal{O}$ thus parameterizes the electric field
dependence of the breakdown, which as we have discussed above does not depend
much on the device specifics. $\mathcal{O}$ can thus be compared between
devices, contrary to the absolute voltage when the breakdown probability reaches
95\%.
But the breakdown probability depends strongly on
whether an avalanche is initiated by electrons or holes and it is, therefore,
expected that $\mathcal{O}$ changes with changing electron/hole breakdown
initiation ratio.
Figure \ref{fig:BD} shows the breakdown probability derived from the PDE
measurements, i.e.\ the PDE divided by $PDE_{\mbox{\footnotesize max}}$.
The solid lines depict the best fit parameterizations of the breakdown
probability, which all yield fit probabilities of 30\% or better.
The fitted values of $\mathcal{O}$ are listed in Table
\ref{tab:ottenumb} together with the corresponding photon absorption lengths.
The 589\,nm light source was not available for the measurement of the
Hamamatsu device.
\begin{figure*}[!tb]
\centering
\subfloat[Hamamatsu LVR2]{\includegraphics*[width=\columnwidth]{LVR2BDProb.pdf}}
\subfloat[KETEK PM3325]{\includegraphics*[width=\columnwidth]{KETEKBDP.pdf}}
\caption{Breakdown probability versus relative overvoltage for the two tested
devices. The lines are fits to the data points with the model described
in the text.
}
\label{fig:BD}
\end{figure*}
The value of $\mathcal{O}$ decreases with increasing photon wavelength for each
device, which is a testimony to the fact that the breakdown probability shifts
from majority electron to majority hole initiated breakdowns.
$\mathcal{O}$ thus shows a clear dependence on the ratio of electron to hole
initiated breakdowns.
\begin{table}
\caption{Values of $\mathcal{O}$ derived from PDE measurements at different
wavelength for the two devices. The second column gives the absorption length of
photons with the wavelength given in the first column.\label{tab:ottenumb}}
\centering
\begin{tabular}{c|c|c|c}
\hline
Wavelength & Absorp. length&\multicolumn{2}{c}{
$\mathcal{O}$}\\\cline{3-4}
$[$nm$]$ & $[\mu$m$]$&Hamamatsu & KETEK \\\hline
400 & 0.082 &$17.7\pm0.6$ & $14.6\pm0.3$ \\\hline
452 & 0.43 & $16.1\pm0.6$ & $12.1\pm0.3$ \\\hline
500 & 0.91 & $12.3\pm0.4$ & $9.9\pm0.2$ \\\hline
589 & 2.0 & N/A & $8.3\pm0.2$ \\\hline
\end{tabular}
\end{table}
The absolute value of $\mathcal{O}$ should also depend on the dimensions of the
avalanche region, which we do not know and thus cannot explore further.
For the time being, we resort to the assumption that the dependence
of $\mathcal{O}$ on the width of the avalanche region is small compared to the
observed change with photon-wavelength and can be neglected.
How valid that assumption is needs to be shown in the future on devices
with known dimensions of the high field region.
The avalanche regions of the two tested devices probably have fairly similar widths, which we infer from
the similarities of their respective breakdown voltages, which are 26.8\,V and
37.5\,V at 0\,$^{\circ}$C for the KETEK and Hamamatsu SiPM, respectively.
For the Hamamatsu device $\mathcal{O}$ is 12 for photon
absorption lengths of $0.9\,\mu$m while the KETEK SiPM yields the same number for
absorption lengths of $0.4\,\mu$m. If the difference in absorption lengths is
taken at face value and $\mathcal{O}$ does not depend strongly on details of the
two structures, it follows that the avalanche region is located $0.5\,\mu$m
deeper in the Hamamatsu SiPM than in the KETEK SiPM.
Two more observations are that a) in between absorption lengths $0.08\,\mu$m and
$0.4\,\mu$m, $\mathcal{O}$ changes little in the Hamamatsu SiPM, while it
changes much more in the KETEK SiPM. And b) $\mathcal{O}$ never reaches as high
a value in the KETEK SiPM as in the Hamamatsu SiPM. Under the assumption that
$\mathcal{O}$ does not depend strongly on details of the two structures, we
interpret both observations as evidence for a location of the avalanche region
in the KETEK SiPM that is right below the surface and that already for 400\,nm
photons a significant fraction of photons absorb after the avalanche region. In
the Hamamatsu SiPM, on the other hand, the passive region right below the
surface and before the drift volume starts is thinner than in the KETEK device.
Thus more photons are absorbed and mostly electrons drift into the high-field
region also for $<400$\,nm photons.
In that scenario it is expected that the spectral response of the Hamamatsu
device is higher below 400\,nm because of the larger active volume above the
high field region and thinner passive area. It is also expected that the response of the KETEK SiPM peaks at lower
wavelengths than in the Hamamatsu SiPM because the breakdowns change to hole
dominate ones for shorter wavelengths in the KETEK device than in the Hamamatsu
one. That is indeed what we
observe. Figure \ref{fig:PDE} shows the spectral response of the two devices
measured with the setup explained in \cite{Otte2016a}.
\begin{figure}[!tb]
\centering
\includegraphics*[width=\columnwidth]{PDEvsWL.pdf}
\caption{
PDE vs.\ wavelengths of the two SiPMs from 200\,nm to 1000\,nm.
For the measurement, the bias voltage for each devices is chosen such that the
breakdown probability for 400\,nm photons is 90\%. The spectral response measurement
is fit to the PDE measurements denoted by the data points.
}
\label{fig:PDE}
\end{figure}
\section{Where Optical Crosstalk Photons enter a Cell}
In this section we discuss how $\mathcal{O}$ can be used to determine
where optical crosstalk photons enter a cell. Optical crosstalk (OC) is caused by photons
that are emitted in the breakdown of one cell and propagate into a neighboring
cell where they initiate an additional breakdown. One distinguishes two types
of OC (see e.g.\ \cite{Buzhan2009} ) In case the photon absorbs in the active (depleted) volume of a cell,
the additional breakdown happens nearly simultaneous to the first breakdown, which is why that type of OC is called prompt or direct OC. If
the photon is absorbed in a non-depleted region, e.g.\ in the bulk, the
generated charges first have to diffuse into the depleted volume before they can
initiate a breakdown . The diffusion time $\Delta t$ can take several tens of
nanoseconds depending on the distance $d$ between the location of the photon
absorption and the border to the active volume of the cell; $\Delta t \propto
\sqrt{d}$. But it can also be just a fraction of a nanosecond if the photon
absorbs close to the border.
How well the two types of OC can be separated depends on how well two subsequent
pulses can be separated in the measurement. Any prompt OC measurement is thus
always a combination of \emph{true} prompt OC events and delayed OC events that
have a time delay, which is below the capability of the measurement setup to
resolve two overlapping pulses. Two pulses can be identified as
such in our setup, if they are more than two nanoseconds apart.
\begin{figure*}[!tb]
\centering
\subfloat[Hamamatsu
LVR2]{\includegraphics*[width=\columnwidth]{DOCLVR2.pdf}}
\subfloat[KETEK PM3325]{\includegraphics*[width=\columnwidth]{DOCKetek.pdf}}
\caption{Prompt optical crosstalk of the two tested SiPMs. The black arrow marks
the relative overvoltage at which both devices yield a 90\% breakdown
probability for 400\,nm photons.
\label{fig:PromptOC}
}
\end{figure*}
Figure \ref{fig:PromptOC} shows the prompt OC of the two devices recorded at
seven temperatures between $-75^{\circ}$C and $40^{\circ}$C. In this and
subsequent measurements, OC is quantified as the probability that the breakdown
of one SiPM cell causes one or more other cells to break down too. For the Hamamatsu
device we discarded the measurement at $40^{\circ}$C because the contamination
from pile-up of uncorrelated dark counts was too large and could not be reliably
subtracted. For all other measurements, the accidental pile-up within a 2\,ns
time window could be subtracted by assuming that the number of dark counts in a
given time interval are Poisson distributed. After the correction, all OC curves of one
device fall on top of each other, as expected.
We now compare the OC of the two devices at the bias where the breakdown
probability for 400\,nm photons is 90\%.\footnote{We do not imply that this
operating point is optimal for an application but it allows for an unbiased
comparison.} The arrow in each panel marks the corresponding relative
overvoltage. The KETEK device has a fairly high optical crosstalk of $\sim20$\%,
which is not surprising because it does not have trenches to prevent photons
from propagating into neighboring cells. The prompt OC in the Hamamatsu device,
on the other hand, is only 1.5\%, which is an impressive improvement compared to
past developments \cite{Otte2016a}.
In \cite{Otte2016a} we showed that a valid model of the optical
crosstalk probability vs.\ relative overvoltage is
\begin{equation}\label{OCf}
OC(U_{\mbox{\footnotesize rel}}) = f\cdot C_{\mbox{\footnotesize eff}}\cdot U_{\mbox{\footnotesize rel}}\cdot U_{\mbox{\footnotesize BD}}\cdot \gamma\cdot
\left[1-e^{\left(-\mathcal{O}\cdot U_{\mbox{\footnotesize rel}}\right)}\right]\,.
\end{equation} where we use $f=3\cdot10^{-5}$ from \cite{NepomukOtte2009a} as the number of photons produced per
charge carrier in the avalanche that can also cause OC. We note that other measurements of the photon
intensity exist, e.g.\ \cite{Lacaita1993,Mirzoyan2009}, but those also include
spectral components, which are irrelevant for OC, either because the photon
absorption lengths are too long (photons do not absorb in the device) or too
short (photons absorb in the same cell they are emitted from). $C_{\mbox{\footnotesize eff}}\cdot
U_{\mbox{\footnotesize rel}}\cdot U_{\mbox{\footnotesize BD}}$ is the gain of
the SiPM, and $\gamma$ is a figure of merit that quantifies what fraction of the
photons produced in a breakdown make it into a neighboring cell. The term in
square brackets is the breakdown probability already discussed in the previous
section.
The OC data in Figure \ref{fig:PromptOC} are fit with that model. For the fit we
fixed the cell capacitance $C_{\mbox{\footnotesize eff}}$ at 84\,fF and 154\,fF
and the breakdown voltage at 26.8\,V and 37.5\,V at $0^\circ$C, for the KETEK
and Hamamatsu SiPM, respectively. The capacitance and breakdown voltages had
been measured as described in \cite{Otte2016a}. The breakdown voltage is found to
increase by about 0.1\%/$^\circ$C in both devices.
The Hamamatsu OC measurements can be fit over the entire measured range with an
acceptable fit probability. For the KETEK device, we had to restrict the upper
end of the fit range to a relative overvoltage of 0.15, i.e.\ OC of less
than 20\%, in order for the fit to yield an acceptable fit probability. It is evident from
the KETEK data points, that the OC data turn over in what seems to be a saturating
behavior. An explanation for this behavior is that for large OC
of more than 20\% and the cell size of the device, the probability of more than one
OC photon being absorbed in the same
cell cannot be neglected anymore. That effect is not included in the fit model.
\begin{table}
\caption{Best Fit Values for $\gamma$ Obtained From Fitting the Prompt Optical Crosstalk Measurements Shown in Fig.\ \ref{fig:PromptOC}.
Also shown is the $\mathcal{O}$ value for each fit. The last three rows give the
values obtained from \cite{Otte2016a}.}
\centering
\begin{tabular}[!htb]{c|c|c|c}
Device&Temp.&$\gamma$&$\mathcal{O}$\\\hline\hline
Hamamatsu&-75$^{\circ}$C&0.012$\pm$0.001&13.9$\pm$1.3\\
LVR2 &-60$^{\circ}$C&0.014$\pm$0.001&9.9$\pm$1.4\\
&-40$^{\circ}$C&0.013$\pm$0.001&10.9$\pm$1.6\\
&-20$^{\circ}$C&0.018$\pm$0.003&6.8$\pm$1.2\\
&0$^{\circ}$C&0.017$\pm$0.001&7.2$\pm$0.9\\
&20$^{\circ}$C&0.014$\pm$0.001&16$\pm$0.8\\\hline
KETEK &-75$^{\circ}$C&0.347$\pm$0.009&12.9$\pm$0.7\\
PM3325 WB &-60$^{\circ}$C&0.355$\pm$0.008&13.7$\pm$0.8\\
&-40$^{\circ}$C&0.378$\pm$0.009&13.5$\pm$0.7\\
&-20$^{\circ}$C&0.384$\pm$0.007&15.3$\pm$0.7\\
&0$^{\circ}$C&0.42$\pm$0.01&12.9$\pm$0.7\\
&20$^{\circ}$C&0.415$\pm$0.008&13.8$\pm$0.6\\
&40$^{\circ}$C&0.442$\pm$0.007&12.8$\pm$0.4\\\hline
Hamamatsu LCT5&&0.077$\pm$0.001&13$\pm$0.2\\
SensL J-Series&&0.126$\pm$0.002&8.5$\pm$0.1\\
FBK NUV-HD&&0.557$\pm$0.002& N/A\\
\end{tabular}
\label{OCvals}
\end{table}
Table \ref{OCvals} lists the values for $\gamma$ from the fits. The
average values from our previously measured devices are also listed
\cite{Otte2016a}. Comparing the numbers it is evident that the structure of the
LVR2 device is 5.5 times better than the LCT5 device in preventing photons from
crossing cells. The value for $\gamma$ is 0.014, i.e.\ 1.4\% of all
photons make it into a neighboring cell where they can cause
optical crosstalk. In the KETEK SiPM, between 35\% and 44\% of the photons cause optical crosstalk.
The second factor that determines the amount of OC is the product of breakdown
voltage and cell capacitance, which is 2.25\,pF$\cdot$V for the KETEK and
5.78\,pF$\cdot$V for the Hamamatsu SiPM. It is a figure of merit that is
proportional to the charge generated in an avalanche. Minimizing the figure of
merit by designing
devices with small breakdown voltage and/or small cell capacitance
minimizes OC while retaining good breakdown characteristics, which are
governed by $U_{\mbox{\tiny rel}}$.
This time it is the KETEK
SiPM that outperforms the Hamamatsu device by a factor of 2.6 because of its smaller cell capacitance. However,
the Hamamatsu SiPM has a two times smaller cell capacitance per cell area. We
would thus expect that the product of cell capacitance and breakdown voltage for
an LVR2 with $25\,\mu$m cells will be two times lower than for the KETEK device.
This assumes that the cell capacitance scales linear with area, which is not
necessarily the case as edge effects become important for small cell sizes.
The fit results also allow us to draw conclusions about the location where the
crosstalk producing photons are absorbed relative to the avalanche region. For the
previously tested Hamamatsu LCT5 SiPM we could show that the majority of these
photons are absorbed above the avalanche region \cite{Otte2016a}. The $\mathcal{O}$
value we obtained then was $\sim26$. That interpretation
was confirmed by Hamamatsu, who found that these photons exit the silicon and
reflect off the boundary between the protective layer and the ambient air back into a
cell.
In the Hamamatsu SiPM studied here that contribution to the prompt OC has been
successfully suppressed by eliminating the protective epoxy layer. The same
conclusion comes from the interpretation of $\mathcal{O}$. The best fit
value for $\mathcal{O}$ is about $10\pm1$ in all fits of the optical crosstalk
but the one for $20^\circ$C, where the fit probability is $10^{-7}$ due to a
contamination from random dark counts and can thus be
safely ignored (see Table \ref{OCvals}).
The average value can be compared with the ones we found
from the different PDE measurements (Table \ref{tab:ottenumb}). A small
$\mathcal{O}$ value like 10 corresponds to heavily hole initiated breakdowns, which
means that the OC photons must be absorbed below the high-field region. According
to Table \ref{tab:ottenumb} that is the case if the OC photons absorb in a depth
$>1\,\mu$m below the surface.
Three scenarios come to mind that can explain how optical crosstalk photons
can be absorbed at such depths. The first scenario is that some photons manage to
penetrate the trench between cells. That scenario is unlikely because
photons would absorb uniformly across the cell, i.e.\ absorb above and
below the avalanche region and, in consequence, result in
values for $\mathcal{O}$ larger than 10 because the occurring breakdowns would
be electron and hole initiated. The second scenario is that some photons with
long absorption lengths still bounce off the air-SiPM interface and are absorbed deep
inside the device, i.e.\ mostly below the avalanche structure. The third, and
our preferred scenario is that photons cross into a neighboring cell below the
trench and are absorbed below the avalanche structure.
In the second and third scenario photons can be absorbed in the bulk and the
generated holes diffuse into the active volume where they cause delayed OC (see
next section). If
the diffusion time is less than 2\,ns and thus below the resolving time of our
setup, the delayed OC would be misidentified as a prompt OC event. If the
photons are absorbed in the active volume below the avalanche region a prompt OC would
be caused.
The fit result for the KETEK SiPM yields an $\mathcal{O}$ number of
$13.6\pm0.7$. Comparing that value with the $\mathcal{O}$ numbers in Table
\ref{tab:ottenumb} lets us conclude that the majority of the photons absorb
equally distributed across the avalanche region and thus produce an equal amount of
electron and hole dominated breakdowns. That result is not surprising as the
device does not have trenches in between cells, which would prevent photons to
travel directly from the avalanche region where they are produced into a
neighboring one.
\begin{figure*}[!tb]
\centering
\subfloat[Hamamatsu
LVR2]{
\includegraphics*[width=\columnwidth]{LVR2AP.pdf}
}
\subfloat[KETEK PM3325]{
\includegraphics*[width=\columnwidth]{KetekAP.pdf}
}
\caption{Afterpulsing probability of the
two devices.
The black arrow marks
the relative overvoltage at which both devices yield a 90\% breakdown
probability for 400\,nm photons.
}
\label{fig:AP}
\end{figure*}
\section{Afterpulsing and Delayed Optical Crosstalk}
If the prompt OC in the Hamamatsu device is indeed dominated by misidentified
delayed OC, a reduction of the minority carrier lifetimes in the bulk with a low
resistivity bulk or a better shielding of the active volume from carriers
diffusing out of the bulk with a potential barrier might be a viable way to
reduce OC further, unless those measures are already implemented. We illustrate
the potential room for improvement by discussing the delayed OC and afterpulsing
characteristics of the two tested SiPMs.
Both quantities are extracted by recording time difference between SiPM pulses
as explained in \cite{Otte2016a}. Afterpulsing events become dominant a few ten
nanoseconds after a breakdown when the corresponding cell is recharged to 50\%
or more of its full capacity. Delayed OC signals dominate at shorter time
differences. For the Hamamatsu LVR2 device the subjective division between the two
contributions is made at 20\,ns, and for the KETEK device at 10\,ns. We note
that our choice of separating the two contributions in the described way results
in a contamination of each measurement with events of the opposite type.
That contamination is
acceptable for our purposes. Figure \ref{fig:AP} shows the afterpulsing and
Figure \ref{fig:DOC} the delayed optical-crosstalk probabilities of both
devices.
The KETEK device has an afterpulsing probability of less than 1\%, whereas the
afterpulsing of the Hamamatsu device is two to three times larger, when
compared at their respective bias, which yields a 90\% breakdown probability
for 400\,nm photons (marked by the arrow in the figures).
The uncertainties in the different fits of the Hamamatsu
afterpulsing data do not allow us to claim a temperature dependence. The
afterpulsing of the KETEK SiPM shows irregular behavior for relative
overvoltages above 0.2 for the two lowest temperatures. We attribute that
behavior to delayed optical crosstalk leaking into the afterpulsing measurement due to our
choice of discriminating between the two by means of applying a simple cut in time.
At the same 90\% breakdown-probability yielding bias, the delayed OC changes from 0.01\% at $40^\circ$C to
1\% at $-75^\circ$C for the KETEK SiPM. The temperature dependence is not that
strong in the Hamamatsu SiPM, where the delayed OC is 3.5\% at $20^\circ$C and
increases by a factor of 1.3 to 4.5\% at $-75^\circ$C. We discard the delayed OC
measurement at $40^\circ$C for the same reason we discarded the prompt OC measurement
at the same temperature.
Below relative overvoltages of 0.15, afterpulsing and delayed OC of the KETEK
device are so low that the measurement is affected by systematic effects. Only
at higher overvoltages is it possible to resolve the expected temperature
dependence of the delayed optical crosstalk. The dependence is due to an
increase of the carrier life times in the bulk with decreasing temperatures.
Comparing the prompt and delayed OC performance of both devices has us speculate
about possible future improvements of both technologies. The about ten times
lower delayed OC of the KETEK device is an indication that it should be in
principle possible to lower the delayed OC in the Hamamatsu technology further.
If a lower delayed OC is achieved in the Hamamatsu technology and our assertion
that the prompt OC in the present Hamamatsu device is due to misidentified
delayed OC events, the \emph{effectively measured} prompt OC should go down as well.
On the other hand, it can be expected that future KETEK developments with
trenches will be able to achieve a similar if not better prompt OC performance
than observed in the Hamamatsu SiPM.
\begin{figure*}[!tb]
\centering
\subfloat[Hamamatsu
LVR2]{
\includegraphics*[width=\columnwidth]{LVR2DOC.pdf}
}
\subfloat[KETEK PM3325]{
\includegraphics*[width=\columnwidth]{KetekDOC.pdf}
}
\caption{Delayed optical crosstalk of the
two devices.
The black arrow marks
the relative overvoltage at which both devices yield a 90\% breakdown
probability for 400\,nm photons.
}
\label{fig:DOC}
\end{figure*}
\section{Discussion}
In this work we characterized one prototype SiPM from Hamamatsu and the PM3325
WB SiPM from KETEK. Both SiPMs have dramatically improved characteristics when
compared to previous devices. The PDE of both devices peaks between 40\% and
50\% and nuisance parameters are significantly reduced. In particular impressive
is the 1.5\% prompt optical crosstalk of the Hamamatsu device, which is four
times lower than in the Hamamatsu LCT5 device \cite{Otte2016a}. Equally
impressive are the low afterpulsing and delayed optical crosstalk of the KETEK
device, which are both less than 1\%. A device that combines the excellent
features of both SiPMs would result in another significant improvement in the
SiPM technology.
Analysis methods that probe the microphysics of SiPMs help to understand how
SiPMs work and ultimately provide input in the design of future SiPM
developments. For that purpose we discussed how the vertical
structure of the high-field region is mapped with bias dependent breakdown probability
measurements and how such a mapping can be utilized to learn about the origin of
charge carriers relative to the avalanche structure. Using the method we could
show that the prompt OC producing photons in the Hamamatsu SiPM must be absorbed below
the avalanche structure contrary to the LCT5 device where the majority of OC
photons enter the avalanche region from the surface side.
In the KETEK device, the optical-crosstalk photons \emph{illuminate} the
avalanche region of a neighboring cell from the side. This information will
help to further improve the prompt OC performance in future devices. We are not
aware of another experimental method that provides the same information.
The $\mathcal{O}$-method could also be used to identify the spatial origin of charge
carriers produced by delayed optical crosstalk, afterpulsing, and dark counts
relative to the avalanche region. However, two requirements need to be
fulfilled first. A valid model has to exist that properly describes the bias
dependence of the characteristic of interest and includes the breakdown
probability. And the measurement cannot be contaminated, like, for example, our
delayed optical crosstalk measurement, which also includes some afterpulsing
events. Unless, of course, the model takes these contaminations into account
too.
The empirical mapping of the $\mathcal{O}$ values obtained in PDE measurements
to the photon absorption length allowed us to determine how far below the
surface the avalanche region is located. However, because we have no access to
the structure of the studied devices, we cannot verify the absolute accuracy of
the mapping and the dependence of $\mathcal{O}$ on the size of the avalanche
region. To verify that assumption and for a more precise probing of the
high-field structure, dedicated test structures are needed for calibration. The
main parameters to vary in these structures are the size of the region and its
location below the surface.
Analytical modeling that links $\mathcal{O}$ to the
microphysics of the breakdown, like the ionization coefficients and the
electron/hole breakdown initiation ratio, would further improve the
understanding of SiPMs and expand the usability of the method. We hope that this
paper inspires future work in that direction.
\section*{Acknowledgment}
We are grateful to Hamamatsu and KETEK, who have provided us with samples of their latest developments. This research was in part supported by the National Science Foundation under grant no.\ PHYS-1505228.
|
1,314,259,996,674 | arxiv |
\section{Introduction}
Many applications of automatic speech recognition (ASR) require both low latency and high accuracy, goals that are not easily attained within a single system. Historically, these were achieved by building separate models oriented at either online or offline ASR. In some deployments, the predictions of a model characterized by good streaming properties, but sub-optimal accuracy, could be refined by a different model with worse latency profile, but higher accuracy.
End-to-end models~\cite{Li2022RecentAI}, in particular those utilizing transformers\cite{Vaswani2017,Zhang2020} as a back-bone, can be trained to operate in different configurations that gracefully trade accuracy for latency or compute gains. Transformer-based models are particularly suitable for such applications thanks to their combined use of self-attention and masking -- components that jointly provide direct control over the model's access to past, present and future information when making localized predictions. When combined with frame-synchronous modeling approaches, such as the neural transducer~\cite{graves2012sequence} or connectionist temporal classification (CTC) \cite{Graves2006icml}, transformer attention masking can be used to exactly mimic the target inference conditions during training and thereby acquire better control over the accuracy vs latency trade-off induced by the model topology.
An example of a neural transducer model with low partial latency, but still benefiting from access to larger context, is the stacked encoder approach \cite{Narayanan2021, Mahadeokar2022}, in which data is encoded using cascaded encoders - an encoder operating in causal mode, followed by a stacked non-causal encoder operating on the causal encoder outputs and optionally updating the causal encoder predictions. Though jointly learned, each encoder has its own set of parameters.
Here, we build on an alternative approach, in which transformer-transducer (TT) models are trained with variable masking~\cite{Anshuman2020}. In this approach, different masking patterns are applied to the acoustic encoder during training. This exposes the encoder to varying amounts of future context, allowing the trained model to work in both streaming and non-streaming modes, while sharing parameters between the two modes. There are a number of works in recent literature investigating similar ideas. \cite{yu2020dual, Moritz2021} focused on learning causal and non-causal convolution filters in the context of ``dual-mode'' transducers with a conformer-based acoustic encoder. Doutre et al. \cite{Doutre2020} explored building multi-mode models with distillation of full-context models to streaming variants. Audhkhasi et al. \cite{Audhkhasi2021} proposed a mixture of softmaxes approach to computing the attention probability density function, where the support for mixture components may be chosen to cover different modeling contexts. In contrast, following~\cite{Anshuman2020}, we adopt a single softmax attention scheme, with masking acting as a form of conditional execution governed by sampled choices.
We also build on recent work describing attention-based causal chunking \cite{Shi2021EmformerEM,Chen2021}. In this approach, a chunk-aware attention mask allows access to past and future frames within a pre-defined chunk, but otherwise disallows information flow between chunks. The trained model can then be treated as causal at the chunk level. Here, we extend the causal chunking approach approach to variable causal chunking, thus allowing the model to work in both streaming and global modes. A similar extension was recently also made for CTC models\cite{Zhuoyuan2021}. Further, we relax the constraints on information flow by allowing the model to access distant past information at each layer in streaming mode.
There are other studies aimed at localizing self-attention for streaming applications, not necessarily via masking \cite{huang2020conv,moritz2020streaming,tian2020synchronous, tsunoo2019transformer,zhang2020streaming}. These works do not address the issue of inference-time configurability, and thus do not directly lend themselves to unified two pass applications.
Contributions of this work include:
\begin{itemize}
\item Systematic comparison of the accuracy and latency impact of transformer masking strategies ~\cite{Anshuman2020,Shi2021EmformerEM,Chen2021} and their variable masking variants within the TT framework.
\item Evaluation of variable masking as a strategy for inference-time configurability.
\item Evaluation of variable masking for two-pass acoustic re-scoring in the large-scale data regime.
\end{itemize}
\input{figures/masking}
\section {Transducers, Transformers, Attention and Masking}
\subsection{Transformer Transducer}
In this work we use ASR systems based on the TT architecture\cite{Zhang2020,Anshuman2020}. These models are trained with the RNN-T loss \cite{graves2012sequence} but do not contain any recurrent layers. The predictive distribution of the transducer model is defined as:
\begin{equation}
P( {\mathbf{y}}_{u+1}|{\mathbf{x}}, t, \;{\mathbf{y}}_{1:u}) = \mbox{Softmax} \left ( \mbox{Linear }( \mbox{tanh} ( \mbox{Joint} ) ) \right ),
\end{equation}
\begin{equation} \label{eq:joiner}
\begin{split}
\mbox{Joint} = & \; \mbox{Linear}(\mbox{AcousticEncoder}({\mathbf{x}}, t)) \\
& + \mbox{Linear}(\mbox{LabelEncoder}({\mathbf{y}}_{1:u})),
\end{split}
\end{equation}
where ${\mathbf{x}}$ is the audio sequence; $\mbox{AcousticEncoder}({\mathbf{x}}, t)$ is the acoustic encoder output at time $t$; ${\mathbf{y}}$ is the label sequence; $\mbox{LabelEncoder}({\mathbf{y}}_{1:u})$ is the label encoder output given the previous non-blank tokens ${\mathbf{y}}_{1:u}$.
This work uses standard transformer layers \cite{Vaswani2017} for label encoder and convolution-augmented transformer (Conformer) \cite{Gulati2020} layers for acoustic encoder. The Linear function is a single affine transform.
\subsection{Transformer Attention Masking}
Different from RNNs, transformers allow precise control of the neighborhood information at each step. For example,
at each time $t$, $\mbox{AcousticEncoder}({\mathbf{x}}, t)$ in Eq. \eqref{eq:joiner} may be derived from an arbitrary subset of features in ${\mathbf{x}}$, as defined by the masking strategy implemented in the self-attention layers \cite{Vaswani2017}.
Given the attention input ${\mathbf{Z}} = ({\mathbf{z}}_1, \ldots, {\mathbf{z}}_{L_z})$, ${\mathbf{z}}_t \in {\mathbb{R}}^{d_z}$ self-attention computes:
\begin{equation}
{\mathbf{Q}}=f^q({\mathbf{Z}}), \; {\mathbf{K}}=f^k({\mathbf{Z}}), \; {\mathbf{V}}=f^v({\mathbf{Z}}),
\end{equation}
\begin{equation}
\mbox{Att}({\mathbf{Q}}, {\mathbf{K}}, {\mathbf{V}}) = \mbox{softmax} \left (\alpha {\mathcal{M}}({\mathbf{Q}}{\mathbf{K}}^\mathrm{T}) \right ) {\mathbf{V}}^\mathrm{T},
\end{equation}
where $\alpha = 1/\sqrt{d}$ is the scaling factor dependent on the attention dimension. Vaswani et al. \cite{Vaswani2017} implemented $f^*({\mathbf{Z}})$ mappings as linear transforms.
However, this work uses asymmetric relative positional encoding~\cite{pham2020relative} in acoustic encoder and relative positional encoding~\cite{dai2019transformerxl} for label encoder. The mask ${\mathcal{M}}\in{\{0,1\}^{L_z\cdot L_k}}$
determines the access of each transformer frame to its past and future contexts \cite{Li2022RecentAI}. We consider several masking patterns as depicted in \figref{fig:masking}, in particular we will refer to these as i) fixed masking \cite{Zhang2020} and ii) chunked masking \cite{Shi2021EmformerEM, Chen2021}.
The masking pattern defines the streaming properties of the model.
In low latency recognition scenarios, average acoustic encoder look-ahead should be smaller than the target partial latency and encoder outputs should be produced as soon as possible. However, computing outputs one by one significantly increases the cost of computation, e.g. power use, on parallel hardware. Hence, it is important to balance the latency with compute by chunking input frames and encoding each chunk as a batch. Generally speaking, the larger the chunk size the lower the compute cost. Exploiting this fact, \cite{Shi2021EmformerEM, Chen2021} proposed a masking strategy that uses maximal valid context within a chunk, and then restricts the model to be causal across chunk boundaries, as illustrated in \figref{fig:mchunked}. Chunked masking offers better average look-ahead when compared to fixed masking and allows to benefit from look-ahead at each layer. The compromise between fixed and chunked masking is illustrated in \figref{fig:compromise}.
In this work, we also investigate a hybrid mode (\figref{fig:mhybrid}), where we use causal chunking for future context, and fixed masking for past context,
which allows each layer to access distant past frames.
\subsection{Variable masking and its applications} \label{ssec:vm}
We extend each of the variants presented in \figref{fig:masking} to variable masking scenario, where during training acoustic and/or label encoders are exposed to multiple masking configurations so the model can learn to integrate different amounts of past and future context, and thus be deployed in desired operating conditions during inference. From training perspective, this can be seen as an additional conditioning variable~\textit{i.e.~}:
\begin{equation}
{\mathcal{L}}_{\texttt{TT}} = -\log P( {\mathbf{y}}|{\mathbf{x}}; \boldsymbol\theta_{\texttt{TT}}, {\mathbf{m}}\sim {\mathbf{M}}),
\end{equation}
\noindent where ${\mathbf{m}} = ({\mathbf{m}}_{\texttt{AE}}, {\mathbf{m}}_{\texttt{LE}})$ is a sample from ${\mathbf{M}}$, a predefined set of allowed mask configurations. ${\mathbf{m}}_{\texttt{AE}}$ is the acoustic encoder mask specifying the look-back and look-ahead frames (or chunk sizes) at each transformer layer. The label encoder mask ${\mathbf{m}}_{\texttt{LE}}$ is strictly causal and specifies look-back frames at each layer. Similar conditioning on external variables or transforms has been previously used for speaker adaptive training\cite{bell2020adaptation}.
One advantage of variable masking models is their ability to obtain encodings for different amounts of context. One can use this in Eq. \eqref{eq:joiner} to re-compute the $\mbox{Joint}$ outputs in second pass with a wide chunk (\texttt{WC}) acoustic encoder, while re-using cached label encodings:
\begin{equation} \label{eq:joiner_rescoring}
\begin{split}
\mbox{Joint} = & \; \mbox{Linear} ( \mbox{AcousticEncoder}_{\texttt{WC}}({\mathbf{x}}, t)) \\
& + \mbox{Linear}(\mbox{LabelEncoder}({\mathbf{y}}_{1:U})).
\end{split}
\end{equation}
Note that the $\mbox{AcousticEncoder}_{\texttt{WC}}({\mathbf{x}}, t)$ encodings are obtained with the same acoustic encoder weights, and different mask configuration.
\section {Experiments}
We carry out experiments on large-scale in-house data comprising of utterances from dictation and assistant tasks. We use both semi-supervised and supervised data. Semi-supervised transcripts were automatically obtained from an auxiliary transcription model applied to around 600,000 hours of randomized and anonymized acoustic data. The supervised data consists of 7,000 hours of US English data. All systems in this work were trained for 3.4M updates, each update using gradients accumulated over 8192 utterances. We use SyncSGD with exponentially decaying learning schedule. The models are evaluated on a test set that is around 60 hours long, with utterance duration varying from a few seconds to around 60 seconds.
Our base TT models operate on 6 times decimated acoustic features by means of convolution and frame concatenation based down-sampling. The acoustic encoder makes use of 12 causal Conformer blocks. For streaming applications with variable masking, we found it crucial for training stability to use layer normalization\cite{ba2016_layernorm} rather than batch-level normalization\cite{ioffe15_batchnorm} as in the original Conformer formulation\cite{Gulati2020}. The label encoder is implemented as a standard 6 layer transformer. In total, all TT models shown have 101M parameters.
In the following, we refer to this model as a CCT transducer, short for \textbf{C}onvolutional downsampling, \textbf{C}onformer acoustic encoder and \textbf{T}ransformer label encoder. Depending on masking configuration, we additionally annotate CCT with one of the following prefixes, \{f,vf,c,vc\}, denoting fixed context-expansion (f) look-aheads / look-backs, variable context-expansion masking (vf), chunking (c) and variable-chunking (vc) CCT models, respectively.
\begin{figure}
\centering
\includegraphics[scale=0.55]{figures/varchunk.pdf}
\vspace{-1cm}
\caption{WERs for variable chunk masking models and a number of different masks settings used in audio encoder for the left and right context. Same trends were observed with fixed variable masking.
}
\label{fig:chunked_decodes}
\vspace{-0.4cm}
\end{figure}
\subsection {Single-masking baselines}
First block in Table~\ref{tab:masking_modes} reports WERs for different masking modes outlined in~\figref{fig:masking}. The models were configured to a similar streaming setting consuming 240ms of audio data (or 4 transformer frames after 6x decimation). Fixed masking on both past and future (\figref{fig:mfixed}) is comparable accuracy-wise to a hybrid approach that uses causal chunking combined with context-expansion (fixed) look-backs (\figref{fig:mhybrid}). Causal chunking as depicted in \figref{fig:mchunked} performed about 0.1\% worse than hybrid chunking; the difference disappears if one can trade-off larger chunks for latency. As we will show later in Section \ref{ssec:partial_latencies}, causal chunking offers better streaming properties wrt partial latencies; it also simplifies decoding and saves computation by not requiring the availability of frames from future chunks.
\begin{table}[!htbp]
\centering
\small
\begin{tabular}{c|c|c|c|c|c}
\multicolumn{2}{p{2cm}|}{\centering Left Context} & \multicolumn{2}{p{2cm}|}{\centering Right Context} & \multicolumn{2}{p{2cm}}{\centering WER [\%]} \\ \hline
Type & \#Masks & Type & \#Masks & Stream. & Global \\ \hline\hline
Chunk & 1 & Chunk & 1 & 3.75 & - \\
Fixed & 1 & Fixed & 1 & 3.61 & - \\
Fixed & 1 & Chunk & 1 & 3.66 & - \\ \hline
Fixed & 1 & Chunk & 2 & 3.66 & 3.4 \\
Fixed & 1 & Chunk & 5 & 3.71 & 3.39 \\
Fixed & 1 & Chunk & 9 & 3.69 & 3.46 \\
Fixed & 2 & Chunk & 9 & 3.7 & 3.44 \\
Fixed & 7 & Chunk & 9 & 3.88 & 3.5 \\
\end{tabular}
\caption{WERs for different masking variants in \figref{fig:masking}.}
\label{tab:masking_modes}
\vspace{-0.5cm}
\end{table}
\subsection {Variable masking}
In this section we expand single-masking systems to training with variable masking configurations. We report results for Fixed-Chunk systems from first portion of Table~\ref{tab:masking_modes}, as Fixed-Fixed follows a very similar pattern. We trained models with a variety of masking settings for chunks and look-backs (LB) in acoustic encoder. We also experimented with varying the left context of label encoders; however, this was found to have a negligible impact on accuracy, in line with other works on limited context label encoders with transducers \cite{ghodsi2020, sainath21_interspeech}. We sampled the mask settings independently from each other during training, allowing the model to be arbitrarily configured to any combination of them at inference. During training, the masks are sampled at batch level\footnote{Per-utterance mask sampling offered similar WERs.}, and applied consistently across all transformer layers.
A decode of the model from the last row in Table~\ref{tab:masking_modes} for all its operating settings is shown in \figref{fig:chunked_decodes}. The model was trained with 9 different chunk configurations, ranging from fully causal (60ms audio chunk, or 1 transformer frame) to full future context.
Likewise, we trained with 7 different LB configurations, spanning the history from last 720ms to the full past context. It is interesting to observe how little past context is needed to obtain reasonable results, i.e. the most constrained model configuration with only the current frame and 720ms of past audio obtains 4.22\% WER, and how little WER improvement full past context brings (less than 0.1\% abs.). A similar trend can be observed for different chunk sizes, though increasing the chunk size (or the look-ahead) has a much larger impact on accuracy. Note that the test set includes audio up to 60s, and when testing the models on different long-form audio test sets (results not reported), we did not observe any dramatic increases in WER. This suggests that limiting the context via masking could be a simple but effective alternative to more sophisticated approaches preventing self-attention from collapse on long speech sequences \cite{Likhomanenko2021_CAPE}.
Since a large number of LB settings seems to hurt performance (compare the last two rows in Table~\ref{tab:masking_modes}), in the remainder we will do the analyses on the model trained with 2 LB and 9 chunk size settings. As the default inference setting, we use 8.6s of LB and a 240ms long chunk.
\subsection {Partial Result Word Latency} \label{ssec:partial_latencies}
Recognizer responsiveness is paramount in interactive recognition applications like dictation. Partial Result Word Latency (PRWL) is defined as the delay between when a word is fully spoken and when it is presented to the user. To compute timestamps of each word in our test set, we use a frame synchronous hybrid phonetic HMM-CNN recognizer to produce a 10ms frame time-aligned transcript. Every partial result of the transducer decoder is recorded along with the amount of audio consumed by the acoustic encoder to compute the \textit{first seen timestamp} of each word.
There are several techniques to encourage end-to-end ASR models to prefer lower latency predictions. One approach is to constrain the frame alignment during training to ground truth time alignment \cite{Senior2015}. These constraints do limit the model's ability to learn certain alignments, which impacts accuracy. Another approach, specific to the transducer loss function, is FastEmit (FE) \cite{yu2021_fastemit}. FastEmit has the advantage of not requiring any alignment training data, although it too can affect accuracy. A novel approach
is self-alignment \cite{Kim2021_selfalign}, which uses Viterbi forced-alignment instead of external alignment.
PRWL results for selected models are reported in Table~\ref{tab:latencies}. Chunked masking reduces PRWL by around 45\% relative when compared to fixed masking models. For both fixed masking (\{f,vf\}-CCT) and chunking configurations (\{c,vc\}-CCT), we find FastEmit to be highly effective, as expected. Chunked models can afford less aggressive FE setting, usually resulting in better accuracy for the same latency constraints (\textit{i.e.~} compare vfCCT and vcCCT models, where the latter obtains 45\% lower PWRL for the same FE constant).
~\figref{fig:alignments} shows example token emission delays of an FE-enabled vcCCT configured to streaming and global modes, as well as single-mask variants from Table~\ref{tab:latencies}.
Variable chunking combined with latency penalizing loss improves PRWL, and decreases token emission delay. As a function of chunk size, token emission latency decreases for larger chunks, around 40ms on average between models configured to 60ms vs 240ms chunks.
We also investigated if the amount of look-back affects token emission delay by measuring the system from ~\figref{fig:chunked_decodes} configured to 8.6s and 720ms LB settings, and we did not find a significant effect.
Chunking, due to batched computation, brings large gains in real-time factor (RTF) or compute-induced latency. We do not report RTFs in this work as they were in-depth studied in \cite{Shi2021EmformerEM, Chen2021}.
\begin{figure}
\centering
\includegraphics[scale=0.45]{figures/alignments.pdf}
\vspace{-0.5cm}
\caption{Token emission alignments for different models.}
\label{fig:alignments}
\end{figure}
\begin{table}[]
\centering
\small
\begin{tabular}{l|c|c}
System & PRWL [ms] & WER [\%] \\ \hline\hline
fCCT & 835 & 3.56 \\
~+FastEmit & 700 & 3.61 \\
vfCCT & 855 & 3.68 \\
~+FastEmit & 710 & 3.71 \\
cCCT & 453 & 3.62 \\
~+FastEmit & 409 & 3.66 \\
vcCCT & 473 & 3.56 \\
~+FastEmit & 389 & 3.66 \\
\end{tabular}
\caption{Latency metrics for selected systems.
}
\label{tab:latencies}
\vspace{-0.2cm}
\end{table}
\subsection{Acoustic Rescoring}
Variable masking models allow for an easy application to acoustic re-scoring, as they do not require auxiliary models or distinct parameters to capture different amounts of context. In particular, in this work we re-score by recomputing acoustic embeddings with an acoustic encoder configured to capture larger amounts of audio context as described in Section~\ref{ssec:vm}, and cached label encodings (thus avoiding an expensive auto-regressive recomputation). Results are depicted in \figref{fig:rescoring}. We obtain first-pass 10-best lists from a vcCCT operating on {60, 120, 180 or 240}ms long chunks, and then re-score with AE configured to 0.36, 1.8, 3.6, and 30 seconds chunks. Note that the 60ms setting is effectively a causal model, where the chunk is just one transformer frame.
Depending on chunking configuration, we obtain up to 8\% relative WER reduction from 2nd pass re-scoring. For comparison, decodes with larger chunks gave WERs of 3.52, 3.49 and 3.46 for chunk sizes of 1.8, 3.6 and 30 seconds, respectively; re-scoring is thus on average within 0.1\% abs. WER.
\begin{figure}
\centering
\includegraphics[scale=0.55]{figures/varchunk_rescore_transposed.pdf}
\vspace{-0.25cm}
\caption{WERs obtained with acoustic re-scoring of narrow-chunk first-pass models with several wider-chunk settings. The left-most column depicts WERs for first-pass only systems.}
\label{fig:rescoring}
\end{figure}
\section{Conclusions}
We have shown that variable masking transformer transducer models are a viable choice for two-pass acoustic speech recognition pipelines, without the necessity of maintaining additional parameters in the form of stacked encoders or additional re-scoring modules such as attention encoder-decoder. We have investigated how different masking strategies impact partial latency, finding chunking to be twice as good when compared to fixed masking and similar accuracy constraints.
Finally, we carried out an extensive experimental evaluation of variable masking strategies in a large-scale data regime, including their application to two-pass speech recognition.
{
\small
\section {Acknowledgements}
We thank Vijay Peddinti, Tatiana Likhomanenko, Sicheng Wang and Russ Webb for useful suggestions on this work.
}
\newpage
\ninept
\bibliographystyle{IEEEBib}
|
1,314,259,996,675 | arxiv | \subsection{Introduction}
Although networked multiagent systems are envisioned to autonomously function in place of humans for repetitive, demanding, and often safety-critical missions the current level of controls technology is incapable of providing the needed usability and resiliency of multiagent systems.
Because, the control algorithms of these systems are computed distributively without having a centralized entity monitoring the activity of agents, and hence, adversaries such as attacks to the communication network and/or failure of agent-wise components can easily result in system instability and prohibit the accomplishment of system-level objectives \cite{bullo2009distributed}.
The fragile nature of multiagent systems has triggered the development of detection and isolation algorithms during the last few years \cite{reference25,reference26,reference27,reference29,reference30,reference31}.
For example, \cite{reference25} and \cite{reference26} make a specific assumption on the network connectivity (other than the standard assumption on the connectedness of networked agents) and \cite{reference27} requires that at most a fraction of any normal agent's neighbors to be adversaries, or misbehaving agents, for achieving resilient multiagent system behavior.
Like \cite{reference25} and \cite{reference26}, a computationally expensive and not scalable algorithm is proposed in \cite{reference29} and \cite{reference30} based on input observers technique, where the effect of misbehaving agents on the overall multiagent system performance is also quantified, and an extension of this work is given in \cite{reference31} also focusing on the detection and isolation of misbehaving agents.
This note develops an adaptive control architecture to ensure resilient coordination of networked multiagent systems in the presence of misbehaving agents.
Specifically, we show that the nominal networked multiagent system behavior can be retrieved with the proposed methodology that utilizes a novel distributed state emulators.
Apart from the existing relevant literature \cite{reference25,reference26,reference27,reference29,reference30,reference31} that make specific assumptions on the graph topology and/or the fraction of misbehaving agents, the proposed framework can achieve performance recovery on an arbitrary but connected communication topology and even if all agents are misbehaving.
\subsection{Mathematical Preliminaries}
The notation used in this note is fairly standard.
Specifically, ${\mathbb R}$ denotes the set of real numbers,
${\mathbb R}^{n \times m}$ denotes the set of $n \times m$ real matrices,
$\overline{{\mathbb S}}^{n\times n}_+$ denotes the set of $n \times n$ symmetric nonnegative-definite real matrices,
$\mathrm{diag}(v)$ denotes a diagonal matrix with scalar (or matrix) entries given by $v$,
$\mathrm{I}_n$ denotes the $n\times n$ identity matrix,
$(\cdot)^\mathrm{T}$ denotes the transpose,
$\otimes$ denotes the Kronecker product,
``$\triangleq$'' denotes the equality by definition, and
$\mathbf{1}_{n}$ denotes a $n \times 1$ vector with 1 in all entries.
In addition, we write
$\norm{\cdot}$ for the euclidean norm,
$\norm{\cdot}_\textrm{F}$ for the Frobenius norm,
$\lambda_i(A)$ for the $i$-th eigenvalue of the matrix A,
$\textrm{spec}(A)$ for the spectrum of matrix A,
$\pi_0(A)$, $\pi_+(A)$, and $\pi_-(A)$ for the number of eigenvalues (counted with algebraic multiplicities) of A having zero, positive, and negative real parts, respectively,
$\mathrm{det}(A)$ for the determinant of A,
$\mathrm{max}(\cdot)$ for the maximum, and
$\mathrm{min}(\cdot)$ for the minimum.
We now introduce several results that are necessary for the development of the main result of this note.
\textbf{Lemma 1} \cite{blockm}\textbf{.} Consider the matrix given by
\begin{eqnarray}
M=\left[\begin{array}{cc}A & B \\ 0 & D\end{array}\right].
\end{eqnarray}
Then, the determinant of $M$ satisfies
\begin{equation}
\textrm{det}(M)=\textrm{det}(A)\textrm{det}(D).
\end{equation}
\textbf{Lemma 2} [\citen{yuc02}]\textbf{.} Suppose $Z(\lambda)=A\lambda^2+B\lambda+C$ denotes the quadratic matrix polynomial where $A\in\mathbb{R}^{n \times n}$ and $C\in\mathbb{R}^{n \times n}$, and $A$ is nonsingular.
If $B\in\mathbb{R}^{n\times n}$ is positive-definite, then $\pi_+(Z)=\pi_-(A)+\pi_-(C),$ $\pi_-(Z)=\pi_+(A)+\pi_+(C),$ and $\pi_0(Z)=\pi_0(C),$ where $\pi_+(Z)+\pi_-(Z)+\pi_0(Z)=2n.$
We next recall some of the basic notions from graph theory and networked multiagent systems [\citen{pre1},\citen{pre2}].
Specifically, graphs are broadly used in networked multiagent systems to encode interactions between a group of agents.
An \emph{undirected} graph $\mathcal{G}$ is defined by a set $\mathcal{V}_\mathcal{G}=\{1,\dots,n\}$ of \emph{nodes} and a set $\mathcal{E}_\mathcal{G}\subset\mathcal{V}_\mathcal{G}\times\mathcal{V}_\mathcal{G}$ of \emph{edges}.
If the \emph{unordered} pair $(i,j)\in\mathcal{E}_\mathcal{G}$, then nodes $i$ are $j$ are \emph{neighbors} and the neighboring relation is indicated with $i\sim j$.
The set of neighbors of node $i$ is denoted by $\mathcal{N}_\mathcal{G}(i)=\{j|(i\sim j)\in\mathcal{E}(\mathcal{G})\}$.
The \emph{degree} of a node is given by the number of its neighbors.
Letting $d_i$ be the degree of node $i$, then the \emph{degree} matrix of a graph $\mathcal{G}$, ${\Delta(\mathcal{G})}\in\mathbb{R}^{n\times n}$, is given by ${\Delta(\mathcal{G})}\triangleq \mathrm{diag}(d), d=[d_1,\dots,d_n]^\textrm{T}$.
The \emph{adjacency} matrix of a graph $\mathcal{G}$, $\mathcal{A(G)}\in\mathbb{R}^{n \times n}$, is given by
\begin{equation}
[ {A(\mathcal{G})} ]_{ij}\triangleq\left\{
\begin{array}{cl}
1, & \mathrm{if}\ (i,j)\in\mathcal{E}_\mathcal{G},\\
0, & \mathrm{otherwise}.
\end{array}
\right.
\end{equation}
The \emph{Laplacian} matrix of a graph, $\mathcal{L(G)}\in\bar{\mathbb{S}}_+^{n \times n}$, plays a central role in many graph theoretic treatments of networked multiagent systems is given by $\mathcal{L(G)}\triangleq{\Delta(\mathcal{G})}-\mathcal{A(G)}$,
where the spectrum of the Laplacian for a connected, undirected graph can be ordered as $0=\lambda_1(\mathcal{L(G)})<\lambda_2(\mathcal{L(G)})\leq\cdots\leq\lambda_n(\mathcal{L(G)})$, with $\mathbf{1}_n$ as the eigenvector corresponding to the zero eigenvalue $\lambda_1(\mathcal{L(G)})$ and $\mathcal{L(G)}\mathbf{1}_n=\mathbf{0}_n$ holds.
Networked multiagent systems can be modeled by a graph $\mathcal{G}$, where nodes and edges, represent agents and interagent information exchange links, respectively.
In particular, let $x_i(t)\in\mathbb{R}^{N}$ denote the state of agent $i$ at time $t\geq0$, whose dynamics are described by
\begin{equation} \label{single_agent}
\dot{x}_i(t)=u_i(t),\quad t\geq0, \quad x_i(t_0)=x_{i0}, \quad i\in\mathcal{V}(\mathcal{G}),
\end{equation}
with $u_i(t)\in\mathbb{R}^N$ being the control input of agent $i$.
We consider agents having dynamics of the form given by (\ref{single_agent}) to focus the main result of this note.
In addition, we focus on the consensus problem without loss of much generality when presenting the main contribution of this note.
In particular, if agent $i$ is allowed to access the relative state information with respect to its neighbors, a solution to the standard consensus problem can be given by
\begin{equation}\label{input}
u_i(t)=-\sum_{i\sim j}(x_i(t)-x_j(t)), \quad i\in\mathcal{V}(\mathcal{G}),
\end{equation}
for a connected and undirected graph (throughout this note, we assume that the graph $\mathcal{G}$ is connected and undirected).
The networked multiagent system given by (\ref{single_agent}) and (\ref{input}) can now be described in the form given by
\vspace{0cm}
\begin{equation}\label{complex}
\dot{x}(t)=-(\mathcal{L(G)}\otimes\mathrm{I}_N)x(t), \quad t\geq0, \quad x(t_0)=x_0,
\end{equation}
where $x(t)=[x^\textrm{T}_1(t),\cdots,x^\textrm{T}_n(t)]^\textrm{T}$ denotes the aggregated state vector.
For ease of exposition, we consider the case of $N=1$.
However, all results presented in this note can be trivially extended to the general case.
Finally, considering ($\ref{complex}$), we note that
\begin{eqnarray}
x(t)\rightarrow[(\mathbf{1}_n\mathbf{1}^\textrm{T}_n/n)\otimes\mathrm{I}_N]x_0 \ \ \textrm{as} \ \ t\rightarrow\infty,
\end{eqnarray}
since the graph $\mathcal{G}$ is assumed to be connected and undirected.
That is, the networked multiagent system is said to reach a consensus since $x_1=x_2=\cdots=x_n$.
\subsection{Resilient Coordination Based on State Emulators and Adaptive Control}
This section introduces the proposed adaptive control approach based on state emulators to enable resilient coordination of networked multiagent systems in the presence of misbehaving agents.
The agent dynamics given by (\ref{single_agent}) are augmented to incorporate the effect of these misbehaviors as
\begin{equation} \label{single_agent_with_constant_disturbance}
\dot{x}_i(t)=u_i(t)+w_i,\ t\geq0,\ x_i(t_0)=x_{i0}, \ i\in\mathcal{V}(\mathcal{G}),
\end{equation}
where $w_i\in\mathbb{R}$ is an unknown disturbance applied to agent $i$.
Notice that we represent adversaries as disturbances similar to \cite{yuc01}.
Specifically, we say that an agent is misbehaving if there exists a time such that $w_i\neq0$, $ i\in\mathcal{V}(\mathcal{G})$.
It should be mentioned here that we only consider the case of constant exogenous disturbances for the ease of exposition (using the results of Section VI of \cite{yuc01}, our following results can be easily extended to the case of time-varying disturbances).
In order to mitigate the effect of these exogenous disturbances, the nominal consensus protocol given by (\ref{input}) is modified as
\begin{equation}\label{input_constant_disturbance}
u_i(t)=-\sum_{i\sim j}(x_i(t)-x_j(t))+\hat{w}_i, \quad i\in\mathcal{V}(\mathcal{G}),
\end{equation}
where $\hat{w}_i$ is an adaptive control signal, which estimates the disturbance of agent $i$, and is updated as
\begin{equation}\label{hatw_update}
\dot{\hat{w}}_i(t)=\alpha(x_i(t)-\hat{x}_i(t)), \quad t\geq0, \quad \hat{w}_i(t_0)=\hat{w}_{i0},
\end{equation}
where $\alpha>0$, $\hat{w}_{i0}=0$, and $\hat{x}_i(t)$ is a state emulator given by
\begin{equation}\label{observer}
\dot{\hat{x}}_i(t)=-\textrm{d}_i\hat{x}_i(t)+\sum_{i\sim j}x_j(t) ,\quad t\geq0, \quad \hat{x}_i(0)=\hat{x}_{i0}.
\end{equation}
The undisturbed response of the system is captured by $\hat{x}_i(t)$.
However, notice that indirect disturbances from neighboring agents still affect the emulator system.
Using (\ref{single_agent_with_constant_disturbance}) and (\ref{observer}), the dynamics of the emulator state estimate error are given by
\begin{equation}\label{observer_error}
\dot{\tilde{x}}_i(t)=-d_i\tilde{x}_i(t)-\tilde{w}_i(t), \quad t\geq0, \quad \tilde{x}_i(t_0)=\tilde{x}_{i0},
\end{equation}
where $\tilde{x}(t)\triangleq x_i(t)-\hat{x}_i(t)$ and $\tilde{w}(t)\triangleq w_i-\hat{w}_i(t)$.
In addition, the exogenous disturbance estimate error dynamics are given by
\begin{equation}\label{estimate_error}
\dot{\tilde{w}}_i(t)=-\alpha \tilde{x}_i(t),\quad t\geq0, \quad \tilde{w}_i(t_0)=\tilde{w}_{i0}.
\end{equation}
Now, the networked multiagent system can be described in a compact form as
\begin{eqnarray}\label{agg_sys}
\dot{\hat{x}}(t)&=&-\Delta(\mathcal{G})\hat{x}(t)+\mathcal{A(G)}{x}(t),\quad \hat{x}(t_0)=\hat{x}_{0},\label{agg_sys1} \ \ \ \\
\dot{\tilde{x}}(t)&=&-{\Delta(\mathcal{G})}\tilde{x}(t)-\tilde{w}(t), \quad \tilde{x}(t_0)=\tilde{x}_{0},\label{agg_sys2} \\
\dot{\tilde{w}}(t)&=&\alpha\tilde{x}(t), \quad \tilde{w}(t_0)=\tilde{W}_{0},\label{agg_sys3}
\end{eqnarray}
where $\hat{x}(t)=[\hat{x}_1(t), \dots, \hat{x}_n(t)]^\textrm{T}\in\mathbb{R}^n$,
$\tilde{x}(t)=[\tilde{x}_1(t), \dots, \tilde{x}_n(t)]^\textrm{T}\in\mathbb{R}^n$,
$\tilde{w}(t)=[\tilde{w}_1(t), \dots, \tilde{w}_n(t)]^\textrm{T}\in\mathbb{R}^n$,
denote the aggregated emulator state, emulator estimate error, and disturbance estimate error, respectively.
Furthermore, (\ref{agg_sys1}) can be equivalently written as
\begin{equation} \label{new_refer}
\dot{\hat{x}}(t)=-\mathcal{L}(\mathcal{G})\hat{x}(t)+\mathcal{A}(\mathcal{G})\tilde{x}(t), \ \ t\geq0, \ \ \hat{x}(t_0)=\hat{x}_{0}.
\end{equation}
Next, we consider the state transformation given by
\begin{equation} \label{state_trans}
\hat{y}(t)=T\hat{x}(t)=[\hat{z}_{12}(t), \hat{z}_{13}(t), \dots, \hat{c}_\mathcal{G}(t)]^\textrm{T}
\end{equation}
where $\hat{z}_{1i}(t)=\hat{x}_1(t)-\hat{x}_i(t)$ and $\hat{c}_\mathcal{G}(t)=\sum_{i\in\mathcal{V}(\mathcal{G})}\hat{x}_i(t)$.
Under this state transformation and using (\ref{new_refer}), it follows that
\begin{equation} \label{trans_new_reference}
\dot{\hat{y}}(t)=-T\mathcal{L}T^{-1}\hat{y}(t)+T\mathcal{A}\tilde{x}(t), \ t\geq0, \ \hat{y}(t_0)=\hat{y}_{0}.
\end{equation}
Furthermore, since the graph is undirected and connected, (\ref{trans_new_reference}) can be partitioned as
\begin{eqnarray}
\hspace{-0.4cm}\dot{\hat{z}}_1(t)&=&A_1\hat{z}_1(t)+A_2\tilde{x}(t),\ t\geq0, \ \hat{z}_1(t_0)=\hat{z}_{10}, \label{nonsemistable_dynamics}\\
\hspace{-0.4cm}\dot{\hat{c}}_\mathcal{G}(t)&=&\sum_{i\in\mathcal{V}(\mathcal{G})}\textrm{d}_i\tilde{x}_i(t),\ \hat{c}_\mathcal{G}(t_0)=\hat{c}_{\mathcal{G}0},\label{centroid_dynamics}
\end{eqnarray}
where $z_1=[z_{12},z_{13},\dots,z_{1n}]^\textrm{T},$ $A_1\in\mathbb{R}^{n-1\times n-1}$ and $A_2\in\mathbb{R}^{n-1\times n-1}$ are the matrices obtained by removing the $n^{\textrm{th}}$ row and column from $T\mathcal{L}(\mathcal{G})T^{-1}$ and the matrix obtain by removing the $n^{\textrm{th}}$ row from $T\mathcal{A}(\mathcal{G})$, respectively.
The estimate update given by (\ref{hatw_update}) requires each agent to have access to its state, $x_i(t)$.
This requirement is not needed for the standard consensus protocol (\ref{input}).
However, as shown in the following result, the modified consensus protocol (\ref{input_constant_disturbance}) allows the network multiagent system to achieve consensus even in the presence of disturbances.
Specifically, the following result shows that the modified consensus protocol given by (\ref{input_constant_disturbance}) results in consensus despite the presence of disturbances.
Furthermore, the estimates of the disturbances converge to the actual exogenous disturbances as $t\rightarrow\infty$.
\textbf{Theorem.} Consider the network multiagent system given by (\ref{nonsemistable_dynamics}), (\ref{agg_sys2}), and (\ref{agg_sys3}).
Then, the solution $(\hat{z}_1(t),\tilde{x}(t),\tilde{w}(t))$ is exponentially stable for all $(\hat{z}_{10}, \tilde{x}_{0}, \tilde{w}_0)\in\mathbb{R}^{n-1}\times\mathbb{R}^{n}\times\mathbb{R}^{n}$.
\textbf{Proof.} Note that the system is equivalently described by $\dot{\xi}(t)=M\xi(t),\quad t\geq0, \quad {\xi}(t_0)={\xi}_{0}$,
where
\begin{eqnarray}\label{aug_agg_sys_matrix}
M=\left[\begin{array}{c|cc}
A_1 & A_2 & 0_{n-1\times n} \\
\hline
0_{n\times n-1} & -{\Delta(\mathcal{G})} & -\textrm{I}_n \\
0_{n\times n-1} & \alpha\textrm{I}_n & 0_{n\times n} \\
\end{array}\right],
\end{eqnarray}
and $\xi(t)=[\hat{z}_1(t)^\textrm{T}, \tilde{x}(t)^\textrm{T}, \tilde{w}(t)^\textrm{T}]^\textrm{T}\in\mathbb{R}^{3n-1}$.
Using Lemma 1 the spectrum of $M$ is described as
\begin{eqnarray}
\textrm{spec}(M)=\textrm{spec}(A_1)\cup\textrm{spec}(\left[\begin{array}{cc}-{\Delta(\mathcal{G})} & -\textrm{I}_n \\ \alpha\textrm{I}_n & 0 \\ \end{array}\right]).
\end{eqnarray}
Since the graph is assumed to be connected the spectrum of the Laplacian is described as
\begin{eqnarray}
\textrm{spec}(-\mathcal{L(G)})&=&\{0\}\cup\{\lambda_2(-\mathcal{L(G)}),\dots, \lambda_n(-\mathcal{L(G)})\}\nonumber\\
&=&\{0\}\cup\textrm{spec}(A_1)
\end{eqnarray}
where $\lambda_i(-\mathcal{L(G)})<0, \forall i\in\{2,\dots,n\}$.
Furthermore, note that the characteristic polynomial of $\left[\begin{array}{cc}-{\Delta(\mathcal{G})} & -\textrm{I}_n \\ \alpha\textrm{I}_n & 0 \\ \end{array}\right]$ is given as $Z(\lambda)=\lambda^2\textrm{I}_n+\lambda \Delta(\mathcal{G}) +\alpha \textrm{I}_n$.
Therefore, it can be concluded from Lemma 2 that $\pi_+(Z)=0$, $\pi_0(Z)=0$ and $\pi_-(Z)=2n$.
Thus, $\lambda_i({M})<0, \forall i\in\{1,\dots,3n-1\}$.
Therefore, the system is exponentially stable for all initial conditions and $t\geq0$.
\hfill $\blacksquare$
Notice from (\ref{agg_sys1}) and (\ref{centroid_dynamics}) that $\tilde{x}(t)$ acts as a vanishing perturbation to an ideal consensus equation.
Furthermore, if $\|\tilde{x}(t)\|_2$ is sufficiently small, then agents not only achieve consensus but the agreement point stays close to the original centroid of the system -- the point that would have been reached in the undisturbed case.
That is, the effect of the disturbances on the overall system performance can be related to $\|\tilde{x}(t)\|_2.$
In addition, $\alpha$ has a direct effect on the bound of $\|\tilde{x}(t)\|_2$.
To see this, consider the energy function $E(\tilde{x}(t),\tilde{w}(t))=\frac{1}{2}\tilde{x}^\textrm{T}\tilde{x}+\frac{1}{2\alpha}\tilde{w}^\textrm{T}\tilde{w}$.
Taking the time derivative yields $\dot{E}(\tilde{x}(t),\tilde{w}(t))=-\tilde{x}^\textrm{T}\Delta(\mathcal{G})\tilde{x}\leq0$.
Therefore, $E(\tilde{x}(t),\tilde{w}(t))\leq E(\tilde{x}(0),\tilde{w}(0))=\frac{\|\tilde{w}\|^2_2}{2\alpha}$ and $\|\tilde{x}\|_2\leq\frac{\|\tilde{w}\|_2}{\sqrt{\alpha}}$.
As $\alpha$ is increased, the magnitude of the vanishing perturbation term $\|\tilde{x}\|_2$ becomes smaller.
Meaning, during transient-time the state emulator system (\ref{agg_sys1}) and the emulator centroid, $\hat{c}(t)$, are effected less by disturbances.
Finally, note that $\hat{c}_\mathcal{G}(t)=\int_0^t\sum_{i\in\mathcal{V}(\mathcal{G})}\textrm{d}_i\tilde{x}_i(t) \ \textrm{d}t + \hat{c}_{0}$ remains bounded since $\tilde{x}(t)$ exponentially converges to $0$.
\subsection{Concluding Remarks}
Control algorithms of networked multiagent systems are generally computed distributively without having a centralized entity monitoring the activity of agents; and therefore, adversaries such as attacks to the communication network and/or failure of agent-wise components can easily result in system instability and prohibit the accomplishment of system-level objectives.
Motivation from this standpoint, we proposed a new adaptive control approach based on distributed state emulators to guarantee a desired system-level performance in the presence of misbehaving agents.
\bibliographystyle{IEEEtran}
\section{Gain Margin Analysis} \label{sec:GM}
Gain margin is an important stability measure originally developed for LTI systems, but the margin can easily be extended to adaptive control. In LTI systems the gain margin is the amount of gain factor that can be added to the feedback loop while maintaining stability. For adaptive systems the gain margin measures the amount of allowable gain between the controlled plant and the adaptive control (\cite{QianSang}).
For our analysis we assume ${\cal{K}}=diag(k_1,k_2,\dots,k_m)$ where $k=k_1=k_2=\dots=k_m > 0$ is the positive gain between the controlled plant and the controller. Notice we assume that the gain is the same for all control channels. Now (\ref{eq:control_law}) becomes $u(t)={\cal{K}}u_{rm}(t)+{\cal{K}}u_{pd}(t)$ and (\ref{eq:edot}) becomes:
\begin{equation}\label{eq:GM_edot}
\dot{e}(t)=A_{rm} e(t) + B{\cal{K}} \tilde{K}^T(t)x(t)+B{\cal{K}} \tilde{K}_r(t)r(t),
\end{equation}
or simply
\begin{equation}\label{eq:GM_edot}
\dot{e}(t)=A_{rm} e(t) + B{k} \tilde{K}^T(t)x(t)+B{k} \tilde{K}_r(t)r(t).
\end{equation}
Consiider the Lyapunov function
\begin{equation} \label{eq:lyap_GM}
V\left(e,\tilde{K},\tilde{K}_r\right)=\frac{1}{2}e^TPe+\frac{1}{2}tr\left(\tilde{K}^T{\bar\Gamma_x}^{-1}\tilde{K}\right)+\frac{1}{2}\tilde{K}^T_r{\bar\Gamma_r}^{-1}\tilde{K}_r.
\end{equation}
where ${\bar\Gamma_x}^{-1}={k}{\Gamma_x}^{-1}$ and ${\bar\Gamma_r}^{-1}={k}{\Gamma_r}^{-1}$.
Differentiating (\ref{eq:lyap_GM}) results:
\begin{equation}\begin{split}
\dot V(e(t),\tilde K(t),\tilde {K}_r(t))=&-\frac{1}{2}e^TQe + e^TPB{k}(\tilde K^Tx+ \tilde K_r^T r) \\&- \mathrm{tr}(\tilde{K}^Tk_1(xe^TPB + \sum\limits_{j = 1}^p x_j x^T_j \tilde K)) \\ &- \tilde{K}_r^Tk(re^T PB + \sum\limits_{j = 1}^p r_j r_j \tilde K_r),
\end{split}\end{equation}
simplfiying:
\begin{equation}\begin{split}
\dot V(e(t),\tilde K(t),\tilde {K}_r(t))=&-\frac{1}{2}e^TQe -\mathrm{tr}(\tilde{K}^Tk\sum\limits_{j = 1}^p x_j x^T_j \tilde K) \\ &- \tilde{K}_r^Tk\sum\limits_{j = 1}^p r_j r_j \tilde K_r.
\end{split}\end{equation}
By following the analysis in \cite{Linear_Chow} the expression can now be reduced to
\begin{equation}\begin{split}\label{eq:exp_bounds_linear}
&\dot V(e,\tilde K,\tilde {K}_r)\leq\\&-\frac{\min\left(\lambda_{\min}(Q),2k\lambda_{\min}(\Omega_K),2k\sum\limits_{j = 1}^p r_j^2\right)}{\max\left(\lambda_{\max}(P),\lambda_{\max}(\bar{\Gamma}_x^{-1}),\bar{\Gamma}_r^{-1}\right)}V(e,\tilde K,\tilde {K}_r).
\end{split}\end{equation}
The system remains exponentially convergent for $k\in(0,+\infty)$.
\section{Time-Delay Margin} \label{sec:TimeDelay}
Unlike the gain margin, the phase margin is not easily extended to adaptive systems. The time-delay margin is the nonlinear analog to phase margin. To our knowledge there are no general methodologies hat have been developed to analyze the time-delay margin of a general adaptive system. Linear approximation techniques have been developed. In this section will be used the findings in \cite{Nguyen_TD} to develop a computable time-delay margin for our adapative system. The main results of the paper will be reviewed first. The first result gives an upper bound on the time-delay margin on a linear system.
\begin{lemma}
Given a time-delay system:
\begin{equation}
\label{eq:delay_system}
\dot{x}(t)=Ax(t)-BKx(t-td)
\end{equation}
where $x(t):[0, \infty) \rightarrow \mathbb{R}^n$ and $A- BK$ is Hurwitz. The time-delay system is asymptotically stable if the following inequalities hold
\begin{eqnarray}\label{eq:timedelay}
t_d<&\frac{1}{\omega}{\cos}^{-1}{\frac{\bar{\mu}(A)+\bar{\mu}(jBK)}{\|BK\|_2}}\\
\omega<&\bar\mu(-jA)+\|BK\|_2
\end{eqnarray}
where
\begin{equation}
\label{eq:mu_functional}
\mu_i(C)=\lambda\left(\frac{C+C^*}{2}\right).
\end{equation}
The proof of this lemma can be found in \cite{Nguyen_TD} and will not be discussed in this paper.
\end{lemma}
The next result can be used to approximate nonlinear adaptive law without linearization. The comparison lemma approximates the nonlinear adaptive law with a bounded linear version. The lemma is stated as:
\begin{lemma} The equilibrium state of $y(t)=0$ of the differential equation
\begin{equation}
\dot{y}(t)=-{\Phi}^T(t) \Gamma \Phi(t) y(t)
\end{equation}
where $y(t):[0, \infty) \rightarrow \mathbb{R}$,$\Phi(t) \in {\cal{L}}_2 :[0,\infty)\to\mathbb{R}$ is a piecewise continuous and bounded function, and $\Gamma>0\in\mathbb{R}^{n \times n}$, is uniformly asymptotically stable, if there exist a constant $\gamma>0$ such that
\begin{equation}
\frac{1}{T_0}\int_t^{t+T_0}\Phi^T(t)\Gamma\Phi(t)\mathrm{d}\tau\geq\gamma
\end{equation}
which implies that $y(t)$ is locally bounded by the solution of a linear differential equation
\begin{equation}
\dot{z}(t)=-\gamma z(t)
\end{equation}
for $t \in [t_i,t_i+T_0)$, where $t_i=t_{i-1}+T_0$ and $i=1,2,...,n\to\infty$.
\end{lemma}
The proof of the lemma can be founded in \cite{Nguyen_TD} and other sources.
Now consider a time delay on (\ref{eq:plant_with_control}):
\begin{equation}
\dot{x}(t)=Ax(t)+B(K^T(t-td)x(t-td)+K_r(t-td)r(t-d)),
\end{equation}
and the error expression (\ref{eq:edot}) now becomes:
\begin{equation}\label{eq:edot_td}\begin{split}
\dot{e}(t)=&A_{rm}e(t)+B(K_r(t-td)r(t-td)-K_r^*r(t)\\&+K^T(t-td)x(t-td) -K^*x(t)).
\end{split}\end{equation}
Consider the following expression:
\begin{equation}\label{eq:dotK_linear}\begin{split}
\dot{K}^T(t)x(t)=-B^TPe(t)x^T(t)\Gamma_xx(t)-{\tilde K}^T(t) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t).
\end{split}\end{equation}
Linearization of the second term produces:
\begin{equation}\begin{split}
{\tilde K}^T(t) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t)=&{\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t_i)\\
&+ {\tilde K}^T(t) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t_i) \\
&-{\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t_i) \\
&+ ({\tilde K}^T(t_i)) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t)\\
&- ({\tilde K}^T(t_i)) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t_i)\\
&+ \ldots
\end{split}\end{equation}
and by approximation:
\begin{equation}\begin{split}
{\tilde K}^T(t) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t)\approx&-{\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t_i)\\
&+ {\tilde K}^T(t) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t_i) \\
&+ {\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x_{rm}(t)\\
&+ {\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x e(t)
\end{split}\end{equation}
for $t\in[t_i,t_i+T_0)$,where $t+0=0,t_i=t_{i-1}+T_0$, and $i=1,2,\dots,n\to\infty$. Returning to (\ref{eq:dotK_linear}) we see that the first term can be approximated with help from lemma 2:
\begin{equation}
-B^TPe(t)x^T(t)\Gamma_xx(t)=-B^TPe(t)\gamma_x
\end{equation}
where $\gamma_x=\mathop{\rm inf}_{t\in[t_i,t_i+T_0)}\left(\frac{1}{T_0}\int_t^{t+T_0}x^T(t)\Gamma_x x(t)\mathrm{d}\tau\right)\ge0\in\mathbb{R}$. Thus an approximation of (\ref{eq:dotK_linear}) is found. An identical analysis is made for $\dot{K_r}^T(t)r(t)$.
Consider the following:
\begin{equation}\begin{split}
K^T(t)\dot{x}(t)=&K^T(t_i)\dot{x}(t_i)+\dot{K}^T(t_i)\dot{x}(t_i)\Delta t\\&+K^T(t_i)(\dot{x}(t)-\dot{x}(t_i))+\dots
\end{split}\end{equation}
and by approximation:
\begin{equation}
K^T(t)\dot{x}(t)\approx K^T(t_i)\dot{e}(t) + K^T(t_i)\dot{x}_{rm}(t)+\dot{K}^T(t_i)\dot{x}(t_i)\Delta t.
\end{equation}
An identical analysis is made for ${K_r}^T(t)\dot{r}(t)$.
Differentiating (\ref{eq:edot_td}) yields:
\begin{eqnarray*}
\ddot{e}(t)=A_{rm} \dot{e}(t)+B(\dot{K}_rr(t-td)+K_r\dot{r}(t-td)\\
-K_r^*\dot{r}(t)+\dot{K}^T(t-td)x(t-td)\\
+K^T(t-td)\dot{x}(t-td)-K^*\dot{x}(t)).
\end{eqnarray*}
Using the linearization and approximations derived above yields:
\begin{eqnarray*}
\dot{x}(t)
=&
\left[ {\begin{array}{cc}
A_{rm}-B K^{{\star}^T} & 0 \\
I & 0 \\
\end{array} } \right]
x(t)+\\&
\left[ {\begin{array}{cc}
BK^T(t_i) & -B{\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x-\gamma_{xr} B B^T P \\
0 & 0 \\
\end{array} } \right]
x(t-t_d)\\&
+
\left[ {\begin{array}{cc}
d_1(t-t_d)+d_2 \\
0 \\
\end{array} } \right]
\end{eqnarray*}
where
$\gamma_{xr}=\gamma_{x}+\gamma_{r}$, $x(t)=
\left[ {\begin{array}{cc}
\dot{e}(t)\\
{e}(t)\\
\end{array} } \right]$,
\begin{equation}\begin{split}
d_2(t-t_d)=&-{\tilde K}_r^T(t-t_d) \sum\limits_{j = 1}^p r_j r^T_j \Gamma_r r(t_i)\\
&-{\tilde K}_r^T(t_i) \sum\limits_{j = 1}^p r_j r^T_j \Gamma_r r(t-t_d)\\
&+ K_r^T(t_i)\dot{r}(t-t_d)-{\tilde K}^T(t-t_d) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_r x(t_i)\\
&-{\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_r x_{rm}(t-t_d)\\
&+K^T(t_i)\dot{x}_{rm}(t-t_d)
\end{split}\end{equation}
\begin{equation}\begin{split}
d_3=& \dot{K}_r^T(t_i)\dot{r}(t_i)\Delta t + {\tilde K}_r^T(t_i) \sum\limits_{j = 1}^p r_j r^T_j \Gamma_r r(t_i)\\
&+ {\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t_i)+\dot{K}^T(t_i)\dot{x}(t_i)\Delta t
\end{split}\end{equation}
are treated as time varying disturbances and do not affect the time-delay margin thus are neglected.
Lemma 2 can be used to determine the time-delay margin of the system:
\begin{eqnarray}
t_{di}<&\frac{1}{\omega}{\cos}^{-1}{\frac{\bar{\mu}(C)+\bar{\mu}(jDK)}{\|DK\|_2}}\\
\omega_i<&\bar\mu(-jC)+\|DK\|_2.
\end{eqnarray}
where
\begin{eqnarray}
C_{i}=&\left[ {\begin{array}{cc}
A_{rm}-BK^{{\star}^T} & 0 \\
I & 0 \\
\end{array} } \right]\\
D_i=&\left[ {\begin{array}{cc}
BK^T(t_i) & -B{\tilde K}^T(t_i) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x-\left(\gamma_{xr}\right) B B^T P \\
0 & 0 \\
\end{array} } \right].
\end{eqnarray}
for for $t\in[t_i,t_i+T_0)$,where $t+0=0,t_i=t_{i-1}+T_0$, and $i=1,2,\dots,n\to\infty$. The time-delay margin is local and is computed for a time window after it has enlapsed.
Another estimation method. We will take $x(t)=
\left[ {\begin{array}{cc}
{e}(t)\\
\tilde{K}^{T}{x}(t)\\
\tilde{K}_r^{T}{x}(t)\\
\end{array} } \right]$, and find a linear estimation for $\dot{x}(t)$. From (\ref{eq:edot_td}) we see that $\dot{x}(t)$ can be written as:
\begin{equation}\begin{split}
\dot{e}(t)=&A_{rm}e(t)+B(\tilde{K}_r(t-td)r(t-td)-K_r^*r(t)\\&+\tilde{K}^T(t-td)x(t-td) -K^*x(t)\\&-{K}^{\star}(t-td)x(t-td)-{K}^{\star}_r(t-td)r(t-td)).
\end{split}\end{equation}
Differentating $\tilde{K}^{T}{x}(t)$ results:
\begin{equation}\begin{split}
\dot{\tilde{K}}^{T}{x}(t)+\tilde{K}^{T}{\dot{x}}(t)=&-B^TPe(t)x^T(t)\Gamma_xx(t)\\&-{\tilde K}^T(t) \sum\limits_{j = 1}^p x_j x^T_j \Gamma_x x(t)\\&+\tilde{K}^{T}(A_{rm}x(t) + B_{rm}r(t)\\& + B \tilde{K}^T(t))x(t)+B \tilde{K}_r(t)r(t),
\end{split}\end{equation}
which can be appromatied by
\begin{equation}\begin{split}
\dot{\tilde{K}}^{T}{x}(t)+\tilde{K}^{T}{\dot{x}}(t)\le&-\gamma_xB^TPe(t)\\&-\lambda_{min}(\sum\limits_{j = 1}^p x_j x^T_j \Gamma_x){\tilde K}^T(t)x(t)\\&+\tilde{K}(t)^{T}(A_{rm}x(t) + B_{rm}r(t))\\& +\tilde{K}(t_i)^{T}( B \tilde{K}^T(t))x(t)+B \tilde{K}_r(t)r(t))).
\end{split}\end{equation}
Differentating $\tilde{K}_r^{T}{r}(t)$ results:
\begin{equation}\begin{split}
\dot{\tilde{K}}_r^{T}{r}(t)+\tilde{K}_r^{T}{\dot{r}}(t)=&-B^TPe(t)r^T(t)\Gamma_r r(t)\\&-{\tilde K}_r^T(t) \sum\limits_{j = 1}^p r_j r^T_j \Gamma_r r(t)\\&+\tilde{K}_r^{T}{\dot{r}}(t),
\end{split}\end{equation}
which can be appromatied by
\begin{equation}\begin{split}
\dot{\tilde{K}}_r^{T}{r}(t)+\tilde{K}_r^{T}{\dot{r}}(t)\le&-\gamma_rB^TPe(t)\\&-\lambda_{min}(\sum\limits_{j = 1}^p r_j r^T_j \Gamma_r){\tilde K}_r^T(t)x(t)\\&+\tilde{K}_r^{T}{\dot{r}}(t).
\end{split}\end{equation}
We can then estimate the adaptive system as a LTI system in a given time window,
\begin{eqnarray*}
\dot{x}(t)
\le &
\left[ {\begin{array}{ccc}
A_{rm} & 0 & 0 \\
-\gamma_x B^{T} P & \tilde{K}^{T}(t_i) B-\lambda_{min}(\sum\limits_{j = 1}^p x_j x^T_j \Gamma_x) & \tilde{K}^{T}(t_i) B \\
-\gamma_r B^{T} P & 0 & -\lambda_{min}(\sum\limits_{j = 1}^p r_j r^T_j \Gamma_r)\\
\end{array} } \right]
x(t)\\&+
\left[ {\begin{array}{ccc}
0& B & B \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array} } \right]
x(t-t_d)\\&
+
\left[ {\begin{array}{c}
-B(K^{\star}x(t)+K_r^{\star}r(t)+K(t-t_d)x(t-t_d)+K_r(t-t_d)r(t-t_d)\\
\tilde{K}^{T}(A_{rm}x(t)+B_{rm}r(t)) \\
\tilde{K}^{T}_r \dot{r}(t)\\
\end{array} } \right]
\end{eqnarray*} |
1,314,259,996,676 | arxiv | \section*{\noindent Editor\hfill}
David Garfinkle\\
\smallskip
Department of Physics
Oakland University
Rochester, MI 48309\\
Phone: (248) 370-3411\\
Internet:
\htmladdnormallink{\protect {\tt{garfinkl-at-oakland.edu}}}
{mailto:garfinkl@oakland.edu}\\
WWW: \htmladdnormallink
{\protect {\tt{http://www.oakland.edu/physics/Faculty/david-garfinkle}}}
{http://www.oakland.edu/physics/Faculty/david-garfinkle}\\
\section*{\noindent Associate Editor\hfill}
Greg Comer\\
\smallskip
Department of Physics and Center for Fluids at All Scales,\\
St. Louis University,
St. Louis, MO 63103\\
Phone: (314) 977-8432\\
Internet:
\htmladdnormallink{\protect {\tt{comergl-at-slu.edu}}}
{mailto:comergl@slu.edu}\\
WWW: \htmladdnormallink{\protect {\tt{http://www.slu.edu/arts-and-sciences/physics/faculty/comer-greg.php}}}
{http://www.slu.edu/arts-and-sciences/physics/faculty/comer-greg.php}\\
\bigskip
\hfill ISSN: 1527-3431
\bigskip
DISCLAIMER: The opinions expressed in the articles of this newsletter represent
the views of the authors and are not necessarily the views of APS.
The articles in this newsletter are not peer reviewed.
\begin{rawhtml}
<P>
<BR><HR><P>
\end{rawhtml}
\end{flushleft}
\pagebreak
\section*{Editorial}
The next newsletter is due December 2018. Issues {\bf 28-51} are available on the web at
\htmladdnormallink
{\protect {\tt {https://files.oakland.edu/users/garfinkl/web/mog/}}}
{https://files.oakland.edu/users/garfinkl/web/mog/}
All issues before number {\bf 28} are available at
\htmladdnormallink {\protect {\tt {http://www.phys.lsu.edu/mog}}}
{http://www.phys.lsu.edu/mog}
Any ideas for topics
that should be covered by the newsletter should be emailed to me, or
Greg Comer, or
the relevant correspondent. Any comments/questions/complaints
about the newsletter should be emailed to me.
A hardcopy of the newsletter is distributed free of charge to the
members of the APS Division of Gravitational Physics upon request (the
default distribution form is via the web) to the secretary of the
Division. It is considered a lack of etiquette to ask me to mail
you hard copies of the newsletter unless you have exhausted all your
resources to get your copy otherwise.
\hfill David Garfinkle
\bigbreak
\vspace{-0.8cm}
\parskip=0pt
\section*{Correspondents of Matters of Gravity}
\begin{itemize}
\setlength{\itemsep}{-5pt}
\setlength{\parsep}{0pt}
\item Daniel Holz: Relativistic Astrophysics,
\item Bei-Lok Hu: Quantum Cosmology and Related Topics
\item Veronika Hubeny: String Theory
\item Pedro Marronetti: News from NSF
\item Luis Lehner: Numerical Relativity
\item Jim Isenberg: Mathematical Relativity
\item Katherine Freese: Cosmology
\item Lee Smolin: Quantum Gravity
\item Cliff Will: Confrontation of Theory with Experiment
\item Peter Bender: Space Experiments
\item Jens Gundlach: Laboratory Experiments
\item Warren Johnson: Resonant Mass Gravitational Wave Detectors
\item David Shoemaker: LIGO
\item Stan Whitcomb: Gravitational Wave detection
\item Peter Saulson and Jorge Pullin: former editors, correspondents at large.
\end{itemize}
\section*{Division of Gravitational Physics (DGRAV) Authorities}
Chair: Emanuele Berti; Chair-Elect:
Gary Horowitz; Vice-Chair: Nicolas Yunes.
Secretary-Treasurer: Geoffrey Lovelace; Past Chair: Peter Shawhan; Councilor: Beverly Berger
Members-at-large:
Kelly Holley-Bockelmann, Leo Stein, Lisa Barsotti, Theodore Jacobson, Michael Lam, Jess McIver.
Student Members: Cody Messick, Belinda Cheeseboro.
\parskip=10pt
\vfill\eject
\section*{\centerline
{we hear that \dots}}
\addtocontents{toc}{\protect\medskip}
\addtocontents{toc}{\bf DGRAV News:}
\addcontentsline{toc}{subsubsection}{
\it we hear that \dots , by David Garfinkle}
\parskip=3pt
\begin{center}
David Garfinkle, Oakland University
\htmladdnormallink{garfinkl-at-oakland.edu}
{mailto:garfinkl@oakland.edu}
\end{center}
Vicky Kalogera has been elected to the National Academy of Sciences.
Nicolas Yunes has been elected Vice-Chair of DGRAV. Jess McIver and Michael Lam have been elected Members-at-large of the DGRAV Executive Committee. Belinda Cheesboro has been elected Student Student Member of the DGRAV Executive Committee.
Hearty Congratulations!
\vfill\eject
\section*{\centerline
{DGRAV Student Travel Grants}
}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it DGRAV Student Travel Grants, by Beverly Berger}
\parskip=3pt
\begin{center}
Beverly Berger, LIGO
\htmladdnormallink{beverlyberger-at-me.com}
{mailto:beverlyberger@me.com}
\end{center}
In the year of the General Relativity Centennial, the APS Topical Group in Gravitation (now the Division of Gravitational Physics) began an effort to raise funds to support the travel of student members to present their work at the April Meeting. This travel grant program, started in 1999 with GGR (now DGRAV) operating funds, has been wildly successful. This year more than 40 students received such grants as part of the largest participation ever by DGRAV (or GGR) members in the April Meeting.
The fund-raising program began because DGRAV operating funds cannot meet the demand from ever increasing numbers of applicants. Students are the future of our field. We would like to assist as many as possible to experience the excitement (and networking opportunities) of the April Meeting.
Since this campaign began, DGRAV has raised \$50,000. Thank you, donors! However, the goal for the campaign is \$75,000. Please donate to help us reach this goal. If your students have benefited from the student travel grant program or you, when a student, received an award, or if you believe it is important to support students this way, please consider a donation.
As a special incentive, DGRAV is offering a General Relativity Centennial commemorative medallion, designed and minted for the 2015 Centennial banquet, to donors who request one to express our gratitude for your support. These are in limited quantity and will be distributed in order of the gift receipt date.
For more information or to contribute on line, please visit the campaign website,
\href{https://www.aps.org/about/support/campaigns/dgrav/index.cfm}{https://www.aps.org/about/support/campaigns/dgrav/index.cfm}, which also lists donors to the campaign.
To contribute by mail, please make your check payable to the American Physical Society, note ``DGRAV Travel'' in the memo, and mail to Irene I. Lukoff, Director of Development, One Physics Ellipse, College Park, MD 20740.
If you would like a medallion, please send a request with your mailing address to Peter Shawhan, \htmladdnormallink{pshawhan@umd.edu}{mailto:pshawhan@umd.edu}, once your donation is received.
\vfill\eject
\section*{\centerline
{Town Hall Meeting}
}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it Town Hall Meeting, by Emanuele Berti}
\parskip=3pt
\begin{center}
Emanuele Berti, University of Mississippi
\htmladdnormallink{eberti-at-olemiss.edu}
{mailto:eberti@olemiss.edu}
\end{center}
Town Hall Meeting on ``Gravitational Wave Theory and Simulations in the Era of Detections''
Gravitational wave astronomy is finally a reality. The first multi-messenger observation of a binary neutron star merger (GW170817) was one of the biggest science stories of 2017, and binary black hole detections are becoming routine. However there is still a lot of work needed to coordinate the efforts of instrumentalists, data analysts, astrophysicists and gravitational theorists.
Gravitational-wave theorists, numerical relativists, astrophysicists and LIGO/Virgo members should work together as a single community to make the best of the wealth of data that will be collected during LIGO/Virgo's O3 run. For this reason, Beverly Berger and Manuela Campanelli proposed to organize a Town Hall Meeting on ``Gravitational Wave Theory and Simulations in the Era of Detections.''
The Town Hall Meeting, sponsored by DGRAV and co-sponsored by DAP, took place immediately after the DGRAV and DAP Business Meetings on Monday, April 16 at the APS April Meeting in Columbus, OH. Slides from both meetings are available online at
\href{https://dgrav.org/2018/04/17/slides-from-the-2018-business-and-town-hall-meetings/}{https://dgrav.org/2018/04/17/slides-from-the-2018-business-and-town-hall-meetings/}
Laura Cadonati (on behalf of the LIGO Scientific Collaboration) presented the anticipated evolution of noise power spectral densities for LIGO/Virgo, as well as current plans for KAGRA, in the next few years. Advanced LIGO (aLIGO) is expected to reach target sensitivity around 2019. The proposed A+ (if funded by NSF) could be operating by 2024. The expected range for a BNS (1.4+1.4 solar masses) would go from 96 Mpc (O2) to 173 Mpc (aLIGO) and 325 Mpc (A+). The expected range for a BBH (30+30 solar masses) would go from 983 Mpc (O2) to 1606 Mpc (aLIGO) and 2563 Mpc (A+).
Laura and Lisa Barsotti discussed current plans to update the Virgo detector. Finally, Laura presented a table with the current list of priorities from the LSC-Virgo White Paper on Gravitational Wave Data Analysis and Astrophysics:
\href{https://dcc.ligo.org/LIGO-T1700214/public}{https://dcc.ligo.org/LIGO-T1700214/public}
Laura's presentation was followed by 7 short presentations on astrophysics science targets and numerical relativity tools (or results) that could be relevant for O3.
Carl Rodriguez showed some results from simulations of BBH mergers in clusters, suggesting that such mergers could produce signals with nonzero eccentricity in the LIGO band.
Davide Gerosa pointed out the importance of measuring precessional effects (beyond the ``effective spin'' parameter) to extract astrophysics from the observations.
Ilias Cholis asked whether merging BHs in globular clusters could experience runaway collisions leading to formation of intermediate mass BHs; if so, LIGO observations could be used to derive limits on the occupation fraction of intermediate mass BHs in globular clusters.
Zach Etienne pointed out that Moore's Law is slowing (at least for CPU-based codes - a point raised in discussions by Maurice van Putten and Kai Staats, among others) and that more efficient numerical relativity algorithms are necessary. He also pointed out the importance of writing software documentation for the benefit of students and postdocs.
Roland Haas gave an introduction to the Einstein Toolkit and its applications.
Antonios Tsokaros presented results on sequences of spinning BNS which suggest that corotating sequences can have low spin ($< 0.3$) even for close binaries.
Milton Ruiz presented simulations of BNS in GRMHD which imply (for causal EOS) that the maximum mass of the remnant should be below ~2.16 solar masses.
After these presentations, Bangalore Sathyaprakash gave an introduction to current plans for ``Third Generation'' (3G) detectors on behalf of the GWIC 3G Committee and 3G Science Case Team. He made the argument that LIGO and Virgo both have facility-imposed limits on sensitivity (implying at most a factor ~3 improvement in strain sensitivity, with gravity gradient noise limiting sensitivity below 10 Hz). Therefore there is a compelling case to build detectors that can observe deeper into the cosmos, with facilities that will be good ~30-40 years after construction. One of the GWIC charges is to ``commission a study of ground-based gravitational wave science from the global scientific community, investigating potential science vs. architecture vs. network configuration vs. cost trade-offs.'' The GWIC subcommittee has constituted five 3G subcommittees: (1) Science Case Team (3G-SCT), (2) R\&D Coordination, (3) Governance, (4) Agency Interfacing, (5) Community Networking. The Science Case will be developed by an international consortium of scientists under the leadership of the 3G-SCT, which consists of 18 members. For more details, see
\href{https://gwic.ligo.org/3Gsubcomm/}{https://gwic.ligo.org/3Gsubcomm/}
The 3G Science Case consortium is open to anyone who wishes to contribute to the development of the science case for 3G. If you are interested, please send a one-page CV and research interests relevant to 3G to either B.S. Sathyaprakash (bss25@psu.edu) or Vicky Kalogera (vicky@northwestern.edu).
Emanuele Berti (on behalf of the Executive Committee of the APS Division of Gravitational Physics) coordinated a lively discussion on how theorists could help LVC members and vice versa.
The discussion concerned the following main points:
1) The astrophysics/GR communities need the full posteriors to optimize the science return of LVC observations. How quickly should the LVC release these posteriors? Most participants agreed that the answer to this question is complex.
Bangalore Sathyaprakash explained that careful data analysis is necessary before releasing posteriors. Jolien Creighton argued that it is important for the LVC to retain some core science in order to maintain a vibrant collaboration, which in turn is vital to the larger community. A discussion involving various people (Will Farr, Emanuele Berti, Chad Galley, Davide Gerosa and many others) followed.
It was argued that the current model for open data release is tuned to the (urgent) needs of observational astronomers. However, astrophysicists and theorists could benefit from a quicker release of the bulk properties of detected events. For example, Chad Galley pointed out that in other fields it is common to release data sets of lower quality which can be improved over time.
As the number of detections increases, one may consider releasing estimates of masses/distance/effective spins and other ``simple'' properties of each event for use by astronomers and modelers, and then refining parameter estimation for more in-depth applications (such as tests of general relativity).
2) Emanuele Berti also proposed that the collaboration between theorists and LVC members could follow the particle-physics model in two ways:
(i) At CERN, experimental physicists work in close collaboration with a ``theory division.'' The level of involvement, access to the data, and authorship details for the members of this ``theory division'' could and should be discussed on a case-by-case basis. It would be healthy to start discussions of this possibility within the LVC. Similar discussions in the recent past led to a more active LVC involvement of numerical relativists.
(ii) A key reference in particle physics is the ``particle data group'', which is updated annually to reflect the state of the art in the knowledge of particle properties, physical constants, etcetera. Various people (including Leo Stein, Emanuele Berti, Vitor Cardoso, Pedro Ferreira and Thomas Sotiriou, among others) have been discussing the possibility of creating and maintaining a ``gravity data group'' that could list the best known limits on (say) the mass of the graviton, Lorentz violation in gravity etcetera, and current constraints on various proposed modifications of general relativity. This effort requires manpower to maintain a website (or a similar ``living'' resource) and a white paper - that could be circulated on the arXiv - explaining how to interpret the constraints and pointing to relevant references.
\vfill\eject
\section*{\centerline {News from the International Society}
\centerline {on General Relativity and Gravitation (ISGRG)}
}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it GRG Society, by Eric Poisson}
\parskip=3pt
\begin{center}
Eric Poisson, President of ISGRG
\htmladdnormallink{epoisson-at-uoguelph.ca}
{mailto:epoisson@uoguelph.ca}
\end{center}
The International Society on General Relativity and Gravitation is the only explicitly international organization for general relativity, gravitational physics, and mathematical gravitation. It was formed under the auspices of the International Union of Pure and Applied Physics (IUPAP) as an Affiliated Commission. The Society's purposes are to promote the study of General Relativity and Gravitation and to exchange information in the interest of its members and the profession.
Times have been exciting at the Society since the great success of GR21 at Columbia University.
Probably the most enjoyable activity at the Society is the awarding of various prizes to meritorious scientists. First and foremost is the \href{http://www.isgrg.org/IUPAPprize.php}{Young Scientist Prize}, sponsored by IUPAP. In 2017 the Prize was given to Aron Wall, for his fundamental contributions to our understanding of gravitational entropy and the generalized second law of thermodynamics. In 2018 the Prize was awarded to Sam Gralla, for his exceptional and broadly varied contributions to general relativity and relativistic astrophysics. The Society also gives out the \href{http://www.isgrg.org/bergwheelprize.php}{Bergmann-Wheeler Thesis Prize} for an outstanding Ph.D.\ thesis in the broad area of quantum gravity, and the \href{http://www.isgrg.org/ehlersprize.php}{J\"urgen Ehlers Thesis Prize} for an outstanding Ph.D. thesis in mathematical and numerical relativity. Nominations for both prizes are now open for the 2019 instalments. And every three years the Society recognizes the leading figures of our field by awarding a number of \href{http://www.isgrg.org/fellows.php}{Fellowships}; nominations for the 2019 batch are also open.
Plans are well underway for the next GR meeting, GR22, which will be held jointly with the Thirteenth Amaldi Conference on Gravitational Waves in Valencia, Spain, in July, 2019. Most of the plenary speakers and session chairs have been identified, and a lot of information is already available on the \href{http://www.gr22amaldi13.com}{conference website}. The Local Organizing Committee is chaired by Jos\'e Antonio Font, and the Scientific Organizing Committee is chaired by Vitor Cardoso. They have been working very hard, and this should be a most exciting conference, in a beautiful part of the world.
The Society's executive structure has changed in the last few months. The Constitution was amended to increase the Executive Committee by one member, to turn it from a three-member team (President, Deputy-President, Secretary) to a four-member team (President, President-Elect, Past-President, Secretary). The motivation for this change is to have the President-Elect learn the ropes for three years before assuming the role of President. With the previous structure the President came in cold, with no prior experience within the Executive. As previously, the President becomes Past-President (formally Deputy-President) after a three-year term, but now with somewhat reduced responsibilities. I am happy to report that this change in the Constitution, which required a two-thirds majority for adoption, was approved by a vast majority of the membership.
Another recent change, also approved by the members, is the possibility for members of a National Society such as DGRAV to join the Society with a substantial reduction in the membership fee. Thus, one-year and three-year membership dues are reduced by 20\% for members of a National Society, and the price of a life membership is reduced by whopping 30\%.
The recent excitement with gravitational waves and the Nobel Prize (two of three laureates are ISGRG members!) is filling our gravitational hearts with much pride and joy. As President of the Society, I am in awe of these achievements, and welcome the enviable place that gravitation is now occupying in world science. For more information about the Society's activities, please visit our \href{http://www.isgrg.org}{website}.
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\section*{\centerline
{What's new in LIGO}}
\addtocontents{toc}{\protect\medskip}
\addtocontents{toc}{\bf Research Briefs:}
\addcontentsline{toc}{subsubsection}{
\it What's new in LIGO, by David Shoemaker}
\parskip=3pt
\begin{center}
David Shoemaker, MIT
\htmladdnormallink{dhs-at-mit.edu}
{mailto:dhs@mit.edu}
\end{center}
While LIGO did not write a contribution for the MOG for December 2017, a number of the articles there made reference to the first detection of a binary neutron start inspiral (GW10817) and some of the happy fallout. We want to complement that with more of a view from the inside of the events and then a sketch of what's coming in the next half-year as we prepare for the O3 observing run.
By July 2017, as the end of O2 approached, planned for the end of August 2017, we could already assess the run as moderately successful; several clear binary black holes had been detected, and we had a solid body of data which could then be re-analyzed after some processing to regress out some excess noise sources, and with the promise of perhaps some additional events. We were very pleased that Advanced Virgo was making progress in its commissioning, and with a rapid increase in their sensitivity, they set 1 August as the goal to join LIGO in O2 observing. The two LIGO detectors had sensitivities comparable to the 2015-16 O1 run, although due probably to a point absorber on an optic, the Hanford (LHO) instrument had a reach for binary neutron stars of about 60 Mpc (SNR of 8, averaged over positions in the sky and polarizations); Livingston (LLO) had a sensitivity of some 100 Mpc. Virgo had just in late July achieved a reach of 26 Mpc after a breakthrough in commissioning, and had just developed enough of an infrastructure, knowledge of the calibration, and understanding of the noise properties to allow the data to be properly interpreted. We had a firm date of the end of August to complete O2 -- there was lots of reinstallation and commissioning planned for O3. Thus, roughly one month of common observation with Virgo appeared feasible. Based on the detection rates in O2 we had only modest hope that an event might be caught during that month. With that backdrop, any event would be considered a wonderful surprise.
On August 14, the first surprise arrived: An unambiguous binary black hole signal triggered the pipelines watching the two LIGO instruments (the flagship detection software was not yet ready to read in the three instruments in real time), leading to the new detection GW170814 [PRL 119, 141101 (2017)]. The system, of 31 and 25 solar masses and at a distance of 540 Mpc, was seen after combining the signals from the three instruments with a false alarm rate of some 1/27,000 years. While the signal in Virgo was not large against the noise floor, the probability of the signal observed being due to chance was less than $3\times 10^{-3}$. From this detection, two major steps forward in our young observational field were possible.
First, the localization of the source was significantly improved. The addition of Virgo enabled a much smaller angular (roughly a factor of 20 times smaller) area than could be determined from the two LIGO instruments alone, and the distance estimate also improved such that the volume determination for the source was reduced by a factor of 34. These data were sent out in the form of a first-ever 3-detector skymap as quickly as possible to the electromagnetic and particle observers with whom we had memoranda of understanding; no signal was identified that was likely associated with the coalescence. But the key feature of the network had been realized.
Second, with the three-detector network, we could say for the first time that the polarization structure of the signals was consistent with GR. The fact that the two LIGO detectors are as close to co-aligned as possible (given that they are placed on a sphere and are some 3000 km separated) had meant that no conclusions on the polarization state of previous signals could be made. Virgo has a significant projection on the $45^{\circ}$ $\times$ polarization state, and by considering the quality of fit to the three signals we can say that for GW170814 purely tensor polarization is strongly favored over purely scalar or vector polarizations – consistent with General Relativity. More signals will allow more tight constraints, but the added dimension of Virgo allowed an added dimension of inference from the signal.
But events of three days later made this first detection even more important. On August 17th, at 12∶41:04 UTC (a very civilized hour for US East Coast scientists for once!), one of the LIGO pipelines -- gstlal -- rang an alert for one of its architects, Chad Hanna, that a very significant single detector trigger from Hanford was seen, with a mass range corresponding to a binary neutron star coalescence. This single-detector trigger would normally be due to a detector noise ‘glitch’ in one instrument and not the other, or a case of only one instrument in observing mode. As we opened up windows to take a closer look, a background task designed to watch for coincidences between gamma ray bursts and GW triggers produced an estimate of a very high significance for this pair of events with a false alarm rate of some $10^{-16}$ Hz. A handful of colleagues inspected the Livingston trace and discovered the automated software discounted the trigger due to very strong ‘glitch’ in the time series in the middle of the otherwise clear trace. It was quickly determined that a simple algorithm to ‘window out’ the glitch left a signal that could be analyzed with low-latency software, and it was clear that very consistent waveforms had been seen in the two LIGO detectors of very high significance and plausibly associated with the GRB (with some 1.7 seconds interval between the inferred GW arrival time and the GRB arrival time). Attention turned to Virgo: Had they been on the air? Did they also see this signal?
Indeed, Virgo was running at its modest O2 sensitivity; however, the usual data pipeline from the Virgo data to the computers in the US being used for the signal analysis was down, and an alternative was pursued to get the Virgo data together with the LIGO data to systems where the three signals could be combined coherently into a sky map. While this work started, the first machine-readable alert was sent to the electromagnetic and particle groups with which there were agreements for low-level and low-latency data sharing, roughly 27 minutes after the signal traversed the Earth.
The combined LIGO and Virgo data were employed to form a sky map, as had been practiced with GW170814 just three days earlier, and the result quickly checked for plausibility and consistency with the GRB signal seen in both Fermi and INTEGRAL. About 5 hours after the signal was first seen, we sent this skymap out to our partners. The next observatories which had a chance to see the signal were in Chile, where the sun had not yet set, but groups put together their observing plan and almost as soon as the sun set 6 groups, with Swope the first among them, very quickly independently identified a new optical signal near a host Galaxy, within the region supplied in the LIGO-Virgo skymap. A follow-up campaign covering all wavelengths of electromagnetic radiation as well as neutrinos followed, with extremely rich results (see for instance ApJ 848:L12).
One interesting side note is that in fact, the signal in the Virgo detector from GW170817 is not visible in a spectrogram of the event time. The `antenna pattern' of the laser interferometer gravitational-wave detectors is rather smooth, with best sensitivity on average overhead and underfoot, and a sensitivity of one-half for signals arriving along one arm or the other. But a signal arriving at $45^{\circ}$ to the arms will not excite the differential arm lengths -- and the signal source for GW170817 was very close to that null in response for the Virgo antenna. The sky map effectively uses the detectors as a phased array, and the very small signal in Virgo was just as meaningful as a large signal would have been, and thus Virgo played a central role in this story. We are grateful that Nature provided a ‘test injection’ three days earlier to verify that the Virgo instrument was working and that we had the analysis tools for forming the sky map also tested so recently.
The O2 Observing run ended just 13 days later, with this `bang'. Since that time, both Virgo and LIGO have been working to improve the sensitivity of the detectors in preparation for O3, planned to start in early 2019. Virgo expects to make a significant step in sensitivity from 26 to 60 Mpc reach for binary neutron star coalescences, and the two LIGO detectors are expected to reach 120 Mpc reach (from 60 and 100 in O2). With this improvement, and remembering that we measure the amplitude of the GW, the rate of detections by the network is expected to increase significantly. The installation work is nearly finished in Virgo and LIGO, and the commissioning starting up, leaving from August 2018 until the start of the run for chasing noise sources and tuning control systems.
In parallel, the Virgo and LSC teams are preparing for the onslaught of signals expected in O3. Work continues on a generous handful of papers still remaining from the previous runs, in parallel with the development of new systems to provide fully automated low-latency alerts -- every minute counts in alerting the non-GW observing community. A very significant change is that the LIGO and Virgo signals will be sent out as public alerts, consistent with our policy to work toward more engagement of a larger scientific community. For our published papers, we continue to release one hour of data around signals, and are catching up on posting posterior samples along with data behind the figures, again with the hope that there will be a growing group outside of the Collaborations looking at the data.
O3 is expected to last about one calendar year (and with a chance that the Japanese KAGRA instrument will join near the end), followed by further commissioning to reach the sensitivity foreseen for Advanced LIGO and Advanced Virgo by design. O4 will be performed with that sensitivity, and then in the early 2020’s we expect to make another improvement to the instrument sensitivity – in the case of LIGO, to add frequency-dependent `squeezed light' to reduced quantum noise and to install mirrors with lower thermal noise. This `A+' upgrade should yield a factor of 1.5-2 in sensitivity for observing into the 2020’s.
Yet further in the future are plans to introduce mild cryogenics in one or more detectors in the existing observatories, along with other techniques that would be used in a new ‘Third Generation’ observatory and instrument. The Gravitational Wave International Committee (\href{https://gwic.ligo.org}{https://gwic.ligo.org}) is helping to coordinate efforts in Europe and in the US to ensure we have a coherent and robust science case, and that we take advantage of the benefits of a global organization for both technology and governance. A success-oriented schedule suggests that we could have instruments a factor of 10 more sensitive (so an event rate a factor of 1000 greater) than Advanced LIGO and Virgo operating by the mid-2030’s. We hope that will be in conjunction with LISA -- the space-based gravitational-wave antenna -- for both multi-messenger and multi-band science.
We have lost a number of the early key contributors to the field of interferometric detection of gravitational waves in recent times: Ron Drever, Adalberto Giazotto, Albrecht R\"udiger, and Roland Schilling. They all were able to see the first fruit of their efforts, happily, but we miss them all. With that, and with thanks to them and the thousand plus persons who made this story possible --
The future of this new field of gravitational wave observation looks very promising; and, happily, some promises were already fulfilled.
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\section*{\centerline {Mathematics, Physics, and their Interaction}
\centerline {Conference in Honor of} \centerline {Demetrios Christodoulou's 65th Birthday}}
\addtocontents{toc}{\protect\medskip}
\addtocontents{toc}{\bf Conference Reports:}
\addcontentsline{toc}{subsubsection}{
\it Christodoulou Conference, by Lydia Bieri}
\parskip=3pt
\begin{center}
Lydia Bieri, University of Michigan
\htmladdnormallink{lbieri-at-umich.edu}
{mailto:lbieri@umich.edu}
\end{center}
This international conference in honor of Demetrios Christodoulou's 65th birthday was held July 10-14, 2017, at ETH Zurich in Switzerland. It brought together mathematicians, physicists and astrophysicists working in general relativity, astrophysics, fluid dynamics and partial differential equations in general. The realm of talks spanned from experiments to mathematical theory. Around 150 participants attended the conference, including experts as well as junior researchers and students.
Physics and mathematics are interwoven and more than ever interdisciplinary research is required to solve the challenging problems. A culmination of this fact is the theory of general relativity, where not only the Einstein equations govern the laws of the universe but also give them a geometric structure. Demetrios' research reflects this interplay of mathematics and physics in a most beautiful way.
The talks were spread over 5 days. The first day featured a discussion session after the reception in the evening, which was well-attended and interesting new questions were addressed. There was also a conference dinner, during which several of Demetrios' students and collaborators shared interesting and also amusing stories.
General relativity (GR) constituted the largest part of the program. Gilbert Weinstein commenced the talks speaking about harmonic maps with prescribed singularities and applications to general relativity. These maps have shown useful in finding lower bounds on the total mass among other things. Gilbert with M. Khuri and S. Yamada use similar ideas to investigate a class of black holes with non-standard topology in 5 dimensions. Mihalis Dafermos reported on the cosmic censorship conjectures. These conjectures, originally put forth by Roger Penrose, remain among the most central unsolved problems in GR. Whereas the strong cosmic censorship conjecture suggests that for generic
Einstein vacuum initial data, the solution spacetime determined by
initial data cannot be extended as a suitably regular Lorentzian
manifold; the weak cosmic censorship conjecture says that for generic asymptotically flat vacuum data, the resulting vacuum spacetime has a complete
future null infinity, thus no naked singularities should occur. Obviously, there is room to formulate what ``generic" data should be. Christodoulou's work in the 1980s and 1990s led him to a more precise formulation of these conjectures. In particular, studying the collapse of a spherically symmetric self-gravitating scalar field for certain classes of initial data, Christodoulou showed that such naked
singularities may occur, but they are unstable, therefore establishing a proof of a definitive version of cosmic censorship in this case. Dafermos has done work investigating cosmic censorship. In his latest results with J. Luk, they prove that
one can extend the maximal Cauchy evolution of appropriate Einstein vacuum data across a piece of the Cauchy horizon of a Kerr black hole as a Lorentzian manifold with $C^0$-metric. Their work suggests that a $C^0$-version of strong cosmic censorship is false, but that a more precise version due to Christodoulou may still hold, in the sense that the Cauchy horizon is $C^0$-stable but not more, that is, at the same time it will be singular in the sense suggested by the vacuum weak null singularities. Exciting new insights about cosmic censorship was given by Robert Wald.
If an extremal or nearly extremal black hole can be made to absorb matter with sufficiently large angular momentum or charge as compared with its energy, one would obtain an apparent contradiction with cosmic censorship.
Wald presented recent work where no over-charging or over-spinning of a black hole can occur, provided only that the non-electromagnetic matter satisfies the null energy condition, taking into account all second order effects.
Claudio Bunster explored gravitational domain walls and the dynamics of $G$. A. Shadi Tahvildar-Zadeh presented work with M. Kiessling on how to remedy the problem of infinities inherent in Maxwell-Lorentz electrodynamics of point charges, which were identified by Hermann Weyl as the main obstacle on the path to a possible symbiosis of GR and QM.
Shing-Tung Yau talked about his crucial work with P. Chen and M.-T. Wang on quasi-local mass. Let us remind ourselves that there are many important physical quantities and questions that
were understood in Newtonian mechanics, however, their
counterparts in GR are not easy to formulate, let alone to understand. Energy and mass are prominent examples, which are understood for certain classes of spacetimes, but the ultimate insights are still lacking. Yau and collaborators have made important contributions to shed light on this problem. In particular, the Wang-Yau quasi-local mass has good properties and shows fruitful to solve other problems in GR. Recently, Yau and collaborators have extended their work to define quasi-local energy and optimal isometric
embeddings in reference to the de-Sitter (dS) and the Anti-de-Sitter
(AdS) spacetimes. Richard Schoen explained recent work with S.-T. Yau on a proof of the positive energy theorem in higher dimensions. Their famous result of 1979 established positivity of total mass for large classes of $4$-dimensional spacetimes. Ed Witten gave a different proof of positivity. Lately, Schoen and Yau considered cases when the dimension is greater than $8$ and the manifold is not spin. The proof is accomplished by extending the minimal hypersurface approach in the presence of singularities and controlling the singular sets which arise.
Gerhard Huisken reported on various foliations of asymptotically-flat $3$-manifolds
arising as spacelike slices in Lorentzian spacetimes, modeling isolated gravitating systems.
These have proven crucial to understand physical concepts such as (quasi-local) mass, and momenta in a geometric way independent of coordinate systems.
Sergiu Klainerman talked about the monumental work he and Demetrios Christodoulou accomplished in the proof of the nonlinear stability of Minkowski space and the impact it has had on research in GR and in nonlinear wave equations. One main outcome of their work has been a complete understanding of the null asymptotics for physically interesting spacetimes. Based on that knowledge Christodoulou derived the null memory effect of gravitational waves. In a linearized theory, ordinary memory had been known since 1974, when Ya.B. Zel'dovich and A.G. Polnarev computed it. Moreover, Christodoulou's fundamental result of black hole formation of 2008 combined techniques developed in the stability proof with new ones. Mathematically, the methods developed by Christodoulou and Klainerman have proven crucial in many other nonlinear hyperbolic problems. The Christodoulou-Klainerman stability proof was generalized to the Einstein-Maxwell
case by Nina Zipser in 2000; then in 2007 Lydia Bieri generalized it for the Einstein vacuum equations obtaining borderline decay of the data at infinity, and less regularity. There has been a lot of work in various directions, that was initiated by the Christodoulou-Klainerman result.
Hans Lindblad reported about a recent result of his and Martin Taylor proving stability for the massive Einstein-Vlasov system in the harmonic gauge.
Thibault Damour reviewed the theoretical developments on the motion and gravitational radiation of binary black holes that have been crucial in interpreting the LIGO events as being emitted by the coalescence of two black holes. In particular, he presented the effective-one-body (EOB) formalism and how EOB and numerical relativity are put to work to compute templates that have been used to search the first coalescence signals, and to measure the masses and spins of the coalescing black holes.
Lydia Bieri explained the gravitational wave memory effect and gave new insights.
GR predicts that gravitational waves change the spacetime permanently, which results in a permanent displacement of test masses, the memory. For a long time, it was believed that there was only one type of memory, and that what Ya. B. Zel'dovich and A. G. Polnarev found in a linearized theory was the ``linear" version of the ``nonlinear" one that D. Christodulou found in 1991 in the fully nonlinear theory. Lydia Bieri and David Garfinkle showed that these are in fact two different memory effects, the former due to fields that do not reach null infinity, the latter due to fields that do reach null infinity. Bieri, Chen and Yau showed that in the Einstein-Maxwell equations a specific component of the electromagnetic field contributes to the null memory, and Bieri and Garfinkle derived the contribution from neutrino radiation to the null memory in GR. Robert Wald and A.Tolish computed interesting examples of ordinary and null memories. It is interesting to see that there is no memory in higher dimensions. See the works by
Garfinkle, S. Hollands, A. Ishibashi, Tolish and Wald as well as G. Satishchandran and Wald. There has been a wealth of work on gravitational memory and on analogs of memory in other theories. Bieri and Garfinkle for the first time outside of GR derived the two analogs of the memories in electromagnetism for the Maxwell equations. A. Strominger and several collaborators, followed by a large group of researchers, have worked on memory analogs in many other theories. In the cosmological setting, for Einstein-de Sitter the memory is enlarged by a factor involving redshift (Bieri and Garfinkle) and the same holds for Friedmann-Lema\^itre-Robertson-Walker (FLRW) (Tolish and Wald). Bieri, Garfinkle and N. Yunes derived that in a $\Lambda$CDM cosmology with large inhomogeneities the memory in the cosmological zone is multiplied by a factor involving not only redshift but also weak gravitational lensing. It is believed that gravitational wave memory will be detected in the near future. See the result by P.D. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen who suggest to detect memory with LIGO by stacking black hole merger events.
Barry Barish reported on the major results and perspectives of LIGO. A major breakthrough of GR happened in 2015 with LIGO's first detection of gravitational waves. Since then several events have been recorded by the detectors. This marks the beginning of a new era where we get information directly from the universe itself. The improved LIGO and future detectors will give a much deeper view into the universe.
Further on cosmology, Ruth Durrer explained that cosmology cannot really be formulated without GR. She gave new insights on the cosmic microwave background (CMB), weak lensing and weak lensing of the CMB. She also explained how planned high precision CMB polarization experiments will be able to measure effects of frame dragging on cosmological scales. David Spergel shared his insights on how to measure the geometry and topology of the universe with CMB observations. In particular, he discussed how precision measurements of the CMB constrain the universe's shape and show that the universe is nearly flat and very large.
As we know, many interesting mathematical and physical features occur in other theories.
Wilhelm Schlag reported on longterm dynamics of dispersive evolution equations. In particular, he talked about
developments dealing with the description of asymptotic states of solutions to semilinear evolution equations, and new results on damped subcritical Klein-Gordon equations.
Joachim Krieger spoke about results on regularity and singularity formation of critical wave maps and the role that by now classical work by Christodoulou-Shatah -Tahvildar-Zadeh played in their genesis.
Going further, the half-wave maps equation is a novel geometric evolution equation, which displays many intriguing mathematical properties with links to minimal surfaces, conformal symmetry, and completely integrable systems.
Enno Lenzmann described how the motivation for this evolution equation comes from exactly solvable quantum systems (Haldane-Shastry models) and completely intergable Calogero-Moser spin systems. He then presented new results of his with P. G\'erard and A. Schikorra about the explicit classification of traveling solitary waves together with their complete spectral analysis.
Carlos Kenig explained the energy channels method
and some of its applications to study singularity formation for nonlinear wave equations.
Sijue Wu talked about water waves modeled by the Euler equations under appropriate assumptions.
Another field of Christodoulou's research combining mathematics and physics are the Euler equations.
They form the core part of many fluid models in endless applications. Within mathematics, the Euler equations belong to one of the hot topics and big challenges of modern research. One would like to understand the dynamics, singularities or stability properties of the solutions. These questions are tackled in the framework of the Cauchy problem, where for certain types of initial data and conditions one would like to solve the equations (locally or globally in time) and derive a full description of the solutions. Among the many applications, water waves are described by the Euler equations for an incompressible, inviscid and
irrotational fluid with air density zero.
Sijue Wu reported on recent results on
$2$-dimensional water waves with angled crests. In particular, she explained the
well-posedness of the Cauchy problem including water waves with non-$C^1$ interfaces.
Jalal Shatah gave exciting insights on weak turbulence.
Andrew Majda talked about
low-dimensional reduced-order models for statistical response and uncertainty quantification in turbulent systems.
Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering including climate, material, and neural science.
The nature of the problem is such that there is a rapid growth of small uncertainties from imperfect modeling equations or perturbations in initial values, requiring naturally a probabilistic characterization for the evolution of the turbulent system.
There are many challenges to overcome. Majda discussed a general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems.
Zhouping Xin explained recent results on transonic shocks in curved nozzles. In particular,
he discussed some steady compressible flows in nozzles with variable cross sections. To be precise, he considered
a nonlinear free boundary value problem with nonlinear boundary conditions for mixed type equations and discussed the existence of single and multiple transonic shocks in terms of the geometry of the nozzle and the given exit pressure.
Many nonlinear partial differential equations (PDE) arising in mechanics, geometry, and other areas naturally are of mixed elliptic-hyperbolic type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others.
Gui-Qiang Chen presented natural connections of nonlinear PDE of mixed elliptic-hyperbolic type with these longstanding problems and discussed some of the most recent developments in the analysis of these nonlinear PDE. He also gave ideas on developing and identifying mathematical approaches, ideas, and techniques for dealing with the mixed-type problems.
Domenico Giulini reported on aspects of $3$-manifold theory motivated by GR.
Many speakers shared personal stories on how they have interacted with Demetrios, some of which were most amusing. And Demetrios himself added a few more, among them he explained how working on problems in physics made him become a mathematician. In particular, John Wheeler, his PhD thesis advisor, gave him the following problem in 1968: the formation of black holes in pure GR, by the focusing of incoming gravitational waves. The more Demetrios studied this problem, the more he realized that geometry and PDE play a crucial role, and that the general realm is far more intricate.
Demetrios then added, smiling, that he solved this problem in full details in 2008; Wheeler had changed the topic of Christodoulou's thesis, so that the latter received his PhD in 1971. Demetrios' 2008 black hole formation result is a masterpiece of physics and mathematics, relying in particular on physical insights combined with new and deep geometric-analytic methods. Someone in the audience pointed out that, being a mathematician and a physicist, Demetrios was indeed quite ``interdisciplinary".
This conference combined most recent research and deep insights from many highly-active fields, and it sparked new ideas for future research. The participants enjoyed the vibrant discussions within and across the different topics. As an especially nice aspect of this conference emerged the interdisciplinary interactions.
The conference website can be found at
\href{https://www.math.ethz.ch/fim/conferences/past-conferences/2017/christodoulou.html}
{https://www.math.ethz.ch/fim/conferences/past-conferences/2017/christodoulou.html}
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\section*{\centerline
{Stephen Hawking (1942-2018)}}
\addtocontents{toc}{\protect\medskip}
\addtocontents{toc}{\bf Obituaries:}
\addcontentsline{toc}{subsubsection}{
\it Stephen Hawking (1942-2018), by Robert M. Wald}
\parskip=3pt
\begin{center}
Robert M. Wald, University of Chicago
\htmladdnormallink{rmwa-at-uchicago.edu}
{mailto:rmwa@uchicago.edu}
\end{center}
Stephen Hawking died peacefully in his home in Cambridge, England on March 14, 2018. His life story of making truly groundbreaking discoveries while succumbing to ALS---and his continuing to lead an extremely active and productive life for more than three decades after he completely lost the ability to speak and was nearly totally paralyzed---will be well known to most readers. Further information about Hawking's life is readily available from many sources, including the movie ``The Theory of Everything,'' for which Eddie Redmayne deservedly received the best actor Academy Award for his portrayal of Hawking. I will therefore not attempt to review anything about his life here. I also will not attempt to summarize his truly major contributions to the singularity theorems, the classical theory of black holes, quantum cosmology, and many other areas. However, I would like to make some remarks on his most famous work: the discovery of thermal particle creation by black holes.
I recently carefully reread his paper ``Particle Creation by Black Holes'' (Commun. Math. Phys. {\bf 43}, 199-220 (1975)) in connection with the closing remarks I gave at Hawking's 75th birthday conference in July, 2017. Everyone knows that this is a groundbreaking paper, but all of the reasons why it is such a truly amazing paper may not be as well known. There are many instances of great discoveries in science where the author makes a bold new hypothesis that miraculously turns out to be right. It is all the more remarkable that this is {\em not} such an example. All of Hawking's results in this paper are derived from general relativity and the basic principles of quantum field theory by careful reasoning and calculations, with no room for any additional hypotheses.
A static (``eternal'') Schwarzschild black hole spacetime is relatively simple to consider on account of its time translation symmetry. However, a past event horizon is present in this spacetime, and its presence introduces an ambiguity in the initial conditions for a quantum field. To avoid this ambiguity, Hawking considered instead a spacetime in which gravitational collapse to a Schwarzschild black hole occurs. This removes the ambiguity---and the gravitational collapse spacetime is much more physically relevant to consider in any case---but the lack of time translation symmetry would seem to make intractable the problem of solving for the dynamical evolution of a quantum field. However, Hawking realized that if one considers the {\em backward in time} evolution, a remarkable simplification occurs: A wave that passes very close to the horizon when evolved backward in time gets highly blueshifted, and the geometric optics approximation can be used to evolve it through the dynamical region.
Particle creation will occur when dynamical evolution results in the mixing of positive and negative frequencies. Thus, the key calculation in the paper consisted of starting with a wave that is purely positive frequency with respect to Killing time in the future and determining its positive and negative frequency parts in the past. In view of the geometric optics behavior, Hawking realized that this amounts to determining the relationship between positive frequency with respect to Killing time and positive frequency with respect to affine time near the horizon of a black hole. Hawking then calculated the positive and negative frequency parts with respect to affine time of a wave with Killing time frequency $\omega$. The form of this expression is not very illuminating except that at the relevant high frequencies the positive and negative frequency parts are very closely related, with the only difference involving a factor containing the logarithm of the affine frequency. Using an analyticity argument, Hawking the showed that the magnitudes of positive an negative parts are simply related by a factor of $\exp(\pi \omega/\kappa)$, where $\kappa$ is the surface gravity of the black hole. This fact is sufficient to calculate the expected number of created particles. However, this number is infinite. Nevertheless, by re-doing the calculations considering wave packets that are nearly of frequency $\omega$ but are localized in time, Hawking showed that this infinity is actually physical---the result of a steady rate of particle creation over all (late) times. When this finite, steady flux of created particles is calculated, the square of the above exponential factor comes in and corresponds precisely to a Boltzmann factor, yielding the astounding result that black holes emit thermal radiation at temperature $T= \kappa/2 \pi$. As Hawking then pointed out, this is just what is needed for the consistency of black hole thermodynamics.
If the paper ended there, it would easily have ranked as one of the most remarkable papers in physics written in the second half of the 20th century. But, in my view, what makes this paper so truly amazing is that Hawking was not content to stop there:
First, Hawking had done the above calculation for a massless scalar field. He then argued that (despite some nontrivial differences) the calculation would continue to hold for electromagnetic and linearized gravitational fields. He then showed that it would also continue to hold for massless Fermi-Dirac fields, but that there would be a sign change because of the nature of the Dirac product versus the Klein-Gordon product, so that one would end up with a Fermi-Dirac distribution rather than a Bose-Einstein distribution. Finally, he argued that the results also would hold for non-zero rest mass. So, one truly gets thermal emission of {\em all} species of particles.
Next, the above calculation had been done for a spherically symmetric spacetime, which is exactly Schwarzschild outside of the collapsing matter. Hawking then argued that, in fact, the late time radiation depends only on the asymptotic final state of the black hole, not the details of the collapse, so his results hold for any collapse that settles down to a Schwarzschild final state. This is a nontrivial argument that occupies about a page of the paper.
Having argued that the late time radiation depends only on the black hole final state, Hawking then derives what the particle creation would be if the final state were a Kerr black hole. This is a very nontrivial generalization on account of superradiance, but Hawking does this in one paragraph. What about charged fields if the final state were a Reissner-Nordstrom black hole? A similar superradiance behavior occurs there, and Hawking shows that this has a similar effect on particle creation.
Hawking then goes on to consider back-reaction of particle creation on the black hole. This was completely new territory. Hawking first argued that an observer freely falling into the black hole should not see any large local effects---thereby making this paper the first anti-firewall paper ever written. He then argued that if there are no large local effects near the horizon, then there must be a flux of negative energy into the black hole corresponding to the positive energy flux at infinity. But this, he argued, would cause the black hole to lose mass and evaporate within a finite time. He then drew a spacetime diagram of an evaporating black hole---a diagram that has been redrawn in literally thousands of papers.
Are there any flaws or errors in the paper? I do not find Hawking's heuristic picture of particle creation involving tunneling to be a good description of the phenomenon, but, as Hawking clearly says, this is just a heuristic picture that is not used to derive any conclusions, so one cannot even call this a flaw. Hawking believed at the time that he wrote that paper that the ambiguities inherent in the definition of ``particles'' near the black hole would give rise to ambiguities in defining the quantum field stress-energy tensor; that the quantum stress energy tensor would therefore have a status similar to that of a gravitational stress-energy pseudotensor; and that therefore one could talk meaningfully only about its averaged/global effects. This turned out not to be the case---the quantum stress-energy tensor is a perfectly good quantum field observable. However, the only effect of Hawking's cautious treatment of the quantum stress-energy tensor was to make his arguments on back-reaction more awkward and difficult to make; all of the arguments are correct. Thus, after careful scrutiny of the paper 43 years after it was written, the only genuine errors I have been able to find are that the word ``gauge'' is misspelled in 4 places, and the name ``Bekenstein'' also is misspelled.
It is a truly amazing paper, written by an even more amazing person!
\vfill\eject
\section*{\centerline
{Joseph Polchinski (1954-2018)}}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it Joseph Polchinski (1954-2018), by Gary Horowitz}
\parskip=3pt
\begin{center}
Gary Horowitz, University of California, Santa Barbara
\htmladdnormallink{gary-at-physics.ucsb.edu}
{mailto:gary@physics.ucsb.edu}
\end{center}
Theoretical physicists lost one of their most brilliant colleagues when Joe Polchinski passed away on February 2, 2018, after a two year battle with brain cancer. Joe was a high energy theorist who made fundamental contributions to quantum field theory and string theory. He literally ``wrote the book" on string theory, publishing a widely used two volume set on the subject in 1998. But for gravitational physicists, he is best known for two contributions that directly apply to black holes.
The first contribution started with a discovery that Joe made (with two graduate students) that, contrary to widespread belief, string theory is not just a theory of strings \cite{Dai:1989ua}. There are also higher dimensional extended objects he called D-branes. (The D stands for Dirichlet, and refers to the boundary conditions at the ends of the string.) These objects had been missed earlier because they are nonperturbative, having a tension that is proportional to $1/g$ where $g$ is the string coupling constant. Joe later showed that these objects carry charges \cite{Polchinski:1995mt} and are closely related to the extended charged black holes (or ``black branes") that Andy Strominger and I had found a few years earlier \cite{Horowitz:1991cd}. More precisely, D-branes can be viewed as the source of the extremal limit of these objects. By combining these D-branes in the right way, Andy and Cumrun Vafa were able to count the microstates of an extremal black hole for the first time, and reproduce the Hawking-Bekenstein entropy \cite{Strominger:1996sh}.
Joe's D-branes turned out to have many other applications as well, and led directly to the discovery of gauge/gravity duality which has dominated work in string theory for the past two decades.
In 2012, Joe touched off an explosion of interest in the black hole information puzzle. Working with Don Marolf and two graduate students, he extended earlier arguments by Mathur \cite{Mathur:2009hf} and showed that three widespread beliefs about black holes were inconsistent \cite{Almheiri:2012rt}. One cannot maintain that nothing happens to an observer that falls into a large black hole, that black hole evaporation is a unitary process, and that ordinary local quantum field theory holds outside the horizon of a macroscopic black hole. They raised the possibility that one might have to give up the first belief and suggested there might be a ``firewall" near the horizon of a black hole that has evaporated down to half its original size. Although this suggestion remains controversial, it has stimulated many new ideas which are being pursued today.
Joe was an avid bicyclist and would often lead visitors to the KITP on strenuous rides up the mountains behind Santa Barbara. As our chancellor said at the time of his death: ``Joe was known for climbing mountains, both intellectual and literal. Many of our colleagues fondly remember epic bike rides with him to the top of Gibraltar and Old San Marcos Pass, discussing life and physics all the way. He set his sights high and navigated fearlessly over all obstacles in order to achieve the extraordinary, encouraging and inspiring others along the way."
Joe was a wonderful colleague and friend. He gave generously of his time to anyone who asked: students, postdocs, and colleagues. He will be deeply missed.
\vskip 1cm
\bibliographystyle{JHEP}
|
1,314,259,996,677 | arxiv | \section{Introduction}
\label{intro-sect}
For conjunctive queries, the query containment problem is NP-complete~\cite{Chandra77-1}.
When we have constants that are numbers (e.g., they may represent prices, dates, weights, lengths, heights) then, often, we want to compare them by checking, e.g., whether two numbers are equal or whether one is greater than the other, etc.
To reason about numbers we want to have a more expressive language than conjunctive queries and, thus, we add arithmetic comparisons to the definition of the query.
We know that the query containment problem for conjunctive queries with arithmetic comparisons is $\Pi^p_2$-complete~\cite{Klug88-1,vdmeyden92,Kolaitis98-1}.
In previous literature \cite{WangTM05,Afrati04-1,AfratiLM06,Afrati06}, it has been noticed that there are classes of CQACs for which the query containment problem remains in NP and these classes can be syntactically characterized. In this paper we find much broader such classes of queries.
Query containment and many problems in answering queries using views are closely related.
In particular, in the framework of answering queries using views, we want to find all certain answers of the query on a given view instance, i.e., all the answers that are provable ``correct.''
A popular way for answering queries using views is by finding rewritings of the query in terms of the views that are contained in the query. There may exist many contained rewriting in a certain query language. We want to find the maximal contained rewriting (MCR for short) that contains all the rewritings, if there exists such a rewriting.
The main results in this paper are the following:
{\bf Query Containment} We solve an open problem mentioned in \cite{Afrati19} by extending significantly the class of CQAC queries that admit an NP containment test.
As concerns closed arithmetic comparisons, we think we are close to the boundary between the problem being in NP and being in $\Pi^p_2$.
The class of queries we consider includes (but is broader than) the following case: The contained query is allowed to have any closed arithmetic comparisons. The containing query is allowed to have any closed arithmetic comparisons that involve the head variables, but not between a head variable and a body variable. Moreover, the comparisons that are allowed in the body variables are the following: Several left semi-interval arithmetic comparisons and at most one right semi-interval arithmetic comparison.
This result is proven via a transformation of the queries to a Datalog query (for the containing query) and a conjunctive query (for the contained query) and reducing checking containment between these two. This result captures all results in \cite{Afrati19} but in a new way that allows us to further use the transformation to compute MCRs and certain answers in the framework of the problem of answering queries using views.
{\bf MCRs} We extend the results in \cite{Afrati04-1} and prove that we can find an MCR in the language of Datalog with arithmetic comparisons in the case where the query has the restrictions of the containing query above and the views use any closed ACs, except ACs between the head and non-head variables. In \cite{Afrati04-1}, only semi-interval arithmetic comparisons were allowed in the query and in the views.
{\bf Computing certain answers} We show for the first time how to compute certain answers in polynomial time using MCRs for the case the conjunctive queries have arithmetic comparisons.
\vspace*{.2cm}
In proving the above main results, we needed to prove intermediate results, which could be extended beyond what was necessary for proving the main results. Those intermediate results are the
following:
a) We defined the class of semi-monadic Datalog queries which include the class of monadic queries. We proved that checking containment of a conjunctive query to a
semi-monadic Datalog query is NP-complete.
b) We proved that if there is an MCR in the language of (possibly infinite) conjunctive queries with arithmetic comparisons, then, even in the presence of dependencies, we can use this MCR to compute all certain answers for any CQAC query and any CQAC views.
The structure of the paper is as follows:
Section 5 proves the result about query containment. The proofs in this section need technical results about the implication problem of arithmetic comparisons that are presented in the early subsections of Section 5. They also need a result about query containment of a conjunctive query to a semi-monadic Datalog query. This is presented in Appendix C. Moreover, in Appendix B, we present the proof of the main technical result of Section 5.
Sections~\ref{sec-6} and~\ref{subsec-mcr-datalogAC} consider the framework of answering queries using views.
Section~\ref{subsec-mcr-datalogAC} uses the findings of Section 5 to compute an MCR using the language Datalog$^{AC}$.
Section~\ref{sec-6} shows the relation between the output after computing an MCR on a view instance and the output after computing the certain answers on the same instance.
Appendix E starts a discussion on a new direction concerning unclean data and MCRs.
We include Appendices A and D, for reasons of completeness.
\subsection{Related work }
\textit{CQ and CQAC containment:} The problem of containment between conjunctive queries (CQs, for short) has been studied in \cite{Chandra77-1}, where the authors show that the problem is NP-complete, and the containment can be tested by finding a containment mapping. As we already mentioned, considering CQs with arithmetic comparisons (CQACs), the problem of query containment is $\Pi^p_2$-complete \cite{vdmeyden92}.
Zhang and Ozsoyoglu, in \cite{Zhang93-1}, showed that the testing containment of two CQACs can be done by checking the containment entailment.
Kolaitis et al. \cite{Kolaitis98-1} studied the computational complexity of the query-containment problem of queries with disequations ($\neq$). In particular, the authors showed that the problem remains $\Pi^p_2$-hard even in the cases where the acyclicity property holds and each predicate occurs at most three times. However, they proved that if each predicate occurs at most twice then the problem is in coNP.
Karvounarakis and Tannen, in \cite{karvounarakis2008conjunctive}, also studied CQs with disequations ($\neq$) and identified special cases where query containment can be tested through checking for a containment mapping (i.e., the containment problem for these cases is NP-complete).
The homomorphism property for query containment of conjunctive queries with arithmetic comparisons was studied in \cite{Klug88-1,Zhang94-4,Afrati04-1,Afrati19}.
In \cite{Afrati04-1}, Afrati et al. investigated cases where the normalization step is not needed. They also identified classes of CQACs where the homomorphism property holds.
In \cite{Afrati19}, Afrati showed that the problem of containment of two CQACs such that the homomorphism property holds is in NP. This work also identifies certain classes of CQACs where the homomorphism property holds.
The containment of certain subclasses of CQACs that the homomorpshim property holds are also identified in \cite{Zhang94-4}.
\vspace{10px}
\textit{Datalog containment:} Since one of our results concerns monadic Datalog queries and the containment problem we briefly list some related results.
Although to test the containment of two Datalog queries is undecidable \cite{Shmueli93-1}, the containment of a CQ in a Datalog query is decidable.
In the general case of non-monadic Datalog query, the problem of containment of a CQ in a recursive Datalog query is EXPTIME-complete \cite{Cosmadakis86-1, Chandra81-1, Sagiv88-1}. As \cite{Cosmadakis88-1} shows, the containment between monadic Datalog queries is decidable. In \cite{Chaudhuri92-1}, the containment problem of a Datalog query in a conjunctive query is proven to be doubly exponential.
The containment of a CQ in a linear monadic Datalog (i.e., each rule has at most one IDB)
is NP-complete~\cite{Chaudhuri94-1}. In this work, we extend this result for any monadic Datalog query (exdending it also to a wider class, called semi-monadic Datalog queries). Recent work on containment problem for monadic Datalog includes \cite{BenediktBS12}.
\vspace{10px}
\textit{Rewritings and finding MCRs:} The problem of answering queries using views has been extensively investigated in the relevant literature (e.g., \cite{Levy95-1, Levy00-1, 2017Afrati}); including finding equivalent and contained rewriting. Algorithms for finding maximally contained rewritings (MCRs) have also been studied in the past \cite{Abiteboul98-1, Grahne99-1, Levy96-1, Pottinger00-1, Mitra99-1, Duschka97-3, Afrati99-1, AfratiLM06}. The authors in \cite{Pottinger00-1} and \cite{Mitra99-1} propose two algorithms, the Minicon and shared variable algorithm, respectively, for finding MCRs in the language of unions of CQs when both queries and views are CQs. \cite{Pottinger00-1} also considers restricted cases of arithmetic comparisons (LSI and RSIs) in both queries and views. In \cite{Abiteboul98-1}, the queries have inequalities ($\neq$), while the views are CQs. As it is also proven in this work, the data complexity of finding certain answers is co-NP hard. The works in \cite{Duschka97-3} and \cite{Afrati99-1} studied the problem where the query is given by a Datalog query, while the views are given by CQs and union of CQs, respectively. In both papers, the language of MCRs is Datalog. The authors in \cite{CaoFGL18} studied the problem of finding MCRs in the framework of bounded query rewriting. They investigated several query classes, such as CQs, union of CQs, and first order queries, and analyzed the complexity in each class. Afrati and Kiourtis in \cite{AfratiK10} proposed an efficient algorithm that finds MCRs in the language of union of CQs in the presence of dependencies. The work in \cite{AfratiLM06} investigated the problem of finding MCRs for special cases of CQACs.
\vspace{10px}
\textit{Certain answers and MCRs:} The problem of finding certain answers has been extensively investigated in the context of data integration and data exchange, the last 20 years (e.g., \cite{AndritsosFFHHHKMNPVVY02, Abiteboul98-1, Grahne99-1, fagin2005data, AfratiK10, KonstantinidisA13}). In \cite{Grahne99-1, fagin2005data}, the authors investigated the problem of finding certain answers in the context of data exchange, considering CQs. The work in \cite{fagin2005data} was extended for arithmetic and linear arithmetic CQs in \cite{CateKO13}, where the authors proved that the problem of finding certain answers into target schema is co$\exists$R-complete in data complexity.
In \cite{Abiteboul98-1}, the authors investigated the relationship between MCRs and certain answers.
In \cite{AfratiK10}, the authors proved that an MCR of a union CQs computes all the certain answers, where MCR is considered in the language of union of CQs.
In many of the works about finding certain answers the chase algorithm is used as a tool.
Recent studies of the chase in the framework of query containment and data integration include \cite{BenediktKMMPST17} and \cite{KonstantinidisA14}.
\vspace{10px}
\textit{Other work with arithmetic comparisons in queries:} As concerns studying other related problems in the presence of arithmetic comparisons recent work can be found
in \cite{FanLLT18}, where the authors propose to extend
graph functional dependencies with linear arithmetic expressions
and arithmetic comparisons. They study the problems of testing satisfiability and related problems over integers (i.e., for non-dense orders).
A thorough study of the complexity of the problem of evaluating conjunctive queries with inequalities ($\neq$) is done in \cite{KoutrisMRS15}. In \cite{PapadimitriouY97} the complexity of evaluating conjunctive queries with
arithmetic comparisons is investigated for acyclic queries, while query containment for acyclic conjunctive queries was investigated in \cite{ChekuriRaj97}.
Recent works \cite{CateKO13,AfratiLP08} have added arithmetic to extend the expressiveness of tuple generating dependencies and data exchange mappings, and studied the complexity of related problems. Queries with arithmetic comparisons on incomplete databases are considered in \cite{ConsoleHL20}.
\section{Preliminaries}
A \textit{relation schema} is a named relation defined by its name (called \textit{relation name} or \textit{relational symbol}) and a vector of attributes. An \textit{instance} of a relation schema is a collection of tuples with values over its attribute set. These tuples are called {\em facts}. The schemas of the relations in a database constitute its \textit{database schema}. A relational \textit{database instance} (database, for short) is a collection of stored relation instances.
A {\em conjunctive query (CQ in short)} $Q$ over a database schema ${\cal S}$ is a query of
the form: $h( \overline{X})\ :-\ e_1( \overline{X}_1),\ldots, e_k( \overline{X}_k)$,
where $h( \overline{X})$ and $ e_i( \overline{X}_i)$ are atoms, i.e.,
they contain a relational symbol (also called \textit{predicate} - here, $h$ and $e_i$ are predicates) and a
vector of variables and constants.
The atoms that contain only constants are called \textit{ground} atoms and they represent \textit{facts}.
The {\em head} $h(\overline{X})$, denoted $head(Q)$,
represents the results of the query, and $e_1 \ldots e_k$ represent
database relations (also called base relations) in ${\cal S}$.
The variables in $\overline{X}$
are called \textit{head} or \textit{distinguished} variables,
while the variables in $\overline{X}_i$ are called \textit{body} or \textit{nondistinguished}
variables of the query.
The part of the conjunctive query on the right of symbol $:-$ is called the {\em body} of the query and is denoted $body(Q)$.
Each atom in the body of a conjunctive query
is said to be a {\em subgoal}.
A conjunctive query is said to be
\textit{safe} if all its distinguished variables also occur in its
body. We only consider safe queries here.
The \textit{result} (or \textit{answer}), denoted $Q(D)$, of a CQ $Q$ when it is applied on a database instance $D$ is the set of atoms such that for each assignment $h$ of variables of $Q$ that makes all the atoms in the body of $Q$ true the atom $h(head(Q))$ is in $Q(D)$.
{\em Conjunctive queries with arithmetic comparisons (CQAC for short)} are conjunctive queries that, besides the
{\em ordinary} relational subgoals, use also builtin subgoals that are arithmetic comparisons (AC for short), i.e., of the form
$X\theta Y$ where $\theta$ is one of the following: $<, >, \leq, \geq, =, \neq$. Also, $X$ is a variable and
$Y$ is either a variable or constant. If $\theta$ is either $<$ or $>$ we say that it is an open arithmetic comparison
and if $\theta$ is either $\leq$ or $\geq$ we say that it is a closed AC. If the AC is either of the form $X<c$ or $X\leq c$ (respectively, either $X>c$ or $X\geq c$, resp.), where $X$ is a variable and $c$ is a constant, then it is called \textit{left semi-interval}, LSI for short (\textit{right semi-interval}, RSI for short, respectively).
In the following, we use the notation $Q=Q_0+\beta$ to describe a CQAC query $Q$, where
$Q_0$ are the relational subgoals of $Q$ and $\beta$ are the arithmetic comparison
subgoals of $Q$. We define the {\em closure} of a set of ACs to be all the ACs that are implied by this set of ACs.
The result $Q(D)$ of a CQAC $Q$, when it is applied on a database $D$, is given by taking all the assignments of variables (in the same fashion as CQs) such that the atoms in the body are included in $D$ and the ACs are true. For each assignment where these conditions are true, we produce a fact in the output $Q(D)$.
All through this paper, we assume the following setting without mentioning it again.
\begin{enumerate}
\item Values for the arguments in the arithmetic comparisons are
chosen from an infinite, totally densely ordered set, such as the
rationals or reals.
\item The arithmetic comparisons are not contradictory (or, otherwisie, we say that they are consistent); that is,
there exists an instantiation of the variables such that all the
arithmetic comparisons are true.
\item All the comparisons are safe, i.e., each variable in the
comparisons also appears in some ordinary subgoal.
\end{enumerate}
A \textit{union of CQs} (resp. CQACs) is defined by a set ${\cal Q}$ of CQs (resp. CQACs) whose heads have the same arity, and its answer ${\cal Q}(D)$ is given by the union of the answers of the queries in ${\cal Q}$ over the same database instance $D$; i.e., ${\cal Q}(D)=\bigcup_{Q_i\in{\cal Q}}Q_i(D)$.
A query $Q_1$ {\em is contained} in a query $Q_2$, denoted
$Q_1 \sqsubseteq Q_2$, if for any database $D$ of the base
relations, the answer computed by $Q_1$ is a subset of the answer
computed by $Q_2$, i.e., $Q_1(D) \subseteq Q_2(D)$. The two
queries are {\em equivalent}, denoted $Q_1 \equiv Q_2$, if $Q_1
\sqsubseteq Q_2$ and $Q_2 \sqsubseteq Q_1$.
A {\em homomorphism} $h$ from a set of relational atoms ${\cal A}$ to another set of relational atoms ${\cal B}$ is a mapping
of variables and constants from one set to variables or constants of the other set that maps
each variable to a single variable or constant and each constant to the same constant.
Each atom of the former set should map to an atom of the latter set with the same relational symbol. We also say that the homomorphism $h'$ from a set ${\cal A}'\supseteq{\cal A}$ is an \textit{extension} of $h$ if for each variable or constant $x$ in ${\cal A}'\cap{\cal A}$ we have $h'(x)=h(x)$.
A {\em containment mapping} from a conjunctive query $Q_1$ to a conjunctive query $Q_2$ is a homomorphism from the atoms in the body of $Q_1$ to the atoms in the body of $Q_2$ that maps
the head of $Q_1$ to the head of $Q_2$. All the mappings we refer to in this paper are containment mappings unless we say otherwise. Chandra and Merlin \cite{Chandra77-1} show that a conjunctive query $Q_1$
is contained in another conjunctive query $Q_2$ if and only if
there is a containment mapping from $Q_2$ to $Q_1$. The query containment problem for CQs is NP-complete.
\subsection{Testing query containment for CQACs}
\label{subsec:canonical-dbs}
In this section, we describe two tests for CQAC query containment; using containment mappings and using canonical databases.
In the rest of the paper, we denote with $Q_1=Q_{10}+\beta_1$ and
$Q_2=Q_{20}+\beta_2$ the containing and contained query, respectively, where $Q_{10}$ denotes the relational atoms in the body of $Q_1$ and $\beta_1$ denotes the ACs. Similarly for query $Q_2$.
First, we present the test using containment mappings (see, e.g., in \cite{2017Afrati}). Although finding a single containment mapping suffices to test query containment for CQs (see the previous section), it is not enough in the case of CQACs. In fact, all the containment mappings from the containing query to the contained one should be considered. Before we describe how containment mappings can be used in order to test query containment between two CQACs, we define the concept of normalization of a CQAC.
\begin{definition}
\label{dfn-normalization}
Let $Q_1$ and $Q_2$ be two conjunctive queries with arithmetic
comparisons (CQACs). We want to test whether $Q_2
\sqsubseteq Q_1$. To do the testing, we first normalize
each of $Q_1$ and $Q_2$ to $Q'_1$ and $Q'_2$, respectively. We {\em normalize} a CQAC query as
follows:
\begin{itemize}
\item For each occurrence of a shared variable $X$ in a normal (i.e., relational) subgoal,
except for the first occurrence, replace the occurrence of $X$ by a fresh
variable $X_i$, and add $X = X_i$ to the comparisons of the
query; and
\item For each constant $c$ in a normal subgoal, replace the constant by a
fresh variable $Z$, and add $Z = c$ to the comparisons of the
query.
\end{itemize}
\end{definition}
Theorem~\ref{thm:cont-CQAC}\cite{Gupta94-1,Zhang94-4} describes how we can test the query containment of two CQACs using containment mappings.
\begin{theorem}
\label{thm:cont-CQAC}
Let $Q_1,Q_2$ be CQACs, and $Q'_1=Q'_{10}+\beta'_1 ,Q'_2=Q'_{20}+\beta'_2$
be the respective queries after normalization.
Suppose there is at least one containment mapping from $Q'_{10}$ to $Q'_{20}$.
Let $\mu_1, \ldots , \mu_k $ be all
the containment mappings from $Q'_{10}$ to $Q'_{20}$. Then $Q_2
\sqsubseteq Q_1$ if and only if the following logical implication $\phi$
is true:
$$\phi: \beta'_2 \Rightarrow \mu_1(\beta'_1) \vee \cdots \vee
\mu_k(\beta'_1).$$
(We refer to $\phi$ as the {\em containment entailment} in the rest of this paper.)
\end{theorem}
The following theorem says that, if the CQACs have only closed ACs, then normalization is not necessary. For the proof see, e.g., \cite{2017Afrati}.
\begin{theorem}
\label{thm:denorm}
Consider two CQAC queries, $Q_1=Q_{10}+\beta_1$ and
$Q_2=Q_{20}+\beta_2$ over densely totally ordered domains. Suppose $\beta_1$
contains only $\leq$ and $\geq $, and each of $\beta_1$ and
$\beta_2$ does not imply any ``='' restrictions. Then $Q_2 \sqsubseteq
Q_1$ if and only if
$$\phi:\beta_2 \Rightarrow \mu_1(\beta_1) \vee \cdots \vee \mu_l(\beta_1),$$
where $\mu_1,\ldots,\mu_l$ are all the containment mappings from
$Q_{10}$ to $Q_{20} $.
\end{theorem}
As mentioned in the beginning of this section, there is another containment test for CQACs, which uses \textit{canonical databases} (see, e.g., in \cite{2017Afrati}). Considering a CQ $Q$, a canonical database is a database instance constructed as follows. We consider an assignment of the variables in $Q$ such that a distinct constant which is not included in any query subgoal is assigned to each variable. Then, the ground subgoals produced through this assignment define a canonical database of $Q$. Note that although there is an infinite number of assignments and canonical databases, depending on the constants selection, all the canonical databases are isomorphic; hence, we refer to such a database instance as the canonical database of $Q$. To test, now, the containment $Q_2\sqsubseteq Q_1$ of the CQs $Q_1$, $Q_2$, we compute the canonical database $D$ of $Q_2$ and check if $Q_2(D)\subseteq Q_1(D)$.
Extending the test using canonical databases to CQACs, a single canonical database does not suffice. We construct a canonical database of a CQAC $Q_2$ with respect to a CQAC $Q_1$ as follows. Consider the set $S=S_V\cup S_C$ including the variables $S_V$ of $Q_2$, and the constants $S_C$ of both $Q_1$ and $Q_2$. Then, we partition the elements of $S$ into blocks such that no two distinct constants are in the same block. Let ${\cal P}$ be such a partition; for each block in the partition ${\cal P}$, we equate all the variables in the block to the same variable and, if there is a constant in the block, we equate all the variable to the constant. For each partition ${\cal P}$, we create a number of {\em canonical databases}, one for each total ordering on the variables and constants that are present (after we have equated appropriately, as explained).
Although there is an infinite number of canonical databases, depending of the constants selected, there is a bounded set of canonical databases such that every other canonical database is isomorphic to one in this set.
Such a set is referred as \textit{the set of canonical databases} of $Q_2$ w.r.t. $Q_1$. To test now the containment $Q_2\sqsubseteq Q_1$ of the CQACs $Q_1$, $Q_2$, we construct all the canonical databases of $Q_2$ w.r.t. $Q_1$ and, for each canonical database $D$, we check if $Q_2(D)\subseteq Q_1(D)$.
\begin{theorem}
A CQAC query $Q_2$ is contained into a CQAC query $Q_1$ if and only if, for each database belonging to the set of canonical databases of $Q_2$ with respect to $Q_1$, the query $Q_1$ computes all the tuples that $Q_2$ computes if applied on it.
\end{theorem}
\subsection{Answering queries using views}
A \textit{view} is a named query which can be treated as a regular relation. The query defining the view is called \textit{definition} of the view (see, e.g., in \cite{2017Afrati}).
Considering a set of views $\mathcal{V}$ and a query $Q$ over a database schema ${\cal S}$, we want to answer $Q$ by accessing only the instances of views~\cite{Levy95-1, Halevy01-1, 2017Afrati}. To answer the query $Q$ using $\mathcal{V}$ we could rewrite $Q$ into a new query $R$ such that $R$ is defined in terms of views in $\mathcal{V}$ (i.e., the predicates of the subgoals of $R$ are view names in $\mathcal{V}$).
We denote by $\mathcal{V}(D)$ the output of applying all the view definitions on a database instance $D$. Thus, $\mathcal{V}(D)$ and any subset of it defines a view instance $\mathcal{I}$ for which there is a database $D$ such that $\mathcal{I} \subseteq \mathcal{V}(D)$.
If, for every database instance $D$, we have $R(\mathcal{V}(D))=Q(D)$ then $R$ is an \textit{equivalent rewriting} of $Q$ using $\mathcal{V}$. If $R(\mathcal{V}(D))\subseteq Q(D)$, then $R$ is a \textit{contained rewriting} of $Q$ using $\mathcal{V}$. To find and check query rewitings we use the concept of expansion which is defined as follows.
\begin{definition}
The \emph{view-expansion},\footnote{In Section \ref{subsec-mcr-datalogAC}, we will need to differentiate between view-expansion and Datalog-expansion which we will define shortly, therefore, when confusion arises we use these prefixes.} $R^{exp}$,
of a rewriting $R$ defined in terms of views in $\mathcal{V}$, is obtained from $R$ as follows. For each subgoal $v_i$ of $R$ and the corresponding view definition $V_i$ in $\mathcal{V}$, if $\mu_i$ is the mapping from the head of $V_i$ to $v_i$ we replace $v_i$ in $R$ with the body of $\mu_i(V_i)$. The non-distinguished variables in each view are replaced with
fresh variables in $R^{exp}$.
\end{definition}
To test whether a query $R$ defined in terms of views set $V$ is a contained (resp. equivalent) rewriting of a query $Q$ defined in terms of the base relations, we check whether $P^{exp} \sqsubseteq Q$ (resp. $P^{exp} \equiv Q$).
There are settings where there is no equivalent rewriting of the query using the views. In such cases, finding a containing rewriting returning as many answers of the query as possible matters. In this context, we define a contained rewriting, called \emph{maximally contained rewriting} (\emph{MCR}, for short), that returns most of the answers of the query.
\begin{definition}
A rewriting $R$ is called a {\em maximally contained rewriting} ({\em MCR}) of query $Q$ using views $\mathcal{V}$ with respect to query language ${\cal L}$ if
\begin{enumerate}
\item $R$ is a contained rewriting of $Q$ using $\mathcal{V}$ in ${\cal L}$, and
\item every contained rewriting of $Q$ using $\mathcal{V}$ in language ${\cal L}$ is contained in $R$.
\end{enumerate}
\end{definition}
A view instance $\mathcal{I}$ is a database with facts of the view relations. It is expected that $\mathcal{I}$ is computed by applying the views on a database over the base relations in terms of which the views are defined. The notion of certain answers is another way to get information from a view instance about the query.
\begin{definition}
We define the certain answers of ($Q,\mathcal{I}$) with respect to $\mathcal{V}$ as follows:
\begin{itemize}
\item Under the Closed World Assumption (CWA):
\[\text{certain}(Q,\mathcal{I})=\bigcap\{Q(D): D \text{ such that } \mathcal{I}=\mathcal{V}(D)\}.\]
\item Under the Open World Assumption (OWA):
\[\text{certain}(Q,\mathcal{I})=\bigcap\{Q(D): D \text{ such that } \mathcal{I}\subseteq\mathcal{V}(D)\}.\]
\end{itemize}
\end{definition}
The relation between what an MCR computes and the set of certain answers on a view instance is not easy to find.
In sections \ref{sec-6} and \ref{subsec-mcr-datalogAC}, we present the way MCRs and certain answers are connected for CQACs, under the OWA.
\subsection{Datalog queries}
A \textit{Datalog query} (a.k.a. Datalog program) is a finite set of Datalog rules, where a \textit{rule} is a CQ whose predicates in the body could either refer to a base relation or to a head of a rule in the query (either the same rule or other rule). Furthermore, there is a designated predicate, which is called \textit{query predicate}, and returns the result of the query.
The predicates in the body of each rule in a Datalog query are of two types; the ones referring to base relations and the ones referring to a head of a rule. The predicates of the former type are called \textit{extensional} (\textit{EDB}, for short) while the predicates of the latter are called intensional (\textit{IDB}, for short). The atom whose predicate is an EDB (resp. IDB) is called \textit{base atom} (resp. \textit{derived atom}). A Datalog query is called \textit{monadic} if all the IDBs are unary.
The evaluation of a Datalog query on a database instance is performed by applying the rules on the database until no more facts (i.e., ground head atoms) are added to the set of the derived atoms. The answer of a Datalog query on a database is the set of facts derived during the computation for the query predicate. Namely, the evaluation follows the fixpoint semantics. A $Datalog^{AC}$ query allows in each rule also arithmetic comparisons (ACs) as subgoals, i.e., each rule is a CQAC. The evaluation process remains the same, only now, the AC subgoals should be satisfied too.
We say that we \emph{unfold a rule} if we replace an IDB subgoal with the body of another rule that has this IDB predicate in its head, and we do that iteratively.
A \textit{partial expansion} of a Datalog query is a conjunctive query that results from unfolding the rules one or more times; the partial expansion may contain IDB predicates. A \textit{datalog-expansion} of a Datalog query is a partial expansion that contains only EDB predicates. Considering all the (infinitely many) expansions of a Datalog query we can prove that a Datalog query is equivalent to an infinite union of conjunctive queries.
An expansion of a $Datalog^{AC}$ query is defined the same way as an expansion of a Datalog query, only now we carry the ACs in the body of each expansion we produce. Thus, in an analog way, a $Datalog^{AC}$ query is equivalent to an infinite union of CQACs.
A \textit{derivation tree} depicts a computation of a Datalog query. Considering a fact $e$ in the answer of the Datalog query, we construct a derivation tree for this fact as follows. Each node in this tree, which is rooted at $e$, is a ground fact. For each non-leaf node $n$ in this tree, there is a rule in the query which has been applied to compute the atom node $n$ using its children facts. The leaves are facts of the base relations. Such a tree is called \textit{derivation tree} of the fact $e$.
During the computation, we use an {\em instantiated rule}, which is a rule where all the variables have been replaced by constants. We say that a rule is {\em fired }
if there is an instantiation of this rule where all the atoms in the body of the rule are in the currently computed database.
\section{The algorithm to check satisfaction of a collection of ACs}
\label{subsec-algo}
We will present{\bf\texttt ~algorithm AC-sat} which, on input a collection of ACs, checks whether there is a
satisfying assignment, i.e., an assignment of real numbers to the variables that makes all the ACs in the collection true. If there is not then we say that the conjunction of ACs is false or that the collection of ACs is {\em contradictory} or is not {\em consistent}.
We define the {\em induced directed graph } of a collection $C$ of ACs of the form $X\theta Y$ where $\theta$ is one of the $<, >, \leq, \geq, =, \neq$. We consider that this collection is divided into two sub-collections, the collection $C_A$ including all the ACs where $\theta$ is one of the $<, >, \leq, \geq, =$ and the collection $C_B$ including all the ACs where $\theta$ is $\neq$.
The induced directed graph is built using the ACs in $C_A$ and has nodes that are variables or constants. There is an edge labeled $\leq$ between two nodes $n_1, n_2$ if there is an AC in the collection $C_A$
which is $n_1 \leq n_2$. There is an edge labeled $<$ between two nodes $n_1, n_2$ if there is an AC in the collection $C_A$
which is $n_1 < n_2$. (We only label edges $<$ or $\leq$ since the other direction, $>$ or $\geq$ is indicated by the direction of the edge.)
We treat each equation $X=Y$ in $C_A$ as two ACs of the form $X\leq Y$ and $X\geq Y$ and we add edges accordingly.
Finally we add edges labeled $<$ between all the pairs of constants depending on their order.\\
{\bf\texttt ~Algorithm AC-sat: }
We consider the induced directed graph $G$ of the collection $C$ of ACs.
We then find all the strongly connected components of $G$. We say that an edge belongs to a strongly connected component if it joins two
nodes in this strongly connected component.
The collection $C$ of ACs is contradictory if either of the following is true.
\begin{description}
\item[Case 1.] There is a strongly connected component with two distinct constants belonging to it.
\item[Case 2.] There is a strongly connected component with an edge labeled $<$.
\item[Case 3.] There is a $A_1\neq A_2$ AC in $C_B$ such that $A_1$ and $A_2$ belong to the same strongly connected component.
\end{description}
\begin{lemma}
\label{lemma-prelim}
The{\bf\texttt ~algorithm AC-sat} is a complete and sound procedure to check that a conjunction of ACs is contradictory.
\end{lemma}
\begin{proof}
First we prove that this procedure is complete; i.e., we prove that if the procedure shows that the conjunction is not false then we can assign
constants to variables to make all ACs true.
Since neither Case 1 nor Case 2 happens, all strongly connected components have $\leq$ labels and at most one constant. Thus,
we assign to each of the elements of a strongly connected component the same constant, which is either a new constant or the constant of the component, as follows: We collapse each strongly connected component to one node and the induced directed graph is reduced to an acyclic directed graph. We consider a
topological sorting of this acyclic graph into a number of levels. We assign constants
following this topological sorting, so that constants in the next level are greater than the constants in the previous levels. This makes all ACs true.
Now we prove that this procedure is sound. Whenever the procedure stops in Cases 1 and 2 then there is no assignment that satisfies all the ACs in this strongly connected component because there is a cycle
with either two distinct constants on it or with an edge labeled $<$. This cycle means that all the variables on it should be the same. The existence of two distinct constants on it or of an edge labeled $<$ means that two variables on the cycle should be distinct.
Whenever the procedure stops in Case 3, then $A_1$ and $A_2$ should be equal according to the
strongly connected component they belong. Thus we cannot find an assignment that satisfies also the
AC $A_1\neq A_2$.
\end{proof}
The above algorithm is used to prove the following lemma, whose full proof can be found in \cite{Afrati19}.
\begin{lemma}
\label{lemma-LSI-appendix-NP}
Consider the following implication:
$$c_1\wedge c_2 \wedge ...\Rightarrow d_1 \vee d_2 \vee ...$$
where the conjunction of ACs $c_1\wedge c_2 \wedge ...$ is consistent
(i.e., it has a satisfying assignment from the set of real numbers)
and the $d_i$'s are all closed SI (i.e., either LSI or RSI) comparisons.
%
Then the implication is true if and only if one of
the following happens:
(i) there is a single $d_i$ from the rhs such that
$$c_1\wedge c_2 \wedge ...\Rightarrow d_i$$
or
(ii) there are two ACs from the rhs from which one is LSI and one is RSI, say $d_i$ and $d_j$ (we call them {\em coupling ACs} for the conjunction $c_1\wedge c_2 \wedge ...$) such that
$$c_1\wedge c_2 \wedge ...\Rightarrow d_i \vee d_j.$$
%
\end{lemma}
\section{Analysing the containment entailment}
\label{sec-analysis}
In this section, the first two subsections serve as an introduction to the containment entailment and its preliminary analysis.
In the end of this section, we define the classes of queries we consider in later sections.
Consider the containment entailment (as in Theorem \ref{thm:cont-CQAC} or Theorem~\ref{thm:denorm}).
$$ \beta_2 \Rightarrow \mu_1(\beta_1) \vee \cdots \vee
\mu_k(\beta_1).$$
\subsection{Containment Implications}
The right hand side (rhs, for short) of the containment entailment is a disjunction of disjuncts, where each disjunct is a
conjunction of ACs. We can turn this, equivalently, to a conjunction of conjuncts, where each conjunct
is a disjunction of ACs. We call each of these last conjuncts a {\em rhs-conjunct} (from right hand side conjunct). Now we can turn the containment entailment, equivalently, into a number of implications. In each implication, we keep the left hand side of the containment entailment the same and have the right hand side be one of the
rhs-conjuncts. We call each such implication a {\em containment implication}.
\begin{example}
\label{ex-firstLSI}
For an example, consider the following normalized CQACs.
\vspace*{-.5cm}
%
\begin{center}
\begin{tabular}{lll}
$Q_1: q()$ & $:-$ &$a(X_1,Y_1,Z_1), X_1=Y_1, Z_1< 5$ \\
$Q_2: q()$ & $:-$ &$a(X,Y,Z'), a(X',Y',Z), X\leq 5, Y\leq X, Z\leq Y, $$ $$X'=Y', Z'<5$\\
\end{tabular}
\end{center}
Testing the containment $Q_2
\sqsubseteq Q_1$, it is easy to see that there are the following containment mappings:
\vspace*{-.5cm}
\begin{itemize}
\item $\mu_1: X_1\rightarrow X, Y_1\rightarrow Y, Z_1\rightarrow Z'$
\item $\mu_2: X_1\rightarrow X', Y_1\rightarrow Y', Z_1\rightarrow Z$
\end{itemize}
Hence, the containment entailment is given as follows:
\begin{center}
\begin{tabular}{l}
$X\leq 5 \wedge Y\leq X\wedge Z\leq Y\wedge X'=Y'\wedge Z'<5\Rightarrow$\\
$\big(\; \mu_1(X_1)\!\!=\!\!\mu_1(Y_1) \;\wedge\; \mu_1(Z_1)\!\!<\!5\;\big)\;\vee$ \\
$\big(\;\mu_2(X_1)\!\!=\!\!\mu_2(Y_1) \;\wedge\; \mu_2(Z_1)\!\!< 5\;\big)$\\
\end{tabular}
\end{center}
which is equivalently written:
\begin{center}
\begin{tabular}{l}
$X\leq 5 \wedge Y\leq X\wedge Z\leq Y\wedge X'=Y'\wedge Z'<5 \Rightarrow$\\
$(X=Y \wedge Z'< 5) \vee (X'=Y' \wedge Z< 5)$\\
\end{tabular}
\end{center}
It is easy to verify that the above implication is true (due to the second part of the disjunction in the right-hand side which is also included in the antecedent).
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%
%
%
%
%
%
%
%
Now we consider the containment entailment we built above. According to
what we analyzed in this section, we can equivalently rewrite this containment entailment by
transforming its right hand side into a conjunction, where each conjunct is a disjunction of ACs. The transformed entailment is the following, where $\beta=X\!\leq \!5 \wedge Y\!\leq \!X\wedge Z\!\leq \!Y\wedge X'\!\!=\!\!Y'\wedge Z'\!<\!5$:
\vspace*{-.5cm}
\begin{center}
\begin{tabular}{lll}
$\beta \Rightarrow \; (X\!\!=\!\!Y\vee X'\!\!=\!\!Y')\wedge(X\!\!=\!\!Y\vee Z\!< \!5)\wedge(Z'\!< \! 5\vee X'\!=\! Y')\wedge(Z'\!< \! 5\vee Z\!<\! 5)$ \\
\end{tabular}
\end{center}
%
%
%
%
%
%
%
%
\end{example}
The following two theorems are proved in \cite{Afrati19}
and serve as an introduction to the results in the present paper (the second theorem is proven based on the first theorem):
\begin{theorem}
The following two are equivalent:
a) One disjunct in the rhs suffices to make the containmnent entailment true.
b) For each containment implication, one disjunct in the rhs suffices to make it true.
\end{theorem}
\begin{theorem}
If the containing query contains only closed LSIs and the contained query any closed AC then the containment problem is in NP.
\end{theorem}
\subsection{ACs over single-mapping variables}
\label{sec:single-mapping}
Suppose two CQACs $Q_1=Q_{10}+\beta_1$ and $Q_2=Q_{20}+\beta_2$.
We consider
the containment entailment:
\begin{equation} \label{eq:cont-entail}
\beta_2\Rightarrow\mu_1(\beta_1)\vee\cdots\vee\mu_k(\beta_1)
\end{equation}
\noindent
where $\mu_1,\dots, \mu_k$ are all the containment mappings from$Q_{10}$ to $Q_{20}$.
Suppose $\beta_1$ is such that $\beta_1 = \beta_{11}\wedge\beta_{12}$ where $\beta_{11}$ is the conjunction of ACs among and on the distinguished variables and $\beta_{12}$ the ACs on the nondistinguished variables, i.e., there is no AC between a head variable and a nondistinguished variable. In this special case, we observe that in the containment entailment, each term on the right hand side becomes:
\begin{align*}
\mu_i(\beta_1)=\mu_i(\beta_{11})\wedge\mu_i(\beta_{12}).
\end{align*}
However, $\mu_i(\beta_{11})$ is the same for every term on the right hand side of the entailment because all the containment mappings $\mu_i$ are the same as concerns the distinguished variables, by definition. Thus, applying the distributive law, we write the containment entailment:
\begin{align*}
\beta_2\Rightarrow\mu_1(\beta_{11})\wedge[\mu_1(\beta_{12})\vee\cdots\vee\mu_k(\beta_{12})].
\end{align*}
Consequently, the containment entailment is equivalent to conjunction of the following two entailments:
\begin{align*}
\beta_{2}&\Rightarrow\mu_1(\beta_{11}). \\
\beta_{2}&\Rightarrow\mu_1(\beta_{12})\vee\cdots\vee\mu_k(\beta_{12}).
\end{align*}
Hereon, we will call the second entailment the \textit{body containment entailment} (or simply containment entailment when confusion does not arise) and the first the \textit{head entailment}.
\subsubsection{Introducing single-mapping variables}
The above analysis is valid because of the fact that the variables in the head of $Q_1$ always map to the same variable in $Q_2$, independently of the containment mapping from $Q_1$ to $Q_2$.
Such a property could be straightforwardly extended to other cases. There may exist more variables (besides the head variables) of the containing query that are
always mapped on the same variables of the contained query, for any containment mapping. We call them single-mapping variables and give the formal definition below.
\begin{definition}(single-mapping variables)
\label{defn:single-map-vars}
Let $Q_1=Q_{10}+\beta_1$, $Q_2=Q_{20}+\beta_2$ be two CQACs, such that there is at least one containment mapping from $Q_{10}$ to $Q_{20}$. Consider the set ${\cal M}$ of all the containment mappings from $Q_{10}$ to $Q_{20}$. Each variable $X$ of $Q_1$ which is always mapped on the same variable of $Q_2$ (i.e., for each $\mu\in{\cal M}$ the $\mu(X)$ always equals the same variable) is called a \textit{single-mapping variable with respect to $Q_2$}.
\end{definition}
Notice that the head variables of $Q_1$ are single-mapping variables with respect to any query. For another example, consider that there is a predicate $r$ such that $Q_1$ has $g_{11}, g_{12},\dots , g_{1n}$ subgoals with predicate $r$ and $Q_2$ has a \textit{single} subgoal $g_2$ with predicate $r$. Since each of the $g_{11}, g_{12},\dots , g_{1n}$ subgoals maps on $g_2$, for every containment mapping from $Q_1$ to $Q_2$, the variables in $g_{11}, g_{12},\dots , g_{1n}$ subgoals are single-mapping variables.
Thus, we extend the previous analysis in the next subsection and show that the containment entailment can be decomposed into two parts in a more general case.
\subsection{The classes of queries}
Here we define what it means for a pair of CQACs to be a disjoint-AC pair wrto a set of single-mapping variables.
Then we state Proposition \ref{prop:single-map-vars} that says, that, for such a pair, the containment entailment can be broken in two entailments. Then we restrict our definition to containing queries that
only allow SI on non-single-mapping variables and, in particular with only one RSI. This is the class of queries for which we prove in Section \ref{ssec:mot} that the containment test is in NP. We also define CQAC queries which we call RSI1+ queries and this is the class of queries for which the results of Section \ref{subsec-mcr-datalogAC} hold.
\begin{definition}
Let $Q_1=Q_{10}+\beta_1$, $Q_2=Q_{20}+\beta_2$ be two CQACs with closed ACs, such that there is at least one containment mapping from $Q_{10}$ to $Q_{20}$. Let ${\cal X}_1$ be the set of variables of $Q_1$. We assume that the set , ${\cal X}_1$, of variables of $Q_1$ can be partitioned into the sets ${\cal X}_1^{sv}$, ${\cal X}_1^{nsv}$, s.t. ${\cal X}_1^{sv}\cap{\cal X}_1^{nsv}=\emptyset$, ${\cal X}_1^{sv}$ contains only single-mapping variables of $Q_1$ with respect to $Q_2$ and there are no ACs of $Q_1$ joining a variable in ${\cal X}_1^{sv}$ with a variable in ${\cal X}_1^{nsv}$. Then we say that ($Q_1$,$Q_2$) is a {\em disjoint-AC pair} with respect to ${\cal X}_1^{sv}$\footnote{When it is obvious from the context, we do not refer to ${\cal X}_1^{sv}$.}.
\end{definition}
\begin{proposition}
\label{prop:single-map-vars}
Let $Q_1=Q_{10}+\beta_1$, $Q_2=Q_{20}+\beta_2$ be two CQACs with closed ACs, such that there is at least one containment mapping from $Q_{10}$ to $Q_{20}$. Let ${\cal X}_1$ be the set of variables of $Q_1$.
Let ($Q_1$,$Q_2$) be a {\em disjoint-AC pair} with respect to ${\cal X}_1^{sv}$, where ${\cal X}_1^{sv}$ contains only single-mapping variables of $Q_1$ with respect to $Q_2$.
Then, the containment entailment $\beta_2\Rightarrow\mu_1(\beta_1)\vee\cdots\vee\mu_k(\beta_1)$ is true if and only if both the following two are true:
\begin{itemize}
\item $\beta_{2}\Rightarrow\mu_1(\beta_{11})$, {\em head containment entailment} and
\item $\beta_{2}\Rightarrow\mu_1(\beta_{12})\vee\cdots\vee\mu_k(\beta_{12})$, {\em body containment entailment}\footnote{We retain the same names as in the simple case above, for simplicity of reference; they are actually {\em single-mapping entailment } and {\em non-single-mapping entailment}.}
\end{itemize}
where $\mu_1,\dots \mu_k$ are all the containment mappings from $Q_{10}$ to $Q_{20}$ and $\beta_1 = \beta_{11}\wedge\beta_{12}$,
where $\beta_{11}$
includes all the ACs of $\beta_1$ over the variables in ${\cal X}_1^{sv}$, and $\beta_{12}$ includes all the ACs of $\beta_1$ over the variables in ${\cal X}_1^{nsv}={\cal X}_1 - {\cal X}_1^{sv}$.
\end{proposition}
The proof of the Proposition~\ref{prop:single-map-vars} is an immediate consequence of the Definition of single-mapping variables and was analysed in details in the previous section.
We define a CQAC CRSI1+ query, or simply RSI1+ query hereon, to be a query that:
\begin{enumerate}
\item It has only closed ACs.
\item There are no ACs between a head variable and a nondistinguished variable.
\item The ACs on nondistinguished variables are semi-interval ACs and there is a single right semi-interval AC.
\end{enumerate}
When there are no ACs on the head variables, then we say that this is a RSI1 query.
Notice that, given a query $Q_1$ which is a RSI1+ and any CQAC query $Q_2$ then the pair $(Q_1,Q_2)$ is a disjoint-AC pair.
The following definitions formally describes the {\em RSI1 disjoint-AC pair}.
\begin{definition} Let $V_{sm}$ be a set of single-mapping variables in $Q_1$ and $V_{Q_1}$ be the set of variables of $Q_1$.
A pair of CQACs ($Q_1$, $Q_2$) is called {\em RSI1 disjoint-AC pair with respect to $V_{sm}$} if the following is true:
\begin{enumerate}
\item Both $Q_1$ and $Q_2$ have only closed ACs.
\item There are no ACs between the variables in $V_{sm}$ and the variables in $V_{Q_1}-V_{sm}$.
\item The ACs in $Q_1$ are such that the following are true:
\begin{enumerate}
\item The ACs on variables in $V_{Q_1}-V_{sm}$ are semi-interval (SI, for short), and
\item there is a single right semi-interval (RSI) AC, among the ACs on variables in $V_{Q_1}-V_{sm}$.
\end{enumerate}
\end{enumerate}
\end{definition}
Notice that, given a query $Q_1$ which is a RSI1+ and any CQAC query $Q_2$ then the pair $(Q_1,Q_2)$ is an RSI1 disjoint-AC pair.
We say that a body containment entailment is an {\em RSI1 entailment} if the ACs in each disjunct on the right hand side include only one RSI AC and the others are LSI ACs.
\begin{itemize}
\item For every RSI1 disjoint-AC pair, the body containment entailment is an RSI1 entailment.
\item In the next section, we consider RSI1 disjoint-AC pairs of queries.
\end{itemize}
Naturally, because of symmetry, we can define LSI1 disjoint-AC pairs of quries where now only one LSI is allowed and all the results are also valid for this class.
\section{CQAC Query Containment Using Datalog}
\label{ssec:mot}
The main result of this section is the following theorem:
\begin{theorem}
\label{thm-np-complete}
Consider a pair ($Q_1$, $Q_2$) which is a RSI1 disjoint-AC pair of queries.
Then testing containment of $Q_2$ to $Q_1$ is NP-complete.
\end{theorem}
A byproduct of the proof of this theorem is a reduction of the CQAC containment problem, in this special case, to a containment problem where we check containment of a CQ to a Datalog query (i.e., both these queries have no ACs, their definitions use only relational atoms). This reduction is also important in other sections of this paper where we use it to construct MCRs for CQAC queries and views and prove that certain answers can be computed in polynomial time for certain cases of queries and views.
Proposition~\ref{prop:single-map-vars} leads us to focus on the body containment entailment of the two CQAC queries. Thus, we ignore the ACs of the containing query that are on the single-mapping variables and call the resulting query the {\em reduced containing query}. For the first three subsections of this section, we will only refer to the reduced containing query, so, we will say simply containing query. Note, here, that we do not ignore any AC from the contained query, since all the ACs of the contained query are required in order to check body containment entailment.
Thus this section has two large parts:
\begin{itemize}
\item Transformation of the reduced containing query $Q_1$ to a Datalog query and transformation of the contained query $Q_2$ into a CQ query. This is presented in the three first subsections of this section.
\item Proving that $Q_2$ is contained in the reduced containing query $Q_1$ if and only if their transformed CQ and Datalog queries, respectively, are contained in each other. The main results of this section are stated formally in
Subsection \ref{subsec-main-proof-sec5}.
\end{itemize}
Theorem \ref{thm-np-complete} extends significantly the corresponding result in \cite{AfratiLM06}.
The transformations and the proof are along similar lines as the transformations and the proof explained in \cite{AfratiLM06} with many modifications to capture the new features.
Algorithm AC-sat presented in Section \ref{subsec-algo} is missing from \cite{AfratiLM06}.
This algorithm offers an elegant way to prove technical preliminary results about implications involving arithmetic comparisons.
The structure of this section is as follows: In Subsection
\ref{subsec-tree-like}, two implications are analyzed that will be met when we prove the main result in this section later on. These implications are simply implications that involve ACs, their relation to the containment problem is that they have the same structure as the containment entailment. Thus, we provide some explanations in Subsection
\ref{subsec-tree-like} as to the reason these results lead towards the idea of using the Datalog transformation of the containing query.
Then in Subsections \ref{subsec-contained-trans} and
\ref{subsec-construct-Datalog} we present the transformations of the reduced containing query and the contained query respectively.
Subsection \ref{subsec-main-proof-sec5} contains the statement of the main results in this section.
Subsection \ref{subsec-examples} contains examples of the transformations presented in Subsections \ref{subsec-contained-trans} and
\ref{subsec-construct-Datalog}.
Finaly, in Subsection
\ref{subsec-simplefacts}, we present preliminary partial results and intuition for the proof of the main technical result. The proof itself is presented in \ref{prf:thm-main123p}.
\subsection{The tree-like structure of the containment entailment}
\label{subsec-tree-like}
First, as Theorem 2.4 shows, the query normalization is not needed for testing containment into this setting.
The following proposition is where the class of RSI1s comes useful.
\begin{proposition}
\label{trick-pro}
Let $\beta$ be a conjunction of closed ACs which is consistent, and each $\beta_1, \beta_2, \ldots ,\beta_k$ be a conjunction of closed
RSI1s. Suppose the following is true:
$$\beta \Rightarrow \beta_1 \vee \beta_2 \vee \cdots \vee \beta_k.$$
Then there is a $\beta_i$ (w.l.o.g. suppose it is $\beta_1$) such that either of the following two happens:
%
%
%
%
%
%
\begin{enumerate}[label=(\roman*)]
\item $\beta \Rightarrow \beta_1,$ \textbf{or}
\item there is an AC $e$ in $\beta_1$ such that the following are true:
\begin{enumerate}[label=(\alph*)]
\item $\beta \wedge \neg e \Rightarrow \beta_2 \vee \cdots \vee \beta_k$
(or equivalently,
$\beta \Rightarrow \beta_2 \vee \cdots \vee \beta_k \vee e$),
\item $\beta \Rightarrow \beta_1 \vee \neg e$, and
\item all the other ACs, besides $e$, in $\beta_1$ are directly implied by $\beta$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
Suppose there is no $\beta_i$ such that $$\beta \Rightarrow \beta_i$$
Then we claim that there is
a $\beta_i$ (w.l.o.g. suppose it is $\beta_1$) such that all the ACs in $\beta_1$ are directly implied by $\beta$ (i.e., $\beta \Rightarrow e_i$ if $e_i$ is an AC in $\beta_1$), except for one AC $e$., i.e., we claim that also the following is true:
$$\beta \Rightarrow \beta_1 \vee \neg e$$
Towards contradiction,
suppose that for all the $\beta_i$s there are at least two ACs that are not directly implied by $\beta$. Since all
the $\beta_i$'s are RSI1s, each $\beta_i$ has at least one LSI that is not directly implied. If we take all these LSI's after applying the distributive law and converting the right-hand side from a disjunction of conjunctions to a conjunction of disjunctions, then we will have a conjunct that contains only LSIs, none of which is directly implied by $\beta$.
We need to show that this is impossible --- i.e., it is not true that $\beta \Rightarrow ac_1 \vee ac_2 \cdots $ if
none of the LSI $ac_i$ is directly implied by $\beta$. This is proved in Lemma \ref{lemma-LSI-appendix-NP}.
Now we write equivalently the implication in the statement of the proposition as:
$$\beta \wedge \neg \beta_1\Rightarrow
\beta_2 \vee \beta_3 \cdots \vee \beta_k,$$ or
equivalently (assuming $\beta_1=e_1 \wedge \cdots \wedge e_t$,
where the $e_i$s are ACs)
$$(\beta \wedge \lnot e_1)
\vee (\beta \wedge \lnot e_2) \vee \cdots \vee
(\beta \wedge \lnot e_t) \Rightarrow \beta_2 \vee \beta_3 \cdots \vee \beta_k.$$
Assume w.l.o.g. that $e=e_1$. Since each $e_i$, with the exception of $e_1$, is
entailed by $\beta$, each disjunct with the exception of the first one in the left-hand side
is always false. Hence, the latter entailment yields:\\
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\beta \wedge \neg e\Rightarrow \beta_2 \vee \beta_3 \cdots
\vee \beta_k.$
\end{proof}
Proposition
\ref{trick-pro} begins to show a tree-like structure of the containment entailment and it gives the first intuition for constructing a Datalog query from the containing query that will help in deciding query containment. The following example gives an illustration of this intuition.
\begin{example}
\label{ex:description-of-prop52}
Let us consider the following two Boolean queries.
\begin{center}
\begin{tabular}{lll}
$Q_1: q()$ & $:-$ &$a(X,Y,Z),X\leq 8,Y\leq 7,Z\geq 6.$ \\
$Q_2: q()$ & $:-$ &$a(X,Y,Z),a(U_1,U_2,X),a(V_1,V_2,Y),$\\
& &$ a(Z,Z_1,Z_2),a(U_1',U_2',U_1),a(V_1',V_2',V_1),$\\
& &$ U_1'\leq 8,U_2'\leq 7,U_2\leq 7,V_1'\leq 8,$\\
& &$ V_2'\leq 7,V_2\leq 7,Z_1\leq 7,Z_2\geq 6.$\\
\end{tabular}
\end{center}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{intuition-bodyQ2.png}
\caption{Illustration of containment entailment of Example~\ref{ex:description-of-prop52}}
\label{fig:intuition-bodyq2}
\end{figure}
The query $Q_2$ is contained in the query $Q_1$. To verify this, notice that there are $6$ containment mappings from\footnote{we always mean containment mappings from the relational subgoals of $Q_1$ to the relational subgoals of $Q_2$} $Q_1$ to $Q_2$. These mappings are given as follows: $\mu_1:$ $(X,Y,Z)\rightarrow(X,Y,Z)$, $\mu_2:$ $(X,Y,Z)\rightarrow(U_1,U_2,X)$, $\mu_3:$ $(X,Y,Z)\rightarrow(V_1,V_2,Y)$, $\mu_4:$ $(X,Y,Z)\rightarrow(Z,Z_1,Z_2)$, $\mu_5:$ $(X,Y,Z)\rightarrow(U_1',U_2',U_1)$, and $\mu_6:$ $(X,Y,Z)\rightarrow(V_1',V_2',V_1)$. After replacing the variables as specified by the containment mappings, the query entailment is $\beta\Rightarrow\beta_1\vee\beta_2\vee\beta_3\vee\beta_4\vee\beta_5\vee\beta_6$, where:
\begin{center}
\begin{tabular}{ll}
\multicolumn{2}{l}{$\beta:$ $U_1'\leq 8\;\wedge U_2'\leq 7\;\wedge U_2\leq 7\;\wedge V_1'\leq 8\;\wedge V_2'\leq 7\;\wedge V_2\leq 7\;\wedge Z_1\leq 7\;\wedge Z_2\geq 6$.}\\
$\beta_1:$ $X\leq 8\;\wedge Y\leq 7\;\wedge Z\geq 6$. & $\beta_4:$ $Z\leq 8\;\wedge Z_1\leq 7\;\wedge Z_2\geq 6$.\\
$\beta_2:$ $U_1\leq 8\;\wedge U_2\leq 7\;\wedge X\geq 6$. & $\beta_5:$ $U_1'\leq 8\;\wedge U_2'\leq 7\;\wedge U_1\geq 6$.\\
$\beta_3:$ $V_1\leq 8\;\wedge V_2\leq 7\;\wedge Y\geq 6$. & $\beta_6:$ $V_1'\leq 8\;\wedge V_2'\leq 7\;\wedge V_1\geq 6$.\\
\end{tabular}
\end{center}
We now refer to Figure~\ref{fig:intuition-bodyq2} to offer some intuition about and visualization on Proposition \ref{trick-pro} using the above queries. The circles in the figure represent the mappings $\mu_1, \dots, \mu_6$, and the dots are the variables of $Q_2$. Notice now the intersections between the circles.
Proposition \ref{trick-pro} refers to these intersections, such as the one between $\mu_3$ and $\mu_6$ (or, the one between $\mu_2$ and $\mu_5$).
The AC $V_1\geq 6$ ($V_1$ is included in the intersection between $\mu_3$ and $\mu_6$) is the one that is not directly implied by $\beta$, as stated in the case (ii) of the Proposition \ref{trick-pro}. In particular, it is easy to verify that the following are true:
\begin{itemize}
\item $\beta\wedge \neg (V_1\geq 6)\Rightarrow\beta_1\vee\beta_2\vee\beta_3\vee\beta_4\vee\beta_5$.
\item $\beta\Rightarrow\beta_6\vee \neg (V_1\geq 6)$ (i.e., $\beta\Rightarrow(V_1'\leq 8\;\wedge V_2'\leq 7\;\wedge V_1\geq 6)\vee \neg (V_1\geq 6)$).
\item $\beta\Rightarrow(V_1'\leq 8)$ and $\beta\Rightarrow(V_2'\leq 7)$.
\end{itemize}
\end{example}
Proposition \ref{trick-pro1} is a generalization of
Proposition
\ref{trick-pro}.
\begin{proposition}
\label{trick-pro1}
Let $\beta$ be a conjunction of closed SI ACs which is consistent, and $\beta_1, \beta_2, \ldots ,\beta_k$ each be a conjunction of closed
RSI1s (i.e., in each conjunct there is only one RSI and the rest are LSI ACs). Suppose the following is true:
$$\beta \Rightarrow \beta_1 \vee \beta_2 \vee \cdots \vee \beta_k \vee e_1 \vee e_2 \vee \cdots$$
where $e_i$s are closed SIs such that the following implication is not true: $\beta \Rightarrow e_1\vee e_2 \vee \cdots$.
Then there is a $\beta_i$ (w.l.o.g. suppose it is $\beta_1$) such that either of the following two happen:
\begin{enumerate}[label=(\roman*)]
\item $\beta \Rightarrow \beta_1 \vee e_1 \vee e_2 \vee \cdots,$ \textbf{or}
\item there is an AC $e$, called {\em special for this mapping}, in $\beta_1$ such that the following are true:
\begin{enumerate}[label=(\alph*)]
\item $\beta \wedge \neg e \Rightarrow \beta_2 \vee \cdots \vee \beta_k \vee e_1 \vee e_2 \vee \cdots$
, or equivalently,
$$\beta \Rightarrow \beta_2 \vee \cdots \vee \beta_k \vee e \vee e_1 \vee e_2 \vee \cdots.$$
\item $\beta \Rightarrow \beta_1 \vee \neg e \vee e_1 \vee e_2 \vee \cdots $ and
\item all the other ACs $ac_j$ in $\beta_1$, with $j=1,2,\ldots $, besides $e$, are either directly implied by $\beta$ or coupled with one of the $e_i$s for $\beta$ i.e., either $\beta \Rightarrow ac_j$ or
$\beta\Rightarrow e_i \vee ac_j$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
Suppose there is no $\beta_i$ such that $$\beta \Rightarrow \beta_i\vee e_1\vee e_2 \vee \cdots $$
Then we claim that there is
a $\beta_i$ (w.l.o.g. suppose it is $\beta_1$) such that all the ACs $a_i$ in $\beta_1$ are such that $a_i\vee e_1\vee \cdots$ is directly implied by $\beta$ (i.e., $\beta \Rightarrow a_i\vee e_1\vee \cdots$ if $a_i$ is an AC in $\beta_1$), except for one AC $a_1=e$ (wlog suppose this is $a_1$), i.e., we claim that the following is true for $e$:
$$\beta \Rightarrow \beta_1 \vee \neg e\vee e_1\vee e_2 \vee \cdots$$
Towards contradiction,
suppose that for all the $\beta_i$s there are at least two ACs (say AC $a^{i12}$ is such an AC) such that the following does not happen:
\begin{equation} \label{eq-1a}
\beta \Rightarrow a^{i12}\vee e_1\vee e_2 \vee \cdots
\end{equation}
\noindent
Since all
the $\beta_i$'s are RSI1s, each $\beta_i$ has at least one LSI for which the implication \ref{eq-1a} is not true. If we take all these LSI's (after applying the distributive law and converting the right-hand side from a disjunction of conjunctions to a conjunction of disjunctions), then we will have a conjunct that contains only LSIs, none of which is such that
the implication \ref{eq-1a} is true.
Then we will have a case like in Lemma \ref{lemma-LSI-appendix-NP}.
According to Lemma \ref{lemma-LSI-appendix-NP}, there are two cases: a) There is a single SI on the rhs which is implied by $\beta$ or b) there are two SI in the rhs whose disjunction is implied, of which one is LSI and one is RSI. Thus, in both cases, we have only one LSI, say it is $a_{LSI}$ such that
$$\beta\Rightarrow a_{LSI} \vee e_1\vee e_2 \vee \cdots.$$
This is a contradiction to our assumption.
We write equivalently the implication in the statement of the proposition as:
$$\beta \wedge \neg [\beta_1\vee e_1\vee e_2 \vee \cdots ]\Rightarrow
\beta_2 \vee \beta_3 \cdots \vee \beta_k$$ or
equivalently (assuming $\beta_1=a_1 \wedge \cdots \wedge a_t$,
where the $e_i$s are ACs)
$$(\beta \wedge \neg a_1\wedge \neg e_1 \wedge \neg e_2 \wedge \cdots)
\vee (\beta \wedge \neg a_2\wedge \neg e_1 \wedge \neg e_2 \wedge \cdots) \vee \cdots \vee
(\beta \wedge \neg a_t\wedge \neg e_1 \wedge \neg e_2 \wedge \cdots)$$ $$ \Rightarrow \beta_2 \vee \beta_3 \cdots \vee \beta_k$$
Assume w.l.o.g. that $e=a_1$. Since each $a_i\vee e_1\vee \cdots$, with the exception of $a_1$, is
entailed by $\beta$, each disjunct with the exception of the first one in the left-hand side
is always false. Hence, the latter entailment yields:\\
~~$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\beta \wedge \neg e\Rightarrow \beta_2 \vee \beta_3 \cdots
\vee \beta_k\vee e_1\vee e_2 \vee \cdots.$
\end{proof}
\begin{example}
\label{ex:trick-pro1}
Continuing Example~\ref{ex:description-of-prop52}, we will use the Proposition \ref{trick-pro1} to see how the AC on the variable $V_1$ is related to other ACs on the same variable in another mapping (here it is the mapping $\mu_4$). To see that, notice that the AC $V_1\geq 6$ is the special AC for $\mu_6$ and it is coupled with the AC $V_1\leq 8$ in $\beta_3$ (i.e., $\beta\Rightarrow (V_1\geq 6) \vee (V_1\leq 8)$). In particular, as we saw in Example~\ref{ex:description-of-prop52}, the following is true.
$$\beta\Rightarrow\beta_1\vee\beta_2\vee\beta_3\vee\beta_4\vee\beta_5\vee (V_1\geq 6).$$
Then, according to the Proposition~\ref{trick-pro1} (where $e_1=V_1\geq 6$), there is $\beta_i$ (in this case, $\beta_3$ is such a $\beta_i$) such that the following are true (case (ii) in the proposition):
\begin{itemize}
\item $\beta\wedge \neg (Y\geq 6)\Rightarrow\beta_1\vee\beta_2\vee\beta_4\vee\beta_5\vee (V_1\geq 6)$.
\item $\beta\Rightarrow\beta_3\vee \neg (Y\geq 6)\vee (V_1\geq 6)$; i.e.,
$$\beta\Rightarrow(V_1\leq 8\;\wedge V_2\leq 7\;\wedge Y\geq 6)\vee \neg (Y\geq 6)\vee (V_1\geq 6).$$
\item $\beta\Rightarrow(V_2\leq 7)$, while $V_1\leq 8$ is coupled with $V_1\geq 6$.
\end{itemize}
\end{example}
We give a first glance of what is going to happen in the rest of this section. In particular, we do the following:
\begin{enumerate}
\item We transform the containing query $Q_1$ into a Datalog query $Q^{Datalog}_{Q_1}$.
\item We transform the contained query into a CQ, $Q^{CQ}_{Q_2}$.
\item The above two transformations are done by keeping the relational subgoals of $Q_1$ ($Q_2$, respectively) and encoding the
arithmetic comparisons into relational predicates.
\item
We prove (Theorem \ref{thm:main}) that $Q_2$ is contained in $Q_1$ if and only if $Q^{CQ}_{Q_2}$ is contained in $Q^{Datalog}_{Q_1}$.
\end{enumerate}
Intuitively, using those transformations we aim to replace the ACs with relations; hence, transform the problem of CQAC containment to a containment problem of a Datalog query in a CQ. One might wonder why the transformation of the containing query to a Datalog query is required. The answer to this question is based on the containment entailment. The disjunction in the right-hand-side implies arbitrary combinations of the ACs, since the contained query can be arbitrarily long independently of the size of the containing query. Hence, the program-expansion of $Q^{Datalog}_{Q_1}$ that verifies the containment can be
arbitrarily long, depending on the size of the contained query.
\subsection{Construction of Datalog Query for Containing Query}
\label{subsec-construct-Datalog}
In this subsection, we describe the construction of a Datalog query for a given RSI1 query $Q$.
The Datalog query has two kinds of rules: The rules that depend only on the containing query, and we call them {\em basic rules}, and the rules that also take into account the contained query, and we call them {\em dependant rules}.
In various places, in order to illustrate the construction, we will use the query in the following running example.
\begin{example}
\label{ex:running41}
The following query $Q_1$ is an RSI1 query:
\begin{center}
\begin{tabular}{lll}
$Q_1({W_1,W_2})$ & $:-$ &$a(W_1,W_2,Y), e(X,Y),e(Y,Z),X\geq 5,Z\leq 8.$ \\
\end{tabular}
\end{center}
For simplicity in the notation we will denote by $\overline{W}$ the vector $W_1,W_2$ of head variables. Thus, we are writing the query as:
\begin{center}
\begin{tabular}{lll}
$Q_1(\overline{W})$ & $:-$ &$a(\overline{W},Y),e(X,Y),e(Y,Z),X\geq 5,Z\leq 8.$ \\
\end{tabular}
\end{center}
\end{example}
{\bf Construction of the basic rules} $Q_1^{Datalog}:$ We construct three kinds
of rules, {\em mapping rules, coupling rules,} and a single {\em query rule}.
First, we introduce the EDB predicates and the IDB predicates that we use and describe how we construct them.
The EDB predicates are all the predicates from the relational subgoals of $Q_1$ and an extra binary predicate $U$. Intuitively, $U(X,Y)$ encodes the AC $X\leq Y$.
Now, the IDB predicates are as follows:
\begin{enumerate}
\item
We introduce new semi-unary IDBs,\footnote{We call them semi-unary for reasons that will become apparent later during the proof.} two pairs of IDBs for each
constant $c$ in $Q_1$ (intuitively, that compares a non-single-mapping variable to this constant), namely $I_{\geq c}$, $I_{\leq c}$ and
$J_{\geq c}$, $J_{\leq c}$. Intuitively, these predicates have as arguments the vector $\overline{W}$ of variables in the head of the query $Q_1$ and another variable $X$.
\item
For each AC $X\theta c$, we construct the IDB predicate atoms $I_{\theta
c}(X,\overline{W})$ and $J_{\theta c}(X,\overline{W})$, where $\theta$ is either $\leq$ or $\geq$.
\item
For each AC $X\theta c$, considering the IDB predicate atom $I_{\theta
c}(X,\overline{W})$ ($J_{\theta c}(X,\overline{W})$, respectively), we refer to
$J_{\theta c}(X,\overline{W})$
($I_{\theta
c}(X,\overline{W})$, respectively),
as the {\em associated $I$-atom } ( {\em associated $J$-atom }
respectively) of $X\theta c$, and we refer to $X\theta c$ as the {\em associated AC} of
$I_{\theta
c}(X,\overline{W})$ ($J_{\theta c}(X,\overline{W})$, respectively). We also refer to $I_{\theta
c}(X,\overline{W})$ as the associated $I$-atom of $J_{\theta c}(X,\overline{W})$ and vice versa.
\item
We have also a query IDB predicate which is denoted $Q_1^{Datalog}(\overline{W})$
\end{enumerate}
Now, we describe the construction of the basic rules of the Datalog query which use the EDB predicates of the containing query and are as follows.
We call them basic because they do not depend on the ACs of the contained query.
\begin{enumerate}
\item
The {\em query rule} copies into its body all the relational subgoals of $Q_1$, and
replaces each AC subgoal of $Q_1$ that compares a non-single-mapping variable to a constant by its associated $I$-atom. The head of this rule is the same as the head of the query $Q_1$.
\item
We
get one {\em mapping rule} for each SI arithmetic comparison $e$ in $Q_1$ which is on a non-single-mapping variable.
The body of each mapping rule is a copy of the body of the query rule, except that the
$I$ atom associated with $e$ is deleted. The head is the $J$ atom
associated with $e$.
\item
For every pair of constants $c_1 \leq c_2$ used in $Q_1$, we
construct three {\em coupling rules}.
First, we construct the following two coupling rules:
\begin{center}
\begin{tabular}{l}
$I_{\leq c_2}(X,\overline{W})~:-~J_{\geq c_1}(X,\overline{W})$ \\
$I_{\geq c_1}(X,\overline{W})~:-~J_{\leq c_2}(X,\overline{W})$
\end{tabular}
\end{center}
Then, we construct a coupling rule which is the following:
$$I_{\leq c_2}(X,\overline{W}):-~J_{\geq c_1}(Y,\overline{W}),U(X,Y).$$
\end{enumerate}
\begin{example}
\label{ex-running-dat}
For the query $Q_1$ of Example~\ref{ex:running41}, the construction we described yields the following basic rules of the Datalog
query $Q_1^{Datalog}$:
\begin{center}
\begin{tabular}{l l l}
$Q_1^{Datalog}(\overline{W})$ $:-$ & $e(X,Y),e(Y,Z),a(\overline{W},Y),I_{\geq 5}(X,\overline{W}),$& \multirow{2}{*}{(query rule)}\\
&$I_{\leq 8}(Z,\overline{W}).$& \\
$J_{\leq 8}(Z,\overline{W})$ $:-$ & $e(X,Y),e(Y,Z),a(\overline{W},Y),I_{\geq 5}(X,\overline{W}).$ & (mapping rule)\\
$J_{\geq 5}(X,\overline{W})$ $:-$ & $e(X,Y),e(Y,Z),a(\overline{W},Y),I_{\leq 8}(Z,\overline{W}).$ & (mapping rule)\\
$I_{\leq 8}(X,\overline{W})$ $:-$ & $J_{\geq 5}(X,\overline{W}).$ & (coupling rule)\\
$I_{\geq 5}(X,\overline{W})$ $:-$ & $J_{\leq 8}(X,\overline{W}).$ & (coupling rule)\\
$I_{\leq 8}(X,\overline{W})$ $:-$ &$J_{\geq 5}(Y,\overline{W}), U(X,Y)$ & (coupling rule)\\
$I_{\geq 5}(X,\overline{W})$ $:-$ &$J_{\leq 8}(Y,\overline{W}), U(Y,X)$ & (coupling rule)\\
\end{tabular}
\end{center}
Intuitively, a coupling rule denotes that a formula $AC_1 \vee AC_2$ ( for two
SI comparisons $AC_1=X\theta_1 c_1$ and $AC_2=Y\theta_2 c_2$) is either true or it is implied by $X\leq Y$ (which is encoded by the predicate $U(X,Y)$). Thus, the first coupling rule in the above query says that
$ X\leq 8 \vee X\geq 5$ is true and the second coupling rule says the same but refering to different
$I$ and $J$-atoms. Moreover, the last coupling rule says that $X\leq Y\Rightarrow X\leq 8 \vee Y\geq 5$.
\end{example}
{\bf Construction of the dependant rules} $Q_1^{Datalog}:$
First, we describe the EDB predicates that we introduce (they all depend on the ACs of the contained query):
\begin{itemize}
\item A unary predicate $U_{\theta c}(X,\overline{W})$, where $\theta$ is either $\leq$ or $\geq$ (the intuition for $\overline{W}$ is that it will carry, during the computation, the head variables of the query rule), for each SI AC $X\theta c$ in the closure of the ACs in the contained query. Note that although $U_{\theta c}$ typically includes $\overline{W}$, in the following, we could ignore it, for simplicity.
\end{itemize}
We have one kind of dependant rules, the \textit{link rules}:
\begin{itemize}
\item
For each pair of constants $(c_1,c_2)$, one in SIs of $Q_1$ and the other in an SI in the closure of ACs of $Q_2$ then, if $c_1\leq c_2$, we add the non-recursive link rule:
$$ I_{\geq c_1}(X,\overline{W}):-U_{\geq c_2}(X,\overline{W}).$$
Similarly, we do in a symmetric way for the $\leq$ ACs in $Q_1$ and $Q_2$.
\end{itemize}
Thus,
each link
rule encodes an
entailment of the form $X\leq 7 \Rightarrow X\leq 8$, i.e., it encodes,
in general, an entailment $X\leq c_1 \Rightarrow X\leq c_2$ where $c_1
\leq c_2$. Intuitively, the link rules are used to link the ACs between the contained query and the containing query, as described through the containment entailment. Typically, the unary predicates represent the ACs of the contained query.
For an example of dependant rules, see below (also analyzed in the next subsections):
\begin{center}
\begin{tabular}{l l l}
$I_{\geq 5}(X,\overline{W})$ & $:-~U_{\geq 6}(X,\overline{W}).$ & (link rule)\\
$I_{\leq 8}(X,\overline{W})$ & $:-~U_{\leq 7}(X,\overline{W}).$ & (link rule)\\
\end{tabular}
\end{center}
\subsection{Construction of CQ for Contained Query}
\label{subsec-contained-trans}
We now describe the construction of the contained query turned into a CQ .
{\bf Construction of } $\mathbf{Q_2^{CQ}:}$ We introduce new unary
EDBs, specifically two of them, by the names $U_{\geq c}$ and $U_{\leq c}$, for each constant $c$ in $Q_2$. In addition, we use the binary predicate $U$ to represent the closed SI ACs between two variables, as we saw in the previous section.
Let us now construct the CQ $Q_2^{CQ}$ from $Q_2$.
We initially copy the regular subgoals of $Q_2$,
and for each SI $X_i\theta c_i$ in the closure of $\beta_2$ we add a
unary predicate subgoal $U_{\theta c_i}(X_i)$. Then, for each AC $X\leq Y$ in the closure of ACs in $Q_2$, we add the unary subgoal $U(X,Y)$ in the body of the rule.
For example, considering the CQAC $Q_2$ with the following definition:
\begin{center}
\begin{tabular}{ll}
$Q_2(W_1,W_2):-$ & $e(A,B),e(B,C),e(C,D),e(D,E), A\geq 6,$\\
&$E\leq 7,
a(W_1,W_2,B), a(W_1,W_2,D).$ \\
\end{tabular}
\end{center}
we construct the $Q_2^{CQ}$ whose definition is:
\begin{center}
\begin{tabular}{ll}
$Q_2^{CQ}(W_1,W_2):-$ & $e(A,B),e(B,C),e(C,D),e(D,E),U_{\geq 6}(A),$\\
&$U_{\leq 7}(E),
a(W_1,W_2,B), a(W_1,W_2,D).$ \\
\end{tabular}
\end{center}
Thus the dependant rules for our running example, query $Q_1$, and the above contained query $Q_2$ are:
\begin{center}
\begin{tabular}{l l l}
$I_{\geq 5}(X,\overline{W})$ & $:-~U_{\geq 6}(X,\overline{W}).$ & (link rule)\\
$I_{\leq 8}(X,\overline{W})$ & $:-~U_{\leq 7}(X,\overline{W}).$ & (link rule)\\
\end{tabular}
\end{center}
Now, we have completed the description of the construction of both $Q_1^{Datalog}$ from $Q_1$ and
$Q_2^{CQ}$ from $Q_2$. We go back to our examples and put all together.
\begin{example}
\label{ex-full}
Our contained query is the one in Subsection \ref{subsec-contained-trans}.
Our containing query is the one in Example \ref{ex:running41}. The transformation of the contained query is shown in Subsection \ref{subsec-contained-trans}. The transformation of the contained query is shown in Example \ref{ex-running-dat}, where we see the basic rules. To complete the Datalog query, we add
the following link rules:
\begin{center}
\begin{tabular}{l l l}
$I_{\geq 5}(X,\overline{W})$ & $:-~U_{\geq 6}(X,\overline{W}).$ & (link rule)\\
$I_{\leq 8}(X,\overline{W})$ & $:-~U_{\leq 7}(X,\overline{W}).$ & (link rule)\\
\end{tabular}
\end{center}
In fact, we constructed the two new link rules in the Datalog query for $Q_1$. One rule links the constant 6 from the ACs of $Q_2$ to the constant 5 from the ACs of $Q_1$. The other link rule links constants 7 and 8 from queries $Q_1$ and $Q_2$, respectively.
\end{example}
\subsection{Proving the main theorem and the complexity}
\label{subsec-main-proof-sec5}
The constructions of the Datalog query and the CQ presented in Sections \ref{subsec-construct-Datalog} and \ref{subsec-contained-trans}, respectively, lead to the following theorem.
\begin{theorem}
\label{thm:main}
Consider two conjunctive queries with arithmetic comparisons, $Q_1$ and $Q_2$ such that ($Q_1,Q_2$) is an RSI1 disjoint-AC pair.
%
%
%
Then, $Q_1$ contains $Q_2$ if and only if the following two happen a) $Q_1^{Datalog}$ contains $Q_2^{CQ}$ and b) the head entailment is true.
\end{theorem}
The challenging part of the Theorem~\ref{thm:main} concerns the part (a) which is restated in the Theorem~\ref{thm:main123}. The part (b) of Theorem~\ref{thm:main} is a straightforward consequence of Proposition~\ref{prop:single-map-vars}.
\begin{theorem}
\label{thm:main123}
Consider two conjunctive queries with arithmetic comparisons, $Q_1$ and $Q_2$ such that ($Q_1,Q_2$) is an RSI1 disjoint-AC pair.
%
Let $Q_1^{Datalog}$ be the transformed Datalog query of $Q_ 1$. Let
$Q_2^{CQ}$ be the transformed CQ query of $Q_2$.
Then, the body containment entailment for containment of $Q_2$ to $Q_1$ is true if and only if $Q_1^{Datalog}$ contains $Q_2^{CQ}$.
\end{theorem}
The proof of Theorem \ref{thm:main123} is in the \ref{prf:thm-main123p}. The following theorem proves that checking body containment entailment is NP-complete.
\begin{theorem}
\label{thm:datalog-np-complete}
Consider two conjunctive queries with arithmetic comparisons, $Q_1$ and $Q_2$ such that ($Q_1,Q_2$) is an RSI1 disjoint-AC pair. Let $Q_1^{Datalog}$ be the transformed Datalog query of $Q_ 1$. Let
$Q_2^{CQ}$ be the transformed CQ query of $Q_2$. Checking whether $Q_2^{CQ}$ is contained in $Q_1^{Datalog}$ is NP-complete.
\end{theorem}
Theorem~\ref{thm:datalog-np-complete} can be generalized to a stronger result, which is presented in Section~\ref{sec:semi-monadic-datalog-cont} in Theorem~\ref{thm-semi-monadic-np}. Theorem \ref{thm-np-complete} is a straightforward consequence of Theorem~\ref{thm:check-head-body-entailments}.
\begin{theorem}
\label{thm:check-head-body-entailments}
Consider two conjunctive queries with arithmetic comparisons, $Q_1$ and $Q_2$ such that ($Q_1,Q_2$) is an RSI1 disjoint-AC pair. Let $\phi_h$ and $\phi_b$ be the head and body entailments, respectively. Then, checking $\phi_h$ is polynomial and checking $\phi_b$ is NP-complete.
\end{theorem}
To prove that checking $\phi_h$ is polynomial, observe that it suffices to compute the closure of a set of ACs. This can be done in polynomial time.
Consider two conjunctive queries with arithmetic comparisons, $Q_1$ and $Q_2$ such that $Q_1$ is an RSI1+ query and $Q_2$ is a CQAC with closed ACs. It is straightforward that ($Q_1$, $Q_2$) is a RSI1 disjoint-AC pair with respect to the set of head variables of $Q_1$.
The following is a corollary of Theorem \ref{thm:check-head-body-entailments}.
\begin{corollary}
\label{cor:check-head-body-entailments}
Consider two conjunctive queries with arithmetic comparisons, $Q_1$ and $Q_2$ such that $Q_1$ is an RSI1+ query and $Q_2$ is a CQAC with closed ACs.
Let $\phi_h$ and $\phi_b$ be the head and body entailments, respectively. Then, checking $\phi_h$ is polynomial and checking $\phi_b$ is NP-complete.
\end{corollary}
\subsection{More examples to illustrate the technique}
\label{subsec-examples}
Another example to use later to illustrate the functionality of the second kind of coupling rules.
\begin{example}
\label{ex:running412}
Consider a relational schema with the binary relations $e$ and $a$, as well as the following two CQACs over this schema.
\begin{center}
\begin{tabular}{lll}
$Q_1: q(W_1,W_2)$ & $:-$ &$a(W_1,W_2,Y),e(X,Y),e(Y,Z),X\geq 5,Z\leq 5$ \\
$Q_2: q(W_1,W_2)$ & $:-$ &$e(A,B),e(B,C_1),e(C_2,D),e(D,E),a(W_1,W_2,B),$\\
& &$ a(W_1,W_2,D), C_1\leq C_2,A\geq 5,E\leq 5$\\
\end{tabular}
\end{center}
Checking the containment $Q_2\sqsubseteq Q_1$, note that there are two containment mappings $\mu_1$, $\mu_2$ from $Q_{10}$ to $Q_{20}$ such that $\mu_1(W_i)=\mu_2(W_i)=W_i$, and
\begin{itemize}
\item $\mu_1:$ $Y\rightarrow B$, $X\rightarrow A$, $Z\rightarrow C_1$.
\item $\mu_2:$ $Y\rightarrow D$, $X\rightarrow C_2$, $Z\rightarrow E$.
\end{itemize}
Then, applying the mappings on the query entailment we conclude the following implication:
$$((C_1\leq C_2)\wedge(A\geq 5)\wedge(E\leq 5))\Rightarrow((A\geq 5)\wedge (C_1\leq 5)) \vee ((C_2\geq 5)\wedge (E\leq 5))$$
Analyzing the aforementioned entailment, it is easy to verify that it is true, since $(C_1\leq C_2)\Rightarrow (C_1\leq c)\vee (C_2\geq c)$ is true for every constant $c$; hence, $Q_2\sqsubseteq Q_1$.
Let us now construct $Q_1^{Datalog}$ from $Q_1$ and $Q_2^{CQ}$ from $Q_2$. To construct $Q_1^{Datalog}$ from $Q_1$ we follow the algorithm in Section~\ref{subsec-construct-Datalog}. In particular, we initially construct the query rule, which is given as follows. For simplicity in the notation, we will denote by $\overline{W}$ the vector of head variables $W_1,W_2$. Note that the subgoals $I_{\geq 5}(X,\overline{W})$, $I_{\leq 5}(Z,\overline{W})$ correspond to the ACs $X\geq 5$ and $Z\leq 5$, respectively.
\begin{center}
\begin{tabular}{ll}
$Q_1^{Datalog}: q(\overline{W})$ $:-$ &$e(X,Y),e(Y,Z),a(\overline{W},Y),I_{\geq 5}(X,\overline{W}),I_{\leq 5}(Z,\overline{W})$\\
\end{tabular}
\end{center}
Then, we construct the basic mapping and coupling rules, which are given by the following rules:
\begin{center}
\begin{tabular}{lll}
$J_{\geq 5}(X,\overline{W})$ $:-$ &$e(X,Y),e(Y,Z),a(\overline{W},Y),I_{\leq 5}(Z,\overline{W})$ & (mapping rule)\\
$J_{\leq 5}(Z,\overline{W})$ $:-$ &$e(X,Y),e(Y,Z),a(\overline{W},Y),I_{\geq 5}(X,\overline{W})$ & (mapping rule)\\
$I_{\leq 5}(X,\overline{W})$ $:-$ &$J_{\geq 5}(X,\overline{W})$ & (coupling rule)\\
$I_{\geq 5}(X,\overline{W})$ $:-$ &$J_{\leq 5}(X,\overline{W})$ & (coupling rule)\\
$I_{\leq 5}(X,\overline{W})$ $:-$ &$J_{\geq 5}(Y,\overline{W}), U(X,Y)$ & (coupling rule)\\
$I_{\geq 5}(X,\overline{W})$ $:-$ &$J_{\leq 5}(Y,\overline{W}), U(X,Y)$ & (coupling rule)\\
\end{tabular}
\end{center}
To find the $Q_2^{CQ}$, we initially copy the head $Q_2$, along with its relational subgoals. Then, we consider the subgoal $U(C_1,C_2)$ representing the AC $C_1\leq C_2$, as well as the unary suboals $U_{\geq 5}(A,\overline{W})$ and $U_{\leq 5}(E,\overline{W})$ to represent the ACs $A\geq 5$ and $E\leq 5$, respectively. Consequently, we end up with the following CQ definition:
\begin{center}
\begin{tabular}{lll}
$Q_2^{CQ}: q(W_1,W_2)$ & $:-$ &$e(A,B),e(B,C_1),e(C_2,D),e(D,E),a(W_1,W_2,B),$\\
& &$ a(W_1,W_2,D), U(C_1,C_2),U_{\geq 5}(A,\overline{W}),U_{\leq 5}(E,\overline{W})$\\
\end{tabular}
\end{center}
Finally, the link rules included in the Datalog query $Q_1^{Datalog}$ are constructed as follows:
\begin{center}
\begin{tabular}{ll}
$I_{\leq 5}(X,\overline{W})$ $:-$ &$U_{\leq 5}(X,\overline{W})$\\
$I_{\geq 5}(X,\overline{W})$ $:-$ &$U_{\geq 5}(X,\overline{W})$\\
\end{tabular}
\end{center}
%
%
%
%
%
\end{example}
{\sl Useful observation:}
Notice that, because of the restrictions we have assumed on our queries, $\overline{W}$ as it appears in the construction of the Datalog query does not contain any of the variables in the first position of a semi-unary predicate.
Finally, it helps with the inuition to obseerve the following: Even if the query $Q_1$ was different but only as concerns AC that involve head variables, the Datalog query would be the same because we do the test for such ACs in the preliminary step. Thus the following CQAC would have been transformed to the same query as above:
\begin{center}
\begin{tabular}{ll}
$Q_1({W_1,W_2}):-$ & $a(W_1,W_2,Y), e(X,Y),e(Y,Z),$\\
& $X\geq 5,Z\leq 8,W_1<W_2,W_1< 4.$\\
\end{tabular}
\end{center}
\subsection{Preliminary partial results and intuition on the proof of Theorem \ref{thm:main123}
}
\label{subsec-simplefacts}
The proof of Theorem \ref{thm:main123} is presented in the \ref{prf:thm-main123p}. Here we give some insight into the technicalities involved in its proof.
In our proof, we will apply the Datalog query $Q_1^{Datalog}$ on the canonical database of the CQ query $Q_2^{CQ}$ constructed from the contained query $Q_2$.
This canonical database uses constants (different from the constants in the ACs) that correspond one-to-one to variables of the query $Q_2$.
Thus, as we compute facts, each fact being either an $I$ fact or a $J$ fact, we do the following observations about the result of firings for each of the two kinds of recursive rules (i.e., the coupling rules and the mapping rules):
(all the $\theta$s represent either $\leq$ or $\geq$ and the $c_i$s are constants from the ACs of the queries.
\begin{itemize}
\item We have two kinds of coupling rules. Consider a coupling rule of the first kind which is of the form: $$I_{\theta_1 c_1}(X,\overline{W}) :-~J_{\theta c_2}(X,\overline{W}).$$
When this rule is fired, its variable $X$ is instantiated to a constant, $y$, in the canonical database, $D$, of $Q_{20}$. The constant $y$ corresponds to the variable $Y$ of $Q_2$ by convention. Then the following is true by construction: $X\theta_1 c_1 \vee X\theta_2 c_2$, and, hence, the following is true:
$$\beta_2\Rightarrow X\theta_1 c_1 \vee X\theta_2 c_2$$
Now consider the other kind of coupling rule, which is of the form:
$$I_{\theta_1 c_1}(X,\overline{W}) :-~J_{\theta c_2}(Y,\overline{W}),U(X,Y).$$
By construction of the rule, the EDB $U(X,Y)$ is mapped in $D$ to two constants/variables
such that there in $Q_2$ an AC which is $X\leq Y$. Thus, by construction of the rule, the following is true again:
$$\beta_2\Rightarrow X\theta_1 c_1 \vee Y\theta_2 c_2$$
We say in both cases of coupling rules that the facts in both sides of the rule are coupled and that the corresponding ACs are coupled.
\item Consider a mapping rule
$$J_{\theta_1 c_1}(Z,\overline{W}) :-~BodyQ_1,I_{\theta_2 c_2}(X,\overline{W}), I_{\theta_3 c_3}(X,\overline{W}),\dots.$$
The $BodyQ_1$ denotes all the relational subgoals of $Q_1$. When a mapping rule is fired, then there is a containment mapping, $\mu$, from the relational subgoals of $Q_1$ to the relational subgoals of $Q_2$ and, moreover, the $I$ facts in the body of the rule have been computed in previous rounds of the computation.
The $I$ facts can be computed either via link rules or via coupling rules. When the $I$ facts in the body of the rule (for the instantiation that fires the rule) are computed via coupling rules using $J$ facts, each $I$ fact is coupled with a $J$ fact.
Notice that each $I$ fact corresponds to an AC in $\mu(\beta_1)$ by construction of a mapping rule.
Putting the implications we derived for coupling rules above together for all $I$ facts in the body of the mapping rule,
we derive the implication:
$$\beta_2 \Rightarrow \mu (\beta_1)\vee e_1 \vee e_2 \vee \cdots , \vee e_t$$
where $ e_1 , e_2, \ldots$ are the ACs corresponding to the $J$ facts from which each $I$ fact was computed. Finally, observe that by construction of the rule, one of the ACs in $\mu(\beta_1)$ is not represented in the body of the rule (it is represented in the head of the rule). This justifies the
presence of $e_t$ in the implication, which represents this special AC in $\mu(\beta_1)$.
\end{itemize}
\section{When U-CQAC MCRs compute certain answers}
\label{sec-6}
\def\mathcal{L}{\mathcal{L}}
\def\mathcal{I}{\mathcal{I}}
\def\mathcal{P}{\mathcal{P}}
\def\mathcal{D}{\mathcal{D}}
\def\mathcal{V}{\mathcal{V}}
\def\mathcal{C}{\mathcal{C}}
\def\mathcal{T}{\mathcal{T}}
\def\mathcal{S}{\mathcal{S}}
\def\mathcal{U}{\mathcal{U}}
\def\mathcal{M}{\mathcal{M}}
\def\mathcal{E}{\mathcal{E}}
\def\mathcal{K}{\mathcal{K}}
\def\textsf{LAV}{\textsf{LAV}}
\def\textsf{Dept}_\textsf{A}{\textsf{Dept}_\textsf{A}}
\def\textsf{Dept}_\textsf{B}{\textsf{Dept}_\textsf{B}}
\def\textsf{Staff}{\textsf{Staff}}
\def\textsf{CoreCover$\C$}{\textsf{CoreCover$\mathcal{C}$}}
In this section we prove that, given CQAC query and views, if there is a maximally contained rewriting (MCR) in the language of (possibly infinite) union of CQACs then this MCR computes all the certain answers on any view instance $\mathcal{I}$. This section extends the results in \cite{AfratiK10} for CQs.
Moreover, we prove this result in a more general setting, in that we also assume that there is a set of constraints $\mathcal{C}$ that the database ought to satisfy.
The set $\mathcal{C}$ contains tuple generating dependencies (tgds) and equality generating dependencies (egds). We assume that the chase algorithm (see description of chase algorithm as well as definitions for tgds and egds in \ref{pre:dependencies-chase}) terminates on $\mathcal{C}$.
We give the definition of certain answers under constraints, as follows.
\begin{definition}
Suppose there exists a database instance $D$ such that $\mathcal{I}\subseteq \mathcal{V}(D)$. Then,
we define the certain answers of ($Q,\mathcal{I}$) with respect to $\mathcal{V}$ as follows:
\begin{itemize}
\item Under the Open World Assumption:
\[\text{certain}(Q,\mathcal{I})=\bigcap\{Q(D): D \text{ such that } \mathcal{I}\subseteq\mathcal{V}(D)\}\] In the presence of a set of constraints $\mathcal{C}$, we also require that all databases $D$ used for certain$(Q,\mathcal{I})$ satisfy $\mathcal{C}$ and denote it by $\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$.
\end{itemize}
If there is no database instance $D$ such that $\mathcal{I}\subseteq \mathcal{V}(D)$, we say that the set $\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$ is undefined.
\end{definition}
\subsection{Preliminaries}
We first define query containment under constraints:
\begin{definition}
Let $\mathcal{C}$ be a set of tdgs and egds, and $Q_1$, $Q_2$ be two conjunctive queries. We say that {\em $Q_1$ is contained in $Q_2$ under the dependencies $\mathcal{C}$,} denoted $Q_1\sqsubseteq_\mathcal{C} Q_2$, if for all databases $D$ that satisfy $\mathcal{C}$ we have that $Q_1(D)\subseteq Q_2(D)$.
\end{definition}
We check CQAC containment under contstraints $\mathcal{C}$ by using the $\mathcal{C}$-canonical databases (see \ref{pre:dependencies-chase1}).
We define contained rewriting under constraints:
\begin{definition}
\label{cont-rewr-dfn}
(Contained rewriting)
Let $Q$ be a query defined on schema $\cal S$, and $\cal{V}$ a set of views defined on $\cal S$. Let $R$ be a query formulated in terms of the view relations in
the set $\cal{V}$.
$R$ is a {\em contained rewriting of $Q$ using $\mathcal{V}$ under the OWA and under the constraints $\mathcal{C}$} if and only if for every view instance $\mathcal{I}$ the following is true:
For any database $D$ such that $\mathcal{I}\subseteq \mathcal{V}(D)$ that satisfies the constraints in $\mathcal{C}$, we have that $R(\mathcal{I})\subseteq Q(D)$.
\end{definition}
\begin{theorem}
Suppose query $Q$, views $\cal{V}$, and rewriting $R$ all belong to the language of CQACs. Then $R$ is a contained rewriting of
$Q$ using views $\cal{V}$ if and only if $R^{exp}\sqsubseteq_\mathcal{C} Q$.
\end{theorem}
\begin{proof} If the expansion is not contained in the query, then we find a counterexample to prove that it is not a contained rewriting as follows: Since $R^{exp}$ is not contained in $Q$, there is a $\mathcal{C}$-canonical database $D$ of $R^{exp}$ such that a tuple $t$ is computed by $Q_2$ on $D$ but not by $Q_1$. We compute $\mathcal{V}$ on $D$ and produce view instance $\mathcal{I}$. Then $t$ is in $R(\mathcal{I})$ (because a subset of $\mathcal{I}$ is isomorphic to the body of $R$) but $t$ is not in $Q(D)$.
If the expansion $R^{exp}$ is contained in the query then, since $\mathcal{I} \subseteq \mathcal{V}(D)$ for any $D$ that satisfies the constraints, we have that
$ R(\mathcal{I}) \subseteq R(\mathcal{V}(D) ) $.
However $R(\mathcal{V}(D) )$ is equal to $R^{exp}(D )$ because to compute the former we first apply the mappings from the view definition to $D$ (to compute $\mathcal{V}(D)$) and then apply the mapping from $R$ to $\mathcal{V}(D)$ thus resulting in a mapping from $R^{exp}$ to $D$ for each tuple that is computed.
Consequently, the following is true:
$$ R(\mathcal{I}) \subseteq R(\mathcal{V}(D) ) \subseteq R^{exp}(D ) \subseteq Q(D)$$
for any $D$ that satisfies the constraints. Hence $R$ is a contained rewriting under the constraints.
\end{proof}
\subsubsection{Database AC-instance with t-instance}
A {\em database AC-instance with ACs} is a database with domain a set of constants and a set of variables that we call {\em labeled nulls} (the two sets are disjoint), i.e., it contains relational atoms that use labeled nulls and constants. It may also contain ACs among the labeled nulls or among labeled nulls and constants.
When the ACs define a total ordering, then we call $\mathcal{I}$ a \textit{t-instance}.
Let $J_1$, $J_2$ be sets of atoms over the schema ${\cal S}$ such that $J_1$ is an AC-instance and $J_2$ is a t-instance. An
\textit{order-homomorphism}p $h : J_1\rightarrow J_2$ is a mapping from the atoms in $J_1$ to the atoms in $J_2$ with the following properties:
\begin{enumerate}
\item For every constant $c$ in $J_1$, we have $h(c) = c$.
\item For every atom $r(X_1,\dots,X_m)$ in $J_1$, we have that $r(h(X_1),\dots,h(X_m))$ is an atom in $J_2$, where $X_1,\dots,X_m$ are either variables or constants.
\item if $(X_1\;\theta\;X_2)$ is true in $J_1$, where $\theta$ is $<, >, \leq, \geq, =$, then $(h(X_1)\;\theta\;h(X_2))$ is implied by the partial order of $J_2$.
\end{enumerate}
\subsection{Representative possible worlds (RPW)}
In this section, we will prove that, for CQAC views, a
maximally contained rewriting $\mathcal{P}$ with respect to U-CQAC \footnote{In the literature, usually, by U-CQAC we define the class of finite unions of CQACs, in this section we assume that it may be also infinite.} of a CQAC query $Q$ under a given set of constraints computes the certain answers of $Q$ under the OWA, i.e., we prove the following theorem.
\begin{theorem}\label{owa_mcrs}
Let $\mathcal{C}$ be a set of constraints that are tgds and egds.
Let $Q$ be a CQAC query, $\mathcal{V}$ a set of CQAC views. Suppose there exists an MCR ${\cal R}_{MCR}$ of $Q$ with respect to U-CQAC and under the constraints $\mathcal{C}$. Let $\mathcal{I}$ be a view instance such that the set $\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$ is defined.
Then, under the open world assumption, ${\cal R}_{MCR}$ computes all the certain answers of $Q$ on any view instance $\mathcal{I}$ under the constraints $\mathcal{C}$, that is: ${\cal R}_{MCR}(\mathcal{I})=\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$.
\end{theorem}
We define the concept of representative possible worlds of a view instance $\mathcal{I}$ in order to analyze how we compute the certain answers.
Given a view instance $\mathcal{I}$, we define a set of {\em representative possible worlds (RPW, for short)} $\mathcal{P}_\mathcal{I}$. A RPW is a AC-instance. The set $\mathcal{P}_\mathcal{I}$ has the following properties: a) for all
$D_\mathcal{I}\subseteq \mathcal{P}_\mathcal{I}$ the following is true: $\mathcal{I}\subseteq\mathcal{V}(D_\mathcal{I})$, b) for each database instance $D$ such that $\mathcal{I}\subseteq\mathcal{V}(D)$ there is a representative possible world $D_{\mathcal{I}}$ in $\mathcal{P}_\mathcal{I}$ such that there is an
order-homomorphism
from $D_{\mathcal{I}}$ to $D$
The set $\mathcal{P}_\mathcal{I}$ of RPWs is finite and we can construct it by the following algorithm, consisting of two main stages:
\paragraph{Stage 1:} In this stage we construct a Boolean query.
Let $\mathcal{I}$ be a view instance. We use $\mathcal{I}$ to produce a Boolean CQAC rewriting, $R_{\mathcal{I}}$, as follows:\footnote{A rewriting is a CQAC query expressed in terms of the views; it stands alone, it does not have to be contained in a specific query.}
\begin{enumerate}
\item We turn all the constants in $\mathcal{I}$ to variables so that distinct constants are turned into distinct variables.
\item We add on the variables the ACs that imply a total ordering, which is the ordering of the constants they came from (recall that constants are from a totally ordered domain).
\end{enumerate}
\paragraph{Stage 2:} The following steps construct the set of RPWs:
\begin{enumerate}
\item We consider the expansion $R_{\mathcal{I}}^{exp}$ of $R_{\mathcal{I}}$. We consider the set ${\cal{R}}_{\mathcal{I}}$ of the canonical databases of $R_{\mathcal{I}}^{exp}$ for which $R_{\mathcal{I}}^{exp}$ computes to true.
Each element of $R_{\mathcal{I}}$ is a database t-instance.
\item For each $D$ in ${\cal{R}}_{\mathcal{I}}$, we do as follows: We apply the chase on $D$ with constraints
$\mathcal{C}$. Thus, if the chase succeeds, we derive $D_{chased}$ and
add it in $\mathcal{P}_\mathcal{I}$ which is the set of representative possible worlds.
\end{enumerate}
This finishes the construction of $\mathcal{P}_\mathcal{I}$. Notice that the databases in $\mathcal{P}_\mathcal{I}$ are exactly all the $\mathcal{C}$-canonical databases of $R_{\mathcal{I}}^{exp}$.
\begin{theorem}
The above procedure finds all representative possible worlds of the view instance $\mathcal{I}$
\end{theorem}
\begin{proof}
Let $D$ be a database instance that satisfies the constraints $\mathcal{C}$ and such that $\mathcal{I}\subseteq \mathcal{V}(D)$. The tuples in $\mathcal{I} \cap \mathcal{V}(D)$ are produced by an
order-homomorphism,
$ h_1$, from $R_{\mathcal{I}}^{exp}$ to $D$. To see that, imagine that we apply the view definitions in $\mathcal{V}$ on $D$ in one step (since we know that $\mathcal{I}\subseteq \mathcal{V}(D)$).
This means that $D$ is contained under $\mathcal{C}$ (we can imagine that $D$ is a Boolean query with no variables, just constants) in $R_{\mathcal{I}}^{exp}$.
Thus, by the containment test, and taking into account Theorem~\ref{thm-chase-main22}, there is a $\mathcal{C}$-canonical database of $R_{\mathcal{I}}^{exp}$ that maps isomorphically on $D$ by $h_2$ according to the following proposition.
\begin{proposition}
\label{pro-RPW-property}
Suppose database instance $D$ which, viewed as a Boolean query, is contained in a CQAC $Q$. Then, there is a canonical database of $Q$ that maps isomorphically on $D$.
\end{proposition}
\begin{proof}
The proof of this proposition results from the observation that, by definition, the canonical databases of $Q$ represent all homomorphic images of the relational atoms of $Q$ that satisfy the ACs in $Q$.
\end{proof}
\vspace*{-.6cm}
\end{proof}
The following is an example showing how we construct $R_{\mathcal{I}}$ and $R_{\mathcal{I}}^{exp}$.
\begin{example}
Consider the query $Q$ and the views $V_1$, $V_2$ with the following definitions.
\begin{center}
\begin{tabular}{ll}
$Q: q()$ $:-$ &$a(X,Y,W), b(Y,Z,W), X\leq 14$\\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ll}
$V_1: v_1(X,Y)$ $:-$ &$a(X,Y,Z),X\leq 9$\\
$V_2: v_2(X,Y)$ $:-$ &$b(X,Y,Z)$\\
\end{tabular}
\end{center}
Now, we consider the following view instance:
$\mathcal{I}:\{v_1(1,2),v_2(2,3),v_1(5,6)\}$.
We build a Boolean rewriting from $\mathcal{I}$ as we explained above, which, in this specific view instance is the following rewriting:
\begin{center}
\begin{tabular}{ll}
$R_{\mathcal{I}}: q()$ $:-$ &$v_1(X_1,X_2), v_2(X_2,X_3), v_1(X_5,X_6),$\\
&$X_1< X_2,X_2< X_3,X_5< X_6,X_3< X_5$\\
\end{tabular}
\end{center}
where, variable $X_1$ represents constant 1, variable $X_2$ represents constant 2, etc. Since $1<2<3<5<6$, we have added in the above query $X_1< X_2,X_2< X_3,X_5< X_6,X_3< X_5$.
This rewriting is a contained rewriting in the query $Q$. However this is not always the case, e.g., imagine a view instance $\mathcal{I}'$ that contained only $v_1(5,6)$; it is easy to verify that the rewriting built based on this view instance $\mathcal{I}'$ would not have been contained in $Q$.
The expansion of the rewriting $R_{\mathcal{I}}$ is the following:
\begin{center}
\begin{tabular}{ll}
$R^{exp}_{\mathcal{I}}: q()$ $:-$ &$a(X_1,X_2,Z_1), b(X_2,X_3,Z_2), a(X_5,X_6,Z_3),$\\
&$X_1< X_2,X_2< X_3,X_5< X_6,X_3< X_5$\\
\end{tabular}
\end{center}
The representative possibe worlds for $\mathcal{I}:\{v_1(1,2),v_2(2,3),v_1(5,6)\}$ are obtained from the canonical databases of the expansion $R^{exp}_{\mathcal{I}}$. Each RPW contains the relational atoms in $R^{exp}_{\mathcal{I}}$ and the variables (labeled nulls) $X_i$ have the total order shown in $R^{exp}_{\mathcal{I}}$. However the variables (labeled nulls) $Z_i$ can have any ordering, thus all their orderings create more than one RPW.
\end{example}
\subsection{When a view instance has at least one representative possible world}
There is a broad class of views where the set $\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$ is always defined independently of the view instance $\mathcal{I}$, as the following proposition shows.
\begin{proposition}\label{repeat_vars_lemma}
Let $\mathcal{V}$ be a set of CQAC views and $Q$ a CQAC query. If there are no egds in the set of constraints $\mathcal{C}$ and, each view definition a) has no repeated variables in the head and b) has no ACs that contain head variables, then the set $\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$ is defined on any view instance $\mathcal{I}$.
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\end{proposition}
\begin{proof}
When we construct the RPWs, for each view tuple in the view instance $\mathcal{I}$, we associate position-wise each variable in the head of the view definition with a constant in the view tuple. This should create an order-homomorphism from the head of the view definition to the view tuple. This is possible because there are no duplicate variables and no ACs on the head variables that could be violated.
\vspace*{-.5cm}
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\end{proof}
\vspace*{-.4cm}
Towards future work, we begin a discussion on it in Section \ref{sec-app-unclean} to argue that
even in the case where certain answers are not defined, an MCR can be used to produce results that ``make sense'', when we assume that we are dealing with
non-clean data.
\subsection{Main result }
We will now prove Theorem \ref{owa_mcrs}, which is the main result of this section and its main ingredients are the following Propositions \ref{pro-qi-certain} and \ref{owa_mcrs_lemma}. The first says that if we take the intersection of all the answers computed by applying the query $Q$ on each of the representative possible worlds we produce all the certain answers of the query. The second one says that there is a CQAC contained rewriting that produces this intersection.
We also need to use the fact that each CQAC contained rewriting computes only certain answers if applied on a view instance $\mathcal{I}$; this is true by the definition of contained rewriting (Definition \ref{cont-rewr-dfn}).
\begin{proposition}
\label{pro-qi-certain}
Let $\mathcal{C}$ be a set of constraints that are tgds and egds. Let $\mathcal{V}$ be a set of CQAC views and $\mathcal{I}$ a view instance such that the set $\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$ is defined.
Let $Q$ be a CQAC query.
Then $\bigcap_{\mathcal{D}_\mathcal{I} \in \mathcal{P}_{\mathcal{I}}} Q(D_\mathcal{I})$ is equal to the certain answers of $Q$ given $\mathcal{V}$ on view instance $\mathcal{I}$ under the constraints $\mathcal{C}$, where $ \mathcal{P}_{\mathcal{I}}$ is the set of representative possible worlds on $\mathcal{I}$.
\end{proposition}
\begin{proof}
Certainly, $\bigcap_{\mathcal{D}_\mathcal{I} \in \mathcal{P}_{\mathcal{I}}} Q(D_\mathcal{I})$ is a superset of the set of certain answers.
We want to prove that it is also a subset of the set of certain answers. By contradiction, suppose not. Then, there is a PW $D$ such that the answers of $Q$ on $D$ do not contain all the tuples in $\bigcap_{\mathcal{D}_\mathcal{I} \in \mathcal{P}_{\mathcal{I}}} Q(D_\mathcal{I})$. This means that there is a tuple $t$ in $\bigcap_{\mathcal{D}_\mathcal{I} \in \mathcal{P}_{\mathcal{I}}} Q(D_\mathcal{I})$
which is not in $Q(D)$. However, according to the definition of RPW, there is a RPW
$D_r$ such that there is an
order-homomorphism
from $D_r$ to $D$, hence $Q(D_r)\subseteq Q(D)$. Since $t$ is in $\bigcap_{\mathcal{D}_\mathcal{I} \in \mathcal{P}_{\mathcal{I}}} Q(D_\mathcal{I})$, $t$ is also in $Q(D_r)$. Hence contradiction.
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\end{proof}
\begin{proposition}\label{owa_mcrs_lemma}
Let $\mathcal{C}$ be a set of constraints that are tgds and egds.
Let $Q$ be CQAC query and $\mathcal{V}$ be a set of CQAC views. Let $\mathcal{I}$ be a view instance such that
the set $\text{certain}_{\mathcal{C}}(Q,\mathcal{I})$ is defined.
Then, given a
tuple $t_0 \in certain(Q,\mathcal{I})$, there is a contained CQAC rewriting $R$
such that $t_0 \in R(\mathcal{I})$.
\end{proposition}
\begin{proof}
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%
We consider as $R$ the Boolean query $R_{\mathcal{I}}$ with the proper variables in the head that are the variables that represent the constants in $t_0$.
Now we need to prove that $R$ is a contained rewriting. $R$ was created from $R_{\mathcal{I}}$ which produces all the RPWs. Since $t_0$ is in the certain answers of the query $Q$, there is a
order-homomorphism
from $Q$ to every RPW and this order-homomorphism produces $t_0$.
All the RPWs are all the canonical databases of $R_{\mathcal{I}}^{exp}$ chased with the constraints. Hence the previously mentioned order-homomorphisms provide the proof for the containment test that proves containment of $R_{\mathcal{I}}^{exp}$ to $Q$ under the constraints $\mathcal{C}$. Since $R$ only differs from $R_{\mathcal{I}}$ as to the head, the same
order-homomorphisms
can be used to prove containment of $R^{exp}$ to $Q$ under the constraints $\mathcal{C}$.
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\end{proof}
We now put all together to finish the proof of Theorem \ref{owa_mcrs}:
\begin{proof}{\bf (Theorem \ref{owa_mcrs})}
We will show the following:
\begin{enumerate}
\item $\mathcal{P}(\mathcal{I}) \subseteq$ certain$(Q,\mathcal{I})$
\item certain$(Q,\mathcal{I})\subseteq \mathcal{P}(\mathcal{I})$
\end{enumerate}
Since $\mathcal{P}$ is a contained rewriting of $Q$, the first is a direct consequence of the definition of a contained rewriting.
To prove (2), we use the two propositions. One proposition says that we can compute all the certain answers by considering only a finite number of possible worlds, $\mathcal{P}(\mathcal{I})$. The other one uses $\mathcal{P}(\mathcal{I})$ to prove that there is CQAC contained rewriting which computes a tuple $t_0$ if this tuple is in certain answers.
\end{proof}
\section{Finding MCR for CQAC-RSI1+ Query and CQAC$^-$ Views}
\label{subsec-mcr-datalogAC}
In this section, we show that for a RSI1+ query and a special case of CQAC views, we can find an MCR in the language of (possibly infinite) union of CQACs.
We will show that this MCR is expressed in
Datalog$^{AC}$. In detail, we consider
the following case of query and views:
\begin{itemize}
\item There are only closed arithmetic comparisons in both query and views.
\item The views are CQAC queries which do not use ACs of the form $X\leq Y$ or $X\geq Y$ where $X$ is a head variable and $Y$ is a nondistinguished variable. We call this class of CQAC queries CQAC$^{-}$.
\item The query is a esRSI1+ query.
%
\end{itemize}
We think of an expansion of a rewriting as having three kinds of variables: a) the head variables, b) the {\em view-head} variables, which are all the variables that are present in the rewriting and c) the {\em view-nondistinguished} variables, which are all the other variables in the expansion of the rewriting (these do not appear in the rewriting). The head variables are also view-head variables.
\subsection{ACs in rewritings}
\label{subsec-rectified}
Consider a CQAC query and a set of CQAC views.
When we have a rewriting $R$ the variables in the rewriting also satisfy some ACs that are in the closure of the ACs in the expansion of the rewriting. We include those ACs in the rewriting $R$ and produce $R'$, which we call the {\em AC-rectified rewriting of $R$}. Thus, the expansions of $R$ and $R'$ are equivalent queries. Hence, we derive the following proposition:
\begin{proposition
Given a set of CQAC views, a rewriting $R$ and its rectified version $R'$, the following is true: For any
view instance $\mathcal{I}$ such that there is a database instance $D$ for which $\mathcal{I}\subseteq \mathcal{V}(D)$, we have that $R(\mathcal{I})=R'(\mathcal{I})$.
\end{proposition}
\begin{definition}
We say that a rewriting $R$ is {\em AC-contained} in a rewriting $R_1$ if the AC-rectified rewriting $R'$ of $R$ is contained in $R_1$ as queries.
\end{definition}
From hereon,
when we refer to a rewriting, we mean the AC-rectified version of it and when we say that a rewriting is contained in another rewriting we mean that it is AC-contained.
An example follows.
\begin{example}
\label{ex:export-nondist1}
Consider query $Q$ and view $V_2$:
\begin{center}
\begin{tabular}{l l}
$Q(A)$ & $~\hbox{\rm :-}~~p(A), A < 4.$\\
$V_2(Y,Z)$ & $~\hbox{\rm :-}~~p(X), s(Y,Z), Y \leq X, X \leq Z.$\\
\end{tabular}
\end{center}
The following rewriting is a contained rewriting of the query in terms of the view in the language CQAC:
$$R(Y_1)~~\hbox{\rm :-}~~V_2(Y_1,Z_1),V_2(Y_2,Z_2), Z_1\leq Y_2, Y_1\geq Z_2, Y_1 < 4.$$
\noindent
Now consider the following contained rewriting:
$R'(X) $ ~\hbox{\rm :-}~ $V_2(X,X), X < 4.$
\noindent
This rewriting uses only one copy of the view. We can show that $R$ is not contained in $R'$ and that
$R'$ is not contained in $R$. However, they compute the same output on any view instance (to see that just include in $R$ the ACs $Y_1\leq Z_1$ and $Y_2\leq Z_2$).
\end{example}
\subsection{Building MCRs for RSI1 queries}
\label{subsec-buildingMCRs}
First, in this subsection, we present the algorithm for building an MCR in the language of (possibly infinite) union of CQACs for the case of CQAC$^-$ views and queries that are RSI1. The algorithm for building an MCR for query $Q$ and view set $\mathcal{V}$ is the following:
{\bf Algorithm MCR-RSI1}:
\begin{enumerate}
\item For the query $Q$, we construct the Datalog query $Q^{Datalog}$.
We use the construction in Subsection \ref{subsec-construct-Datalog}. The link rules will use the constants present in the views and in the query.
\item For each view $v_i$ in $\mathcal{V}$, we construct a new view $v_i^{CQ}$. We use the construction in Subsection \ref{subsec-contained-trans}.
\item Consider the EDB predicates introduced in Section \ref{ssec:mot} (and used in Steps 2 and 3 above) which encode ACs. We call them AC-EDB predicates and use them to construct a new set of {\em auxiliary} views as follows: a) Views with head $u_{\theta c}$, one for each
semi-unary predicate $U_{\theta c}$. The definition is $u_{\theta c}(\overline{W},X)~:-~
U_{\theta c}(\overline{W},X)$. b) A single view $u$, whose definition is $u(X,Y)~:-~
U(X,Y)$. We will refer to those EDB predicates (i.e., the $U(X,Y)$ predicate and the semi-unary predicates) as {\em AC-predicates} or {\em AC-subgoals}.
\item We consider now the view set $\mathcal{V}{^{CQ}}$ that contains the views as constructed in the two previous steps above.
\item We find an MCR $R^{CQ}_{MCR}$ for the Datalog query $Q^{Datalog}$ using the views
in $\mathcal{V}{^{CQ}}$. For building the MCR we use the inverse rule algorithm \cite{Duschka97-3}.
\item To obtain an MCR $R_{MCR}$ for $Q'$, we replace in the found MCR $R^{CQ}_{MCR}$, each $v_i^{CQ}$ by
$v_i$, each $u_{\theta c}(X)$ by arithmetic comparison $X \theta c$ and each $u(X,Y)$ by arithmetic comparison $X\leq Y$.
\end{enumerate}
\begin{example}
\label{ex-prime-old1}
In this example,
the reverse rule algorithm produces an MCR without including the auxiliary views, hence, we have not written these views, in order to keep things simple.
Consider the query $Q_1$ and the views:
\begin{center}
\begin{tabular}{ll}
$Q_1()$ & $\mathrm{:-}~e(X,Z),e(Z,Y),X\geq 5,Y\leq 8.$\\
$V_1(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),Z\geq 5.$\\
$V_2(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),Z\leq 8.$\\
$V_3(X,Y)$ & $\mathrm{:-}~e(X,Z_1),e(Z_1,Z_2),e(Z_2,Z_3),e(Z_3,Y).$\\
\end{tabular}
\end{center}
We have already built the Datalog program $Q_1^{Datalog}$ in Example~\ref{ex-running-dat} in a more general setting, where we assume that the query is not Boolean. Here, we use the same $Q_1^{Datalog}$ only that we delete $\overline{W}$ from all the rules. We need to add the link rules which will be with respect to constants 5 and 8 (these are the only constants that appear in the definitions).
The views that will be used to apply the inverse-rule algorithm are:
\begin{center}
\begin{tabular}{ll}
$V'_1(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),U_{\geq 5}(Z).$\\
$V'_2(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),U_{\leq 8}(Z).$\\
$V'_3(X,Y)$ & $\mathrm{:-}~e(X,Z_1),e(Z_1,Z_2),e(Z_2,Z_3),e(Z_3,Y).$\\
\end{tabular}
\end{center}
Notice that we conveniently did not add any auxiliary views here because we guessed that they will not be needed.
In this example, it is relatively easy to anticipate the result of applying the inverse-rule algorithm, by observing the simple form of the expansions of $Q_1^{Datalog}$. Each expansion of $Q_1^{Datalog}$ is a simple path with two unary predicates, one at one end of the path and the other at the other end. Thus, the output of the inverse-rule algorithm is the following program. It is an MCR of $Q_1^{Datalog}$ using the views $V'_1(X,Y)$, $V'_2(X,Y)$, and $V'_3(X,Y)$.
\begin{center}
\begin{tabular}{ll}
$R'() $ & $\mathrm{:-}~v'_1(X,W),T(W,Z),v'_2(Z,Y).$\\
$T(W,W)$ & $\mathrm{:-}~.$\\
$T(W,Z)$ & $\mathrm{:-}~T(W,U),v'_3(U,Z).$\\
\end{tabular}
\end{center}
The following is an MCR of the input query $Q_1$ (rather than of $Q_1^{Datalog}$) using the views $V_1(X,Y)$, $V_2(X,Y)$ and $V_3(X,Y)$:
\begin{center}
\begin{tabular}{ll}
$R() $ & $\mathrm{:-}~V_1(X,W),T(W,Z),V_2(Z,Y).$\\
$T(W,W)$ & $\mathrm{:-}~.$\\
$T(W,Z)$ & $\mathrm{:-}~T(W,U),V_3(U,Z).$\\
\end{tabular}
\end{center}
\end{example}
\subsection{Proof that the algorithm {\bf MCR-RSI1} is correct}
The proof of the following proposition is a straightforward consequence of the construction of $R_{MCR}$ from $R^{CQ}_{MCR}$.
\begin{proposition}
\label{pro-conncect}
Consider the Datalog programs $R^{CQ}_{MCR}$ and $R_{MCR}$. For each CQAC Datalog-expansion $E$ of $R_{MCR}$, there is a CQ Datalog-expansion $E^{CQ}$ of $R^{CQ}_{MCR}$ (and vice versa), where the following is true: The relational subgoals of $E$ are isomorphic to the purely relational subgoals of $E^{CQ}$ (by ``purely relational subgoals we mean those that do not encode ACs) and each AC in $E$ corresponds to a subgoal in $E^{CQ}$ that encodes this AC.
\end{proposition}
\begin{theorem}
\label{thm-mcr-cont}
Given a query $Q$ which is RSI1 and views $\mathcal{V}$ which are
CQAC$^{-}$s, the following is true:
Let $R$ be a CQAC contained rewriting of $Q$ in terms of $\mathcal{V}$. Then $R$ is contained in the one found by the algorithm in Subsection~\ref{subsec-buildingMCRs} Datalog$^{AC}$ program $R_{MCR}$.
\end{theorem}
\begin{proof}
Let $R$ be a CQAC contained rewriting of $Q$ using $\mathcal{V}$ and let $R_{exp}$ be the view-expansion of $R$.
We assume $R$ is AC-rectified (see Subsection \ref{subsec-rectified}). Let $Q^{Datalog}$ be the transformed query of $Q$ as in Subsection \ref{subsec-construct-Datalog}. We argue using the following rewritings and their expansions:
\begin{itemize}
\item $R$ is a CQAC query which is a contained rewriting of $Q$ using $\mathcal{V}$.
\item $R_{exp}$ is the view-expansion of $R$ with respect to $\mathcal{V}$.
\item $R'$ is $R$ with ACs in the closure of ACs in $R$ now being relational predicates.
\item $R'_{exp}$ is the view-expansion of $R'$ with respect to $\mathcal{V}^{CQ}$.
\end{itemize}
We also consider:
\begin{itemize}
\item $R_{exp}^{CQ}$ is $R_{exp}$ transformed into a CQ as in Subsection \ref{subsec-contained-trans}.
\item $R^{CQ}$ is the rewriting that we prove can be created from $R_{exp}^{CQ}$ (we mean contained rewriting of $Q^{Datalog}$ using $\mathcal{V}^{CQ}$).
\end{itemize}
First, we observe that because we have the auxiliary views in $\mathcal{V}^{CQ}$, $R'$ is a rewriting in terms of $\mathcal{V}^{CQ}$ (we do not know yet whether it is a contained rewriting to the Datalog query).
The closure of ACs in $R_{exp}$ may contain: a) ACs carried over from the views definitions, b) ACs that involve only view-head variables and c) ACs that involve view-nondistinguished variables and are not in class (a) or (b).
Because of the constraint on the views to be only CQAC$^-$, the third class (c) does not exist.
The reason is that, in this class, belong ACs that are implied from at least two ACs, each one carried over from views definitions of two different view atoms in $R$. For this to happen, these two ACs should, each, relate a nondistinguished variable in a view definition with a head variable in the view definition (this is represented as view-head variable in $R_{exp}$).
Because $R$ is AC-rectified the second class (b) of ACs appear in $R$ too. Thus $R'_{exp}$ can be viewed as resulting from $R_{exp}$ by a CQ-transformation (i.e., according to the
Subsection \ref{subsec-contained-trans}). Thus we argue as follows:
Since $R$ is a contained rewriting to $Q$, $R_{exp}$ is a contained query to $Q$, and according to the results of Section 5, $R'_{exp}$ is a contained query to $Q^{Datalog}$, hence $R'$ is a contained rewriting to $Q^{Datalog}$. Hence $R'$ is contained in $R^{CQ}_{MCR}$.
Now, we use Proposition \ref{pro-conncect}.
$R$ and $R'$ differ only in that the ACs of one are AC-predicates of the other, in one to one fashion. For any Datalog-expansion of $R_{MCR}$ there is a Datalog-expansion of $R^{CQ}_{MCR}$ (and vice versa) that differ in the same way. Hence the Datalog-expansion of $R^{CQ}_{MCR}$ that proves $R'$ is in
$R^{CQ}_{MCR}$ can be used to derive a Datalog-expansion of $R_{MCR}$ that proves $R$ is in
$R_{MCR}$.
\end{proof}
\begin{theorem}
\label{thm-mcr-in-query}
Given a query $Q$ which is CQAC-RSI1 and views $\mathcal{V}$ which are
CQAC$^{-}$s, the following is true:
The found by the algorithm in Subsection~\ref{subsec-buildingMCRs} Datalog$^{AC}$ program, $R_{MCR}$, is a contained rewriting.
\end{theorem}
\begin{proof}
Consider a CQAC Datalog-expansion, $R$, of the found Datalog$^{AC}$ program, $R_{MCR}$. Take the view-expansion, $R_{exp}$, of $R$. Transform $R_{exp}$ into a CQ, $R_{exp}^{CQ}$, using the construction in Subsection \ref{subsec-contained-trans}. Now, we argue in the same way as we argued in the proof of Thorem \ref{thm-mcr-cont} to prove that $R_{exp}^{CQ}$ is the view-expansion of a Datalog-expansion of $R^{CQ}_{MCR}$.
According to the reverse-rule algorithm, $R_{exp}^{CQ}$ is contained in the $Q^{Datalog}$ program, hence according to the results in Section \ref{ssec:mot}, $R_{exp}$ is contained in the query $Q$. Consequently, $R$ is a contained rewriting of $Q$.
\end{proof}
Thus we have proved:
\begin{theorem}
\label{thm-mainsec6}
Given a query $Q$ which is CQAC-SI1 and views $\mathcal{V}$ which are
CQAC$^{-}$s, the algorithm in Subsection~\ref{subsec-buildingMCRs} finds an MCR of $Q$ using $\mathcal{V}$ in the language of (possibly infinite) union of CQACs.
\end{theorem}
\subsection{Building MCR for RSI1+ query}
Now, we present the algorithm for building an MCR in the language of (possibly infinite) union of CQACs for the case of CQAC$^-$ views and queries that are RSI1+. The algorithm for building an MCR for query $Q$ and viewset $\mathcal{V}$ is the following:
{\bf Algorithm MCR-RSI1+}:
\begin{enumerate}
\item We consider query $Q'$ which results from the given query $Q$ after we have removed the ACs that contain only head variables.
\item We apply the algorithm for building MCR for query $Q'$ and views $\mathcal{V}$ (from previous subsection). Let this MCR be
$R'_{MCR}$.
\item We add a new rule in $R'_{MCR}$ (and obtain $R_{MCR}$ ) to compute the query predicate $Q$ as follows:
$$Q(\overline{W}):- Q'(\overline{W}), ac_1, ac_2, \ldots$$
where $ac_1, ac_2, \ldots$ are the ACs that we removed in the first step of the present algorithm.
\end{enumerate}
\subsection{Proof that the algorithm {\bf MCR-RSI1+} is correct}
We consider the found by the {\bf Algorithm MCR-RSI1+} Datalog$^{AC}$ program, $R_{MCR}$.
Theorem \ref{thm-mcr-cont1} below proves that every CQAC contained rewriting is contained in $R_{MCR}$ and Theorem
\ref{thm-mcr-in-query1} proves that $R_{MCR}$ is a contained rewriting.
\begin{theorem}
\label{thm-mcr-cont1}
Given a query $Q$ which is RSI1+ and views $\mathcal{V}$ which are
CQAC$^{-}$s, the following is true:
Let $R$ be a CQAC contained rewriting to $Q$ in terms of $\mathcal{V}$. Then $R$ is contained in the one found by the {\bf Algorithm MCR-RSI1+} Datalog$^{AC}$ program, $R_{MCR}$.
\end{theorem}
\begin{proof}
Let $R$ be a contained rewriting to query $Q$. Since $Q'$ contains $Q$, $R$ is a contained rewriting of $Q'$ too. Hence, according to the results Theorem \ref{thm-mcr-cont}, $R$ is contained to $R'_{MCR}$.
Since $R$ is contained to $Q$, we consider the view-expansion of $R$, let it be $R_{exp}$ and we know that this is contained in $Q$, hence the containment entailment is true. However, $Q$ is a RSI1+ query, hence we can, according to Section \ref{sec:single-mapping} break the containment entailment in two as follows:
\begin{center}
\begin{tabular}{ll}
$\beta_2\Rightarrow \mu_1(\beta_{Q'}) \vee \cdots $&\\
$\beta_2\Rightarrow \mu_1(\beta_{Q-head}) $ & eq. (1)\\
\end{tabular}
\end{center}
where $\beta_2 $ is the conjunction of ACs in the closure of ACs in $R_{exp} $ and $\beta_{Q'}$ is the conjunction of ACs in $Q'$, $\beta_{Q-head}$ is the conjunction of ACs that use only head variables, and $mu_i$'s are all the mappings from $Q$ to $R_{exp} $.
Observe that in equation (1), we can replace $\beta_2$ with only those ACs in the closure of $\beta_2$ that involve head variables. Because $R$ is AC-rectified, all these ACs appear in $R$; let us denote them by $\beta_{head}$
Thus $\beta_{head}$ logically implies $ \beta_{Q-head}$.
Now, $R'_{MCR}$ and $R_{MCR}$ have the same expansions, except that the latter has the ACs in $ \beta_{Q-head}$ as well.
Hence we have concluded that a) $R$ is contained to $R'_{MCR}$ and b) the ACs in $R$ imply the added ACs in each expansion of $R'_{MCR}$ to make an expansion of $R_{MCR}$.
Now we only need to prove the following claim:
Suppose a CQAC $Q_2$ is contained in CQAC $Q_1'$. Let $Q_1$ be $Q_1'$ with some more ACs on the head variables such that these ACs are implied by the ACs in $Q_2$. Then $Q_2$ is contained in $Q_1$.
Proof of the claim: When a tuple of $Q_2$ is computed, then the same tuple is computed for $Q'_1$. However, the constants in the tuple are such that the ACs in $Q_2$ are satisfied. Since the added ACs to make $Q_1$ are implied by the ACs in $Q_2$, those ACs are satisfied too, and hence, the tuple is computed for $Q_1$ too.
\end{proof}
\begin{theorem}
\label{thm-mcr-in-query1}
Given a query $Q$ which is CQAC-RSI1 and views $\mathcal{V}$ which are
CQAC$^{-}$s, the following is true:
The found by the algorithm in Subsection~\ref{subsec-buildingMCRs} Datalog$^{AC}$ program, $R_{MCR}$, is a contained rewriting.
\end{theorem}
\begin{proof}
Let $E$ be a CQAC query which is a Datalog-expansion of $R_{MCR}$. Let $E'$ be the CQAC that results from
$E$ by removing the head ACs. By Theorem \ref{thm-mcr-in-query}, $E'$ is a contained rewriting in $Q'$.
Hence if we consider the view-expansion, $E'_{exp}$, of $E'$, the containment entailment is true for $E'_{exp}$ and $Q'$.
Moreover, trivially we have $\beta_{E}\Rightarrow \beta_{Q-head}$
and using the distributive law, we derive the containment entailment that shows containment of the view-expansion
$E_{exp}$ of $E$ to $Q$.
\end{proof}
A straightforward consequence of the above two theorems is the following theorem which is the main result of this section
\begin{theorem}
\label{thm-mainsec6}
Given a query $Q$ which is CQAC-SI1+ and views $\mathcal{V}$ which are
CQAC$^{-}$s, the algorithm in Subsection~\ref{subsec-buildingMCRs} finds an MCR of $Q$ using $\mathcal{V}$ in the language of (possibly infinite) union of CQACs which is expressed by a Datalog$^{AC}$ query.
\end{theorem}
A straightforward consequence of the above theorem and the main result in Section \ref{sec-6} is the following theorem:
\begin{theorem}
Given a query $Q$ which is CQAC-SI1+ and views $\mathcal{V}$ which are
CQAC$^{\;-}$s, we can find all certain answers of $Q$ using $\mathcal{V}$ on a given view instance $\mathcal{I}$ in time polynomial on the size of $\mathcal{I}$.
\end{theorem}
\subsection{Another example}
\begin{example}
\label{ex-prime-new}
This is similar to Example \ref {ex-prime-old1}
with slight alterations to make the point that we need auxiliary views. The alterations are as follows: We have added a new relational subgoal $a(U)$ and a new AC on the variable of this relational subgoal in the query and we have added a relational subgoal on the same predicate on the first view. Again auxiliary views that are not used in building the MCR and are not written here.
Thus, we consider the query $Q_1$ and the views:
\begin{center}
\begin{tabular}{ll}
$Q_1()$ & $\mathrm{:-}~e(X,Z),e(Z,Y),X\geq 5,Y\leq 8,a(U),U\geq 46.$\\
$V_1(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),Z\geq 5,a(Y).$\\
$V_2(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),Z\leq 8.$\\
$V_3(X,Y)$ & $\mathrm{:-}~e(X,Z_1),e(Z_1,Z_2),e(Z_2,Z_3),e(Z_3,Y).$\\
\end{tabular}
\end{center}
The Datalog program $Q_1^{Datalog}$ is exactly the same as the one in Example~\ref{ex-prime-old1} with the only alternation that the body of each mapping rule includes $U_{\geq 46}(U)$.
The views that will be used to apply the inverse-rule algorithm are (now we have added one auxiliary view which we guessed will be needed):
\begin{center}
\begin{tabular}{ll}
$V'_1(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),U_{\geq 5}(Z),a(Y),U_{\geq 46}(Y).$\\
$V'_2(X,Y)$ & $\mathrm{:-}~e(X,Z),e(Z,Y),U_{\leq 8}(Z).$\\
$V'_3(X,Y)$ & $\mathrm{:-}~e(X,Z_1),e(Z_1,Z_2),e(Z_2,Z_3),e(Z_3,Y).$\\
$V'_4(Y)$ & $\mathrm{:-}~U_{\geq 46}(Y).$\\
\end{tabular}
\end{center}
The new view is view $V_4'$.
The Datalog program now for the query $Q$ is\footnote{Some link rules are ommitted since they are not used by the inverse rule algorithm to produce an MCR. For the same reason some coupling rules are ommitted.}:
\begin{center}
\begin{tabular}{l l l}
$Q_1^{Datalog}()$ & $:-~e(X,Y),e(Y,Z),a(U),U_{\geq 46}(U).$&\\
&$I_{\geq 5}(X),I_{\leq 8}(Z).$ & (query rule)\\
$J_{\leq 8}(Z)$ & $:-~e(X,Y),e(Y,Z),a(\overline{W},Y),$&\\&$I_{\geq 5}(X).$ & (mapping rule)\\
$J_{\geq 5}(X)$ & $:-~e(X,Y),e(Y,Z),a(\overline{W},Y),$&\\&$I_{\leq 8}(Z).$ & (mapping rule)\\
$I_{\leq 8}(X)$ & $:-~J_{\geq 5}(X).$ & (coupling rule)\\
$I_{\geq 5}(X)$ & $:-~J_{\leq 8}(X).$ & (coupling rule)\\
$I_{\geq 5}(X)$ & $:-~U_{\geq 5}(X).$ & (link rule)\\
$I_{\leq 8}(X)$ & $:-~U_{\leq 8}(X).$ & (link rule)\\
\end{tabular}
\end{center}
This is the output of the inverse rule algorithm:
\begin{center}
\begin{tabular}{ll}
$R'() $ & $\mathrm{:-}~v'_1(X,W),T(W,Z),v'_2(Z,Y),v'_4(W).$\\
$T(W,W)$ & $\mathrm{:-}~.$\\
$T(W,Z)$ & $\mathrm{:-}~T(W,U),v'_3(U,Z).$\\
\end{tabular}
\end{center}
The following is an MCR of the input query $Q_1$ using the views $V_1(X,Y)$, $V_2(X,Y)$ and $V_3(X,Y)$. Notice that we replace view $v'_4(W)$ by $W\geq 46$.
\begin{center}
\begin{tabular}{ll}
$R() $ & $\mathrm{:-}~V_1(X,W),T(W,Z),V_2(Z,Y),W\geq 46.$\\
$T(W,W)$ & $\mathrm{:-}~.$\\
$T(W,Z)$ & $\mathrm{:-}~T(W,U),V_3(U,Z).$\\
\end{tabular}
\end{center}
\end{example}
\subsection{Extending the result}
The following example shows that the result of Theorem~\ref{thm-mcr-cont} (hence the result of Theorem \ref{thm-mainsec6}) cannot be extended to include views that are in the language of CQACs.
\begin{example}
Suppose we have the following query and views:
\begin{center}
\begin{tabular}{l}
$Q$: $q(Y):- a(Y,X),b(X,Z),X\geq 5, Z\leq 6$\\
$V_1$: $v_1(Y):- a(Y,X),b(X,X'),X\geq Y, X'\leq 6$\\
$V_2$: $v_2(Y):- a(Y,Z'),b(Z',Z),Y\geq Z, Z'\geq 5$\\
\end{tabular}
\end{center}
Here, the view definitions violate the constraint that no AC should be included between a head variable and a nondistinguished variable.
The following is a contained rewriting, for which we will argue that the technique in this section does not work:
\begin{center}
\begin{tabular}{l}
$R$: $q(Y):-v_1(Y),v_2(Y)$\\
\end{tabular}
\end{center}
The view-expansion of $R$ is:
\begin{center}
\begin{tabular}{l}
$R_{exp}$: $q(Y):- a(Y,X),b(X,X'), X'\leq 6, a(Y,Z'),b(Z',Z), Z'\geq 5$\\
\end{tabular}
\end{center}
and after we transform it to a CQ it becomes:
\begin{center}
\begin{tabular}{l}
$R^{CQ}_{exp}$: $q(Y):- a(Y,X),b(X,X'), U_{\leq 6}(X'), a(Y,Z'),b(Z',Z), U_{\geq 5}(Z')$\\
\end{tabular}
\end{center}
Suppose we transform $Q$ to $Q^{Datalog}$, then the following is the only Datalog-expansion of the program that could be used to prove that $Q^{Datalog}$ contains $R^{CQ}_{exp}$.
\begin{center}
\begin{tabular}{l}
$E$: $a(Y,X),b(X,X'), U_{\leq 6}(X'), a(Y,Z'),b(Z',Z), U_{\geq 5}(Z'),U(Z,X)$\\
\end{tabular}
\end{center}
We see that the body of $R^{CQ}_{exp}$ and $E$ will be isomorphic if we append $U(Z,X)$ to $R^{CQ}_{exp}$, which is equivalent to appending $Z\leq X$ to $R_{exp}$, which, further means appending $Z\leq X$ to $R$. This is not possible however, since in $R$ there is only one variable. This remark highlights the reason for failing to extend the result beyond Theorem \ref{thm-mainsec6} and probably, in future work, complexity results will prove that it is rather impossible to be extended.
%
%
%
%
\end{example}
\section{Conclusions}
In this paper we have investigated into the computational complexity of query containment for CQACs and of computing certain answers in the framework of answering CQAC queries using CQAC views. Our results point to several directions for future research.
Candidates for which the problem of CQAC query containment may be $\Pi^p_2$-complete are the following:
a) The containing query contains only LSI (both open and closed). b) The containing query contains two closed LSI and two closed RSI. Probably it is still $\Pi^p_2$-complete if the contained query is restricted, e.g., with only SI arithmetic comparisons.
We know that there are classes of queries where the CQ query containment problem is polynomial e.g., when the containing query is acyclic.
We conjecture that, for many such classes of queries, if we consider the classes of queries as in Section 5 but with the containing query having relational subgoals as the CQ queries that have a polynomial time algorithm, then
the containment problem is polynomial. E.g., if the relational subgoals of the containing query form an acyclic hypergraph and there are only several closed LSIs and one closed RSI on the nondistinguished variables, then
testing containment may be done in polynomial time.
In the framework of answering queries using views, besides using any new results for query containment to extend the results in Section 7 further, we have started a discussion in \ref{sec-app-unclean} for CQs to see whether MCRs can be useful for handling certain types of unclean data.
This discussion may be extended to include CQACs.
\bibliographystyle{abbrv}
|
1,314,259,996,678 | arxiv | \section{Introduction}
Deep reinforcement learning~(RL) can learn near-optimal policies to perform low-level actions with pre-defined reward functions. While the low-level actions are adequate for the specified tasks, it remains challenging to reuse this learned knowledge to accomplish new tasks efficiently. By contrast, when humans perform multiple tasks, the low-level muscle movements are naturally summarized into high-level action representations such as ``pick up,'' or ``turn left'', which can be reused in novel tasks with slight modifications. As a result, though we are aware of our intent to perform a task, we carry out the most complicated movements without thinking about the detailed joint motions or muscle contractions~\citep{kandel2021principles}. By analogy with such abilities of humans, we ask the question: can RL agents have action representations of low-level motor controls, which can be reused, modified, or composed to perform new tasks?
As pointed out in \cite{kandel2021principles}, ``the task of the motor systems is the reverse of the task of the
sensory systems. Sensory processing generates an internal representation in the brain of the outside world or of the state of the body. Motor processing begins with an internal representation: the desired purpose of movement''. In the past decade, representation learning has made tremendous progress in representing high-dimensional sensory signals such as images and audios for revealing the geometric and semantic structures hidden in the raw signals~\citep{bengio2013representation,chen2018sparse, kornblith2019better, chen2020simple, baevski2020wav2vec, radford2021learning, bardes2021vicreg, bommasani2021opportunities, he2022masked}. With the help of the generalization ability of sensory representation learning, downstream control tasks can be accomplished efficiently~\citep{nair2022r3m, xiao2022masked, yuan2022pre}. In contrast to the advancement in sensory representation learning, action representation learning is yet under-explored. Thus, we make a step to study this topic with an aim to find generalizable action representations which can be reused or efficiently adapted to perform new tasks. A key idea in sensory representation learning is pretraining with a comprehensive task (or a set of tasks) and reusing the emergent latent representation.
Inspired by this simple and powerful idea, we train a multi-task policy network, whereas a set of different tasks share the same latent action representation space. Further, the time-variant sensory and time-invariant action representations are decoupled and concatenated as the sensory-action representation, which is transformed by a policy network to form the low-level action control. Surprisingly, when trained on a comprehensive set of tasks, we discover that such a simple structure learns an emergent self-organized action representation. The emergent action representation can be reused for intra-action interpolation, inter-action composition, and efficient task adaptation. In particular, we demonstrate the effectiveness of the emergent action representations with Mujoco locomotion environments and show zero-shot composition and few-shot task adaptation in the representation space. The agents adapt to new tasks in fewer steps than strong meta RL baselines. Besides the effectiveness of the action representation, we find the decoupled time-variant sensory representation exhibits equivariant properties. The evidence elucidates that reusable and generalizable action representations may lead to efficient, adaptable, and composable RL, thus forming the basis of abstract action planning and understanding motor signal space. The primary contributions in this work are listed as follows:
\begin{enumerate}
\item We put forward the idea of leveraging emergent action representations from multi-task learners to better understand motor action space and accomplish task generalization.
\item We decouple the state-related and task-related information of the sensory-action representations and reuse them to conduct action planning more efficiently.
\item Our approach is a strong adapter, which achieves higher rewards with fewer steps than strong meta RL baselines when adapting to new tasks.
\item Our approach supports intra-action interpolation as well as inter-action composition by modifying and composing the learned action representations.
\end{enumerate}
Next, we begin our technical discussion right below and leave the discussion of many valuable and related literature to the end.
\section{Preliminaries}
{\bf Soft Actor-Critic. }
In this paper, our approach is built on Soft Actor-Critic~(SAC)~\citep{haarnoja2018soft}. SAC is a stable off-policy actor-critic algorithm based on the maximum entropy reinforcement learning framework, in which the actor maximizes both the returns and the entropy.
In SAC, there are three types of parameters to update during optimization: the policy parameter $\omega$, the soft Q-function parameter $\theta$ and a learnable temperature $\alpha$. The objectives are:
\begin{equation}
J(\alpha)=\mathbb{E}_{\mathbf{a}_t \sim \pi_\omega}\left[-\alpha \log \pi_\omega\left(\mathbf{a}_t \mid \mathbf{s}_t\right)-\alpha \overline{\mathcal{H}}\right] \label{sac alpha objective}
\end{equation}
\begin{equation}
J_\pi(\omega)=\mathbb{E}_{\mathbf{s}_t \sim \mathcal{D}}\left[\mathbb{E}_{\mathbf{a}_t \sim \pi_\omega}\left[\alpha \log \pi_\omega\left(\mathbf{a}_t \mid \mathbf{s}_t\right)-Q_\theta\left(\mathbf{s}_t, \mathbf{a}_t\right)\right]\right] \label{sac policy objective}
\end{equation}
\begin{equation}
J_Q(\theta)=\mathbb{E}_{\left(\mathbf{s}_t, \mathbf{a}_t\right) \sim \mathcal{D}}\left[\frac{1}{2}\left(Q_\theta\left(\mathbf{s}_t, \mathbf{a}_t\right)-\left(r\left(\mathbf{s}_t, \mathbf{a}_t\right)+\gamma \mathbb{E}_{\mathbf{s}_{t+1} \sim p}\left[V_{\bar{\theta}}\left(\mathbf{s}_{t+1}\right)\right]\right)\right)^2\right] \label{sac q objective}
\end{equation}
where $\overline{\mathcal{H}}$ in Equation~\ref{sac alpha objective} is a pre-defined minimum expected entropy. The value function in Equation~\ref{sac q objective} is parameterized by the weights $V_{\bar{\theta}}$ in the target Q-network and can be defined by:
\begin{equation}
V\left(\mathbf{s}_t\right)=\mathbb{E}_{\mathbf{a}_t \sim \pi}\left[Q\left(\mathbf{s}_t, \mathbf{a}_t\right)-\alpha \log \pi\left(\mathbf{a}_t \mid \mathbf{s}_t\right)\right] \label{sac v}
\end{equation}
\paragraph{Task Distribution.}\label{task distribution}
We assume the tasks that the agent may meet are drawn from a pre-defined task distribution $p(\mathcal{T})$. Each task in $p(\mathcal{T})$ corresponds to a Markov Decision Process~(MDP). Therefore, a task $\mathcal{T}$ can be defined by a tuple $(\mathcal{S}, \mathcal{A}, p(\mathbf{s}_{t+1}|\mathbf{s}_t, \mathbf{a}_t), p(\mathbf{s}_0), R(\mathbf{s}_t, \mathbf{a}_t))$, in which $\mathcal{S}$ and $\mathcal{A}$ are respectively the state and action space, $p(\mathbf{s}_{t+1}|\mathbf{s}_t,\mathbf{a}_t)$ the transition probability distribution, $p(\mathbf{s}_0)$ the initial state distribution and $R(\mathbf{s}_t, \mathbf{a}_t)$ the reward function.
The concept of task distribution is frequently employed in meta RL problems, but we have made some modifications and extensions on it to better match with the setting in this work. We divide all the task distributions into two main categories, the ``uni-modal" task distributions and the ``multi-modal" task distributions. Concretely, the two scenarios are defined as follows:
\begin{itemize}[leftmargin=*]
\item \textit{Definition 1 (Uni-modal task distribution): }In a uni-modal task distribution, there is only one modality among all the tasks in the task distribution.
For example, in HalfCheetah-Vel, a Mujoco locomotion environment, we train the agent to run at different target velocities. Therefore, running is the only modality in this task distribution.
\item \textit{Definition 2 (Multi-modal task distribution): }In contrast to uni-modal task distribution, there are multiple modalities among the tasks in this task distribution. A multi-modal task distribution includes tasks of several different uni-modal task distributions.
For instance, we design a multi-modal task distribution called HalfCheetah-Run-Jump, which contains two modalities including HalfCheetah-BackVel and HalfCheetah-BackJump. The former has been defined in the previous section, and the latter contains tasks that train the agent to jump with different reward weight. In our implementation, we actually train four motions in this environment, running, walking, jumping ans standing. We will leave more details in Section \ref{exp} and Appendix \ref{environments}.
\end{itemize}
\section{Emergent Action Representations from Multi-Task Training}
In this section, we first introduce the sensory-task decoupled policy network architecture. Next, we discuss the multitask policy training details, along with the additional constraints to the task embedding for the emergence of action representations. Lastly, we demonstrate the emergence of action representations through various phenomena and applications.
\subsection{Multitask Policy Network and Training}
{\bf Decoupled embedding and concatenated decoding.} An abstract high-level task, e.g., ``move forward'', typically changes relatively slower than the transient sensory states. As a simplification, we decouple the latent representation into a time-variant sensory embedding $\mathcal{Z}_{\mathbf{s}_t}$ and a time-invariant task embedding $\mathcal{Z}_\mathcal{T}$, which is shown in Figure~\ref{idea}. These embeddings concatenate to form a sensory-action embedding $\mathcal{Z}_\mathcal{A}(\mathbf{s}_t,\mathcal{T}) = [\mathcal{Z}_{\mathbf{s}_t}, \mathcal{Z}_\mathcal{T}]$, which is transformed by the policy network (action decoder) $\psi$ to output a low-level action distribution $p(\mathbf{a}_t) = \psi(\mathbf{a}_t | \mathcal{Z}_{\mathbf{s}_t}, \mathcal{Z}_\mathcal{T})$, e.g., motor torques. The action decoder $\psi$ is a multi-layer perceptron (MLP) that outputs a Gaussian distribution in the low-level action space $\mathcal{A}$.
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{assets/main_final.png}
\caption{\textbf{Emergent action representations from multi-Task training.} The sensory information and task information are encoded separately. When both are concatenated, an action decoder decodes them into a low level action.}
\label{idea}
\end{figure}
\textbf{Latent sensory embedding (LSE).} The low-level sensory state information is encoded by an MLP state encoder $\phi$ into a latent sensory embedding $\mathcal{Z}_{\mathbf{s}_t} = \phi(\mathbf{s}_t) \in \mathbb{R}^m$. It includes the proprioceptive information of each time step. LSE is time-variant in an RL trajectory, and the state encoder is shared among different tasks. We use LSE and sensory representation interchangeably in this paper.
\textbf{Latent task embedding (LTE).} A latent task embedding $\mathcal{Z}_\mathcal{T}\in \mathbb{R}^d$ encodes the time-invariant knowledge of a specific task. Let's assume we are going to train $N$ different tasks, and their embeddings form an LTE set $\{ \mathcal{Z}_{\mathcal{T}_N} \}$. These $N$ different tasks share the same state encoder $\phi$ and action decoder $\psi$; in other words, these $N$ tasks share the same policy network interface, except for their task embeddings being different. Additionally, the LTE vectors are trainable during learning, and we can always add a new task by adding a new trainable LTE vector to the set $\{ \mathcal{Z}_{\mathcal{T}_N} \}$. After training, the LTE interface can be reused as a high-level action interface. Hence, we use LTE and action representation interchangeably in this paper.
\paragraph{Training of the multi-task policy networks.}
A detailed description of the multi-task training is demonstrated in Algorithm~\ref{alg1}. When computing objectives and their gradients, we still use policy $\pi$ parameterized by $\omega$ to indicate all the parameters in the encoders and the decoder.
The overall training procedure is based on SAC. The only difference is that the policy network and Q networks additionally take as input the LTE $\mathcal{Z}_\mathcal{T}$ and a one-hot task label, respectively.
During training, we also apply two techniques to constrain this space: 1)~we normalize the LTEs so that they lie on a hypersphere; 2)~we inject a random noise to the LTEs to enhance the smoothness of the space.
\begin{algorithm} [h]
\caption{Multi-task Training}
\label{alg1}
\begin{algorithmic}
\REQUIRE Training task set $\{\mathcal{T}_N\} \sim p(\mathcal{T})$, $\theta_1$, $\theta_2$, $\omega$
\STATE $ \overline \theta_1 \leftarrow \theta_1, \overline \theta_2 \leftarrow \theta_2, \mathcal{B} \leftarrow \emptyset $
\FOR{each pre-train epoch}
\FOR{$\mathcal{T}_i$ in $\{\mathcal{T}_n\}$}
\STATE Sample a batch $\mathcal{B}_i$ of multi-task RL transitions with $\pi_\omega$
\STATE $\mathcal{B}\leftarrow\mathcal{B}\cup\mathcal{B}_i$
\ENDFOR
\ENDFOR
\FOR{each train epoch}
\STATE Sample RL batch $b\sim\mathcal{B}$
\FORALL{transition data in $b$}
\STATE $\mathcal{Z}_{\mathbf{s}_t}=\phi(\mathbf{s}_t)$
\STATE $\mathcal{\widetilde Z}_{\mathcal{T}_i}=normalize(\mathcal{Z}_{\mathcal{T}_i}+n), n\sim\mathcal{N}(0,\sigma^2)$
\STATE Sample action $\mathbf{a}_{t}\sim\psi(\cdot | \mathcal{Z}_{\mathbf{s}_t}, \mathcal{\widetilde Z}_{\mathcal{T}_i})$
\ENDFOR
\FOR{each optimization step}
\STATE Compute SAC objectives $J(\alpha),J_\pi(\omega),J_Q(\theta)$ with $b$
\STATE Update SAC parameters
\ENDFOR
\ENDFOR
\ENSURE The optimal model of state encoder $\phi^*$ and action decoder $\psi^*$ and a set of LTEs $\{\mathcal{Z}_{\mathcal{T}_N}\}$
\end{algorithmic}
\end{algorithm}
\subsection{The Emergence of Action Representation}
After we train the multi-task policy network with a comprehensive set of tasks, where the LTE vectors in $\{\mathcal{Z}_{\mathcal{T}_N} \}$
share the same embedding space, we find that $\{\mathcal{Z}_{\mathcal{T}_N} \}$ self-organizes into a geometrically and semantically meaningful structure. Tasks with the same modality are embedded in a continuous fashion, which facilitates intra-task interpolation. Surprisingly, the composition of task embeddings from different modalities leads to novel tasks, e.g., ``run'' $+$ ``jump'' $=$ ``jump run''. Further, the action representation can be used for efficient task adaptation. Visualization also reveals interesting geometric structures in task embedding and sensory representation spaces. In this subsection, we dive into these intriguing phenomena, demonstrating the emergence of action representation and showing the generalization of the emergent action representation.
{\bf Task interpolation \& composition.}
After training the RL agent to accomplish multiple tasks, we select two pre-trained tasks and generate a new LTE through linear integration between the LTEs of the two chosen tasks. The newly-generated task embedding is expected to conduct the agent to perform another different task.
The generated LTE is defined by
\begin{equation}
\mathcal{Z'}=f(\beta\mathcal{Z}_{\mathcal{T}_i}+(1-\beta)\mathcal{Z}_{\mathcal{T}_j})
\label{interpolation equation}
\end{equation}
where $i,j$ are the indices of the selected tasks and $\mathcal{Z}_{\mathcal{T}_i},\mathcal{Z}_{\mathcal{T}_j}$ are their corresponding LTEs. $\beta$ is a hyperparameter ranging in (0,1). The function $f(\cdot)$ is a regularization function related to the pre-defined quality of the LTE Space. For instance, in this paper, $f(\cdot)$ is a normalization function to extend or shorten the result of interpolation to a unit sphere.
A new task is interpolated by applying the aforementioned operation on the LTEs of tasks sampled from a uni-modal distribution. The interpolated task usually has the same semantic meaning as the source tasks while having different quantity in specific parameters, e.g., running with different speeds. A new task is composed by applying the same operation on tasks sampled from a multi-modal distribution. The newly composed task is usually lie in a new modality between the source tasks. For example, when we compose ``run'' and ``jump'' together, we will have a combination of an agent running while trying to jump.
{\bf Efficient adaptation.}
We find that an agent trained with the multi-task policy network can adapt to unseen tasks quickly by only optimizing the LTEs.
This shows that the LTEs learn a general pattern of the overall task distribution. When given a new task after pre-training, the agent explores in the LTE Space to find a suitable LTE for the task. Specifically, we perform a gradient-free cross-entropy method~(CEM)~\citep{de2005tutorial} in the LTE space for accomplishing the desired task. Detailed description can be found in Algorithm~\ref{alg2}.
\paragraph{Geometric Structures of LTEs and the LSEs} We visualize the LSEs and and LTEs to understand their geometric structures. The results are deferred to Section~\ref{sec: viz}.
\begin{algorithm}
\caption{Adaptation via LTE Optimization}
\label{alg2}
\begin{algorithmic}
\REQUIRE Adaptation task $\mathcal{T} \sim p(\mathcal{T})$, $\phi^*,\psi^*$, capacity of elite set $m$, number of sampling $n$
\STATE Initialize the elite set $\mathbb{Z}_e$ with $m$ randomly sampled LTEs from the LTE Space
\FOR{each adapt epoch}
\STATE Initialize the overall test set by $\mathbb{Z}\leftarrow\emptyset$
\FOR{$\mathcal{Z}_i$ in $\mathbb{Z}_e$}
\STATE Sample $n$ LTEs $\mathcal{Z}_{i1},\ldots,\mathcal{Z}_{in}$ near $\mathcal{Z}_{i}$
\STATE $\mathbb{Z}\leftarrow\mathbb{Z}\cup\{\mathcal{Z}_{i}, \mathcal{Z}_{i1},\ldots,\mathcal{Z}_{in}\}$
\ENDFOR
\FOR{$\mathcal{Z}_j$ in $\mathbb{Z}$}
\WHILE{not done}
\STATE $\mathcal{Z}_{\mathbf{s}_t}=\phi^*(\mathbf{s}_t),$
\STATE $\mathbf{a}_{t}\sim\psi^*(\cdot | \mathcal{Z}_{\mathbf{s}_t}, \mathcal{Z}_j)$
\STATE $r_t=R(\mathbf{s}_t,\mathbf{a}_t | \mathcal{T})$
\STATE $\mathbf{s}_{t+1}\sim p(\mathbf{s}_{t+1}|\mathbf{s}_t,\mathbf{a}_t)$
\ENDWHILE
\ENDFOR
\STATE Sort the task embeddings in $\mathcal{Z}$ by high cumulative reward in the trajectory
\STATE Select the top $m$ LTEs in $\mathcal{Z}$ to update $\mathbb{Z}_e$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{experiments}\label{exp}
In this section, we first demonstrate the training process and performance of the multi-task policy network. Then, we use the LTEs as a high-level action interface to instruct the agents to perform unseen skills through interpolation without any training. After that, we conduct experiments to evaluate the effectiveness of the LTEs in task adaptation. Lastly, we visualize the LSEs and LTEs to further understand the structure of the state and action representation. We use \textbf{\underline{e}}mergent \textbf{\underline{a}}ction \textbf{\underline{r}}epresentation~(EAR) to refer to the policy using the LTEs.
\subsection{experimental setups}
\textbf{Environments. }
We evaluate our method on five continuous control environments~(HalfCheetah-Vel, Ant-Dir, Hopper-Vel, Walker-Vel, HalfCheetah-Run-Jump) based on OpenAI Gym and the Mujoco simulator. Detailed descriptions of these RL benchmarks are listed in Appendix~\ref{environments}.
\textbf{Baselines. }
We compare EAR-SAC, the emergent action representation based SAC with several multi-task RL and meta RL baselines. For multi-task RL baselines, we use multi-head multi-task SAC~(MHMT-SAC) and one-hot embedding SAC~(OHE-SAC; for ablation). For meta RL baselines, we use MAML~\citep{finn2017model} and PEARL~\citep{rakelly2019efficient}. Detailed descriptions of these baselines are listed in Appendix~\ref{baselines}.
\subsection{Multi-task Training for Action Representations}
In this section, we train the multi-task network and evaluate whether sensory-action representations can boost training efficiency. EAR-SAC is compared with all the multi-task RL and meta RL baselines on final rewards in Table~\ref{final rewards of training} and additionally compared with off-policy baselines on the training efficiency in Figure~\ref{efficiency of training}.
We find that EAR-SAC outperforms all the baselines in terms of training efficiency. In environments with high-dimensional observations~(e.g., Ant-Dir), EAR-SAC achieves large performance advantage against the baselines. We attribute this to that the learned action representation space may provide meaningful priors when the policy is trained with multiple tasks.
\begin{figure}[t]
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/HalfCheetah-Vel_Running_Average_Rewards_train.pdf}
\end{minipage}
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/Ant-Dir_Running_Average_Rewards_train.pdf}
\end{minipage}
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/Hopper-Vel_Running_Average_Rewards_train.pdf}
\end{minipage}
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/Walker-Vel_Running_Average_Rewards_train.pdf}
\end{minipage}
\caption{\textbf{Training performance of our method and baselines.} The x-axis represents the total training steps~(in million steps) and the y-axis represents the average reward of all the training tasks. All the training are based on 3 seeds. The shaded area is one standard deviation.}
\label{efficiency of training}
\end{figure}
\begin{table}[t]
\centering
\begin{tabular}[width=10cm]{c|cccc}
\toprule
Method & HalfCheetah-Vel & Ant-Dir & Hopper-Vel & Walker-Vel \\
\midrule
\textbf{EAR-SAC (Ours)} & \textbf{-136.9$\pm$1.29} & \textbf{1216.7$\pm$54.97} & \textbf{173.5$\pm$11.09} & \textbf{172.7$\pm$4.05}\\
MAML & -500.9$\pm$47.33 & 422.8$\pm$29.40 & 34.0$\pm$41.92 & -4.9$\pm$0.58 \\
PEARL & -155.0$\pm$15.50 & 635.9$\pm$19.21 & 170.0$\pm$7.80 & 163.9$\pm$1.07\\
MHMT-SAC & -145.2$\pm$2.60 & 1020.0$\pm$54.90 & 172.9$\pm$1.17 & 160.6$\pm$5.21\\
OHE-SAC & -609.0$\pm$8.10 & 195.7$\pm$0.29 & 90.7$\pm$2.39 & 65.0$\pm$20.83 \\
\bottomrule
\end{tabular}
\caption{\textbf{Comparison with baselines on the final performance.} The metric is the return of the last epoch, and the mean and standard deviation is calculated among 3 seeds.}
\label{final rewards of training}
\end{table}
\subsection{Action Representation as a High-level Control Interface}\label{interpolate}
\begin{wrapfigure}[14]{r}{0.4\textwidth}
\centering
\includegraphics[width=0.37\textwidth]{assets/interpolation.pdf}
\caption{\textbf{Visualization of the interpolated tasks.}}
\label{interpolation result}
\end{wrapfigure}
In this section, we control the agent by recomposing the LTEs.
We perform intra-action interpolation and inter-action composition in uni-modal task distributions and multi-modal task distributions respectively.
\textbf{Intra-action interpolation.}
Intra-action interpolation is conducted in HalfCheetah-Vel, Ant-Dir, Hopper-Vel, and Walker-Vel. We interpolate the the action representations between two tasks using Equation~\ref{interpolation equation}. The coefficient $\beta$ is searched to better fit the target task.
An interpolation example in HalfCheetah-Vel is demonstrated in Figure~\ref{interpolation result}.
We select the tasks of running at 1 m/s and 2 m/s to be interpolated and get three interpolated tasks: run at 1.2 m/s, 1.5 m/s, 1.7 m/s. We perform evaluation on each task for a trajectory, and visualize them.
In each task, we make the agent start from the same point (not plotted in the figure) and terminate at 100 time steps. Only part of the whole scene is visualized in the figure. We find that through task interpolation, the agent manages to accomplish these interpolated tasks without any training. We leave other results in all environments to Appendix~\ref{details}.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{assets/composition.pdf}
\caption{\textbf{Visualization of task compositions.} The first row represents part of a trajectory of run-jumping, and the second row represents part of a trajectory of walk-standing. Animated results are in the project page.}
\label{composition}
\end{figure}
\textbf{Inter-action composition.}
Inter-action composition is conducted in HalfCheetah-Run-Jump, in which we merge two uni-modal task distributions. Taking HalfCheetah-BackVel and HalfCheetah-BackJump as an example, we find that the agent has learned four motions: walking backward, running backward, standing with its front leg, and jumping with its front leg. We select walk-standing as a composition task pair and run-jumping as the other and compose them with Equation~\ref{interpolation equation}. These two generated representations are evaluated in 1000 time steps and part of their evaluation trajectories are visualized in Figure~\ref{composition}. We find that the walk-standing representation enables the agent to walk backward with a standing posture despite some pauses to keep itself balance, while the run-jumping representation helps the agent to jump when running backward after some trials.
These empirical results indicate that the LTEs can be used as action representations by composing them for different new tasks.
\subsection{Task Adaptation with Action Representations}\label{adapt}
In this section, we assess how well an agent can adapt to new tasks by only updating the LTEs.
We compare the agent's adaptation ability with the meta RL baselines~(MAML,PEARL) and the ablation baseline~(OHE-SAC).
The results in Figure~\ref{efficiency of adaptation} demonstrate that EAR can adapt to new tasks and achieve to the converged performance in no more than three epochs, outperforming the baseline meta RL methods. We note that, with the help of the LTE space, we can adapt to new tasks using zero-th order optimization method.
\begin{figure}[t]
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/HalfCheetah-Vel_Adaptation_Rewards_adapt.pdf}
\end{minipage}
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/Ant-Dir_Adaptation_Rewards_adapt.pdf}
\end{minipage}
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/Hopper-Vel_Adaptation_Rewards_adapt.pdf}
\end{minipage}
\centering
\begin{minipage}{0.245\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/Walker-Vel_Adaptation_Rewards_adapt.pdf}
\end{minipage}
\caption{\textbf{Adaptation results.} The x-axis represents the adaptation steps and the y-axis represents the average reward of all the adaptation tasks. We fix the adaptation task set for all algorithms in the same environment.
}
\label{efficiency of adaptation}
\end{figure}
\subsection{Visualization of State and Action Representations}
\label{sec: viz}
In this section, we further analyze and visualize the sensory-action representations based on the HalfCheetah environment.
\begin{figure}[h]
\centering
\begin{minipage}{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/rss_vs_sr.pdf}
\caption{\textbf{Comparison between raw state signals~(left) and sensory representations~(right) in HalfCheetah-Vel.} To make the period of running motion easier to observe, we reduce the time interval between adjacent time steps. For now a period of the motion is 15 steps, while in the original setting it is only 6 steps.}
\label{rss_vs_sr}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{assets/ar_space.pdf}
\caption{\textbf{Visualization of the action representation space in HalfCheetah-Vel.} Each grid on the unit sphere represents the terminal of an action representation and is colored based on how the agent performs when the policy is conditioned on the action representation. The redder the grid is, the faster the agent runs. }
\label{LTE Space}
\end{minipage}
\end{figure}
\textbf{The sensory representations. }
In HalfCheetah-Vel, the halfcheetah agent is encouraged to run at a specific velocity, thus making its motion periodical. Therefore, the ideal raw signals and the LSEs in the trajectory should be periodical as well. After conducting Principal Components Analysis~(PCA)~\citep{abdi2010principal} to reduce the dimension of all the raw signals and the sensory representations collected in a trajectory, we visualize them for different time steps in Figure~\ref{rss_vs_sr}. We find that the raw state signals appear to be only roughly periodical due to the noise in the states. However, the LSEs show stronger periodicity than the raw signals, indicating that the sensory representations can reduce the influence of the noise in raw signals, thus helping the agent better understand and react to the environment.
\textbf{The action representations. }
To better understand the intriguing fact that the agent can adapt to new tasks using the emergent action representations when only sparse training tasks are provided, we plot the LTE space in Figure~\ref{LTE Space}. We find that the LTEs on the hypersphere automatically form a continuous space, constructing the basis of the composition of the representations in it. This also explains why the interpolation and composition of the LTEs result in meaningful new behaviors: when interpolated and normalized, the new LTE would still lie on this hypersphere, leading to new and meaningful action sequences.
\subsection{Comparison with one-hot representation}
Readers might be curious about whether the learned, manipulable LTEs better than a naive one-hot embedding. In this section, we compare EAR with a different policy network that takes in an one-hot embedding rather then the LTE.
Specifically, in multi-task training, we concatenate the raw state vector with the one-hot embedding as the input of the policy network to form a multi-task RL architecture. We find that the one-hot embedded policy performs unsatisfactorily in all environments and fails completely in Ant-Dir environment. We attribute this to the fact that, without the sensory-action representation, the agent fails to understand the underlying relation among multiple tasks, especially in those environments with high-dimensional observations and a large training task set.
We also compare both methods in the task adaptation setting. When optimizing the one-hot embedding to adapt to new tasks, the agent can barely reach a reasonable performance in any of the new tasks. These facts echo with the idea that the simple emergent action representations are meaningful and effective in task learning and generalization.
\section{Related Work}
\textbf{Representation learning in reinforcement learning.}
Representation learning has been widely applied in RL to generate representations of sensory information~\citep{pmlr-v119-laskin20a,pmlr-v97-chandak19a,pmlr-v139-yarats21a}, policies~\citep{pmlr-v97-edwards19a}, dynamics~\citep{DBLP:journals/corr/WatterSBR15}.
In recent years, action representation has attracted more attention. In previous works~\citep{dulac2015reinforcement},
the action embeddings have been proposed to compress discrete actions to a continuous space based on prior knowledge. Recently, many researchers focus on mapping between the original continuous action space and a continuous manifold to facilitate and accelerate policy learning~\citep{allshire2021laser} or simplify teleoperation~\citep{losey2020controlling}.
Moreover, such action representations are extended to high-dimensional real-world robotics control problems~\cite{jeon2020shared}.
In this area, a closely related work is \citet{chandak2019learning},
which enables a policy to output a latent action representation with reinforcement learning and then uses supervised learning to construct a mapping from the representation space to the discrete action space.
However, in our work, we discover emergent self-organized sensory-action representations from multi-task policy training without additional supervised learning, which can be further leveraged to generalize to new tasks.
\textbf{Multi-task reinforcement learning.}
Multi-task reinforcement learning~(multi-task RL) is an important problem in RL, which trains an RL agent to learn multiple tasks simultaneously with a single policy. A direct idea is to share parameters and learn a joint policy with multiple objectives~\citep{wilson2007multi,pinto2017learning}, but the gradients of different tasks may conflict with each other. Some works tackle this problem by directly reducing gradient conflicts~\citep{yu2020gradient}, designing specific loss weighting ~\citep{hessel2019multi,kendall2018multi}, or leveraging regularization methods~\citep{duong2015low}.
Modular networks has also been proposed to perform multi-task learning by constructing the policy network with a combination of smaller sub-networks~\citep{heess2016learning,yang2020multi}. In our work, we adopt the multi-task RL setting and train a multi-task policy network with a sensory-action representation layer.
\textbf{Meta reinforcement learning.}
The adaptation part of this work also shares a similar goal as meta reinforcement learning ~(meta RL).
~\citep{finn2017model,pmlr-v80-xu18d,rothfuss2018promp, DBLP:journals/corr/WangKTSLMBKB16, DBLP:journals/corr/abs-2005-01643}, aiming to quickly adapt to new tasks.
Meta RL executes meta-learning~\citep{schmidhuber1987evolutionary,thrun2012learning} algorithms under the framework of reinforcement learning, which aims to train an agent to quickly adapt to new tasks. A large number of meta RL methods are gradient-based, performing gradient descent on the policy~\citep{finn2017model,pmlr-v80-xu18d,rothfuss2018promp}, the hyperparameters~\citep{xu2018meta}, or loss functions~\citep{sung2017learning,NEURIPS2018_7876acb6} based on the collected experience. Other meta RL algorithms are context-based methods~\citep{DBLP:journals/corr/WangKTSLMBKB16, rakelly2019efficient}. They embed context information into a latent variable and condition the meta-learning policy on such variables. Besides, recently offline meta RL algorithms~\citep{DBLP:journals/corr/abs-2005-01643,pmlr-v139-mitchell21a} have also been widely explored, which enables the agent to leverage the existing offline data to perform meta-training. However, we exploit the emergent high-level representations to abstract low-level actions for adaptation rather than introduce complicated new techniques.
\section{Conclusion}
In this paper, we present our finding that the task embeddings in a multi-task policy network can automatically form a space where action representations reside. The emergent action representations abstract information of a sequence of actions and can serve as a high-level interface to instruct an agent to perform motor skills. Specifically, we find the action representations can be interpolated, composed, and optimized to generate novel action sequences for unseen tasks.
Along with this work, a promising direction is to learn the action representation via self-supervised learning. Another intriguing direction is to learn hierarchical action representations that may capture different levels of semantics and geometric information in motor signals, thus facilitating future applications such as hierarchical planning~\citep{lecun2022path}.
\section*{Acknowledgement}
Yubei would like to thank Yann LeCun and Rob Fergus for sharing their visionary thoughts on action representation learning.
\section{Experimental Details}
\subsection{Environments}\label{environments}
The detailed descriptions of the reinforcement learning benchmarks in our experiments are listed as follows:
\begin{itemize}[leftmargin=*]
\item \textbf{HalfCheetah-Vel~(Uni-modal):} In this environment, we train the halfcheetah agent to run at a target velocity. The training task set contains 10 velocities. The target velocities during training range from 1 m/s to 10 m/s. For every 1 m/s, we set a training task. The adaptation task set contains 3 velocities that are uniformly sampled from [1,10]. The agent is penalized with the $l_1$ error between its velocity and the target velocity.
\item \textbf{Ant-Dir~(Uni-modal):} In this environment, we train the Ant agent to run in a target direction. The training task set contains 24 directions which are consecutive integers from 1 to 10. The adaptation task set contains 5 integer directions which are uniformly sampled from [0,360) degrees. The agent is rewarded with the velocity along the target direction and penalized with the velocity along the direction perpendicular to our target.
\item \textbf{Hopper/Walker-Vel~(Uni-modal):} Similar to HalfCheetah-Vel, Hopper/Walker-Vel contains 10 velocities in training task set and 3 in adaptation task set. The target velocities when training range from 0.2 m/s to 2 m/s and every 0.2 m/s we set a training task. The Hopper/Walker agent is penalized with the $l_1$ error between its velocity and the target velocity.
\item \textbf{HalfCheetah-Run-Jump~(Multi-modal):} There are two different uni-modal task distributions in this environment, respectively HalfCheetah-BackVel and HalfCheetah-BackJump. HalfCheetah-BackVel trains the halfcheetah agent to run backward at a target velocity, and the agent is penalized with the $l_1$ error between its velocity and the target velocity. HalfCheetah-BackJump trains the agent to raise its hind leg to jump, and the agent is rewarded with the height of its hind leg. HalfCheetah-Run-Jump contains 7 velocities for running backward (from 1 m/s to 7 m/s and every 1 m/s a training task is set) and 3 tasks for jumping (3 different weights for the leg height in the total reward function) in the training task set. In our implementation, the jumping task with smallest weight finally turns out to be a motion of ``standing" instead of jumping.
\end{itemize}
\subsection{Baselines}\label{baselines}
The detailed descriptions of the multi-task RL and meta RL baselines are listed as follows:
\begin{itemize}[leftmargin=*]
\item \textbf{MAML.} MAML~\citep{finn2017model} is a gradient-based meta-RL algorithm based on Trust Region Policy Optimization~\citep{pmlr-v37-schulman15}. It aims to learn easily adaptable model parameters for tasks in a specific task distribution. MAML conducts explicit training on model parameters so that a small number of training data and gradient steps are required to adapt to a new task. In our implementation, we modify the range and density of task sampling, which will be discussed in the following section.
\item \textbf{PEARL.} PEARL~\citep{rakelly2019efficient} is a context-based off-policy meta-RL algorithm based on Soft Actor Critic~\citep{haarnoja2018soft} and variational inference method~\citep{kingmaauto}. PEARL tackles the problem of limited sample efficiency in meta-RL domain. PEARL integrates off-policy RL algorithms with a latent task variable which infers how to adapt to a new task with small amount of history contexts.
\item \textbf{Multi-head multi-task SAC~(MHMT-SAC).}
MHMT-SAC is a multi-task RL architecture, in which each training task has an independent head input, and the body of the policy network is shared among all the tasks. We compare our method with MHMT-SAC to evaluate the performance of an agent with emergent action representation in multi-task training.
\item \textbf{One-hot embedding SAC~(OHE-SAC).} The sensory-action representation is introduced when training the agent to learn multiple tasks in the training set. In OHE-SAC, We omit the representation layer to make the policy network a common multi-task RL structure, which concatenates a one-hot embedding and the raw state vector as the input of the policy network. Through comparison between the final performance of the two methods, OHE-SAC helps identify the effect of action representations in training and adaptation.
\end{itemize}
\subsection{Multi-task training}\label{details}
\subsubsection{Task sampling density}
In the experiments, we fix the range of task sampling to be for different algorithms in the same environment. We find that in HalfCheetah-Vel environment, multi-task RL methods get different final rewards for different tasks~(in Figure \ref{cheetah-vel}). The reason is that when the agent is trained well enough, the penalty of velocity in the reward will become slight, thus making the control cost of the agent dominant in the reward function. Therefore, it is natural that the faster the agent runs, the larger cost it should pay, leading to a relatively low reward. Thus to make our comparison valid and persuasive, the range of task sampling should be fixed. The detailed settings of the implementation in our paper are demonstrated in Table \ref{sampling density}.
Similarly, the adaptation tasks are also shared among all the algorithms in the same environment in Section \ref{adapt}.
\begin{figure}[t]
\centering
\begin{minipage}{0.48\linewidth}
\centering
\includegraphics[width=0.8\linewidth]{assets/HalfCheetah-Vel_Rewards_appendix_ear.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centering
\includegraphics[width=0.8\linewidth]{assets/HalfCheetah-Vel_Rewards_appendix_mhmt.pdf}
\end{minipage}
\caption{Training curves of tasks with different target velocities with multi-task RL method.}
\label{cheetah-vel}
\end{figure}
\begin{table}[t]
\centering
\begin{tabular}{c|ccc}
\toprule
Method & HalfCheetah-Vel & Hopper/Walker-Vel & Ant-Dir\\
\midrule
\textbf{EAR-SAC (Ours)} & \textbf{10 tasks in [0,10]} & \textbf{10 tasks in [0,2]} & \textbf{24 tasks in [0,360)}\\
MAML & 100 tasks in [0,10] & 40 tasks in [0,2] & 40 tasks in [0,360)\\
PEARL & 100 tasks in [0,10] & 20 tasks in [0,2] & 100 tasks in [0,360)\\
MTMH-SAC & 10 tasks in [0,10] & 10 tasks in [0,2] & 24 tasks in [0,360)\\
OHE-SAC & 10 tasks in [0,10] & 10 tasks in [0,2] & 24 tasks in [0,360)\\
\bottomrule
\end{tabular}
\caption{Density of task sampling in the training procedure.}
\label{sampling density}
\end{table}
\subsubsection{Hyperparameters}
Environment:
\begin{itemize}
\item Max environment steps per trajectory: 200
\item Reward scale: 5.0
\end{itemize}
Multi-task training:
\begin{itemize}
\item Sensory representation size: 16
\item Task embedding size: 3
\item State encoder architecture: 2-layer MLP
\item Action decoder architecture: 4-layer MLP
\item Learning rates for SAC update: 3e-4
\end{itemize}
Adaptation:
\begin{itemize}
\item Elite set capacity $m$: 6
\item Number of sampling $n$: 15
\end{itemize}
\subsection{Task interpolation and composition. }
\textbf{Intra-action interpolation. }
We have performed task interpolation between adjacent tasks in HalfCheetah-Vel in Section~\ref{interpolate}, where we can interpolate the LTEs of running at 1m/s and 2m/s to control the agent to run at 1.2, 1.5, 1.7m/s. The coefficient $\beta$ of the task interpolation is demonstrated in Table \ref{cheetah interpolation detail}. We can find that as the coefficient of interpolation decreases, the cheetah tends to run faster, which indicates that the LTEs between the embeddings of running at 1m/s and 2m/s show continuity in the LTE Space.
Meanwhile, to fully evaluate the performance of task interpolation, we conduct another experiment on other environments. We aim to adjust the interpolation coefficients to achieve 1:1 interpolation between pairs of tasks. For instance, in the experiment aforementioned, we find that when $\beta=0.42$, the agent can run forward at 1.5m/s, which is a 1:1 interpolation of running at 1m/s and 2m/s. The results of 1:1 interpolation are shown in Table \ref{cheetah interpolate},\ref{hopper interpolate},\ref{walker interpolate},\ref{ant interpolate}.
\begin{table}[htbp]
\centering
\begin{tabular}{c|c|c}
\toprule
Task & $\beta$ & Evaluation \\
\midrule
Vel-1.0~(pre-train)& 1 & 0.98m/s \\
\bf{Vel-1.2~(interpolate)}& 0.52 & 1.20m/s \\
\bf{Vel-1.5~(interpolate)}& 0.42 & 1.49m/s \\
\bf{Vel-1.7~(interpolate)}& 0.35 & 1.70m/s \\
Vel-2.0~(pre-train)& 0 & 1.98m/s \\
\bottomrule
\end{tabular}
\caption{\textbf{Coefficients and evaluation results of task interpolation in HalfCheetah-Vel.}}
\label{cheetah interpolation detail}
\end{table}
\textbf{Inter-action composition.}
We have performed task composition in the HalfCheetah-Run-Jump environment. The coefficient $\beta$ of the task composition is demonstrated in Table \ref{cheetah composition detail}.
\begin{table}[h]
\centering
\begin{tabular}{c|c}
\toprule
Task & $\beta$ \\
\midrule
Run-jumping& 0.27 \\
Walk-standing& 0.45 \\
\bottomrule
\end{tabular}
\caption{\textbf{Coefficients of task composition in HalfCheetah-Run-Jump.}}
\label{cheetah composition detail}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{c|ccccccccc}
\toprule
Target & 1.5 & 2.5 & 3.5 & 4.5 & 5.5 & 6.5 & 7.5 & 8.5 & 9.5 \\
\midrule
$\beta$ & 0.42 & 0.42 & 0.38 & 0.39 & 0.43 & 0.45 & 0.46 & 0.51 & 0.48 \\
Evaluation & 1.51 & 2.50 & 3.50 & 4.50 & 5.50 & 6.50 & 7.50 & 8.50 & 9.50 \\
\bottomrule
\end{tabular}
\caption{\textbf{1:1 interpolation in HalfCheetah-Vel.}}
\label{cheetah interpolate}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{c|ccccccccc}
\toprule
Target & 0.3 & 0.5 & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 \\
\midrule
$\beta$ & 0.59 & 0.50 & 0.47 & 0.39 & 0.51 & 0.64 & 0.55 & 0.64 & 0.60 \\
Evaluation & 0.30 & 0.50 & 0.70 & 0.90 & 1.11 & 1.30 & 1.50 & 1.70 & 1.90 \\
\bottomrule
\end{tabular}
\caption{\textbf{1:1 interpolation in Hopper-Vel.}}
\label{hopper interpolate}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{c|ccccccccc}
\toprule
Target & 0.3 & 0.5 & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 \\
\midrule
$\beta$ & 0.54 & 0.36 & 0.49 & 0.38 & 0.45 & 0.44 & 0.66 & 0.82 & 0.77 \\
Evaluation & 0.30 & 0.50 & 0.70 & 0.90 & 1.10 & 1.30 & 1.50 & 1.70 & 1.90 \\
\bottomrule
\end{tabular}
\caption{\textbf{1:1 interpolation in Walker-Vel.}}
\label{walker interpolate}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{c|cccccccc}
\toprule
Target & 7.5 & 22.5& 37.5& 52.5& 67.5& 82.5& 97.5& 112.5 \\
\midrule
$\beta$ & 0.76 & 0.56 & 0.37 & 0.20 & 0.59 & 0.31 & 0.40 & 40 \\
Evaluation & 7.27 & 22.54 & 37.51 & 52.32 & 67.54 & 82.58 & 95.52 & 112.52 \\
\bottomrule
\toprule
Target & 127.5 & 142.5& 157.5& 172.5& 187.5& 202.5& 217.5& 232.5 \\
\midrule
$\beta$ & 0.34 & 0.22 & 0.52 & 0.48 & 0.41 & 0.47 & 0.38 & 0.27 \\
Evaluation & 127.69 & 142.80 & 157.56 & 172.39 & 187.50 & 202.50 & 217.44 & 232.61 \\
\bottomrule
\toprule
Target & 247.5 & 262.5& 277.5& 292.5& 307.5& 322.5& 337.5& 352.5 \\
\midrule
$\beta$ & 0.35 & 0.56 & 0.40 & 0.35 & 0.86 & 0.52 & 0.66 & 0.48 \\
Evaluation & 247.50 & 262.33 & 277.49 & 292.42 & 307.37 & 322.55 & 337.32 & 352.82 \\
\bottomrule
\end{tabular}
\caption{\textbf{1:1 interpolation in Ant-Dir.}}
\label{ant interpolate}
\end{table}
|
1,314,259,996,679 | arxiv | \section{Experimental Devices and Measurement Procedures}
\label{sec:expDevices}
The double layer graphene devices were fabricated by using procedures
previously described in detail in Refs.~\onlinecite{Ponomarenko11} and
\onlinecite{Gorbachev12A}. The maximum size of our double-layer Hall
bars is currently limited to, typically, 2\,$\mu$m $\times$ 10\,$\mu$m
because of the formation of pockets of a hydrocarbon residue at the
interface between graphene and BN (for details, see Supplementary
Material in Ref.~\onlinecite{Ponomarenko11}). These pockets (or
bubbles) appear randomly, and our Hall bars are made to fit inside
clean patches between bubbles. This restricts the width of
double-layer devices to 1.5-2\,$\mu$m. Making smaller and narrower
Hall bars is impractical because of reduced mobility and difficulties
associated with alignment of two Hall bars exactly on top of each
other. Therefore, at present it is impossible to study magnetodrag in
devices of different widths. The reported Hall bars had a width of 2
$\mu$m, and the overlapping area for top and bottom Hall bars was
$\approx 15$\,$\mu$m$^2$.
All experimental results presented in this work were obtained by using
DC measurements with current commutation at each data point. We have
chosen DC over AC measurements because in the latter case a
significant out-of-phase signal (up to ~30\%) appears near the
neutrality point even at frequencies as lows as $\sim$ 10-30\,Hz. This
signal originates probably from capacitance coupling between the
closely spaced graphene layers. Each data point was measured for 12
seconds (6 sec for each polarity of the current), which corresponds to
an effective frequency of 0.2 Hz, low enough to avoid the capacitive
coupling. Further increase in the acquisition time improved accuracy
but did not affect the reported curves.
To ensure that the measurements are done in the linear response
regime, applied current was kept low. To this end, I-V curves were
measured at representative points of the reported maps. . We have
found empirically that a current of 50 nA can be used near the
neutrality point where the nonlinearity is largest. At higher carrier
densities, it is possible to increase the current by a factor of
10. The main reason for the nonlinearity is a voltage drop along the
Hall bar. This voltage can be comparable to the difference between
Fermi levels in the two graphene layers and, if the current is high,
the voltage drop can result in a gradient of carrier concentration
along the direction of current.
Furthermore, there is a finite tunnelling resistance through 3 atomic
layers of BN separating top and bottom graphene Hall bars. At low
biases used in all our measurements, the interlayer resistance $R_T$
exceeded 300\,k$\Omega$ over the whole range of gate voltages. This
value translates into a small contribution to the measured drag
resistivity estimated as \cite{Halperin}
\begin{equation}
\label{estimate}
\delta \rho_{xx}^D = \frac{L}{W} \frac{\rho^2}{12 R_T}
\end{equation}
for the case of equivalent layers. At the neutrality point at $T=240$\,K we have
$R_T = 500$\, k$\Omega$, $\rho=1$\,k$\Omega$ is the intralayer resistivity (resistance per square)
and the aspect ratio $L/W\approx 2$ for our sample. This yields the insignificant contribution
$\delta \rho_{xx}^D \approx 0.33$\,$\Omega$.
However, to ensure that the direct electrical coupling
does not affect the measured drag resistance, we have also compared
measurements with the passive graphene layer (the one where we measure
drag voltage) being floated and grounded. In latter case, several
different contacts were connected to the ground. If the tunnelling
current is significant the grounded contacts should act as a sink and
affect drag measurements. The difference was found insignificant for
trilayer BN devices, as expected from the estimate (\ref{estimate}).
In contrast, devices with bilayer BN as a spacer exhibited very
significant changes indicating that bilayers are too transparent to be
used as an insulating layer for drag measurements.
Another important consideration in the reported measurements was to
ensure that there was no additional AC current flowing through the
devices due to external radiation at high frequencies, which could
increase electron temperature in graphene with respect to cryostat
temperature. To this end, radio frequency filters were fitted. Their
efficiency was judged by the observation of strong suppression of a
rectification signal (voltage in the absence of applied current),
which could be comparable to drag voltage if no filters were used.
\section{Thickness of BN determined from transport measurements}
\label{sec:expBN}
The thickness of BN separating graphene layers was routinely
determined during the device fabrication. To this end, we could use
several techniques including atomic force microscopy (AFM) \cite{Dean10}, optical contrast \cite{Gorbachev11,Golla13}, Raman spectroscopy \cite{Gorbachev11} and tunnelling \cite{Britnell12}. In
practice, the first two were found sufficient to determine the
thickness with single-layer accuracy. The tunnelling resistance for
the known device area was then used as an additional cross-check of the
BN thickness. In this section we show that transport measurements
provide yet another way to measure the thickness of the BN layer.
The approach is based on converting the Hall resistance measured in
weak magnetic fields into carrier concentration and then finding the
thickness as a fitting parameter from the gate voltage dependence of
carrier concentration. In general the density is nonlinear complicated
function of both gate voltages. However, if one of the layers is kept
at the neutrality point, the density in the other layer acquires a
relatively simple dependence on gate voltage. Our approach is similar
to the one recently used by Kim et al. \cite{Kim12}.
Consider a 4-plate capacitor in Fig.~\ref{fig:Set1}, which consists of
two grounded graphene sheets separated by a BN layer with thickness
$d_\textrm{int}$ and two gates, top and bottom, isolated by relatively thick BN,
with thicknesses $d_T$ and $d_B$, respectively. The charge densities
in both graphene layers $n_T$ and $n_B$ are related to the applied
gate voltages $V_T$ and $V_B$ through the system of nonlinear
equations:
\begin{eqnarray}
&&
E_T\frac{d_T}{\varepsilon}+\frac{1}{e}E_F(n_T)=V_T,
\label{ex1}
\\
&&
E_\textrm{int}\frac{d_\textrm{int}}{\varepsilon}+\frac{1}{e}\left(E_F(n_T)-E_F(n_B)\right)
=0,
\label{ex2}
\\
&&
E_B\frac{d_B}{\varepsilon}+\frac{1}{e}E_F(n_B)=-V_B,
\label{ex3}
\\
&&
E_T-E_\textrm{int}=\frac{en_T}{\varepsilon_0},
\label{ex4}
\\
&&
E_\textrm{int}-E_B=\frac{en_B}{\varepsilon_0},
\label{ex5}
\end{eqnarray}
where $E_T$, $E_B$ and $E_\textrm{int}$ are the electric fields in the
top, bottom and middle BN layers, respectively, $\varepsilon = 3.2$ is
the dielectric constant of BN \cite{Kim12}, $\varepsilon_0$ the
electric constant, $e$ the charge of electron and
$E_F(n)=\sign(n)\hbar v_F \sqrt{\pi |n|}$ the Fermi energy in graphene
($v_F \approx 10^6$\,m$/$s is the Fermi velocity; for simplicity we
assume $T= 0$ and no external doping). The Fermi energy is positive
for electrons ($n>0$ ) and negative for holes
($n<0$). Eqs. (\ref{ex1}-\ref{ex3}) describe the potential drop across
the top, middle and bottom BN, whereas Eqs. (\ref{ex4}-\ref{ex5})
follow from the Gauss law for the top and bottom graphene layers.
\begin{figure}
\centerline{
\includegraphics[width=0.5\columnwidth]{figS1.eps}
}
\caption{Sketch for 4-plate capacitor model.}
\label{fig:Set1}
\end{figure}
If one of the graphene layers is at the neutrality point (e.g. for
$n_T=0$), Eqs. (\ref{ex1}-\ref{ex5}) reduce to a one-line expression
that links gate voltage directly to carrier density in the
corresponding layer:
\begin{equation}
\label{ex6}
V_B=\frac{qd_B}{\varepsilon\ep_0}n_B+\sign(n_B)
\frac{\hbar v_F\sqrt{\pi |n_B|}}{q}\left(1+\frac{d_B}{d_\textrm{int}}\right).
\end{equation}
The first (linear) term gives the slope of $V_B(n_B)$ at high
densities and depends only on the thickness of the insulator
separating graphene from the gate (this term is just the classical
capacitance). The second (root-squared) term comes from the quantum
capacitance of graphene and is responsible for nonlinear behaviour of
the double-layer system close to the NP. The term depends on ratio
$d_B/d_\textrm{int}$.
\begin{figure}
\centerline{
\includegraphics[width=0.8\columnwidth]{figS2.eps}
}
\caption{BN thickness extracted from transport measurements. (a)
Density in the top graphene layer as a function of gate voltage
applied to this layer, provided the bottom graphene layer is kept
neutral. Open symbols -- experiment; solid curve -- best fit to
Eq.~(\ref{ex7}). Different points correspond to different voltages
applied to the bottom gate. (b) Same type of measurements as in (a)
but with top and bottom layers interchanged. Both (a) and (b) give
the same value of $d_\textrm{int}$.}
\label{fig:exp2}
\end{figure}
Similar expression can be derived for $n_B=0$:
\begin{equation}
\label{ex7}
V_T=\frac{qd_T}{\varepsilon\ep_0}n_T+
\sign(n_T)\frac{\hbar v_F\sqrt{\pi |n_T|}}{q}\left(1+\frac{d_T}{d_\textrm{int}}\right).
\end{equation}
Therefore the thickness $d_\textrm{int}$ can be determined from two
independent sets of measurements, presented in
Figs.~\ref{fig:exp2}a and \ref{fig:exp2}b. The experiment is carried
out as the following. We fix $V_B$ and ramp $V_T$ recording Hall
resistance for both layers. When the bottom layer reaches its
neutrality point we record values of $V_T$ and $n_T$. This gives one
data point in Fig.~\ref{fig:exp2}a. The same procedure is repeated for
several $V_B$ until enough data are collected. Then the same procedure
is carried out for fixed $V_T$, that is, the top and bottom layers are
effectively swapped in these measurements. Within experimental
accuracy, Figs.~\ref{fig:exp2}a and \ref{fig:exp2}b yield the same
value of $d_\textrm{int}$ which is $\approx 4$ times larger than interlayer
spacing for bulk graphite and BN. This agrees well with the expected
separation for the top and bottom graphene sheets if three BN layers
are in between. This value is also in agreement with the AFM
measurements carried on the same device.
\section{Phenomenological theory of double-layer devices}
\label{sec:theory}
Here we present the full analytic solution of the theoretical
model given by Eqs.~(6) and (4) of the main text.
\subsection{Model equations within linear response}
Under the assumptions of linear response we have to re-write the
equations~(6) of the main text as follows [the frictional force is taken in the form (4) of the main text; for brevity we write
equations for one layer only]
\begin{subequations}
\label{phm}
\begin{eqnarray}
\label{qeq}
&&
- \frac{\kappa_1}{en_1} \boldsymbol{\nabla} \rho_1+\boldsymbol{E}_1 + R_H^{(1)}[\boldsymbol{j}_1\times \boldsymbol{e}_z] =
e \frac{\rho_0}{n_1} \Gamma (\boldsymbol{P}_{1}-\boldsymbol{P}_2) + e R_0^{(1)}\boldsymbol{P}_1,
\\
&&
\nonumber\\
\label{jeq}
&&
\frac{\rho_0}{n_1}\boldsymbol{E}_1 + e R_H^{(1)}[\boldsymbol{P}_1\times \boldsymbol{e}_z]
= e \Gamma (\boldsymbol{P}_{1}-\boldsymbol{P}_2) + R_0^{(1)} \boldsymbol{j}_1
\\
&&
\nonumber\\
\label{ceq}
&&
\boldsymbol{\nabla}\cdot\boldsymbol{P}_1 = -\frac{\rho_1-\rho_0}{\tau_{\textrm{ph}}}
-\frac{\rho_1-\rho_2}{2\tau_\textrm{Q}}.
\end{eqnarray}
\end{subequations}
Here $R_0=1/(enM)$ and $R_H=B/(en)$ are the Drude and Hall resistances
of the single-layer graphene far from the Dirac point; and
$\Gamma=\gamma/e^2$ is the drag resistance far from the Dirac point
and in zero field.
The above equations should be supplemented by the following boundary conditions:
the quasiparticle currents must vanish at the edge of the sample
\begin{subequations}
\label{bc}
\begin{equation}
\boldsymbol{P}_i(y=0) = \boldsymbol{P}_i(y=W)=0,
\end{equation}
and on average no electric current is allowed in the passive layer
\begin{equation}
\label{av}
\overline{\boldsymbol{j}_2} = \frac{1}{W} \int\limits_0^W\!\! dy\, \boldsymbol{j}_2 = 0.
\end{equation}
\end{subequations}
The model (\ref{phm}) with the boundary conditions (\ref{bc}) admits a
full analytic solution. The results simplify in the case of identical
layers. The corresponding solutions are given below.
The results for inequivalent layers are qualitatively similar to the
below solutions. We illustrate the dependence of magnetodrag in the
double neutrality point on magnetic field in Fig.~\ref{fig:B}b for the case of
different mobilities in the layers: $M=7$\, m$^2/Vs$ in the active
layer and $M=3$\, m$^2/Vs$ and in the passive layer.
\subsection{Drag between identical layers}
Given the nonuniform current flow, the drag resistivity is defined
as the ratio of the averaged induced voltage in the passive layer to
the driving current in the active layer
\begin{equation}
\label{rd-def}
\rho_{xx}^D = \frac{\overline{E_{2x}}}{\overline{j_{1x}}},
\end{equation}
where the averaging was defined in Eq.~(\ref{av}).
\subsubsection{General expression}
The resulting expression reads
\begin{equation}
\label{res-rd}
\rho_{xx}^D = \frac{r_0}{2}\left[
\frac{1}{1 - \left[1-\frac{n^2}{\rho_0^2}\right] \frac{R_H^2}{R_H^2+R_0^2} \bar{f}_+}
-
\frac{1}{1 -\frac{n^2}{\rho_0^2} \frac{2\Gamma}{2\Gamma +r_0}
- \left[1-\frac{n^2}{\rho_0^2}\right]
\frac{R_H^2}{R_H^2+R_0^2+\frac{\rho_0^2-n^2}{n^2}2\Gamma r_0} \bar{f}_-}
\right],
\end{equation}
where $r_0=1/(e\rho_0 M)$ is the residual resistance of single-layer
graphene at the Dirac point in the absence of magnetic field, and
\begin{equation}
\bar{f}_\pm = 1 - \frac{\tanh(W/L_\pm)}{W/L_\pm},
\end{equation}
with
\begin{equation}
L_+^{-2} = \frac{e^2 n}{4\kappa\tau_{\rm ph}} R_0 \left[1+\frac{R_H^2}{R_0^2}\right],
\quad
L_-^{-2} = \frac{e^2 n}{4\kappa} R_0
\left[ \frac{1}{\tau_{\rm ph}} + \frac{1}{\tau_Q} \right]
\left[\frac{2\Gamma}{r_0}\left(1-\frac{n^2}{\rho_0^2}\right) + 1+\frac{R_H^2}{R_0^2}\right].
\end{equation}
Note, that $R_H/R_0 = BM$.
\subsubsection{Drude limit}
Far away from the Dirac point (i.e. for $\mu\gg T$) only one band
contributes (such that $n=\rho_0$) and the result (\ref{res-rd}) simplifies
to the standard Drude form which is independent of magnetic field
\begin{equation}
\rho_{xx}^D (\mu\gg T) = - \Gamma.
\end{equation}
\subsubsection{Zero field limit}
In the absence of magnetic field $R_H=0$ and the result (\ref{res-rd})
simplifies to
\begin{equation}
\label{zfrd}
\rho_{xx}^D (B=0) = \frac{r_0}{2}
\left[1 - \frac{1}{1-\frac{n^2}{\rho_0^2}\frac{2\Gamma}{2\Gamma+r_0}} \right].
\end{equation}
Setting $n=\rho_0$ we recover the above Drude result. On the other
hand at the Dirac point $n=0$ and drag vanishes
\begin{equation}
\rho_{xx}^D (B=0; n=0) = 0.
\end{equation}
\subsubsection{Neutrality point}
At the Dirac point ($n=0$) in the presence of magnetic field we find
\begin{equation}
\rho_{xx}^D (n=0) = \frac{r_0}{2}
\left[
\frac{1}{1-\bar{f}_+ \frac{(BM)^2}{1+(BM)^2}}
-
\frac{1}{1-\bar{f}_- \frac{(BM)^2}{1+(BM)^2+2\Gamma/r_0}}
\right].
\end{equation}
Clearly, this result vanishes in the absence of the field.
\subsubsection{Neutrality point in an infinite sample}
Consider now the above result in the limit of an infinite sample $W\gg
L_+$. (Note that $L_+ > L_-$ by definition.) Then $\bar f_\pm = 1$
and we find (in agreement with Eq.~(5) of the main text)
\begin{equation}
\rho_{xx}^D (n=0; W\rightarrow \infty) = \frac{r_0}{2} \left[
\frac{1}{1-\frac{(BM)^2}{1+(BM)^2}}
-
\frac{1}{1-\frac{(BM)^2}{1+(BM)^2+2\Gamma/r_0}}
\right] =
\frac{\Gamma r_0}{2\Gamma + r_0} (BM)^2 >0.
\end{equation}
The resulting drag is {\it positive}.
\subsubsection{Neutrality point in a narrow sample}
For completeness, let us now consider now a limit of a
vanishing width $W\ll L_-$. Here $\bar f_\pm \approx
W^2/(24L_\pm^2)$ and expanding in this small factor we find
\begin{equation}
\rho_{xx}^D (n=0; W\rightarrow 0) \approx
- r_0(BM)^2 \frac{e W^2}{24 M \kappa \tau_Q} < 0.
\end{equation}
The resulting drag is {\it negative}, independent of phonon scattering
time and mobility (this can be seen by taking into account that the
resistance $r_0$ is inversely proportional to the mobility).
\subsection{Hall drag between identical layers}
The Hall drag resistance is defined as
\begin{equation}
\label{rh-def}
\rho_{xy}^D = \frac{\overline{E_{2y}}}{\overline{j_{1x}}},
\end{equation}
where the averaging was defined in Eq.~(\ref{av}).
The resulting expression for the Hall drag resistance is
\begin{equation}
\label{res-rh}
\rho_{xy}^D = \frac{r_0}{2}\frac{n}{\rho_0} BM \left[
\frac{1}{1 - \left[1-\frac{n^2}{\rho_0^2}\right] \frac{R_H^2}{R_H^2+R_0^2} \bar{f}_+}
-
\frac{\frac{r_0}{2\Gamma+r_0} - \left(1-\frac{n^2}{\rho_0^2}\right)
\frac{2\Gamma \bar{f}_-}{r_0\left[1+(BM)^2 +
\left(1-\frac{n^2}{\rho_0^2}\right)\frac{2\Gamma}{r_0}\right]}
}
{1 -\frac{n^2}{\rho_0^2} \frac{2\Gamma}{2\Gamma +r_0}
- \left[1-\frac{n^2}{\rho_0^2}\right]
\frac{R_H^2}{R_H^2+R_0^2+\frac{\rho_0^2-n^2}{n^2}2\Gamma r_0} \bar{f}_-}
\right],
\end{equation}
where it is clear that Hall drag vanishes in the absence of the
magnetic field and at the Dirac point as it should. It is also clear
that Hall drag vanishes in the Drude limit where $n=\rho_0$.
\begin{figure}
\centerline{
\includegraphics[width=\columnwidth]{figS3.eps}}
\caption{Left Panel: Fit of the zero-field drag resistivity. This fit is used to determine the phenomenological friction coefficient
$\gamma$. Note, that the theoretical curve in this plot is shifted vertically to fit the shape of the experimental data. The obtained $\gamma$ is subsequently used to calculate magnetodrag. Right Panel: magneto-drag at charge neutrality calculated for different sample widths in the case of inequivalent layers.}
\label{fig:B}
\end{figure}
\section{Relaxation rates}
In order to determine how the above results depend on carrier density,
we need to estimate the dependence of the scattering rates
$\tau_Q^{-1}$, $\tau_{\rm ph}^{-1}$ and the phenomenological
coefficient $\gamma$ on the chemical potential and temperature. While
comparing our theory to the experimental data, we use the
interpolation formulas listed below, such that the only remaining
fitting parameters are the electron-electron and electron-phonon
coupling constants ($\alpha$ and $g_{\rm ph}$, respectively).
\subsection{Momentum relaxation rate}
Since the drag at zero magnetic field is entirely determined by the
"friction coefficient" $\gamma$ we can use earlier theoretical work \cite{Narozhny12A,Schuett13A}
and actual experimental data to fit
$\gamma$. The result is a function of a single parameter that has the
following general properties: (i) $\gamma\propto\alpha^2$, (ii)
$\gamma\rightarrow 0$ for $\mu\gg T$, and (iii) $\gamma$ remains
finite at the Dirac point.
For $T=240\,$K, the system can be approximately regarded as
ballistic. In this case, microscopic calculations show that at the
Dirac point $\gamma(\mu=0)\sim\alpha^2$, while far away from the Dirac
point $\gamma(\mu\gg T)\sim\alpha^2 T^2/\mu^2$. In the intermediate
region $\mu\sim T$ the momentum relaxation rate is given by a
complicated integral, but effectively it just interpolates between the
two limits. Given that the phenomenological theory is only accurate in
those two limits, we may use a simple interpolation
\begin{equation}
\gamma = \frac{\alpha^2}{1+\mu^2/2T^2}.
\end{equation}
For $T=160$\,K, the system enters the diffusive regime which complicates
the theory. We use our earlier estimates of the drag resistance in the diffusive
regime \cite{Narozhny12A} and fit the dimensionless relaxation rate $\gamma$
to the drag data at zero magnetic field, see Fig.~\ref{fig:B}. Within the phenomenological theory, the zero-field
drag resistance is given by Eq.~(\ref{zfrd}). Using this result and respecting the above general restrictions on $\gamma$ we arrive at the empirical expression, which is applicable for not too high values of the chemical potential:
\begin{equation}
\gamma\approx\frac{\alpha^2}{\sqrt{4+(\mu/T)^2 (1+0.35\ln^2(\mu/T))}}.
\end{equation}
\subsection{Energy relaxation rate}
The lowest order phonon contribution to the electron-hole
recombination in graphene is kinematically forbidden (within the same
valley). There are, however, many possible mechanisms of recombination
\cite{Foster09A,Song12}, all of them involving phonons. Therefore,
we phenomenologically regard the time and length scale of electron and
hole recombination as $\tau_{\textrm{ph}}$ and $\ell_{\textrm{ph}}$.
The microscopic theory \cite{Schuett13A} includes
thermoelectric effects formulated in terms of energy currents. In
graphene the energy current is equal to the total momentum
\cite{Schuett13A}. Therefore, the corresponding relaxation processes do
not require recombination and can be directly attributed to phonons.
The quasiparticle relaxation rate due to the scattering on phonons can
be estimated with the help of the Fermi golden rule. The result is
\begin{equation}
\frac{1}{\tau_{\rm ph}} = \frac{g_{\rm ph}^2(T) T}{\cosh(\mu/T)}.
\end{equation}
This estimate includes the disorder-assisted electron-phonon
scattering processes \cite{Song12} as well as phonon-induced
intervalley scattering and is valid for temperatures below the Debye
temperature in hBN.
\begin{figure}
\centerline{
\includegraphics[width=0.5\columnwidth]{figS4.eps}}
\caption{Sketch of the quasi-particle relaxation between two layers}
\label{fig:C}
\end{figure}
\subsection{Imbalance relaxation}
The quasiparticle recombination rate due to the energy transfer
between the graphene layers can also be estimated by means of the
Fermi Golden Rule in the two limits $\mu\ll T$ and $\mu\gg
T$. Interpolating between the two limits, we obtain the following
estimate
\begin{equation}
\frac{1}{\tau_Q}=\frac{\alpha^2\sqrt{T^2+\mu^2}}{4\cosh(\mu/T)}.
\end{equation}
The corresponding process is illustrated in Fig.~\ref{fig:C}.
|
1,314,259,996,680 | arxiv | \section{Introduction}
Low-rank information extraction from a noisy data matrix is a crucial statistical challenge. The spiked random matrix models have recently gained extensive interest in the fields of statistics, probability, and machine learning, serving as a valuable platform for exploring this issue \cite{donoho1995adapting,peche2014deformed,BBP,lesieur2017constrained}. A prominent example is the spiked Wigner model, where a rank one matrix is observed through a component-wise homogeneous noise.
Heterogeneity being a fundamental part of many real-world problems, we consider here an inhomogeneous version of the Wigner spike model, discussed in \cite{AJFL_inhomo,behne2022fundamental}, where the signal is observed through an inhomogeneous, block-constant noise. Consider a partition $\{ 1, \ldots, N \} \!=\! [N]$ into $q$ disjoint groups $C_{1}^{N} \cup \cdots \cup C_{q}^{N}=[N]$. This partition is encoded by a function $g:\! [N]\! \mapsto \![q]$ which maps each index $i \in [N]$ into its group $g(i) \!\in\! [q]$. Let $\tilde{\boldsymbol{\Delta}} \!\in\! \mathbb{R}^{q \times q}$ be a
symmetric matrix encoding a block-constant symmetric matrix $\boldsymbol{\Delta}\! \in\! \mathbb{R}^{N \times N}$
\begin{equation}\label{eq:varprofile}
\boldsymbol{\Delta}_{ij} = \tilde{\boldsymbol{\Delta}}_{g(i)g(j)}.
\end{equation}
We observe the signal $\boldsymbol{x}^{\star} \in \mathbb{R}^{N}$ which is assumed to have independent identically distributed coordinates generated from some prior distribution $\mathbb{P} _{0}$ (i.e. $\mathbb{P} ( \boldsymbol{x}^{\star} = \boldsymbol{x} ) = \prod_{i = 1}^{N}\mathbb{P}_{0} (x_{i}^{\star} = x_{i})$) through noisy measurements:
\begin{equation} \label{eqn: low-rank inhomogenous}
\boldsymbol{Y} = \sqrt{\frac{1}{N}}\boldsymbol{x}^{\star}(\boldsymbol{x}^{\star})^{T} + \boldsymbol{A} \odot \sqrt{\boldsymbol{\Delta}}.
\end{equation}
Here and throughout the article $\odot$ denotes the Hadamard product, $\sqrt{\boldsymbol{\Delta}}$ is the Hadamard square-root of $\boldsymbol{\Delta}$ and $\boldsymbol{A}$ is a real-valued symmetric GOE matrix with off-diagonal elements of unit variance. The Bayes-optimal performance of this model in the asymptotic limit $N \to \infty$ was studied rigorously in \cite{AJFL_inhomo,behne2022fundamental,MourratXia-tensor, MourratXiaChen-tensor} who characterized the fundamental information-theoretic limit of reconstruction in this model. Here we focus instead on the algorithmic problem of reconstructing the (hidden) spike. {\bf Our contributions are many-fold}:
\begin{itemize}[wide=1pt, topsep=0pt,nosep]
\item We show how one can construct an Approximate Message Passing (AMP) algorithm for the inhomogeneous Wigner problem, whose asymptotic performance can be tracked by a rigorous state evolution, generalizing the homogeneous version of the algorithm for low-rank factorization \cite{BayatiMontanari,DBLP:journals/corr/DeshpandeAM15,lesieur2017constrained}.
\item AMP is shown to give Bayes optimal performances, as characterized in \cite{AJFL_inhomo}, for a wide choice of parameters. There exists, however, a region of parameters where AMP differs from Bayes performances, yielding a computational-to-statistical gap \cite{bandeira2018notes,celentano2020estimation}. In this region, we conjecture that the problem is hard for a large class of algorithms.
\item Finally, we present a linear version of AMP \cite{maillard2022construction}, that turns out to be equivalent to a spectral method, which is optimal in the sense that it can detect the presence of the spike in the same region as AMP. This is quite remarkable since, as shown in \cite[Section~2.4]{AJFL_inhomo}, the standard spectral method (PCA) applied to a simple renormalization of the matrix fails to do so.
\end{itemize}
\paragraph{\textbf{Related work}}
The class of approximate message passing algorithms (AMP) has attracted a lot of attention in the high-dimensional statistics and machine learning community, see e.g. \cite{donoho2009message,BayatiMontanari,rangan2011generalized,DBLP:journals/corr/DeshpandeAM15,lesieur2017constrained,gerbelot2021graph,feng2022unifying}. The ideas behind this algorithm have roots in physics of spin glasses \cite{mezard1987spin,bolthausen2014iterative,zdeborova2016statistical}. AMP algorithms are optimal among first order methods \cite{celentano2020estimation}, thus their reconstruction threshold provides a bound on the algorithmic complexity in our model. Our approach to the inhomogeneous version of AMP relies on several refinements of AMP methods to handle the full complexity of the problem, notably the
spatial coupling technique \cite{krzakala2012probabilistic,donoho2013information,8205391,gerbelot2021graph}.
Factorizing low-rank matrices is a ubiquitous problem with many applications in machine learning and statistics, ranging from sparse PCA to community detection and sub-matrix localization. Many variants of the homogeneous problem have been studied in the high-dimensional limit \cite{DBLP:journals/corr/DeshpandeAM15,lesieur2017constrained,barbier2018rank,lesieur2017constrained,10.1214/19-AOS1826,lelargemiolanematrixestimation,barbier2020information}. The inhomogeneous version was discussed in details in \cite{AJFL_inhomo,behne2022fundamental}. Spectral methods are a very popular tool to solve rank-factorization problems \cite{donoho1995adapting,peche2014deformed,BBP}. Using AMP as an inspiration for deriving new spectral methods was discussed, for instance, in \cite{saade2014spectral,lesieur2017constrained,aubin2019spiked,mondelli2018fundamental,mondelli2022optimal,maillard2022construction,venkataramanan2022estimation}.
\section{Main results}
\paragraph{\textbf{Message passing algorithm}}
For each $t \geq 0$, let $(f_t^a)_{a \in [q]}$ be a collection of Lipschitz functions from $\mathbb{R} \to \mathbb{R}$, and define $f_{t}: \mathbb{R}^{N} \times \mathbb{N} \mapsto \mathbb{R}^{N}$ by
\[
f_t(\bm{x}) := ( f_t^{g(i)} (x_i) )_{i \in [N]} \in \mathbb{R}^N.
\]
These linear functions are often called denoiser functions and can be chosen amongst several options, such as the Bayes optimal denoisers for practical applications (see Section~\ref{sec:Bayes}), or even linear denoisers (see Section~\ref{sec:linear}). We shall consider the following AMP recursion for an estimator in the inhomogeneous spiked Wigner problem
\begin{equation}
\label{eqn: AMP spike}
\boldsymbol{x}^{t+1} = \left(\frac{1}{\sqrt{N}\bm{\Delta}} \odot \bm{Y}\right) f_{t}\left(\boldsymbol{x}^{t}\right)-\bm{\mathrm{b}}_{t} \odot f_{t-1}\left(\boldsymbol{x}^{t-1}\right)
\end{equation}
with the so-called Onsager term $\bm{\mathrm{b}}_{t} = \frac{1}{\bm{\Delta}} f_{t}^{\prime}(x^{t}) \in \mathbb{R}^N$
where $\frac{1}{\bm{\Delta}}$ is the Hadamard inverse of $\bm{\Delta}$ and $f_{t}^{\prime}$ is the vector of coordinate wise derivatives.
In practical implementations, we initialize the algorithm with some non-null $\boldsymbol{x}^{0}$ and let it run for a certain number of iterations. One efficient way to do this is the spectral initialization \cite{mondelli2021approximate} with the method described in sec. \ref{sec:spec}. In Figure~\ref{fig:dense_general} we provide an example of the performance of the AMP together with the Bayes-optimal estimator predicted by the asymptotic theory. Even at very moderate sizes, the agreement is clear.
\paragraph{\textbf{State evolution}}
Our first main contribution is the generalisation of the state evolution characterization of the behaviour of AMP \cite[Theorem~1]{8205391} in the inhomogeneous setting. To state a well-defined limit of the AMP, we have the following assumptions.
\begin{assumption} \label{ass:limit}
To ensure that our inhomogeneous AMP has a well-defined limit, we assume that
\begin{enumerate}
\item For each $a \in [q]$, we have
\[
\lim_{N \to \infty} \frac{|C_a^N|}{N} \to c_a \in (0,1).
\]
\item The family of real valued functions such that $(f_t^a)_{a \in [q]}$ and $(f_t^a)^\prime_{a \in [q]}$ are Lipschitz.
\item For each $a \in [q]$, there exists $\left(\sigma^0_a\right)^{2} \in \mathbb{R}$ such that, in probability,
\[
\lim_{N \to \infty} \frac{1}{|C_a^N|} \sum_{i \in C_a^N} f_0^a(x^0_i) f_0^a(x^0_i) = \left(\sigma^0_a\right)^{2} .
\]
\end{enumerate}
\end{assumption}
Our first result describes the distribution of the iterates in the limit. Our mode of convergence will be with respect to $L$-pseudo-Lipschitz test functions $\phi: \mathbb{R}^M \to \mathbb{R}$ satisfying
\begin{equation}\label{eq:pseudoLip}
| \phi(x) - \phi(y)| \leq L (1 + \|x\| + \| y\|) \| x- y\| \qquad \text{for all $x,y \in \mathbb{R}^M$}.
\end{equation}
We define the following state evolution parameters $\mu_{b}^{t}$ and $\sigma_{b}^{t}$ for $b \in [q]$ through the recursion
\begin{equation}\label{eq:state}
\begin{aligned}
& \mu^{t + 1}_{b} = \sum_{a \in [q]} \frac{c_{a}}{\tilde{\bm{\Delta}}_{ab}} \mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star} f_{t}^{a}\left(\mu^{t}_{a}x_{0}^{\star} + \sigma^{t}_{a}Z\right)] ~\text{with}~ x_{0}^{\star} \sim \mathbb{P}_{0}, Z \sim \mathcal{N} (0, 1) \\
& (\sigma_{b}^{t + 1})^{2} = \sum_{a=1}^{q} \frac{c_{a}} {\tilde{\bm{\Delta}}_{ab}}\mathbb{E}_{x_{0}^{\star}, Z}\left[ (f_{t}^{a}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z))^{2}\right] ~\text{with}~ x_{0}^{\star} \sim \mathbb{P}_{0}, Z \sim \mathcal{N} (0, 1),
\end{aligned}
\end{equation}
where $x_0^\star$ and $Z$ are independent. We use the initialization $\bm{\mu}^0 = \bm{\sigma}^0 = 0$. We prove that the iterates $x_i^{t}$ are asymptotically equal in distribution to $\mu_{g(i)}^{t}x_{0}^{\star} + \sigma_{g(i)}^{t}Z$ where $x^\star_0$ and $Z$ are independent.
\begin{wrapfigure}{rt}{0.5\textwidth}
\begin{center}
\includegraphics[width=0.47\textwidth]{AMP_2.png}
\end{center}
\vspace{-0.5cm}
\caption{Performance of the inhomogeneous AMP algorithm against the information-theoretical optimal {\rm MMSE}. The variance profile is proportional to $\tilde{\bm{\Delta}} \!=\! \begin{bmatrix}
1 & 3\\
3 & 2
\end{bmatrix}$
with two equally sized blocks with standard Gaussian prior when $N \!=\! 500$ at various {\rm snr}. }
\label{fig:dense_general}
\vspace{-1.cm}
\end{wrapfigure}
\begin{theorem}[State evolution of AMP iterates in the inhomogeneous setting] \label{theorem: AMP inhomogeneous}
Suppose that Assumption~\ref{ass:limit} holds, and that $\mathbb{P}_0$ has bounded second moment. Let $\phi: \mathbb{R}^2 \to \mathbb{R}$ be a $L$-pseudo-Lipschitz test functions satisfying \eqref{eq:pseudoLip}. For any $a \in [q]$, the following limit holds almost surely
\[
\hspace{-9cm}\lim_{N \to \infty} \frac{1}{|C_a^N|} \sum_{i \in C_a^N} \phi( x_i^t, x_i^\star ) = \mathbb{E}_{x_{0}^{\star}, Z} \phi( \mu_{a}^{t}x_{0}^{\star} + \sigma_{a}^{t}Z, x_0^\star )
\]
where $Z$ is a standard Gaussian independent from all other variables.
\end{theorem}
\begin{remark}
The notion of convergence under the pseudo-Lipschitz test functions induces a topology that is equivalent to the one generated by the $2$-Wasserstein topology \cite[Remark~7.18]{feng2022unifying}. We can weaken the second moment assumption on $\mathbb{P}_0$ to finite $k$th moment, but the induced topology will then change to the $k$-Wasserstein topology, see \cite[Theorem~1]{8205391}.
\end{remark}
Even though the theoretical result above applies in the high-dimensional limit, numerical simulations show that even for medium-sized $N$ (around $500$), the behaviour of the iterates is well described by the state evolution parameters. Through the state evolution equations \eqref{eq:state} we are able to track the iterates of the AMP iteration with just two vectors of parameters obeying the state evolution recursion: the overlap with the true signal $(\mu_a^{t})_{a \in [q]}$ and its variance $(\sigma^{t}_a)_{a \in [q]}$.
For the inhomogeneous AMP \eqref{eqn: low-rank inhomogenous} iteration we obtain the following necessary and sufficient condition for the overlaps of a fixed point of the iteration:
\begin{theorem}[Bayes-Optimal fixed point]\label{theorem: AMP fixed point}
Assume AMP satisfies Assumption~\ref{ass:limit} and let the denoising functions be the Bayes ones \eqref{eqn: Bayes-optimal functions}. Then the overlaps $\bm{\mu} = (\mu_a)_{a \in [q]}$ in \eqref{eq:state} satisfy the following fixed point equation
\begin{equation}
\mu_{b} = \sum_{a \in[q]} \frac{c_{a}}{\tilde{\boldsymbol{\Delta}}_{ab}}\mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star}\mathbb{E}_{posterior}[x_{0}^{\star}|\mu_{a}x_{0}^{\star} + \sqrt{\mu_{a}}Z]].
\end{equation}
\end{theorem}
\begin{remark}
The state evolution fixed point equation above coincides with the fixed point equation satisfied by the Bayes optimal estimator in \cite[Equation~2.14]{AJFL_inhomo}.
\end{remark}
\begin{figure}[t]
\begin{center}
\includegraphics[width = 10cm]{MMSE_GAP.png}
\caption{The information-theoretic optimal mean squared error for a sparse prior. The AMP detectability transition occurs at $\lambda(\bm{\Delta}) = 1$ in Section~\ref{sec:linear}, which will yield a statistical-to-computational gap. This is in contrast to the continuous phase transition in Figure~\ref{fig:dense_general}, where the AMP phase transition agrees with the optimal one}
\label{fig:optimal}
\end{center}
\vspace{-0.5cm}
\end{figure}
\paragraph{\textbf{A spectral method}}
The spectrums of matrices with variance profiles are difficult to analyze because standard tools to compute the BBP transition result in complicated systems of equations. Given the matrix $\bm{Y}$ defined in \eqref{eqn: low-rank inhomogenous} we consider the transformed matrix
\begin{equation}\label{eq:transformedYgenvariance}
\tilde {\bm{Y}} := \frac{\mathbb{E}_{x_0^\star} [(x_0^\star)^2]}{\sqrt{N}\bm{\Delta}} \odot \bm{Y} - \mathbb{E}_{x_0^\star} [(x_0^\star)^2]^2 \operatorname{diag}\left(\frac{1}{\bm{\Delta}} \begin{bmatrix}
1 \\
\vdots \\
1
\end{bmatrix} \right) .
\end{equation}
Using AMP tools, we are able to analyze the spectral method. In particular, we are able to recover the phase transition for the top eigenvalue of spiked matrices with covariance profiles through the inhomogeneous AMP. Let $\bm{c} = (c_a)_{a \in [q]}$. We define the inhomogeneous signal-to-noise (SNR) ratio of such a model by
\begin{equation} \label{eq:SNR}
{\rm SNR}({\bm \Delta}) := \lambda(\bm{\Delta}) = \mathbb{E}_{x_0^\star} [(x_0^\star)^2]^2 \left\| \operatorname{diag}(\sqrt{\bm{c}}) \frac{1}{\tilde{\bm{\Delta}}} \operatorname{diag}(\sqrt{\bm{c}}) \right\|_{op}.
\end{equation}
\begin{conjecture}\label{conjecture:BBP}
The top eigenvalue of $\tilde {\bm{Y}}$ separates from the bulk if and only if the signal to noise ratio $\lambda(\bm{\Delta}) > 1$. In particular, if $\bm{\hat{x}}$ is the top eigenvector of $\tilde{\bm{Y}}$ then
\[
\lim_{N \to \infty} \frac{|\bm{\hat{x}} \cdot \bm{x}^\star|}{ \|\bm{\hat{x}} \| \|\bm{x}^\star \| } = 0
\]
if and only if $\lambda(\bm{\Delta}) < 1$.
\end{conjecture}
This matches precisely the recovery transition in \cite[Lemma~2.15 Part (b)]{AJFL_inhomo}. In this paper, we rigorously show that with ${\rm SNR}({\bm \Delta}) < 1$ our proposed spectral method fails to recover the signal. We conjecture that this is the sharp transition for spectral methods.
We postpone a full mathematical analysis of this spectral method for future studies. However, we provide indications for validity of the result by using the linear AMP formalism. The connection between the two is a standard phenomenon \cite{saade2014spectral,mondelli2018fundamental,maillard2022construction}. We illustrate the eigenvalue BBP-like transition in Fig.\ref{fig:spectrum}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm]{Bulk_not_separated.png}
\includegraphics[width=7cm]{Bulk_separated.png}
\caption{Illustration of the spectrum of $\bm{\tilde Y} \in \mathbb{R}^{10^3 \times 10^3}$ evaluated at noise profiles with {\rm snr} $\lambda(\bm{\Delta}) = 0.7$ (left, before the transition) and $1.8$ (right, after the transition), with the outlying eigenvector correlated with the spike arises at eigenvalue one.}
\label{fig:spectrum}
\end{center}
\end{figure}
\paragraph{\textbf{Statistical to computational gaps}}As the linear AMP transition arises at $\lambda\!>\!1$, the linear stability analysis of AMP initialized close to a trivial fixed point will recover an identical transition. As in the homogeneous case, the inhomogeneous problem is thus algorithmically tractable only for $\lambda\!>\!1$. However, it was shown that, for sparse enough priors, the Bayes estimate (that is possibly NP-hard) can achieve a positive correlation for $\lambda\!<\!1$. This illustrates the statistical-to-computational gap as in e.g. \cite{bandeira2018notes,barbier2018rank,aubin2019spiked,lelargemiolanematrixestimation,celentano2020estimation}. In this situation, the spectral method described in this work should thus be optimal. Interestingly, this is {\it not} the case of the standard PCA analysis based on the matrix $\bm{Y}$, which fails to achieve a transition at $\lambda\!=\!1$ \cite[Proposition~2.18]{AJFL_inhomo}.
\section{The inhomogeneous AMP algorithm}\label{sec:AMP}
In this section, we derive the formula for the inhomogeneous AMP iteration \eqref{eqn: AMP spike}. We first recall the general matrix framework of AMP from \cite{8205391}:
\paragraph{\textbf{Matrix AMP}}\label{sec:matrixAMP}
In the matrix setting an AMP algorithm operates on the vector space $\mathcal{V}_{q, N} \equiv\left(\mathbb{R}^{q}\right)^{N} \simeq \mathbb{R}^{N \times q}$. Each element of $\bm{v} = (v_1, \dots, v_N) \in \mathcal{V}_{q, N}$ will be regarded as $N$ - vector with entries $v_i \in \mathbb{R}^{q}$.
\begin{definition}[AMP]\label{def:AMP}
A matrix AMP acting on this space is represented by $(\boldsymbol{A}, \mathcal{F}, \boldsymbol{v}^{0})$, where:
\begin{enumerate}
\item $\boldsymbol{A} = \boldsymbol{G} + \boldsymbol{G}^\mathsf{T}$, where $\boldsymbol{G} \in \mathbb{R}^{N \times N}$ has iid entries $G_{ij} \sim N(0,\frac{1}{2})$.
\item $\mathcal{F}$ is a family of $N$ Lipschitz functions $f_t^i: \mathbb{R}^q \mapsto \mathbb{R}^q$ indexed by time $t$. The family $\mathcal{F}$ encodes a function $f_{t}: \mathcal{V}_{q, N} \rightarrow \mathcal{V}_{q, N}$ that acts separately on each coordinate $v_{j} \in \mathbb{R}^{q}$,
\begin{equation}
f_{t}(\boldsymbol{v}) = (
f_{t}^{1}(v_{1}), \dots, f_{t}^{N}(v_{N})
\in \mathcal{V}_{q, N}.
\end{equation}
\item $\boldsymbol{v}^{0} \in \mathcal{V}_{q, N}$ is a starting condition.
\end{enumerate}
\end{definition}
The algorithm itself is a sequence of iterates generated by:
\begin{equation} \label{eqn: Matrix MAMP}
\boldsymbol{v}^{t+1}= \frac{\boldsymbol{A}}{\sqrt{N}} f_{t}\left(\boldsymbol{v}^{t}\right) - f_{t-1}\left(\boldsymbol{v}^{t-1}\right)\bm{\mathrm{B}}_{t}^{T}
\end{equation}
where $\bm{\mathrm{B}}_{t}$ is the $q \times q$ Onsager matrix given by
\begin{equation} \label{eqn: Onsager matrix}
\bm{\mathrm{B}}_{t} = \frac{1}{N} \sum_{j = 1}^{N} \partial f_{t}^{j} (\boldsymbol{x}_{j}^{t}).
\end{equation}
where $\partial f_{t}^{j}$ denotes the Jacobian matrix of $f_t^j$. The limiting properties of the AMP sequences are well known and can be found in \cite[Theorem~1]{8205391}.
\paragraph{\textbf{The inhomogeneous AMP}} \label{sec:inhomoAMP}
We now define an inhomogeneous AMP iteration which takes into account the block-constant structure of the noise:
\begin{definition}\label{def:inhomoAMP}
An inhomogeneous AMP on $\mathcal{V}_{1,N} = \mathbb{R}^N$ is represented by $( \boldsymbol{A}, \mathcal{F}, \boldsymbol{x}^{0},\bm{\Delta})$, where the terms $\boldsymbol{A}$, $\mathcal{F}$, $\boldsymbol{x}^{0}$ are defined in Definition~\ref{def:AMP} and $\bm{\Delta}$ is the $N \times N$ variance profile encoded by $\tilde{ \bm{\Delta}} \in \mathbb{R}^{q \times q}$ and grouping $g: [N] \to [q]$ defined by \eqref{eq:varprofile}. We further assume that the family of functions $\mathcal{F}$ is encoded by functions $f_t^a: \mathbb{R} \mapsto \mathbb{R} $ for $a \in [q]$ which define the group dependent function
\begin{equation}
f_{t}(\boldsymbol{x}) = (
f_{t}^{g(1)}(x_{1}), \dots, f_{t}^{g(N)}(x_{N}) )
\in \mathbb{R}^N.
\end{equation}
\end{definition}
The sequence of iterates $\boldsymbol{x}^{t} \in \mathbb{R}^{N}$ of the $( \boldsymbol{A}, \mathcal{F}, \boldsymbol{v}^{0},\bm{\Delta})$ are defined as follows:
\begin{equation} \label{eqn: version}
\boldsymbol{x}^{t+1}= \left( \frac{1}{\sqrt{N} \sqrt{\boldsymbol{\Delta}}} \odot \boldsymbol{A} \right) f_{t} \left(\boldsymbol{x}^{t}\right)-\mathrm{\boldsymbol{b}}_{t} \odot f_{t-1} \left(\boldsymbol{x}^{t-1}\right),
\end{equation}
where $\frac{1}{\sqrt{\bm{\Delta}}}$ is the Hadamard inverse square root of the noise, and the Onsager term $\boldsymbol{\mathrm{b}}_{t}$ has the following form
\begin{equation} \label{eqn: Onsager vector}
\boldsymbol{\mathrm{b}}_{t} = \frac{1}{N} \begin{pmatrix}
\frac{1}{\Delta_{11}} (f_{t}^{g(1)})^{\prime} (x_{1}^{t}) &+& \ldots &+& \frac{1}{\Delta_{1N}} (f_{t}^{g(N)})^{\prime} (x_{N}^{t})\\
\vdots && \vdots && \vdots \\
\frac{1}{\Delta_{N1}} (f_{t}^{g(1)})^{\prime} (x_{1}^{t}) &+& \ldots &+& \frac{1}{\Delta_{NN}} (f_{t}^{g(N)})^{\prime} (x_{N}^{t})
\end{pmatrix} = \frac{1}{N\boldsymbol{\Delta}} f_{t}^{'} (x^{t}) \in \mathbb{R}^N .
\end{equation}
In order to track the iterates of the recursion \eqref{eqn: version} we reduce this recursion to the matrix setting with an embedding of the inhomogeneous AMP into the matrix AMP.
\paragraph{\textbf{State evolution of the inhomogeneous AMP}} Through a continuous embedding, we will reduce our inhomogeneous AMP to the matrix AMP framework, and recover the state evolution of the inhomogeneous AMP. We define the diagonal matrix operator $\operatorname{blockdiag}: \mathbb{R}^N \mapsto \mathcal{V}_{q, N}$ which outputs a block diagonal matrix according to the block structure of our discretization of $[N]$:
\[
\operatorname{blockdiag}(\bm{v}) = \bm{M} \quad \text{where} \quad M_{ij} = \begin{cases}
v_{j} & g(j) = i
\\ 0 & \text{otherwise} .
\end{cases}
\]
Likewise, we define the projection operator $\operatorname{blockproj}: \mathcal{V}_{q, N} \mapsto \mathbb{R}^N$ which extracts a vector of size $N$ from a $N \times q$ according to the block structure of $[N]$ by
$$\operatorname{blockproj}(\boldsymbol{M}) = (M_{ig(i)})_{i \leq N} \in \mathbb{R}^N.$$
Under these changes of variables, we define
\[
\boldsymbol{r}^{t} = \operatorname{blockdiag}(\boldsymbol{x}^{t}) \in \mathcal{V}_{q, N} \quad \text{ for $t \geq 0$}
\]
and $\tilde{f_{t}}$: $(\mathbb{R}^{q})^{N} \mapsto (\mathbb{R}^{q})^{N}$ by
\begin{equation}
\left(\tilde{f}_{t}(\boldsymbol{r}^{t})\right)_{ij} = \frac{1}{\sqrt{\tilde{\Delta}_{g(i)j}}}f_{t}^{g(i)}(x_{i}^{t}) \qquad \text{for $i,j \in [N] \times [q]$.}
\end{equation}
We encode the family of functions $\tilde f_t$ by $\tilde{\mathcal{F}}(\bm{\Delta})$.
\begin{lemma}
Let $\bm{x}^t$ be iterates from the AMP $(\boldsymbol{A}, \mathcal{F}, \boldsymbol{v}^{0},\bm{\Delta})$. Then the iterates $\bm{r}^t :=\operatorname{blockdiag}(\bm{x}^t)$ follow the generalized matrix AMP $(\bm{A}, \tilde{\mathcal{F}}(\bm{\Delta}), \bm{r}^0)$.
\end{lemma}
\begin{proof}
We will show that the projection of the iterates $\bm{r}^t$ from $(\bm{A}, \tilde{\mathcal{F}}(\bm{\Delta}), \bm{r}^0)$ are the iterates from $(\boldsymbol{A}, \mathcal{F}, \boldsymbol{v}^{0},\bm{\Delta})$. It is easy to check that
\begin{equation}
\frac{\boldsymbol{A}}{\sqrt{N} \sqrt{\boldsymbol{\Delta}}} f_{t}(\boldsymbol{x}^{t}) = \operatorname{blockproj}\left(\frac{\boldsymbol{A}}{\sqrt{N} }\tilde{f}_{t}(\boldsymbol{r}^{t})\right) .
\end{equation}
Next, notice that Jacobian is given by
$$\partial \tilde{f}_{t}^{i}(\bm{r}_i^t) =
\left[\begin{array}{ccccc}
0 & \cdots & \frac{1}{\sqrt{\tilde{\Delta}_{g(i)1}}}(f^{g(i)}_{t})^{\prime} (x_{i}^{t}) & \cdots & 0 \\
\vdots & \ddots & \vdots & \ddots &\vdots \\
0 & \cdots & \frac{1}{\sqrt{\tilde{\Delta}_{g(i)q}}}(f^{g(i)}_{t})^{\prime} (x_{i}^{t}) &\cdots & 0
\end{array}\right],$$
which is a matrix where only the column number $g(i)$ has non-zero elements. Applying \eqref{eqn: Onsager matrix}, we thus for $a, b \in [q] \times [q]$
\begin{equation} \label{eqn: Onsager term change of variables}
\left(\boldsymbol{B}_{t}\right)_{ab} = \frac{1}{N\sqrt{\tilde{\boldsymbol{\Delta}}_{ab}}} \sum_{i: g(i) = a}(f_{t}^{a})^{\prime}(x_{i}^{t}) .
\end{equation}
It follows that
\[
\mathrm{\boldsymbol{b}}_{t} \odot f_{t - 1}(\boldsymbol{x}^{t - 1}) = \operatorname{blockproj}\left(\tilde{f}_{t - 1}(\boldsymbol{r}^{t - 1})\boldsymbol{B}_{t}^{T}\right).
\]
\end{proof}
As a consequence, the inhomogeneous state evolution equations in Theorem~\ref{theorem: AMP inhomogeneous} follow immediately from the state evolution equations of $(\bm{A}, \tilde{\mathcal{F}}(\bm{\Delta}), \bm{r}^0)$ discussed in \cite[Section~2.1]{8205391}.
This follows from the observation that given the law of $\bm{r}^t$ in the high dimensional limit, the law of $\bm{x}^t = \operatorname{blockproj}( \bm{r}^t )$ is straightforward to compute. We define
\begin{equation}\label{eq:sigma_centered}
(\sigma_b^{t + 1})^2 := \sum_{a=1}^{q} \frac{c_{a}}{\tilde{\boldsymbol{\Delta}}_{ab}}\mathbb{E}\left[ (f^b_{t}(Z_{b}^{t}))^{2}\right] .
\end{equation}
We will show that the distribution of the iterate $\bm{x}_i^t$ is asymptotically normal with mean $0$ and variance $(\sigma_{g(i)}^t)^2$.
\begin{lemma}[Behavior of the AMP iterates in the inhomogeneous setting with no spike] \label{lem:state_evoluton_centered}
Suppose that Assumption~\ref{ass:limit} holds, and that $\mathbb{P}_0$ has a bounded second moment. Let $\phi: \mathbb{R}^2 \to \mathbb{R}$ be a $L$-pseudo-Lipschitz test functions satisfying \eqref{eq:pseudoLip}. For any $a \in [q]$, then the following limit holds almost surely
\[
\lim_{N \to \infty} \frac{1}{|C_a^N|} \sum_{i \in C_a^N} \phi( x_i^t, x_i^\star ) = \mathbb{E}_{x_{0}^{\star}, Z} \phi( \sigma_{a}^{t}Z, x_0^\star )
\]
where $Z$ is an independent standard Gaussian.
\end{lemma}
\begin{proof}
In matrix-AMP \cite[Th.~1]{8205391}, the marginals of the iterates $\bm{r^t}$ from $(\bm{A}, \tilde{\mathcal{F}}(\bm{\Delta}), \bm{r}^0)$ are approximately Gaussian and encoded by the positive definite matrices
\begin{eqnarray}
\widehat{\boldsymbol{\Sigma}}_{a}^{t}=\mathbb{E}\left[\tilde{f}_{t}^{i}\left(\boldsymbol{Z}^{t}\right) \tilde{f}_{t}^{i}\left(\boldsymbol{Z}^{t}\right)^{\top}\right] \qquad \text{for all $i \in C_a^N$},\\
{\rm with}\,\,\boldsymbol{Z}^{t} \sim N(0, \boldsymbol{\Sigma}^{t})\,\, \rm{and}\,\,
\label{eqn: linear_comb}
\boldsymbol{\Sigma}^{t + 1}=\sum_{a=1}^{q} c_{a} \widehat{\boldsymbol{\Sigma}}_{a}^{t}.
\end{eqnarray}
We now show that $a \in [q]$ and $i \in C_a^N$, $\widehat{\boldsymbol{\Sigma}}_{a}^{t}$ depends only on $(\sigma_{a}^{t})^{2} = \boldsymbol{\Sigma}_{aa}^{t}$. Indeed, by the definition of $\tilde{f}_{t}^{i}$ we have
\begin{equation} \label {eqn: sigma_hat}
\widehat{\boldsymbol{\Sigma}}_{a}^{t} (k, l) = \mathbb{E}\left[(\tilde{f}_{t}^{i}\left(\boldsymbol{Z}^{t}\right))_{k} (\tilde{f}_{t}^{i}\left(\boldsymbol{Z}^{t}\right))_{l}\right] = \mathbb{E}\left[ \frac{1}{\sqrt{\tilde{\boldsymbol{\Delta}}_{ak}}} \frac{1}{\sqrt{\tilde{\boldsymbol{\Delta}}_{al}}} (f_{t}^{a}(Z_{a}^{t}))^{2}\right],
\end{equation}
where $Z_{a}^{t} \sim N(0, (\sigma_{a}^{t})^{2})$ is the $a$th component of the Gaussian vector $\boldsymbol{Z}_{t}$. The key observation here is that by construction our function $\tilde{f}_{t}^{i}, \mathbb{R}^{q} \mapsto \mathbb{R}^{q}$ depends only on the ith component $Z_{i}^{t}$ of the Gaussian vector $\boldsymbol{Z}^{t}$.
To characterize the limiting distribution of $\boldsymbol{x}^{t} = \operatorname{blockproj}(\boldsymbol{r}^{t})$, we only need to keep track of the variances $(\sigma_{j}^{t})^{2}, j \in [N]$. Using \eqref{eqn: sigma_hat}, for a given $a \in [q]$ and $i \in C_{a}^{N}$ we write
\begin{equation}
\widehat{\boldsymbol{\Sigma}}_{a}^{t} (g(j), g(j)) = \mathbb{E}\left[(\tilde{f}_{t}^{i}\left(\boldsymbol{Z}^{t}\right))_{g(j)} (\tilde{f}_{t}^{i}\left(\boldsymbol{Z}^{t}\right))_{g(j)}\right] = \frac{1}{\tilde{\boldsymbol{\Delta}}_{ag(j)}}\mathbb{E}\left[ (f_{t}^{i}(Z_{i}^{t}))^{2}\right].
\end{equation}
Finally, with \eqref{eqn: linear_comb} we get that for any $b \in [q]$ and any $j \in C_b^N$,
using $Z_{a}^{t} \sim N(0, (\sigma_{a}^{t})^{2})$,
\begin{equation} \label{eqn: state_evolution}
(\sigma_{j}^{t + 1})^{2} = (\sigma_b^{t + 1})^2 = \boldsymbol{\Sigma}_{bb}^{t + 1} = \sum_{a=1}^{q} c_{a} \widehat{\boldsymbol{\Sigma}}_{a}^{t}(g(j), g(j)) = \sum_{a=1}^{q} \frac{c_{a}}{\tilde{\boldsymbol{\Delta}}_{ag(j)}}\mathbb{E}\left[ (f_{t}^{a}(Z_{a}^{t}))^{2}\right]
\end{equation}
\end{proof}
\paragraph{\textbf{The inhomogeneous spiked Wigner model in the light of the AMP approach}}\label{sec:spiked}
We now generalize the state evolution equations from Lemma~\ref{lem:state_evoluton_centered} to spiked matrices with an inhomogenous noise profile as was stated in Theorem~\ref{theorem: AMP inhomogeneous}. This reduction via a change of variables is standard, see for example \cite[Lemma 4.4]{DBLP:journals/corr/DeshpandeAM15}. Remember that in the inhomogeneous version of the spiked Wigner model we observe the signal $\boldsymbol{x}^{\star}$ through an inhomogeneous channel:
\begin{equation} \label{eq:spikedWigner}
\boldsymbol{Y} = \sqrt{\frac{1}{N}}\boldsymbol{x}^{\star}(\boldsymbol{x}^{\star})^{T} + \boldsymbol{A} \odot \sqrt{\boldsymbol{\Delta}}.
\end{equation}
Our AMP algorithm is defined with the following recursion:
\begin{equation}
\boldsymbol{x}^{t+1}= \bigg( \frac{1}{\sqrt{N}\boldsymbol{\Delta}} \odot \bm{Y} \bigg) f_{t}\left(\boldsymbol{x}^{t}\right)-\boldsymbol{\mathrm{b}}_{t} \odot f_{t-1}\left(\boldsymbol{x}^{t-1}\right)
\end{equation}
where $\boldsymbol{\mathrm{b}}_{t} = \frac{1}{\boldsymbol{\Delta}} f_{t}^{'} (\boldsymbol{x}^{t})$ and $f$ is encoded by the family of functions in Definition~\ref{def:inhomoAMP}. The main difference in contrast to the iteration \eqref{eqn: version} is that our data matrix $\bm{Y}$ is no longer a centered matrix, while $\frac{1}{\sqrt{\bm{\Delta}}} \odot \bm{A}$ is. We would like to reduce \eqref{eqn: AMP spike} to an iteration of the form \eqref{eqn: version} with respect to a different parameter $\bm{s}^t$ which is uniquely determined by $\bm{x}^t$
\[
\boldsymbol{s}^{t+1}=\bigg(\frac{1}{\sqrt{N} \sqrt{\boldsymbol{\Delta}}} \odot \boldsymbol{A} \bigg) f_{t}\left(\boldsymbol{s}^{t}\right)-\boldsymbol{\mathrm{b}}_{t} \odot f_{t-1}\left(\boldsymbol{s}^{t-1}\right).
\]
Doing so will allow us to recover the limiting laws of the iterates from Lemma~\ref{lem:state_evoluton_centered}. This is done via a standard change of variables to recenter $\bm{Y}$. We will sketch the argument in this section, and defer the full proof of Theorem~\ref{theorem: AMP inhomogeneous} to the Appendix~\ref{app:theostate}.
To simplify notation, we will often denote $f_{t}(\boldsymbol{x}^{t}) := \boldsymbol{\hat{x}}^{t}$.
We proceed following the approach of \cite[Lemma~4.4]{DBLP:journals/corr/DeshpandeAM15}. We will rewrite \eqref{eqn: AMP spike} using the definition of Y to get
\begin{equation}\label{eq:xiterates1}
\begin{aligned}
\boldsymbol{x}^{t+1} &= \bigg( \frac{1}{\sqrt{N}\boldsymbol{\Delta}} \odot \boldsymbol{Y} \bigg) f_{t}\left(\boldsymbol{x}^{t}\right)-\boldsymbol{\mathrm{b}}_{t} \odot f_{t-1}\left(\boldsymbol{x}^{t-1}\right) \\
&= \bigg( \frac{1}{\sqrt{N}\boldsymbol{\Delta}} \odot \boldsymbol{Y} \bigg) \boldsymbol{\hat{x}}^{t}-\boldsymbol{\mathrm{b}}_{t} \odot \boldsymbol{\hat{x}}^{t-1}\\
&=\bigg( \frac{1}{N \boldsymbol{\Delta}} \odot \boldsymbol{x^{\star}}\boldsymbol{(x^{\star})^{T}} \bigg) \boldsymbol{\hat{x}}^{t} + \bigg( \frac{1}{\sqrt{N} \sqrt{\boldsymbol{\Delta}}} \odot \bm{A} \bigg) \boldsymbol{\hat{x}}^{t}-\boldsymbol{\mathrm{b}}_{t} \odot \boldsymbol{\hat{x}}^{t-1}.
\end{aligned}
\end{equation}
If indices are independent, then by the strong law of large numbers one would expect that
\begin{equation}\label{eqn: SLLN}
\left(\bigg( \frac{1}{N \boldsymbol{\Delta}} \odot \boldsymbol{x^{\star}}\boldsymbol{(x^{\star})^{T}} \bigg) \boldsymbol{\hat{x}}^{t} \right)_{j} = x_{j}^{\star} \sum_{a \in [q]} \sum_{i \in C_{a}^{N}} \frac{1}{N} \frac{x_{i}^{\star}\hat{x}_{i}^{t}}{\boldsymbol{\Delta}_{ji}} \rightarrow x_{j}^{\star} \sum_{a \in [q]} \frac{c_{a}}{\boldsymbol{\Delta}_{ji_{a}}} \mathbb{E}[x_{0}^{\star}\hat{x}_{i_{a}}^{t}],
\end{equation}
where $i_{a}$ is some index belonging to the group $C_{a}^{N}$ and $x_{0}^{\star}$ is a random variable distributed according to the prior distribution $\mathbb{P}_{0}$. For $b \in [q]$ and $i \in C_b^N$, we define the block overlap $\mu_{b}^{t}$ using the recursion
\begin{equation} \label{eqn: mu}
\mu^{t + 1}_{i} = \mu^{t + 1}_{b} = \sum_{a \in [q]} \frac{c_{a}}{\tilde{\boldsymbol{\Delta}}_{ab}} \mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star} f_{t}^{a}\left(\mu^{t}_{a}x_{0}^{\star} + \sigma^{t}_{a}Z\right)],
\end{equation}
where $Z$ is a standard Gaussian random variable independent from all others sources of randomness. Notice that \eqref{eqn: mu} is precisely the asymptotic behavior of the summation appearing in \eqref{eqn: SLLN} by Lemma~\ref{lem:state_evoluton_centered}, which is how we control \eqref{eqn: SLLN} in the rigorous proof.
We now make a change of variables and track the iterates
\begin{equation}\label{eq:s_iterates}
\boldsymbol{s}^{0} = \boldsymbol{x}^{0} - \boldsymbol{\mu}^{0} \odot \boldsymbol{x}^{\star} \qquad \boldsymbol{s}^{t} = \boldsymbol{x}^{t} - \boldsymbol{\mu}^{t} \odot \boldsymbol{x}^{\star}, \quad t \geq 1
\end{equation}
where $\boldsymbol{\mu}^{0}$ is the vector of block overlaps of the initial condition $\boldsymbol{x}^{0}$ with the truth. We reduced the \eqref{eqn: AMP spike} iteration to the following iteration in which we easily recognize a version of \eqref{eqn: version}:
\begin{equation} \label{eqn: AMP change spike}
\boldsymbol{s}^{t+1}= \left( \frac{1}{\sqrt{N} \sqrt{\boldsymbol{\Delta}}} \odot \boldsymbol{A}\right) f_{t} \left(\boldsymbol{s}^{t} + \boldsymbol{\mu^{t}} \odot \boldsymbol{x}^{\star}\right)-\bm{\mathrm{b}}_{t} \odot f_{t-1} \left(\boldsymbol{s}^{t-1} + \boldsymbol{\mu^{t - 1}} \odot \boldsymbol{x}^{\star}\right)
\end{equation}
with the initial condition $\boldsymbol{s}^{0} = \boldsymbol{x}^{0} - \boldsymbol{\mu^{0}} \odot \boldsymbol{x}^{\star}$ and the Onsager term taken from \eqref{eqn: Onsager vector} is given by
\begin{equation}
\bm{\mathrm{b}}_{t} = \frac{1}{\boldsymbol{\Delta}} f_{t}^{'} (\boldsymbol{s}^{t} + \boldsymbol{\mu}^{t} \odot \boldsymbol{x}^{\star}).
\end{equation}
Using Lemma~\ref{lem:state_evoluton_centered}, we can recover the asymptotic behavior of the iterates $\bm{x}^t$ given in \eqref{eqn: AMP spike} by computing the iterates $\boldsymbol{s}^{t} + \boldsymbol{\mu}_{t} \odot \boldsymbol{x}^{\star}$ where $\boldsymbol{s}^{t}$ follows \eqref{eqn: AMP change spike} and $\bm{\mu}_t$ satisfies \eqref{eqn: mu}. From this reduction we obtain the following state evolution equations describing the behaviour of \eqref{eqn: AMP spike}:
\begin{enumerate}
\item $x_{j}^{t} \approxeq \mu^{t}_{g(j)}x_{0}^{\star} + \sigma_{g(j)}^{t}Z$ for $j \in [N]$, where $Z \sim \mathcal{N}(0, 1)$
\item $\mu^{t + 1}_{b} = \sum_{a \in [q]} \frac{c_{a}}{\tilde{\Delta}_{ab}} \mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star} f_{t}^{a}\left(\mu^{t}_{a}x_{0}^{\star} + \sigma^{t}_{a}Z\right)]$ with $x_{0}^{\star} \sim \mathbb{P}_{0}, Z \sim \mathcal{N} (0, 1)$
\item $(\sigma_{b}^{t + 1})^{2} = \sum_{a=1}^{q} \frac{c_{a}} {\tilde{\Delta}_{ab}}\mathbb{E}_{x_{0}^{\star}, Z}\left[ (f_{t}^{a}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z))^{2}\right]$ with $x_{0}^{\star} \sim \mathbb{P}_{0}$, $Z \sim \mathcal{N} (0, 1).$
\end{enumerate}
This (informally) characterizes the limiting distribution of the state evolution of the iterates from the inhomogeneous AMP stated in Theorem~\ref{theorem: AMP inhomogeneous}. The main technical difficulty is the equality in \eqref{eqn: SLLN} and \eqref{eqn: mu} already uses the asymptotic distribution of the overlaps at finite $N$. This technical difficulty is dealt with in the full proof of Theorem~\ref{theorem: AMP inhomogeneous} in Appendix~\ref{app:theostate}.
\paragraph{\textbf{Fixed-point equation of state evolution in the Bayes-optimal setting}}\label{sec:Bayes}
Suppose that we know the prior distribution $\mathbb{P}_{0}$ of $x_{0}^{\star}$. The Bayes-optimal choice for the denoising functions $f_{t}^{j}, j \in [N]$ is simply the expectation of $x_{0}^{\star}$ with respect to the posterior distribution,
\begin{equation} \label{eqn: Bayes-optimal functions}
f_{t}^{j} (r) = f_{t}^{g(j)} (r) = \mathbb{E}_{posterior}[x_{0}^{\star}| \mu^{t}_{g(j)}x_{0}^{\star} + \sigma_{g(j)}^{t}Z = r].
\end{equation}
Under this Bayes-optimal setting, we can simplify the equations obtained in the previous section and see that AMP estimator is indeed an optimal one by studying its fixed point.
\begin{proof}[Theorem~\ref{theorem: AMP fixed point}]
For this choice of $f_{t}^{j}$ the Nishimori identity (see for example \cite[Proposition~16]{lelargemiolanematrixestimation}) states that for $a \in [q]$ and $j \in C_a^N$,
\begin{equation}
\tilde{\mu}_{a}^{t}:= \mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star}f_{t}^{a}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z)] = \mathbb{E}\left[ (f_{t}^{a}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z))^{2}\right].
\end{equation}
In this setting, the state evolution equations from Theorem~\ref{theorem: AMP inhomogeneous} initialized at $\bm{\mu}^0 = \bm{\sigma}^0 = 0$ reduce to
\begin{equation} \label{eqn: State evolution Bayes-optimal}
\begin{cases}
\tilde{\mu}_{a}^{t} = \mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star}f_{t}^{a}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z)] \\
\mu^{t + 1}_{b} = \sum_{a \in [q]} \frac{c_{a}}{\tilde{\Delta}_{ab}}\ \tilde{\mu}_{a}^{t}, b \in [q] \\
(\sigma_{b}^{t + 1})^{2} = \sum_{a \in [q]} \frac{c_{a}}{\tilde{\Delta}_{ab}}\ \tilde{\mu}_{a}^{t}, b \in [q].
\end{cases}
\end{equation}
Remarkably with the Bayes-optimal choice of the denoising functions we have that for $t \geq 1$ for each block $b \in [q]$, $\mu^{t + 1}_{b} = (\sigma_{b}^{t + 1})^{2}$. Therefore a necessary and sufficient condition for an estimator to be a fixed point of the state evolution is to simply have its overlaps $\mu^{t}_{b}$ unchanged by an iteration of the state evolution. This translates into the following equation for the overlaps $\mu_{b}, b \in [q]$
\begin{equation} \label{eqn: fixed point}
\mu_{b} = \sum_{a \in[q]} \frac{c_{a}}{\tilde{\Delta}_{ab}}\mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star}\mathbb{E}_{posterior}[x_{0}^{\star}|\mu_{a}x_{0}^{\star} + \sqrt{\mu_{a}}Z]].
\end{equation}
The result of Theorem~\ref{theorem: AMP fixed point} now follows immediately.
\end{proof}
\section{A spectral method adapted to the inhomogeneous spiked Wigner model}
\label{sec:spec}
In this section, we will describe how one can use the convergence of the inhomogeneous AMP with a simple choice of denoiser to recover the BBP transition of spiked Wigner matrices.
\paragraph{\textbf{From AMP to a spectral method}}\label{sec:linear}
Remarkably, AMP and the state evolution machinery associated with it can help us design a simple spectral algorithm that matches the information-theoretic phase transition \cite[Remark~2.16]{AJFL_inhomo}. Recall that Theorem~\ref{theorem: AMP inhomogeneous} does not require the denoising functions $f_{t}$ to be Bayes-optimal, but can be applied to any Lipschitz family of functions. In this section, we analyze the state evolution for the family of identity functions, $f_{t}(x) = x$. By Remark~\ref{rem:scaling}, we can assume that the entries of $\bm{x}^\star$ have unit variance. With this choice of denoising functions the AMP iteration will simply become:
\begin{align} \label{eqn: AMP identity denosing}
\bm{x}^{t + 1} = \left( \frac{1}{\sqrt{N}\bm{\Delta}} \odot \bm{Y} \right) \bm{x}^{t} - \bm{\mathrm{b}}_{t} \odot \bm{x}^{t - 1} \quad \text{ where } \quad
\bm{\mathrm{b}}_{t} = \frac{1}{\bm{\Delta}} f_{t}^{'} = \frac{1}{\bm{\Delta}} \begin{bmatrix}
1 \\
\vdots \\
1
\end{bmatrix}.
\end{align}
If we denote $\bm{B}_{t} = \operatorname{diag}( \bm{\mathrm{b}}_{t})$, it is easy to see that the fixed point of this iteration yields
\begin{equation}
\bm{x} = \left( \frac{1}{\sqrt{N}\bm{\Delta}} \odot \bm{Y} \right) \bm{x} - \bm{B}_{t} \bm{x}
\end{equation}
so any $\bm{x}$ fixed by the AMP iteration \eqref{eqn: AMP identity denosing} must be an eigenvector of the matrix
\begin{equation}\label{eq:transformedY}
\bm{\tilde{Y}} = \left( \frac{1}{\sqrt{N}\bm{\Delta}} \odot \bm{Y} \right) - \bm{B}_{t} = \left(\frac{1}{\sqrt{N}\bm{\Delta}} \odot \bm{Y} \right) - \operatorname{diag}\left(\frac{1}{\bm{\Delta}} \begin{bmatrix}
1 \\
\vdots \\
1
\end{bmatrix} \right) .
\end{equation}
A simple spectral method consists in taking the principal eigenvector (associated to the largest eigenvalue) of the matrix $\frac{\bm{Y}}{\sqrt{N}\bm{\Delta}} - \bm{B}_{t}$, and this linear AMP is a quick way to find such an eigenvector.
\paragraph{\textbf{Analysis of the spectral method using state evolution}} It is expected that the spectral algorithm described above behaves as the AMP iteration with identity denoising functions around its fixed point. Therefore we can analyze this spectral algorithm using state evolution machinery for the AMP iteration. In the case of identity functions we have $f_{t}^{a}(x) = x$ for all $a \in [q]$, so
\begin{align}
\mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star}f_{t}^{a}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z)] &= \mathbb{E}_{x_{0}^{\star}, Z}[x_{0}^{\star}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z)] = \mu^{t}_{a} \\
\mathbb{E}_{x_{0}^{\star}, Z}[(f_{t}^{a}(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z))^{2}] &=
\mathbb{E}_{x_{0}^{\star}, Z}[(\mu^{t}_{a}x_{0}^{\star} + \sigma_{a}^{t}Z)^{2}] = \ (\mu^{t}_{a})^{2} + (\sigma^{t}_{a})^{2} \label{eq:variancesigma}
\end{align}
which transforms state evolution equations \eqref{eq:state} into the following simple form:
\begin{align}
\mu^{t + 1}_{b} = \sum_{a \in [q]} \frac{c_{a}}{\Delta_{ba}} \mu^{t}_{a} \quad \text{and} \quad
(\sigma^{t + 1}_{b})^{2} = \sum_{a \in [q]} \frac{c_{a}}{\Delta_{ba}} (\mu^{t}_{a})^{2} + (\sigma^{t}_{a})^{2}).
\end{align}
Rewriting the overlap state evolution in a matrix form we get for $\bm{c} = (c_a)_{a \in [q]}$ that
\begin{equation}
\operatorname{diag}(\sqrt{\bm{c}}) \bm{\mu}^{t + 1} = \operatorname{diag}(\sqrt{\bm{c}}) \frac{1}{\bm{\Delta}} \operatorname{diag}(\sqrt{\bm{c}}) \left(\operatorname{diag}(\sqrt{\bm{c}})\bm{\mu}^{t}\right).
\end{equation}
If $\lambda(\bm{\Delta}) = \left\|\operatorname{diag}(\sqrt{\bm{c}}) \frac{1}{\bm{\Delta}} \operatorname{diag}(\sqrt{\bm{c}})\right\|_{op} < 1,$ then this is a contraction with respect to the Euclidean norm, so there is a unique fixed point at $0$ when $\lambda(\bm{\Delta}) < 1$. There is instability if $\lambda(\bm{\Delta}) > 1$, so we conjecture that this transition corresponds to the BBP transition for \eqref{eq:transformedY} as stated in Conjecture~\ref{conjecture:BBP}.
\begin{remark}\label{rem:scaling}
In general, we can let $\gamma = \mathbb{E}_{x_0^\star} [(x_0^\star)^2]$ and consider the normalized matrix
\[
\bm{\bar{Y}} = \frac{\bm{Y}}{\gamma} = \sqrt{\frac{1}{N}} \frac{\boldsymbol{x}^{\star}(\boldsymbol{x}^{\star})^{T}}{\gamma} + \boldsymbol{A} \odot \frac{\sqrt{\boldsymbol{\Delta}}}{\gamma} = \sqrt{\frac{1}{N}}\boldsymbol{\bar x}^{\star}(\boldsymbol{\bar x}^{\star})^{T} + \boldsymbol{A} \odot \sqrt{\boldsymbol{\bar \Delta}}
\]
for $\bm{\bar{x}} = \frac{\bm{x}}{\sqrt{\gamma}}$ and $\boldsymbol{\bar \Delta} = \frac{\bm{\Delta}}{\gamma^2}$. Notice that the entries of $\bm{\bar{x}}$ now have unit variance. Under this setting, the transition of the transformation in \eqref{eq:transformedY} applied to $\bar{Y}$, which appears in \eqref{eq:transformedYgenvariance}, has transition at
\[
\lambda(\bm{\bar \Delta}) = \left\|\operatorname{diag}(\sqrt{\bm{c}}) \frac{1}{\bm{\bar \Delta}} \operatorname{diag}(\sqrt{\bm{c}})\right\|_{op} = \mathbb{E}_{x_0^\star} [(x_0^\star)^2]^2 \left\|\operatorname{diag}(\sqrt{\bm{c}}) \frac{1}{\bm{ \Delta}} \operatorname{diag}(\sqrt{\bm{c}})\right\|_{op} < 1
\]
which is the generalized SNR defined in \eqref{eq:SNR}.
\end{remark}
\section{Acknowledgments}
We thank Alice Guionnet \& Lenka Zdeborov\'a for valuable discussions. We acknowledge funding from the ERC Project LDRAM: ERC-2019-ADG Project 884584, and by the Swiss National Science Foundation grant SNFS OperaGOST, $200021\_200390$.
\newpage
|
1,314,259,996,681 | arxiv | \section{Introduction}\label{s:intro}
Type Ia supernovae (SNe~Ia) are the energetic end of C/O white dwarfs
in binary systems that reach temperatures and densities high enough to
cause a runaway chain of nuclear reactions \citep[for a review,
see][]{Hillebrandt00}. Despite having collected observations of
thousands of SNe~Ia, we still do not know what progenitor systems
produce SNe~Ia and do not fully understand how the star explodes.
Measurements of SNe~Ia led to the discovery of the accelerating
universe \citep{Riess98:Lambda, Perlmutter99}, and observing SNe~Ia is
still one of the best ways to constrain cosmological parameters. It
behooves us to understand the physics behind SN~Ia explosions, and
with additional knowledge of the progenitors and explosions, we hope
to further improve the utility of SNe~Ia for measuring cosmic
distances.
SN~Ia UV spectra are dominated by a forest of overlapping lines from
iron-group elements (IGEs). UV photons are repeatedly absorbed and
re-emitted in those lines and gradually scattered redward where lower
opacities allow them to escape. The UV is crucial to the formation of
the optical spectral-energy distribution (SED) of SNe~Ia
\citep{Sauer08}, and extremely sensitive to both the progenitor
composition and explosion mechanism \citep{Hoflich98, Lentz01}.
Observations of the UV directly probe the composition of the outermost
layers of ejecta, which are transparent at optical wavelengths soon
after explosion.
After light-curve shape correction \citep[e.g.,][]{Phillips93}, the
luminosity of a SN~Ia still depends significantly on host-galaxy
environment \citep{Kelly10, Lampeitl10:host, Sullivan10}. This may
indicate that there are subtle environmental effects imprinted on the
progenitor stars that directly affect our luminosity calibration.
Different progenitor metallicity may affect the outcome of the
explosion \citep{Timmes03}, the relation between luminosity and
light-curve shape \citep{Howell07, Mazzali01, Mazzali06}, and the
observed UV spectrum especially at $\lambda < 2600$~\AA\
\citep{Hoflich98, Lentz00}.
At optical wavelengths SNe~Ia have remarkably uniform luminosity ($\sigma
\approx 0.16$~mag; e.g., \citealt{Hicken09:lc}), after correcting for
light-curve shape and color. This relationship extends to the $U$
band, but with larger scatter \citep{Jha06:lc}. The scatter can be
further reduced to 0.11~mag after making a correction based on a
measurement of the ejecta velocity \citep{Foley11:vel, Foley11:vgrad,
Foley12:vel}. The intrinsic $B-V$ color of SNe~Ia correlates strongly
with ejecta velocity, with redder SNe having higher velocity. This is
explained as additional line blanketing in the $B$ band of the
high-velocity SNe, and this trend should extend to the UV
\citep{Foley11:vel}.
High-quality UV spectra for low-$z$ SNe~Ia are of particular importance
to the calibration of high-$z$ SNe~Ia. Since the observed SN light
originates as rest-frame UV, most $z > 1$ SNe~Ia only have rest-frame
UV light curves, and low-$z$ UV spectra are critical for understanding
these data. Our current lack of UV spectra limits K-corrections and
even SN classification at $z > 1$
\citep{Riess07}.
SNe~Ia bright enough for the \textit{International Ultraviolet
Explorer} were rare \citep{Foley08:uv}, and only one SN~Ia has a
published high signal-to-noise ratio (S/N) \textit{Hubble Space
Telescope} (\textit{HST}) spectrum near maximum light covering
wavelengths $\lesssim 2900$~\AA\ -- SN~1992A \citep[hereafter,
K93]{Kirshner93}. \textit{Swift} has obtained spectra of several
SNe~Ia, but except for a few SNe \citep[e.g.,
SN~2009ig;][]{Foley12:09ig}, the spectra are generally of such low S/N
that detailed spectral analysis is not feasible \citep{Bufano09}. A
Cycle 13 \textit{HST} program (GO-10182; PI Filippenko) attempted to
observe several SNe~Ia in the UV; unfortunately, the STIS spectrograph
failed before any data could be obtained. The program was executed
using the ACS prism, which does not provide sufficient resolution to
distinguish spectral features \citep{Wang12}. In Cycle 17, 30 SN~Ia
were observed at a single epoch by \textit{HST} (GO-11721; PI Ellis);
however, these spectra did not probe below 2900~\AA\ \citep{Cooke11}.
It has been 20 years since the last high-S/N \textit{true}-UV
maximum-light SN~Ia spectrum has been observed and published.
SN~2011iv was discovered at an unfiltered magnitude of 12.8 on 2011
December 2.57 (UT dates are used throughout) by \citet{Drescher11}.
There was no object detected on 2011 October 29.57 to a limit of
18.8~mag. It was discovered in NGC~1404, an elliptical galaxy in the
Fornax cluster at $cz = 1947$~km~s$^{-1}$ \ \citep{Graham98} that also hosted
SN~2007on \citep[e.g.,][]{Stritzinger11} and is at $D = 20.4$~Mpc
($\mu = 31.53 \pm 0.07$~mag\footnote{This measurement is consistent
with the \citet{Blakeslee10} weighted average distance modulus for the
Fornax cluster of $\mu = 31.54 \pm 0.02$~mag, but is larger than other
distance estimates that have higher uncertainty.} from a surface
brightness fluctuation measurement; \citealt{Blakeslee10}).
\citet{Chen11} and \citet{Stritzinger11:11iv} obtained optical spectra
of SN~2011iv on 2011 December 3.7 and 4.1, respectively, and
determined that it was a young SN~Ia. We triggered multiple programs
to study the photometric and spectroscopic evolution of the SN, its
circumstellar environment, its polarization, its energetics, and other
aspects. In particular, we triggered our \textit{HST}
target-of-opportunity program to obtain UV spectra of SNe~Ia
(GO-12592; PI Foley; \citealt{Foley11:11iv}). The first of seven UV
spectra was obtained on 2011 December 11.08 and covered wavelengths of
1615--10,230~\AA. Nearly simultaneously (2011 December 11.24), we
obtained a near-infrared (NIR) spectrum with the Folded Port Infrared
Echellette (FIRE) spectrograph \citep{Simcoe08, Simcoe10} on the 6.5~m
Magellan Baade Telescope. In this \textit{Letter}, we focus on this
first UV-optical-NIR (UVOIR) spectrum spanning 0.16--2.5~$\mu$m.
\section{Observations}\label{s:obs}
Optical photometry of SN~2011iv was collected with the Panchromatic
Robotic Optical Monitoring and Polarimetry Telescopes (PROMPT) 3 and 5
\citep{Reichart05}. Basic data reduction (bias and flat-field
correction) was performed using standard routines in IRAF. Local
standard stars were measured by \citet{Stritzinger11}. The complex
background around SN~2011iv was modeled with a low-order polynomial
surface. Once the galaxy contamination was removed, photometry was
performed using the point-spread-function fitting technique. We
present our $B$ and $V$ light curves in Figure~\ref{f:lc}.
\begin{figure}
\begin{center}
\epsscale{1.1}
\rotatebox{0}{
\plotone{sn2011iv_lc_landolt.ps}}
\caption{$B$ (blue points) and $V$ (green points) light curves of
SN~2011iv. The dashed curves correspond to the $\Delta m_{15} =
1.69$~mag \citet{Hamuy96:temp} light-curve templates.}\label{f:lc}
\end{center}
\end{figure}
SN~2011iv was observed by \textit{HST} using the STIS spectrograph on
2011 December 11.08, corresponding to $t = 0.6$~days relative to $B$
maximum (see Section~\ref{s:results}). The observations were obtained
over two orbits with three different gratings, all with the $52\hbox{$^{\prime\prime}$}
\times 0.\arcsec2$ slit. Exposures of 2200 and 1350~s utilized the
near-UV MAMA detector and the G230L grating. Two exposures of 100~s
were taken with both the CCD/G430L and CCD/G750L setups, respectively.
The three setups yield a combined wavelength range of
1615--10,230~\AA.
The data were reduced using the standard \textit{HST} Space Telescope
Science Data Analysis System (STSDAS) routines to bias subtract,
flat-field, extract, wavelength-calibrate, and flux-calibrate each SN
spectrum.
Almost immediately after obtaining the STIS spectrum (on 2011 December
11.24), we obtained a NIR spectrum (0.83--2.5~$\mu$m) of SN~2011iv
with FIRE. SN~2011iv was observed in the high-resolution echellette
mode with the 0.\arcsec6 slit, yielding a resolution of $R = 6000$ or
about 50~km~s$^{-1}$ . Four frames were taken on source with 240~s exposures
using ABBA nodding.
The data were reduced using a custom-developed IDL pipeline
(FIREHOSE), which evolved from the MASE pipeline used for optical
echelle reductions \citep{Bochanski09}. The NIR sky flux was modeled
using off-source pixels as described by \citet{Kelson03} and
subtracted from the science frame before extraction, which was
weighted by a simple boxcar profile. An A0V star was observed for
telluric corrections, following the procedures outlined by
\citet{Vacca03}. The \texttt{xtellcor} procedure, derived from the
Spextool package \citep{Cushing04}, performs both telluric correction
and relative flux calibration. The corrected echelle orders are then
combined into a single one-dimensional spectrum.
There is substantial wavelength overlap between the STIS and FIRE
spectra (8295--10,230~\AA). Using this overlap region, we scaled the
FIRE spectrum to the STIS spectrum and combined the two spectra. The
combined spectrum (1615--24,880~\AA) is presented in
Figures~\ref{f:spec} and \ref{f:model}; it has been corrected for
Milky Way reddening of $E(B-V) = 0.011$~mag
\citep{Schlegel98}.
\begin{figure*}
\begin{center}
\epsscale{1.1}
\rotatebox{0}{
\plotone{sn2011iv_spec.ps}}
\caption{UVOIR \textit{HST}/STIS and Magellan/FIRE maximum-light spectrum
of SN~2011iv.}\label{f:spec}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\epsscale{1.1}
\rotatebox{0}{
\plotone{sn2011iv_spec_log.ps}}
\caption{UVOIR maximum-light spectrum of SN~2011iv (black curve).
This spectrum is the same as that shown in Figure~\ref{f:spec}, but
with a different scale to highlight the UV and NIR portions. The
blue, red, and orange curves represent model spectra generated from a
zero-metallicity delayed-detonation N100 model, the W7 model, and the
solar-metallicity polluted W7$_{Z_{\sun}}$ model,
respectively.}\label{f:model}
\end{center}
\end{figure*}
\section{Results}\label{s:results}
Our spectrum of SN~2011iv is the first high-S/N, near-maximum, true-UV
SN~Ia spectrum since that of SN~1992A \citepalias{Kirshner93}; the
earliest high-S/N UV SN~Ia spectrum yet published; and the first
contemporaneous and continuous UVOIR spectrum of a SN~Ia. This unique
dataset provides the opportunity to examine the full SED, producing
strong constraints on SN~Ia models.
Fitting the $B$ and $V$ light curves (Figure~\ref{f:lc}) with a
third-order polynomial function, we estimate that SN~2011iv reached
maximum light in the $B$ band on 2,455,$906.0 \pm 0.3$ JD and peaked
at $B = 12.53 \pm 0.03$~mag and $V = 12.51 \pm 0.03$~mag. The
$B_{\max}-V_{\max}$ pseudo-color is consistent with negligible
reddening in the host galaxy. Assuming a Milky Way extinction of
$A_{V} = 0.038$~mag \citep{Schlegel98} and our preferred distance
modulus, SN~2011iv peaked at $M_{V} = -19.06 \pm 0.08$~mag. The
optical light curves of SN~2011iv indicate that it is a relatively
fast-declining SN with $\Delta m_{15} (B) = 1.69 \pm 0.05$~mag; the
light curves are well matched by the $\Delta m_{15} = 1.69$~mag
\citet{Hamuy96:temp} templates (Figure~\ref{f:lc}).
As seen in Figure~\ref{f:spec}, optical emission dominates the SEDs of
SNe~Ia at maximum light. The median flux over 100~\AA\ bins centered
at 2500, 3000, and 3500~\AA\ is 2.4, 18.1, and 50.0\% that of the peak
flux (in $f_{\lambda}$ units), respectively. Similarly, the median
flux over 500~\AA\ bins centered at 1, 1.5, and 2~$\mu$m is 3.6, 0.8,
and 0.4\% that of the peak flux (in $f_{\lambda}$ units),
respectively.
SN~2011iv has a relatively strong silicon ratio, $\mathcal{R}({\rm
Si~II})$ \citep{Nugent95}, of $0.50 \pm 0.05$. This measurement is
consistent with its decline rate. Optical spectra of SN~2011iv are
extremely similar to those of SNe~1992A \citepalias{Kirshner93} and
2004eo \citep{Pastorello07:04eo}. Both SNe~1992A and 2004eo also had
relatively fast decline rates ($\Delta m_{15} (B) = 1.48$ and
1.46~mag, respectively), but were still spectroscopically ``normal.''
Using the method of \citet{Blondin06}, we determine the velocity of
the \ion{Si}{2} $\lambda 6355$ feature to be $v_{\rm Si~II} = -10,630
\pm 130$~km~s$^{-1}$ . Using the relationships between \protect\hbox{$v_{\rm Si~II}$}\ and
maximum-light velocity, \protect\hbox{$v_{\rm Si~II}^{0}$}\ \citep{Foley11:vgrad}, we derive
$v_{\rm Si~II}^{0} = -10,700 \pm 300$~km~s$^{-1}$ ; therefore, \protect\hbox{$v_{\rm Si~II}^{0}$}\ of
SN~2011iv is $\sim\!\!$~ 800~km~s$^{-1}$ \ lower than the median value for the
\citet{Foley11:vgrad} sample (it has a lower velocity than 84\% of all
SNe~Ia in the sample).
\subsection{Comparison to Other Supernovae}
In Figure~\ref{f:uv}, we compare our SN~2011iv spectrum to the
near-maximum UV spectra of SNe~1992A \citepalias{Kirshner93} and
2009ig \citep{Foley12:09ig}, which have both been dereddened by their
Milky Way values of $E(B-V) = 0.018$ and 0.032~mag \citep{Schlegel98},
respectively. At $\lambda < 3000$~\AA, the features in the SN~1992A
and SN~2011iv spectra are remarkably similar, showing only minor
differences. There is a slight wavelength offset in the features at
2300--2700~\AA, with the features being more blueshifted in the
SN~1992A spectrum. Although SN~1992A was observed at an epoch
4.5~days later than SN~2011iv, SN~1992A had higher \protect\hbox{$v_{\rm Si~II}^{0}$}, and the
higher ejecta velocity is likely the cause for the small differences
in the UV. SN~2009ig is significantly different from the other two
SNe; its spectrum lacks the prominent features at 2380 and 2560~\AA\
seen in both SN~1992A and SN~2011iv. There is perhaps an indication
that the features are blueshifted even further than in SN~1992A, but
the S/N is too low to be conclusive. Similarly, the SN~2009ig
spectrum does not have a sufficiently high S/N at $\lambda <
2300$~\AA\ to say anything definitive about features in the far UV.
\begin{figure*}
\begin{center}
\epsscale{1.1}
\rotatebox{0}{
\plotone{sn2011iv_uv.ps}}
\caption{Near-maximum-light spectra of SNe~2011iv (black), 1992A (red;
\citetalias{Kirshner93}), and 2009ig (blue; \citealt{Foley12:09ig}).
The SN~2009ig spectrum is the combination of the $t = -2.1$ and 1.5~d
spectra (to increase the S/N). The spectra have been scaled to have
similar flux at 3250~\AA.}\label{f:uv}
\end{center}
\end{figure*}
Examining the three SNe in the near UV (2800--3800~\AA), there is even
more diversity in their spectra. The spectra of SNe~1992A and 2011iv
are still similar in this region, but SN~1992A has more pronounced
features, especially surrounding \ion{Fe}{2} $\lambda 3250$ that is
observed at $\sim\!\!$~ 3000~\AA. Again, SN~2009ig has a very different
spectrum from the other two SNe. Besides the clearly different Ca
H\&K line profile, the biggest difference is in the continuum shape;
SN~2009ig is relatively flat in the near-UV, while the other two SNe
are much steeper. SN~2009ig has a very large \protect\hbox{$v_{\rm Si~II}^{0}$}, and the
interpretation of \citet{Foley11:vel} is that the large velocity range
for absorbing lines, which correlates with \protect\hbox{$v_{\rm Si~II}^{0}$}, causes more line
blanketing in the near-UV. Additional line blanketing, either from
velocity or differences in the ejecta composition, may also explain
the relatively flat spectral slope in the near-UV. Similar to what
was seen with extremely low-resolution UV spectra \citep{Wang12}, this
sample of SNe~Ia appears to have a large diversity in near-UV
continua.
Over 2000--2500~\AA, SN~2011iv has a S/N of 28 per 1.5~\AA\ pixel,
while SN~1992A has a S/N of 2.6 per 2~\AA\ pixel. The SN~2011iv
spectrum has a S/N that is $\sim\!\!$~ 15 times that of the SN~1992A
spectrum. This exquisite spectrum shows additional features that are
presumably hidden in the noise of the SN~1992A spectrum.
Specifically, there is an absorption feature at $\sim\!\!$~ 2120~\AA\ in
the SN~2011iv spectrum that is not seen in the SN~1992A spectrum.
\subsection{Comparison to Models}
The majority of the SN luminosity is emitted at optical
wavelengths. Most of this flux is redistributed from shorter
wavelengths where IGEs provide significant line opacity due to
millions of atomic line transitions. Although optical spectra are not
particularly good at differentiating between models (R\"{o}pke et~al.,
submitted), all models must reproduce the main optical signatures of
SNe~Ia to be credible.
It has been proposed that UV spectra of SNe~Ia are a good tool to
constrain different explosion models and progenitor metallicities
\citep[e.g.,][]{Hoflich98, Lentz01}. Our unique spectrum provides the
possibility to test this assertion. For that purpose we compare two
different explosion models to our data. Specifically, we take the 1D
deflagration model W7 \citep{Nomoto84:w7} and the state-of-the-art 3D
delayed-detonation model N100 (R\"{o}pke et~al., submitted). Both
models yield $\sim\!\!$~ $0.6 M_{\sun}$ of $^{56}$Ni, which is typical for
normal SNe~Ia \citep{Stritzinger06}. For both models we obtained a
spectral time sequence with the Monte Carlo radiative transfer code
ARTIS \citep{Kromer09}.
To study the influence of progenitor metallicity, we created an
additional model W7$_{Z_{\sun}}$, in which we polluted the outer
layers of W7 (zones where the sum of the IGE abundances was lower than
solar IGE content, i.e., $v < -12$,500~km~s$^{-1}$ ) with solar abundances. A
comparison of maximum-light spectra of our three models and the UVOIR
spectrum of SN~2011iv is shown in Figure~\ref{f:model}. Both the N100
and W7$_{Z_{\sun}}$ models have similar SEDs to that of SN~2011iv from
the UV to the NIR. In contrast, the original W7 model has excess UV
flux.
The differences in the W7 spectra are mostly limited to the far-UV
($\lambda < 2800$~\AA), indicating that only this wavelength regime is
strongly sensitive to progenitor metallicity (see also
\citealt{Lentz01}). If we want to use SNe~Ia as distance indicators
at $z \gtrsim 1.5$, for which the observed optical data correspond to
the rest-frame far-UV, light-curve fitting algorithms have to account
for a range of different progenitor metallicities.
In contrast, our two metallicity points for W7 indicate that in the
near-UV ($2800 < \lambda < 3800$~\AA), the influence of different
progenitor metallicities is weaker. This region might be better
suited to differentiate between explosion models. Indeed, the N100
delayed-detonation model shows different characteristics than W7 in
this regime and reproduces the near-UV features of SN~2011iv somewhat
better.
\section{Discussion \& Conclusions}\label{s:conc}
We presented a UVOIR spectrum of SN~2011iv, a relatively normal SN~Ia.
These data were obtained 0.6~days after $B$-band maximum light with
\textit{HST}/STIS and Magellan/FIRE. This spectrum is the first
contemporaneous and continuous UVOIR spectrum of a SN~Ia. It is also
the first published high-S/N, near-maximum, true-UV SN~Ia spectrum
since that of SN~1992A \citepalias{Kirshner93} and the earliest
high-S/N UV SN~Ia spectrum yet published.
We compared the UV spectrum of SN~2011iv to the near-maximum light UV
spectra of SNe~1992A and 2009ig. SNe~1992A and 2011iv have very
similar UV spectra. However, SN~2009ig has a spectrum that is
significantly different from that of the other two SNe. Specifically,
SN~2009ig lacks two prominent features at 2300--2700~\AA\ (though
this may be the result of its higher ejecta velocity), and it has a
relatively flat near-UV continuum.
Although the specific reasons for the differences between the UV
spectra of the three SNe are not yet clear, the SNe have several
additional observational differences. SNe~1992A, 2009ig, and 2011iv
have S0, Sa, and E1 host galaxies; $\Delta m_{15} (B) = 1.48$, 0.89,
and 1.69~mag; and $v_{\rm Si~II}^{0} = -12$,900, $-13$,500, and
$-10,$700~km~s$^{-1}$ , respectively \citep{Foley11:vgrad, Foley12:09ig}. All
three SNe have early-type host galaxies, so there is no clear
difference in progenitor age or metallicity; additional SNe must be
observed to probe the effects of environment on SN~Ia UV spectra.
Nonetheless, \citet{Sauer08} were able to produce different near-UV
continua (qualitatively similar to what is seen in Figure~\ref{f:uv})
by varying the amount of Ti and Cr generated in the explosion.
Specifically, models with less Ti/Cr had flatter near-UV continua.
SNe~1992A and 2009ig have similar \protect\hbox{$v_{\rm Si~II}^{0}$}, while SN~2011iv has
significantly lower \protect\hbox{$v_{\rm Si~II}^{0}$}; however, SNe~1992A and 2009ig have
different UV spectra, while SNe~1992A and 2011iv have very similar UV
spectra. Finally, SNe~1992A and 2011iv are relatively fast-declining
SNe~Ia, while SN~2009ig is a relatively slow-declining SN~Ia. It
appears that the characteristics of SN~Ia UV spectra are better
predicted by light-curve shape than ejecta velocity. We hypothesize
that $^{56}$Ni generation, which regulates the decline rate
\citep{Kasen07:wlr}, is dominant over the kinetic energy per unit mass
in the formation of the UV spectra of SNe~Ia near maximum light.
UV spectra are particularly sensitive to progenitor metallicity and
the details of the explosion. The maximum-light spectrum of SN~2011iv
is similar to both the delayed-detonation N100 model and the
solar-metallicity polluted W7$_{Z_{\sun}}$ model spectra; the standard
W7 model has significantly more far-UV flux than SN~2011iv.
Additional data will be useful in determining the full density
structure of the explosion.
This first spectrum shows the power of UV spectroscopy for
understanding the progenitors and explosions of SNe~Ia. Our full
set of UV, optical, and NIR data for SN~2011iv, including seven
\textit{HST} spectra, will be analyzed in a later paper.
SN~2011iv is one of the best-observed SNe, and detailed analysis of
this full dataset should provide even stronger constraints on
explosion models.
\begin{acknowledgments}
\textit{Facilities:}
\facility{HST(STIS), Magellan:Baade(FIRE), PROMPT}
\bigskip
This research was supported by a Clay Fellowship (R.J.F.), an NSF
Graduate Research Fellowship (E.R.N.), the TABASGO Foundation
(A.V.F.), and NASA/{\it HST} grant GO-12592. G.P.\ acknowledges
support by the Proyecto FONDECYT 11090421, proyecto regular UNAB
DI-28-11/R, and by the grant ICM P10-064-F (Millennium Center for
Supernova Science), with input from ``Fondo de Innovaci\'{p}n para la
Competitividad, del Ministerio de Econom\'{i}a, Fomento y Turismo de
Chile.'' S.T.\ is supported by the DFG (Transregional Collaborative
Research Center TRR~33). The simulations were performed at JSC,
J\"{u}lich, Germany (grants PRACE042, HMU14/20).
\end{acknowledgments}
\bibliographystyle{fapj}
|
1,314,259,996,682 | arxiv | \section{Introduction}
\label{sec-1}
In this paper we study a time-dependent family of smooth, embedded, closed curves $\gamma_t = F_t(\mathbb{S}^1) \subset \mathbb{S}^2$ on the unit sphere, evolving by the curve shortening flow:
\begin{equation}
\label{eq:csf}
\pd[F_t]{t} = -\curvecurv \nor
\end{equation}
where $\curvecurv$ is the (signed) geodesic curvature of $\gamma_t$ with respect to a choice of smooth unit normal vector field $\nor$. Our aim is to classify convex ($\curvecurv > 0$) ancient solutions, which by definition exist on the maximal time interval $(-\infty,T)$. If $T\ne \infty$, we will assume from now on that $T=0$. We prove the following theorem:
\begin{theorem}
[Classification of Ancient Solutions]
\label{thm:classification_ancient}
Let the family of curves $\gamma_t$ be a closed, convex, embedded ancient solution to the curve shortening flow on the sphere $S^2$. Then $\gamma_t$ is either a fixed equator for all $t \in (-\infty, \infty)$, or a family of shrinking geodesic circles, existing on $(-\infty, 0)$, converging to an equator as $t \to -\infty$ and shrinking to a point at $T=0$.
\end{theorem}
The curve shortening flow has been studied extensively in the plane. The principal result is the Gage-Hamilton-Grayson theorem \cite{MR840401, MR906392}, stating that arbitrary, smooth, embedded, closed solutions $\gamma_t$ shrink to round points in finite time $T<\infty$. On surfaces, the curves shortening flow either collapses to a round point in finite time (as in the plane case), or exists for all time, converging to a closed geodesic as $t\to\infty$ \cite{MR979601,MR1630194,MR2668967}.
Ancient solutions to the curve shortening flow in the plane have been classified in \cite{MR2669361} as precisely the contracting circles and the contracting Angenent ovals. The former is a Type I ancient solution ($\lim_{t\to-\infty} \sup_{\gamma_t}\abs{\curvecurv} < \infty$), while the latter is a Type II ancient solution ($\lim_{t\to-\infty} \sup_{\gamma_t}\abs{\curvecurv} = \infty$). Our theorem shows that the only Type I ancient solution on the sphere are the "obvious" ones, and that no Type II ancient solutions exist.
To begin in section \ref{sec-2}, we introduce some notation and preliminary results. Then in \ref{sec-3}, we obtain a Harnack inequality for convex curves evolving by curve shortening on $\mathbb{S}^2$ in Theorem \ref{thm:harnack}. This allows us to show that for convex curves, the curvature is monotonically increasing, hence bounded on any time interval $(-\infty, t_0]$. A standard bootstrapping argument then furnishes us with bounds on all higher derivatives. Next, in section \ref{sec-4} we use the Gauss-Bonnet theorem to show that $\int_{\gamma_t}\curvecurv \to 0$ as $t\to -\infty$. Combining this with the curvature estimates, we are readily able to show that $\gamma_t$ converges smoothly to an equator as $t\to-\infty$. To complete the theorem, in section \ref{sec-5}, we use a perturbed, parabolic version of Aleksandrov reflection inspired by \cite{MR1846204,MR1386736}. This shows that $\gamma_t$ reflects "above" (the precise definition is given in section \ref{sec-5}) itself for the perturbed reflection and hence so too in the limit for all reflections preserving the equator. It is then easy to show that each $\gamma_t$ is preserved under all equator preserving reflections and therefore is a round circle.
\section*{Acknowledgements}
Both authors would like to thank Professor Bennett Chow for suggesting this problem and providing much useful guidance on laying out the program. The second author is especially thankful, this paper arising from her Ph.D. thesis under Professor Chow's supervision. This paper was completed while the first author was a SEW Visiting Assistant Professor at UCSD, acting as an informal Ph.D. advisor to the second author's Ph.D. research at UCSD.
\section{Notation and Preliminaries}
\label{sec-2}
\label{sec:notation}
\subsection{Convex Curves on $\mathbb{S}^2$.}
\label{sec-2-1}
A closed, embedded curve $\gamma$ divides $\mathbb{S}^2$ into two open, disjoint regions. If one region has area strictly small than $2\pi$, we label that region $\interior{\Omega}$ and call it the interior of $\gamma$. The other region $\exterior{\Omega}$ is the exterior. Let $\nor$ denote the interior pointing unit normal (so that for $x\in\gamma$, $\exp_x^{\mathbb{S}^2}(\epsilon\nor) \in \interior{\Omega}$ for small $\epsilon > 0$). Note that if the area of both regions equals $2\pi$, it is not possible in general to single out one region as interior and one as exterior; consider for instance when $\gamma$ is an equator. The issue is equivalent to defining a unit normal vector field on $\gamma$ and declaring it be either interior or exterior pointing. In such a case, we will choose a unit normal vector field $\nor$ and designate it interior pointing.
On a Riemannian manifold $M$, there are several notions of convexity in common use. We will use the following definitions: A subset $K \subset M$ is \emph{(geodesically) convex} if every two points $x,y \in K$ can be joined by a length minimising geodesic (of $M$) entirely contained within $K$. Note that we don't require this geodesic to be unique so that a closed hemisphere of $\mathbb{S}^n$ is geodesically convex. $K$ is \emph{weakly (geodesically) convex} if any two points in $K$ may be connected by a length minimizing geodesic of $K$. The difference between the two notions is that the length minimising geodesic in a weakly convex set need not be length minimising in $M$. It is well known that weakly convex is equivalent to non-negative boundary curvature. For example, if $M = \mathbb{S}^1 \times \fld[R]$ is a flat cylinder, then a geodesic disc of radius greater than $\pi/2$ is weakly convex, but not convex. On the sphere however (as in the plane), weakly convex sets are convex (Proposition \ref{prop:convex_sets}). Since geodesics in $\mathbb{S}^2$ are great circles, and length minimising geodesics are half great circles, the length of any minimizing geodesic joining $x$ to $y$ is at most $\pi$. The following proposition characterises convex regions $K$ of $\mathbb{S}^2$ with boundary a smooth embedded curve. The results (and arguments) are standard and well known, but we could not find a good single reference, so give the details here.
\begin{prop}
\label{prop:convex_sets}
Let $K \subset \mathbb{S}^2$ be a connected open set with boundary $\bdry{K} = \gamma$ a smooth, closed embedded curve. Let $\nor$ be the interior unit normal vector field along $\gamma$ and $\curvecurv$ the geodesic curvature with respect to $\nor$. Then the following are equivalent:
\begin{enumerate}
\item $K$ is convex,
\item for every $x \in \gamma$, $K \subset H^+_x$ where $H^+_x$ is the hemisphere with boundary the tangent great circle $E_x$ to $\gamma(x)$ and interior normal $\nor(x)$,
\item $\gamma$ may be written as the graph over an equator of a smooth function $f$ such that
\[
f''(\theta) + 2 \tan(f(\theta)) (f'(\theta))^2 + \cos (f(\theta)) \sin (f(\theta)) \geq 0, \quad \theta \in \mathbb{S}^1.
\]
\item The boundary curvature, $\curvecurv \geq 0$.
\end{enumerate}
\end{prop}
\begin{proof}
\begin{itemize}
\item (1) $\Rightarrow$ (2):
Let $x \in \gamma$ and let $E_x$ be the tangent great circle to $\gamma = \bdry{K}$ at $x$. With $\nor(x)$ pointing interior to $K$, let $H^+(x)$ denote the hemisphere with boundary $E_x$ and interior normal $\nor(x)$. Then $H^+(x) \intersect K \ne \emptyset$ and we need to show that in fact $K \subset H^+(x)$.
Suppose otherwise, so that there is a $y$ in the interior of $H^-$, and let $\sigma$ be the unique (since $y \notin E_x$) length minimising geodesic joining $x$ to $y$. $\sigma$ lies in $H^-$, but also does not intersect $K$ in a neighbourhood of $x$ since it points exterior to $K$ ($\ip{\sigma'(x)}{\nor(x)} < 0$). On the other hand, since $K$ is convex and both $x,y\in K$ we must have $\sigma \subset K$, a contradiction.
\item (2) $\Rightarrow$ (3):
We work in polar coordinates $(\cos\phi\cos\theta, \cos\phi\sin\theta, \sin\phi)$ with the equator given by $\phi = 0$ (as opposed to the usual convention of $\phi=0$ at the north pole). For graphs $(\theta, f(\theta))$ the curvature is given by
\[
\curvecurv = \frac{\cos(f)}{((f')^2 + \cos^2(f))^{3/2}} \left(f'' + 2(f')^2 \tan(f) + \sin (f) \cos(f)\right).
\]
So here, we prove that (2) implies $\gamma$ may be written as a graph and that $\curvecurv \geq 0$.
\emph{$\gamma$ is a graph}: Take any tangent great circle $E$, $H^+$ the hemisphere with $K \subset H^+$, and $p$ the center of $H^+$. If $p \in \gamma$, then we can rotate $E$ along the geodesic joining $p$ to $x$ to obtain a new $E$ with $K \subset H^+$ and $p \in K$ (remember $K$ is open). Then by convexity, for any $y \in E$, the geodesic ray joining $p$ to $y$ intersects $\gamma$ in precisely one point, which we may denote by $f(y)$, expressing $\gamma$ as the graph of $f$ (which must be smooth).
$\curvecurv \geq 0$: This follows since $K$ lies on one side of every tangent great circle, so the local Taylor expansion of $\gamma$ shows that the curvature vector is interior pointing everywhere.
\item (3) $\Rightarrow$ (4):
As above, the condition on $f$ is precisely that $\curvecurv \geq 0$.
\item (4) $\Rightarrow$ (1):
We prove the contrapositive. Suppose that $K$ is not convex. We need to show that there is a $y \in \gamma$ such that $\curvecurv(y) < 0$.
Since $K$ is not convex, there is a length minimising geodesic $\alpha$ meeting $\clsr{K}$ only at it's endpoints. Moreover, we can choose $\alpha$ to have length strictly less than $\pi$: if not, then any arbitrary great circle must intersect $\cmplt{K}$ in a connected arc of length at least $\pi$ and so intersects $K$ in a connected arc of length at most $\pi$. Therefore any $x,y \in K$ may be connected by a length minimising geodesic contradicting that $K$ is not convex.
Let $\sigma_z$ be the continuous family of geodesic rays of length $\pi$ starting at $z \in \alpha$, perpendicular to $\alpha$. There are precisely two such families, and we choose $\sigma_z$ so that $\sigma_z$ intersects $\gamma$ at distance less than $\pi$ for $z$ near the endpoints of $\alpha$. Note that on the sphere, for each $z \in \alpha$, we have $\sigma_z \intersect \gamma \ne \emptyset$ since the end points of $\alpha$ (which also lie on $\gamma)$ lie on either side of the equator containing $\sigma_z$. Let $y = y(z) \in \gamma$ be the first point where $\gamma_z$ meets $\gamma$ and let $\rho(z) = d(z,y)$.
Then $\rho$ is continuous and attains it's maximum at a point $z_0$ in the interior of $\alpha$ since $\rho = 0$ on the end points of $\alpha$ and by assumption $\rho(z) > 0$ for some $z \in \alpha$. Now we have $\rho_{z_0}$ a geodesic meeting $\gamma$ orthogonally at $y_0$, hence we can solve for $z$ as a function of $y$ near $y_0$. Then the second variation formula (varying $y$)) shows that $\curvecurv(y_0) \leq 0$.
\end{itemize}
\end{proof}
From here on we will freely use the results of the proposition without further comment and by a \emph{convex curve} we will mean a closed, embedded curve $\gamma$ satisfying any of the four conditions. Lastly, let us note that for $\gamma$ convex, approximating $\gamma$ by convex polygons (with geodesic arcs), it is possible to show that the length $L(\gamma) \leq 2\pi$, a fact we will employ in section 4. See \cite[Problem 1.10.1]{MR2208981} for details. In section 5, we will find it very useful to write $\gamma$ as the graph over an equator.
\subsection{Evolution of basic quantities}
\label{sec-2-2}
Let us now record the evolution of various quantities under the curve shortening flow. This is very similar to the plane case \cite{MR840401,MR742856}. We make use of the Serret-Frenet formulae,
\[
\conx_{\tang} \tang = \curvecurv \nor, \quad \conx_{\tang} \nor = -\curvecurv \tang
\]
with $\tang$ the unit tangent to $\gamma$ and $\nor$ the interior pointing normal.
Let $s = s_t$ denote the arc-length parameter of $\gamma_t$. The commutator of $\pd{s}$ and $\pd{t}$ is
\begin{equation}
\label{eq:commutator}
\left[\pd{t}, \pd{s}\right] = -\curvecurv^2 \pd{s}.
\end{equation}
Under the curve shortening flow on $\mathbb{S}^2$, the curvature evolves according to
\begin{equation}
\label{eq:curvature_evolution}
\curvecurv_t = \curvecurv_{ss} + \curvecurv^3 + \curvecurv
\end{equation}
where subscripts denote partial derivatives. The maximum principle now ensures that if $\curvecurv > 0$ at some time $t_0$, then this holds for all $t\geq t_0$.
Lastly, the element of arc-length $ds$ evolves according to
\begin{equation}
\label{eq:arclength_evolution}
\pd{t} ds = -\curvecurv^2 ds.
\end{equation}
\subsection{Aleksandrov reflection}
\label{sec-2-3}
In section 5 we will make use of an Aleksandrov reflection argument, so give the preliminaries here. Embed $\mathbb{S}^2$ in $\fld[R]^3$ via the standard embedding, and let $E$ denote the equator $\{z = 0\}$.
The argument rests on a "tilted" Aleksandrov reflection. Let $\reflectionvector$ be a vector in $\fld[R]^3$ such that $\ip{\reflectionvector}{\vec{e}_z} < 0$ where $\vec{e}_z = (0,0,1)$, and let $\reflectionplane$ be the plane through the origin with normal vector $\reflectionvector$. Then $\reflectionplane$ intersects $E$ in two antipodal points. Let $\delta(V) \in (0, \pi/2)$ be the angle between $\reflectionvector$ and the plane $\{z=0\}$. Notice that for each fixed $\delta$, the set of $\reflectionvector$ with $\delta(\reflectionvector)=\delta$ is a compact set parameterised by $\mathbb{S}^1$ acting as rotations about the $z$-axis. We consider the Aleksandrov reflection across the plane $P$,
\[
\reflectionmap(X) = X - 2\ip{X}{V}V
\]
which is an isometry of $\fld[R]^3$ preserving $\mathbb{S}^2$, hence is also an isometry of $\mathbb{S}^2$. See figure \ref{fig:tilted_reflection}.
\begin{figure}[htb]
\centering
\includegraphics[width=.9\linewidth]{./img/tilted_reflection.pdf}
\caption{\label{fig:tilted_reflection}Reflection in the $(\vec{e_z}, V)$-plane showing the reflected equator and geodesics through the north pole (dotted lines).}
\end{figure}
Let $\reflectionhalfspace^+ = \{X \in \fld[R]^3: \ip{X}{\reflectionvector}>0\}$ be the open half space lying on the side of $\reflectionplane$ into which $\reflectionvector$ points, and $\reflectionhalfspace^- = \{X \in \fld[R]^3: \ip{X}{\reflectionvector} < 0\}$ be the open half space on the other side of $\reflectionplane$. For any set $S\subset \fld[R]^3$, let $\reflectionset{S}^{\pm} = S \intersect \reflectionhalfspace^{\pm}$. In particular $\reflectionset{(\mathbb{S}^2)}^{\pm}$ are hemispheres with boundary equator equal to $\reflectionplane \intersect \mathbb{S}^2$. Notice that $\reflectionmap$ takes $\reflectionhalfspace^{\pm}$ to $\reflectionhalfspace^{\mp}$.
Next we will need the nearest point projection $\pi(X)$ to the equator $E$. For a point $X\in\mathbb{S}^2$, let $\rho(X) = d(X, E)$ denote the distance from $X$ to $E$ and let $\pi(X)$ denote the set of nearest points on $E$ to $X$. If $X$ is not either pole $(0,0,1), (0,0,-1)$, then $\pi(X)$ is a single point. Otherwise, $\pi(X) = E$. For each point $y\in \pi(X)$, there is a unique, length minimising geodesic joining $X$ to $y$ with length equal to $\rho(X)$. By the first variation formula, this geodesic meets $E$ orthogonally, hence lies on the unique great circle passing through $X$ and the north pole $(0,0,1)$. The dotted lines in figure \ref{fig:tilted_reflection} show some geodesics passing through the north pole.
For any two curves $\alpha,\beta$ on $\mathbb{S}^2$, and any $X \in E$, let us write $\alpha \geq_X \beta$ (resp. $\alpha >_X \beta$) if
\[
\inf \{\rho(Y) : Y \in \pi^{-1}(X) \intersect \alpha\} \geq (\text{resp. } >) \> \sup \{\rho(Y) : Y \in \pi^{-1}(X) \intersect \beta\}
\]
whenever both sets are non-empty. The $\inf$ and $\sup$ are required since $\alpha$ and $\beta$ need not be graphs over the equator and so the fibres $\pi^{-1}(X) \intersect \alpha$ and $\pi^{-1}(X) \intersect \beta$ may have multiple points. Loosely speaking, we say $\alpha$ lies above $\beta$ over the point $X$ in the equator $\{z=0\}$. We will also write $\alpha \geq (\text{resp. } >) \> \beta$ if $\alpha \geq_X (\text{resp. } >_X) \> \beta$ for every $X \in E$. Notice in particular that we require strict inequality to hold for \emph{every} $X$.
\begin{remark}
\label{rem:partial_order}
The relations $\leq_X$ and $\leq$ are not partial orders in general since they are not reflexive. In fact, $\alpha \leq_X \alpha$ if and only if the fibre $\pi^{-1} (X)$ intersects $\alpha$ in a single point. The relation $\leq$ is only a partial order when restricted to curves that are graphs over the equator: $\alpha \leq \alpha$ if and only if $\alpha$ is a graph over the equator.
\end{remark}
Using polar coordinates as above, we may also rewrite $\alpha \geq_X \beta$ if and only if $\theta(X) \in \{\theta(\alpha)\} \intersect \{\theta(\beta)\}$ and
\[
\inf\{\phi(\alpha) : \theta(\alpha) = \theta(X)\} \geq \sup\{\phi(\beta) : \theta(\beta) = \theta(X)\}.
\]
That is, $\alpha \geq_X \beta$ if and only if there is at least one point on $\alpha$ and at least one point on $\beta$ with azimuthal angle $\theta$ equal to the azimuthal angle of $X$ and so that the smallest polar angle $\phi$ of $\alpha$ is greater than or equal to the greatest polar angle of $\beta$.
\section{Harnack Inequality and Curvature Estimates}
\label{sec-3}
Just as for the curve shortening flow in the plane, there is a Harnack inequality for the curve shortening flow on $\mathbb{S}^2$. This is the fundamental result of this section, from which everything else follows.
\begin{theorem}
[Harnack Inequality]
\label{thm:harnack}
For any immersed solution to the curve shortening flow defined on the time interval $[-\alpha, 0)$ and with $\curvecurv > 0$, we have
\[
(\log k)_{ss} + k^2 + \frac{1}{2(t-\alpha)} \geq 0.
\]
\end{theorem}
\begin{proof}
From the evolution of the curvature in equation \eqref{eq:curvature_evolution} and the commutator equation \eqref{eq:commutator}, we deduce
\begin{align*}
k_{st} &= k_{ts} + k^2k_s = (k_{sss} + 3k^2k_s + k_s) + k^2k_s\\
&= k_{sss} + 4k^2k_s + k_s \\
k_{sst} &= k_{tss} + 2kk_s^2 + 2k^2k_{ss} = (k_{ssss} + 3k^2k_{ss} + 6kk_s^2 + k_{ss}) + 2kk_s^2 + 2k^2k_{ss} \\
&= k_{ssss} + 5k^2k_{ss} + 8kk_s^2 + k_{ss}.
\end{align*}
Let $Q$ be the quantity
\[
Q = (\log k)_{ss} + k^2.
\]
Computing the time derivative, we get
\begin{align*}
Q_t &= -\frac{k_{ss}}{k^2}k_t + \frac{k_{sst}}{k} + 2k^{-3}k_t k_s^2 - k^{-2}(2k_sk_{st}) + 2kk_t \\
&= -\frac{k_{ss}^2}{k^2} + 6kk_{ss} + \frac{k_{ssss}}{k} + 2k_s^2 + \frac{2k_s^2k_{ss}}{k^3} - \frac{2k_sk_{sss}}{k^2} + 2k^4 + 2k^2 \\
&= Q_{ss} + \left(\frac{2k_s}{k}\right)Q_s +2Q^2 +2k^2 \\
&\ge Q_{ss} + \left(\frac{2k_s}{k}\right)Q_s + 2Q^2.
\end{align*}
An ODE comparison with $q(t) = -1/2(t-\alpha)$ which satisfies $q_t = 2q^2$ and $\lim_{t\to\alpha} q(t) = -\infty$ shows that $Q(s,t) \geq q(t)$ completing the result.
\end{proof}
\begin{cor}
\label{cor:curvature_time_increasing}
For an ancient solution $\gamma_t$, with $\curvecurv > 0$, we have
\[
\curvecurv_t \geq 0.
\]
\end{cor}
\begin{proof}
As $\gamma_t$ is an ancient solution with $\curvecurv>0$, the Harnack inequality holds for any $\alpha < 0$, supplying us with
\begin{align*}
0 &\leq (\log k)_{ss} + k^2 + \frac{1}{2(1-\alpha)} \\
&= \frac{\curvecurv_{ss}}{\curvecurv} - \frac{\curvecurv_s^2}{\curvecurv} + \curvecurv^2 + \frac{1}{2(1-\alpha)} \\
& \leq \frac{\curvecurv_{ss} + \curvecurv^3 + \curvecurv}{\curvecurv} + \frac{1}{2(1-\alpha)} \\
&= \frac{\curvecurv_t}{\curvecurv} + \frac{1}{2(1-\alpha)} \\
\end{align*}
for $t \in [\alpha, 0)$. Taking the limit $\alpha \to -\infty$ gives the result.
\end{proof}
We are now able to obtain a curvature bound, and by standard bootstrapping arguments, we also obtain higher derivative bounds.
\begin{cor}
\label{cor:derivative_bounds}
For any $t_0 < 0$ and any integer $j\geq 0$, there exists a constant $C_j(t_0)$ such that
\[
\abs{\curvecurv^{(j)}} \leq C_j(t_0)
\]
on $(-\infty, t_0)$.
\end{cor}
\begin{proof}
By Corollary \ref{cor:curvature_time_increasing}, $\curvecurv$ is increasing in $t$, and since $\curvecurv > 0$, we may take $C_0(t_0) = \sup \{\curvecurv(x,t_0) : x\in \mathbb{S}^1\}$.
The higher derivative estimates follow by standard bootstrapping arguments similar to those described in \cite{MR1375255}. For example, we obtain $C_1(t_0)$ from the evolution equation $(k_s)_t = k_{sss} + 4k^2k_s + k_s$ by applying the maximum principle to the evolution of $(t-(t_0-1))k_s^2 + C_0(t_0) k^2$ and using the fact that $\abs{\curvecurv} \leq C_0(t_0)$.
\end{proof}
\section{Backwards Convergence}
\label{sec-4}
Armed with the curvature bounds, we can prove that any ancient solution $\gamma_t$ converges smoothly to an equator as $t\to-\infty$. We begin with a lemma.
\begin{lemma}
\label{lem:intcurvetozero}
Let $\gamma_t$ be an ancient solution to the curve shortening flow. Then
\begin{align*}
\lim_{t\to -\infty} \int_{\gamma_t} k ds = 0
\end{align*}
exponentially fast.
\end{lemma}
\begin{proof}
By the Gauss-Bonnet theorem we have
\[
\int_{\gamma_t} \curvecurv ds = 2\pi - A
\]
where $A$ is the area of $\interior{\Omega_t}$. Recalling that $\curvecurv_t = \curvecurv_{ss} + \curvecurv^3 + \curvecurv$ and that $(ds)_t = - \curvecurv^2 ds$, we obtain
\[
\pd{t} \int_{\gamma_t} \curvecurv ds = \int_{\gamma_t} \curvecurv_{ss} + \curvecurv ds = \int_{\gamma_t} \curvecurv ds = 2\pi - A.
\]
Therefore
\[
A_t = A - 2\pi,
\]
and hence
\begin{equation}
\label{eq:At}
A = 2\pi[1 - (1 - A(0)/2\pi)e^t].
\end{equation}
Then letting $t\to-\infty$, we find that $A(t) \to 2\pi$ exponentially fast and the Gauss-Bonnet formula implies that
\[
\int_{\gamma_t} \curvecurv ds \to 0
\]
exponentially fast.
\end{proof}
Combining our estimates so far, we now obtain the following important proposition:
\begin{prop}
\label{prop:dk_curv_tozero}
Let $\gamma_t$ be an ancient solution to the curve shortening flow. Then for every integer $j\geq 0$, we have
\[
\max_{s \in \mathbb{S}^1} \abs{\curvecurv^{(j)}} (s, t) \to 0
\]
as $t \to -\infty$.
\end{prop}
\begin{proof}
First, let us prove the case $j=0$. We argue by contradiction. Suppose the proposition is false. Then there exists $\epsilon>0$, a sequence $t_i \to -\infty$ and, a sequence $s_i$ such that $\curvecurv(s_i,t_i) \ge \epsilon$ for all $i$. Since we also have $\abs{\curvecurv_s} \le C_1(1)$ for $t\leq 1$, we find that
\[
\curvecurv(s, t_i) \geq \epsilon -C_1(1)\abs{s-s_i} \geq \epsilon/2\]
for all $s$ such that $|s-s_i| \leq \frac{\epsilon}{2C_1(1)}$. But this implies that for every $i$,
\[
\int_{\gamma_{t_i}} \curvecurv (s,t) ds \geq \int_{|s-s_i| \leq \tfrac{\epsilon}{2C_1(1)}} \curvecurv(s,t)ds \geq \frac{\epsilon^2}{4C_1(1)}
\]
contradicting the fact that $\int_{\gamma_t} \curvecurv \to 0$ as $t\to -\infty$ by lemma \ref{lem:intcurvetozero}.
The result for $j>0$ follows by a bootstrapping argument similar to the proof of Corollary \ref{cor:derivative_bounds}.
\end{proof}
Now we have all the ingredients to prove that the backwards limit is an equator. First we have sub-sequential convergence.
\begin{lemma}
\label{lem:subsequential_backward_limit}
Let $\gamma_t$ be an ancient, embedded, convex solution to the curve shortening flow on $\mathbb{S}^2$. Then there is a sequence $t_k \to -\infty$ with $\gamma_{t_k} \to_{C^{\infty}} \gamma_{-\infty}$ as $k\to\infty$, with $\gamma_{-\infty}$ an equator.
\end{lemma}
\begin{proof}
Since $\pd{t} L = -\int \curvecurv^2 ds < 0$, $L(t)$ is bounded below by $L(-1) > 0$ for all $t\leq-1$. By the paragraph following Proposition \ref{prop:convex_sets}, $L \leq 2\pi$.
Proposition \ref{prop:dk_curv_tozero} shows that the curvature and all derivatives converge to $0$. Since $L$ is bounded, as $t\to -\infty$, $\abs{\gamma'}$ is bounded (say in a constant speed parametrisation) above and away from zero. The Arzela-Ascoli theorem then provides us with a sequence $t_k \to -\infty$ such that $\gamma_{t_k}$ converges smoothly to a closed, immersed curve with zero curvature, i.e. to an equator $\gamma_{-\infty}$.
\end{proof}
Next we show that the limit is unique and that the flow remains in a fixed hemisphere.
\begin{cor}
\label{cor:hemisphere}
The equator $\gamma_{-\infty}$ is unique and $\gamma_t$ lies in one of the hemispheres $H^{\pm}_{-\infty}$ defined by $\gamma_{-\infty}$ for all $t \in (-\infty,0)$.
\end{cor}
\begin{proof}
Since $\curvecurv > 0$, Gauss-Bonnet implies that $A(t) < 2\pi$ for all $t$ and that the curvature vector points inward (recall that the interior $\interior{\Omega_t}$ is the region enclosed by $\gamma_t$ with area less than $2\pi$). Thus $\interior{\Omega_{t_1}} \subsetneq \interior{\Omega_{t_2}}$ whenever $t_2 < t_1$.
In particular, for our sequence $(t_k)$, $\interior{\Omega_{t_j}} \subsetneq \interior{\Omega_{t_k}}$ for $k>j$. If $\interior{\Omega_{t_j}}$ is not wholly contained in either hemisphere $H_{-\infty}^{\pm}$ for some $j$, then it contains points in both hemispheres, $x_j^{\pm} \in H_{-\infty}^{\pm}$. We obtain a contradiction by choosing $k<j$ with $\gamma_{t_k}$ sufficiently close to $\gamma_{-\infty}$ so that $x_j^+$ lies on the opposite side of $\gamma_{-\infty}$ to $x_j^-$ contradicting $\interior{\Omega_{t_k}} \subset \interior{\Omega_{t_j}}$ has points on both sides of $\gamma_{t_k}$. Thus $\interior{\Omega_{t_k}}$ lies entirely in one or the other hemisphere $H_{-\infty}^{\pm}$ for every $k$.
Now for any $t \in (-\infty,0)$, choose $k$ such that $t_k < t$. Then $\interior{\Omega_{t}} \subsetneq \interior{\Omega_{t_k}}$, the latter lying in a hemisphere. Lastly, suppose there is a sequence $t_k'$ with $\gamma_{t_k'}$ converging to different equator. This equator has points lying in both hemispheres defined by $\gamma_{-\infty}$ and hence $\gamma_{t_k'}$ also has points in both hemispheres for $t_k'$ sufficiently negative, a contradiction.
\end{proof}
\begin{remark}
Any closed, embedded curve on $\mathbb{S}^2$ with $\curvecurv \geq 0$ must lie in a closed hemisphere. The result above shows that under the flow, an embedded, convex, ancient solution remains in a fixed hemisphere for all time.
\end{remark}
Now we can extend the sub-sequential convergence to full convergence.
\begin{theorem}
\label{thm:backward_limit}
Let $\gamma_t$ be an ancient, embedded, convex solution to the curve shortening flow on $\mathbb{S}^2$. Then $\gamma_{t} \to_{C^{\infty}} \gamma_{-\infty}$ up to diffeomorphism as $t \to -\infty$.
\end{theorem}
\begin{proof}
First, we have $C^0$ convergence: for any $\epsilon>0$, choose $t_k$ with $\gamma_{t_k}$ $\epsilon$-close to $\gamma_{-\infty}$ in $C^{0}$ norm. Then for any $t < t_k$, both $\gamma_t$ and $\gamma_{t_k}$ lie in the same hemisphere with $\interior{\Omega_{t_k}} \subsetneq \interior{\Omega_{t}}$. Thus $\gamma_t$ lies between $\gamma_{-\infty}$ and $\gamma_{t_k}$ and so is also $\epsilon$-close to $\gamma_{-\infty}$ in $C^0$ norm.
$C^1$ convergence follows since the total length $L(t_k) \to 2\pi$ as $k\to\infty$. But also $\inpd[L]{t} = - \int k^2 ds < 0$ so that $L$ is monotone increasing backwards in time hence $L(t) \to 2\pi$ as $t\to-\infty$. But now parametrising $\gamma_t$ on $[0,1]$ with constant speed gives $\abs{\gamma_t'} = L(t) \to 2\pi$ and so $C^1$ convergence up to diffeomorphism follows.
Smooth convergence up to diffeomorphism now follows since the curvature and all derivatives of curvature converge to $0$.
\end{proof}
\section{Ancient solutions are shrinking round circles}
\label{sec-5}
In this section, we prove that ancient, convex solutions to the curve shortening flow are shrinking round circles.
\begin{lemma}
[Backwards approximate symmetry]
\label{lem:backward_approximate_symmetry}
For any $\delta \in (0,\pi/2)$, there exists a $t_{\delta} \in (-\infty, 0)$ such that for every $V$ with $\delta(V) = \delta$ and all $t\leq t_{\delta}$, we have $\reflectionmap(\reflectionset{(\gamma_t)}^+) \geq \reflectionset{(\gamma_t)}^-$.
\end{lemma}
\begin{proof}
We use polar coordinates as in section \ref{sec:notation}. From Corollary \ref{cor:hemisphere} and Proposition \ref{prop:convex_sets}, we can assume that on $(-\infty, 0)$, $\gamma_t$ lies in the upper hemisphere $\{z>0\}$, written as a graph $\phi = f_t(\theta)$ of a smooth family of positive, smooth functions $f_t [0,2\pi] \to \fld[R]$. Since $\gamma_t$ smoothly converges uniformly to the equator $\{z=0\}$, we have $\npd{\theta}{k} f_t \to 0$ uniformly for each $k\geq 0$.
Provided that $\delta<\pi/4$, the reflected equator $\reflectionmap(\{z=0\})$ can be written as a graph $(\theta, g_{-\infty}(\theta))$. Since $\reflectionmap$ is an isometry, $\reflectionmap(\gamma_t)$ converges smoothly and uniformly to the reflected equator $\reflectionmap(\{z=0\})$. As the latter is a graph, possibly by choosing $t_0 < 0$ \emph{independently of $\delta$}, we can assume that $\reflectionmap(\gamma_t)$ may be written as a graph $(\theta, g_t(\theta))$ for $t < t_0$ with $g_t \to g_{-\infty}$ smoothly as $t\to-\infty$.
In spherical polar coordinates, for $X,Y \in \mathbb{S}^2$ the nearest-point projection is $(\theta(X), \phi(X)) \mapsto (\theta(X), 0)$. If $\theta(X) = \theta(Y)$, the statement $X\geq Y$ is equivalent to $\phi(X) \geq \phi(Y)$. Thus to show that $\reflectionmap(\gamma_t)^+ \geq \gamma_t^-$ it is enough to show that $g_t(\theta) \geq f_t(\theta)$.
The proof is composed of estimates for \emph{interior points} (i.e. points away from $P \cap \mathbb{S}^2$) and for \emph{boundary points} (i.e. points near $P \cap \gamma_t$).
\emph{Interior Points}
For $\delta < \pi/4$, the reflected equator $(\theta, g_{-\infty}(\theta))$, $\theta \in [0,\pi]$ is given by a non-negative, smooth, concave function $g_{-\infty}$ symmetric about $\pi/2$ and strictly positive for $\theta \in (0, \pi)$. Given any $\epsilon \in (0, \pi/2)$, let $G = g_{-\infty} (\epsilon) = g_{-\infty}(\pi-\epsilon)$. Then $G < g_{\infty}(\theta)$ for any $\theta \in (\epsilon, \pi-\epsilon)$. Choose $t_1<t_0$ such that for $\theta \in (\epsilon, \pi-\epsilon)$ and $t<t_1$, we have $f_t(\theta) < G/2$ and $\abs{g_t(\theta) - g_{-\infty}(\theta)} < G/2$. This is possible since $f_t \to 0$ uniformly, and $g_t \to g_{-\infty}$ uniformly. Then, since $g_{-\infty} > G$ on $(\epsilon, \pi - \epsilon)$, for any $\epsilon > 0$, there is a $t_1 = t_1(\epsilon)$ such that $g_t(\theta) > f_t(\theta)$ for $\theta \in (\epsilon, \pi-\epsilon)$ and $t\leq t_1$.
\emph{Boundary Points}
Choose an orientation on $\theta$ so that $P \intersect \{z>0\}$ lies in the region with $\theta \in (-\pi, 0)$. Then recalling that $\gamma_t$ is a graph over the equator, $\gamma_t \intersect P = \{\theta_0(t), \theta_1(t)\}$ with $\theta_0(t) \in (-\pi, 0)$ and $\theta_1(t) \in (\pi, 2\pi)$. Moreover as $t \to -\infty$ we have $\theta_0(t) \to 0$ and $\theta_1(t) \to \pi$. The aim is to show that given $\tilde{\epsilon}>0$, there is a $t_{\tilde{\epsilon}}$ such that $g_t > f_t$ on $(\theta_0(t), \tilde{\epsilon}) \union (\pi-\tilde{\epsilon}, \theta_1(t))$ for every $t<t_{\tilde{\epsilon}}$. It's enough to prove it on $(\theta_0(t), \tilde{\epsilon})$. The proof on $(\pi-\tilde{\epsilon}, \theta_1(t))$ is similar.
We use that $f_t \to 0$, and $g_t \to g_{-\infty}$ smoothly and uniformly, and that $\theta_0(t) \to 0$ as $t\to-\infty$. Notice that $\reflectionmap(\{z=0\})$ lies above the equator $\{z=0\}$ for $\theta \in (0,\pi)$ and lies below for $\theta \in (-\pi,0)$. Thus, $g_{-\infty}$ is odd about $\theta=0$ and increasing near $\theta = 0$ so that $g_{-\infty}'(0) > 0$ (in fact equal to $\tan(2\delta)$) and $g_{-\infty}''(0) = 0$.
Choose $t_1 < t_0$ such that $g_t'(\theta_0(t)) > f_t'(\theta_0(t))$ for all $t<t_1$ which we can do since $f_t' \to 0$ uniformly and $g_t'(\theta_0) \to g_{-\infty}'(0) = \tan(2\delta) > 0$. We also have that $g_t(\theta_0) = f_t(\theta_0)$ since $\theta_0$ is the point about which $f_t$ is reflected across $P$. Expand $g_t$ and $f_t$ in a Taylor series about $\theta_0(t)$ to get that for $\theta > \theta_0$, $g_t > f_t$ if and only if
\[
g_t'(\theta_0) - f_t'(\theta_0) > - \frac{1}{2} (g_t''(c) - f_t''(c)) (\theta - \theta_0)
\]
where $c = c(\theta, t) \in (\theta_0(t), \theta)$. As $t\to -\infty$ the left hand side converges to $\tan(2\delta)$ whilst the right hand side converges to $0$ hence there is a $t_2 = t_2(\tilde{\epsilon}) < t_1$ such that the inequality $g_t - f_t > 0$ is satisfied for any $\theta \in (\theta_0(t), \tilde{\epsilon})$ and $t<t_2$.
\emph{Combined estimates}
To finish the proof, fix any $\epsilon>0$ and use the interior estimates to obtain $g_t > f_t$ for $\theta \in (-\epsilon, \pi-\epsilon)$ and $t < t_1$. Then choose $\tilde{\epsilon} > \epsilon$ to obtain $g_t > f_t$ for $\theta \in (\theta_0(t), \tilde{\epsilon}) \union (\pi-\tilde{\epsilon}, \theta_1(t))$ and $t < t_2$ from the boundary estimates. Then let $t_{\delta} = \min\{t_1, t_2\}$ to get $g_t > f_t$ for all $\theta \in (\theta_0(t), \theta_1(t))$ and all $t<t_{\delta}$.
\end{proof}
\begin{lemma}
[Approximate symmetry preserved]
\label{lem:approximate_symmetry_preserved}
There is a $T \in (-\infty, 0)$ such that $\reflectionmap(\gamma_t)^+ \geq \gamma_t^-$ for $t \in (-\infty, T)$ and all $\delta \in (0,\pi/4)$.
\end{lemma}
\begin{proof}
Recall that both the $\gamma_t^-$ and $\reflectionmap(\gamma_t)^+$ may be written as graphs over the equator for $t\in(-\infty, t_0)$. Since both the equator $\gamma_{-\infty}$ and the reflected equator $\reflectionmap(\gamma_{-\infty})$ meet $P$ transversely, $\gamma_t$ smoothly converges to the equator, and $\reflectionmap$ is an isometry, there is a $t_1 = t_1(\delta) \in (-\infty,t_0)$ such that both $\gamma_t^-$ and $\reflectionmap(\gamma_t)^+$ meet $P$ transversely for all $t \in (-\infty, t_1)$. Thus $\gamma_t^-$ and $\reflectionmap(\gamma_t)^+$ are connected curves meeting $P$ transversely in precisely two points for each $t$.
Now we apply the maximum principle. Since $\reflectionmap$ is an isometry, $\reflectionmap(\gamma_t^+)$ evolves by curve shortening. Since $P\intersect \mathbb{S}^2$ is a great circle, it is stationary under the curve shortening flow so we can think of it too evolving by curve shortening. Therefore, as both $\gamma_t^-$ and $\reflectionmap(\gamma_t)^+$ meet $P$ transversely, the maximum principle ensures that both curves do not intersect $P$ at any \emph{other} points, hence remain in $\reflectionset{\mathbb{S}^2}^-$ for all $t\in(-\infty, t_1)$.
The above allows us to set up a maximum principle argument: we have two connected curves $\gamma_t^-, \reflectionmap(\gamma_t^+)$ evolving by curve shortening and they agree at their end points which remain on $P$. By Lemma \ref{lem:backward_approximate_symmetry}, for all $t \in (-\infty, t_{\delta})$ we have $d(x,y,t) > 0$ for any $x\in\gamma_t^-$ and $y\in\reflectionmap(\gamma_t^+)$ away from the end points. We also obtain that at the end points, the angle $\reflectionmap(\gamma_t^+)$ makes with the $\{z=0\}$ plane is strictly bigger that the angle $\gamma_t^-$ makes with the $\{z=0\}$ plane. By the parabolic Hopf boundary point lemma (see e.g. \cite{MR1483984}), this positive lower bound is preserved under the flow and so $d(x,y,t) > 0$ for $(x,y)$ near both end points. Now, in the usual way (e.g. \cite{MR1656553}) a contradiction is obtained if $d(x,y,t) \leq 0$ at some time $t$ for some $(x,y)$ since this must occur at a first time $t>t_{\delta}$ at an interior point $(x,y)$.
This furnishes us with a $T_{\delta}$ for each $\delta$ such that $\reflectionmap(\gamma_t)^+ \geq \gamma_t^-$ for $t \in (-\infty, T_{\delta})$. Let $T = \inf\{T_{\delta}: \delta \in (0,\pi/2)\}$. We need to show that $T>-\infty$. To see this, observe that the above argument is valid provided both $\gamma_t^-$ and $\reflectionmap(\gamma_t)^+$
\begin{enumerate}
\item are graphs over the equator,
\item meet $P$ transversely,
\item are non-empty
\end{enumerate}
for $t \in (-\infty, T)$.
\begin{enumerate}
\item Recall that $\reflectionmap(\gamma_{-\infty})$ is a graph $g_{-\infty}^{\delta}$ for each $\delta\in(0,\pi/4)$ with maximum derivative at the end points $\theta = \{0, \pi\}$. As $\delta \to 0$, the derivative $(g_{-\infty}^{\delta})'(0)$ monotonically decreases to $0$. For each fixed $t$ then $[\reflectionmap(\gamma_t)+]'$ becomes more horizontal as $\delta$ decreases hence if $\reflectionmap(\gamma_t)^+$ is a graph (so does not have a vertical tangent) for some $\delta_0$, then it remains a graph for every $\delta < \delta_0$. Of course whether $\gamma_t^-$ is a graph or not is independent of $\delta$, and by convexity we know that $\gamma_t$ is a graph for all $t \in (-\infty,0)$.
\item The angle $P$ makes with the $\{z=0\}$ plane increases monotonically as $\delta \to 0$. If $\gamma_t^-$ and $\reflectionmap(\gamma_t)^+$ meet $P$ transversely for some $\delta_0$ then they continue to do so for every $\delta<\delta_0$.
\item Provided $\gamma_t^-$ lies below the maximum $\phi$ coordinate of $P \intersect \mathbb{S}^2$, both curves $\gamma_t^-$ and $\reflectionmap(\gamma_t)^+$ are non-empty. But now just observe that this $\phi$ monotonically increases to $\pi/2$ as $\delta \to 0$.
\end{enumerate}
\end{proof}
Next we characterise those curves $\alpha$ with maximal approximate symmetry for every $\delta>0$ as round circles.
\begin{prop}
[Exact symmetry]
\label{prop:exact_symmetry}
Let $\alpha \subset (\mathbb{S}^2)^+ = \mathbb{S}^2 \intersect \{z \geq 0\}$ be a smooth curve. If $\reflectionmap(\reflectionset{\alpha}^+) \geq \reflectionset{\alpha}^-$ for every $\reflectionvector$ such that $\ip{\reflectionvector}{\vec{e}_z} < 0$ and $\delta(\reflectionvector) \in (0,\pi/4)$, then $\alpha$ is a round circle with center the north pole $(0,0,1)$.
\end{prop}
\begin{proof}
Let $\reflectionvector_0$ be a vector in $\fld[R]^3$ such that $\ip{\reflectionvector_0}{\vec{e}_z} = 0$ (so that $\delta(\reflectionvector_0) = 0$). Choose any $X \in \reflectionset{(\gamma_{-\infty})}^-$. Then by assumption, we have
\[
\reflectionmap(\reflectionset{\alpha}^+) \geq_X \reflectionset{\alpha}^-
\]
for all $\reflectionvector$ lying in the plane spanned by $\vec{e}_z$ and $\reflectionvector_0$, and with $\ip{\reflectionvector}{\vec{e}_z} < 0$ and $\delta(\reflectionvector) \in (0,\pi/4)$. By continuity, letting $\reflectionvector \to \reflectionvector_0$ we obtain $\reflectionmap[\reflectionvector_0](\reflectionset[\reflectionvector_0]{\alpha}^+) \geq_X \reflectionset[\reflectionvector_0]{\alpha}^-$ for each $X \in \gamma_{-\infty}^-$ and hence
\[
\reflectionmap[\reflectionvector_0](\reflectionset[\reflectionvector_0]{\alpha}^+) \geq \reflectionset[\reflectionvector_0]{\alpha}^-
\]
for every $\reflectionvector_0$ with $\ip{\reflectionvector_0}{\vec{e}_z} = 0$.
Now, we need some simple properties of $\reflectionmap[\reflectionvector_0]$ following from the fact that $\ip{\reflectionvector_0}{\vec{e}_z} = 0$:
\begin{itemize}
\item $\reflectionmap[\reflectionvector_0]^2 = \id$,
\item $\alpha \geq \beta \Rightarrow \reflectionmap[\reflectionvector_0](\alpha) \geq \reflectionmap[\reflectionvector_0](\beta)$,
\item $\reflectionmap[\reflectionvector_0] = \reflectionmap[-\reflectionvector_0]$, and
\item $\alpha^{\pm}_{\reflectionvector_0} = \alpha^{\mp}_{-\reflectionvector_0}$.
\end{itemize}
Thus we obtain,
\begin{align*}
\reflectionset[\reflectionvector_0]{\alpha}^+ &= \reflectionmap[\reflectionvector_0]^2(\reflectionset[\reflectionvector_0]{\alpha}^+) = \reflectionmap[\reflectionvector_0](\reflectionmap[\reflectionvector_0](\reflectionset[\reflectionvector_0]{\alpha}^+)) \\
&\geq \reflectionmap[\reflectionvector_0](\reflectionset[\reflectionvector_0]{\alpha}^-) = \reflectionmap[-\reflectionvector_0](\reflectionset[-\reflectionvector_0]{\alpha}^+) \\
&\geq \reflectionset[-\reflectionvector_0]{\alpha}^- = \reflectionset[\reflectionvector_0]{\alpha}^+.
\end{align*}
We must have equality all the way through and hence
\begin{equation}
\label{eq:reflection_invariant}
\reflectionmap[\reflectionvector_0](\reflectionset[\reflectionvector_0]{\alpha}^-) = \reflectionset[\reflectionvector_0]{\alpha}^+
\end{equation}
for any $\reflectionvector_0$.
To finish, equation \eqref{eq:reflection_invariant} implies that $\alpha$ must have a horizontal (i.e. no $\vec{e}_z$ component) tangent at $\reflectionplane[\reflectionvector_0] \intersect \alpha$. But every point of $\alpha$ lies on $\reflectionplane[\reflectionvector_0]$ for some $\reflectionvector_0$ hence $\alpha$ has a horizontal tangent everywhere and hence is a round circle.
\end{proof}
\begin{theorem}
Let $\gamma_t$ be a convex, ancient solution to the curve shortening flow. Then $\gamma_t$ is the unique up to isometry of $\mathbb{S}^2$, family of shrinking circles ancient solution.
\end{theorem}
\begin{proof}
The approximate symmetry preserved lemma \ref{lem:approximate_symmetry_preserved}, implies that for every $\reflectionvector$ with $\delta(\reflectionvector) \in (0,\pi/4)$, $\reflectionmap(\reflectionset{(\gamma_t)}^+) \geq \reflectionset{(\gamma_t)}^-$ for all $t\in (-\infty,T)$. The exact symmetry proposition \ref{prop:exact_symmetry} applies at each such $t \in (-\infty, T)$, showing that $\gamma_t$ is a round circle for every $t\in(-\infty,T)$. Uniqueness of solutions ensures that $\gamma_t$ is a round circle for every $t \in (-\infty,0)$.
\end{proof}
\printbibliography
\end{document}
|
1,314,259,996,683 | arxiv | \section{INTRODUCTION}
Despite the maturity of the field of stochastic optimal control theory, the majority of the theoretical and computational work considers stochastic systems with Gaussian stochastic disturbances. This observation is valid if one considers the lack of scalable and real time algorithms for control of high dimensional stochastic systems with disturbances that are far from being Gaussian and zero mean. Motivated by this lack of theory and algorithm on control of systems with more complex stochasticity, we consider dynamics with Gaussian and the more general compound Poisson noise.
Compound Poisson process, also known as the marked-jump process, is a doubly stochastic process where the stochasticity arises from both the jump time and amplitude \cite{hanson2007applied}. For simplicity, we use the term jump noise for compound Poisson noise. Processes with jumps have been widely used to describe the random evolution of, e.g., brain dynamics \cite{anvari2016disentangling}, of soil moisture dynamics \cite{daly2006probabilistic}, or of financial figures such as stock prices, market indices, and interest rates \cite{tankov2003financial}. For application on dynamical systems, jump stochastic terms in the dynamics can capture the discontinuities that arise due to phenomena such as gust or due to interactions of the system in consideration with the environment. Therefore, it is important that methods that deal with jump terms in the dynamics are developed.
The contributions of this paper are as follows:
\begin{itemize}
\item We derive a novel algorithm for control of systems with jump noise from an information theoretic point of view. The new algorithm extends the capability of a previous scheme to handle a more complex form of stochasticity than the common Gaussian noise \cite{williams2016aggressive}.
\item We show the connection of the resulting scheme with an alternative approach to stochastic optimization that does not rely on importance sampling. With this equivalence established, convergence of the algorithm can be shown using techniques common in optimization literature \cite{zhou2014gradient}.
\item We present an iterative MPC algorithm. The algorithm can utilize the parallel computing capabilities of the GPU, which means a large number of sampling trajectories can be propagated simultaneously and the algorithm can be implemented in real time. We implement the algorithm in simulation on a cartpole and quadrotor system with Gaussian and jump noise added, and we compare the performance of the algorithm against the path integral control based information theoretic MPC algorithm \cite{williams2016aggressive} that doesn't account for the jump noise.
\end{itemize}
The information theoretic approach we take in this paper is based on the path integral framework, which originated from Kappen and Theodorou's work \cite{kappen2005linear, theodorou2010generalized}. An iterative path integral control method, developed by Williams \cite{williams2016aggressive}, has been implemented for autonomous racing. This method uses the information theoretic notions of free energy and relative entropy, and obtains optimal control policy distribution through minimization of the Kullback-Leibler divergence (KL-Divergence) between a control induced probability measure and the optimal control policy induced probability measure. This approach allows for a solution to the stochastic optimal control problem using an importance sampling scheme.
The stochastic optimization approach is motivated by the lack of computational methods for stochastic optimal control. This approach is based on the stochastic approximation method \cite{borkar2009stochastic} where noisy observations are used to approximate stochastic functions. Optimization is then performed based on this noisy approximation to iteratively improve the solution. The derivation of this approach assumes a parameterized sampling distribution of entire trajectories based on system dynamics, reformulates the original problem with respect to the parameters, and obtains an update rule for these parameters through gradient descent.
The rest of this paper is organized as follows: in section II we provide the problem formulation. In section III we introduce the information theoretic approach to the stochastic optimal control problem. In section IV we introduce the stochastic optimization approach to the problem. Then we provide the MPC algorithm in section V. The simulation results are included in section VI. Finally, we conclude this paper in section VII.
\section{PROBLEM FORMULATION}
Consider a stochastic system with state $\mathbf{x}_t \in \mathbb{R}^n$ and control $\mathbf{u}_t \in \mathbb{R}^m$ at time t. We assume the dynamics also has additive noise from Brownian motion $\mathrm{d}\mathbf{w} \in \mathbb{R}^p$ and marked-jump process $\mathrm{d}\mathbf{P} \in\ \mathbb{R}^q$ with constant jump rate. We define $U \in \mathbb{R}^{m\times T}$ as the control sequence and $X \in \mathbb{R}^{n\times T}$ as the state trajectory over the time horizon $T$. We can formulate our stochastic optimal control problem as:
\begin{equation}
U^* = \arg\min_{U \in \mathcal{U}}\mathbb{E}_{\mathbb{Q}}\Big[\phi(\mathbf{x}_T,T)+\int^T_{t_0} \mathcal{L}(\mathbf{x}_t,\mathbf{u}_t,t)\mathrm{d}t \Big]
\end{equation}
where $\mathcal{U}$ is the set of admissible control sequences, and the expectation is taken with respect to the probability measure $\mathbb{Q}$ induced by the controlled dynamics:
\begin{equation}
\mathrm{d}\mathbf{x}_t = \mathbf{F}(\mathbf{x}_t,\mathbf{u}_t,t)\mathrm{d}t + \mathbf{B}(\mathbf{x}_t,t)\mathrm{d}\mathbf{w}^{(1)} + \mathbf{H}(\mathbf{x}_t,\mathcal{Q},t)\mathrm{d}\mathbf{P}^{(1)}
\end{equation}
with $\mathbb{E}[\mathrm{d}\mathbf{P}^{(1)}]=\nu^{(1)}\mathrm{d}t$ and $\nu^{(1)}$ is the jump rate. We assume zero mean normal distribution for the mark distribution, $\phi_{\mathcal{Q}}(q;t)\sim\mathcal{N}(0,\bm{\Sigma}_J)$. For the cost function we consider a state-dependent cost and a quadratic control cost:
\begin{equation}
\mathcal{L}(\mathbf{x}_t,\mathbf{u}_t,t) = q(\mathbf{x}_t,t) + \frac{1}{2} \mathbf{u}_t^\mathrm{T}\mathbf{R}(\mathbf{x}_t,t)\mathbf{u}_t
\end{equation}
We consider dynamics affine in control:
\begin{equation}
\mathbf{F}(\mathbf{x}_t,\mathbf{u}_t,t) = \mathbf{f}(\mathbf{x}_t,t) + \mathbf{G}(\mathbf{x}_t,t)\mathbf{u}_t
\end{equation}
The proof for existence and uniqueness of solution to the problem we are considering can be found in \cite{oksendal2005applied}.
\section{Information Theoretic Approach}
In this section we present the derivation of our sampling based stochastic trajectory optimization method for jump diffusion processes using an information theoretic approach.
Before we start the derivation we need to introduce two quantities from information theory that are the foundation to our derivation. First we define the \textit{Free Energy} of a system as:
\begin{equation}
\mathcal{F}(S(X)) = -\log\Big(\mathbb{E}_\mathbb{P}\Big[\exp\big(-\frac{1}{\lambda}S(X)\big)\Big]\Big)
\end{equation}
where $\lambda \in \mathbb{R}^+$ is called the inverse temperature, and $S(X)$ is the state-dependent cost of a trajectory, $S(X) = \phi(\mathbf{x}_T,T) + \int^T_{t_0} q(\mathbf{x}_t,t)\mathrm{d}t$. The expectation is taken with respect to $\mathbb{P}$, which is the probability measure induced by the uncontrolled dynamics:
\begin{equation}
\mathrm{d}\mathbf{x}_t = \mathbf{f}(\mathbf{x}_t,t)\mathrm{d}t + \mathbf{B}(\mathbf{x}_t,t)\mathrm{d}\mathbf{w}^{(0)} + \mathbf{H}(\mathbf{x}_t,\mathcal{Q},t)\mathrm{d}\mathbf{P}^{(0)}
\end{equation}
with $\mathbb{E}[\mathrm{d}\mathbb{P}^{(0)}] = \nu^{(0)}\mathrm{d}t$ and the same mark distribution as the controlled dynamics. Next let $\mathbb{M},\mathbb{N}$ be two probability distributions that are absolutely continuous with each other. Then the \textit{KL-Divergence} between them is:
\begin{equation}
\mathbb{D}_{KL}(\mathbb{M}\parallel\mathbb{N}) = \mathbb{E}_\mathbb{M}\Big[\log\Big(\frac{\mathrm{d}\mathbb{M}}{\mathrm{d}\mathbb{N}}\Big)\Big]
\end{equation}
The KL-Divergence provides a measure of how one probability distribution diverges from a second and can be roughly thought of as the distance between two probability distributions, although it is not symmetric. The KL-Divergence is useful for defining optimization objectives.
Now suppose probability distributions $\mathbb{Q}$ and $\mathbb{P}$ as defined previously are absolutely continuous with each other, we can make the following observation:
\begin{equation}
\begin{split}
\mathcal{F}(S(X)) &= -\log\Big(\mathbb{E}_\mathbb{P}\Big[\exp\big(-\frac{1}{\lambda}S(X)\big)\Big]\Big) \\
&= -\log\Big(\mathbb{E}_{\mathbb{Q}}\Big[\exp\big(-\frac{1}{\lambda}S(X)\big)\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} \Big] \Big)
\end{split}
\end{equation}
where we changed the expectation by multiplying by $1=\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{Q}}$. Since the negative logarithm is a convex function, we can apply Jensen's inequality and obtain:
\begin{equation}
\mathcal{F}(S(X)) \leq -\mathbb{E}_{\mathbb{Q}}\Big[\log\Big(\exp\big(-\frac{1}{\lambda}S(X)\big)\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} \Big) \Big]
\end{equation}
The right hand side can be simplified as:
\begin{equation}
\begin{split}
RHS&= \frac{1}{\lambda}\mathbb{E}_{\mathbb{Q}}\Big[S(X) + \lambda\log\Big(\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} \Big) \Big] \\
&= \frac{1}{\lambda} \Big(\mathbb{E}_{\mathbb{Q}}[S(X)] + \lambda \mathbb{D}_{KL}(\mathbb{Q}\parallel\mathbb{P}) \Big)
\end{split}
\end{equation}
Substituting the terms back to (9):
\begin{equation}
\lambda\mathcal{F}(S(X)) \leq \mathbb{E}_{\mathbb{Q}}[S(X)] + \lambda \mathbb{D}_{KL}(\mathbb{Q}\parallel\mathbb{P})
\end{equation}
To find the KL-Divergence between $\mathbb{Q}$ and $\mathbb{P}$, we need $\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}$, which can be found using Girsanov's theorem \cite{hanson2007applied}:
\begin{equation}
\begin{split}
\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}} &= \exp\Big(\frac{1}{2}\int^T_{t_0}\mathbf{u}_t^\mathrm{T}\mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{G}(\mathbf{x}_t,t)\mathbf{u}_t\mathrm{d}t \\
&+ \int^T_{t_0}\mathbf{u}_t^\mathrm{T}\mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{B}(\mathbf{x}_t,t)\mathrm{d}\mathbf{w}^{(1)}\\
&-\int^T_{t_0}((\gamma^J-1)\nu^{(0)})\mathrm{d}t\Big)\cdot\prod_{k=1}^{\mathbf{P}^{(0)}(t)}\gamma^J(T_k^-)\gamma^M(\mathcal{Q}_k,T_k^-)
\end{split}
\end{equation}
where $\Sigma(\mathbf{x}_t,t) = \mathbf{B}(\mathbf{x}_t,t)\mathbf{B}(\mathbf{x}_t,t)^T$, $\gamma^J(t)$ is the ratio of jump rates in the two dynamics, $\int_0^T\nu^{(1)}\mathrm{d}t=\int_0^T\gamma^J(t)\nu^{(0)}\mathrm{d}t$, and $\gamma^M(q;t)$ is the scaling between the mark distributions, $\int_{\mathcal{Q}_1}\phi_{\mathcal{Q}}^{(1)}(q;t)\mathrm{d}q=\int_{\mathcal{Q}_0}\gamma^M(q;t)\phi_{\mathcal{Q}}^{(0)}(q;t)\mathrm{d}q=1$.
Here we consider the case where the change of measure only includes changes in drift, and the jump rates and mark distributions are the same. Therefore, both $\gamma^J$ and $\gamma^M$ have the value 1, and the last two terms can be dropped. Additionally, since $\mathrm{d}\mathbf{w}^{(1)}$ is a Brownian motion with respect to $\mathbb{Q}$, we get $\mathbb{E}_{\mathbb{Q}}\Big[\int_0^T\mathrm{d}\mathbf{w}^{(1)}\Big]=0$. The KL-Divergence then simplifies to:
\begin{equation}
\begin{split}
&\mathbb{D}_{KL}(\mathbb{Q}\parallel\mathbb{P}) =\\
&\mathbb{E}_\mathbb{Q}\Big[\frac{1}{2}\int^T_{t_0}\mathbf{u}_t^\mathrm{T}\mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{G}(\mathbf{x}_t,t)\mathbf{u}_t\mathrm{d}t\Big]
\end{split}
\end{equation}
Using this result, if we assume the control cost matrix has the form:
\begin{equation}
\mathbf{R}(\mathbf{x}_t,t) = \lambda\mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{G}(\mathbf{x}_t,t)
\end{equation}
we get the following form on the right hand side of equation (11):
\begin{equation}
\begin{split}
\mathbb{E}_\mathbb{Q}[S(X)]&+\lambda \mathbb{D}_{KL}(\mathbb{Q}\parallel\mathbb{P}) =\\
&\mathbb{E}_\mathbb{Q}\Big[S(X)+\frac{1}{2}\int^T_{t_0}\mathbf{u}_t^\mathrm{T}\mathbf{R}(\mathbf{x}_t,t)\mathbf{u}_t\mathrm{d}t\Big]
\end{split}
\end{equation}
Note that this is equivalent to the cost function in (1). With this we have shown that the free energy serves as the lower bound for our stochastic optimal control problem, and we can rewrite (11) as a minimization problem:
\begin{equation}
\lambda\mathcal{F}(S(X)) = \inf_\mathbb{Q}\Big[\mathbb{E}_\mathbb{Q}[S(X)]+\lambda \mathbb{D}_{KL}(\mathbb{Q}\parallel\mathbb{P})\Big]
\end{equation}
In this minimization problem we have a state cost and a control cost in the form of KL-Divergence, which penalizes deviation from the uncontrolled distribution. We now define the optimal measure that achieves the lower bound as:
\begin{equation}
\frac{\mathrm{d}\mathbb{Q}^*}{\mathrm{d}\mathbb{P}} = \frac{\exp(-\frac{1}{\lambda}S(X))}{\mathbb{E}_\mathbb{P}[\exp(-\frac{1}{\lambda}S(X))]}
\end{equation}
This result can be easily verified by plugging it into (11) and is derived in [9]. With this we can solve the minimization problem defined by (16) by moving the probability distribution $\mathbb{Q}$ induced by some control as close to the optimal distribution as possible. The distance can be represented by the KL-Divergence between the two distributions and the problem becomes:
\begin{equation}
U^* = \arg\min_{U\in\mathcal{U}} \mathbb{D}_{KL}(\mathbb{Q}^*\parallel\mathbb{Q})
\end{equation}
\subsection{KL-Divergence Minimization}
Applying the definition of KL-Divergence we have:
\begin{equation}
\begin{split}
\mathbb{D}_{KL}(\mathbb{Q}^*\parallel\mathbb{Q}) &= \mathbb{E}_{\mathbb{Q}^*}\Big[\log\Big(\frac{\mathrm{d}\mathbb{Q}^*}{\mathrm{d}\mathbb{Q}}\Big)\Big] \\
&= \mathbb{E}_{\mathbb{Q}^*}\Big[\log\Big(\frac{\mathrm{d}\mathbb{Q}^*}{\mathrm{d}\mathbb{P}}\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\Big)\Big]
\end{split}
\end{equation}
We already have $\frac{\mathrm{d}\mathbb{Q}^*}{\mathrm{d}\mathbb{P}}$ from its definition. For $\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}$, we can use Girsanov's theorem:
\begin{equation}
\begin{split}
\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} &= \exp\Big(\frac{1}{2}\int^T_{t_0}\mathbf{u}_t^\mathrm{T}\mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{G}(\mathbf{x}_t,t)\mathbf{u}_t\mathrm{d}t \\
&- \int^T_{t_0}\mathbf{u}_t^\mathrm{T}\mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{B}(\mathbf{x}_t,t)\mathrm{d}\mathbf{w}^{(0)}\Big)
\end{split}
\end{equation}
Setting the terms inside the exponential as $\mathcal{D}(X,U)$ and plugging the results back in (19) we have:
\begin{equation}
\begin{split}
&\mathbb{D}_{KL}(\mathbb{Q}^*\parallel\mathbb{Q}) =\\
& \mathbb{E}_{\mathbb{Q}^*}\Big[-\frac{1}{\lambda}S(X)-\log(\mathbb{E}_\mathbb{P}[\exp(-\frac{1}{\lambda}S(X))])+\mathcal{D}(X,U)\Big]
\end{split}
\end{equation}
Since $S(X)$ is not dependent on the control we can drop the first two terms from the minimization. Now we discretize the control as step functions $\mathbf{u}_t=\mathbf{u}_j$ if $j\Delta t\leq t < (j+1)\Delta t$ with $j=\{0,1,\cdots,N-1\}$. Then we have:
\begin{equation}
\begin{split}
&\mathcal{D}(X,U) =\\
&\sum_{j=0}^{N-1}\Bigg(\frac{1}{2}\mathbf{u}_j^\mathrm{T}\int^{t_{j+1}}_{t_j} \mathcal{G}(\mathbf{x}_t,t)\mathrm{d}t \mathbf{u}_j - \mathbf{u}_j^\mathrm{T}\int^{t_{j+1}}_{t_j} \mathcal{B}(\mathbf{x}_t,t) \mathrm{d}\mathbf{w}^{(0)}\Bigg)
\end{split}
\end{equation}
where
\begin{equation}
\mathcal{G}(\mathbf{x}_t,t) = \mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{G}(\mathbf{x}_t,t)
\end{equation}
\begin{equation}
\mathcal{B}(\mathbf{x}_t,t) = \mathbf{G}(\mathbf{x}_t,t)^\mathrm{T}\Sigma(\mathbf{x}_t,t)^{-1}\mathbf{B}(\mathbf{x}_t,t)
\end{equation}
\begin{equation}
N = T/\Delta t
\end{equation}
Note that each $\mathbf{u}_j$ does not depend on the trajectory taken, so we can taken them out of the expectation:
\begin{equation}
\begin{split}
\mathbb{E}_{\mathbb{Q}^*}\Big[\mathcal{D}(X,U)\Big] &= \sum_{j=0}^{N-1}\Bigg(\frac{1}{2}\mathbf{u}_j^\mathrm{T}\mathbb{E}_{\mathbb{Q}^*}\Big[\int^{t_{j+1}}_{t_j} \mathcal{G}(\mathbf{x}_t,t)\mathrm{d}t\Big] \mathbf{u}_j \\
&- \mathbf{u}_j^\mathrm{T}\mathbb{E}_{\mathbb{Q}^*}[\int^{t_{j+1}}_{t_j} \mathcal{B}(\mathbf{x}_t,t) \mathrm{d}\mathbf{w}^{(0)}]\Bigg)
\end{split}
\end{equation}
We can approximate the two integrals for small enough $\Delta t$ as:
\begin{equation}
\int^{t_{j+1}}_{t_j} \mathcal{G}(\mathbf{x}_t,t)\mathrm{d}t \approx \mathcal{G}(\mathbf{x}_{t_j},t_j) \Delta t
\end{equation}
\begin{equation}
\int^{t_{j+1}}_{t_j} \mathcal{B}(\mathbf{x}_t,t) \mathrm{d}\mathbf{w}^{(0)} \approx \mathcal{B}(\mathbf{x}_{t_j},t_j) \epsilon^{(0)}_j\sqrt{\Delta t}
\end{equation}
where $\epsilon^{(0)}_j$ is a vector with standard normal variable in each entry, $\epsilon^{(0)}_j\sim\mathcal{N}(0,\bm{\Sigma}_D)$. Then we can find $\mathbf{u}_j^*$ by taking the gradient with respect to $\mathbf{u}_j$, setting it to zero and solving for $\mathbf{u}_j$. The optimal control is found as:
\begin{equation}
\mathbf{u}^*_j = \frac{1}{\Delta t} \mathbb{E}_{\mathbb{Q}^*}\Big[\mathcal{G}(\mathbf{x}_{t_j},t_j)\Big]^{-1}\mathbb{E}_{\mathbb{Q}^*}\Big[\mathcal{B}(\mathbf{x}_{t_j},t_j)\epsilon^{(0)}_j\sqrt{\Delta t}\Big]
\end{equation}
\subsection{Importance Sampling}
We have obtained the optimal control in the form of expectation with respect to the optimal distribution. We can't sample from the optimal distribution, but we can sample from the uncontrolled distribution $\mathbb{P}$ to approximate the controls. Therefore, we need to change the expectation through multiplying by $\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{P}}$ and using the Radon-Nikodym derivative $\frac{\mathrm{d}\mathbb{Q}^*}{\mathrm{d}\mathbb{P}}$:
\begin{equation}
\begin{split}
\mathbf{u}^*_j &= \frac{1}{\Delta t} \mathbb{E}_\mathbb{P}\Big[\frac{\exp(-\frac{1}{\lambda}S(X))\mathcal{G}(\mathbf{x}_{t_j},t_j)}{\mathbb{E}_\mathbb{P}[\exp(-\frac{1}{\lambda}S(X))]}\Big]^{-1}\\
&\cdot\mathbb{E}_\mathbb{P}\Big[\frac{\exp(-\frac{1}{\lambda}S(X)) \mathcal{B}(\mathbf{x}_{t_j},t_j)\epsilon^{(0)}_j\sqrt{\Delta t}}{\mathbb{E}_\mathbb{P}[\exp(-\frac{1}{\lambda}S(X))]}\Big]
\end{split}
\end{equation}
The equation can be further simplified since $\mathcal{G}(\mathbf{x}_{t_j},t_j)$ and $\mathcal{B}(\mathbf{x}_{t_j},t_j)$ are deterministic at time $t_j$:
\begin{equation}
\begin{split}
&\mathbf{u}^*_j =\\
&\frac{1}{\Delta t}\mathcal{G}(\mathbf{x}_{t_j},t_j)^{-1} \mathcal{B}(\mathbf{x}_{t_j},t_j) \mathbb{E}_\mathbb{P}\Big[\frac{\exp(-\frac{1}{\lambda}S(X)) \epsilon^{(0)}_j\sqrt{\Delta t}}{\mathbb{E}_\mathbb{P}[\exp(-\frac{1}{\lambda}S(X))]}\Big]
\end{split}
\end{equation}
Note that the expectations are taken with respect to the uncontrolled dynamics. This is not ideal since it means waiting for random Gaussian and jump noise to generate a meaningful trajectory. Therefore, we need to change the sampling distribution to the control induced distribution. In addition, we can also change the sampling variance to $\bm{\Sigma}_D^{(1)} = c \bm{\Sigma}_D^{(0)}$ to increase the state space explored. To perform importance sampling we multiply by $\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{Q}}$ and change from the zero mean $\epsilon^{(0)}_j \sqrt{\Delta t}$ to the non zero mean $\mathbf{G}(\mathbf{x})\mathbf{u}_j\Delta t + \epsilon^{(1)}_j \sqrt{\Delta t}$:
\begin{equation}
\begin{split}
\mathbf{u}^*_j &= \mathbf{u}_j + \frac{1}{\Delta t}\mathcal{G}(\mathbf{x}_{t_j},t_j)^{-1} \mathcal{B}(\mathbf{x}_{t_j},t_j)\\
&\cdot \mathbb{E}_\mathbb{Q}\Big[\frac{\exp(-\frac{1}{\lambda}S(X)) \epsilon^{(1)}_j\sqrt{\Delta t}\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}}{\mathbb{E}_\mathbb{Q}[\exp(-\frac{1}{\lambda}S(X))\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}]}\Big]
\end{split}
\end{equation}
We can use Girsanov's theorem again to get $\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}$:
\begin{equation}
\begin{split}
\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} &= \exp\Bigg(-\frac{1}{2} \sum_{j=0}^{N-1}\Big(\mathbf{u}_j^\mathrm{T}\mathbf{G}(\mathbf{x}_{t_j})^\mathrm{T}\Sigma^{-1}\mathbf{G}(\mathbf{x}_{t_j})\mathbf{u}_j \Delta t \\
&+ \mathbf{u}_j^\mathrm{T}\mathbf{G}(\mathbf{x}_{t_j})^\mathrm{T}\Sigma^{-1}\mathbf{B}(\mathbf{x}_{t_j})\epsilon_j^{(1)}\sqrt{\Delta t}\\
&+ (1-c^{-1})\epsilon_j^{(1)\mathrm{T}}\mathbf{B}(\mathbf{x}_{t_j})^\mathrm{T}\Sigma^{-1}\mathbf{B}(\mathbf{x}_{t_j})\epsilon_j^{(1)} \Delta t \Big)\Bigg)
\end{split}
\end{equation}
The last terms comes from the change of sampling variance and the detailed derivation can be found in \cite{williams2015model}. The addition of these terms can be added into the state cost:
\begin{equation}
\tilde{S}(X) = \phi(\mathbf{x}_{t_N}, t_N) + \sum_{j=0}^{N-1} \tilde{q}(\mathbf{x}_{t_j},\mathbf{u}_j,t_j) \Delta t
\end{equation}
\begin{equation}
\begin{split}
&\tilde{q}(\mathbf{x}_{t_j},\mathbf{u}_j,t_j) = q(\mathbf{x}_{t_j},t_j) + \frac{1}{2} \mathbf{u}_j^\mathrm{T}\mathbf{R}\mathbf{u}_j + \lambda \mathbf{u}_j^\mathrm{T}\mathcal{B}\frac{\epsilon_j^{(1)}}{\sqrt{\Delta t}} \\
&+ \frac{1}{2} \lambda (1-c^{-1})\epsilon_j^{(1)\mathrm{T}}\mathbf{B}(\mathbf{x}_{t_j})^\mathrm{T}\Sigma^{-1}\mathbf{B}(\mathbf{x}_{t_j})\epsilon_j^{(1)}/\Delta t
\end{split}
\end{equation}
With the new state cost we can obtain the final expression of optimal control update rule:
\begin{equation}
\mathbf{u}^*_j = \mathbf{u}_j + \mathcal{G}(\mathbf{x}_{t_j},t_j)^{-1}\mathcal{B}(\mathbf{x}_{t_j},t_j) \Big(\frac{\mathbb{E}_\mathbb{Q}[\exp(-\frac{1}{\lambda}\tilde{S}(X)) \frac{\epsilon^{(1)}_j}{\sqrt{\Delta t}}]}{\mathbb{E}_\mathbb{Q}[\exp(-\frac{1}{\lambda}\tilde{S}(X))]} \Big)
\end{equation}
The term inside the square brackets is approximated as:
\begin{equation}
\frac{\sum_{m=1}^M\exp(-\frac{1}{\lambda}\tilde{S}(X^m)) \frac{\epsilon_j^m}{\sqrt{\Delta t}}}{\sum_{m=1}^M\exp(-\frac{1}{\lambda}\tilde{S}(X^m))}
\end{equation}
using $M$ sample trajectories.
\section{Stochastic Optimization Approach}
\subsection{Problem Reformulation}
Consider a system with the same definition of state, control and dynamics as in the previous section, the optimal control problem can be defined in the same way:
\begin{equation}
U^* = \arg\min_{U\in\mathcal{U}}\mathbb{E}[J(X,U)]\label{eq:1}
\end{equation}
where $J:\mathbb{R}^{n \times T} \times \mathbb{R}^{m \times T}\rightarrow\mathbb{R}$ is an arbitrary cost function. We can introduce an exponential shape function $L(y)=\exp(y)$ to redefine the optimal control problem as a maximization problem:
\begin{equation}
U^* = \arg\max_{U\in\mathcal{U}}\mathbb{E}\Big[L\Big(-\frac{1}{\lambda} J(X,U)\Big)\Big]
\end{equation}
The expectation is taken over the control policy, which is parameterized by a set of parameters $\theta\in\Theta$ that we have control over. Finally, since $\ln:\mathbb{R}^+\rightarrow\mathbb{R}$ is a strictly increasing function, reformulating the maximization problem does not change the solution to the original optimal control problem:
\begin{equation}
\begin{split}
\theta^* &= \arg\max_{\theta\in\Theta}\ln\Big(\mathbb{E}\Big[L\Big(-\frac{1}{\lambda} J(X,U)\Big)\Big]\Big)\\
&= \arg\max_{\theta\in\Theta} l(\theta)
\end{split}
\end{equation}
\subsection{Probability Distribution Parameterization}
Assume a time discretization of control policy with step functions, and the stochastic control policy at each time instant has additive Gaussian and jump noise around some mean, $\mathbf{u}_j=\bm{\mu}_j+\epsilon_{D,j}+\epsilon_{J,j}\Delta\mathbf{P}$. We have the diffusion term $\epsilon_{D,j}\sim\mathcal{N}(0,\bm{\Sigma}_D)$ and the jump term $\epsilon_{J,j}\sim\mathcal{N}(0,\bm{\Sigma}_J)$ with known variances, and $\mathbb{E}[\Delta P]=\nu\Delta t$ with known jump rate $\nu$. Assuming stochasticity enters the system through the control channels, the probability density/mass function of each trajectory can be expressed as $p(X,U;\theta)=p(U;\theta)$ since the dynamics is deterministic. Assume the noise at each time instant is i.i.d., then the pdf/pmf of each trajectory can be expressed as:
\begin{equation}
p(U;\theta) = \prod^{N-1}_{j=0} p(\mathbf{u}_j;\theta_j)
\end{equation}
Although there is no closed form expression for the pdf/pmf of the entire trajectory, the pdf/pmf at each time instant can be written out explicitly. For small enough $\Delta t$ such that $\nu\Delta t \ll 1$, the zero-one jump law \cite{hanson2007applied} applies and the jump noise has Bernoulli distribution with probability of jump being $\nu\Delta t$. In addition, when jump occurs, $\mathbf{u}_j\sim\mathcal{N}(\bm{\mu}_j,\bm{\Sigma}_D+\bm{\Sigma}_J)$ is normally distributed with the variance as the sum of diffusion and jump noise since the sum of two normally distributed random variable is still normal. Therefore the pdf/pmf can be expressed as:
\begin{equation}
\begin{split}
p(\mathbf{u}_j;\theta_j) &= I_j(\nu\Delta t)\Bigg(\frac{1}{\sqrt{(2\pi)^n|\bm{\Sigma_D}+\bm{\Sigma_J}|}}\\
&\exp\Big(-\frac{1}{2}(\mathbf{u}_j-\bm{\mu}_j)^\mathrm{T}(\bm{\Sigma}_D+\bm{\Sigma}_J)^{-1}(\mathbf{u}_j-\bm{\mu}_j)\Big)\Bigg)\\
&+ (1-I_j)(1-\nu\Delta t)\Bigg(\frac{1}{\sqrt{(2\pi)^n|\bm{\Sigma_D}|}}\\
&\exp\Big(-\frac{1}{2}(\mathbf{u}_j-\bm{\mu}_j)^\mathrm{T}\bm{\Sigma}_D^{-1}(\mathbf{u}_j-\bm{\mu}_j)\Big)\Bigg)\\
&= h(\mathbf{u}_j)\exp\Big(\theta_j^\mathrm{T}T(\mathbf{u}_j)-A(\theta_j)\Big)
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
h(\mathbf{u}_j) &= I_j(\nu\Delta t)\Bigg(\frac{1}{\sqrt{(2\pi)^n|\bm{\Sigma}_D+\bm{\Sigma}_J|}}\\
&\exp\Big(-\frac{1}{2}\mathbf{u}_j^\mathrm{T}(\bm{\Sigma}_D+\bm{\Sigma}_J)^{-1}\mathbf{u}_j\Big)\Bigg) \\
&+ (1-I_j)(1-\nu\Delta t)\Bigg(\frac{1}{\sqrt{(2\pi)^n|\bm{\Sigma}_D|}}\\
&\exp\Big(-\frac{1}{2}\mathbf{u}_j^\mathrm{T}\bm{\Sigma}_D^{-1}\mathbf{u}_j\Big)\Bigg)\\
A(\theta_j) &= \frac{1}{2} \bm{\mu}_j^\mathrm{T}(\bm{\Sigma}_D+I_j\bm{\Sigma}_J)^{-1}\bm{\mu}_j \\
T(\mathbf{u}_j) &= (\bm{\Sigma}_D+I_j\bm{\Sigma}_J)^{-\frac{1}{2}} \mathbf{u}_j \\
\theta_j &= (\bm{\Sigma}_D+I_j\bm{\Sigma}_J)^{-\frac{1}{2}}\bm{\mu}_j
\end{split}
\end{equation}
The term $I_j$ term is an indicator function with $I_j=1$ when jump occurs and $I_j=0$ when there is no jump.
\subsection{Gradient Descent}
Now the gradient can be taken for the step direction at each time step j:
\begin{equation}
\begin{split}
\nabla_{\theta_j} l(\theta) &= \frac{\int L(-\frac{1}{\lambda}J(X,U))\nabla_{\theta_j} p(U;\theta)\mathrm{d}U}{\int L(-\frac{1}{\lambda}J(X,U))p(U;\theta)\mathrm{d}U} \\
&= \frac{\int L(-\frac{1}{\lambda}J(X,U))p(U;\theta) \nabla_{\theta_j} \ln p(U;\theta)\mathrm{d}U}{\int L(-\frac{1}{\lambda}J(X,U))p(U;\theta)\mathrm{d}U}
\end{split}
\end{equation}
The term $\nabla_{\theta_j} \ln p(U;\theta)$ can be calculated as:
\begin{equation}
\begin{split}
\nabla_{\theta_j} \ln p(U;\theta) &= \nabla_{\theta_j} \sum_{j=0}^{N-1}\big(\ln h(\mathbf{u}_j) + \theta_j^\mathrm{T}T(\mathbf{u}_j)-\frac{1}{2} \theta_j^\mathrm{T}\theta_j \big) \\
&= T(\mathbf{u}_j) - \theta_j
\end{split}
\end{equation}
Plug it back into the gradient we have:
\begin{equation}
\nabla_{\theta_j} l(\theta) = \frac{\int L(-\frac{1}{\lambda}J(X,U))p(U;\theta) (T(\mathbf{u}_j) - \theta_j) \mathrm{d}U}{\int L(-\frac{1}{\lambda}J(X,U))p(U;\theta)\mathrm{d}U}
\end{equation}
With the gradient the update rule for $\theta$ can be found as:
\begin{equation}
\begin{split}
\theta_j^{k+1} &= \theta_j^k + \\
&\alpha^k\Bigg(\frac{\int L(-\frac{1}{\lambda}J(X,U))p(U;\theta) (T(\mathbf{u}_j) - \theta_j^k) \mathrm{d}U}{\int L(-\frac{1}{\lambda}J(X,U))p(U;\theta)\mathrm{d}U}\Bigg) \\
&= \theta_j^k + \alpha^k\Bigg(\frac{\mathbb{E}\Big[L(-\frac{1}{\lambda}J(X,U)) (T(\mathbf{u}_j) - \theta_j^k) \Big]}{\mathbb{E} \Big[L(-\frac{1}{\lambda}J(X,U))\Big]}\Bigg)
\end{split}
\end{equation}
Then $\bm{\mu}_j$ can be substituted in for $\theta_j$:
\begin{equation}
\begin{split}
&(\bm{\Sigma}_D+I_j\bm{\Sigma}_J)^{-\frac{1}{2}}\bm{\mu}_j^{k+1} = (\bm{\Sigma}_D+I_j\bm{\Sigma}_J)^{-\frac{1}{2}}\bm{\mu}_j^k \\
&+ \alpha^k\Bigg(\frac{\mathbb{E}\Big[L(-\frac{1}{\lambda}J(X,U)) (\bm{\Sigma}_D+I_j\bm{\Sigma}_J)^{-\frac{1}{2}} (\mathbf{u}_j - \bm{\mu}_j^k) \Big]}{\mathbb{E} \Big[L(-\frac{1}{\lambda}J(X,U))\Big]}\Bigg) \\
\end{split}
\end{equation}
The final update law can be obtained as:
\begin{equation}
\begin{split}
\bm{\mu}_j^{k+1} &= \bm{\mu}_j^k + \alpha^k\Bigg(\frac{\mathbb{E}\Big[L(-\frac{1}{\lambda}J(X,U)) (\mathbf{u}_j - \bm{\mu}_j^k) \Big]}{\mathbb{E} \Big[L(-\frac{1}{\lambda}J(X,U))\Big]}\Bigg) \\
&= \bm{\mu}_j^k + \alpha^k\Bigg(\frac{\mathbb{E}\Big[\exp(-\frac{1}{\lambda}J(X,U)) (\epsilon_{D,j}^k+I_j\epsilon_{J,j}^k) \Big]}{\mathbb{E} \Big[\exp(-\frac{1}{\lambda}J(X,U))\Big]}\Bigg)
\end{split}
\end{equation}
The update law (49) is very close to the one (36) obtained from the information theoretic approach. In the case of noise entering the system through control channels only, $\mathbf{B}$ and $\mathbf{H}$ matrices are the same as $\mathbf{G}$, and the matrix transform $\mathcal{G}^{-1}\mathcal{B}$, which maps from state space to control space, goes to identity. Taking step size $\alpha=1$, the two control laws differ only in the extra terms resulted from importance sampling.
With this alternative derivation, convergence of the update law can be shown using the ODE method as described in \cite{zhou2014gradient}.
\begin{algorithm} [!hb]
\caption{MPPI Control on Jump Diffusion}
\begin{algorithmic}
\STATE $\textbf{Given:}$
\STATE $\text{M: Number of samples;}$
\STATE $\text{N: Number of timesteps;}$
\STATE $(\mathbf{u}_0,\mathbf{u}_1,\cdots,\mathbf{u}_{N-1}) \text{: Initial control sequence;}$
\STATE $\mathbf{x}_0 \text{: Initial states;}$
\STATE $\Delta t, \mathbf{f}, \mathbf{G}, \mathbf{B}, \mathbf{H} \text{: System/sampling dynamics;}$
\STATE $\phi, q, \lambda, \mathbf{R} \text{: Cost function parameters;}$
\STATE $c,\bm{\Sigma}_D,\bm{\Sigma}_J, \nu\text{: Noise parameters}$
\STATE $\mathbf{u}_{init} \text{: Value for new control initialization;}$
\WHILE {\textit{task not completed}}
\FOR {$m=0$ \text{to} $M-1$}
\STATE $\text{Update } \mathbf{x}_0 \text{;}$
\STATE $\text{Sample } \mathbf{\varepsilon}^m=\big(\epsilon_0^m,\cdots,\epsilon_{N-1}^m\big), \epsilon_i^m \in \mathcal{N}(0,c \bm{\Sigma}_D);$
\FOR {$i=0$ \text{to} $N-1$}
\STATE $p=U(0,1);$
\IF {$p< \nu \Delta t$}
\STATE $\epsilon_j=\mathcal{N}(0,\bm{\Sigma}_J)$
\STATE $\mathbf{x}_{i+1} = \mathbf{x}_i + (\mathbf{f}+\mathbf{G}\mathbf{u}_i)\Delta t + \mathbf{B} \epsilon_i^m\sqrt{\Delta t} + \mathbf{H}\epsilon_j\sqrt{\Delta t};$
\STATE $\epsilon_i^m = \epsilon_i^m + \epsilon_j;$
\ELSE
\STATE $\mathbf{x}_{i+1} = \mathbf{x}_i + (\mathbf{f}+\mathbf{G}\mathbf{u}_i)\Delta t + \mathbf{B} \epsilon_i^m\sqrt{\Delta t};$
\ENDIF
\STATE $\tilde{S}(X^m) = \tilde{S}(X^m) + \tilde{q}(\mathbf{x}_i,\mathbf{u}_i,\epsilon_i^m);$
\ENDFOR
\ENDFOR
\FOR {$i=0$ to $N-1$}
\STATE $\mathbf{u}_i = \mathbf{u}_i + \frac{\sum_{m=1}^M\exp{\big(-\frac{1}{\lambda}\tilde{S}(X^m)\big)}\frac{\epsilon_i^m}{\sqrt{\Delta t}}}{\sum_{m=1}^M\exp{\big(-\frac{1}{\lambda}\tilde{S}(X^m)\big)}};$
\ENDFOR
\STATE $\text{Execute control policy }\mathbf{u}_0;$
\FOR {$i=0$ to $N-2$}
\STATE $\mathbf{u}_i = \mathbf{u}_{i+1};$
\ENDFOR
\STATE $\mathbf{u}_{N-1} = \mathbf{u}_{init};$
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\section{Model Predictive Control Algorithm}
From both approaches, we get an iterative update law for the optimal control policy at each timestep. This allows for the algorithm to be implemented in a MPC fashion. In the MPC setting, after the optimal control sequence is obtained, only the first control action is executed and re-optimization occurs from the new initial states. Since an entire optimal control sequence is given at every timestep, we can keep the un-executed control sequence to warm start optimization for the next iteration. This is very important for increasing the performance of the algorithm as we are reusing information from previous optimization iterations.
Another key aspect of the algorithm is that its computationally involved parts, namely trajectory propagation and cost computation for each sampled trajectory, can be done in parallel on a GPU. Parallel computation allows us to sample thousands of trajectories at the same time rather than in sequence, with the computation time for each iteration of less than 20 miliseconds in our simulation, which satisfies the real time requirement for a 50 Hz controller. The description of the new Model Predictive Path Integral (MPPI) algorithm is given in Algorithm 1.
The algorithm is based on the assumption that both Gaussian and jump noise affect the states through the control channels. Jump noise is simulated using the zero-one jump law, which states that if $\nu\Delta t \ll 1$, the probability of more than one jump occuring at each timestep can be neglected. A jump timer $p$ is sampled from a uniform distribution to check whether jump occurs at each timestep. When a jump occurs, a zero mean Gaussian vector determines the magnitude of jump noise in each control channel.
\section{Simulation Results}
We compare the MPPI algorithm for jump diffusion processes against the old MPPI algorithm in \cite{williams2016aggressive} that doesn't account for jump noise on a cart pole and quadrotor in simulation with artificial Gaussian and jump noise. To avoid confusion, we refer to the algorithm presented in this paper as the \textcolor{green}{new MPPI} algorithm and the algorithm without jump noise in sampling as the \textcolor{blue}{old MPPI} algorithm. In the trajectory plots the mean trajectory and 95\% confidence interval are plotted. Note that the confidence intervals are not labeled explicitly but shaded with the same color as mean trajectories. The red line indicates the target state.
\subsection{Cart Pole}
We applied the new and old MPPI algorithm on a standard cart pole system in simulation. The task is to swing up and stabilize the cart pole. We used 1000 trajectories during sampling and ran each algorithm for 100 trials. We tested the robustness of both algorithms by varying the jump amplitude and rate while keeping Gaussian noise the same. In Table. \ref{table:1}, we demonstrate the simulation results. The new MPPI algorithm has a higher success rate in stabilizing the cart pole. Specifically, with only small jump noise, both algorithms managed to balance the cart pole. As the jump amplitude increased, both algorithms started to fail, but the new algorithm has a higher success rate of stabilizing the pole than the old algorithm. For a fixed jump amplitude, increasing the jump rate results in lower success rates in both algorithms and vice versa.
Fig. \ref{figure:1} demonstrates the responses of both algorithms in a trial when the old MPPI algorithm failed. The pole angle plot and the Poisson noise plot show that the old MPPI algorithm failed after a noise spike and had to restabilize the pole. On the other hand, the new algorithm experienced noise spikes of similar magnitude and maintained balance. The cart position plot shows that the new algorithm managed to maintain balance efficiently around the origin.
\begin{table} [!ht]
\hspace{-0.5cm}
\caption{Success rates of \textcolor{blue}{old} and \textcolor{green}{new} algorithm on a cart pole}
\label{table:1}
\begin{center}
\begin{tabular}{l*{2}{c}}
Jump noise & \color{green}New MPPI & \color{blue}Old MPPI \\
\hline
$\nu=0.25,\bm{\Sigma}_J=1$ & \color{green}100\% & \color{blue}100\% \\
$\nu=0.25,\bm{\Sigma}_J=1.5$ & \color{green}96\% & \color{blue}91\% \\
$\nu=0.25,\bm{\Sigma}_J=2$ & \color{green}96\% & \color{blue}81\% \\
$\nu=0.25,\bm{\Sigma}_J=3$ & \color{green}88\% & \color{blue}61\% \\
$\nu=0.1, \bm{\Sigma}_J=2$ & \color{green}97\% & \color{blue}92\% \\
$\nu=0.5, \bm{\Sigma}_J=2$ & \color{green}91\% & \color{blue}73\%
\end{tabular}
\end{center}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=8.75 cm]{CartPole_Comparison_SingleTraj.png}
\caption{Comparison of \textcolor{blue}{old} and \textcolor{green}{new} algorithm on a cartpole. \textit{Top left: cart position; Top right: pole angle; Botton left: Gaussian noise; Botton right: Poisson noise}.}
\label{figure:1}
\vspace{-0.5cm}
\end{figure}
\subsection{Quadrotor}
We also applied both algorithms on a quadrotor system in simulation. The task is to fly from an initial position to a target position. Since it is a more complex system we increased the number of sampling trajectories to 3000 and ran each algorithm for 100 trials. Again we varied the jump amplitude and rate while keeping Gaussian noise the same. Table. \ref{table:2} lists the simulation results. Similar to the cartpole simulation, we found that the new MPPI algorithm has a higher success rate in completing the task. Specifically, with only small jump noise, both algorithms could carry out the task perfectly. As we increased the jump noise amplitude, the failure rate of the old MPPI algorithm increased while the new algorithm maintained perfect task completion rate. Additionally, for a jump amplitude large enough that the old MPPI algorithm has a non zero failure rate, increasing the jump rate further increases the failure rate of the old MPPI algorithm and vice versa.
\begin{table} [t]
\hspace{-0.5cm}
\caption{Success rates of \textcolor{blue}{old} and \textcolor{green}{new} algorithm on a quadrotor}
\label{table:2}
\begin{center}
\begin{tabular}{l*{2}{c}}
Jump noise & \color{green}New MPPI & \color{blue}Old MPPI \\
\hline
$\nu=0.2,\bm{\Sigma}_J=5$ & \color{green}100\% & \color{blue}100\% \\
$\nu=0.2,\bm{\Sigma}_J=10$ & \color{green}100\% & \color{blue}98\% \\
$\nu=0.2,\bm{\Sigma}_J=20$ & \color{green}100\% & \color{blue}97\% \\
$\nu=0.2,\bm{\Sigma}_J=30$ & \color{green}100\% & \color{blue}87\% \\
$\nu=0.1, \bm{\Sigma}_J=20$ & \color{green}100\% & \color{blue}98\% \\
$\nu=0.5, \bm{\Sigma}_J=20$ & \color{green}100\% & \color{blue}91\%
\end{tabular}
\end{center}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=8.75 cm]{Quad_plot_0.png}
\caption{Comparison of \textcolor{blue}{old} and \textcolor{green}{new} algorithm on a quadrotor with 3000 trajectories in sampling. \textit{Top left: x position; Top right: y position; Botton left: z position; Botton right: roll angle}.}
\label{figure:2}
\vspace{-0.5cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=8.75 cm]{Quad_plot_1.png}
\caption{Comparison of \textcolor{blue}{old} and \textcolor{green}{new} algorithm on a quadrotor with 3000 trajectories in sampling. \textit{Top left: x velocity; Top right: y velocity; Botton left: z velocity; Botton right: roll rate}.}
\label{figure:3}
\end{figure}
In Fig. \ref{figure:2} and \ref{figure:3}, we compare the response of the two algorithms for one test case ($\nu=0.2,\bm{\Sigma}_J=20$). The plots include the mean and 95\% confidence region of the responses. The x and y position plots show that the mean of trajectories resulted from both algorithms follow a similar path to the target, but the variance of trajectories resulted from the old MPPI algorithm is much larger. From the z position plot, we observe that the variance of trajectories resulted from the old MPPI algorithm is significantly larger than the new MPPI algorithm since there are three crash runs. From the x and y velocity plots we find distinct areas where the variance of both algorithms increase. These areas correspond to the high variance regions of the position plots.
We also took two test cases ($\nu=0.2,\bm{\Sigma}_J=5$ and $\nu=0.2,\bm{\Sigma}_J=20$) and ran both algorithms with 6000 sampling trajectories. Fig. \ref{figure:4} and \ref{figure:5} show the response of both algorithms with high jump noise amplitude. Doubling the sampling trajectories resulted in one less crash run for the old MPPI algorithm, while the new MPPI algorithm maintained perfect success rate. From the x and y position plots, we observe that the variance for both algorithms are smaller than the case with fewer sampling trajectories. There is one region in the z position plot where the variance for the old MPPI algorithm increases significantly due to the crash runs. The results suggest that increasing the number of sampling trajectories correspond to a decrease in variance in generated trajectories. The decrease is resulted from better approximation of the expectation with more samples.
\begin{figure}[t]
\centering
\includegraphics[width=8.75 cm]{Quad_plot_high0.png}
\caption{Comparison of \textcolor{blue}{old} and \textcolor{green}{new} algorithm on a quadrotor with 6000 trajectories in sampling. \textit{Top left: x position; Top right: y position; Botton left: z position; Botton right: roll angle}.}
\label{figure:4}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=8.75 cm]{Quad_plot_high1.png}
\caption{Comparison of \textcolor{blue}{old} and \textcolor{green}{new} algorithm on a quadrotor with 3000 trajectories in sampling. \textit{Top left: x velocity; Top right: y velocity; Botton left: z velocity; Botton right: roll rate}.}
\label{figure:5}
\end{figure}
In Fig. \ref{figure:6} we compare the total variance (sum of variance in all states over the entire time horizon for all trajectories) of both algorithms with two jump noise levels using 3000 and 6000 sampling trajectories. We find that with low jump noise amplitude, the old MPPI algorithm results in slightly lower variance than the new MPPI algorithm. The new MPPI algorithm tends to generate trajectories that oscillate around the target location since the dynamics is perturbed more during sampling. For the case of high jump noise amplitude, the difference in variance between the two algorithms is significantly reduced with increased sampling trajectories. This is due to the benefit of better exploration by including jump noise is reduced with increased sampling trajectories.
\begin{figure}[t]
\centering
\includegraphics[width=8.75 cm]{variance.png}
\caption{Comparison of total variance of trajectories resulted from \textcolor{blue}{old} and \textcolor{green}{new} algorithm under high ($\bm{\Sigma}_J$=20) and low ($\bm{\Sigma}_J$=5) levels of jump noise amplitude with 3000 and 6000 trajectories in sampling ($\nu$=0.2).}
\label{figure:6}
\end{figure}
\section{CONCLUSIONS}
We presented an information theoretic model predictive control algorithm that obtains the optimal control through sampling with Gaussian and compound Poisson noise. We provided an alternative stochastic optimization derivation, from which convergence of the control update law can be proven. We applied the algorithm on cart pole and quadrotor systems with artificially introduced compound Poisson noise and compared its performance to the previously developed algorithm that doesn't include compound Poisson noise in sampling. We demonstrated superior performance of our new algorithm than the old algorithm under large Poisson noise level and comparable performance under low Poisson noise level. Our results suggest that it is important to consider the statistical characteristics of stochastic disturbances in the computation of the optimal control policies.
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\bibliographystyle{unsrt}
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1,314,259,996,684 | arxiv |
\section{Introduction}
\subsection{Computing with Determinants}
The \emph{determinantal complexity} of a polynomial $f \in \mathbb{F}[x_1, \ldots, x_n]$, denoted $\dc(f)$, is the minimal integer $m$ such that there exists an affine map $L : \mathbb{F}^n \to \mathbb{F}^{m \times m}$ such that $f(\mathbf{x}) = \mathsf{Det}(L(\mathbf{x}))$, where for every square matrix $M$, $\mathsf{Det}(M)$ denotes the determinant of $M$.
This notion was first implicitly defined by Valiant \cite{V79}, and it is tightly related to the $\VP$ vs.\ $\VNP$ problem, the algebraic analog of the $\P$ vs. $\NP$ problem. The essence of the $\VP$ vs.\ $\VNP$ problem is showing that some explicit polynomials are hard to compute. By defining natural notions of reductions and completeness, Valiant showed that this problem is in fact equivalent to showing that, for fields of characteristic different than two, the determinantal complexity of the permanent polynomial,
\[
\mathsf{Perm}_n(X) = \sum_{\sigma \in S_n} \prod_{i=1}^n x_{i,\sigma(i)},
\]
doesn't grow like a polynomial function in $n.$\footnote{Strictly speaking, the $\VP$ vs. $\VNP$ question is equivalent to showing that the determinantal complexity of the $\mathsf{Perm}_n$ is at least $n^{\omega(\log n)}$, but we skip over this fine grained detail for now.}
This fact is a consequence of the \emph{completeness} property of the determinant: Valiant showed that if $f$ has an algebraic formula of size $s$, then the determinantal complexity of $f$ is at most $s$. This remains true even if $f$ has an \emph{algebraic branching program} (ABP) of size $s$: ABPs are a natural and more powerful model of computation than formulas. We refer to \cite{SY10} and \cite{S15} for more background on algebraic complexity theory and for proofs of these statements.
Thus, Valiant also established en passant the non-obvious fact that the determinantal complexity of every polynomial is finite, and it's at most roughly $\binom{n+d}{n}$ for every $n$-variate polynomial of degree $d$. Standard counting arguments also show that this estimate is close to being tight for almost every such polynomial.
The benefit of this reformulation of the $\VP$ vs.\ $\VNP$ problem is that it appears to strip away altogether the notion of ``computation'': indeed, this problem can be stated without even defining a computational model in any standard sense of the word, and thus it can potentially be proved without having to argue about the topology or structure of every possible arithmetic computation.
In practice, however, proving lower bounds on determinantal complexity is (unsurprisingly) difficult. Currently, for $n$-variate polynomials, there are no known lower bounds which are super-linear in $n$ (see \autoref{sec:previous results} for more details on previous work). Due to the completeness property mentioned above, a lower bound of $s$ on the determinantal complexity of $f$ will imply the same lower bound for algebraic formulas and even algebraic branching programs. However, super-linear lower bounds for formulas are well-known for decades \cite{K85}, and super-linear lower bounds for ABPs were recently established in \cite{CKSV19}, so there doesn't seem to be any major complexity-theoretic barrier for proving such lower bounds for determinantal complexity: the main obstacle is seemingly lack of techniques for reasoning about computations using determinants, and hence it is important to study this model and to develop techniques to understand it and to prove lower bounds, for the permanent as well as for other explicit polynomials.
Even for the purpose of separating $\VP$ and $\VNP$, one need not necessarily prove a lower bound on the determinantal complexity of the permanent; the same conclusion will hold if the lower bound on determinantal complexity is shown for any ``explicit'' polynomial (formally, in the class $\VNP$, which we don't define here) in lieu of the permanent.
Before we describe the previous work concerning determinantal complexity, we provide a brief remark about the notion of a ``trivial'' lower bound in this context which is worth remembering when evaluating the previous results (and our result). Unlike most standard computational models, observe that for an $n$-variate polynomial of degree $d$, even a lower bound of $n$ is non-trivial for determinantal complexity. This is because every coordinate of the affine map $L$ can depend on all $n$ variables. Nevertheless, since the determinant of an $m \times m$ matrix is a degree $m$ polynomial, and thus $\mathsf{Det}(L(\mathbf{x}))$ is a polynomial of degree at most $m$ for every affine map $L$, the degree $d$ \emph{is} a trivial lower bound on the determinantal complexity of $f$. Therefore, it is natural to consider polynomial families in which $d \le n$ or alternatively to hope to prove lower bounds stronger than $\max\{n,d\}$.
\subsection{Previous work}\label{sec:previous results}
The early work on determinantal complexity mostly focused on proving lower bounds for the permanent. Recall that the $n \times n$ permanent, $\mathsf{Perm}_n$, is a degree $n$ polynomial, so the trivial lower bound is $\dc(\mathsf{Perm}_n) \ge n$. Since over characteristic 2 the permanent and determinant coincide, the results described here hold for characteristic not equal to 2.
Already in 1913, Szeg\H{o} \cite{Szego13}, answering a question of P\'{o}lya \cite{Polya13}, showed that there's no way to generalize the $2 \times 2$ identity
\[
\mathsf{Perm}
\begin{pmatrix}
x_{1,1} & x_{1,2} \\
x_{2,1} & x_{2,2}
\end{pmatrix}
=
\mathsf{Det}
\begin{pmatrix}
x_{1,1} & x_{1,2} \\
-x_{2,1} & x_{2,2}
\end{pmatrix}
\]
by affixing $\pm$ signs to an $n\times n$ matrix of variables for $n \ge 3$.
Marcus and Minc \cite{MM61} strengthened this result by showing that for every $n$, $\dc(\mathsf{Perm}_n) > n$. Subsequent work by von zur Gathen \cite{vzG87}, Babai and Seress (see \cite{vzG87}), Cai \cite{Cai90} and Meshulam \cite{Meshulam} obtained the slightly stronger lower bound $\dc(\mathsf{Perm}_n) \ge \sqrt{2}n$.
Mignon and Ressayre \cite{MR04} greatly improved the lower bound by proving $\dc(\mathsf{Perm}_n) \ge n^2/2$, over the complex numbers. Cai, Chen and Li \cite{CCL10} extended this lower bound to fields of positive characteristic different than two, and Landsberg, Manivel and Ressayre \cite{LMR13} extended this result to the \emph{border} version of determinantal complexity, that is, they showed that the permanent is not even in the closure of polynomials with determinantal complexity less than $n^2/2$. Finally, Yabe \cite{Yabe} obtained an improved lower bound of $(n-1)^2+1$ over the real numbers.
However, while these lower bounds are quadratic in the degree, $\mathsf{Perm}_n$ is a polynomial with $n^2$ many variables, and notably none of these lower bounds is larger than the number of variables. In particular, these results don't even recover a weak form of the $n^3$ formula lower bound of Kalorkoti for $\mathsf{Perm}_n$ \cite{K85}.
Landsberg and Ressayre \cite{LR17} considered determinantal representations that respect certain symmetries (which they called \emph{equivariant determinantal complexity} and denoted $\edc$), and proved that $\edc(\mathsf{Perm}_n)$ is exponential in $n$. It's unclear how stringent the symmetry requirement is; Ladnsberg and Ressayre put forward the ambitious conjecture that $\edc$ and $\dc$ are polynomially related, which, if true, would imply $\VP \neq \VNP$. To the best of our knowledge, this conjecture remains open, but it's worth mentioning that in the context of \emph{regular determinantal complexity}, another notion defined and studied by \cite{LR17}, it can be shown unconditionally that requiring symmetry may result in a super-polynomial blow-up \cite{IL17}.
The question of lower bounds for other explicit polynomial was also considered: Mignon and Ressayre \cite{MR04} proved that the determinantal complexity of quadratic polynomials of rank $r$ is \emph{exactly} $\lceil (r+1)/2 \rceil$ (this, of course, cannot give a lower bound beyond $\lceil (n +1)/2 \rceil$). Chen, Kayal and Wigderson \cite{CKW11} observed that the technique of Mignon and Ressayre implies an $n/2$ lower bound on the determinantal complexity of the elementary symmetric polynomial of degree 2, $\sum_{1 \le i < j \le n} x_i x_j$. Kumar \cite{K19} used a different technique to prove a similar lower bound for the power symmetric polynomials $\sum_i x_i^d$ for $d \ge 2$ over $\mathbb{C}$.
The last lower bound was improved in a recent work of Alper, Bogart and Velasco \cite{ABV17}: an immediate corollary of their main theorem is that $\dc \left( \sum_i x_i^d \right) \ge n+1$, for every $d \ge 2$. Note that this lower bound is (only slightly) larger than the number of variables $n$, which is the first lower bound we are aware of with this feature. The results of Alper et al.\ are more general, and are stated as a function of the co-dimension of the singular locus of the polynomial, a notion we use as well (see \autoref{sec:main-lb}). In particular they are able to prove that $\dc(\mathsf{Perm}_3)=7$, but their main statement can't imply any lower bound stronger than $n+1$ for an $n$-variate polynomial.
\subsection{Our result}\label{sec:our result}
Our main result is the following theorem.
\begin{theorem}\label{thm:main}
For every natural number $n \ge 6$, the determinantal complexity of the polynomial $\sum_{i = 1}^n x_i^n$ over the field of complex numbers is at least $1.5n - 3$.
\end{theorem}
Although for simplicity we state our results for the complex numbers, all the results in this paper also hold for algebraically closed fields of positive characteristic $p$, as long as $p$ doesn't divide $n$. This assumption is not only an artifact of the proof. For example, when $n=p^k$, and over characteristic $p$,
\[
\sum_{i=1}^{p^k} x_i^{p^k} = \left( \sum_{i=1}^{p^k} x_i \right)^{p^k}
\]
has determinantal complexity at most $n=p^k$; it is also a polynomial of degree $n$, so its determinantal complexity is at least, and hence equals, $n$.
As discussed in \autoref{sec:previous results}, this is the first non-trivial\footnote{This means that the degree of the polynomials is at most the number of variables.} lower bound of the form $(1 + \epsilon) n$, for any $\epsilon > 0$ for any explicit $n$ variate polynomial family, and improves the previous best bound of $n + 1$ by Alper, Bogart and Velasco \cite{ABV17} by a constant factor.
This result, of course, is not fully satisfactory. The best upper bound we're aware of for $\dc(\sum_{i=1}^n x_i^n)$ is $O(n^2)$, which follows from converting the natural algebraic formula or ABP computing this polynomial to a determinantal expression. We suspect that the true complexity might be $\Omega(n^2)$ or at the very least $\omega(n)$.
Quantitatively, the situation here is somewhat similar to the case of lower bounds on the rank of 3-dimensional tensors, where the best lower bounds are only a constant factor away from the trivial lower bound, and proving super-linear lower bounds remains a challenging open problem (cf.\ \cite{AFT11,BD80,Blaser99,Shpilka01}, among others).
We now give an outline of the main ideas in our proof.
\subsection{Overview of the proof}
Let $M \in \mathbb{F}[x_1, x_2, \ldots, x_n]^{m \times m}$ matrix of affine functions such that $\sum_{i = 1}^n x_i^n = \mathsf{Det}(M(\mathbf{x}))$. \autoref{thm:main} shows a lower bound of $1.5n - 3$ on $m$. There are essentially three main ingredients to the proof of \autoref{thm:main}, and we now discuss them in some more detail.
\subsubsection*{Converting the matrix $M$ into a normal form}
Let $M_0 \in \mathbb{F}^{m \times m}$ be the \emph{constant part} of the matrix $M$, i.e. $M_0 = M(\mathbf{0})$. As a first step of our proof, we show (in \autoref{lem: rearranging constant terms}) that without loss of generality, $M_0$ can be assumed to be a diagonal matrix of rank equal to $m-1$. We a say that a matrix $M$ is in \emph{normal form} if it has this additional structure.
It is quite easy to observe that the rank of $M_0$ is at most $m-1$. However, for technical reasons, we actually need the lower bound on the rank as well, and this fact is a consequence of comparing the dimensions (as algebraic varieties) of the singular locus (which is just the the set of zeroes of a polynomial of multiplicity at least two) of the determinant and that of the polynomial $\sum_{i = 1}^n x_i^n$. Observations of this nature have been used in the context of determinantal complexity lower bounds before, and indeed, we crucially rely on a well known lemma of von~zur~Gathen (see \autoref{fact:singular locus of det}) for the proof. The details can be found in \autoref{sec: redn to normal form}.
\subsubsection*{Determinantal complexity of higher degree polynomial maps}
As the key ingredient of our proof, we show that for any matrix $M(\mathbf{x}) \in \mathbb{F}[\mathbf{x}]^{m\times m}$ where the entries of $M$ are polynomials of degree at most $n-1$ and $M$ is in normal form, if $\mathsf{Det}(M(\mathbf{x})) =\sum_{i = 1}^n x_i^n $, then $m \geq n/2$. Moreover, roughly the same lower bound continues to hold as long as $\det(M) = \left(\sum_{i = 1}^n x_i^n\right)(1 + Q)$ for any polynomial $Q$, with $Q(\mathbf{0}) = 0$.
Thus, this is a significant generalization of the $n/2$ lower bound on the standard notion determinantal complexity (where the entries of $M$ are affine functions) of $\sum_{i = 1}^n x_i^n$ as shown in \cite{K19}: this shows that roughly the same lower bound continues to hold even when the entries of the matrix are arbitrary polynomials of degree as high as $n-1$ and the determinant of the matrix equals an arbitrary multiple of $\sum_{i = 1}^n x_i^n$ with a non-zero constant term.
The proof of the lemma relies on the observation that the polynomial $\sum_{i = 1}^n x_i^n$ does not vanish with multiplicity at least two very often. This seemingly simple observation has been previously used in the context of lower bounds on algebraic branching programs computing this polynomial \cite{K19, CKSV19} in a crucial way. See \autoref{sec:lb for higher deg maps} for further details.
\subsection*{Trading dimension of the matrix for degree}
As the final ingredient of our proof, we use a well known property of determinants (\autoref{lem:reducing size}) to show that if there is an $m \times m$ matrix $M$ whose entries are affine functions and $\mathsf{Det}(M) = \sum_{i = 1}^n x_i^n$, then there is an $(m - n + 2)\times (m-n + 2)$ matrix $N$ whose entries are polynomials of degree at most $n-1$ and $\mathsf{Det}(N) = (\sum_{i = 1}^n x_i^n)(1 + Q)$ for a polynomial $Q$ which vanishes at zero. Moreover, if the matrix $M$ is in normal form, then the matrix $N$ continues to be in normal form.
Thus, we are in a setup where we can invoke the lower bound in \autoref{lem: n/2 lb for higher deg entries} discussed earlier and we get that the dimension of $N$ which equals $m - n + 2$ must be at least $n/2-1$, thereby implying that $m$ is at least $1.5n - 3$. The details of this step can be found in \autoref{sec:completing the proof}.
\section{Preliminaries}\label{sec:prelim}
In this paper $\mathbb{F}$ always denotes an algebraically closed field. We use $\mathbf{x}$ to denote a tuple of $n$ variables $x_1, \ldots, x_n$, where $n$ is understood from the context (or is otherwise explicitly mentioned).
We consider polynomial maps $M : \mathbb{F}^n \to \mathbb{F}^{m \times m}$ given by $m^2$ polynomials $(M_{i,j})_{i,j\in [m]}$. The same object can be thought of as a matrix of polynomials $M(\mathbf{x}) \in \mathbb{F}[\mathbf{x}]^{m \times m}$ and we use both points of view interchangeably. The degree of $M$ is the maximum degree of its coordinates, i.e., $\deg M = \max_{i,j} \deg M_{i,j}$.
Each $M(\mathbf{x}) \in \mathbb{F}[\mathbf{x}]^{m \times m}$ can be uniquely written as $M(\mathbf{x}) = M'(\mathbf{x}) + M_0$, where $M_0 \in \mathbb{F}^{m \times m}$ and in all $m^2$ coordinates of $M'$, the constant term is zero. We then call $M_0$ the \emph{constant part} of the map. A polynomial in which the constant term is zero is called \emph{constant free}, and a polynomial map is called constant free if all of its coordinates are constant free, i.e., in the above decomposition, $M_0=0$.
We denote the determinant polynomial by $\mathsf{Det}$. In cases where it is important to emphasize the dimension of the matrices in question we write it in the subscript, so for example the $m \times m$ determinant polynomial is denoted by $\mathsf{Det}_m$.
We assume knowledge of basic concepts in algebraic geometry such as affine varieties $V \subseteq \mathbb{C}^n$ and their dimension, which we denote $\dim(V)$. We encourage readers unfamiliar with those terms to consult the excellent textbook \cite{CLO07}.
\subsection*{Determinantal Complexity}
We now formally define the notion of determinantal complexity, which is the focus of this paper.
\begin{definition}[Determinantal Complexity]
The determinantal complexity of a polynomial $P \in \mathbb{F}[\mathbf{x}]$ is defined as the minimum $m \in \mathbb{N}$ such that there is a $m \times m$ matrix $M \in \mathbb{F}[\mathbf{x}]$ whose entries are polynomials of degree at most one such that
\[
P = \mathsf{Det}(M) \, . \qedhere
\]
\end{definition}
\begin{remark}
The above definition naturally generalizes to a family of polynomials in the following sense. A family $\{P_n\}_{n\in \mathbb{N}}$ of polynomials is said to have determinantal complexity at most $f(n): \mathbb{N} \to \mathbb{N}$ if there exists an $n_0 \in \mathbb{N}$, such that for every $n \geq n_0$, the determinantal complexity of $P_n$ is at most $f(n)$.
\end{remark}
\section{A lower bound on determinantal complexity}\label{sec:main-lb}
This section will be devoted for a proof of \autoref{thm:main}. We begin with the following lemma, which was instrumental in the recent proofs of lower bounds for algebraic formulas and algebraic branching programs.
\begin{lemma}[\cite{CKSV19, K19}]\label{lemma: width lb}
Let $d \ge 2$ be a natural number. Let $P_1, P_2, \ldots, P_t, Q_1, \ldots, Q_t, \allowbreak L \in \mathbb{C}[\mathbf{x}]$ be polynomials such that $\deg(P') < d$, $P_1, \ldots, P_t, Q_1, \ldots, Q_t$ have a common zero and
\[
\sum_{i = 1}^n x_i^d = \sum_{j = 1}^t P_j(\mathbf{x})Q_j(\mathbf{x}) + P' \, .
\]
Then, $t \geq n/2$.
\end{lemma}
We now show that without loss of generality, the constant part of every polynomial map $M$ such that $\sum_{i=1}^n x_i^d = \mathsf{Det}_m(M(\mathbf{x}))$ has a very special form: is it an $m \times m$ diagonal matrix with $0$ in the $(1,1)$ coordinate and $1$ in all diagonal entries.
\subsection{Reducing the matrix \texorpdfstring{$M$}{M} to a normal form}\label{sec: redn to normal form}
This claim is not entirely new and very similar statements were proved, for example, in \cite{MR04, ABV17}. For completeness, and since the exact statement we need is slightly more general, we provide a proof.
\begin{lemma}\label{lem: rearranging constant terms}
Let $d \ge 2$ be a natural number and let $M(\mathbf{x}) \in \mathbb{F}[\mathbf{x}]^{m \times m}$ be a polynomial map such that
\[
\mathsf{Det}_m(M(\mathbf{x})) = \sum_{i = 1}^n x_i^d \, .
\]
Then, there exists a matrix $\tilde{M}(\mathbf{x})\in \mathbb{F}[\mathbf{x}]^{m \times m}$ with $\deg(\tilde{M}) \leq \deg(M)$,
\[
\mathsf{Det}_m(\tilde{M}(\mathbf{x})) = \sum_{i = 1}^n x_i^d \, ,
\]
and the constant part of $\tilde{M}$ is a diagonal $m \times m$ matrix $\tilde{M}_0$ such that $(\tilde{M}_0)_{1,1}=0$ and $(\tilde{M}_0)_{i,i}=1$, for $2 \le i \le m$.
\end{lemma}
To prove \autoref{lem: rearranging constant terms} we require a few preliminaries. We begin with the definition of a singular locus of a polynomial (or a hypersurface).
\begin{definition}\label{def:singular locus}
Let $f \in \mathbb{F}[\mathbf{x}]$ be a polynomial. The \emph{singular locus} of $f$, denoted $\Sing(f)$, is the variety defined by
\[
\Sing(f) = \set{\mathbf{a} : \frac{\partial f }{\partial x_i} (\mathbf{a}) = 0, 1 \le i \le n}. \qedhere
\]
\end{definition}
The singular locus of the determinant was studied by von zur Gathen, who proved the following fact.
\begin{fact}[\cite{vzG87}]\label{fact:singular locus of det}
Let $\mathbb{F}$ be an algebraically closed field and let $\mathsf{Det}_m $ denote the $m \times m$ determinant polynomial. Then $\Sing(\mathsf{Det}_m) \subseteq \mathbb{F}^{m \times m}$ is precisely the set of matrices of rank at most $m-2$, and $\dim \Sing(\mathsf{Det}_m) = m^2 - 4$.
\end{fact}
The following is a slight generalization of a lemma of von zur Gathen (cf.\ also \cite{ABV17}).
\begin{lemma}\label{lem:image misses singular locus}
Let $f \in \mathbb{F}[\mathbf{x}]$ be a polynomial, and let $M : \mathbb{F}^n \to \mathbb{F}^{m \times m}$ be a polynomial map such that $f(\mathbf{x}) = \mathsf{Det}_m(M(\mathbf{x}))$. Suppose further that $\dim(\Sing(f)) < n-4$. Then $\Img(M) \cap \Sing(\mathsf{Det}_m) = \emptyset$. Furthermore, all matrices in $\Img(M)$ have rank at least $m-1$.
\end{lemma}
\begin{proof}
Let $y_{i,j}$ denote the coordinates of $\mathbb{F}^{m \times m}$ and write $M = (M_{i,j})_{i ,j \in [m]}$. Using the chain rule, we compute
\begin{equation}\label{eq:chain rule}
\frac{\partial f}{\partial x_k} = \sum_{i, j \in [m]} \frac{\partial \mathsf{Det}_m}{\partial y_{i,j}} (M(\mathbf{x})) \cdot \frac{\partial M_{i,j}}{\partial x_k} (\mathbf{x}), \quad k \in [n].
\end{equation}
Suppose $A \in \Img(M) \cap \Sing(\mathsf{Det}_m)$, and let $B$ be such that $A = M(B)$. By definition of $\Sing(\mathsf{Det}_m)$, $\frac{\partial \mathsf{Det}_m}{\partial y_{i,j}} (M(B)) = 0$ for all $i, j \in [m]$, and by \eqref{eq:chain rule} we get that $B \in \Sing (f)$. Thus $M^{-1} (\Sing(\mathsf{Det}_m)) \subseteq \Sing(f)$, and $\dim (M^{-1} (\Sing(\mathsf{Det}_m))) \le \dim \Sing(f) < n-4$. On the other hand, using a standard lower bound on the dimension of pre-images of polynomial maps (see Theorem 17.24 of \cite{HarrisAlgebraicGeometry}), if $\Img(M)$ and $\Sing(\mathsf{Det}_m)$ aren't disjoint,
\[
\dim(M^{-1} (\Sing(\mathsf{Det}_m))) \ge n + (m^2 - 4) - m^2 = n-4.
\]
This contradiction implies that $\Img(M) \cap \Sing(\mathsf{Det}_m) = \emptyset$. The ``furthermore'' part of the theorem follows from \autoref{fact:singular locus of det}.
\end{proof}
We will also need the following easy fact which shows that $\sum_{i=1}^n x_i ^d$ satisfies that assumption of \autoref{lem:image misses singular locus}.
\begin{fact}[\cite{K19, CKSV19}]\label{fact:sing locus of power-sym}
For every $d \ge 2$, $\dim(\Sing(\sum_{i=1}^n x_i^d)) = 0$.
\end{fact}
We are now ready to prove \autoref{lem: rearranging constant terms}.
\begin{proof}[Proof of \autoref{lem: rearranging constant terms}]
Let $f = \sum_{i=1}^n x_i^d$ and let $M : \mathbb{F}^n \to \mathbb{F}^{m \times m}$ be a polynomial map such that $f(\mathbf{x}) = \mathsf{Det}_m(M(\mathbf{x}))$, and write $M=M'+M_0$ where $M_0$ is the constant part of $M$.
First, observe that
\[
0 = f(\mathbf{0}) = \mathsf{Det}_m(M(\mathbf{0})) = \mathsf{Det}_m(M_0),
\]
which implies that $\mathsf{rank}(M_0) < m$. By \autoref{lem:image misses singular locus} and \autoref{fact:sing locus of power-sym}, we also know that $\mathsf{rank}(M_0) = \mathsf{rank}(M(\mathbf{0})) \ge m-1$, so $\mathsf{rank}(M_0) = m-1$.
By performing Gaussian elimination on the rows and on the columns, we can find two $m \times m$ matrices $G_1, G_2$ such that $\det(G_i) = \pm 1$ for $i=1,2$ and $N_0 := G_1 M_0 G_2$ is a diagonal matrix such that $(N_0)_{1,1} = 0$ and $(N_0)_{i,i} \neq 0$ for $2 \le i \le m$.
Now define a diagonal $m\times m$ matrix $\Delta$ such that $\Delta_{i,i} = 1/(N_0)_{i,i}$ for $2 \le i \le m$, and
\[
\Delta_{1,1} = \mathsf{Det}(G_1) \cdot \mathsf{Det}(G_2) \cdot \prod_{i=2}^m (N_0)_{i,i}.
\]
It readily follows that $\mathsf{Det}(\Delta) = \mathsf{Det}(G_1) \cdot \mathsf{Det}(G_2)$, and that $\tilde{M_0} := (G_1 M_0 G_2) \Delta$ is a diagonal matrix such that $(\tilde{M_0})_{1,1} = 0$ and $(\tilde{M_0})_{i,i} = 1$ for all $2 \le i \le m$.
Finally, define $\tilde{M} = G_1 M G_2 \Delta$. We verify that indeed
\begin{align*}
\mathsf{Det}(\tilde{M}(\mathbf{x})) &= \mathsf{Det}(G_1) \cdot \mathsf{Det}(M(\mathbf{x})) \cdot \mathsf{Det}(G_2) \cdot \mathsf{Det}(\Delta) \\
&= \mathsf{Det}(M(\mathbf{x})) \cdot (\mathsf{Det}(G_1) \cdot \mathsf{Det}(G_2))^2 = \mathsf{Det}(M(\mathbf{x})) = f(\mathbf{x}).
\end{align*}
We also have that
\[
\tilde{M} = G_1 (M' + M_0) G_2 \Delta = G_1 M' G_2 \Delta + G_1 M_0 G_2 \Delta = G_1 M' G_2 \Delta + \tilde{M_0}.
\]
Since $G_1, G_2, \Delta \in \mathbb{F}^{m \times m}$, it also holds that $\tilde{M}' := G_1 M' G_2 \Delta$ is a matrix of constant-free polynomials, and that $ \deg \tilde{M} \le \deg M$.
\end{proof}
We will also use the following simple and well known property of the determinant of a block matrix.
\begin{lemma}\label{lem:reducing size}
Let $M \in \mathbb{F}^{m \times m}$ be a matrix, and let $A \in \mathbb{F}^{t\times t}, B \in \mathbb{F}^{t\times m-t}, C \in \mathbb{F}^{m-t\times t}, D \in \mathbb{F}^{m-t\times m-t}$ be its submatrices as follows:
\[ M = \begin{pmatrix}
A & B\\\
C & D
\end{pmatrix}
\]
If $D$ is invertible, then
\[
\mathsf{Det} (M) = \mathsf{Det} (A - BD^{-1}C ) \cdot \mathsf{Det} (D) \, .
\]
\end{lemma}
\begin{proof}
Follows directly from the decomposition
\[ \begin{pmatrix}
A & B\\\
C & D
\end{pmatrix}
=
\begin{pmatrix}
A - BD^{-1}C & BD^{-1}\\\
0 & I_{m-t}
\end{pmatrix}
\cdot
\begin{pmatrix}
I_t & 0\\\
C & D
\end{pmatrix}
\]
and the multiplicativity of the determinant.
\end{proof}
\subsection{Determinantal complexity of higher degree polynomial maps}\label{sec:lb for higher deg maps}
In the following lemma we prove a lower bound of $n/2$ on the determinantal complexity in a more general model than the standard model. This is a generalization with respect to two properties. First, the entries of the matrix are no longer constrained to be polynomials of degree at most $1$, and can have degree as high as $d-1$, while computing the degree $d$ polynomial $\left(\sum_{i = 1}^n x_i^d\right)$. Moreover, the determinant of the matrix $M$ does not even have to compute the candidate hard polynomial $\left(\sum_{i = 1}^n x_i^d\right)$ exactly. It suffices if the determinant is equal to a polynomial of the form $\left(\sum_{i = 1}^n x_i^d\right) \cdot (\beta + Q)$ where $\beta$ is a non-zero field constant and $Q$ is an arbitrary polynomial (of potentially very high degree!) which is constant free, i.e. $Q(\mathbf{0}) = 0$.
\begin{lemma}\label{lem: n/2 lb for higher deg entries}
Let $d \ge 2$ be a natural number and let $M(\mathbf{x}) \in \mathbb{F}[\mathbf{x}]^{m \times m}$ such that $\deg(M) \le d-1$, and the constant part of $M$ is a diagonal matrix $M_0$ such that $(M_0)_{1,1}=0$ and $(M_0)_{i,i}=1$ for $2 \le i \le m$. Suppose that
\[
\mathsf{Det}(M) = \left(\sum_{i = 1}^n x_i^d\right) \cdot (\beta + Q) \, ,
\]
where $\beta \in \mathbb{F}$ is non-zero and $Q$ is a constant free polynomial. Then $m \geq n/2-1$.
\end{lemma}
\begin{proof}
Using the Laplace expansion of $\mathsf{Det}(M)$ along the first row, we get
\[
\mathsf{Det}(M) = \sum_{j=1}^m (-1)^{(j + 1)}M_{1, j}\cdot \mathsf{Det} (N_{1, j}) \, ,
\]
where $N_{i, j}$ is the submatrix of $M$ obtained by deleting the $i$-th row and the $j$-th column. For every $j \in [m]$, $j > 1$, we claim that $\mathsf{Det}(N_{1, j})$ is a constant free polynomial, i.e. \[
\mathsf{Det}(N_{1, j})(\mathbf{0}) = \mathsf{Det}\left(N_{1, j}(\mathbf{0})\right) = 0 \, .
\]
To see this, we observe that for every $j\in [m]\setminus \{1\}$, $N_{1, j}(\mathbf{0})$ is a $(m-1)\times (m-1)$ matrix, which has at most $m-2$ non-zero entries. This follows since $M_0$ has at most $m-1$ non-zero entries and in obtaining $N_{1, j}$ from $M$, we drop the entry $M_{j, j}$, which is one of the $(m-1)$ entries of $M$ with a non-zero constant term, and hence one of the $(m-1)$ non-zero entries of $M_0$. However, we note that $N_{1,1}(\mathbf{0})$ is the $(m-1) \times (m-1)$ identity matrix, so the constant term of $\mathsf{Det}(N_{1,1})$ is $1$, and we write $\mathsf{Det}(N_{1,1}) = 1 + P(\mathbf{x})$ where $P$ is constant free. Therefore, we have
\[
\left(\sum_{i = 1}^n x_i^d\right)\cdot (\beta + Q) = \mathsf{Det}(M) = M_{1,1}(1 + P) + \sum_{j = 2}^m (-1)^{(j + 1)}M_{1, j}\cdot \mathsf{Det}(N_{1, j}) \,
\]
In other words,
\[
\left(\sum_{i = 1}^n x_i^d\right)\cdot (\beta + Q) = \mathsf{Det}(M) = M_{1,1} + M_{1,1}\cdot P + \sum_{j = 2}^m (-1)^{(j + 1)}M_{1, j}\cdot \mathsf{Det}(N_{1, j}) \,
\]
Slightly rearranging (and using $\beta \neq 0$), we get
\[
\sum_{i = 1}^n x_i^d = \frac{1}{\beta}\left(-\left(\sum_{i = 1}^n x_i^d\right)\cdot Q + M_{1,1} + M_{1,1}\cdot P + \sum_{j = 2}^m (-1)^{(j + 1)}M_{1, j}\cdot \mathsf{Det}(N_{1, j}) \right)\,
\]
Since, $\deg(M_{1,1}) < d$ and $M_{1,1}, P, M_{1, 2}, \mathsf{Det}(N_{1, 2}), \ldots, M_{1, k}, \mathsf{Det}(N_{1, k}), Q$ are all constant free (and hence share a common zero, namely $\mathbf{0}$), we have from \autoref{lemma: width lb} that $m \geq n/2-1$.
\end{proof}
\subsection{Completing the proof of Theorem \ref{thm:main}}\label{sec:completing the proof}
We are now ready to complete the proof of \autoref{thm:main}.
\begin{proof}[Proof of \autoref{thm:main}]
Let $M$ be an $m \times m$ matrix with $\deg(M) \leq 1$ such that
\[
\sum_{i = 1}^n x_i^n = \mathsf{Det}(M) \, .
\]
From \autoref{lem: rearranging constant terms}, we can assume without loss of generality that the constant part $M_0$ of $M$ is a diagonal matrix such that $(M_0)_{1,1} = 0$ and $(M_0)_{i,i}=1$ for $2 \le i \le m$. In particular, all the off diagonal entries of $M$ and $M_{1,1}$ are homogeneous linear forms or zero, and $M_{j, j} \neq 0$ for $j > 1$.
Observe that for every $t \leq m-1$, the principal minor $D_t$ of $M$ which is obtained by deleting the first $m -t$ rows and columns of $M$ is invertible over the field of rational functions $\mathbb{F}(\mathbf{x})$. To see this, observe that the matrix $D_t(\mathbf{0})$ is the identity matrix, which implies that $\mathsf{Det}(D_t)$ is a non-zero polynomial. Moreover, since every entry of $M$ has degree at most $1$, and $\mathsf{Det}(M)$ has degree $n$, we know that $m \geq n$. So, we conclude that the principal minor $D := D_{(n-2)}$ of $M$ is invertible over $\mathbb{F}(\mathbf{x})$. Thus, if $B$ and $C$ are respectively the submatrices of $M$ defined as
\[ M = \begin{pmatrix}
A & B\\\
C & D
\end{pmatrix}
\]
then by \autoref{lem:reducing size} we have
\begin{equation}\label{eq:block determinant}
\mathsf{Det}(M) = \mathsf{Det}(A - BD^{-1}C)\cdot \mathsf{Det}(D) \, .
\end{equation}
Since $D^{-1} = \adj(D)/\det(D)$, where $\adj(D)$ is the adjugate matrix of $D$, the entries of $D^{-1}$ can be written as as a ratio of two polynomials, where the numerator has degree at most $n-3$ and the denominator, which is equal to $\mathsf{Det}(D)$, has degree at most $n-2$. Moreover, as discussed earlier in the proof,
the constant part of $D$ is the identity matrix, so there is a constant free polynomial $R \in \mathbb{F}[\mathbf{x}]$
such that
\[
\mathsf{Det}(D) = 1 + R \, .
\]
Thus, every entry of the $(m-n+2) \times (m-n+2)$ matrix $A - BD^{-1}C$ can be written as a ratio of two polynomials with the numerator being a polynomial of degree at most $n-1$ and the denominator being equal to $\mathsf{Det}(D)= 1 + R$. Therefore, by clearing the denominators and using \eqref{eq:block determinant}, we get that
\[
\mathsf{Det}(M) \cdot (1 + R)^{m - n + 2} = \mathsf{Det}(N)\cdot (1 + R) \, ,
\]
where $N$ is the matrix with polynomial entries of degree at most $n-1$ obtained by multiplying every entry of $A - BD^{-1}C$ by $1 + R$. Simplifying further, we get
\[
\left(\sum_{i=1}^n x_i^n \right) \cdot (1+R)^{m-n+1} = \mathsf{Det}(M) \cdot (1 + R)^{m - n + 1} = \mathsf{Det}(N)\, .
\]
We are almost ready to invoke \autoref{lem: n/2 lb for higher deg entries} to obtain a lower bound on the size of $N$ (and hence $M$), but to do that we need to ensure that the constant part of $N$, $N_0$, is a diagonal matrix with $(N_0)_{1,1}=0$ and $(N_0)_{i,i}=1$ for $2 \le i\le m-n+2$. We now verify that this is indeed the case.
Recall that by the structure of the constant part $M_0$ of $M$, all the entries of $B$ and $C$ and the $(1,1)$ entry of $A$ are constant free, and the constant term of $A_{i,i}$ is $1$ for $2 \le i \le m-n+2$. Thus, every entry of the matrix $BD^{-1}C$ is a rational function with a constant free numerator, and hence all the off-diagonal entries in $A - BD^{-1}C$ as well as its $(1,1)$ entry are rational functions with a constant free numerator. Moreover, the denominator of all the entries $\left(A - BD^{-1}C\right)$ equals $\mathsf{Det}(D) = 1 + R$, for a constant free polynomial $R$. So, expressing each entry of $A-BD^{-1}C$ as a quotient of polynomials, the constant term of each numerator on the diagonal is $1$ except for the $(1,1)$ entry, which has a constant free numerator. Finally, observe that eliminating the denominator of the entries of $\left(A - BD^{-1}C\right)$ by multiplying every entry by $(1 + R)$ gives us the matrix $N$.
Thus the matrix $N$ satisfies the hypothesis of \autoref{lem: n/2 lb for higher deg entries}, and hence $(m - n + 2) \geq n/2 -1$. This gives us $m \geq 1.5n - 3$ and completes the proof of \autoref{thm:main}.
\end{proof}
\section*{Acknowledgment}
Mrinal thanks Ramprasad Saptharishi for various discussions on determinantal complexity over the years, and in particular for explaining the proof of the result of Mignon and Ressayre to him.
\bibliographystyle{customurlbst/alphaurlpp}
|
1,314,259,996,685 | arxiv | \section{Introduction}
\label{intro}
We say that a connected surface immersed in
a Lorentzian 3-manifold $(M^3,g)$
is of {\it mixed type}
if both the space-like and time-like point sets
are non-empty.
In general, the mean curvature of
such surfaces
diverges: for example,
the graph of a smooth function
$t=f(x,y)$ in
the Lorentz-Minkowski space-time
$({\vect{R}}^3_1;t,x,y)$
gives a space-like (resp. time-like)
surface if $B>0$ (resp. $B<0$),
where
\begin{equation}\label{eq:B}
B:=1-f_x^2-f_y^2.
\end{equation}
In this situation,
the unit normal vector is given by
\begin{equation}\label{eq:nu}
\nu=\frac{1}{\sqrt{|B|}}
(1, f_x,f_y),
\end{equation}
and the mean curvature function is
computed as
\begin{equation}\label{eq:H0}
H=
\frac{\left(f_x^2-1\right)f_{yy} -2 f_x f_y f_{xy}
+\left(f_y^2-1\right) f_{xx}}
{2 |B(x,y)|^{3/2}},
\end{equation}
which is unbounded around the
set $\{B(x,y)=0\}$, in general.
On the other hand, several zero
mean curvature surfaces
of mixed type in ${\vect{R}}^3_1$
were found in
\cite{K1}, \cite{G}, \cite{Kl}, \cite{ST}, \cite{KKSY},
\cite{FRUYY2}, \cite{FKKRSUYY1} and \cite{FKKRSUYY2}.
Moreover, such examples can be found in other
space-times:
in fact, a zero mean curvature surface of
mixed type in the
de Sitter 3-space (resp. in the anti-de Sitter
3-space) is given
in this paper (cf. Example \ref{ex:fds} and
Example \ref{ex:fads}).
It is known that zero mean
curvature surfaces in ${\vect{R}}^3_1$
change types across their fold singularities,
except for the special case as in \cite{FKKRSUYY1}.
On the other hand, in \cite{HKS}, it was shown that
space-like non-zero constant mean curvature surfaces
do not admit fold singularities,
which suggests that
space-like non-zero constant mean curvature surfaces
never change types. More precisely,
the following questions naturally arise:
\begin{itemize}
\item[(a)] {\it Is there a mixed type
surface with non-zero constant mean curvature? }
\item[(b)] {\it Is there a mixed type
surface whose mean curvature vector field
is smooth and does not vanish
along the curve of type change?}
\end{itemize}
In this paper, we show that
the answer to Question (a)
is negative. This is a consequence of
the following assertion:
\begin{Thm}\label{thm:main}
Let $U$ be a connected domain in ${\vect{R}}^2$, and
$f:U\to (M^3,g)$ a real analytic
immersion into an oriented real analytic
Lorentzian manifold
$(M^3,g)$.
We denote by $U_+$ $($resp. $U_-)$
the set of points where
$f$ is space-like $($resp. time-like$)$.
Suppose that $U_+,U_-$ are both non-empty,
and the mean curvature function $H$
on $U_+\cup\, U_-$ is bounded.
Then for each $p\in \overline{U_+}\cap \overline{U_-}$,
there exists a sequence $\{p_n\}_{n=1,2,3,..}$
in $U_+$ $($resp. $U_-)$
converging to $p$
so that $\lim_{n\to \infty}H(p_n)=0$, where $\overline{U_+},\overline{U_-}$
are the closures of $U_+,U_-$ in $U$.
\end{Thm}
There exist space-like and time-like
constant mean curvature
immersions in ${\vect{R}}^3_1$ which are
not of mixed type although
their induced metrics degenerate
along certain smooth curves
(cf. Examples \ref{ex:fp} and \ref{ex:fh}
in Section 2).
Also, there are similar such examples of
space-like constant mean curvature one
surfaces in
the
de Sitter 3-space $S^3_1$
with singularities which are not of mixed type
(\cite{FKKRUY}).
The existence of such examples implies that
we cannot drop the assumption that
both
$U_+,U_-$ are non-empty.
The proof of Theorem \ref{thm:main} is given
in Section 2.
On the other hand, we show that
the answer to Question
(b)
is affirmative.
In fact, we show in Section 3 that
the mean curvature vector fields
of real analytic surfaces of mixed type
with bounded mean curvature functions
can be analytically
extended across the sets of type change
under a suitable genericity assumption
(cf. Proposition \ref{prop:tc}).
Moreover, we show the following:
\begin{Thm}\label{thm:existence}
There exists a real analytic function $g(x,y)$
on ${\vect{R}}^2$ whose graph realized in ${\vect{R}}^3_1$
satisfies the following properties:
\begin{enumerate}
\item the set $\Sigma_g$ of non-degenerate points of
type change of the graph of $g$ is non-empty,
and the induced metric of the graph of $g$
is non-degenerate on ${\vect{R}}^2\setminus \Sigma_g$,
\item the mean curvature function of
the graph of $g$ is bounded on ${\vect{R}}^2\setminus \Sigma_g$.
\item
the mean curvature vector field can be
extended to $\Sigma_g$ real analytically,
and does not vanish at each point of $\Sigma_g$.
\end{enumerate}
\end{Thm}
This suggests that surfaces with smooth mean
curvature vector fields
form an important sub-class of
the set of mixed type surfaces.
\section{Behavior of mean curvature
along curves of type change
}
\label{sec:1}
Let $(M^3,g)$ be an oriented real analytic
Lorentzian $3$-manifold.
Then, the vector product
$\mb v\times_g \mb w$
is defined
for linearly independent tangent vectors $\mb v,\mb w$
at $p\in M^3$, satisfying the
following three properties:
\begin{enumerate}
\item $\mb v\times_g \mb w$
is orthogonal to $\mb v$ and
$\mb w$,
\item $\{\mb v, \mb w, \mb v\times_g \mb w\}$
is a basis of the tangent space $T_pM$
which is compatible with the
orientation of $M^3$,
\item it holds that
$$
g_p(\mb v\times_g \mb w,\mb v\times_g \mb w)
=-g_p(\mb v,\mb v)g_p(\mb w,\mb w)+g_p(\mb v,\mb w)^2.
$$
\end{enumerate}
For each tangent vector
$\mb v \in T_pM^3$ ($p\in M^3$),
we set
$$
|\mb v|:=\sqrt{|g_p(\mb v,\mb v)|}.
$$
We fix a domain $U$ in ${\vect{R}}^2$.
Let $f:U\to M^3$ be a real analytic immersion.
Set
$
f_{u}:=df(\partial_u)$,
$
f_{v}:=df(\partial_v)
$,
where
$\partial_u:=\partial/\partial u$,
$\partial_v:=\partial/\partial v$.
Using three real analytic functions
$$
g_{11}:=g(f_u,f_u),\quad
g_{12}=g_{21}= g(f_u,f_v),\quad
g_{22}:=g(f_v,f_v)
$$
on $U$, we define
a function $\beta:U\to {\vect{R}}$ by
\begin{equation}\label{def:beta}
\beta:=g_{11}g_{22}-g_{12}^2.
\end{equation}
Then
$$
U_+:=\{p\in U\,;\,\beta(p)>0\},\qquad
U_-:=\{p\in U\,;\,\beta(p)<0\}
$$
give the set of space-like points
and
the set of time-like points, respectively.
The unit normal vector field
\begin{equation}\label{eq:omega}
\nu=\frac{f_u\times_g f_v}{|f_u\times_g f_v|}
\end{equation}
of $f$
is well-defined on $U_+\cup U_-$.
Using this, we set
$$
h_{11}:=g(f_{uu},\nu),\quad
h_{12}=h_{21}=g(f_{uv},\nu),\quad
h_{22}:=g(f_{vv},\nu),
$$
where
$$
f_{uu}=\nabla_{\partial_u}f_u,
\quad
f_{uv}=\nabla_{\partial_v}f_u=\nabla_{\partial_u}f_v,
\quad
f_{vv}=\nabla_{\partial_v}f_v,
$$
and $\nabla$ is the Levi-Civita connection
of the Lorentzian manifold $(M^3,g)$.
Each $h_{ij}$ ($i,j=1,2$)
is a function defined on $U_+\cup U_-$.
The mean curvature function $H$
is also defined on $U_+\cup U_-$,
and is given by
\begin{equation}\label{eq:H}
H:=
\frac{g_{11} h_{22}-2 g_{12}
h_{12}+g_{22} h_{11}}{2|\beta|}
=\frac{\alpha}{2|\beta|^{3/2}},
\end{equation}
where
\begin{equation}\label{def:alpha}
\alpha:=\sqrt{|\beta|}(g_{11} h_{22}-2 g_{12}
h_{12}+g_{22} h_{11}).
\end{equation}
Then the following assertion holds:
\begin{Lemma}\label{lem:A}
The function $\alpha:U_+\cup U_-\to {\vect{R}}$
can be analytically extended to $U$.
\end{Lemma}
\begin{proof}
We set
$\tilde \nu:=f_u\times_g f_v$.
Then
$$
\beta=-g(f_u\times_g f_v,f_u\times_g f_v)
$$
and
$\nu=\tilde \nu/\sqrt{|\beta|}$ holds
(cf. \eqref{eq:omega}).
Therefore, we have that
\begin{align*}
\alpha& =
\sqrt{|\beta|}
\biggl(g(f_{vv},\nu)g_{11}
-2g(f_{uv},\nu)g_{12}
+g(f_{uu},\nu)g_{22}\biggr) \\
&=
g(f_{vv},\tilde \nu)g_{11}
-2g(f_{uv},\tilde \nu)g_{12}
+g(f_{uu},\tilde \nu)g_{22},
\end{align*}
proving the assertion.
\end{proof}
Using the lemma,
we now give the proof of
Theorem \ref{thm:main}:
\begin{proof}[Proof of Theorem \ref{thm:main}]
We may assume that the mean curvature function $H$ is not identically zero.
Let $(x^1,x^2)$ be the coordinates of $U$.
We fix a point
$p\in \overline{U_+} \cap \overline{ U_-}$.
Let $\epsilon>0$ be an arbitrary positive number
and $V$ a neighborhood of $p$.
It is sufficient to show that
there exist points $q_+\in V_+$
and $q_-\in V_-$ such that
$|H(q_+)|$ and $|H(q_-)|$ are both
less than $\epsilon$.
We may assume that $V$ is connected.
If $\beta\ge 0$ or $\beta \le 0$
on $V$, this contradicts
the fact that
$p\in \overline{U_+} \cap \overline{ U_-}$.
So, we can take two points $q_0,q_1\in V$
such that $\beta(q_0)>0$ and $\beta(q_1)<0$.
We then take a smooth curve $\gamma(s)$
($s\in [0,2\pi]$) on $V$
such that $\gamma(0)=q_0$ and
$\gamma(2\pi)=q_1$.
Since the image of $\gamma$ lies in $V$,
we can write $\gamma=(\gamma^1,\gamma^2)$
and each $\gamma^i$ ($i=1,2$)
has the following Fourier series
expansion:
$$
\gamma^i(s)=u^i_0+\sum_{k=1}^\infty
\left(
u_k^i \cos k s+v_k^i \sin k s
\right) \qquad (i=1,2).
$$
We then set
$$
\gamma^i_N(s)=u^i_0+\sum_{k=1}^N
\left(
u_k^i \cos k s+v_k^i \sin k s
\right)\qquad (i=1,2),
$$
where $N$ is a
sufficiently
large positive integer.
Then the real analytic curve
defined by $\gamma_N(s):=(\gamma^1_N(s),\gamma^2_N(s))$
satisfies
\begin{equation}\label{eq:pm}
\beta(\gamma_N(0))>0,
\qquad
\beta(\gamma_N(2\pi))<0.
\end{equation}
Since
$$
\hat\beta(s):=\beta(\gamma_N(s)) \qquad (0\le s \le 2\pi)
$$
is a real analytic function
defined on $[0,2\pi]$,
the set of zeros of the function $\hat \beta(s)$
consists of a finite set of points
$$
0<s_1<\cdots<s_n<2\pi.
$$
By \eqref{eq:pm},
we can choose the number $j$
such that the sign of $\hat\beta(s)$ changes
from positive to negative
at $s=s_j$.
Then there exists a positive integer $m$
such that
$$
\lim_{s\to s_j}\frac{\hat\beta(s)}{(s-s_j)^{m}}=b \,(\ne 0),
$$
where $b$ is a non-zero real number.
Since $\hat\beta(s)$ changes sign at $s=s_j$,
the integer $m$ is odd.
By Lemma \ref{lem:A}, we
may regard $\alpha$ as a real analytic
function on $U$.
So we set
$$
\hat\alpha(s):=\alpha(\gamma_N(s)).
$$
By \eqref{eq:H},
we have that
$$
H(\gamma_N(s)):=\frac{\hat \alpha(s)}{2|\hat \beta(s)|^{3/2}}
$$
for $s\ne s_1,...,s_n$.
Since $H$ is bounded,
we have
$
\hat\alpha(s_j)=0.
$
Since $\hat \alpha(s)$ is a real analytic function,
there exists a positive integer $\ell$
such that
$$
\lim_{s\to s_j}\frac{\hat \alpha(s)}{(s-s_j)^{\ell}}=a\, (\ne 0),
$$
where $a$ is a non-zero real number.
Then it holds that
$$
\lim_{s\to s_j}|s-s_j|^{(3m/2)-\ell}
|H(\gamma_N(s))|=
\frac{|a|}{|b|^{3/2}}\,(\ne 0).
$$
Since $H$ is bounded, we have
$2\ell\ge 3m$.
Moreover, since $m$ is odd,
we have
$
2\ell> 3m.
$
Then we have
$
\lim_{s\to s_j}
|H(\gamma_N(s))|=0.
$
In particular, if we set
$$
q_+:=\gamma_N(s_j-\delta),\qquad
q_-:=\gamma_N(s_j+\delta),
$$
then $|H(q_+)|$ and $|H(q_-)|$
are less than $\epsilon$ for
sufficiently small $\delta>0$.
So we get the assertion.
\end{proof}
As a consequence,
we get the following corollary:
\begin{Cor}\label{cor:main}
Under the assumption of Theorem \ref{thm:main},
the function $\alpha:U_+\cup U_-\to {\vect{R}}$
can be analytically extended to $U$
and vanishes on $\overline{U_+}\cap \overline{U_-}$.
\end{Cor}
\begin{proof}
By Lemma \ref{lem:A},
the function $\alpha$ can be
analytically extended to $U$.
Suppose that
$\alpha(p)\ne 0$ for
$p\in \overline{U_+}\cap \overline{U_-}$.
Then the mean curvature function
cannot be bounded, since $\beta(p)=0$.
\end{proof}
We give here several examples:
\begin{Exa}[A space-like CMC
surface with parabolic symmetry]\
\label{ex:fp}
Consider the map
$f_P:{\vect{R}}^2\to {\vect{R}}^3_1$ such that
$$
f_P(u,v):=\left(-\eta(v) +u^2 v+v,
-\eta(v) +u^2 v-v,2 u v\right),
$$
where
$$
\eta(v):=
\frac{1}{2} \left(\arctan(v)-\frac{v}{v^2+1}\right),
\qquad \left|\arctan(v)\right|<\frac{\pi}2.
$$
This surface has singularities on
the $u$-axis. Moreover, the inverse image
$f^{-1}_P(\{\mb 0\})$ coincides with
the $u$-axis, where $\mb 0:=(0,0,0)$.
One can easily check that
$f_P$ gives a space-like immersion
of constant mean curvature $1/2$
on ${\vect{R}}^2\setminus \{v=0\}$.
Moreover, the image of $f_P$
is contained
in the set (cf. Figure 1, left)
$$
\mathcal P:=\left\{
(t,x,y)\in {\vect{R}}^3_1\,;\,
-t^2+x^2+y^2=2(t-x)\eta\left(\frac{t-x}2\right)
\right\}.
$$
The light-like line
$$
L:=\{(c,c,0)\,;\,c\in {\vect{R}} \}
=\left\{\lim_{u\to \infty}f_P(u,\frac{c}{u^2})\,;\,
c\in{\vect{R}}\right\}
$$
is contained in $\mathcal P$,
and the image of $f$ coincides with
$\mathcal P\setminus L$.
The set $\mathcal P$ itself is
a surface in ${\vect{R}}^3_1$
without self-intersections
which has a cone-like singular point
at the origin $\mb 0$, and
has bounded mean curvature function
on $\mathcal P\setminus\{\mb 0\}$.
Moreover, the induced metric on $\mathcal P$
degenerates only on the line $L$.
This implies that
we cannot drop the assumption that
$U_+,U_-$ are non-empty
in the statement of Theorem \ref{thm:main}.
This example is an analogue of
the maximal surface
called {\it the Enneper surface
of the 2nd kind}
or
{\it parabolic catenoid}
(cf. \cite{K1}, \cite{FKKRSUYY1}).
\end{Exa}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{{c@{\hspace{20mm}}c}}
\resizebox{3.8cm}{!}{\includegraphics{FIG0.eps}}&
\resizebox{5.2cm}{!}{\includegraphics{FIG1.eps}}
\end{tabular}
\caption{
The figure of
$\mathcal{P}$ (left) and $\mathcal{H}$ (right).
}
\label{fig:P_H}
\end{center}
\end{figure}
\begin{Exa}[A space-like CMC surface
with hyperbolic symmetry]\
\label{ex:fh}
We next consider the map
defined by
$$
f_H(u,v):=(v \cosh u,v \sinh u,\phi(v))
\qquad ((u,v)\in {\vect{R}}\times (-1,1)),
$$
where
$$
\phi(v):=\log\left(\frac{1+v}{1-v}\right)-v.
$$
Like the case of $f_P$, this surface
has singularities on the $u$-axis
and $f^{-1}_H(\{\mb 0\})$ coincides with
the $u$-axis.
One can easily check that
$f_H$ gives a space-like immersion
of constant mean curvature $1/2$
on ${\vect{R}}^2\setminus \{v=0\}$.
Moreover, the image of $f_H$
is contained
in the set (cf. Figure 1, right)
$$
\mathcal H:
=\left\{
(t,x,y)\in {\vect{R}}^3_1\,;\,
y=\phi(\pm\sqrt{t^2-x^2})
\right\}
=
\left\{
(t,x,y)\in {\vect{R}}^3_1\,;\,
t^2=x^2+\psi(y)^2
\right\},
$$
where $\psi:{\vect{R}}\to (-1,1)$
is the inverse function of
$\phi:(-1,1)\to {\vect{R}}$.
Two light-like lines
$$
L_\pm:=\{(c,\pm c,0)\,;\,c\in {\vect{R}} \}
$$
are contained in $\mathcal H$
and
$$
\mathcal H=L_+\cup L_- \cup
(\mbox{Image of }f_H)\cup (\mbox{Image of }f'_H),
$$
where
$$
f'_H(u,v):=(-v \cosh u,v \sinh u,\phi(v))
\qquad ((u,v)\in {\vect{R}}\times (-1,1)).
$$
Like as in the case of $\mathcal P$,
the set $\mathcal H$
has no self-intersections,
and has bounded mean curvature function
on $\mathcal H\setminus\{\mb 0\}$.
The origin $\mb 0$ is
a cone-like singular point.
Moreover, its induced metric
degenerates along the lines $L_\pm$.
This example is an analogue of
the maximal surface
called {\it the catenoid of the
2nd kind}
or {\it hyperbolic catenoid}
(cf. \cite{K1}, \cite{FKKRSUYY1}).
\end{Exa}
Similar examples, that is, a family of
space-like surfaces
with constant mean curvature
one containing light-like lines
in the de Sitter 3-space $S^3_1$
have recently been found in \cite{FKKRUY}.
The following is one typical mixed type surface
whose mean curvature vanishes
identically.
\begin{Exa}\label{eq:Kob}
\label{ex:fk}
Consider the function
$$
f_K(x,y):=x \tanh y.
$$
Then the graph of $f_K$ in ${\vect{R}}^3_1$
gives a zero mean curvature surface,
which is space-like on the set
$U_+:=\{(x,y)\in {\vect{R}}^2\,;\, x^2>\cosh^2 y\}$
and time-like
on the set
$U_-:=\{(x,y)\in {\vect{R}}^2\,;\, x^2<\cosh^2 y\}$.
This example is called the {\it helicoid of the 2nd kind},
which was found by Kobayashi \cite{K1}.
\end{Exa}
On the other hand, we
can find a similar example
in another space form:
\begin{Exa}
\label{ex:fds}
Consider the
map $f_Z : {\vect{R}} \times S^1 \rightarrow S^3_1$
given by
\[
f_Z(u,v):=\left( \sinh u \sin v,\, \cos u \cos v,\, \sin u \cos v,\, \cosh u \sin v \right),
\]
where
$$
S^3_1:=\{(t,x,y,z)\in {\vect{R}}^4_1\,;\, -t^2+x^2+y^2+z^2=1\}
$$
is the de Sitter 3-space, which is
the space-time of
constant sectional curvature $1$.
Then the first fundamental form of
$f_Z$ is given by
$ds^2=\cos 2v\, du^2 + dv^2$.
In particular, $f_Z$
is space-like (resp. time-like)
if $\cos 2v>0$
(resp. $\cos 2v<0$).
Moreover, the mean curvature function of
$f_Z$ vanishes identically.
\end{Exa}
\begin{Exa}
\label{ex:fads}
We define an immersion $f_{\rm ads}:{\vect{R}}^2 \rightarrow H^3_1$ by
\[
f_{\rm ads}(u,v)=
\left( \cosh u \cosh v,
\,\sinh a u\sinh v ,\,\cosh a u\sinh v ,
\,\sinh u \cosh v \right),
\]
where $a= 1/\tanh \alpha$ $(\alpha\neq0)$
is a constant, and
\begin{align*}
H^3_1&=\left\{ (t,x,y,z)\in {\vect{R}}^4_2\,;\,
-t^2-x^2+y^2+z^2=-1 \right\}
\end{align*}
is the anti-de Sitter 3-space, which is
the space-time of
constant sectional curvature $-1$.
Then the first fundamental form
of $f_{\rm ads}$ is given by
$$
\dfrac{\cosh 2 \alpha - \cosh 2 v}{2\sinh^2\alpha} du^2 + dv^2.
$$
In particular,
$f_{\rm ads}$ is
space-like (resp. time-like)
if $\cosh 2\alpha>\cosh 2v$
(resp. $\cosh 2\alpha<\cosh 2v$).
\end{Exa}
\section{Properties of points where
surfaces change type}
\label{sec:2}
In this section, we shall investigate
the properties of functions $t=f(x,y)$
whose graphs induce mixed type surfaces
in
${\vect{R}}^3_1$
with bounded mean curvature.
\begin{Def}[cf. {\cite[Definition 2.3]{FKKRSUYY2}}]
\label{def:nondegtc}
Let $U$ be a domain
in the $xy$-plane ${\vect{R}}^2$,
and $f:U\to {\vect{R}}$ a
$C^\infty$-function.
We set
$$
B:=1-f_x^2-f_y^2.
$$
A point $p \in U$
is called a {\it non-degenerate point of type change}
if
\begin{equation}\label{eq:NB}
B(p)=0,\qquad \nabla B(p)\ne0
\end{equation}
hold, where $\nabla B:=(B_x,B_y)$.
\end{Def}
By definition, the first fundamental
form of the graph of $f$ is degenerate
at a non-degenerate point of type change.
We set
$$
A := (f_x^2-1)f_{yy} -2 f_xf_y f_{xy}+(f_y^2-1)f_{xx}.
$$
Then the functions $A,B$ can be
considered as a
special case of
the functions (cf.
\eqref{def:beta} and
\eqref{def:alpha})
$\alpha, \beta$
by setting $(u,v)=(x,y)$.
By \eqref{eq:H0}, we have
\begin{equation}\label{eq:H2}
H=\frac{A}{2|B|^{3/2}}.
\end{equation}
\begin{Prop}[cf. Proposition 2.4 in \cite{FKKRSUYY2}]
\label{prop:Gu}
\label{prop:equiv}
Suppose that the mean curvature function of the
graph of $f$ is bounded.
Let
$p\in U$ be a point satisfying
$B(p)=0$. Then the following two
assertions are equivalent:
\begin{enumerate}
\item\label{item:equiv:1}
the point $p$ is a
non-degenerate point of type change.
\item\label{item:equiv:2}
$p$ is a
dually regular point in the sense of
\cite{G}, that is,
$p$ is a point where
$f_{xx}(p)f_{yy}(p)-f_{xy}(p)^2\ne 0$ .
\end{enumerate}
\end{Prop}
\begin{proof}
The proof is almost parallel
to that of
Proposition 2.4 in \cite{FKKRSUYY2}.
It holds that
\begin{equation}\label{eq:Bxy}
\nabla B
=\op{Hess}(f)\pmt{f_x \\ f_y},
\qquad \op{Hess}(f)
:=\pmt{f_{xx} & f_{xy}\\ f_{xy} & f_{yy}}.
\end{equation}
Now suppose that (2) holds.
Then $\op{Hess}(f)$ is a regular matrix at $p$.
Since $B(p)=0$, $(f_x,f_y)\ne 0$ at $p$.
Thus, \eqref{eq:Bxy} implies that $\nabla B\ne 0$
at $p$, that is, (1) holds.
We next suppose on the contrary that (2)
does not hold. By a suitable linear
coordinate change of $(x, y)$,
we may assume without loss of generality that
$f_{xy}(p) = 0$.
Then either $f_{xx}(p) = 0$ or $f_{yy}(p) = 0$.
By \eqref{eq:H2}, and
Theorem \ref{thm:main}, we have
$A(p)=0$.
This with $B(p) = 0$ and $f_{xy}(p) = 0$
implies that
$$
f_x(p)^2f_{xx}(p) + f_y(p)^2f_{yy}(p) = 0.
$$
This with $f_{xx}(p)=0$ or $f_{yy}(p)=0$ implies that
$$
\op{Hess}(f)\pmt{f_x \\ f_y}
=\pmt{
f_x(p)f_{xx}(p)\\
f_y(p)f_{yy}(p)
}
=\pmt{0 \\ 0}.
$$
So (1) does not hold.
\end{proof}
A regular curve $\Gamma:(a,b)\to {\vect{R}}^3_1$
is called
{\it null\/} or {\it isotropic\/}
if
$\dot \Gamma(t):=d\Gamma(t)/dt$
is a light-like vector
for each $t\in (a,b)$.
\begin{Def}\label{def:n-deg}
A null curve $\Gamma:(a,b)\to {\vect{R}}^3_1$
is called non-degenerate at
$t=c$ if
$\dot \Gamma(c)$ and $\ddot \Gamma(c)$
are linearly independent.
If $\Gamma(t)$ is non-degenerate
for all $t\in (a,b)$,
the curve
$\Gamma$
is called a {\it non-degenerate} null curve.
\end{Def}
Let $p\in U$
be a non-degenerate point of type change.
Then, by the implicit function theorem,
there exists a regular curve
$
\gamma:(-\epsilon,\epsilon)\to U
$
such that
$B\circ \gamma(t)=0$
and
$\gamma(0)=p$, where $\epsilon$ is a
positive number.
We call this curve $\gamma$
{\it the characteristic curve}
of type change.
The following assertion
is a generalization of
\cite[Proposition 2.5]{FKKRSUYY2}
for zero-mean curvature surfaces.
\begin{Prop}\label{prop:1}
Suppose that the graph $t=f(x,y)$
over a domain $U$
has bounded mean curvature function.
If the graph changes type
along a regular curve $\gamma(t)$
$(|t|<\epsilon)$
such that $f\circ \gamma(t)$
is a non-degenerate null curve in ${\vect{R}}^3_1$,
then $\gamma(t)$
consists of non-degenerate points
of type change.
\end{Prop}
\begin{proof}
The proof is completely parallel
to that of \cite[Proposition 2.5]{FKKRSUYY2}.
\end{proof}
The converse assertion is given
as follows,
which is a generalization of
\cite[Proposition 2.6]{FKKRSUYY2}
for zero-mean curvature surfaces.
\begin{Prop}
Suppose that the graph $t=f(x,y)$
over a domain $U$
has bounded mean curvature function.
Let $p\in U$ be a
non-degenerate point of type change
and $\gamma(t)$
$(|t|<\epsilon)$
the characteristic curve
of type change such that $\gamma(0)=p$.
Then $f\circ \gamma(t)$
is a non-degenerate null curve.
\end{Prop}
\begin{proof}
Using the fact that
$A(\gamma(t))=0$ holds,
the proof of this assertion
is completely parallel to
that of \cite[Proposition 2.6]{FKKRSUYY2}.
\end{proof}
Moreover, the following assertion holds:
\begin{Prop}\label{prop:tc}
Let $t=f(x,y)$
be a real analytic
function over the domain $U$
which gives a graph with
bounded mean curvature function.
Suppose that the zeros of $B(x,y)$ are all
non-degenerate points of type change.
Then, the mean curvature
vector $H\nu$
can be analytically
extended
to all of
$U$.
\end{Prop}
\begin{proof}
Let $p\in U$ be
a non-degenerate point of type
change. Then we can take
a real analytic local coordinate system
$(u,v)$ centered at $p$
such that the $u$-axis is the
characteristic curve
of type change.
By the condition $\nabla B(u,0)\ne (0,0)$
(cf. \eqref{eq:NB}),
there exists a real analytic function
$b(u,v)$ defined near the $u$-axis
such that $B(u,v)=v b(u,v)$ and
$b(u,0)\ne 0$.
On the other hand, Theorem \ref{thm:main}
yields that
there exists a real analytic function
$a(u,v)$ defined near the $u$-axis
such that
\begin{equation}\label{eq:Aa}
A(u,v)=v^2 a(u,v).
\end{equation}
By \eqref{eq:H2}, we have
$$
H(u,v)=\frac{\sqrt{|v|} a(u,v)}{2 |b(u,v)|^{3/2}}.
$$
By \eqref{eq:nu}, we have that
\begin{equation}\label{eq:Hnu}
H\nu
=
\frac{\sqrt{|v|}a(u,v)}{2 |b(u,v)|^{3/2}}
\frac{1}{\sqrt{|v| |b(u,v)|}}(1,f_x,f_y)
=
\frac{a(u,v)}{2b(u,v)^{2}}
(1,f_x,f_y),
\end{equation}
proving the assertion.
\end{proof}
Finally, we prove
Theorem \ref{thm:existence}
in the introduction:
\begin{proof}[Proof of Theorem \ref{thm:existence}]
Let $f:{\vect{R}}^2\to {\vect{R}}$ be a real analytic function whose graph gives
a zero-mean curvature surface, with
function $B:=1-f_x^2-f_y^2$
satisfying $\nabla B\ne (0,0)$ if $B=0$.
Take a real analytic function $\psi:{\vect{R}}\to {\vect{R}}$
such that
\begin{equation}\label{eq:psi}
\psi(0)=\psi'(0)=\psi''(0)=0.
\end{equation}
We then set
\begin{equation*}
g(x,y):= f(x,y) + \psi (B(x,y)),
\end{equation*}
and
$$
\tilde B:=1-g_x^2-g_y^2.
$$
Since
$$
g_x = f_x + \psi'(B)B_x, \quad
g_y = f_y + \psi'(B)B_y,
$$
we have that
\begin{equation}\label{eq:BtB}
\tilde B=B-2\psi'(B)(f_xB_x+f_yB_y)-\psi'(B)^2(B_x^2+B_y^2).
\end{equation}
Here, the relation $C_1\equiv C_2 \mod B$ for
two real analytic functions
$C_i(x,y)$ ($i=1,2$)
means that $(C_1-C_2)/B$
is a real analytic function on ${\vect{R}}^2$.
Since $\psi'(B)\equiv 0 \mod B$,
$\tilde B$ can be divided by $B$.
Thus, to show the mean curvature vector field
can be smoothly extended across the set $B=0$,
it is sufficient to show
that
$$
\tilde A:=
(g_y^2-1)g_{xx}
-2g_xg_yg_{xy}
+(g_x^2-1)g_{yy}
$$
can be divided by $B^2$.
Since
\begin{align*}
g_{xx}
&= f_{xx} + \psi''(B)B_x^2 + \psi'(B)B_{xx},
\\
g_{xy}&
= f_{xy} + \psi''(B)B_xB_y
+ \psi'(B)B_{xy} , \\
g_{yy} &
= f_{yy} + \psi''(B)B_y^2
+ \psi'(B)B_{yy},
\end{align*}
the fact that $A=0$ yields
\begin{equation}\label{eq:tA3}
\tilde A\equiv \psi''(B)\Gamma+\psi'(B)\Delta \mod B^3,
\end{equation}\label{eq:tA}
where
\begin{align*}
\Gamma&:=( f_y^2-1)B_{x}^2
-2f_x f_yB_xB_y
+(f_x^2-1)B_y^2, \\
\Delta&:=
2 (B_x f_x f_{yy}
-B_x f_{xy} f_y-B_y f_x f_{xy}+
B_y f_{xx} f_y) \\
&\phantom{aaaaaaaaaaaaaaaaaa}
+B_{xx} \left(f_y^2-1\right)
-2B_{xy} f_x f_y+
B_{yy}
\left(f_x^2-1\right).
\end{align*}
Since
\begin{align*}
\Gamma&=
(-B-f_x^2)B_{x}^2
-2f_xf_yB_xB_y
+(-B-f_y^2)B_y^2 \\
&=-B(B_x^2+B_y^2)
-(f_xB_x+f_yB_y)^2
\end{align*}
and
\begin{align}
f_xB_x+f_yB_y &=-2 \nonumber
(f_x(f_xf_{xx}+f_yf_{xy})
+f_y(f_xf_{xy}+f_y f_{yy}))\\
&=2A +2B (f_{xx}+f_{yy})
=2B (f_{xx}+f_{yy}) \label{eq:tB},
\end{align}
we have that
\begin{equation}\label{eq:Gamma}
\Gamma\equiv -B(B_x^2+B_y^2) \mod B^2.
\end{equation}
Since
$$\psi'(B)\equiv 0\mod B^2,\qquad
\psi''(B)\equiv 0\mod B,
$$
\eqref{eq:tA3} and \eqref{eq:Gamma}
yield that
$
\tilde A
$
can be divided by $\tilde B^2$.
To give an explicit example,
we consider the function $f_K(x,y):=x \tanh y$
given in Example \ref{eq:Kob}.
Then, we have
$$
B(x,y)=(\cosh^2 y - x^2)\text{sech}^4 y
$$
and $x=\pm \cosh y$
give the characteristic curves of type change.
We consider the new function
\begin{equation}\label{eq:g0}
g(x,y):=x \tanh y+ c\tanh^3(B(x,y))\qquad (0< c\le 1),
\end{equation}
where $c$ is a constant.
Then the mean curvature vector field
is real analytic along the set
of type change
$\Sigma_f:=\{(\pm \cosh y,y)\,;\, y\in {\vect{R}}\}$.
By \eqref{eq:Gamma},
$$
\Gamma\equiv
-4B \text{sech}^4y
\mod B^2
$$
holds. By a straightforward calculation,
$$
\Delta=2(B+1)\op{sech}^4 y
$$
holds. Since $\psi(B)=c \tanh^3(B)$, we have
$$
\psi'(B)\equiv 3cB^2,\quad \psi''(B)\equiv 6cB
\mod B^3.
$$
Thus, \eqref{eq:tA3} yields that
\begin{equation}\label{eq:ratio}
\left.
\frac{\tilde A}{\tilde B^2}
\right|_{(x,y)=(\pm \cosh y,y)}
=
\left.
\frac{\tilde A}{B^2}
\right|_{(x,y)=(\pm \cosh y,y)}
=
\frac{-18c}{\cosh^4 y}
\qquad (y\in {\vect{R}}),
\end{equation}
which never vanishes on the set $\Sigma_f$.
To complete the proof, it is
sufficient to show that $\tilde B/B$
has no zeros if $c$ is sufficiently small.
We shall now compute $\tilde B/B$
using \eqref{eq:BtB}.
We set
$$
\phi(t):=\frac{\tanh t}t
$$
which is a real analytic bounded function.
We set
$$
U:=x \,\op{sech}^2y,\quad V:=\op{sech}\, y,\quad
S:=\op{sech}(V^2-U^2).
$$
Here $U$ is unbounded, but $V, S$ are bounded on
${\vect{R}}^2$. By a straight-forward calculation,
one can get that
$$
\frac{\tilde B}{B}
=1-12 c B \phi(B)^2 S^2 (C_1+C_2),
$$
where
\begin{align*}
C_1&:=2 U(U^2-V^2)\tanh y, \\
C_2&:=3c B^2 \phi(B)^2S^2
\biggl(U^2V^4+(-2U^2+V^2)^2\tanh^2 y\biggr).
\end{align*}
Since
$$
\frac{\cosh(V^2-U^2)}{\cosh(U^2)}
=\cosh(V^2)-\sinh(V^2)\tanh(U^2),
$$
using the fact that $|V|\le 1$, we have
$$
e^{-1}\le \exp(-V^2)=\cosh(V^2)-\sinh(V^2)<
\frac{\cosh(V^2-U^2)}{\cosh(U^2)}.
$$
In particular
$$
S|U|^m
=\frac{|U|^m}{\cosh(U^2)}\frac{\cosh(U^2)}{\cosh(V^2-U^2)}
<\frac{e|U|^m}{\cosh(U^2)}
$$
is a bounded function for $m\ge 0$.
Then we can write
$$
\frac{\tilde B}{B}
=1-12 c \phi(B)^2 SB (SC_1+SC_2).
$$
Since
$\tanh y$, $\phi(B)$, and $SB=B\op{sech} B$
are all bounded,
there exists a positive constant $m$ which
does not depend on the choice of $c\in (0,1]$
such that
$\phi(B)^2SB(SC_1+SC_2)<m$ holds
for all $(x,y)\in {\vect{R}}^2$, and so
$$
\left|\frac{\tilde B}{B}-1\right|<12 mc.
$$
If $0<c<1/(12 m)$, then the zero set of
$\tilde B$ coincides with that of $B$, proving the
assertion.
\end{proof}
\begin{ack}
The first, the fourth and the fifth authors
thank Udo Hertrich-Jeromin and Kosuke Naokawa
for fruitful conversations at TU-Wien.
The authors thank Wayne Rossman for
valuable comments.
\end{ack}
|
1,314,259,996,686 | arxiv | \section{Introduction}\label{sec:intro}
Fortin operators are a critical tool for the stability analysis of mixed finite element schemes, cf.~\cite{BoffiBrezziFortin}.
The discontinuous Petrov--Galerkin (DPG) method with optimal test functions, on the other hand, is a framework that aims at automatic inf-sup stability. In practice, optimal test functions have to be approximated and the question of existence of Fortin operators re-appears. In this case, local (element-wise) operators are sufficient. A first answer was given in~\cite{practicalDPG}, with subsequent studies in~\cite{breakSpace,constrFortin,DemkowiczZanotti20,KLove2}.
In the case of singularly-perturbed problems, uniform discrete stability, or robustness of the method, requires the existence of uniformly bounded Fortin operators. This has been an open problem.
In this paper, we present local Fortin operators for $H^1$ and $\ensuremath{{\boldsymbol{H}}}(\mathrm{div})$, on simplices in arbitrary dimension and arbitrary polynomial degree. In contrast to previous results, our constructions are explicit (not needed in applications) and require fewer degrees of freedom. More importantly, we include parameter-dependent exponential layers that guarantee uniform boundedness of our operators for parameter-dependent norms (the $\ensuremath{{\boldsymbol{H}}}(\mathrm{div})$-case is restricted to two and three space dimensions). We illustrate their application to a DPG method for a reaction-dominated diffusion problem, leading to robustness, i.e., uniform stability, error control, and convergence. In this case, we consider the energy-norm induced by the problem. We have not analyzed the case of balanced norms as proposed in~\cite{DPGrefusion}. This and possible extensions to other singularly-perturbed problems like advection-dominated diffusion are left to future research.
Let us shortly discuss the abstract setting of the DPG method: Consider the variational formulation
\begin{align*}
u\in U\colon \qquad b(u,v) = L(v) \quad\forall v\in V,
\end{align*}
where $U$, $V$ are Hilbert spaces with norms $\norm\cdot{U}$, $\norm{\cdot}V$, $b(\cdot,\cdot)$ is a bounded bilinear form and induces a boundedly invertible operator $B\colon U\to V'$, $u\mapsto b(u,\cdot)$.
Choosing finite dimensional spaces $U_h\subset U$, $V_h\subset V$, the fully discrete DPG method reads:
\begin{align}\label{eq:DPG:practical}
u_h \in U_h\colon \qquad b(u_h,v) = L(v) \quad\forall v\in \Theta_h(U_h),
\end{align}
where $\Theta_h\colon U\to V_h$ is defined through ($\ip\cdot\cdot_V$ being the inner product on $V$)
\begin{align*}
\ip{\Theta_h u}{v_h}_V = b(u,v_h) \quad\forall v_h \in V_h.
\end{align*}
Well-posedness of discrete DPG is ensured if there exists a Fortin operator $\Pi_{\mathrm{F}}\colon V\to V_h$ such that
\begin{align}\label{eq:fortin:abstract}
\norm{\Pi_{\mathrm{F}} v}V &\leq C_{\Pi_{\mathrm{F}}} \norm{v}V, \quad
b(u_h,v-\Pi_{\mathrm{F}} v) = 0 \quad\forall u_h\in U_h, \, v\in V,
\end{align}
see,~\cite{practicalDPG}. Furthermore, the existence of a Fortin operator also implies quasi-optimality,
\begin{align*}
\norm{u-u_h}U \leq C C_{\Pi_{\mathrm{F}}} \min_{w_h\in U_h} \norm{u-w_h}U
\end{align*}
with $C=C_b/c_b$ where $C_b$ and $c_b$ are the boundedness and $\inf$-$\sup$ constants of $b(\cdot,\cdot)$, respectively.
It also plays an important role in the a posteriori error control, see~\cite[Theorem~2.1]{DPGaposteriori},
\begin{align*}
\norm{u-u_h}U \eqsim \norm{Bu_h-L}{V'} + \osc(L),
\end{align*}
where $\osc(L) = \sup_{0\neq v\in V} L(v-\Pi_{\mathrm{F}} v)/\norm{v}V$.
One of the main motivations that had driven the development of the DPG method was to derive robust numerical schemes for singularly perturbed problems, see, e.g.,~\cite{DPGconfusion,DPGrefusion}.
All these problems have in common that they naturally lead to parameter dependent trial and test norms. For a concrete example consider the test space $H^1(T)$ (here $T$ as an element of the mesh $\ensuremath{\mathcal{T}}$) equipped with the norm
\begin{align*}
\norm{v}{T,\alpha} = \norm{v}T + \alpha \norm{\nabla v}T \quad\text{for some fixed } \alpha>0.
\end{align*}
In~\cite{practicalDPG,breakSpace,DemkowiczZanotti20} Fortin operators are constructed (resp. their existence is shown).
Let us write $\Pi_{\mathrm{F}}\colon H^1(T)\to P^k(T)$ (a polynomial space). Besides some conditions they satisfy the boundedness estimates
\begin{align*}
\norm{\Pi_{\mathrm{F}} v}T \lesssim \norm{v}T + h_T\norm{\nabla v}T, \quad\norm{\nabla \Pi_{\mathrm{F}} v}T \lesssim \norm{\nabla v}T.
\end{align*}
Combining the latter two estimates yields
\begin{align*}
\norm{\Pi_{\mathrm{F}} v}{T,\alpha} \lesssim \max\{1,\alpha^{-1}h_T\} \norm{v}{T,\alpha}.
\end{align*}
If $h_T\lesssim \alpha$ it is clear that $\norm{\Pi_{\mathrm{F}} v}{T,\alpha} \lesssim \norm{v}{T,\alpha}$ uniformly.
However, when $\alpha\lesssim h_T$ --- a case often encountered with singularly perturbed problems --- then we get $\norm{\Pi_{\mathrm{F}} v}{T,\alpha} \lesssim \alpha^{-1}h_T \norm{v}{T,\alpha}$. This means that, particularly on coarse meshes, the Fortin operator is not uniformly bounded. By our prior considerations, this means that quasi-optimality as well as a posteriori error control is spoiled.
One of the main objectives of this work is to define discrete spaces $V_h$ and construct corresponding Fortin operators $\Pi_{\mathrm{F}}\colon V\to V_h$ with boundedness constant $C_{\Pi_{\mathrm{F}}}\eqsim 1$ for small parameters ($\alpha\lesssim h_T$ in the previous example).
We do this by first revisiting the construction of Fortin operators for the spaces $H^1(T)$ and $\Hdivset{T}$ in the case $h_T\lesssim \alpha$.
Contrary to prior works we construct our Fortin operators in an explicit manner. This allows to precisely write down a basis for the discrete test spaces yielding --- compared with the operators from~\cite{practicalDPG,breakSpace,DemkowiczZanotti20} --- smaller dimensions.
The novel idea of definition also requires a different analysis which we present in detail.
Our constructions are valid for any polynomial degree (this statement will be made precise below) and arbitrary dimension (except for some operators from Section~\ref{sec:fortinDiv}).
However, main advantage is that the definition and construction can be extended to the case $\alpha\lesssim h_T$.
Specifically, we use modified face bubble functions $\eta_{\alpha,F}$ instead of polynomial face bubble functions $\eta_F$. They are defined in such a way that their volume norm $\norm{\eta_{\alpha,F}}T$ resp. $\norm{\nabla \eta_{\alpha,F}}T$ scales differently than $\norm{\eta_F}T$ resp. $\norm{\nabla \eta_F}T$ depending on the ratio $\alpha/h_T$, though $\eta_{\alpha,F}|_{\partial T} = \eta_F|_{\partial T}$.
The analysis requires some additional tools and steps.
We also consider low order polynomial cases which allow for even smaller dimensions in the test space.
As mentioned above, Fortin operators for the DPG method have been constructed in various works:
The first one was~\cite{practicalDPG}. Other articles that analyze the existence of Fortin operators for second-order PDEs include~\cite{breakSpace,constrFortin,DemkowiczZanotti20}.
The latter references consider all arbitrary fixed polynomial orders. For low order methods with smaller test space dimensions we refer to~\cite{CarstensenGHW14,CarstensenHellwig16}.
For a fourth-order PDE model problem we have shown existence of Fortin operators in~\cite{KLove2}.
The remainder of this work is organized as follows: In Section~\ref{sec:basis} we introduce some notation and define basis functions as well as novel face bubble functions.
Section~\ref{sec:fortin} and Section~\ref{sec:fortinDiv} discuss the construction of Fortin operators for $H^1$ and $\ensuremath{{\boldsymbol{H}}}({\rm div\,})$, respectively.
Section~\ref{sec:num} concludes this article with a short description of a DPG method for a singularly perturbed reaction-diffusion problem and numerical examples.
\section{Preliminaries}\label{sec:basis}
The notation $a\lesssim b$ ($a\gtrsim b$) for $a,b>0$ means that there exists $C>0$ such that $a\leq C\,b$ ($C\,a\geq b$). We write $a\eqsim b$ for $a,b>0$ if $a\lesssim b \lesssim a$.
The generic constant $C$ is independent of involved functions, the diameter of elements, and parameters like $\alpha$ and $\varepsilon$, where present.
\subsection{Mesh and spaces}
Let $\ensuremath{\mathcal{T}}$ denote a shape-regular simplicial mesh of a Lipschitz domain $\Omega$ with $\mathrm{diam}(\Omega)\leq 1$.
Throughout, $T\in\ensuremath{\mathcal{T}}$ is some fixed element, ${\widehat T}$ is the reference element given as the convex hull of the origin and the $n$ coordinate axis vectors. E.g., for $n=2,3$ it reads
\begin{align*}
{\widehat T} = \begin{cases}
\operatorname{conv}\{(0,0)^\top,(1,0)^\top,(0,1)^\top\}, & n=2,\\
\operatorname{conv}\{(0,0,0)^\top,(1,0,0)^\top,(0,1,0)^\top,(0,0,1)^\top\}, & n=3.
\end{cases}
\end{align*}
Here, we understand $\operatorname{conv}$ as the interior of the convex hull of a set.
We adopt the standard notation for Lebesgue and Sobolev spaces, $L^2(\omega)$, $\ensuremath{\boldsymbol{L}}^2(\omega) := L^2(\omega;\ensuremath{\mathbb{R}}^n)$,
\begin{align*}
H^1(\omega) = \set{v\in L^2(\omega)}{\nabla v\in \ensuremath{\boldsymbol{L}}^2(\omega)},
\quad
\Hdivset\omega = \set{{\boldsymbol\tau}\in \ensuremath{\boldsymbol{L}}^2(\omega)}{{\rm div\,}{\boldsymbol\tau}\in L^2(\omega)}
\end{align*}
for a Lipschitz domain $\omega\subset \ensuremath{\mathbb{R}}^n$.
With $\ensuremath{{\boldsymbol{n}}}_\omega$ we denote the normal vector on the boundary of $\omega$ pointing from $\omega$ to its complement $\ensuremath{\mathbb{R}}^n\setminus\omega$.
Recall that traces of $H^1(\omega)$ elements are well defined (in the sense of trace operators) and the canonic trace space is $H^{1/2}(\partial\omega)$.
Normal traces of $\Hdivset\omega$ elements are well defined (in a duality sense) and the canonic trace space is $H^{-1/2}(\partial\omega)$. We simply write ${\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_\omega$ for the normal trace of ${\boldsymbol\tau}\in \Hdivset\omega$.
We denote by $\|\cdot\|_\omega$ the canonical $L^2(\omega)$ norm induced by the $L^2(\omega)$ inner product $\ip{\cdot}{\cdot}_\omega$ for a Lipschitz domain $\omega\subset \ensuremath{\mathbb{R}}^n$. The volume measure of $\omega$ is given by $|\omega|$. The same notation for norm and inner product is used for $L^2(\omega;\ensuremath{\mathbb{R}}^n)$.
The surface measure of $\gamma\subseteq \partial \omega$ is denoted by $|\gamma|$ and $\|v\|_\gamma$ is the $L^2(\gamma)$ norm induced by the inner product $\dual{\cdot}{\cdot}_\gamma$. We also use the same notation for the duality between $H^{-1/2}(\partial \omega)$ and $H^{1/2}(\partial \omega)$,
\begin{align*}
\dual{\phi}{v}_{\partial \omega} \quad\text{for } \phi \in H^{-1/2}(\partial \omega), v\in H^{1/2}(\partial \omega).
\end{align*}
Recall the following relation between traces of $H^1(\omega)$ and $\Hdivset\omega$,
\begin{align}\label{eq:dualityrelation}
\dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_\omega}{v}_{\partial \omega} = \ip{{\rm div\,}{\boldsymbol\tau}}v_\omega + \ip{{\boldsymbol\tau}}{\nabla v}_\omega
\end{align}
for all ${\boldsymbol\tau}\in \Hdivset\omega$, $v\in H^1(\omega)$. Obviously, for sufficiently regular functions this is just the integration by parts formula.
Let $\Pi_T^q\colon L^2(T)\to P^q(T)$ denote the $L^2(T)$ orthogonal projection on $P^q(T)$, the space of polynomials on $T$ of degree less than or equal to $q\in\ensuremath{\mathbb{N}}_0$.
For vector-valued polynomials (each component is a polynomial of degree less than or equal to $q$) we use the symbol $\ensuremath{\boldsymbol{P}}^q(T)$.
Recall the first-order approximation property
\begin{align*}
\|v-\Pi_T^qv\|_T\leq\|v-\Pi_T^0v\|_T \lesssim h_T \|\nabla v\|_T \quad\text{for all }v\in H^1(T).
\end{align*}
An important tool is the following (multiplicative version) of the trace inequality. It can be derived from~\cite[Theorem~1.6.6]{BrennerScott08} with a scaling argument (using the reference element ${\widehat T}$) and the approximation property of $\Pi_T^0$.
\begin{lemma}\label{lem:traceineq}
For any $v\in H^1(T)$ we have
\begin{align}\label{eq:traceineq}
\norm{v-\Pi_T^0v}{\partial T} \lesssim \norm{v-\Pi_T^0v}T^{1/2}\norm{\nabla v}{T}^{1/2} \lesssim h_T^{1/2}\norm{\nabla v}T
\end{align}
with hidden constants only depending on the shape of $T$.
\end{lemma}
We denote by $\ensuremath{\mathcal{V}}_T$ the set of the $n+1$ vertices of $T$, $\ensuremath{\mathcal{F}}_T$ is the set of $n+1$ faces of $T$ and $\ensuremath{\mathcal{V}}_F$ denotes the set of $n$ vertices of $F$.
For $z\in \ensuremath{\mathcal{V}}_T$ let $F_z\in\ensuremath{\mathcal{F}}_T$ be the face opposite to $z$, i.e., $F_z = \operatorname{conv}\big(\ensuremath{\mathcal{V}}_T\setminus \{z\}\big)$.
Similarly, for $F\in\ensuremath{\mathcal{F}}_T$ let $z_F\in\ensuremath{\mathcal{V}}_T$ be the vertex opposite to $F$.
For $F\in\ensuremath{\mathcal{F}}_T$ let $P^q(\ensuremath{\mathcal{F}}_T)\subset L^2(\partial T)$ denote face-wise polynomials of degree less than or equal to $q\in\ensuremath{\mathbb{N}}_0$ and $P_c^q(\ensuremath{\mathcal{F}}_T):=P^q(\ensuremath{\mathcal{F}}_T)\cap C^0(\partial T)$.
Note that $\ensuremath{{\boldsymbol{n}}}_T$ is face-wise constant. For the fixed element $T\in\ensuremath{\mathcal{T}}$ and any $F\in\ensuremath{\mathcal{F}}_T$ we abbreviate $\ensuremath{{\boldsymbol{n}}}_F = \ensuremath{{\boldsymbol{n}}}_T|_F$.
For a vertex $z\in\ensuremath{\mathcal{V}}_T$ we denote by $\ensuremath{\mathcal{E}}_z$ the set of $n$ edges that share the same vertex $z$, i.e., for each $E\in\ensuremath{\mathcal{E}}_z$ there is a $z'\in\ensuremath{\mathcal{V}}_T\setminus\{z\}$ with $E = \operatorname{conv}\{z,z'\}$. To each $E=\operatorname{conv}\{z,z'\}\in\ensuremath{\mathcal{E}}_z$ we associate the (tangential) vector $\ensuremath{{\boldsymbol{t}}}_E = z'-z$ (the orientation does not matter for our analysis nor for implementation).
Furthermore, $h_T = \mathrm{diam}(T)$, $h_T \eqsim \mathrm{diam}(F)$ for all $F\in\ensuremath{\mathcal{F}}_T$ with hidden constants only depending on the shape of $T$.
Some other relations that we frequently use without further notice are $|T|\eqsim h_T^n$, $|F|\eqsim h_T^{n-1}$, $|\partial T|\eqsim |F|$, $|T|\eqsim |F|h_T$ for any $F\in\ensuremath{\mathcal{F}}_T$.
In the following subsections we define special functions that will be used for the construction of the Fortin operators. For ease of reading and reference they are listed, together with their relevant properties, in Table~\ref{tab:basis} below.
\subsection{Low-order basis functions}\label{sec:low}
The functions $\eta_z\in P^1(T)$ are canonical basis functions with $\eta_z(z') = \delta_{z,z'}$ for $z,z'\in \ensuremath{\mathcal{V}}_T$ and $\delta_{\cdot,\cdot}$ denoting the Kronecker-$\delta$,
\begin{align*}
\eta_F = \prod_{z\in\ensuremath{\mathcal{V}}_F} \eta_z \in P^{n}(T)
\end{align*}
are the face bubble functions, and $\eta_{T} = \prod_{z\in\ensuremath{\mathcal{V}}_T} \eta_{z}\in P^{n+1}(T)$ is the element bubble function.
Clearly, $\linhull\{\eta_z\,:\,z\in \ensuremath{\mathcal{V}}_T\}= P^1(T)$.
Alternatively, we may use the following basis for $P^1(T)$:
First, we abbreviate $d_F = \eta_{z_F}$. Then, for $F\in\ensuremath{\mathcal{F}}_T$ we define
\begin{align*}
\nu_F = \sum_{z\in\ensuremath{\mathcal{V}}_F}\eta_z - (n-1)d_F.
\end{align*}
For space $P^0(\ensuremath{\mathcal{F}}_T)$ we use the characteristic functions $\chi_F|_{F'} = \delta_{F,F'}$ for $F'\in \ensuremath{\mathcal{F}}_T$ as basis functions.
\begin{lemma}\label{lem:dualbasis}
We have
\begin{align}\label{eq:basisdual}
\dual{\nu_F}{\chi_{F'}}_{\partial T} = |F| \delta_{F,F'} \quad\forall F,F'\in\ensuremath{\mathcal{F}}_T
\end{align}
and
\begin{align*}
\linhull\set{\nu_F}{F\in\ensuremath{\mathcal{F}}_T} = P^1(T), \quad \linhull\set{\nu_F|_{\partial T}}{F\in\ensuremath{\mathcal{F}}_T} = P_c^1(\ensuremath{\mathcal{F}}_T).
\end{align*}
\end{lemma}
\begin{proof}
Note that~\eqref{eq:basisdual} implies that $\nu_F|_{\partial T}$, $F\in\ensuremath{\mathcal{F}}_T$ are linearly independent. This implies the last two assertions because $\dim (P^1(T)) = n+1 = \dim (P_c^1(\ensuremath{\mathcal{F}}_T))$. It only remains to prove~\eqref{eq:basisdual}: Let $F\in\ensuremath{\mathcal{F}}_T$. From its definition we see that $\nu_F|_F = 1$, thus, $\dual{\nu_F}{\chi_{F}}_{\partial T} = |F|$. Let $F'\in \ensuremath{\mathcal{F}}_T\setminus\{F\}$.
Using $\int_{F'} \eta_z \,\mathrm{d} x = |F'|n^{-1}$ for $z\in \ensuremath{\mathcal{V}}_{F'}$ one verifies that
\begin{align*}
\dual{\nu_F}{\chi_{F'}}_{\partial T} &= \sum_{z\in\ensuremath{\mathcal{V}}_F\cap\ensuremath{\mathcal{V}}_{F'}}\int_{F'} \eta_z\,\mathrm{d} x - (n-1)\int_{F'}d_F \,\mathrm{d} x
\\&= (n-1)|F'|n^{-1} - (n-1)|F'|n^{-1} = 0,
\end{align*}
finishing the proof.
\end{proof}
Let $\ensuremath{\boldsymbol{RT}}^0(T) =\set{{\boldsymbol\psi}\in \ensuremath{\boldsymbol{L}}^2(T)}{{\boldsymbol\psi} = \boldsymbol{\alpha} + \beta {\boldsymbol{x}}, \, \boldsymbol{\alpha}\in\ensuremath{\mathbb{R}}^n, \beta\in \ensuremath{\mathbb{R}}}$ denote the lowest-order Raviart--Thomas space where ${\boldsymbol{x}}\colon T\to\ensuremath{\mathbb{R}}^n$, $z\mapsto z$.
Let ${\boldsymbol\psi}_{F}\in \ensuremath{\boldsymbol{RT}}^0(T)$ denote the canonical Raviart--Thomas basis function with
\begin{align*}
{\boldsymbol\psi}_F\cdot\ensuremath{{\boldsymbol{n}}}_T|_{F'} = \delta_{F,F'} \quad\forall F,F'\in \ensuremath{\mathcal{F}}_T
\end{align*}
and $\norm{{\boldsymbol\psi}_F}{T} \eqsim |T|^{1/2}$. One verifies the explicit representation ${\boldsymbol\psi}_F(z) = \frac{|F|}{n|T|}(z-z_F)$.
Note that by Lemma~\ref{lem:dualbasis} we have that
\begin{align*}
\dual{{\boldsymbol\psi}_F\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{F'}}_{\partial T} = |F|\delta_{F,F'} \quad\forall F,F'\in\ensuremath{\mathcal{F}}_T.
\end{align*}
We also use Bernardi--Raugel elements, see~\cite{BernardiRaugel85},
\begin{align*}
\eeta_F := \eta_F\ensuremath{{\boldsymbol{n}}}_F, \quad F\in\ensuremath{\mathcal{F}}_T
\end{align*}
for which we get
\begin{align*}
\dual{\eeta_F\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{F'}}_{\partial T} = \dual{\eeta_F\cdot\ensuremath{{\boldsymbol{n}}}_T}{1}_F \delta_{F,F'}
\eqsim |F| \delta_{F,F'} \quad\forall F,F'\in\ensuremath{\mathcal{F}}_T.
\end{align*}
We also define edge based functions: Fix a vertex $z_*\in\ensuremath{\mathcal{V}}_T$ and set $\ensuremath{\mathcal{E}}_* = \ensuremath{\mathcal{E}}_{z_*}$.
Clearly, $\ensuremath{{\boldsymbol{t}}}_E$ ($E\in\ensuremath{\mathcal{E}}_*$) are linearly independent and span $\ensuremath{\mathbb{R}}^n$.
Let ${\boldsymbol\sigma}_E$, ($E\in\ensuremath{\mathcal{E}}_*$) denote a basis of $\ensuremath{\boldsymbol{P}}^0(T)$ such that (for any $z\in T$)
\begin{align*}
{\boldsymbol\sigma}_E(z)\cdot\ensuremath{{\boldsymbol{t}}}_{E'} = \delta_{E,E'} \quad\forall E,E'\in\ensuremath{\mathcal{E}}_*.
\end{align*}
We have that $|{\boldsymbol\sigma}_E(z)|\eqsim 1$ with constants only depending on the shape of $T$.
With these preparations we define edge functions by $\eta_E = \prod_{z\in\ensuremath{\mathcal{V}}_T\cap\overline{E}}\eta_{z}$ and tangential edge functions by
\begin{align*}
\eeta_E := \eta_E\ensuremath{{\boldsymbol{t}}}_E \quad\forall E\in \ensuremath{\mathcal{E}}_*.
\end{align*}
These functions play the role of element bubble functions in $\Hdivset{T}$ as can be seen from the next result.
\begin{lemma}\label{lem:rotatedBR}
We have $\eeta_E\in \ensuremath{\boldsymbol{P}}^{2}(T)$ and
\begin{align*}
\eeta_E\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = 0, \quad
\ip{{\boldsymbol\sigma}_E}{\eeta_{E'}}_T = \ip{{\boldsymbol\sigma}_E}{\eeta_{E}}_T\delta_{E,E'}\eqsim |T|\delta_{E,E'}\quad\forall E,E'\in\ensuremath{\mathcal{E}}_*.
\end{align*}
\end{lemma}
\begin{proof}
Clearly, $\eta_E\in P^2(T)$, thus, $\eeta_E\in \ensuremath{\boldsymbol{P}}^2(T)$. Note that $\eta_E|_{\partial T}$ is supported on $n-1$ faces, say $F_j$, $j=1,\dots,n-1$. For these faces we also have $\ensuremath{{\boldsymbol{t}}}_E \subseteq F_j$ yielding $\ensuremath{{\boldsymbol{t}}}_E\cdot\ensuremath{{\boldsymbol{n}}}_{F_j}=0$. We conclude $\eeta_E\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = 0$.
The final assertion follows from the definition of ${\boldsymbol\sigma}_E$ and $\eeta_E$ and scaling arguments.
\end{proof}
\subsection{Higher-order basis functions}\label{sec:ho}
For $F\in\ensuremath{\mathcal{F}}_T$ let $\widetilde\chi_{F,j}\in P^p(T)$, $j=1,\dots,\dim(P^p(F))$ be such that $\widetilde\chi_{F,j}|_F$ is a basis of $P^p(F)$ with $\norm{\widetilde\chi_{F,j}}\infty = \norm{\widetilde\chi_{F,j}|_F}\infty \eqsim 1$ and define
\begin{align*}
\eta_{F,j} &= \eta_F\widetilde\chi_{F,j}, \quad j=1,\ldots,\dim P^p(F), \, F\in\ensuremath{\mathcal{F}}_T.
\end{align*}
Let $\chi_{F,j}\in P^p(\ensuremath{\mathcal{F}}_T)$, $j=1,\dots,\dim(P^p(F))$, be such that $\chi_{F,j}|_{F'}=0$ for $F'\in\ensuremath{\mathcal{F}}_T\setminus\{F\}$ and
\begin{align*}
\dual{\chi_{F,j}}{\eta_{F,k}}_F = \delta_{j,k}|F|, \quad j,k=1,\ldots,\dim(P^p(F)).
\end{align*}
Let $\widetilde\chi_{T,j}\in P^p(T)$, $j=1,\dots,\dim(P^p(T))$, denote a basis of $P^p(T)$ with $\norm{\widetilde\chi_{T,j}}\infty\eqsim 1$ and define
\begin{align*}
\eta_{T,j} = \eta_T\widetilde\chi_{T,j}, \quad j=1,\dots,\dim(P^p(T)).
\end{align*}
Furthermore, let $\chi_{T,j}$, $j=1,\dots,\dim(P^p(T))$ be such that
\begin{align*}
\ip{\chi_{T,j}}{\eta_{T,k}}_T = |T|\delta_{j,k}, \quad j,k = 1,\dots,\dim(P^p(T)).
\end{align*}
By scaling arguments one verifies that $\norm{\chi_{F,j}}\infty\eqsim 1$ and $\norm{\chi_{T,j}}\infty \eqsim 1$.
Let $\widetilde P^{p}(T)$ denote the orthogonal complement of $P_b^{p}(T) = \set{v\in P^{p}(T)}{v|_{\partial T}=0}$ in $P^{p}(T)$.
Let $\widetilde\nu_{\partial T,j}$, $j=1,\dots,\dim(\widetilde P^{p+1}(T))$, denote a basis of $\widetilde P^{p+1}(T)$ with $\norm{\widetilde \nu_{\partial T,j}}\infty \eqsim 1$.
Furthermore, let $\nu_{\partial T,j}\in \widetilde P^{p+1}(T)$, $j=1,\dots,\dim(\widetilde P^{p+1}(T))$, denote a basis with
\begin{align*}
\dual{\nu_{\partial T,j}}{\widetilde\nu_{\partial T,k}}_{\partial T} = |\partial T| \delta_{j,k} \quad
j,k = 1,\dots,\dim(\widetilde P^{p+1}(T)).
\end{align*}
One verifies that $\norm{\nu_{\partial T,j}}\infty \eqsim 1$ by scaling arguments.
Define ${\boldsymbol\psi}_{\partial T} = \sum_{F\in\ensuremath{\mathcal{F}}_T}{\boldsymbol\psi}_F$ and note that ${\boldsymbol\psi}_{\partial T}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = 1$.
Define
\begin{align*}
{\boldsymbol\psi}_{\partial T,j} = {\boldsymbol\psi}_{\partial T} \widetilde\nu_{\partial T,j} \quad j=1,\dots,\dim(\widetilde P^{p+1}(T)).
\end{align*}
Thus, by our previous considerations, $\dual{{\boldsymbol\psi}_{\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,k}}_{\partial T} = |\partial T| \delta_{j,k}$, $j,k=1,\dots,\dim(\widetilde P^{p+1}(T))$.
We also define higher order edge functions: For $E\in\ensuremath{\mathcal{E}}_*$, $j=1,\dots,\dim(P^{p}(T))$, define
\begin{align*}
\eeta_{E,j} &= \eeta_E\widetilde\chi_{T,j}.
\end{align*}
For $E\in\ensuremath{\mathcal{E}}_*$ let $\chi_{E,j}\in P^p(T)$, $j=1,\dots,\dim(P^p(T))$, denote a basis with
\begin{align*}
\ip{\chi_{E,j}}{\widetilde\chi_{T,k}\eta_{E}}_T = |T|\delta_{j,k} \quad j,k=1,\dots,\dim(P^p(T)).
\end{align*}
One verifies that $\norm{\chi_{E,j}}\infty \eqsim 1$.
Furthermore, define
\begin{align*}
{\boldsymbol\sigma}_{E,j} = {\boldsymbol\sigma}_E \chi_{E,j} \quad j=1,\dots,\dim(P^p(T)), \, E\in\ensuremath{\mathcal{E}}_*.
\end{align*}
The proof of the next result follows the arguments given in Lemma~\ref{lem:rotatedBR} together with the aforegoing definitions and is thus omitted.
\begin{lemma}
We have that $\eeta_{E,j}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = 0$, $\eeta_{E,j}\in\ensuremath{\boldsymbol{P}}^{p+2}(T)$, and
\begin{align*}
\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E',k}}_T = \delta_{E,E'}\delta_{j,k} |T|
\end{align*}
for all $E,E'\in\ensuremath{\mathcal{E}}_*$, $j,k=1,\dots,\dim(P^{p}(T))$.
\qed
\end{lemma}
\subsection{Modified face bubble functions}\label{sec:modBubble}
We introduce modified face bubble functions. Before we come to their definition and analysis we state the following result:
\begin{lemma}\label{lem:phiKappa}
Let $R>0$.
Consider $R\geq\kappa>0$ and the function $\phi_\kappa\colon [0,1]\to [0,1]$, $t\mapsto e^{-t/\kappa}$.
Then,
\begin{align*}
\norm{\phi_\kappa}{L^2(0,1)} \eqsim \kappa^{1/2}, \quad \norm{\phi_\kappa'}{L^2(0,1)} \eqsim \kappa^{-1/2},
\quad \phi_\kappa(0) = 1.
\end{align*}
The hidden constants only depend on $R$.
\end{lemma}
\begin{proof}
The results follow from straightforward calculations.
\end{proof}
Recall that $d_F=\eta_{z_F}$. We can interpret $d_F$ as a relative distance function that is $0$ when restricted to $F$ and $1$ when evaluated at the vertex opposite to $F$. Considering $\phi:= \phi_{\alpha/h_T}$, i.e.,
\begin{align*}
\phi(t) = e^{-h_Tt/\alpha},
\end{align*}
define for $F\in\ensuremath{\mathcal{F}}_T$ the modified face bubble function by
\begin{align}\label{eq:def:modbubble}
\eta_{\alpha,F} := (\phi\circ d_F)\eta_F.
\end{align}
Some basic properties of this modified function are given in the next result. A visualization of $\eta_{\alpha,F}$ is presented in Figure~\ref{fig:modbubble}.
\begin{figure}
\begin{center}\includegraphics[width=0.8\textwidth]{data/figModBubbles.png}\end{center}
\caption{Visualization of the face bubble functions $\eta_F$ and $\eta_{\alpha,F}$ on the reference element ${\widehat T}$ and face $F=(0,1)\times\{0\}$.}
\label{fig:modbubble}
\end{figure}
\begin{lemma}\label{lem:modbubble}
Suppose that $0<\alpha\lesssim h_T$.
For any $F\in\ensuremath{\mathcal{F}}_T$ we have that
\begin{align*}
\norm{\eta_{\alpha,F}}T \lesssim |T|^{1/2} \left(\frac\alpha{h_T}\right)^{1/2}, \quad
\norm{\nabla \eta_{\alpha,F}}T \lesssim h_T^{-1}|T|^{1/2} \left(\frac\alpha{h_T}\right)^{-1/2}, \quad
\eta_{\alpha,F}|_{\partial T} = \eta_F|_{\partial T}.
\end{align*}
\end{lemma}
\begin{proof}
The identity $\eta_{\alpha,F}|_{\partial T} = \eta_F|_{\partial T}$ follows since $\phi(0) = 1$ and $\eta_F|_{\partial T\setminus F'} = 0$ for $F'\in\ensuremath{\mathcal{F}}_T\setminus\{F\}$.
We show the details for $n=2$. For $n\geq3$ we may argue similarly.
Let $A_T\colon {\widehat T}\to T$ denote the affine element mapping.
W.l.o.g. let $F\in\ensuremath{\mathcal{F}}_T$ be the face such $A_T\colon {\widehat T}\to T$ maps $F$ to the edge $(0,1)\times\{0\}$.
Then, $d_F\circ A_T(\widehat x,\widehat y) = \widehat y$ for $(\widehat x,\widehat y)\in {\widehat T}$.
Moreover,
\begin{align*}
\eta_{\alpha,F} \circ A_T(\widehat x,\widehat y) &= \widehat\eta_\alpha (\widehat x,\widehat y) := e^{-\widehat y h_T/\alpha} (1-\widehat x-\widehat y)\widehat x.
\end{align*}
The remaining assertions follow by standard calculations, e.g.,
\begin{align*}
\norm{\eta_{\alpha,F}}T^2 = 2|T| \norm{\widehat\eta_\alpha}{{\widehat T}}^2 \lesssim \frac\alpha{h_T} |T|.
\end{align*}
\end{proof}
With the same logic as for the definition of $\eta_{F,j}$ we define
\begin{align*}
\eta_{\alpha,F,j} = \eta_{\alpha,F}\widetilde\chi_{F,j}, \quad j=1,\dots,\dim(P^p(F)), \, F\in\ensuremath{\mathcal{F}}_T.
\end{align*}
The same proof as for Lemma~\ref{lem:modbubble} shows
\begin{lemma}\label{lem:modbubble:ho}
Suppose that $\alpha\lesssim h_T$. Then, for any $F,j$ the function $\eta_{\alpha,F,j}$ satisfies the assertions of Lemma~\ref{lem:modbubble} (replacing $\eta_{\alpha,F}$ by $\eta_{\alpha,F,j}$ and $\eta_F$ by $\eta_{F,j}$).
\end{lemma}
Define the modified Bernardi--Raugel elements by
\begin{align*}
\eeta_{\alpha,F} := \eta_{\alpha,F}\ensuremath{{\boldsymbol{n}}}_F = (\phi\circ d_F)\eeta_F.
\end{align*}
Some important properties of $\eeta_{\alpha,F}$ follow directly from its definition and are summarized in the next result.
Its proof follows the same ideas as the proof of Lemma~\ref{lem:modbubble}.
\begin{lemma}\label{lem:modRB}
For $F\in\ensuremath{\mathcal{F}}_T$ we have $\eeta_{\alpha,F}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = \eeta_F\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T}$
and if $\alpha\lesssim h_T$, then,
\begin{align*}
\norm{\eeta_{\alpha,F}}T \lesssim |T|^{1/2} \left(\frac{\alpha}{h_T}\right)^{1/2},
\quad \norm{{\rm div\,}\eeta_{\alpha,F}}{T}\lesssim |T|^{1/2}h_T^{-1} \left(\frac{\alpha}{h_T}\right)^{-1/2}.
\end{align*}
\end{lemma}
We also need higher order variants: Set $\eeta_{\alpha,\partial T} = \sum_{F\in\ensuremath{\mathcal{F}}_T}\eeta_{\alpha,F}$ and
\begin{align*}
\eeta_{\alpha,\partial T,j} = \eeta_{\alpha,\partial T}\widetilde\nu_{\partial T,j} \quad j=1,\dots,\dim(P_c^{p+1}(\ensuremath{\mathcal{F}}_T)).
\end{align*}
Let $\chi_{\partial T,j}\in P_c^{p+1}(\ensuremath{\mathcal{F}}_T)$, $j=1,\dots,\dim(P_c^{p+1}(\ensuremath{\mathcal{F}}_T))$ denote a basis of $P_c^{p+1}(\ensuremath{\mathcal{F}}_T)$ with
\begin{align*}
\dual{\eeta_{\alpha,\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,k}}_{\partial T} = |\partial T|\delta_{j,k}, \quad
j,k = 1,\dots,\dim(P_c^{p+1}(\partial T)).
\end{align*}
The proof of the next result follows the proof of Lemma~\ref{lem:modRB}.
\begin{lemma}\label{lem:modRB:ho}
The boundedness estimates of Lemma~\ref{lem:modRB} hold with $\eeta_{\alpha,F}$ replaced by $\eeta_{\alpha,\partial T,j}$.
\qed
\end{lemma}
To close this section and to have a better overview we summarize the most important basis functions used in the remainder of this work in Table~\ref{tab:basis}.
\begin{table}[htbp]
\centering
\begin{tabular}{|c|c|c|c|c|c|}\hline
& function & space & basis & definition & property \\ \hline\hline
\multirow{ 6}{*}{\rotatebox[origin=c]{90}{low order}} & $\chi_F$ & $P^0(\ensuremath{\mathcal{F}}_T)$ & yes & $\chi_F|_{F'}=\delta_{F,F'}$ & -- \\ \cline{2-6}
& $\nu_F$ & $P^1(T)$ & yes & $\sum_{z\in\ensuremath{\mathcal{V}}_F}\eta_z-(n-1)d_F$ & $\dual{\nu_F}{\chi_{F'}}_{\partial T} = |F|\delta_{F,F'}$ \\ \cline{2-6}
& ${\boldsymbol\psi}_F$ & $\ensuremath{\boldsymbol{RT}}^0(T)$ & yes & ${\boldsymbol\psi}_F\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = \chi_F$ & $\dual{{\boldsymbol\psi}_F\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{F'}}_{\partial T} = |F|\delta_{F,F'}$ \\ \cline{2-6}
& $\eeta_F$ & $\ensuremath{\boldsymbol{P}}^p(T)$ & no & $\eta_F\ensuremath{{\boldsymbol{n}}}_F$ & $\dual{\eeta_F\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{F'}}_{\partial T} = \dual{\eta_F}1_F\delta_{F,F'}$ \\ \cline{2-6}
& ${\boldsymbol\sigma}_E$ & $\ensuremath{\boldsymbol{P}}^0(T)$ & yes &-- & ${\boldsymbol\sigma}_E\cdot\ensuremath{{\boldsymbol{t}}}_{E'} = \delta_{E,E'}$ \\ \cline{2-6}
& $\eeta_E$ & $\ensuremath{\boldsymbol{P}}^2(T)$ & no & $\eta_E\ensuremath{{\boldsymbol{t}}}_E$ & $\ip{{\boldsymbol\sigma}_E}{\eeta_{E'}}_T = \ip{{\boldsymbol\sigma}_E}{\eeta_{E}}_T\delta_{E,E'}$ \\
\hline\hline
\multirow{ 10}{*}{\rotatebox[origin=c]{90}{higher order}} & $\chi_{T,j}$, $\widetilde\chi_{T,j}$ & $P^p(T)$ & yes & -- & -- \\ \cline{2-6}
& $\eta_{T,j}$ & $P^{p+n+1}(T)$ & no & $\eta_{T}\widetilde\chi_{T,j}$ & $\ip{\eta_{T,j}}{\chi_{T,k}}_T=|T|\delta_{j,k}$ \\ \cline{2-6}
& $\widetilde\chi_{F,j}$ & $P^p(T)$ & yes & -- & -- \\ \cline{2-6}
& $\chi_{F,j}$ & $P^p(\ensuremath{\mathcal{F}}_T)$ & yes & -- & -- \\ \cline{2-6}
& $\eta_{F,j}$ & $P^{p+n}(T)$ & no & $\eta_F\widetilde\chi_{F,j}$ & $\dual{\eta_{F,j}}{\chi_{F',k}}_{\partial T} = \delta_{j,k}\delta_{F,F'}|F|$ \\ \cline{2-6}
& $\nu_{\partial T,j}$, $\widetilde\nu_{\partial T,j}$ & $\widetilde P^{p+1}(T)$ & yes &-- & $\dual{\nu_{\partial T,j}}{\widetilde\nu_{\partial T,k}}_{\partial T} = |\partial T| \delta_{j,k}$ \\ \cline{2-6}
& ${\boldsymbol\psi}_{\partial T,j}$ & $\ensuremath{\boldsymbol{P}}^{p+2}(T)$ & no & $\widetilde\nu_{\partial T,j}\sum_{F\in\ensuremath{\mathcal{F}}_T}{\boldsymbol\psi}_F$ & $\dual{{\boldsymbol\psi}_{\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,k}}_{\partial T} = \delta_{j,k}|\partial T|$ \\ \cline{2-6}
& $\chi_{E,j}$ & $P^p(T)$ & yes & -- & $\ip{\chi_{E,j}}{\widetilde\chi_{T,k}\eta_E}_{T} = \delta_{j,k}|T|$ \\ \cline{2-6}
& ${\boldsymbol\sigma}_{E,j}$ & $\ensuremath{\boldsymbol{P}}^p(T)$ & yes & ${\boldsymbol\sigma}_E\chi_{E,j}$ & -- \\ \cline{2-6}
& $\eeta_{E,j}$ & $\ensuremath{\boldsymbol{P}}^{p+2}(T)$ & no & $\eeta_E\widetilde\chi_{T,j}$ & $\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E',k}}_{T} = \delta_{j,k}\delta_{E,E'}|T|$ \\
\hline\hline
\multirow{ 6}{*}{\rotatebox[origin=c]{90}{modified}} & $\phi$ & -- & -- & $t\mapsto e^{-h_T/\alpha t}$ & -- \\ \cline{2-6}
& $\eta_{\alpha,F}$ & -- & -- & $(\phi\circ d_F)\eta_F$ & $\eta_{\alpha,F}|_{\partial T} = \eta_F|_{\partial T}$ \\ \cline{2-6}
& $\eta_{\alpha,F,j}$ & -- & -- & $\eta_{\alpha,F}\widetilde\chi_{F,j}$ & $\eta_{\alpha,F,j}|_{\partial T} = \eta_{F,j}|_{\partial T}$ \\ \cline{2-6}
& $\eeta_{\alpha,F}$ & -- & -- & $\eta_{\alpha,F}\ensuremath{{\boldsymbol{n}}}_F$ & $\eeta_{\alpha,F}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = \eeta_{F}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T}$ \\ \cline{2-6}
& $\chi_{\partial T,j}$ & $P^{p+1}_c(\ensuremath{\mathcal{F}}_T)$ & yes & -- & -- \\ \cline{2-6}
& $\eeta_{\alpha,\partial T,j}$ & -- & -- & $\eeta_{\alpha,F}\widetilde\nu_{\partial T,j}$ & $\dual{\eeta_{\alpha,F,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,k}}_{\partial T} = |\partial T|\delta_{j,k}$ \\
\hline
\end{tabular}
\caption{Overview of special functions together with some of their main properties.
Here, $\eta_z$ is the canonical Lagrange basis function of $P^1(T)$, $\eta_E$, $\eta_F$, $\eta_T$ are edge, face, and element bubble functions, respectively, and $d_F = \eta_{z_F}$. In column \textit{basis} we indicate whether the respective family of functions generates the indicated space.
For a detailed description we refer to Sections~\ref{sec:low}--\ref{sec:modBubble}.}
\label{tab:basis}
\end{table}
\section{Fortin operator in $H^1(T)$}\label{sec:fortin}
\noindent
We consider a fixed parameter $\alpha>0$ and space $H^1(T)$ equipped with the (squared) norm
\begin{align*}
\norm{v}{T,\alpha}^2 := \norm{v}{T}^2 + \alpha^2 \norm{\nabla v}T^2.
\end{align*}
The idea of this section is to construct Fortin operators, say $\Pi_{\mathrm{F}}^\nabla\colon H^1(T)\to V_h^\nabla$ (with $V_h^\nabla\subset H^1(T)$ being some finite-dimensional subspace) such that, for a fixed $p\in\ensuremath{\mathbb{N}}_0$ and for all $v\in H^1(T)$,
\begin{subequations}\label{eq:fortinprop}
\begin{align}\label{eq:fortinprop:bound}
\norm{\Pi_{\mathrm{F}}^\nabla v}{T,\alpha} &\leq C_\mathrm{F} \norm{v}{T,\alpha}, \\
\label{eq:fortinprop:a}
\dual{\sigma}{v-\Pi_{\mathrm{F}}^\nabla v}_{\partial T} &= 0 \quad\forall \sigma\in P^p(\ensuremath{\mathcal{F}}_T), \\
\label{eq:fortinprop:b}
\ip{u}{v-\Pi_{\mathrm{F}}^\nabla v}_T &= 0 \quad\forall u\in P^p(T)
\end{align}
with $C_\mathrm{F}>0$ independent of $\alpha$, $h_T$ (but possibly dependent on $p$).
Note that~\eqref{eq:fortinprop:a}--\eqref{eq:fortinprop:b} imply
\begin{align}\label{eq:fortinprop:c}
\ip{{\boldsymbol\sigma}}{\nabla(1-\Pi_{\mathrm{F}}^\nabla)v}_T = 0 \quad\forall {\boldsymbol\sigma}\in\ensuremath{\boldsymbol{P}}^{p}(T).
\end{align}
\end{subequations}
This can be seen from integration by parts: Take ${\boldsymbol\sigma}\in \ensuremath{\boldsymbol{P}}^p(T)$, $v\in H^1(T)$. Then, ${\rm div\,}{\boldsymbol\sigma}\in P^{p-1}(T)$ and
\begin{align*}
\ip{{\boldsymbol\sigma}}{\nabla \Pi_{\mathrm{F}}^\nabla v}_T &= -\ip{{\rm div\,}{\boldsymbol\sigma}}{\Pi_{\mathrm{F}}^\nabla v}_T + \dual{{\boldsymbol\sigma}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\Pi_{\mathrm{F}}^\nabla v}_{\partial T}
\\
&= -\ip{{\rm div\,}{\boldsymbol\sigma}}{v}_T + \dual{{\boldsymbol\sigma}\cdot\ensuremath{{\boldsymbol{n}}}_T}{v}_{\partial T} = \ip{{\boldsymbol\sigma}}{\nabla v}_T.
\end{align*}
From the last identities we also see that the weaker condition
\begin{align}\tag{\ref{eq:fortinprop:b}'}\label{eq:fortinprop:b:alt}
\ip{u}{v-\Pi_{\mathrm{F}}^\nabla v}_T &= 0 \quad\forall u\in P^{p-1}(T)
\end{align}
would be sufficient to conclude~\eqref{eq:fortinprop:c}.
However, depending on the problem, condition~\eqref{eq:fortinprop:b} is needed, e.g., in the presence of reaction terms as in the DPG method in Section~\ref{sec:num}.
We stress that our Fortin operator can be easily modified to satisfy~\eqref{eq:fortinprop:b:alt} only.
\subsection{Constructions for moderate parameter}\label{sec:grad:moderate}
Define the space
\begin{align*}
\widetilde V_{hp}^\nabla = P^0(T) + \linhull\set{\eta_{F,j}}{j=1,\dots,\dim(P^p(F)), F\in\ensuremath{\mathcal{F}}_T}
\end{align*}
and operator $\widetilde{\Pi}_{\mathrm{F},hp}^\nabla \colon H^1(T)\to \widetilde V_{hp}^\nabla$ for $v\in H^1(T)$ by
\begin{align*}
\widetilde{\Pi}_{\mathrm{F},hp}^\nabla v = \Pi_T^0 v + \sum_{F\in\ensuremath{\mathcal{F}}_T} \sum_{j=1}^{\dim(P^p(F))} \frac{\dual{\chi_{F,j}}{(1-\Pi_T^0)v}_F}{\dual{\chi_{F,j}}{\eta_{F,j}}_F} \eta_{F,j}.
\end{align*}
The following result collects its main properties.
\begin{lemma}\label{lem:fortin:normal}
Operator $\Pi_{\mathrm{F}}^\nabla=\widetilde{\Pi}_{\mathrm{F},hp}^\nabla$ is idempotent on $P^0(T)$, satisfies property~\eqref{eq:fortinprop:a} and
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp}^\nabla v}T &\lesssim \norm{v}T + h_T\norm{\nabla v}T, \quad
\norm{\nabla \widetilde{\Pi}_{\mathrm{F},hp}^\nabla v}T \lesssim \norm{\nabla v}T,
\quad\norm{(1-\widetilde{\Pi}_{\mathrm{F},hp}^\nabla)v}T \lesssim h_T\norm{\nabla v}T
\end{align*}
for all $v\in H^1(T)$.
\end{lemma}
\begin{proof}
Idempotency can be seen from the definition, since $v\in P^0(T)$ implies that $\Pi_T^0 v =v$, thus, $(1-\Pi_T^0)v = 0$.
Let $v\in H^1(T)$. To see~\eqref{eq:fortinprop:a} we employ the orthogonality property $\dual{\eta_{F,j}}{\chi_{F',k}}_{\partial T} = \delta_{F,F'}\delta_{j,k}|F|$ to get
\begin{align*}
\dual{\chi_{F',k}}{v-\widetilde{\Pi}_{\mathrm{F}}^\nabla v}_{\partial T} &= \dual{\chi_{F',k}}{v-\Pi_T^0v}_{\partial T} - \sum_{F\in\ensuremath{\mathcal{F}}_T} \sum_{j=1}^{\dim(P^p(F))}
\frac{\dual{\chi_{F,j}}{(1-\Pi_T^0)v}_F}{\dual{\chi_{F,j}}{\eta_{F,j}}_F}\dual{\chi_{F',k}}{\eta_{F,j}}_{\partial T}
\\
&= \dual{\chi_{F',k}}{v-\Pi_T^0v}_{\partial T} - \dual{\chi_{F',k}}{(1-\Pi_T^0)v}_{\partial T} = 0
\end{align*}
Since $F'$ and $k$ were arbitrary, condition~\eqref{eq:fortinprop:a} follows.
The boundedness follows from the triangle inequality, the Cauchy--Schwarz inequality and boundedness of $\Pi_T^0$, i.e.,
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp}^\nabla v}{T} \leq \norm{v}T + \sum_{F\in\ensuremath{\mathcal{F}}_T} \sum_{j=1}^{\dim(P^p(T))}\frac{\norm{\chi_{F,j}}F\norm{v-\Pi_T^0 v}F}{\dual{\chi_{F,j}}{\eta_{F,j}}_F} \norm{\eta_{F,j}}T.
\end{align*}
Note that $\dual{\chi_{F,j}}{\eta_{F,j}}_F = |F|$, $\norm{\eta_{F,j}}T \eqsim |T|^{1/2}$, $\norm{\chi_{F,j}}F\eqsim |F|^{1/2}$ which follows by standard scaling arguments and the properties of the basis functions discussed in Section~\ref{sec:basis}.
Applying the trace inequality~\eqref{eq:traceineq} we see that
\begin{align*}
\frac{\norm{\chi_{F,j}}F\norm{v-\Pi_T^0 v}F}{\dual{\chi_{F,j}}{\eta_{F,j}}_F} \norm{\eta_{F,j}}T
&\eqsim |F|^{-1/2}|T|^{1/2} \norm{v-\Pi_T^0 v}{F} \lesssim h_T^{1/2} h_T^{1/2} \norm{\nabla v}T.
\end{align*}
Thus, we conclude that $\norm{\widetilde{\Pi}_{\mathrm{F},hp}^\nabla v}T \lesssim \norm{v}T + h_T\norm{\nabla v}T$.
Then, with similar arguments but using the inverse estimate $\norm{\nabla \eta_{F,j}}T \lesssim h_T^{-1}|T|^{1/2}$, we see that ($\nabla \Pi_T^0 v=0$)
\begin{align*}
\norm{\nabla \widetilde{\Pi}_{\mathrm{F},hp}^\nabla v}T \leq \sum_{F\in\ensuremath{\mathcal{F}}_T} \sum_{j=1}^{\dim(P^p(F))} \frac{\norm{\chi_{F,j}}F\norm{v-\Pi_T^0 v}F}{\dual{\chi_{F,j}}{\eta_{F,j}}_F} \norm{\nabla \eta_{F,j}}T \lesssim \norm{\nabla v}T.
\end{align*}
Finally, the approximation property is derived by using the idempotency, and the established boundedness estimates, i.e.,
\begin{align*}
\norm{(1-\widetilde{\Pi}_{\mathrm{F},hp}^\nabla)v}T = \norm{(1-\widetilde{\Pi}_{\mathrm{F},hp}^\nabla)(v-\Pi_T^0v)}T \lesssim \norm{v-\Pi_T^0v}T + h_T \norm{\nabla(v-\Pi_T^0v)}T
\lesssim h_T\norm{\nabla v}T.
\end{align*}
This concludes the proof.
\end{proof}
To obtain an operator that also satisfies property~\eqref{eq:fortinprop:b} we consider slight modifications by adding a correction term based on element bubbles.
Define the space
\begin{align*}
V_{hp}^{\nabla} = \widetilde V_{hp}^{\nabla} + \linhull\set{\eta_{T,j}}{j=1,\dots,\dim(P^p(T))}
\end{align*}
and the operator $\Pi_{\mathrm{F},hp}^\nabla\colon H^1(T)\to V_{hp}^\nabla$ for all $v\in H^1(T)$ by
\begin{align*}
\Pi_{\mathrm{F},hp}^\nabla v = \widetilde{\Pi}_{\mathrm{F},hp}^\nabla v + \sum_{j=1}^{\dim(P^p(T))} \frac{\ip{\chi_{T,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^\nabla)v}_T}{\ip{\chi_{T,j}}{\eta_{T,j}}_T}\eta_{T,j}.
\end{align*}
\begin{theorem}\label{thm:fortin:normal}
Operator $\Pi_{\mathrm{F}}^\nabla = \Pi_{\mathrm{F},hp}^\nabla$ is idempotent on $P^0(T)$, satisfies~\eqref{eq:fortinprop:a}--\eqref{eq:fortinprop:b} and
\begin{align*}
\norm{\Pi_{\mathrm{F},hp}^\nabla v}T &\lesssim \norm{v}T + h_T\norm{\nabla v}T, \quad
\norm{\nabla \Pi_{\mathrm{F},hp}^\nabla v}T \lesssim \norm{\nabla v}T,
\quad \norm{(1-\Pi_{\mathrm{F},hp}^\nabla)v}T \lesssim h_T\norm{\nabla v}T
\end{align*}
for all $v\in H^1(T)$.
\end{theorem}
\begin{proof}
The idempotency on $P^0(T)$ follows from the idempotency of $\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla}$ (Lemma~\ref{lem:fortin:normal}).
Statement~\eqref{eq:fortinprop:a} follows also from Lemma~\ref{lem:fortin:normal} since the element bubbles $\eta_{T,j}$ vanish on the boundary and, therefore, $\Pi_{\mathrm{F},hp}^\nabla v|_{\partial T} = \widetilde{\Pi}_{\mathrm{F},hp}^{\nabla}v|_{\partial T}$.
To see~\eqref{eq:fortinprop:b} a simple calculation using the orthogonality $\ip{\chi_{T,j}}{\eta_{T,k}}_T = |T|\delta_{j,k}$ yields
\begin{align*}
\ip{\chi_{T,k}}{(1-\Pi_{\mathrm{F}}^\nabla)v}_T &= \ip{\chi_{T,k}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla})v}_T - \sum_{j=1}^{\dim(P^p(T))}\frac{\ip{\chi_{T,k}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla})v}_T}{\ip{\chi_{T,j}}{\eta_{T,j}}_T}\ip{\chi_{T,k}}{\eta_{T,j}}_T \\
&= \ip{\chi_{T,k}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla})v}_T -\ip{\chi_{T,k}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla})v}_T = 0.
\end{align*}
It remains to prove the boundedness estimates which follow --- besides standard arguments --- from the boundedness estimates of $\widetilde{\Pi}_{\mathrm{F}}^{\nabla,j}$ (see Lemma~\ref{lem:fortin:normal}).
First, using the triangle inequality, Cauchy--Schwarz inequality and scaling arguments we estimate
\begin{align*}
\norm{\Pi_{\mathrm{F},hp}^\nabla v}T &\lesssim \norm{\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla}v}T + \sum_{j=1}^{\dim(P^p(T))}|T|^{-1/2}\norm{v-\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla}v}T \norm{\eta_{T,j}}T
\\
&\lesssim \norm{\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla}v}T + \norm{v-\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla}v}T \lesssim \norm{v}T + h_T\norm{\nabla v}T.
\end{align*}
The gradient contribution is estimated by employing the inverse estimate $\norm{\nabla \eta_{T,j}}{T}\lesssim h_T^{-1}\norm{\eta_{T,j}}T$ together with Lemma~\ref{lem:fortin:normal} to give
\begin{align*}
\norm{\nabla\Pi_{\mathrm{F},hp}^\nabla v}{T} \lesssim \norm{\nabla\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla} v}T + h_T^{-1}\norm{v-\widetilde{\Pi}_{\mathrm{F},hp}^{\nabla}v}T
\lesssim \lesssim \norm{\nabla v}T.
\end{align*}
The final assertion $\norm{(1-\Pi_{\mathrm{F},hp}^\nabla)v}T \lesssim h_T\norm{\nabla v}T$ follows as in Lemma~\ref{lem:fortin:normal}.
\end{proof}
\begin{corollary}
Suppose that $h_T\lesssim \alpha$.
From Lemma~\ref{lem:fortin:normal} and Theorem~\ref{thm:fortin:normal} it follows that $\Pi_{\mathrm{F}}^\nabla = \widetilde{\Pi}_{\mathrm{F},hp}^\nabla$ and $\Pi_{\mathrm{F}}^\nabla = \Pi_{\mathrm{F},hp}^\nabla$ satisfy~\eqref{eq:fortinprop:bound}.
Particularly, $\Pi_{\mathrm{F}}^\nabla = \Pi_{\mathrm{F},hp}^\nabla$ has properties~\eqref{eq:fortinprop}.
\end{corollary}
\begin{remark}\label{rem:supconv:grad}
In general, discrete test spaces are chosen such that a Fortin operator exists, which not necessarily implies approximation results of the form
\begin{align*}
\min_{v_h\in V_h} \norm{v-v_h}{T} + \norm{\nabla(v-v_h)}T \lesssim h_T |v|_{2,T} \quad\text{for } v\in H^2(T).
\end{align*}
Here, $|\cdot|_{2,T}$ denotes the $H^2(T)$ seminorm.
For certain supercloseness results in the DPG method the latter approximation property is needed, see~\cite{SupConv2}.
To ensure this property one can simply require $P^1(T)\subset V_h$.
\end{remark}
\subsection{Constructions for small parameter}
In this section we focus on the case $\alpha\lesssim h_T$.
Let $\Pi_{\mathrm{F}}^\nabla=\widetilde{\Pi}_{\mathrm{F},hp}^\nabla$ or $\Pi_{\mathrm{F}}^\nabla=\Pi_{\mathrm{F},hp}^\nabla$. By Lemma~\ref{lem:fortin:normal} resp. Theorem~\ref{thm:fortin:normal} we have the boundedness
\begin{align}\label{eq:bound:fortin}
\norm{\Pi_{\mathrm{F}}^\nabla v}T \lesssim \norm{v}T + h_T \norm{\nabla v}T \lesssim \max\{1,h_T\alpha^{-1}\} \left(\norm{v}T + \alpha\norm{\nabla v}T\right).
\end{align}
We conclude that $\norm{\Pi_{\mathrm{F}}^\nabla v}{T,\alpha} \lesssim \max\{1,h_T\alpha^{-1}\} \norm{v}{T,\alpha}$.
This tells us that $\Pi_{\mathrm{F}}^\nabla$ is only conditionally uniformly bounded. Particularly, for small parameters $\alpha$ and coarse meshes huge boundedness constants are expected so that robustness of the numerical methods is likely to be lost. This can actually be observed in our numerical experiments presented in Section~\ref{sec:num}.
Let us remark that the operators constructed in~\cite{practicalDPG,breakSpace,DemkowiczZanotti20} also satisfy~\eqref{eq:bound:fortin} and are not suited for small parameters, $\alpha\lesssim h_T$.
To overcome this problem we construct an operator based on the modified face bubble functions $\eta_{\alpha,F,j}$ instead of $\eta_{F,j}$.
The construction of the novel Fortin operators follows the definition of $\widetilde{\Pi}_{\mathrm{F},hp}^\nabla$ and $\Pi_{\mathrm{F},hp}^\nabla$ replacing $\eta_{F,j}$ with the modified face bubble functions $\eta_{\alpha,F,j}$. First, set
\begin{align*}
\widetilde V_{hp,\alpha}^\nabla &:= P^0(T) + \linhull\{\eta_{\alpha,F,j}\,:\, j=1,\dots,\dim(P^p(F)), F\in\ensuremath{\mathcal{F}}_T\},
\end{align*}
and define $\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla \colon H^1(T)\to \widetilde V_{hp,\alpha}^\nabla$ for all $v\in H^1(T)$ by
\begin{align*}
\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v &:= \Pi_T^0 v + \sum_{F\in\ensuremath{\mathcal{F}}_T} \sum_{j=1}^{\dim(P^p(F))} \frac{\dual{\chi_{F,j}}{(1-\Pi_T^0)v}_F}{\dual{\chi_{F,j}}{\eta_{\alpha,F,j}}_F} \eta_{\alpha,F,j}.
\end{align*}
Its main properties are given in
\begin{lemma}\label{lem:fortin}
Operator $\Pi_{\mathrm{F}}^\nabla = \widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla$ satisfies~\eqref{eq:fortinprop:a} and is idempotent on $P^0(T)$.
If $\alpha\lesssim h_T$, then
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}{T,\alpha} \lesssim \norm{v}{T,\alpha}, \quad
\norm{v-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}T &\lesssim h_T \norm{\nabla v}{T}
\end{align*}
for all $v\in H^1(T)$.
\end{lemma}
\begin{proof}
The idempotency on $P^0(T)$ can be seen directly from the definition of the operator.
Noting that $\eta_{\alpha,F,j}|_{\partial T} = \eta_{F,j}|_{\partial T}$ for any $F,j$ the proof of Fortin property~\eqref{eq:fortinprop:a} follows as in Lemma~\ref{lem:fortin:normal}.
It remains to prove the boundedness estimates. Suppose that $\alpha\lesssim h_T$.
First, using the triangle inequality and the Cauchy--Schwarz inequality together with the properties of the modified bubble (Lemma~\ref{lem:modbubble:ho}), $\norm{\chi_{F,j}}F \eqsim |F|^{1/2}$,
$\dual{\chi_{F,j}}{\eta_{\alpha,F,j}}_F = |F|$ we infer
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}T &\leq \norm{v}T + \sum_{F\in\ensuremath{\mathcal{F}}_T} \sum_{j=1}^{\dim(P^p(F))} \frac{\norm{\chi_{F,j}}F\norm{v-\Pi_T^0v}F}{\dual{\chi_{F,j}}{\eta_{\alpha,F,j}}} \norm{\eta_{\alpha,F,j}}T
\\ &\lesssim \norm{v}T + \sum_{F\in\ensuremath{\mathcal{F}}_T} \norm{v-\Pi_T^0v}F |F|^{-1/2} |T|^{1/2} \alpha^{1/2}h_T^{-1/2}
\eqsim \norm{v}T + \sum_{F\in\ensuremath{\mathcal{F}}_T} \norm{v-\Pi_T^0v}F \alpha^{1/2}.
\end{align*}
Then, with the multiplicative trace inequality~\eqref{eq:traceineq} and Young's inequality we further get
\begin{align*}
\norm{v-\Pi_T^0v}F \alpha^{1/2} &\lesssim \norm{v-\Pi_T^0 v}T^{1/2}\alpha^{1/2}\norm{\nabla v}T^{1/2}
\lesssim \norm{v-\Pi_T^0 v}T + \alpha \norm{\nabla v}T.
\end{align*}
Putting all the estimates together we infer that
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}T \lesssim \norm{v}T + \sum_{F\in\ensuremath{\mathcal{F}}_T}\norm{v-\Pi_T^0v}F \alpha^{1/2} \lesssim \norm{v}T + \alpha \norm{\nabla v}T.
\end{align*}
We are left with the gradient contribution of the norm. With similar arguments as before we get
\begin{align*}
\alpha\norm{\nabla \widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}T &\leq \alpha\sum_{F\in\ensuremath{\mathcal{F}}_T} \sum_{j=1}^{\dim(P^p(F))} \frac{\norm{\chi_{F,j}}F\norm{v-\Pi_T^0v}F}{\dual{\chi_{F,j}}{\eta_{\alpha,F,j}}} \norm{\nabla \eta_{\alpha,F,j}}T
\\
&\lesssim \alpha\sum_{F\in\ensuremath{\mathcal{F}}_T} \alpha^{-1/2} \norm{v-\Pi_T^0 v}F
\\
&\lesssim \alpha^{1/2} \norm{v-\Pi_T^0v}T^{1/2}\norm{\nabla v}T^{1/2} \lesssim \norm{v}T + \alpha\norm{\nabla v}T.
\end{align*}
Combining all the estimates from above we conclude the boundedness of the Fortin operator.
Finally, to see the approximation property note that $\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla 1 = 1$, thus,
\begin{align*}
\norm{(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla)v}T &= \norm{(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla)(v-\Pi_T^0v)}T \lesssim \norm{v-\Pi_T^0v}T + \alpha \norm{\nabla v}T \lesssim h_T \norm{\nabla v}T.
\end{align*}
Note that we have used that $\alpha\lesssim h_T$.
\end{proof}
To ensure property~\eqref{eq:fortinprop:b} we follow the idea of the definition of $\Pi_{\mathrm{F},hp}^\nabla$ by adding correction terms based on element bubble functions. Contrary to the face bubble functions, the element bubble functions do not need to be modified.
Set
\begin{align*}
V_{hp,\alpha}^\nabla = \widetilde V_{hp,\alpha}^\nabla+\linhull\set{\eta_{T,j}}{j=1,\dots,\dim(P^p(T))}
\end{align*}
and define $\Pi_{\mathrm{F},hp,\alpha}^\nabla\colon H^1(T)\to V_{hp,\alpha}^\nabla$ for all $v\in H^1(T)$ by
\begin{align*}
\Pi_{\mathrm{F},hp,\alpha}^\nabla v:= \widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v + \sum_{j=1}^{\dim(P^p(T))} \frac{\ip{\chi_{T,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla)v}_T}{\ip{\chi_{T,j}}{\eta_{T,j}}_T}\eta_{T,j}.
\end{align*}
The following theorem is one of our main results.
\begin{theorem}\label{thm:fortin}
Suppose that $\alpha\lesssim h_T$.
Then, $\Pi_{\mathrm{F}}^\nabla = \Pi_{\mathrm{F},hp,\alpha}^\nabla$ is idempotent on $P^0(T)$ and satisfies~\eqref{eq:fortinprop}.
\end{theorem}
\begin{proof}
Idempotency follows from the definition.
Verifying~\eqref{eq:fortinprop:a}--\eqref{eq:fortinprop:b} follows as in Theorem~\ref{thm:fortin:normal}.
For the boundedness estimate we also use the arguments displayed in the proof of Theorem~\ref{thm:fortin:normal}. Note that with Lemma~\ref{lem:fortin} and $\norm{\nabla\eta_{T,j}}T\lesssim h_T^{-1}|T|^{1/2}$ we see that, for $v\in H^1(T)$,
\begin{align*}
\norm{\Pi_{\mathrm{F},hp,\alpha}^\nabla v}{T,\alpha} &\lesssim \norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}{T,\alpha} +
\sum_{j=1}^{\dim(P^p(T))}\frac{\ip{\chi_{T,j}}{v-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}_T}{\ip{\chi_{T,j}}{\eta_{T,j}}_T} (\norm{\eta_{T,j}}T + \alpha\norm{\nabla \eta_{T,j}}{T})
\\
&\lesssim \norm{v}{T,\alpha} + |T|^{-1/2}\norm{v-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}T |T|^{1/2} + |T|^{-1/2}\norm{v-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}T \alpha h_T^{-1}|T|^{1/2}
\\
&\lesssim \norm{v}{T,\alpha} + \norm{v-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla v}T.
\end{align*}
In the last step we used $\alpha\lesssim h_T$. Boundedness of $\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^\nabla$ finishes the proof.
\end{proof}
\subsection{Alternative operator for lowest-order spaces and moderate parameter}
In this section we construct a Fortin operator $\Pi_{\mathrm{F}}^{\nabla}\colon H^1(T)\to V_{h}^{\nabla,0}$ such that~\eqref{eq:fortinprop} is satisfied for the lowest-order case $p=0$. In~\cite{CarstensenGHW14} it is shown that for a low-order DPG method for the Poisson problem, test space $V_h^\nabla = P^1(T)$ for the scalar test functions is sufficient to guarantee well-posedness. The authors of~\cite{CarstensenGHW14} did use different techniques.
Here, we complement their results by the construction of a Fortin operator. Define the spaces
\begin{align*}
\widetilde V_h^{\nabla,0} = P^1(T), \quad V_h^{\nabla,0} = \widetilde V_h^{\nabla,0} + \linhull\{\eta_T\}
\end{align*}
and operators $\widetilde{\Pi}_{\mathrm{F}}^{\nabla,0}\colon H^1(T)\to \widetilde V_h^{\nabla,0}$, $\Pi_{\mathrm{F}}^{\nabla,0}\colon H^1(T)\to V_h^{\nabla,0}$ for all $v\in H^1(T)$ by
\begin{align*}
\widetilde{\Pi}_{\mathrm{F}}^{\nabla,0} v &= \Pi_T^0v + \sum_{F\in\ensuremath{\mathcal{F}}_T} \frac{\dual{1}{(1-\Pi_T^0)v}_F}{\dual{1}{\nu_F}_F}\nu_F, \\
\Pi_{\mathrm{F}}^{\nabla,0} v&= \widetilde{\Pi}_{\mathrm{F}}^{\nabla,0} v + \frac{\ip{1}{(1-\widetilde{\Pi}_{\mathrm{F}}^{\nabla,0})v}_T}{\ip{1}{\eta_T}_T} \eta_T.
\end{align*}
The analysis of these operators can be done as in Section~\ref{sec:grad:moderate}. The details are left to the reader.
\begin{theorem}
Let $p=0$.
Operator $\Pi_{\mathrm{F}}^\nabla = \widetilde{\Pi}_{\mathrm{F}}^{\nabla,0}$ resp. $\Pi_{\mathrm{F}}^\nabla = \Pi_{\mathrm{F}}^{\nabla,0}$ is idempotent on $P^0(T)$, satisfies~\eqref{eq:fortinprop:a} and
\begin{align*}
\norm{\Pi_{\mathrm{F}}^\nabla v}T \lesssim \norm{v}T + h_T\norm{\nabla v}T, \quad
\norm{\nabla \Pi_{\mathrm{F}}^\nabla v}T \lesssim \norm{\nabla v}T
\end{align*}
for all $v\in H^1(T)$. Operator $\Pi_{\mathrm{F}}^\nabla = \Pi_{\mathrm{F}}^{\nabla,0}$ additionally satisfies~\eqref{eq:fortinprop:b}.
\qed
\end{theorem}
\subsection{Comparison with existing Fortin operators}\label{sec:fortin:existing}
As mentioned in the introduction, several works have already dealt with the construction of Fortin operators on a simplex that satisfy~\eqref{eq:fortinprop}, see, e.g.,~\cite{practicalDPG,breakSpace,DemkowiczZanotti20}.
These works all have in common that they construct resp. prove the existence of a Fortin operator $\Pi_{\mathrm{F}}^\nabla\colon H^1(T) \to P^{p+n}(T)$ which satisfy~\eqref{eq:fortinprop:bound}-\eqref{eq:fortinprop:a} (for $\alpha\gtrsim h_T$) and
\begin{align*}
\ip{u}{(1-\Pi_{\mathrm{F}}^\nabla)v}_T = 0 \quad\forall u\in P^{p-1}(T), \, v\in H^1(T).
\end{align*}
The latter condition is not the same as~\eqref{eq:fortinprop:b}. In order to satisfy~\eqref{eq:fortinprop:b} one needs to increase the polynomial degree by one, i.e., $P^{p+1+n}(T)$ instead of $P^{p+n}(T)$.
Comparing the dimension of $P^{p+1+n}(T)$ and the space $V_{hp}^\nabla$, we get
\begin{align*}
\dim(P^{p+1+n}(T)) &= \frac{\prod_{j=1}^n(p+1+n+j)}{n!}, \\
\dim(V_{hp}^\nabla) &= 1 + (n+1)\frac{\prod_{j=1}^{n-1}(p+j)}{(n-1)!} + \frac{\prod_{j=1}^n(p+j)}{n!}.
\end{align*}
For the lowest-order case $p=0$ we thus find
\begin{align*}
\dim(P^{n+1}(T)) &= \begin{cases}
10 & n=2, \\
35 & n=3,
\end{cases}
\quad\text{and}\quad
\dim(V_{h0}^\nabla) = \begin{cases}
5 & n=2, \\
6 & n=3.
\end{cases}
\end{align*}
For operator $\Pi_{\mathrm{F}}^{\nabla,0}\colon H^1(T)\to V_h^{\nabla,0}$ we even have a reduction to
\begin{align*}
\dim(V_{h}^{\nabla,0}) = n+1+1 =
\begin{cases}
4 & n=2, \\
5 & n=3.
\end{cases}
\end{align*}
In conclusion, our test spaces are systematically smaller than previously used ones, and guarantee robustness contrary to the previous cases.
\section{Fortin operator in $\Hdivset{T}$}\label{sec:fortinDiv}
We consider a fixed parameter $\alpha>0$ and space $\Hdivset{T}$ equipped with the (squared) norm
\begin{align*}
\norm{{\boldsymbol\tau}}{T,\alpha}^2 = \norm{{\boldsymbol\tau}}{T}^2 + \alpha^2 \norm{{\rm div\,}{\boldsymbol\tau}}{T}^2 \quad\text{for } {\boldsymbol\tau}\in\Hdivset{T}.
\end{align*}
The motivation for this section is the construction of Fortin operators, say $\Pi_{\mathrm{F}}^{{\rm div\,}}\colon \Hdivset{T}\to V_h$ (where $V_h\subseteq\Hdivset{T}$ is some discrete space) such that, for all ${\boldsymbol\tau}\in \Hdivset{T}$,
\begin{subequations}\label{eq:fortinDiv}
\begin{align}
\norm{\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}{T,\alpha} &\leq C_\mathrm{F} \norm{{\boldsymbol\tau}}{T,\alpha}, \label{eq:fortinDiv:bound}\\
\dual{({\boldsymbol\tau}-\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau})\cdot\ensuremath{{\boldsymbol{n}}}_T}{u}_{\partial T} &= 0 \quad\forall u\in P_c^{p+1}(\ensuremath{\mathcal{F}}_T), \label{eq:fortinDiv:a}\\
\ip{{\boldsymbol\sigma}}{{\boldsymbol\tau}-\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}_{T} &= 0 \quad\forall {\boldsymbol\sigma} \in \ensuremath{\boldsymbol{P}}^p(T) \label{eq:fortinDiv:b}
\end{align}
with $C_\mathrm{F}>0$ independent of $\alpha$, $h_T$ (but possibly dependent on $p\in\ensuremath{\mathbb{N}}_0$).
The latter two identities also imply
\begin{align}
\ip{u}{{\rm div\,}({\boldsymbol\tau}-\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau})}_T = 0 \quad\forall u\in P^{p+1}(T), \label{eq:fortinDiv:c}
\end{align}
\end{subequations}
which can be seen from integration by parts: Let $u\in P^{p+1}(T)$ be given, then
\begin{align*}
\ip{u}{{\rm div\,}\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}_T &= -\ip{\nabla u}{\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}_T + \dual{\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{u|_{\partial T}}_{\partial T}
\\
&=-\ip{\nabla u}{\boldsymbol\tau}_T + \dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{u|_{\partial T}}_{\partial T} = \ip{u}{{\rm div\,}{\boldsymbol\tau}}_T \quad\forall {\boldsymbol\tau}\in \Hdivset{T}.
\end{align*}
\subsection{Construction for moderate parameter}
Define
\begin{align*}
\widetilde V_{hp}^{{\rm div\,}} = \linhull\set{{\boldsymbol\psi}_{\partial T,j}}{j=1,\dots,\dim(\widetilde P^{p+1}(T))} \subset \ensuremath{\boldsymbol{P}}^{p+2}(T)
\end{align*}
and operator $\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}\colon \Hdivset{T}\to \widetilde V_{hp}^{{\rm div\,}}$ by
\begin{align*}
\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau} = \sum_{j=1}^{\dim(P_c^{p+1}(\partial T))} \frac{\dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,j}}_{\partial T}}{\dual{{\boldsymbol\psi}_{\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,j}}_{\partial T}} {\boldsymbol\psi}_{\partial T,j}.
\end{align*}
We collect its main properties.
\begin{lemma}\label{lem:fortinDiv:normal}
Operator $\Pi_{\mathrm{F}}^{{\rm div\,}} = \widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}$ satisfies~\eqref{eq:fortinDiv:a} and
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}}T &\lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}{T} \quad \forall {\boldsymbol\tau}\in\Hdivset{T}.
\end{align*}
\end{lemma}
\begin{proof}
To see~\eqref{eq:fortinDiv:a} a simple computation yields
\begin{align*}
\dual{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,k}}_{\partial T}
&= \dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,k}}_{\partial T} -
\sum_{j}\frac{\dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,j}}_{\partial T}}{\dual{{\boldsymbol\psi}_{\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,j}}_{\partial T}} \dual{{\boldsymbol\psi}_{\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,k}}_{\partial T}
\\
&= \dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,k}}_{\partial T} - \dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,k}}_{\partial T} = 0.
\end{align*}
Resolving the duality term, the Cauchy--Schwarz inequality and properties of basis functions give
\begin{align*}
|\dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,j}}_{\partial T}| &= |\ip{{\boldsymbol\tau}}{\nabla \nu_{\partial T,j}}_T + \ip{{\rm div\,}{\boldsymbol\tau}}{\nu_{\partial T,j}}_T|
\\
&\lesssim \norm{{\boldsymbol\tau}}Th_T^{-1}|T|^{1/2} + \norm{{\rm div\,}{\boldsymbol\tau}}T|T|^{1/2}.
\end{align*}
With the triangle inequality we conclude that
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}}T &\leq \sum_{j} \frac{|\dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,j}}_{\partial T}|}{\dual{{\boldsymbol\psi}_{\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_{\partial T,j}}_{\partial T}} \norm{{\boldsymbol\psi}_{\partial T,j}}T
\\
&\lesssim |\partial T|^{-1} (\norm{{\boldsymbol\tau}}Th_T^{-1}|T|^{1/2} + \norm{{\rm div\,}{\boldsymbol\tau}}T|T|^{1/2})|T|^{1/2} \lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}T
\end{align*}
which finishes the proof.
\end{proof}
For the definition of operators that ensure~\eqref{eq:fortinDiv:b} we add a correction term based on the functions $\eeta_{E,j}$. We set
\begin{align*}
V_{hp}^{{\rm div\,}} = \widetilde V_{hp}^{{\rm div\,}} + \linhull\set{\eeta_{E,j}}{j=1,\dots,\dim P^p(T),\, E\in \ensuremath{\mathcal{E}}_*}
\end{align*}
and define $\Pi_{\mathrm{F},hp}^{{\rm div\,}}\colon \Hdivset{T}\to V_{hp}^{{\rm div\,}}$ for all ${\boldsymbol\tau}\in \Hdivset{T}$ by
\begin{align*}
\Pi_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau} = \widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau} + \sum_{E\in\ensuremath{\mathcal{E}}_*} \sum_{j=1}^{\dim(P^p(T))} \frac{\ip{{\boldsymbol\sigma}_{E,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T}{\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E,j}}_T} \eeta_{E,j}.
\end{align*}
\begin{theorem}\label{thm:fortinDiv:normal}
Operator $\Pi_{\mathrm{F}}^{{\rm div\,}} = \Pi_{\mathrm{F},hp}^{{\rm div\,}}$ satisfies~\eqref{eq:fortinDiv:a}--\eqref{eq:fortinDiv:b} and
\begin{align*}
\norm{\Pi_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}}T \lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}T \quad\forall {\boldsymbol\tau}\in \Hdivset{T}.
\end{align*}
Furthermore, ${\rm div\,}\circ\Pi_{\mathrm{F},hp}^{{\rm div\,}} = \Pi_T^{p+1}\circ{\rm div\,}$.
If $h_T\lesssim \alpha$, then $\Pi_{\mathrm{F}}^{{\rm div\,}} = \Pi_{\mathrm{F},hp}^{{\rm div\,}}$ satisfies~\eqref{eq:fortinDiv:bound}.
\end{theorem}
\begin{proof}
Since $\eeta_{E,j}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = 0$, we get $\Pi_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = \widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T}$, thus, condition~\eqref{eq:fortinDiv:a} follows from Lemma~\ref{lem:fortinDiv:normal}.
To see~\eqref{eq:fortinDiv:b} we compute for any $E',k$
\begin{align*}
\ip{{\boldsymbol\sigma}_{E',k}}{(1-\Pi_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T &= \ip{{\boldsymbol\sigma}_{E',k}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T - \sum_{E\in\ensuremath{\mathcal{E}}_*}\sum_{j} \frac{\ip{{\boldsymbol\sigma}_{E,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T}{\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E,j}}_T} \ip{{\boldsymbol\sigma}_{E',k}}{\eeta_{E,j}}_T
\\
&= \ip{{\boldsymbol\sigma}_{E',k}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T -\ip{{\boldsymbol\sigma}_{E',k}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T =0.
\end{align*}
Noting that properties of the basis functions and boundedness by Lemma~\ref{lem:fortinDiv:normal} give
\begin{align*}
\frac{|\ip{{\boldsymbol\sigma}_{E,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T|}{\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E,j}}_T} \norm{\eeta_{E,j}}T
\lesssim \norm{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}T \lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}T,
\end{align*}
we see that
\begin{align*}
\norm{\Pi_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}}T \leq \norm{\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}}T + \sum_{E\in\ensuremath{\mathcal{E}}_*} \sum_{j} \frac{|\ip{{\boldsymbol\sigma}_{E,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,}}){\boldsymbol\tau}}_T|}{\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E,j}}_T} \norm{\eeta_{E,j}}T \lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}T.
\end{align*}
The commutativity property follows from~\eqref{eq:fortinDiv:c} and the fact that $V_{hp}^{{\rm div\,}}\subseteq \ensuremath{\boldsymbol{P}}^{p+2}(T)$ and, therefore, ${\rm div\,}\Pi_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}\in P^{p+1}(T)$.
For the final assertion note that $\norm{{\rm div\,}\Pi_{\mathrm{F},hp}^{{\rm div\,}}{\boldsymbol\tau}}T=\norm{\Pi_T^{p+1}{\rm div\,}{\boldsymbol\tau}}T \leq \norm{{\rm div\,}{\boldsymbol\tau}}T$ by the commutativity property. Together with boundedness in the $L^2(T)$ norm established above, this finishes the proof.
\end{proof}
\begin{remark}\label{rem:supconv:div}
In general, discrete test spaces are chosen such that a Fortin operator exists, which not necessarily implies approximation results of the form
\begin{align*}
\min_{{\boldsymbol\tau}_h\in V_h} \norm{{\boldsymbol\tau}-{\boldsymbol\tau}_h}{T} + \norm{{\rm div\,}({\boldsymbol\tau}-{\boldsymbol\tau}_h)}T \lesssim h_T \norm{\nabla{\boldsymbol\tau}}{T} + \norm{(1-\Pi_T^0){\rm div\,}{\boldsymbol\tau}}T \quad\text{for } {\boldsymbol\tau} \in \ensuremath{{\boldsymbol{H}}}^1(T).
\end{align*}
For certain supercloseness results in the DPG method the latter approximation property is required, see~\cite{SupConv2}.
To ensure this property one can simply require $\ensuremath{\boldsymbol{RT}}^0(T)\subset V_h$.
\end{remark}
\subsection{Construction for small parameter}
For a small parameter, i.e., $\alpha\lesssim h_T$, we build a Fortin operator quite a bit different to the ones presented in the previous section.
The proof of boundedness requires as in the scalar case the multiplicative version of the trace inequality, but also the following Helmholtz decomposition together with elliptic regularity.
\begin{lemma}\label{lem:helmholtz}
Let ${\boldsymbol\tau}\in \Hdivset{T}$. There exist $r\in H_0^1(T)$, ${\boldsymbol{q}}\in \ensuremath{{\boldsymbol{H}}}(\ccurl;T)$ (for $n=2$ we have ${\boldsymbol{q}}\in H^1(T)$)
such that
${\boldsymbol\tau} = \nabla r + \ccurl{\boldsymbol{q}}$ and
\begin{align*}
\norm{\ccurl {\boldsymbol{q}}}T^2 + \norm{\nabla r}T^2 = \norm{{\boldsymbol\tau}}{T}^2, \quad \norm{D^2r}{T} \lesssim \norm{{\rm div\,}{\boldsymbol\tau}}T
\end{align*}
with constants independent of $T$.
\end{lemma}
\begin{proof}
Define $r\in H_0^1(T)$ as the weak solution of $\Delta r = {\rm div\,}{\boldsymbol\tau}$, $r|_{\partial T} = 0$. Elliptic regularity implies (note that ${\rm div\,}{\boldsymbol\tau}\in L^2(T)$ and $T$ is convex) that $r\in H^2(T)$. Moreover,~\cite[Theorem~3.1.1.2]{grisvard} implies that
\begin{align*}
\sum_{|\beta|=2} \norm{D^\beta r}{T} \leq C \norm{\Delta r}{T}
\end{align*}
with a constant independent of $T$.
Finally, ${\rm div\,}({\boldsymbol\tau}-\nabla r) = 0$ by construction implies that ${\boldsymbol\tau}-\nabla r = \ccurl {\boldsymbol{q}}$ for some ${\boldsymbol{q}}\in \ensuremath{{\boldsymbol{H}}}(\ccurl;T)$ for $n=3$ resp. ${\boldsymbol{q}}\in H^1(T)$ for $n=2$.
\end{proof}
The definition and analysis of our Fortin operator is based on the Helmholtz decomposition ${\boldsymbol\tau} = \nabla r + \ccurl{\boldsymbol{q}}$. For the $\ccurl{\boldsymbol{q}}$ contribution we use the Fortin operator defined in the previous section. For $\nabla r$ we consider the following definitions and analysis: Define the space
\begin{align*}
\widetilde V_{hp,\alpha}^{{\rm div\,},\mathrm{aux}} = \ensuremath{\boldsymbol{P}}^0(T) + \linhull\set{\eeta_{\alpha,F,j}}{j=1,\dots,\dim(P_c^{p+1}(\partial T))}
\end{align*}
and operator $\widetilde{\Pi}_{\mathrm{F},hp}^{{\rm div\,},\mathrm{aux}}\colon \Hdivset{T}\to \widetilde V_{hp,\alpha}^{{\rm div\,},\mathrm{aux}}$ for all ${\boldsymbol\tau}\in \Hdivset{T}$ by
\begin{align*}
\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}{\boldsymbol\tau} &= \Pi_T^0{\boldsymbol\tau} + \sum_{j=1}^{\dim(P_c^{p+1}(\ensuremath{\mathcal{F}}_T))}
\frac{\dual{(1-\Pi_T^0){\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,j}}_{\partial T}}{\dual{\eeta_{\alpha,\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,j}}_{\partial T}} \eeta_{\alpha,\partial T,j}.
\end{align*}
\begin{lemma}\label{lem:fortinDivAux}
Operator $\Pi_{\mathrm{F}}^{{\rm div\,}} = \widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}$ satisfies~\eqref{eq:fortinDiv:a}.
If $\alpha\lesssim h_T$, then
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}\nabla r}{T,\alpha} \lesssim \norm{\nabla r}{T,\alpha}
\end{align*}
for all $r\in H_0^1(T)\cap H^2(T)$.
\end{lemma}
\begin{proof}
The verification of~\eqref{eq:fortinDiv:a} follows similarly as in Lemma~\ref{lem:fortinDiv:normal}. We leave the details to the reader and focus on details for the proof of boundedness.
Let $r\in H_0^1(T)\cap H^2(T)$.
Note that elliptic regularity yields $\norm{D^2r}{T} \lesssim \norm{{\rm div\,}\nabla r}{T}$ with constant independent of $T$.
Using standard norm estimates, the multiplicative version of the trace inequality (see Lemma~\ref{lem:traceineq}) and Lemma~\ref{lem:modRB:ho}, we get
\begin{align*}
\frac{|\dual{(1-\Pi_T^0)\nabla r\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,j}}_{\partial T}|}{\dual{\eeta_{\alpha,\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,j}}_{\partial T}} \norm{\eeta_{\alpha,\partial T,j}}T
&\lesssim \norm{(1-\Pi_T^0)\nabla r}{\partial T} |\partial T|^{-1/2}|T|^{1/2} (\alpha/h_T)^{1/2}
\\
&\lesssim \norm{(1-\Pi_T^0)\nabla r}T^{1/2} \alpha^{1/2} \norm{D^2r}{T}^{1/2}
\\
&\lesssim \norm{(1-\Pi_T^0)\nabla r}T + \alpha\norm{\Delta r}{T} \leq \norm{\nabla r}T + \alpha\norm{\Delta r}T.
\end{align*}
By summing over all indices and bounding $\norm{\Pi_T^0\nabla r}T \leq \norm{\nabla r}T$ we find that
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}\nabla r}T \lesssim \norm{\Pi_T^0\nabla r}T + \norm{\nabla r}T + \alpha \norm{\Delta r}T \lesssim \norm{\nabla r}T + \alpha \norm{{\rm div\,}\nabla r}T.
\end{align*}
The same argumentation also proves
\begin{align*}
\alpha\frac{|\dual{(1-\Pi_T^0)\nabla r\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,j}}_{\partial T}|}{\dual{\eeta_{\alpha,\partial T,j}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\chi_{\partial T,j}}_{\partial T}} \norm{{\rm div\,}\eeta_{\alpha,\partial T,j}}T
&\lesssim \alpha \norm{(1-\Pi_T^0)\nabla r}{\partial T} |\partial T|^{-1/2}|T|^{1/2} h_T^{-1}(\alpha/h_T)^{-1/2}
\\
&\lesssim \norm{(1-\Pi_T^0)\nabla r}T^{1/2} \alpha^{1/2} \norm{D^2r}{T}^{1/2}
\\
&\lesssim \norm{(1-\Pi_T^0)\nabla r}T + \alpha\norm{{\rm div\,}\nabla r}{T}
\\ &\leq \norm{\nabla r}T + \alpha\norm{{\rm div\,}\nabla r}T.
\end{align*}
Summing over all indices and using ${\rm div\,}\Pi_T^0\nabla r = 0$ we conclude that
\begin{align*}
\alpha\norm{{\rm div\,}\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}\nabla r}T \lesssim \norm{\nabla r}T + \alpha \norm{{\rm div\,}\nabla r}T.
\end{align*}
Combining all estimates finishes the proof.
\end{proof}
For the definition of operators that ensure~\eqref{eq:fortinDiv:b} we add --- as before --- a correction term based on the functions $\eeta_{E,j}$. We stress that these edge functions do not need to be modified. Set
\begin{align*}
V_{hp,\alpha}^{{\rm div\,}} = V_{hp}^{{\rm div\,}} + \widetilde V_{hp}^{{\rm div\,},\mathrm{aux}}
\end{align*}
and define $\Pi_{\mathrm{F},hp}^{{\rm div\,}}\colon \Hdivset{T}\to V_{hp}^{{\rm div\,}}$ for all ${\boldsymbol\tau} = \nabla r + \ccurl{\boldsymbol{q}}\in \Hdivset{T}$ by
\begin{align*}
\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}{\boldsymbol\tau} = \Pi_{\mathrm{F},hp}^{{\rm div\,}}\ccurl{\boldsymbol{q}} + \widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}\nabla r + \sum_{E\in\ensuremath{\mathcal{E}}_*} \sum_{j=1}^{\dim(P^p(T))} \frac{\ip{{\boldsymbol\sigma}_{E,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}})\nabla r}_T}{\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E,j}}_T} \eeta_{E,j}.
\end{align*}
The following is one of our main results.
\begin{theorem}\label{thm:fortinDiv}
Operator $\Pi_{\mathrm{F}}^{{\rm div\,}} = \Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}$ satisfies~\eqref{eq:fortinDiv:a}--\eqref{eq:fortinDiv:b}.
If $\alpha\lesssim h_T$, then
\begin{align*}
\norm{\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}{\boldsymbol\tau}}{T,\alpha} \lesssim \norm{{\boldsymbol\tau}}{T,\alpha} \quad\forall {\boldsymbol\tau}\in \Hdivset{T}.
\end{align*}
\end{theorem}
\begin{proof}
The verification of~\eqref{eq:fortinDiv:a}--\eqref{eq:fortinDiv:b} follows from the arguments already seen in Theorem~\ref{thm:fortinDiv:normal} and Lemma~\ref{lem:fortinDivAux}.
It only remains to prove boundedness.
By the triangle inequality we get
\begin{align*}
\norm{\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}{\boldsymbol\tau}}{T,\alpha} \leq \norm{\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}\ccurl{\boldsymbol{q}}}{T,\alpha} + \norm{\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}\nabla r}{T,\alpha}
\end{align*}
for ${\boldsymbol\tau}=\nabla r + \ccurl {\boldsymbol{q}}\in \Hdivset{T}$ where $r,{\boldsymbol{q}}$ are defined as in Lemma~\ref{lem:helmholtz}.
Note that $\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}\ccurl{\boldsymbol{q}} = \Pi_{\mathrm{F},hp}^{{\rm div\,}}\ccurl{\boldsymbol{q}}$ and applying Theorem~\ref{thm:fortinDiv:normal} and Lemma~\ref{lem:helmholtz} we see that
\begin{align*}
\norm{\Pi_{\mathrm{F},hp}^{{\rm div\,}}\ccurl{\boldsymbol{q}}}T \lesssim \norm{\ccurl{\boldsymbol{q}}}T + h_T\norm{{\rm div\,}\ccurl{\boldsymbol{q}}}T = \norm{\ccurl{\boldsymbol{q}}}T \leq \norm{{\boldsymbol\tau}}T
\end{align*}
as well as ${\rm div\,}\Pi_{\mathrm{F},hp}^{{\rm div\,}}\ccurl{\boldsymbol{q}} = \Pi_T^{p+1}{\rm div\,}\ccurl{\boldsymbol{q}} = 0$. We conclude that
\begin{align*}
\norm{\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}\ccurl{\boldsymbol{q}}}{T,\alpha} \lesssim \norm{{\boldsymbol\tau}}T \leq \norm{{\boldsymbol\tau}}{T,\alpha}.
\end{align*}
It remains to estimate $\norm{\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}\nabla r}{T,\alpha}$. Using the triangle inequality, the Cauchy--Schwarz inequality, estimates for basis functions and Lemma~\ref{lem:fortinDivAux}, we get
\begin{align*}
\norm{\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}\nabla r}T &\leq \norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}\nabla r}T + \sum_{E\in\ensuremath{\mathcal{E}}_*}\sum_{j} \frac{|\ip{{\boldsymbol\sigma}_{E,j}}{(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}})\nabla r}_T|}{\ip{{\boldsymbol\sigma}_{E,j}}{\eeta_{E,j}}_T} \norm{\eeta_{E,j}}T
\\
& \lesssim \norm{\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}}\nabla r}T + |T|^{1/2}\norm{(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}})\nabla r}T |T|^{-1}|T|^{1/2}
\lesssim \norm{\nabla r}{T,\alpha}.
\end{align*}
For the divergence part in the norm, recall that $\eeta_{E,j} \in \ensuremath{\boldsymbol{P}}^{p+2}(T)$. Therefore, ${\rm div\,}\eeta_{E,j}\in P^{p+1}(T)$ and~\eqref{eq:fortinDiv:c} implies
\begin{align*}
\ip{{\rm div\,}\eeta_{E,j}}{{\rm div\,}(1-\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}})\nabla r}_T = 0, \quad j=1,\dots,\dim(P^p(T)), \, E\in\ensuremath{\mathcal{E}}_*.
\end{align*}
We conclude that
\begin{align*}
\norm{{\rm div\,}(1-\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}})\nabla r}T^2 &= \ip{{\rm div\,}(1-\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}})\nabla r}{{\rm div\,}(1-\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}})\nabla r}_T
\\
&= \ip{{\rm div\,}(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}})\nabla r}{{\rm div\,}(1-\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}})\nabla r}_T
\\
&\leq \norm{{\rm div\,}(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}})\nabla r}T\norm{{\rm div\,}(1-\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}})\nabla r}T.
\end{align*}
It follows that $\norm{{\rm div\,}(1-\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}})\nabla r}T\leq \norm{{\rm div\,}(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}})\nabla r}T$ and with the triangle inequality and Lemma~\ref{lem:fortinDivAux} we get that
\begin{align*}
\alpha\norm{{\rm div\,}\Pi_{\mathrm{F},hp,\alpha}^{{\rm div\,}}\nabla r}T \leq \alpha\norm{{\rm div\,}\nabla r}T + \alpha\norm{{\rm div\,}(1-\widetilde{\Pi}_{\mathrm{F},hp,\alpha}^{{\rm div\,},\mathrm{aux}})\nabla r}T \lesssim \norm{\nabla r}{T,\alpha}
\end{align*}
which finishes the proof together with $\norm{\nabla r}{T,\alpha}\leq \norm{{\boldsymbol\tau}}{T,\alpha}$ (Lemma~\ref{lem:helmholtz}).
\end{proof}
\subsection{Alternative operator for lowest order and moderate parameter}\label{sec:fortinDiv:lo:normal}
First, consider the spaces
\begin{align*}
\widetilde V_h^{{\rm div\,},1} = \ensuremath{\boldsymbol{RT}}^0(T), \quad \widetilde V_h^{{\rm div\,},2} = \ensuremath{\boldsymbol{P}}^0(T) + \linhull\set{\eeta_F}{F\in\ensuremath{\mathcal{F}}_T}
\end{align*}
and operators $\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},j}\colon \Hdivset{T}\to \widetilde V_h^{{\rm div\,},j}$, $j=1,2$, for all ${\boldsymbol\tau}\in\Hdivset{T}$ by
\begin{subequations}\label{eq:def:PiDivTilde}
\begin{align}
\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},1}{\boldsymbol\tau} &= \sum_{F\in\ensuremath{\mathcal{F}}_T} \frac{\dual{{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}}{\dual{{\boldsymbol\psi}_F\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}} {\boldsymbol\psi}_F,
\\
\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},2}{\boldsymbol\tau} &= \Pi_T^0{\boldsymbol\tau} + \sum_{F\in\ensuremath{\mathcal{F}}_T} \frac{\dual{(1-\Pi_T^0){\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}}{\dual{\eeta_F\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}} \eeta_F.
\end{align}
\end{subequations}
One verifies that $\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},1}$ is a projector whereas $\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},2}$ is idempotent on $\ensuremath{\boldsymbol{P}}^0(T)$.
Defining the spaces
\begin{align*}
V_h^{{\rm div\,},j} = \widetilde V_h^{{\rm div\,},j} + \linhull\set{\eeta_E}{E\in\ensuremath{\mathcal{E}}_*}, \quad j=1,2,
\end{align*}
we introduce operators $\Pi_{\mathrm{F}}^{{\rm div\,},j}\colon V\to V_h^{{\rm div\,},j}$, $j=1,2$, for all ${\boldsymbol\tau}\in\Hdivset{T}$ by
\begin{align*}
\Pi_{\mathrm{F}}^{{\rm div\,},j} {\boldsymbol\tau} = \widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},j}{\boldsymbol\tau} + \sum_{E\in\ensuremath{\mathcal{E}}_*} \frac{\ip{{\boldsymbol\sigma}_E}{(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},j}){\boldsymbol\tau}}_T}{\ip{{\boldsymbol\sigma}_E}{\eeta_E}_T} \eeta_E.
\end{align*}
\begin{theorem}\label{thm:fortinDiv:normal:lo}
Operator $\Pi_{\mathrm{F}}^{{\rm div\,}}\in\set{\Pi_{\mathrm{F}}^{{\rm div\,},j}}{j=1,2}$ satisfies~\eqref{eq:fortinDiv:a}--\eqref{eq:fortinDiv:b}
and
\begin{align*}
\norm{\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}T &\lesssim \norm{{\boldsymbol\tau}}T + h_T \norm{{\rm div\,}{\boldsymbol\tau}}T, \quad
\norm{{\rm div\,}\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}T \lesssim \norm{{\rm div\,}{\boldsymbol\tau}}T \quad\forall {\boldsymbol\tau}\in \Hdivset{T}.
\end{align*}
Moreover, ${\rm div\,}\circ \Pi_{\mathrm{F}}^{{\rm div\,},1} = \Pi_T^1\circ {\rm div\,}$.
Furthermore, operator $\Pi_{\mathrm{F}}^{{\rm div\,},1}$ is idempotent on $\ensuremath{\boldsymbol{RT}}^0(T)$ and $\Pi_{\mathrm{F}}^{{\rm div\,},2}$ is idempotent on $\ensuremath{\boldsymbol{P}}^0(T)$.
\end{theorem}
\begin{proof}
Idempotency of the operators follows from their definitions.
Let $\Pi_{\mathrm{F}}^{{\rm div\,}}\in\set{\Pi_{\mathrm{F}}^{{\rm div\,},j}}{j=1,2}$ and $\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}\in\set{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},j}}{j=1,2}$.
First, we check condition~\eqref{eq:fortinDiv:a}. It holds for $\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}$ as can be seen with the same arguments as in Lemma~\ref{lem:fortinDiv:normal}. Since $\eeta_E\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = 0$ by Lemma~\ref{lem:rotatedBR} we have $\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T} = \widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T|_{\partial T}$. We conclude that~\eqref{eq:fortinDiv:a} follows.
Second, condition~\eqref{eq:fortinDiv:b} can be seen as follows,
\begin{align*}
\ip{{\boldsymbol\sigma}_{E'}}{\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}_T &= \ip{{\boldsymbol\sigma}_{E'}}{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}_T +
\sum_{E\in\ensuremath{\mathcal{E}}_*} \frac{\ip{{\boldsymbol\sigma}_E}{(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}){\boldsymbol\tau}}_T}{\ip{{\boldsymbol\sigma}_E}{\eeta_E}_T} \ip{{\boldsymbol\sigma}_{E'}}{\eeta_E}_T
\\
&= \ip{{\boldsymbol\sigma}_{E'}}{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}_T + \ip{{\boldsymbol\sigma}_{E'}}{(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}){\boldsymbol\tau}}_T = \ip{{\boldsymbol\sigma}_{E'}}{{\boldsymbol\tau}}_T
\quad\forall E'\in\ensuremath{\mathcal{E}}_*.
\end{align*}
Next we prove boundedness. Estimate $\norm{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},1}{\boldsymbol\tau}}T\lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}T$ follows as in Lemma~\ref{lem:fortinDiv:normal}.
For the second operator we stress that
\begin{align*}
|\dual{(1-\Pi_T^0){\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_T| = |\ip{{\rm div\,}{\boldsymbol\tau}}{\nu_F}_T + \ip{(1-\Pi_T^0){\boldsymbol\tau}}{\nabla \nu_F}_T|
= |\ip{{\rm div\,}{\boldsymbol\tau}}{\nu_F}_T| \lesssim \norm{{\rm div\,}{\boldsymbol\tau}}T|T|^{1/2}.
\end{align*}
The second identity follows since $\nabla\nu_F\in\ensuremath{\boldsymbol{P}}^0(T)$.
This allows us to estimate
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},2}{\boldsymbol\tau}}T \lesssim \norm{\Pi_T^0{\boldsymbol\tau}}T + \sum_{F\in\ensuremath{\mathcal{F}}_T} |F|^{-1}|T| \norm{{\rm div\,}{\boldsymbol\tau}}T \lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}T.
\end{align*}
Then, by the triangle inequality, the Cauchy--Schwarz inequality, norm estimates of basis functions and the previously established boundedness estimates, we see that
\begin{align*}
\norm{\Pi_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}T &\leq \norm{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}T +
\sum_{E\in\ensuremath{\mathcal{E}}_*} \frac{|\ip{{\boldsymbol\sigma}_E}{(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}){\boldsymbol\tau}}_T|}{\ip{{\boldsymbol\sigma}_E}{\eeta_E}_T} \norm{\eeta_E}T
\\
&\lesssim \norm{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}T + |T|^{1/2}\norm{(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}){\boldsymbol\tau}}T|T|^{-1}|T|^{1/2}
\\
&\lesssim \norm{\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}{\boldsymbol\tau}}T + \norm{(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,}}){\boldsymbol\tau}}T \lesssim \norm{{\boldsymbol\tau}}T + h_T\norm{{\rm div\,}{\boldsymbol\tau}}T.
\end{align*}
Furthermore, the same arguments and ${\rm div\,}\Pi_T^0{\boldsymbol\tau}=0$, $\norm{{\rm div\,}\eeta_F}T\eqsim h_T^{-1}|T|^{1/2}$ show that
\begin{align*}
\norm{{\rm div\,}\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},2}{\boldsymbol\tau}}T \lesssim \norm{{\rm div\,}{\boldsymbol\tau}}T.
\end{align*}
Note that $\Pi_{\mathrm{F}}^{{\rm div\,},1}{\boldsymbol\tau}\in \ensuremath{\boldsymbol{P}}^2(T)$, thus, ${\rm div\,}\Pi_{\mathrm{F}}^{{\rm div\,},1}{\boldsymbol\tau}\in P^1(T)$. Consequently,~\eqref{eq:fortinDiv:c} implies the commutativity property of $\Pi_{\mathrm{F}}^{{\rm div\,},1}$.
Clearly, this also yields $\norm{{\rm div\,}\Pi_{\mathrm{F}}^{{\rm div\,},1}{\boldsymbol\tau}}T\leq \norm{{\rm div\,}{\boldsymbol\tau}}T$. It thus remains to prove that $\norm{{\rm div\,}\Pi_{\mathrm{F}}^{{\rm div\,},2}{\boldsymbol\tau}}T \lesssim \norm{{\rm div\,}{\boldsymbol\tau}}T$.
To do so we argue as in the proof of Theorem~\ref{thm:fortinDiv} to derive $\norm{{\rm div\,}(1-\Pi_{\mathrm{F}}^{{\rm div\,},2}){\boldsymbol\tau}}T\leq \norm{{\rm div\,}(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},2}){\boldsymbol\tau}}T$. Together with the triangle inequality and $\norm{{\rm div\,}\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},2}{\boldsymbol\tau}}T \lesssim \norm{{\rm div\,}{\boldsymbol\tau}}T$ we get that
\begin{align*}
\norm{{\rm div\,}\Pi_{\mathrm{F}}^{{\rm div\,},2}{\boldsymbol\tau}}T \leq \norm{{\rm div\,}{\boldsymbol\tau}}T + \norm{{\rm div\,}(1-\widetilde{\Pi}_{\mathrm{F}}^{{\rm div\,},2}){\boldsymbol\tau}}T \lesssim \norm{{\rm div\,}{\boldsymbol\tau}}T.
\end{align*}
This finishes the proof.
\end{proof}
\subsection{Alternative operator for lowest order and small parameter}\label{sec:fortinDiv:lo}
In this section we construct a simpler Fortin operator for $\alpha\lesssim h_T$ and the lowest-order case ($p=0$ in~\eqref{eq:fortinDiv}), based on $\Pi_{\mathrm{F}}^{{\rm div\,},2}$ from the previous section.
Define the spaces
\begin{align*}
\widetilde V_{h,\alpha}^{{\rm div\,}} &= \ensuremath{\boldsymbol{P}}^0(T) + \linhull\set{\eeta_{\alpha,F}}{F\in\ensuremath{\mathcal{F}}_T}, \\
V_{h,\alpha}^{{\rm div\,}} &= \widetilde V_{h,\alpha}^{{\rm div\,}} + \linhull\set{\eeta_E}{E\in\ensuremath{\mathcal{E}}_*}
\end{align*}
and operators $\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}\colon \Hdivset{T}\to \widetilde V_{h,\alpha}^{{\rm div\,}}$,
$\Pi_{\mathrm{F},\alpha}^{{\rm div\,}}\colon \Hdivset{T}\to V_{h,\alpha}^{{\rm div\,}}$ for all ${\boldsymbol\tau}\in \Hdivset{T}$ by
\begin{align*}
\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau} &= \Pi_T^0{\boldsymbol\tau} + \sum_{F\in\ensuremath{\mathcal{F}}_T} \frac{\dual{(1-\Pi_T^0){\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}}{\dual{\eeta_{\alpha,F}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}} \eeta_{\alpha,F}, \\
\Pi_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau} &= \widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau} + \sum_{E\in\ensuremath{\mathcal{E}}_*} \frac{\ip{{\boldsymbol\sigma}_E}{(1-\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}){\boldsymbol\tau}}_T}{\ip{{\boldsymbol\sigma}_E}{\eeta_E}_T} \eeta_E.
\end{align*}
\begin{theorem}\label{thm:fortinDiv:lo}
Operator $\Pi_{\mathrm{F}}^{{\rm div\,}} = \Pi_{\mathrm{F},\alpha}^{{\rm div\,}}$ satisfies~\eqref{eq:fortinDiv:a}--\eqref{eq:fortinDiv:b} for $p=0$.
If $\alpha\lesssim h_T$ then
\begin{align*}
\norm{\Pi_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau}}{T,\alpha} \lesssim \norm{{\boldsymbol\tau}}{T,\alpha} \quad\forall {\boldsymbol\tau}\in\Hdivset{T}.
\end{align*}
\end{theorem}
\begin{proof}
The verification of~\eqref{eq:fortinDiv:a}--\eqref{eq:fortinDiv:b} follows as in Theorem~\ref{thm:fortinDiv:normal}.
It remains to prove boundedness. First, we show boundedness of $\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}$. Let ${\boldsymbol\tau} = \nabla r + \ccurl{\boldsymbol{q}}\in \Hdivset{T}$ with $r,{\boldsymbol{q}}$ as in Lemma~\ref{lem:helmholtz} be given. Using $\nabla\nu_F\in \ensuremath{\boldsymbol{P}}^0(T)$, ${\rm div\,}\ccurl{\boldsymbol{q}} = 0$ we obtain
\begin{align*}
\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}\ccurl{\boldsymbol{q}} &= \Pi_T^0\ccurl{\boldsymbol{q}} + \sum_{F\in\ensuremath{\mathcal{F}}_T}\frac{\ip{{\rm div\,}(1-\Pi_T^0)\ccurl{\boldsymbol{q}}}{\nu_F}_T + \ip{(1-\Pi_T^0)\ccurl{\boldsymbol{q}}}{\nabla \nu_F}_T}{\dual{\eeta_{\alpha,F}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}}\eeta_{\alpha,F}
\\&= \Pi_T^0 \ccurl{\boldsymbol{q}}.
\end{align*}
Then, $\norm{\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}\ccurl{\boldsymbol{q}}}{T,\alpha}= \norm{\Pi_T^0\ccurl{\boldsymbol{q}}}{T} \leq \norm{\ccurl{\boldsymbol{q}}}T \leq \norm{{\boldsymbol\tau}}{T,\alpha}$.
With the multiplicative trace inequality (Lemma~\ref{lem:traceineq}), properties of the basis functions, Lemma~\ref{lem:modRB} and Lemma~\ref{lem:helmholtz} we get
\begin{align*}
\frac{|\dual{(1-\Pi_T^0)\nabla r\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}|}{\dual{\eeta_{\alpha,F}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}}\norm{\eeta_{\alpha,F}}T
&\lesssim |F|^{-1}|\partial T|^{1/2} \norm{(1-\Pi_T^0)\nabla r}{\partial T} \norm{\eeta_{\alpha,F}}T
\\
&\lesssim |\partial T|^{-1/2}\norm{(1-\Pi_T^0)\nabla r}T^{1/2}\norm{D^2r}T^{1/2}\alpha^{1/2}h_T^{-1/2}|T|^{1/2}
\\
&\lesssim \norm{\nabla r}{T}^{1/2}\alpha^{1/2}\norm{{\rm div\,}{\boldsymbol\tau}}T^{1/2} \lesssim \norm{{\boldsymbol\tau}}T + \alpha\norm{{\rm div\,}{\boldsymbol\tau}}T.
\end{align*}
The last estimates yield $\norm{\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}\nabla r}T\lesssim \norm{{\boldsymbol\tau}}{T,\alpha}$. For the divergence contribution the same arguments prove
\begin{align*}
\alpha\frac{|\dual{(1-\Pi_T^0)\nabla r\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}|}{\dual{\eeta_{\alpha,F}\cdot\ensuremath{{\boldsymbol{n}}}_T}{\nu_F}_{\partial T}}\norm{{\rm div\,}\eeta_{\alpha,F}}T
&\lesssim |F|^{-1}|\partial T|^{1/2} \norm{(1-\Pi_T^0)\nabla r}{\partial T} h_T^{-1}|T|^{1/2}\alpha^{1/2}h_T^{1/2} \\
&\lesssim \norm{\nabla r}{T}^{1/2}\alpha^{1/2}\norm{{\rm div\,}{\boldsymbol\tau}}T^{1/2} \lesssim \norm{{\boldsymbol\tau}}T + \alpha\norm{{\rm div\,}{\boldsymbol\tau}}T.
\end{align*}
We conclude that $\alpha\norm{{\rm div\,}\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}\nabla r}T \lesssim \norm{{\boldsymbol\tau}}{T,\alpha}$.
Putting all estimates together we have shown that
\begin{align*}
\norm{\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau}}{T,\alpha} \leq \norm{\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}\ccurl{\boldsymbol{q}}}{T,\alpha} + \norm{\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}\nabla r}{T,\alpha}
\lesssim \norm{{\boldsymbol\tau}}{T,\alpha}.
\end{align*}
Arguing as in the proof of Theorem~\ref{thm:fortinDiv:normal:lo} we find that
\begin{align*}
\norm{\Pi_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau}}T \lesssim \norm{\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau}}T + \norm{(1-\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}){\boldsymbol\tau}}T
\end{align*}
giving us $\norm{\Pi_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau}}T \lesssim \norm{{\boldsymbol\tau}}{T,\alpha}$.
Finally, arguing as in the proof of Theorem~\ref{thm:fortinDiv} we find that
\begin{align*}
\norm{{\rm div\,}(1-\Pi_{\mathrm{F},\alpha}^{{\rm div\,}}){\boldsymbol\tau}}T \leq \norm{{\rm div\,}(1-\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}){\boldsymbol\tau}}{T}.
\end{align*}
Together with the boundedness of $\widetilde{\Pi}_{\mathrm{F},\alpha}^{{\rm div\,}}$ we conclude that $\alpha\norm{{\rm div\,}\Pi_{\mathrm{F},\alpha}^{{\rm div\,}}{\boldsymbol\tau}}T \lesssim \norm{{\boldsymbol\tau}}{T,\alpha}$ which finishes the proof.
\end{proof}
\subsection{Comparison with existing Fortin operators}
Fortin operators that satisfy~\eqref{eq:fortinDiv} are constructed in~\cite{practicalDPG,breakSpace,DemkowiczZanotti20}.
In~\cite[Lemma~3.3]{practicalDPG}, it is shown that there exists a Fortin operator, mapping into the discrete test space $\ensuremath{\boldsymbol{P}}^{p+2}(T)$ and in~\cite{DemkowiczZanotti20}, the authors impose the minimal condition $\ensuremath{\boldsymbol{RT}}^{p+1}(T)\subset V_h^{{\rm div\,}}$ to ensure the existence of a Fortin operator satisfying~\eqref{eq:fortinDiv}.
Here, $\ensuremath{\boldsymbol{RT}}^{p+1}(T) = \ensuremath{\boldsymbol{P}}^{p+1}(T)+{\boldsymbol{x}} P^{p+1}(T)$ denotes the Raviart--Thomas space.
We stress that all these mentioned operators are uniformly bounded only if $h_T\lesssim \alpha$.
Computing dimensions we get
\begin{align*}
\dim(\ensuremath{\boldsymbol{P}}^{p+2}(T)) &= n \frac{\prod_{j=1}^n(j+p+2)}{n!}, \\
\dim(\ensuremath{\boldsymbol{RT}}^{p+1}(T)) &= n\frac{\prod_{j=1}^n(j+p)}{n!} + (n+1)\frac{\prod_{j=1}^{n-1}(j+p+1)}{(n-1)!}.
\end{align*}
To compute the dimension of $V_{hp}^{{\rm div\,}}$ we note that $\dim(P_c^{p+1}(\ensuremath{\mathcal{F}}_T)) = \dim(P^{p+1}(T))-\dim(P_b^{p+1}(T))$ where we recall that $P_b^p(T)$ denotes the space of element bubbles.
This yields
\begin{align*}
\dim(V_{hp}^{{\rm div\,}}) &= \dim(P_c^{p+1}(\ensuremath{\mathcal{F}}_T))+\dim(\ensuremath{\boldsymbol{P}}^p(T)) \\
&= \frac{\prod_{j=1}^n (j+p+1)}{n!} - \frac{\prod_{j=1}^n(j+p-n)}{n!} + n\frac{\prod_{j=1}^n(j+p)}{n!}.
\end{align*}
For the lowest-order case $p=0$ we thus get
\begin{align*}
\dim(\ensuremath{\boldsymbol{P}}^2(T)) = \begin{cases}
12 & n=2, \\
30 & n=3,
\end{cases}\quad
\dim(\ensuremath{\boldsymbol{RT}}^1(T)) = \begin{cases}
8 & n=2, \\
15 & n=3,
\end{cases}
\end{align*}
and
\begin{align*}
\dim(V_{h0}^{{\rm div\,}}) = \begin{cases}
5 & n=2, \\
7 & n=3.
\end{cases}
\end{align*}
As in Section~\ref{sec:fortin:existing} we conclude that our test spaces are systematically smaller than previously used ones, and guarantee robustness, contrary to the previous cases.
\section{Numerical experiment}\label{sec:num}
In this section we consider the reaction-diffusion problem
\begin{align*}
-\varepsilon^2\Delta u + u = f \quad\text{in } \Omega, \quad u|_{\partial\Omega} = 0.
\end{align*}
First, we give a brief overview of a DPG method for the latter problem and, then, discuss results of our numerical experiment.
\subsection{DPG method for reaction-diffusion problem}\label{sec:DPG}
We introduce the trace operators
\begin{align*}
\tr^{\nabla}\colon H^1(\Omega) \to (\Hdivset{\ensuremath{\mathcal{T}}})', \quad
\tr^{{\rm div\,}}\colon \Hdivset\Omega \to (H^1(\ensuremath{\mathcal{T}}))',
\end{align*}
defined for $u\in H^1(\Omega)$, ${\boldsymbol\sigma}\in\Hdivset\Omega$ by
\begin{alignat*}{2}
\dual{\tr^{\nabla} u}{{\boldsymbol\tau}}_{\partial\ensuremath{\mathcal{T}}} &\,=\,& \ip{u}{{\rm div\,}_\ensuremath{\mathcal{T}}{\boldsymbol\tau}}_\Omega + \ip{\nabla u}{{\boldsymbol\tau}}_\Omega &\quad\forall {\boldsymbol\tau}\in \Hdivset\ensuremath{\mathcal{T}} = \prod_{T\in\ensuremath{\mathcal{T}}}\Hdivset{T}, \\
\dual{\tr^{{\rm div\,}}{\boldsymbol\sigma}}v_{\partial\ensuremath{\mathcal{T}}} &\,=\,& \ip{{\boldsymbol\sigma}}{\nabla_\ensuremath{\mathcal{T}} v}_\Omega + \ip{{\rm div\,}{\boldsymbol\sigma}}{v}_\Omega &\quad\forall v\in H^1(\ensuremath{\mathcal{T}})=\prod_{T\in\ensuremath{\mathcal{T}}} H^1(T).
\end{alignat*}
Here, ${\rm div\,}_\ensuremath{\mathcal{T}}\colon \Hdivset{\ensuremath{\mathcal{T}}}\to L^2(\Omega)$ is given by ${\rm div\,}_\ensuremath{\mathcal{T}}{\boldsymbol\tau}|_T = {\rm div\,}({\boldsymbol\tau}|_T)$ for $T\in\ensuremath{\mathcal{T}}$, ${\boldsymbol\tau}=({\boldsymbol\tau}_T)_{T\in\ensuremath{\mathcal{T}}}\in\Hdivset\ensuremath{\mathcal{T}}$ and $\nabla_\ensuremath{\mathcal{T}}\colon H^1(\ensuremath{\mathcal{T}})\to\ensuremath{\boldsymbol{L}}^2(\Omega)$ is defined similarly.
The trace spaces $H_{00}^{1/2}(\partial\ensuremath{\mathcal{T}}) = \tr^{\nabla}(H_0^1(\Omega))$, $H^{-1/2}(\partial\ensuremath{\mathcal{T}}) = \tr^{{\rm div\,}}(\Hdivset\Omega)$ are closed with respect to the canonical norms (see~\cite{breakSpace})
\begin{align*}
\norm{\widehat u}{1/2,\varepsilon} &= \inf\set{\norm{u}{\Omega,\varepsilon}}{u\in H^1(\Omega), \,\tr^{\nabla} u =\widehat u}, \\
\norm{\widehat \sigma}{-1/2,\varepsilon} &= \inf\set{\norm{{\boldsymbol\sigma}}{\Omega,\varepsilon}}{{\boldsymbol\sigma}\in\Hdivset\Omega, \,\tr^{{\rm div\,}} {\boldsymbol\sigma} =\widehat\sigma}.
\end{align*}
Introducing the spaces
\begin{align*}
U &= L^2(\Omega)\times \ensuremath{\boldsymbol{L}}^2(\Omega)\times H_{00}^{1/2}(\partial\ensuremath{\mathcal{T}})\times H^{-1/2}(\partial\ensuremath{\mathcal{T}}), \\
V &= H^1(\ensuremath{\mathcal{T}})\times \Hdivset\ensuremath{\mathcal{T}},
\end{align*}
where $U$ is equipped with the canonical product norm and $V$ with the (squared) norm
\begin{align*}
\norm{(v,{\boldsymbol\tau})}V^2 = \norm{v}{\ensuremath{\mathcal{T}},\varepsilon}^2 + \norm{{\boldsymbol\tau}}{\ensuremath{\mathcal{T}},\varepsilon}^2 :=
\sum_{T\in\ensuremath{\mathcal{T}}} \norm{v}{T,\varepsilon}^2 + \norm{{\boldsymbol\tau}}{T,\varepsilon}^2
\end{align*}
we obtain the ultraweak formulation of the reaction-diffusion problem by defining ${\boldsymbol\sigma} = \varepsilon\nabla u$ and element-wise integration by parts. This yields
\begin{align}\label{eq:DPG}
\boldsymbol{u}=(u,{\boldsymbol\sigma},\widehat u,\widehat\sigma)\in U\colon
\qquad b(\boldsymbol{u},\ensuremath{\boldsymbol{v}}) = L(\ensuremath{\boldsymbol{v}}) \quad\forall \ensuremath{\boldsymbol{v}}=(v,{\boldsymbol\tau})\in V,
\end{align}
where for $\boldsymbol{u}\in U$, $\ensuremath{\boldsymbol{v}}\in V$ and given $f\in L^2(\Omega)$ we define
\begin{align*}
b(\boldsymbol{u},\ensuremath{\boldsymbol{v}}) &= \ip{u}{\varepsilon{\rm div\,}_\ensuremath{\mathcal{T}}{\boldsymbol\tau}+v}_\Omega + \ip{{\boldsymbol\sigma}}{\varepsilon\nabla_\ensuremath{\mathcal{T}} v+ {\boldsymbol\tau}}_\Omega - \varepsilon\dual{\widehat u}{{\boldsymbol\tau}}_{\partial \ensuremath{\mathcal{T}}} -\varepsilon\dual{\widehat\sigma}{v}_{\partial \ensuremath{\mathcal{T}}}, \\
L(\ensuremath{\boldsymbol{v}}) &= \ip{f}{v}_\Omega.
\end{align*}
The following result contains well-posedness of the ultraweak formulation~\eqref{eq:DPG}.
It can be derived by following the abstract theory presented in~\cite{breakSpace} together with our discussions on the fully-discrete scheme~\eqref{eq:DPG:practical} from the introduction.
\begin{proposition}\label{prop:dpg}
The ultraweak formulation~\eqref{eq:DPG} admits a unique solution $\boldsymbol{u}\in U$ with $\norm{\boldsymbol{u}}U\lesssim \norm{f}\Omega$.
Let $U_h\subset U$, $V_h\subset V$ denote finite-dimensional subspaces and suppose that there exists a Fortin operator $\Pi_{\mathrm{F}}\colon V\to V_h$ satisfying~\eqref{eq:fortin:abstract}.
Then, with $\boldsymbol{u}_h\in U$ denoting the solution of~\eqref{eq:DPG:practical},
\begin{align*}
\norm{\boldsymbol{u}-\boldsymbol{u}_h}U \lesssim \min_{\ensuremath{\boldsymbol{v}}\in U_h} \norm{\boldsymbol{u}-\ensuremath{\boldsymbol{v}}_h}{U}.
\end{align*}
The hidden constants are independent of $\varepsilon$ and the mesh-size.
\end{proposition}
\subsection{Results for reaction-diffusion problem}
In this section we consider the manufactured solution
\begin{align*}
u(x,y) = v(x)v(y) \quad\text{for } (x,y)\in \Omega = (0,1)^2,
\end{align*}
where
\begin{align*}
v(x) = 1-(1-e^{-1/(\sqrt{2}\varepsilon)})\frac{e^{-(1-x)/(\sqrt2\varepsilon)}+e^{-x/(\sqrt2\varepsilon)}}{1-e^{-2/(\sqrt{2}\varepsilon)}}.
\end{align*}
One verifies that $u$ is the solution of
\begin{align*}
-\varepsilon^2 \Delta u + u = f, \quad u|_{\partial \Omega} = 0 \quad\text{with } f(x,y) = \tfrac12(v(x)+v(y)).
\end{align*}
\begin{figure}
\begin{center}
\input{Experiment1}
\end{center}
\caption{Errors in the field variables compared with DPG estimator for $\varepsilon=10^{-3}$ and $\varepsilon=10^{-4}$. The left resp. right column shows the results using test space $V_{h,\mathrm{pol}}$ resp. $V_{h,\varepsilon}$.}
\label{fig:comp}
\end{figure}
\begin{figure}
\begin{center}
\input{Experiment2}
\end{center}
\caption{Ratio $\rho = \norm{u-u_h}{}/\operatorname{est}$ for the two different test spaces $V_{h,\mathrm{pol}}$ and $V_{h,\varepsilon}$ on a fixed mesh. The black dotted line corresponds to $\ensuremath{\mathcal{O}}(\varepsilon^{-1/2})$.}
\label{fig:ratios}
\end{figure}
We use the DPG method from Section~\ref{sec:DPG} with test spaces
\begin{align*}
V_{h,\mathrm{pol}} = \prod_{T\in\ensuremath{\mathcal{T}}} V_{h,\mathrm{pol}}(T), \quad V_{h,\varepsilon} = \prod_{T\in\ensuremath{\mathcal{T}}} V_{h,\varepsilon}(T)
\end{align*}
where
\begin{align*}
V_{h,\mathrm{pol}}(T) = P^3(T)\times \ensuremath{\boldsymbol{P}}^2(T), \quad
V_{h,\varepsilon}(T) = \begin{cases}
V_{h0}^\nabla \times V_{h}^{{\rm div\,},2}, & \varepsilon>h_T, \\
V_{h0,\varepsilon}^\nabla \times V_{h,\varepsilon}^{{\rm div\,}}, & \varepsilon\leq h_T.
\end{cases}
\end{align*}
The trial space is
\begin{align*}
U_h = P^0(\ensuremath{\mathcal{T}})\times \ensuremath{\boldsymbol{P}}^0(\ensuremath{\mathcal{T}}) \times \tr^{\nabla}(P^1(\ensuremath{\mathcal{T}})\cap H_0^1(\Omega)) \times \tr^{{\rm div\,}}(\ensuremath{\boldsymbol{RT}}^0(\ensuremath{\mathcal{T}})),
\end{align*}
where $\ensuremath{\boldsymbol{RT}}^0(\ensuremath{\mathcal{T}}) = \set{{\boldsymbol\tau}\in\ensuremath{\boldsymbol{L}}^2(\Omega)}{{\boldsymbol\tau}|_T\in \ensuremath{\boldsymbol{RT}}^0(T), \, T\in\ensuremath{\mathcal{T}}}\cap \Hdivset\Omega$.
We stress that by our constructions from Sections~\ref{sec:fortin} and~\ref{sec:fortinDiv}, $V_h = V_{h,\varepsilon}$ is a test space that allows for a uniformly bounded Fortin operator~\eqref{eq:fortin:abstract}.
On the other hand, test space $V_h = V_{h,\mathrm{pol}}$ allows for a Fortin operator whose norm depends on $\varepsilon$, cf.~\cite{practicalDPG}.
We also define the DPG error estimator by
\begin{align*}
\operatorname{est} = \sup_{0\neq \ensuremath{\boldsymbol{v}}_h\in V_h} \frac{b(\boldsymbol{u}_h,\ensuremath{\boldsymbol{v}}_h)-L(\ensuremath{\boldsymbol{v}}_h)}{\norm{\ensuremath{\boldsymbol{v}}_h}V}.
\end{align*}
Clearly, this estimator depends on the choice of the test space.
In~\cite{DPGaposteriori} it is shown that $\operatorname{est}$ is, up to an oscillation term, equivalent to the error
\begin{align*}
\norm{\boldsymbol{u}-\boldsymbol{u}_h}U \eqsim \operatorname{est} + \osc(f) = \operatorname{est} + \sup_{0\neq \ensuremath{\boldsymbol{v}}=(v,{\boldsymbol\tau})\in V} \frac{L(\ensuremath{\boldsymbol{v}}-\Pi_{\mathrm{F}}\ensuremath{\boldsymbol{v}})}{\norm{\ensuremath{\boldsymbol{v}}}V},
\end{align*}
provided there exists a uniformly bounded Fortin operator $\Pi_{\mathrm{F}}\colon V\to V_h$.
Figure~\ref{fig:comp} shows the errors of the field variables for $\varepsilon\in \{10^{-3},10^{-4}\}$ and estimator $\operatorname{est}$.
We observe differences when using $V_h = V_{h,\mathrm{pol}}$ or $V_h=V_{h,\varepsilon}$ as test space:
For coarse meshes and $V_h = V_{h,\mathrm{pol}}$ the estimator $\operatorname{est}$ underestimates the errors in the field variables. This effect is more severe for smaller parameters.
This can also be seen in Figure~\ref{fig:ratios}. There, we fix a mesh with four elements and only vary $\varepsilon$. We plot the index $\rho = \|u-u_h\|/\operatorname{est}$. When using $V_h = V_{h,\mathrm{pol}}$ we observe that $\rho=\ensuremath{\mathcal{O}}(\varepsilon^{-1/2})$ for $\varepsilon\to 0$ whereas $\rho=\ensuremath{\mathcal{O}}(1)$ when using $V_h = V_{h,\varepsilon}$.
We conclude that the DPG method with $V_h = V_{h,\mathrm{pol}}$ is not robust, whereas with the new test space $V_h = V_{h,\varepsilon}$ it is.
\subsection{Discrete stability}\label{sec:stability}
Let $\Omega = {\widehat T}$, $\ensuremath{\mathcal{T}} = \{{\widehat T}\}$. In this section we want to study stability of the method by investigating the norm equivalence constants $\lambda_\mathrm{min}$, $\lambda_\mathrm{max}$ in
\begin{align}\label{eq:normequiv}
\lambda_\mathrm{min} \norm{\boldsymbol{u}_h}U^2 \leq b(\boldsymbol{u}_h,\Theta_h\boldsymbol{u}_h) \leq \lambda_\mathrm{max} \norm{\boldsymbol{u}_h}U^2 \quad\forall \boldsymbol{u}_h\in U_h.
\end{align}
Here, $U_h$ is defined as in the previous section. We use two different test spaces, $V_{h,\varepsilon}$ (defined in the previous section) and
\begin{align*}
\widetilde V_h = V_{h0}^{\nabla}\times V_{h0}^{{\rm div\,},2}.
\end{align*}
The difficulty in checking the norm equivalence is the implementation of the trace norms. Note that due to inclusion of boundary conditions and the fact that all nodes of $\ensuremath{\mathcal{T}}$ are on the boundary, we do not have to consider $\norm{\widehat u_h}{1/2,\varepsilon}$.
To calculate $\norm{\widehat\sigma_h}{-1/2,\varepsilon}$ we generate a submesh $\widetilde\ensuremath{\mathcal{T}}$ of $\ensuremath{\mathcal{T}}$ such that all elements that have a boundary face have diameter less than or equal to $\varepsilon/2$. This, heuristically, resolves possible boundary layers. To evaluate $\norm{\widehat \sigma_h}{-1/2,\varepsilon}$ we approximate the PDE
\begin{align*}
{\boldsymbol\tau}\in\Hdivset{\Omega}\colon \qquad -\varepsilon^2\nabla{\rm div\,}{\boldsymbol\tau} +{\boldsymbol\tau} = 0, \quad {\boldsymbol\tau}\cdot\ensuremath{{\boldsymbol{n}}}_{\Omega}|_{\partial\Omega} = \widehat\sigma_h
\end{align*}
by a standard FEM on $\widetilde\ensuremath{\mathcal{T}}$ using lowest-order Raviart--Thomas elements.
The $\Hdivset\Omega$ norm $\norm{{\boldsymbol\tau}_h}{\Omega,\varepsilon}$ of the approximation ${\boldsymbol\tau}_h\in\ensuremath{\boldsymbol{RT}}^0(\ensuremath{\mathcal{T}})$ is taken as $\norm{\widehat \sigma_h}{-1/2,\varepsilon}$.
\begin{figure}
\begin{center}
\input{Experiment3}
\end{center}
\caption{Constants $\lambda_\mathrm{max}$ and $\lambda_\mathrm{min}$ from~\eqref{eq:normequiv} for the test spaces $\widetilde V_h$ and $V_{h,\varepsilon}$ (see Section~\ref{sec:stability}). The black dotted line corresponds to $\ensuremath{\mathcal{O}}(\varepsilon^{-1})$.}
\label{fig:normequiv}
\end{figure}
In view of norm equivalence~\eqref{eq:normequiv} we stress that $\lambda_\mathrm{max}\lesssim 1$ independent of $\varepsilon$ and test space $V_h$ since the bilinear form is uniformly bounded. However, the lower bound is directly related to the stability of the DPG method, i.e., $\lambda_\mathrm{min}$ depends on the discrete $\inf$-$\sup$ constant which for the DPG method is related to the norm of Fortin operators as we already discussed in the introduction.
Figure~\ref{fig:normequiv} visualizes $\lambda_\mathrm{min}$ and $\lambda_\mathrm{max}$ for $V_h = V_{h,\varepsilon}$ and $V_h = \widetilde V_h$.
We observe that $\lambda_\mathrm{max}$ is uniformly bounded for both test spaces (for small $\varepsilon$ we can not even distinguish them in the plot), whereas $\lambda_\mathrm{min}$ deteriorates for $V_h = \widetilde V_h$ and is essentially constant for $V_h=V_{h,\varepsilon}$.
This illustrates that the DPG method with test space $V_{h,\varepsilon}$ is uniformly stable, in contrast to the canonical method with test space $\widetilde V_h$.
\bibliographystyle{abbrv}
|
1,314,259,996,687 | arxiv | \section{Introduction}
\label{Sec:i}
Let $G$ be a graph with adjacency matrix $A$, and let $\lambda_1,\dots,\lambda_n$ be the
eigenvalues of $A$. The $k$-th spectral moment of $G$ is defined as
\begin{equation}
\label{Eq:EMk}
\M_k(G)=\sum_{k=0}^n\lambda_i^k.
\end{equation}
A \emph{walk} of length $k$ in a graph $G$ is any sequence $w_1w_2\dots w_{k+1}$ of vertices of $G$ such that $w_iw_{i+1}$ is an edge in $G$ for $i=1,\dots,k$. Since $\tr(A^k)=\M_k(G)$, where $\tr(A^k)$ is the trace of the $k$-th power of $A$,
$\M_k(G)$ is (see \cite{Book_Spect}) exactly the number of closed walks (walks that start and end at the same vertex) of length $k$ in $G$.
The spectral moments of $G$ are also closely related to the so-called \emph{Estrada index} \cite{EstEst}, which is defined as
\begin{equation}
\label{Eq:EE}
\EE(G)=\sum_{i=1}^ne^{\lambda_i}.
\end{equation}
It follows from \eqref{Eq:EMk}, \eqref{Eq:EE} and the power series expansion of the exponential function that
\begin{equation}
\EE(G)=\sum_{i=1}^n\sum_{k=0}^{\infty}\frac{\lambda_i^k}{k!}=\sum_{k=0}^{\infty}\frac{\M_k(G)}{k!}.
\end{equation}
Ernesto Estrada \cite{EstradaIntro} introduced the parameter $\EE$ in 2000 and showed how it can be used to study aspects of molecular structures such as the degree of folding of proteins, see also \cite{EstradaIntro2,EstradaInt4}. Applications of $\EE$ expanded quickly to the study of complex networks \cite{EsCompl} and quantum chemistry \cite{QUA:QUA20850}. See \cite{Gutman2011Estrada} for a recent survey on the Estrada index.
Let us also define a generalization of the graph invariant $\EE$: for any function $f: \R \to \R$, we set
\begin{equation}\label{Eq:efdefi}
\E_f(G)=\sum_{i=1}^nf(\lambda_i).
\end{equation}
Obviously, we obtain the $k$-th spectral moment for $f(x) = x^k$, the Estrada index for $f(x) = e^x$ and the graph energy (see \cite{li2012graphenergy} and the references therein) for $f(x) = |x|$. More examples will be discussed at a later stage. If we assume that $f$ has a power series expansion around $0$ that converges everywhere, i.e.,
\begin{equation}
\label{Eq:f}
f(x)=\sum_{k=0}^{\infty}a_kx^k,
\end{equation}
then $\E_f$ satisfies the relation
\begin{equation}
\label{Eq:ef}
\E_f(G)=\sum_{i=1}^n\sum_{k=0}^{\infty}a_k\lambda_i^k=\sum_{k=0}^{\infty}a_k\M_k(G).
\end{equation}
Let $\mathbb{T}_D$ denote the set of trees with degree sequence $D$. The class of trees with fixed degree sequence is very popular in extremal graph theory.
For example, it has been studied with regards to the Wiener index \cite{WangHua2008,WangHua2009} and other distance-based invariants \cite{schm12}, spectral radius and Laplacian spectral radius \cite{biyikoglu2008graphs,zhang08b,biyi09},
the energy and the number of independent subsets \cite{Andriantiana2012}, and the number of subtrees \cite{zhang12,zhang12b}.
The greedy tree $G(D)$ is the tree obtained from a ``greedy algorithm'' that we will describe
in detail in the following section. Roughly speaking, it is obtained by assigning the highest degree in $D$ to the root, the largest degrees that are left to its neighbors, and so on.
For any degree sequence $D$, we prove that $G(D)$ has maximum $k$-th spectral moment for any $k\geq 0$, and for sufficiently large $k$, it is unique with this property. Consequently, the greedy tree also maximizes $\E_f$ for any $f$ as in \eqref{Eq:f} among all elements of $\mathbb{T}_D$, provided that the coefficients $a_k$ are nonnegative for even $k$ (the odd spectral moments are $0$ for all bipartite graphs, thus in particular for trees). Details of the proof are provided in Section~\ref{Sec:Main}. Furthermore, in Section~\ref{Sec:Diff} we show that if
two degree sequences $D=(d_1,\dots,d_n)$ and $B=(b_1,\dots,b_n)$ satisfy
\begin{equation}
\label{Eq:Major}
\sum_{i=1}^lb_i \leq \sum_{i=1}^ld_i
\end{equation}
for all $1\leq l \leq n$ (i.e., $D$ \emph{majorizes} $B$), then $\M_k(G(B))\leq \M_k(G(D))$ for any $k\geq 0.$ A number of corollaries can be deduced from these results. In particular a
conjecture of Ili\'c and Stevanovi\'c follows as a corollary to our theorems, which reads as follows:
\begin{conj}[Ili\'c/Stevanovi\'c \cite{Conj}]
\label{Conj:1}
For any $k\geq 2$, the Volkmann tree (see Figure~\ref{fig:volkmann}) has maximum spectral moment $\M_{2k}$ among trees of $n$ vertices with maximum degree $\Delta$.
\end{conj}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=0.7]
\node[fill=black,circle,inner sep=1.5pt] (v) at (8,6) {};
\node[fill=black,circle,inner sep=1.5pt] (v1) at (4,4) {};
\node[fill=black,circle,inner sep=1.5pt] (v2) at (8,4) {};
\node[fill=black,circle,inner sep=1.5pt] (v3) at (12,4) {};
\node[fill=black,circle,inner sep=1.5pt] (v11) at (3,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v12) at (5,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v21) at (7,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v22) at (9,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v31) at (11,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v32) at (13,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v111) at (2.5,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v112) at (3.5,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v121) at (4.5,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v122) at (5.5,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v211) at (7,0) {};
\draw (v)--(v1);
\draw (v)--(v2);
\draw (v)--(v3);
\draw (v1)--(v11);
\draw (v1)--(v12);
\draw (v2)--(v21);
\draw (v2)--(v22);
\draw (v3)--(v31);
\draw (v3)--(v32);
\draw (v11)--(v111);
\draw (v11)--(v112);
\draw (v12)--(v121);
\draw (v12)--(v122);
\draw (v21)--(v211);
\end{tikzpicture}
\caption{The Volkmann tree for $\Delta = 3$, $n = 15$.}
\label{fig:volkmann}
\end{figure}
This, in turn, implies an older conjecture of Gutman, Furtula, Markovi\'c and Gli\v{s}i\'c \cite{gutman2007alkanes}, stating that the Volkmann tree has greatest Estrada index among all trees with maximum degree $\Delta$, see also \cite[p.168]{Gutman2011Estrada}. The Volkmann tree, shown in Figure~\ref{fig:volkmann} in the case $\Delta = 3$ and $n=15$, is essentially a complete $\Delta$-ary tree, and a special case of a greedy tree whose degree sequence is $(\Delta,\Delta,\ldots,\Delta,r,1,1,\ldots,1)$ for some $r$ between $1$ and $\Delta$. Gutman et al. provide an argument supporting their conjecture, which however is not fully rigorous. The Volkmann tree is well-known to be extremal for other graph invariants, notably for the Wiener index \cite{fischermann2002wiener}.
The conjecture of Ili\'c and Stevanovi\'c was proved by Zhang, Zhou and Li \cite{zhang2011estrada} in the case that the maximum degree $\Delta$ is large (greater than $n/3$). See \cite{du2011estrada1,du2011estrada2,du2012estrada1,du2012estrada2} for further recent extremal results concerning the Estrada index, in particular the Estrada index of trees.
\section{Preliminaries}
\label{Sec:p}
We start with formal definitions of specific terminologies and certain types of trees which will be of central interest in this paper, compare also \cite{schm12,zhang12,andriantiana2013greedy}.
\begin{defn}
Let $F$ be a rooted forest where the maximum height of any component is $k-1$. The \emph{leveled degree sequence} of $F$ is the sequence
\begin{equation}
\label{Eq:D}
D=(V_1, \dots,V_k),
\end{equation}
where, for any $1\leq i \leq k$, $V_i$ is the non-increasing sequence formed by the degrees of the
vertices of $F$ at the $i^{\text{th}}$ level (i.e., vertices of distance $i-1$ from the root in the respective component).
\end{defn}
\begin{defn}
\label{Def:lgf}
The \emph{level greedy forest} with leveled degree sequence
\begin{equation}\label{eq:ldegseq}
D=((i_{1,1},\dots,i_{1,k_1}), (i_{2,1},\dots,i_{2,k_2}),\dots,(i_{n,1},\dots,i_{n,k_n}))
\end{equation}
is obtained using the following ``greedy algorithm'':
\begin{enumerate}
\item[(i)] Label the vertices of the first level $g_1^1,\dots,g_{k_1}^1$, and assign degrees
to these vertices such that $\deg g_j^1 = i_{1,j}$ for all $j$.
\item[(ii)] Assume that the vertices of the $h^{\text{th}}$ level have been labeled
$g_1^h,\dots,g_{k_h}^h$ and a degree has been assigned to each of them. Then for all
$1\leq j \leq k_h$ label the neighbors of $g^h_j$ at the $(h+1)^{\text{th}}$ level, if any,
by $$g_{1+\sum_{m=1}^{j-1}(i_{h,m}-1)}^{h+1},\dots,g^{h+1}_{\sum_{m=1}^{j}(i_{h,m}-1)},$$ and assign
degrees to the newly labeled vertices such that $\deg g_j^{h+1} = i_{h+1,j}$ for all $j$.
\end{enumerate}
\end{defn}
The level greedy forest with leveled degree sequence $D$ is denoted by $G(D)$. We will use the labeling described in the definition throughout this paper, for level greedy trees and forests as well as related trees.
\begin{defn}
\label{Def:gt}
A connected level greedy forest is called a \emph{level greedy tree}.
\end{defn}
We will also encounter an edge-rooted version of the level greedy tree.
\begin{defn}
The \emph{edge-rooted level greedy tree} with leveled degree sequence
$$D=((i_{1,1},i_{1,2}), (i_{2,1},\dots,i_{2,k_2}),\dots,(i_{n,1},\dots,i_{n,k_n}))$$
is obtained from the two-component level greedy forest with leveled degree sequence
$$((i_{1,1}-1,i_{1,2}-1), (i_{2,1},\dots,i_{2,k_2}),\dots,(i_{n,1},\dots,i_{n,k_n}))$$
by joining the two roots.
\end{defn}
Now, we are ready to define greedy trees:
\begin{defn}
\label{Def:gf}
If a root in a tree can be chosen such that it becomes a level greedy tree whose
leveled degree sequence, as given in \eqref{eq:ldegseq},
satisfies
$$\min (i_{j,1},\dots,i_{j,k_j})\geq \max (i_{j+1,1},\dots,i_{j+1,k_{j+1}})$$
for all $1\leq j\leq n-1$, then it is called a \emph{greedy tree}.
\end{defn}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=0.7]
\node[fill=black,circle,inner sep=1.5pt] (v) at (10,6) {};
\node[fill=black,circle,inner sep=1.5pt] (v1) at (4,4) {};
\node[fill=black,circle,inner sep=1.5pt] (v2) at (8,4) {};
\node[fill=black,circle,inner sep=1.5pt] (v3) at (12,4) {};
\node[fill=black,circle,inner sep=1.5pt] (v4) at (16,4) {};
\node[fill=black,circle,inner sep=1.5pt] (v42) at (17,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v12) at (4,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v41) at (15,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v32) at (13,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v22) at (8,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v31) at (11,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v23) at (9,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v21) at (7,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v13) at (5,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v11) at (3,2) {};
\node[fill=black,circle,inner sep=1.5pt] (v111) at (3.3,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v112) at (2.7,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v121) at (4.3,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v122) at (3.7,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v131) at (5.3,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v132) at (4.7,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v211) at (7.3,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v212) at (6.7,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v221) at (8.3,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v222) at (7.7,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v231) at (9,0) {};
\node[fill=black,circle,inner sep=1.5pt] (v311) at (11,0) {};
\draw (v)--(v1);
\draw (v)--(v2);
\draw (v)--(v3);
\draw (v)--(v4);
\draw (v1)--(v11);
\draw (v1)--(v12);
\draw (v1)--(v13);
\draw (v2)--(v21);
\draw (v2)--(v22);
\draw (v2)--(v23);
\draw (v3)--(v31);
\draw (v3)--(v32);
\draw (v4)--(v41);
\draw (v4)--(v42);
\draw (v11)--(v111);
\draw (v11)--(v112);
\draw (v12)--(v121);
\draw (v12)--(v122);
\draw (v13)--(v131);
\draw (v13)--(v132);
\draw (v21)--(v211);
\draw (v21)--(v212);
\draw (v22)--(v221);
\draw (v22)--(v222);
\draw (v23)--(v231);
\draw (v31)--(v311);
\node at (10,6.4) {$g_1^1$};
\node at (3.5,4) {$g_1^2$};
\node at (7.5,4) {$g_2^2$};
\node at (12.5,4) {$g_3^2$};
\node at (16.5,4) {$g_4^2$};
\node at (2.5,2) {$g_1^3$};
\end{tikzpicture}
\caption{A greedy tree (only the labels of the first six vertices are shown).}
\label{fig:greedy_tree}
\end{figure}
We denote the set of all permutations of $\{1,\dots,n\}$ by $\mathcal{S}_n$. Let $A=(a_1,\dots,a_n)$ and $B=(b_1,\dots,b_n)$ be sequences of nonnegative numbers.
We say that $A$ \emph{majorizes} $B$ if for all $1\leq k \leq n$ we have
$$
\sum_{i=1}^ka_i \geq \sum_{i=1}^kb_i.
$$
If for any $\sigma\in \mathcal{S}_n$ the sequence $A$ majorizes
$(b_{\sigma(1)},\dots,b_{\sigma(n)})$, then we write
\begin{equation}
\label{Eq:maj}
B \preccurlyeq A.
\end{equation}
\begin{rem}
\label{Rem:1}
Let $\sigma \in \mathcal{S}_n$ be such that $b_{\sigma(1)}\geq\dots \geq b_{\sigma(n)}$.
It is easy to see that $(b_{\sigma'(1)},\dots, b_{\sigma'(n)})\preccurlyeq (b_{\sigma(1)},\dots, b_{\sigma(n)})$
for any $\sigma'\in \mathcal{S}_n$. Relation \eqref{Eq:maj} is equivalent to the statement that $A$ majorizes $(b_{\sigma(1)},\dots,b_{\sigma(n)})$. Furthermore, \eqref{Eq:maj} is equivalent to the statement that for any $k\in\{1,\dots,n\}$
we have
$$
(b_{\sigma'(1)},\dots,b_{\sigma'(k)}) \preccurlyeq (a_1,\dots,a_k)
$$
for all $\sigma'\in \mathcal{S}_n$.
\end{rem}
The rest of this section consists of a series of lemmas describing properties of
sequences, which will then be applied to degree sequences in the following sections.
\begin{lem}[cf. \cite{schm12}]
\label{Lem:mj1}
Suppose that $(b_1,\dots,b_n)\preccurlyeq(a_1,\dots,a_n)$ and
$(b'_1,\dots,b'_n)\preccurlyeq(a'_1,\dots,a'_n)$. Then we have
$$
b'_1b_1+\dots+b'_nb_n\leq a'_1a_1+\dots+a'_na_n.
$$
\end{lem}
The next, stronger looking, lemma is in fact equivalent to Lemma~\ref{Lem:mj1}.
\begin{lem}
\label{Lem:mj2}
Suppose that $(b_1,\dots,b_n)\preccurlyeq(a_1,\dots,a_n)$ and
$(b'_1,\dots,b'_n)\preccurlyeq(a'_1,\dots,a'_n)$. Then we have
$$
(b'_1b_1,\dots,b'_nb_n)\preccurlyeq(a'_1a_1,\dots,a'_na_n).
$$
\end{lem}
\begin{proof}
Let $\sigma$ be the element of $\mathcal{S}_n$ for which $b_{\sigma(1)}b'_{\sigma(1)}\geq\dots \geq b_{\sigma(n)}b'_{\sigma(n)}$.
Using Remark~\ref{Rem:1}, we know that for any $k\in \{1,\dots,n\}$, we have
$$(b_{\sigma(1)},\dots,b_{\sigma(k)})\preccurlyeq(a_1,\dots,a_k)$$
and
$$(b'_{\sigma(1)},\dots,b'_{\sigma(k)})\preccurlyeq(a'_1,\dots,a'_k).$$
By Lemma~\ref{Lem:mj1} this implies
$$b_{\sigma(1)}b_{\sigma(1)}'+\dots+b_{\sigma(k)}b'_{\sigma(k)}\leq a_1a'_1+\dots+a_ka'_k.$$
Hence, $(a_1a'_1,\dots,a_na'_n)$ majorizes $(b_{\sigma(1)}b'_{\sigma(1)},\dots, b_{\sigma(n)}b'_{\sigma(n)})$,
and the lemma follows from Remark~\ref{Rem:1}.
\end{proof}
Let $(k_1,\dots,k_n)$ be a sequence of integers. For any sequence $(a_1,\dots,a_n)$, we define
$$
(a_1,\dots,a_n)*(k_1,\dots,k_n)=\big(b_1,\dots,b_{\sum_{i=1}^nk_i}\big),
$$
where $b_j=a_{\ell}$ whenever $\sum_{i=1}^{\ell-1}k_i< j \leq \sum_{i=1}^{\ell}k_i$ (i.e., each $a_{\ell}$ is repeated $k_{\ell}$ times). For example, $(1,3,2)*(2,3,4)=(1,1,3,3,3,2,2,2,2).$
\begin{rem}
\label{Rem:2}
It is easy to see that if the sequences $(k_1,\dots,k_n)$ and $(a_1,\dots,a_n)$ are non-increasing,
then for any $\sigma$ and $\pi$ in $\mathcal{S}_n$ we have
$$
(a_{\sigma(1)},\dots,a_{\sigma(n)})*(k_{\pi(1)},\dots,k_{\pi(n)}) \preccurlyeq (a_1,\dots,a_n)*(k_1,\dots,k_n).
$$
\end{rem}
\begin{lem}
\label{Lem:mj3}
Assume that $B=(b_1,\dots,b_n)\preccurlyeq(a_1,\dots,a_n)=A$ and let $C=(c_1,\dots,c_n)$ be a
non-increasing sequence of positive integers. Then for any $\sigma \in \mathcal{S}_n$ we have
$
B*(c_{\sigma(1)},\dots,c_{\sigma(n)}) \preccurlyeq A*C.
$
\end{lem}
\begin{proof}
Let $\pi\in\mathcal{S}_n$ be such that $b_{\pi(1)}\geq \dots \geq b_{\pi(n)}$, and let
$B_{\pi}=(b_{\pi(1)}, \dots,$ $b_{\pi(n)})$. By Remark~\ref{Rem:2}, we know that
$B*(c_{\sigma(1)},\dots,c_{\sigma(n)}) \preccurlyeq B_{\pi}*C.$ Since $B_{\pi}*C$ is a
non-increasing sequence, we can prove the lemma by showing that $A*C$ majorizes $B_{\pi}*C$.
The case $n=1$ is trivial. Assume that the statement holds for $n=k$. For $n=k+1$, the relation $B\preccurlyeq A$ implies that
$(b_{\pi(1)},\dots,b_{\pi(k)})\preccurlyeq(a_1,\dots,a_{k})$. By the induction hypothesis we deduce that
\begin{equation}
\label{Eq:8}
(b_{\pi(1)},\dots,b_{\pi(k)})*(c_1,\dots,c_k)\preccurlyeq(a_1,\dots,a_k)*(c_1,\dots,c_k).
\end{equation}
Now we reason by induction with respect to $c_{k+1}$. For any two sequences $S=(s_1,\dots,s_l)$
and $S'=(s'_1,\dots,s'_{l'})$, let $S:S'$ denote the sequence obtained by concatenation, i.e., $(s_1,\dots,s_l,s'_1,$ $\dots,s'_{l'})$.
If $c_{k+1}=1$, then $$(b_{\pi(1)},\dots,b_{\pi(k+1)})*C=((b_{\pi(1)},\dots,b_{\pi(k)})*(c_1,\dots,c_k)):(b_{\pi(k+1)})$$ and
$A*C=((a_1,\dots,a_k)*(c_1,\dots,c_k)):(a_{k+1})$. Using Lemma~\ref{Lem:mj1} we know that
$$
\ssum (B_{\pi}*C)=\sum_{i=1}^{k+1}b_{\pi(i)}c_i\leq \sum_{i=1}^{k+1}a_ic_i =\ssum (A*C),
$$
where $\ssum (B_{\pi}*C)$ and $\ssum (A*C)$ are the sums of the entries in $B_{\pi}*C$
and $A*C$, respectively. With \eqref{Eq:8}, this implies that $A*C$ majorizes $(b_{\pi(1)},\dots,b_{\pi(k+1)})*C$.
The (second) induction step follows from the relations
\begin{align*}
B_{\pi}*(c_1,\dots,c_{k+1})&=(b_{\pi(1)},\dots,b_{\pi(k+1)})*(c_1,\dots,c_{k+1}-1):(b_{\pi(k+1)}),\\
A*(c_1,\dots,c_{k+1})&=(a_1,\dots,a_{k+1})*(c_1,\dots,c_{k+1}-1):(a_{k+1}).
\end{align*}
\end{proof}
\section{Trees with given degree sequence}
\label{Sec:Main}
Let $T$ be a tree and $v$ one of its vertices. We denote by $\mathcal{W}_v(k;T)$ the set of all walks of length $k$ in $T$ starting at $v$, and by $\mathcal{C}_v(k;T)$ the set of all closed walks of length $k$ in $T$ starting and ending at $v$. We also write
\begin{equation}
\label{Eq:op}
\mathcal{W}(k;T)=\bigcup_{v\in V(T)}\mathcal{W}_v(k;T)
\end{equation}
for the set of all walks of length $k$ in $T$ and
\begin{equation}
\label{Eq:cl}
\mathcal{C}(k;T)=\bigcup_{v\in V(T)}\mathcal{C}_v(k;T)
\end{equation}
for the set of all closed walks of length $k$. Note that $\mathcal{C}(k;T) = \emptyset$ whenever $k$ is odd.
\subsection{Vertex rooted trees}
\label{Sec:vr}
Let $W=w_1\dots w_k$ be a walk in a rooted tree $T$. We say that $(i_1,i_2,\ldots,i_k)$ is the \emph{level sequence} of $W$ if $w_l$ is at the $i_l^{\text{th}}$
level in $T$, i.e., at distance $i_l-1$ from the root, for all $l \leq k$. We denote by $\mathcal{W}(i_1,\dots,i_k;T)$ the set of walks with level sequence
$(i_1,\dots,i_k)$ in $T$. For any vertex $v$ of $T$ we define
$$
\mathcal{W}_{v}(i_1,\dots,i_k;T)=\{w_1\dots w_k\in \mathcal{W}(i_1,\dots,i_k;T): w_1=v\}.
$$
The sets $\mathcal{C}(i_1,\dots,i_k;T)$ and $\mathcal{C}_{v}(i_1,\dots,i_k;T)$ are defined analogously. Moreover, we denote the cardinalities of $\mathcal{W}(k;T)$ and $\mathcal{C}(k;T)$ by $W(k;T)$ and $C(k;T)$ respectively, the cardinality of $\mathcal{W}_{v}(i_1,\dots,i_k;T)$ by $W_{v}(i_1,\dots,i_k;T)$, etc. This convention will be kept even if not mentioned explicitly. Finally, the set of rooted forests with leveled degree sequence $D$ is denoted by $\mathcal{T}_D$.
\begin{lem}
\label{Lem:open_wa_ver}
Let $T\in \mathcal{T}_D$ for some leveled degree sequence $D$ of a vertex-rooted forest, and let $G = G(D)$ be the associated greedy forest. Let
$v_1^i,\dots,v_{d_i}^i$ be the vertices of $T$ at the $i^{\text{th}}$ level.
Then for any level sequence of walks $(i_1,\dots,i_l)$, the following relations hold for all $i$:
\begin{align}
\label{Eq:th1}
&(W_{v^{i}_1}(i_1,\dots,i_l;T),\dots,W_{v^{i}_{d_{i}}}(i_1,\dots,i_l;T))\preccurlyeq (W_{g^{i}_1}(i_1,\dots,i_l;G),\dots,W_{g^{i}_{d_{i}}}(i_1,\dots,i_l;G))
\end{align}
and
\begin{equation}
\label{Eq:th11}
W_{g^{i}_1}(i_1,\dots,i_l;G)\geq W_{g^{i}_2}(i_1,\dots,i_l;G)\geq \dots\geq W_{g^{i}_{d_{i}}}(i,\dots,i_l;G).
\end{equation}
\end{lem}
\begin{proof}
The situation where $i\neq i_1$ is not interesting, since we get
$$W_{v^{i}_j}(i_1,\dots,i_l;T)=W_{g^{i}_j}(i_1,\dots,i_l;G)=0$$
for any $j$. So we assume that $i=i_1$ and proceed by induction with respect to $l$.
The initial case $l=1$ is trivial, since we know that
$$W_{v^{i_1}_j}(i_1;T)=W_{g^{i_1}_j}(i_1;G)=1$$
for all $i_1$ and $j$. Assume that the relations \eqref{Eq:th1} and \eqref{Eq:th11} hold whenever $l\leq k$ for some integer $k\geq 1$. Now consider a longer level sequence $(i_1,\dots,i_l)$ where $l=k+1$. There are two cases: $i_2 = i_1 - 1$ or $i_2 = i_1 + 1$ (in all other cases, the number of walks is $0$).
\begin{itemize}
\item \textbf{Case 1:} Assume that $i_2 = i_1+1 = i+1$.
For $1\leq j \leq d_{i}$, we use $a_j$ as an abbreviation for the number of children of $v^i_j$ and $b_j$ for the number of children of $g^i_j$. Clearly, $a_j = \deg v_j^{i} -1$ and $b_j = \deg g_j^i - 1$ if $i \neq 1$, and $a_j = \deg v_j^i$, $b_j = \deg g_j^i$ if $i =1$. In view of the construction of greedy trees, we have
$$b_1 \geq b_2 \geq \cdots \geq b_{d_i},$$
and since $(a_1,a_2,\ldots,a_{d_i})$ is a permutation of $(b_1,b_2,\ldots,b_{d_i})$, it is clear that
\begin{equation}\label{Eq:Deg}
(a_1,a_2,\ldots,a_{d_i}) \preccurlyeq (b_1,b_2,\ldots,b_{d_i}).
\end{equation}
We also write $r_j$ and $s_j$ for the sums
$$r_j = \sum_{t=1}^j a_t \quad \text{and} \quad s_j = \sum_{t=1}^j b_t,$$
and $r_0 = s_0 = 0$. Now note that
$$
W_{v^{i}_j}(i_1,\dots,i_l;T) = \sum_{v^{i+1}_h \sim v_j^{i}} W_{v^{i+1}_h}(i_2,\dots,i_l;T) =
\sum_{h = r_{j-1}+1}^{r_j} W_{v^{i+1}_h}(i_2,\dots,i_l;T),
$$
since every walk with level sequence $(i_1,i_2,\ldots,i_l)$ starting at $v^i_j$ has to go to one of the children $v^{i+1}_h$ ($r_{j-1}+1 \leq h \leq r_j$) first.
Likewise,
$$
W_{g^{i}_j}(i_1,\dots,i_l;G) = \sum_{g^{i+1}_h \sim g_j^{i}} W_{g^{i+1}_h}(i_2,\dots,i_l;G) =
\sum_{h = s_{j-1}+1}^{s_j} W_{g^{i+1}_h}(i_2,\dots,i_l;G).
$$
Now the relation
$$(W_{v^{i}_1}(i_1,\dots,i_l;T),\dots,W_{v^{i}_{d_{i}}}(i_1,\dots,i_l;T))
\preccurlyeq (W_{g^{i}_1}(i_1,\dots,i_l;G),\dots,W_{g^{i}_{d_{i}}}(i_1,\dots,i_l;G))$$
as well as \eqref{Eq:th11}, follow from the induction hypothesis applied to the level sequence $(i_2,\dots,i_l)$ and the majorization inequality \eqref{Eq:Deg}, which also implies that $r_j \leq s_j$ for all $j$.
\item \textbf{Case 2:} Assume that $i_2 = i_1 - 1 = i-1$. This time, we write $a_j$ for the number of children of $v^{i-1}_j$ (which is either $\deg v^{i-1}_j$ or $\deg v^{i-1}_j - 1$) and $b_j$ for the number of children of $g^{i-1}_j$. The relation~\eqref{Eq:Deg} is still valid. Now we have
\begin{multline*}
(W_{v^{i}_1}(i_1,\dots,i_l;T),\dots,W_{v^{i}_{d_{i}}}(i_1,\dots,i_l;T))\\
= (W_{v^{i-1}_1}(i_2,\dots,i_l;T),\dots,W_{v^{i-1}_{d_{i-1}}}(i_2,\dots,i_l;T))*
(a_1,\ldots,a_{d_{i-1}}),
\end{multline*}
since if $v^i_h$ is one of the $a_j$ children of $v^{i-1}_j$, a walk with level sequence $(i_1,\ldots,i_l)$ starting at $v^i_h$ has to start with a step to $v^{i-1}_j$, which means that
$$W_{v^i_h}(i_1,\dots,i_l;T) = W_{v^{i-1}_j}(i_2,\dots,i_l;T).$$
Likewise,
\begin{multline*}
(W_{g^{i}_1}(i_1,\dots,i_l;G),\dots,W_{g^{i}_{d_{i}}}(i_1,\dots,i_l;G))\\
= (W_{g^{i-1}_1}(i_2,\dots,i_l;G),\dots,W_{g^{i-1}_{d_{i-1}}}(i_2,\dots,i_l;G))*
(b_1,\ldots,b_{d_{i-1}}).
\end{multline*}
So \eqref{Eq:th1} and \eqref{Eq:th11} follow from \eqref{Eq:Deg} and the induction hypothesis by means of Lemma~\ref{Lem:mj3}.
\end{itemize}
\end{proof}
Next we study closed walks: it turns out that a completely analogous statement holds.
\begin{lem}
\label{Lem:closed_wa_ver}
Let $T\in \mathcal{T}_D$ for some leveled degree sequence $D$ of a vertex-rooted forest, and let $G = G(D)$ be the associated greedy forest. Let
$v_1^i,\dots,v_{d_i}^i$ be the vertices of $T$ at the $i^{\text{th}}$ level.
Then for any level sequence of walks $(i_1,\dots,i_l)$, the following relations hold for all $i$:
\begin{align}
\label{Eq:th2}
&(C_{v^{i}_1}(i_1,\dots,i_l;T),\dots,C_{v^{i}_{d_{i}}}(i_1,\dots,i_l;T))\preccurlyeq (C_{g^{i}_1}(i_1,\dots,i_l;G),\dots,C_{g^{i}_{d_{i}}}(i_1,\dots,i_l;G))
\end{align}
and
\begin{equation}
\label{Eq:th22}
C_{g^{i}_1}(i_1,\dots,i_l;G)\geq\dots\geq C_{g^{i}_{d_{i}}}(i,\dots,i_l;G).
\end{equation}
\end{lem}
\begin{proof}
As in the proof of Lemma~\ref{Lem:open_wa_ver}, we only need to prove the lemma for $i=i_1$.
The case when $l$ is even is trivial: in this case,
\begin{equation*}
\mathcal{C}_{v^{i}_j}(i_1,\dots,i_l;T)=\mathcal{C}_{g^{i}_j}(i_1,\dots,i_l;G)=\emptyset
\end{equation*}
for all $j$, since there are no closed walks of odd length in a forest.
For the case of odd $l$, say $l=2l'-1$, the proof is similar to that of Lemma~\ref{Lem:open_wa_ver}:
We reason by induction with respect to $l'$. The case $l'=1$ is again trivial. Assume that the lemma holds
for all $l'\leq k$ for some $k\geq 1$. Now consider a level sequence $(i_1,\dots,i_{2k+1})$.
We must have $i_1 = i_{2k+1} = i$ and $i_2 = i \pm 1$ as well as $i_{2k} = i \pm 1$, the other possibilities are trivial.
\medskip
\noindent \textbf{Case 1:} If $i_2 = i_{2k} = i-1$, then, writing $a_j$ for the number of children of $v^{i-1}_j$ and $b_j$ for the number of children of $g^{i-1}_j$, we have
\begin{align*}
&(C_{v^{i}_1}(i_1,\dots,i_{2k+1};T),\dots,C_{v^{i}_{d_{i}}}(i_1,\dots,i_{2k+1};T)) \\
&= (C_{v^{i-1}_1}(i_2,\dots,i_{2k};T),\dots,C_{v^{i-1}_{d_{i-1}}}(i_2,\dots,i_{2k};T))*(a_1,\dots,a_{d_{i-1}})
\end{align*}
and
\begin{align*}
&(C_{g^{i}_1}(i_1,\dots,i_{2k+1};G),\dots,C_{g^{i}_{d_{i}}}(i_1,\dots,i_{2k+1};G)) \\
&= (C_{g^{i-1}_1}(i_2,\dots,i_{2k};G),\dots,C_{g^{i-1}_{d_{i-1}}}(i_2,\dots,i_{2k};G))*(b_1,\dots,b_{d_{i-1}})
\end{align*}
for the same reason as in Case 2 of Lemma~\ref{Lem:open_wa_ver}.
Hence \eqref{Eq:th2} and \eqref{Eq:th22} can be
obtained using Lemma~\ref{Lem:mj3}, the induction hypothesis and \eqref{Eq:Deg}.
\medskip
\noindent \textbf{Case 2:} Assume that $i_2 = i_1+1 = i+1$. Let $h$ be the smallest integer such that $h > 1$ and $i_h=i_1 = i$. If $h$ does not exist, then there is no closed walk with level sequence $(i_1,\dots,i_{2k+1})$, so we can ignore this case. By the definition of $h$ and the assumption that $i_2 = i+1$, we know that $i=\min (i_1,\dots,i_h)$. Clearly,
any walk with level sequence $(i_1,\dots,i_h)$ is closed. Hence for all $j$, any element of
$\mathcal{C}_{g^{i}_j}(i_1,\dots,i_{2k+1};G)$ can be decomposed (uniquely) into a first part that is an element of $\mathcal{C}_{g^{i}_j}(i_1,\dots,i_h;G)$ and a second part that is an element of $\mathcal{C}_{g^{i}_j}(i_h,\dots,i_{2k+1};G)$.
Similarly, an element of
$\mathcal{C}_{v^{i}_j}(i_1,\dots,i_{2k+1};T)$ splits (uniquely) into two parts:
a first part in $\mathcal{C}_{v^{i}_j}(i_1,\dots,i_h;T)$ and a second part in
$\mathcal{C}_{v^{i}_j}(i_h,\dots,i_{2k+1};T)$. This implies that
\begin{align*}
C_{g^{i}_j}(i_1,\dots,i_{2k+1};G)
&= C_{g^{i}_j}(i_1,\dots,i_{h};G)C_{g^{i}_j}(i_h,\dots,i_{2k+1};G)\\
&= W_{g^{i}_j}(i_1,\dots,i_{h};G)C_{g^{i}_j}(i_h,\dots,i_{2k+1};G)
\end{align*}
and
\begin{align*}
C_{v^{i}_j}(i_1,\dots,i_{2k+1};T)
&= C_{v^{i}_j}(i_1,\dots,i_{h};T)C_{v^{i}_j}(i_h,\dots,i_{2k+1};T)\\
&= W_{v^{i}_j}(i_1,\dots,i_{h};T)C_{v^{i}_j}(i_h,\dots,i_{2k+1};T).
\end{align*}
Therefore, we can use Lemma~\ref{Lem:open_wa_ver}, the induction hypothesis and Lemma~\ref{Lem:mj2}
to deduce \eqref{Eq:th2} and \eqref{Eq:th22}; the argument remains valid even if $h=2k+1$, since then the second factor in the formulas above is simply $1$.
\medskip
\noindent \textbf{Case 3:} Assume that $i_{2k} = i_{2k+1}+1 = i+1$. Then the sequence $(i_{2k+1},\dots,i_1)$ satisfies the condition of Case 2. Hence, for this case, \eqref{Eq:th2} and \eqref{Eq:th22} follow from the fact that for any $j$ we have
$$C_{g^{i}_j}(i_1,\dots,i_{2k+1};G) = C_{g^{i}_j}(i_{2k+1},\dots,i_1;G)$$
and
$$C_{v^{i}_j}(i_1,\dots,i_{2k+1};T) = C_{v^{i}_j}(i_{2k+1},\dots,i_1;T).$$
This completes the proof, since there are no closed walks in any other cases.
\end{proof}
The following theorem is a direct consequence of the two Lemmas~\ref{Lem:open_wa_ver} and
\ref{Lem:closed_wa_ver} and the relations \eqref{Eq:op} and \eqref{Eq:cl}.
\begin{thm}
\label{Thm:Main_ver_root}
Let $D$ be a leveled degree sequence of a vertex-rooted forest and $G(D)$ the associated level greedy forest. Then for any nonnegative integer $k$ and all $T\in \mathcal{T}_D$, we have
$$W(k;T)\leq W(k;G(D))$$
and
$$\M_k(T)=C(k;T)\leq C(k;G(D))=\M_k(G(D)).$$
\end{thm}
It turns out that one has strict inequality for sufficiently large even $k$, which is shown in the following lemma:
\begin{lem}\label{Lem:Strict}
Let $D$ be a leveled degree sequence of a vertex-rooted forest and $G = G(D)$ the associated level greedy forest. If $T \in \mathcal{T}_D$ is not isomorphic (as a rooted forest) to $G$, then there exists an integer $k_0$ such that
$$\M_k(T)=C(k;T) < C(k;G(D))=\M_k(G(D))$$
for all even $k \geq k_0$.
\end{lem}
\begin{proof}
It suffices to find one specific level sequence for which we have strict inequality. We take $h_2$ to be the smallest positive integer such that $T$, restricted to the first $h_2$ levels, is not isomorphic to a level greedy rooted forest. Then let $h_1$ be the largest positive integer such that the restriction of $T$ to levels $h_1,h_1+1,\ldots,h_2$ (which we denote by $P$) is still not isomorphic to a greedy rooted forest.
From now on, we only work with the restricted forest $P$. Let $r$ be the number of its roots and
$P_1,P_2,\ldots,P_r$ the components of $P$. Each of them is a level greedy tree: if not, we could remove the root to obtain a rooted forest that is not level greedy, contradicting the maximality of $h_1$. However, by assumption, their union is not a level greedy forest.
Now let $p_1,p_2,\ldots,p_r$ be the number of descendants of the $r$ roots at level $h_2$ ($p_j$ descendants in component $P_j$). The analogous numbers for the greedy tree are $q_1,q_2,\ldots,q_r$, and we call the corresponding components of the restriction of $G$ to the same levels $Q_1,Q_2,\ldots,Q_r$.
We assume, without loss of generality, that $p_1 \geq p_2 \geq \cdots \geq p_r$ and $q_1 \geq q_2 \geq \cdots \geq q_r$. From the construction of level greedy forests, we know that
$$(p_1,p_2,\ldots,p_r) \preccurlyeq (q_1,q_2,\ldots,q_r).$$
In fact, this is a special case of Lemma~\ref{Lem:open_wa_ver}, since $p_1,\ldots,p_r$ and $q_1,\ldots,q_r$ also count walks with level sequence $(h_1,h_1+1,\ldots,h_2)$. The number of closed walks with level sequence
$$(h_2,h_2-1,\ldots,h_1+1,h_1,h_1+1,\ldots,h_2-1,h_2,h_2-1,\ldots,h_1+1,h_1,h_1+1,\ldots,h_2)$$
in $T$ and $G$ are
$$p_1^2 + p_2^2 + \cdots + p_r^2\quad \text{and}\quad q_1^2 + q_2^2 + \cdots + q_r^2$$
respectively: such walks start at level $h_2$, move up to the root, return to level $h_2$, then back to the root, and back to the starting point. They are thus completely determined by the two vertices at level $h_2$ (not necessarily distinct), which have to have the same root.
We suppose first that $p = (p_1,p_2,\ldots,p_r) \neq (q_1,q_2,\ldots,q_r) = q$. Let $i$ be the first index and $j$ the last index where the two differ. Since $q$ majorizes $p$ and the two have the same sums, we must have $q_i > p_i$ and $q_j < p_j$. Let $\epsilon = \min(q_i-p_i,p_j-q_j)$, and replace $p_i$ by $p_i+\epsilon$ and $p_j$ by $p_j-\epsilon$. Then the sum of squares increases by
$$(p_i+\epsilon)^2 - p_i^2 + (p_j-\epsilon)^2 - p_j^2 = 2\epsilon(p_i-p_j+\epsilon) > 0.$$
Repeating this process, we can transform $p$ into $q$, which shows that
$$p_1^2 + p_2^2 + \cdots + p_r^2 < q_1^2 + q_2^2 + \cdots + q_r^2,$$
and we are done in that we have found a level sequence such that $G$ has strictly more closed walks than $T$. The same argument applies (mutatis mutandis) to level sequences of the form
\begin{multline*}
(h_2,h_2-1,\ldots,h_1+1,h_1,h_1+1,h_1,h_1+1,h_1,\ldots, \\
h_1,h_1+1,\ldots,h_2,h_2-1,\ldots,h_1,h_1+1,\ldots,h_2),
\end{multline*}
completing the proof in the case that $p$ and $q$ are not identical (with $k_0 = 4(h_2-h_1)$).
Let us now assume that $p = (p_1,p_2,\ldots,p_r) = (q_1,q_2,\ldots,q_r) = q$, and let $l$ be the last index such that $p_l = q_l \neq 0$. By our choice of $h_2$, the restrictions of $T$ and $G$ to levels $h_1,h_1+1,\ldots,h_2-1$ are isomorphic: they are both level greedy forests consisting of $r$ components. If one component is larger than another, then the number of vertices at level $h_2-1$ is greater as well, and if two components have the same number of vertices at level $h_2-1$, then they are isomorphic by the construction of greedy trees.
Let $m$ be the number of vertices at level $h_2-1$ in the largest component. Then $q_1$ is the sum of the highest $m$ degrees at level $h_2-1$. The only way how $p_1$ can be equal to $q_1$ is thus that $P_1$ and $Q_1$ have the same number of vertices at level $h_2-1$, so they have to be isomorphic (both are known to be level greedy as well!). Likewise, $P_2$ and $Q_2$ have to be isomorphic, etc. The only possible exception are $P_l$ and $Q_l$, the last components with vertices at level $h_2$: here, some vertices in $Q_l$ at level $h_2-1$ might be leaves, so $P_l$ could be smaller than $Q_l$.
Now let $p_1',p_2',\ldots,p_r'$ and $q_1',q_2',\ldots,q_r'$ be the number of vertices at level $h_2-1$ in $P_1,P_2,\ldots,P_r$ and $Q_1,Q_2,\ldots,Q_r$ respectively. The number of closed walks with level sequence
$$(h_2-1,\ldots,h_1+1,h_1,h_1+1,\ldots,h_2-1,h_2,h_2-1,\ldots,h_1+1,h_1,h_1+1,\ldots,h_2-1)$$
in $T$ and $G$ are
$$p_1p_1' + p_2p_2' + \cdots + p_rp_r'\quad \text{and}\quad q_1q_1' + q_2q_2' + \cdots + q_rq_r'$$
respectively, by the same reasoning as before. We know that $p_ip_i' = q_iq_i'$ for $i < l$ and $p_ip_i' = q_iq_i' = 0$ for $i > l$, thus the difference between the two is
$$(q_1q_1' + q_2q_2' + \cdots + q_rq_r') - (p_1p_1' + p_2p_2' + \cdots + p_rp_r') = q_lq_l' - p_lp_l' = q_l(q_l'-p_l').$$
If $q_l' = p_l'$, then the components $P_l$ and $Q_l$ up to level $h_2-1$ have to be isomorphic, and since both are level greedy up to level $h_2$ as well, they must be isomorphic. But then $T$ and $G$, restricted to levels $h_1,h_1+1,\ldots,h_2$, are isomorphic, contradicting our choice of $h_1$ and $h_2$. Thus $q_l' > p_l'$, which means that we have again found a suitable level sequence. Once again, one can generalize to
\begin{multline*}
(h_2-1,\ldots,h_1+1,h_1,h_1+1,h_1,h_1+1,h_1,\ldots, \\
h_1,h_1+1,\ldots,h_2,h_2-1,\ldots,h_1,h_1+1,\ldots,h_2-1),
\end{multline*}
to show that we have strict inequality $C(k;T) < C(k;G)$ for all even $k \geq k_0$, now with $k_0 = 4(h_2-h_1)-2$.
\end{proof}
\subsection{Edge rooted trees}
\label{Sec:edge_r_t}
As we will see at the end of this subsection, Theorem~\ref{Thm:Main_ver_root} still holds if we
consider edge-rooted trees instead of vertex-rooted trees.
For any set $\mathcal{A}$ of walks in a graph and any vertex $v$ and edge $e$ of the same graph, we denote by $\mathcal{A}^e$ and by $\mathcal{A}^v$ the subsets of $\mathcal{A}$ that only contain walks passing through $e$ and $v$, respectively.
Instead of $(\mathcal{A}^e)^{e'}$ we simply write $\mathcal{A}^{e,e'}$. Similarly, $(\mathcal{A}^e)^v=(\mathcal{A}^v)^e=\mathcal{A}^{v,e}=\mathcal{A}^{e,v}.$
For any two adjacent vertices $u$ and $v$ in a graph $G$, we define $\mathcal{C}_{u,v}(k;G)$ to be the set and $C_{u,v}(k;G)$ the number of all closed
walks of length $k$ starting from the edge $uv$ in direction from $u$ to $v$.
Different combinations of these notations are possible. For example, for some edge $uv$ in a graph $G$ and another edge $e$, $\mathcal{C}_{u,v}^{e}(k;G)$ stands for the set of closed walks of length $k$ in $G$ starting at $u$, using the edge $uv$ at the first step and passing through $e$ at a later stage.
\begin{lem}
\label{Lem:Adj_uv}
Let $u$ and $v$ be two adjacent vertices in a graph $G$, and let $e$ be an edge in $G$. Then for all nonnegative integers $k$ we
have
$$
C_{u,v}(k;G)=C_{v,u}(k;G)
\quad\text{and}\quad
C^e_{u,v}(k;G)=C^e_{v,u}(k;G).
$$
\end{lem}
\begin{proof}
Both $C_{u,v}(k;G)$ and $C_{v,u}(k;G)$ are equal to the number of walks of length $k-1$ starting from $u$ and ending at $v$ (which is clearly the same as the number of walks of length $k-1$ starting from $v$ and ending at $u$).
If $e\neq uv$, then both $C^e_{u,v}(k;G)$ and
$C^e_{v,u}(k;G)$ are equal to the number of walks of length $k-1$ starting
from $u$, passing through $e$ and ending at $v$. If $e=uv$, then clearly $C^{e}_{u,v}(k;G)=C_{u,v}(k;G)$, and we are done.
\end{proof}
For any (edge- or vertex-) rooted tree $T$ we denote by $\ro(T)$ the root of $T$. We extend the
notation $\mathcal{C}_{v}(k;T)$ and denote by $\mathcal{C}_{e}(k;T)$ the set of walks of
length $k$ in $T$ which start with the edge $e$ (in either direction). As usual, $C_{v}(k;T)$ and $C_{e}(k;T)$ denote their cardinalities. If $T$ is an edge-rooted tree such that $u$ and
$v$ are the ends of $\ro(T)$, we know by Lemma~\ref{Lem:Adj_uv} that
$$
C_{\ro(T)}(k;T) = C_{u,v}(k;T) + C_{v,u}(k;T)=2 C_{u,v}(k;T)=2 C_{v,u}(k;T).
$$
\begin{lem}
\label{Lem:Start_Root}
Let $D$ be a leveled degree sequence of an edge-rooted tree and $G = G(D)$ the associated edge-rooted greedy tree. For any element $T\in \mathbb{T}_D$ we have
$$C_{\ro(T)}(k;T)\leq C_{\ro(G)}(k;G)$$ for any nonnegative integer $k$.
\end{lem}
\begin{proof}
Let $G_1$ and $G_2$ be the components of $G-\ro(G)$, and let $T_1$ and $T_2$ be the
components of $T-\ro(T)$. Since for odd $k$ we trivially have
$C_{\ro(T)}(k;T) = C_{\ro(G)}(k;G)=0$, we are only interested in even $k=2l.$
Let us reason by induction on $l$. The cases where $l=1,2$ are easy to check, since the closed walks of
length at most $4$ starting with the root edge cannot reach beyond the first two levels, but these parts
of $T$ and $G$ are isomorphic edge-rooted trees. Assume that the lemma holds whenever $l\leq m$ for
some integer $m\geq 2$. Now consider the case where $l=m+1$. The level sequences of
the elements in $\mathcal{C}_{\ro(T)}(k;T)$ and $\mathcal{C}_{\ro(G)}(k;G)$ are of the form $(1,1,i_1,i_2,\dots,i_{k-1})$, and $i_{k-1}$ also has to be $1$.
We first consider walks that do not return immediately to the starting point after the first step. For any $j$ with $2 \leq j\leq k-1$, let $\mathcal{C}^j_{\ro(T)}(k;T)$ and $\mathcal{C}^j_{\ro(G)}(k;G)$ be respectively the subsets of $\mathcal{C}_{\ro(T)}(k;T)$ and $\mathcal{C}_{\ro(G)}(k;G)$ whose elements are the walks with level sequences $(1,1,i_1,i_2,\dots,i_{k-1})$, where $i_j=1$ and
$1\notin \{i_1,i_2,\dots,i_{j-1}\}$. Their cardinalities are denoted by $C^j_{\ro(T)}(k;T)$ and $C^j_{\ro(G)}(k;G)$ respectively. These walks start with the edge root, then go on to higher levels, return to level $1$ for the first time after $j$ steps, and then continue with $k-j-1$ more steps until they return to the starting point. We can uniquely split each of these walks into the $j$ steps from the first step to level $2$ to the first return to level $1$ and the rest. Set
$$\mathbb{S}_j=\{(1,i_1,i_2,\dots,i_{j-1},1): 1\notin \{i_1,i_2,\dots,i_{j-1}\}\}.$$
From Lemma~\ref{Lem:closed_wa_ver}, Lemma~\ref{Lem:Adj_uv} and the induction hypothesis, we now obtain
\begin{align*}
C^j_{\ro(T)}(k;T) &=C_{\ro(T_1),\ro(T_2)}(k-j;T) \sum_{S\in \mathbb{S}_j}C_{\ro(T_2)}(S;T_2)+C_{\ro(T_2),\ro(T_1)}(k-j;T)\sum_{S\in \mathbb{S}_j}C_{\ro(T_1)}(S;T_1)\\
&=\frac{1}{2}C_{\ro(T)}(k-j;T)\big(\sum_{S\in \mathbb{S}_j}C_{\ro(T_2)}(S;T_2)+\sum_{S\in \mathbb{S}_j}C_{\ro(T_1)}(S;T_1)\big)\\
&\leq \frac{1}{2}C_{\ro(G)}(k-j;G)\big(\sum_{S\in \mathbb{S}_j}C_{\ro(G_2)}(S;G_2)+\sum_{S\in \mathbb{S}_j}C_{\ro(G_1)}(S;G_1)\big)\\
&=C^j_{\ro(G)}(k;G)
\end{align*}
for any $j\geq 2$. This covers all the cases where $i_1\neq 1$. Next, consider the subsets
$\mathcal{C}^*_{\ro(T)}(k;T)$ and $\mathcal{C}^*_{\ro(G)}(k;G)$ of
$\mathcal{C}_{\ro(T)}(k;T)$ and $\mathcal{C}_{\ro(G)}(k;G)$, respectively; their elements are
closed walks with level sequence $(1, 1, i_1, i_2,\dots,$ $i_{k-1})$, where $i_1=1$ and for any $h\in \{1,2,\dots,k-2\}$ we always have
$(1,1)\neq (i_h,i_{h+1})$. In words, these walks move forwards and backwards along the edge root for the first two steps, then never use the edge root again, thus they stay in one of the two branches. From Lemma~\ref{Lem:closed_wa_ver}, we now get
\begin{align}
C^*_{\ro(T)}(k;T)
&=C_{\ro(T_1)}(k-2;T_1)+C_{\ro(T_2)}(k-2;T_2)\nonumber\\
&\leq C_{\ro(G_1)}(k-2;G_1)+C_{\ro(G_2)}(k-2;G_2)\nonumber\\
\label{Eq:C_P}
&=C^*_{\ro(G)}(k;G).
\end{align}
We are left with walks that use the edge root, return immediately, and use the edge root again at some stage. The set of these walks is divided further, depending on the first time that the edge root is used again. For any $j\geq 1$, we consider the subsets $\mathcal{C}'^j_{\ro(T)}(k;T)$ and $\mathcal{C}'^j_{\ro(G)}(k;G)$
of $\mathcal{C}_{\ro(T)}(k;T)$ and $\mathcal{C}_{\ro(G)}(k;G)$ whose elements are the closed
walks with level sequence $(1,1,$ $i_1, i_2,\dots,i_{k-1})$, where $i_1=i_j=i_{j+1}=1$ and $(1,1)\neq (i_h,i_{h+1})$
for any $h\in \{1,\dots,j-1\}$. Such a walk can be split uniquely into a walk of length $j+1$ in
$\mathcal{C}^*_{\ro(T)}(j+1;T)$ ($\mathcal{C}^*_{\ro(G)}(j+1;G)$, respectively) and a closed walk of length $k-j-1$ starting with the edge root. From \eqref{Eq:C_P} and Lemma~\ref{Lem:Adj_uv}, we obtain
\begin{align*}
C'^j_{\ro(T)}(k;T)
&= C^*_{\ro(T)}(j+1;T) \cdot \frac{1}{2}C_{\ro(T)}(k-j-1;T)\\
&\leq C^*_{\ro(G)}(j+1;G) \cdot \frac{1}{2}C_{\ro(G)}(k-j-1;G)\\
&=C'^j_{\ro(G)}(k;G)
\end{align*}
for any $j\geq 1$. We see that the greedy tree $G$ has more or at least equally many walks of each type as $T$, which completes the proof.
\end{proof}
\begin{lem}
\label{Lem:Closed_Pass}
Let $D$ be a leveled degree sequence of an edge-rooted tree and $G = G(D)$ the associated edge-rooted greedy tree. For any element
$T\in \mathbb{T}_D$ and for any nonnegative integer $k$ we have
$$
C^{\ro(T)}(k;T)\leq C^{\ro(G)}(k;G).
$$
\end{lem}
\begin{proof}
Any element, say $W$, in $\mathcal{C}^{\ro(T)}(k;T)$ or $\mathcal{C}^{\ro(G)}(k;G)$ has a
unique decomposition as
$
W=W_1W_2W_3
$
for some $W_1,W_2$ and $W_3$ satisfying the following conditions:
\begin{itemize}
\item[$i)$] $W_2$ is a closed walk starting from the edge root, chosen to have maximal length.
\item[$ii)$] $W_1$ and $W_3$ do not use the edge root but can possibly have length zero. By merging the end of $W_1$ with the beginning of $W_3$, we obtain a closed walk $W'$.
\end{itemize}
Under the conditions $i)$ and $ii)$, $W_3$ visits an end of the edge root only once (at its starting point),
otherwise we could extend $W_2$. This means that $W$ can be uniquely recovered from $W'$ and $W_2$ by inserting $W_2$ into $W'$ at the last appearance of an end vertex of the edge root. So the number of possible walks $W$ is the number of possible walks $W'$ times the number of possible walks $W_2$.
Let $T_1$ and $T_2$ be the components
of $T - \ro(T)$, and $G_1$ and $G_2$ those of $G-\ro(G)$. By Lemma~\ref{Lem:closed_wa_ver}, we know that
\begin{align*}
C^{\ro(T_1)}(l;T_1)+C^{\ro(T_2)}(l;T_2)\leq C^{\ro(G_1)}(l;G_1)+C^{\ro(G_2)}(l;G_2)
\end{align*}
for any nonnegative integer $l$. Hence, using Lemma~\ref{Lem:Start_Root} we have
\begin{align*}
C^{\ro(T)}(k;T)
&= \sum_{k_1+k_2=k}\big(C^{\ro(T_1)}(k_1;T_1)C_{\ro(T_1),\ro(T_2)}(k_2;T)+C^{\ro(T_2)}(k_1;T_2)C_{\ro(T_2)\ro(T_1)}(k_2;T)\big)\\
&= \sum_{k_1+k_2=k}\big(C^{\ro(T_1)}(k_1;T_1)+C^{\ro(T_2)}(k_1;T_2)\big) \cdot \frac{1}{2}C_{\ro(T)}(k_2;T)\\
&\leq \sum_{k_1+k_2=k}\big(C^{\ro(G_1)}(k_1;G_1)+C^{\ro(G_2)}(k_1;G_2)\big) \cdot \frac{1}{2}C_{\ro(G)}(k_2;G)\\
&=C^{\ro(G)}(k;G).
\end{align*}
\end{proof}
The next theorem combines Theorem~\ref{Thm:Main_ver_root} and Lemma~\ref{Lem:Closed_Pass}.
\begin{thm}
\label{Th:Main_closed_wa_edge}
Let $D$ be a leveled degree sequence of an edge-rooted tree. For any nonnegative integer $k$ and all $T\in \mathbb{T}_D$, we have
$$
\M_k(T)=C(k;T)\leq C(k;G(D))=\M_k(G(D)).
$$
For sufficiently large even $k$, the inequality is strict unless $T$ and $G(D)$ are isomorphic.
\end{thm}
\begin{proof}
Use Theorem~\ref{Thm:Main_ver_root} to compare the number of closed walks of length $k$ not using
the edge root, and Lemma~\ref{Lem:Closed_Pass} for those which pass through the edge root. The fact that the inequality in Theorem~\ref{Thm:Main_ver_root} is strict for sufficiently large $k$ by Lemma~\ref{Lem:Strict} implies that this is also the case here.
\end{proof}
\subsection{Main result}
The main result of this section is the fact that if we fix a degree sequence $D$, then among
all trees with degree sequence $D$, the greedy tree $G(D)$ has the maximum number of closed walks of any given length. $G(D)$ is not always the unique element of $\mathbb{T}_D$ which reaches the maximum number of fixed
length closed walks: for instance, for any $T\in \mathbb{T}_D$, we have $\mathcal{C}(2;T)=2|E(T)|$, which only depends on $D$.
\begin{thm}
\label{Th:Main}
Let $D$ be a degree sequence of a tree. For any element
$T\in \mathbb{T}_D$ and any $k \geq 0$, we have
$$\M_k(T)=C(k;T)\leq C(k;G(D))=\M_k(G(D)).$$
Moreover, the inequality is strict for sufficiently large even $k$ if $T$ and $G(D)$ are not isomorphic.
\end{thm}
\begin{proof}
If it is possible to choose an edge or a vertex as root such that $T$ is not level greedy, then we let
$T_1$ be the level greedy tree with the same leveled degree sequence as $T$. We iterate this process: if an edge or vertex root can be chosen such that $T_l$ is not level greedy, replace it by the corresponding level greedy tree, which we denote by $T_{l+1}$. Then $\M_k(T_{l+1}) \geq \M_k(T_l)$ for all $k \geq 0$, and for sufficiently large even $k$, the inequality is strict. Therefore, no infinite loops are possible in this process.
Hence there exists an integer $m$
such that $T_m$ is level greedy with respect to any choice of vertex or edge root. This tree $T_m$ satisfies the ``semi-regular'' property defined in \cite{sze11}, and hence it is a greedy tree. From Theorems~\ref{Thm:Main_ver_root} and \ref{Th:Main_closed_wa_edge}, we obtain
$$
C(k;T)\leq C(k;T_1)\leq \dots\leq C(k;T_m) = C(k;G(D))
$$
for any $k\geq 0$, with strict inequality for sufficiently large even $k$.
\end{proof}
\begin{rem} While the inequality in Theorem~\ref{Th:Main} is strict for sufficiently large $k$, there is no ``universal'' $k$ with this property: for every $k$, there exists some degree sequence $D$ and a tree $T$ with degree sequence $D$ that is not isomorphic to the greedy tree $G = G(D)$ such that
$$\M_{\ell}(T) = \M_{\ell}(G),\qquad \ell=0,1,\ldots,k.$$
Consider, for instance, the degree sequence $D=(3,3,2,2,\ldots,2,1,1,1,1)$, where the number of $2$s is $4r-2$ for some integer $r \geq 1$. The greedy tree $G = G(D)$ consists of two neighboring vertices of degree $3$ to which paths are attached: two paths of length $r$ to one of the two, two paths of length $r+1$ to the other. Now let $T$ be the tree where one of the paths of length $r$ in $G$ is interchanged with one of the paths of length $r+1$.
$T$ and $G$ have the same number of (closed) walks of any length that do not contain the vertices of degree $3$, since the forests resulting when the two are removed are isomorphic. Moreover, the subtrees of $T$ and $G$ consisting of vertices whose distance from the degree $3$ vertices is at most $r$ are isomorphic as well. Thus
$$\M_{\ell}(T) = \M_{\ell}(G),\qquad \ell \leq 2r.$$
\end{rem}
\subsection{Consequences of the main result}
Several corollaries follow immediately from our main theorem. In particular, in view of \eqref{Eq:ef}, we obtain the following corollary:
\begin{cor}
For any function $f(x) = \sum_{k=0}^{\infty} a_kx^k$ with nonnegative coefficients and for any tree $T$ with degree sequence $D$, we have
$$\E_f(T)\leq \E_f(G(D)),$$
where $E_f$ is defined as in~\eqref{Eq:efdefi}. If the even part of $f$ is not a polynomial (i.e., $a_k > 0$ for infinitely many even values of $k$), then the inequality is strict unless $T$ is isomorphic to $G(D)$. In particular,
$$\EE(T) < \EE(G(D))$$
for all $T\in \mathbb{T}_D$ that are not isomorphic to $G(D)$.
\end{cor}
Moreover, we also obtain one of the main results of \cite{biyikoglu2008graphs} as another corollary, since the spectral radius $\rho(T)$ of a tree $T$ is equal to the limit $\lim_{\ell \to \infty} \sqrt[2\ell]{\M_{2\ell}(T)}$.
\begin{cor}\label{cor:specrad}
Among all trees with degree sequence $D$, the greedy tree $G(D)$ has the largest spectral radius $\rho(G(D))$.
\end{cor}
In \cite{biyikoglu2008graphs}, it was also shown that the greedy tree is unique with this property.
The Estrada index is just one of in principle infinitely many graph invariants of the form $E_f$. One could certainly conceive of a ``Hyper-Estrada index'', for example:
$$\operatorname{EEE}(G) = \sum_{i=1}^n e^{e^{\lambda_i}}.$$
A somewhat more natural example is the following: note that the characteristic polynomial of a graph $G$ is given by
$$P_G(x) = \prod_{i=1}^n (x-\lambda_i) = x^n \prod_{i=1}^n \left( 1 - \frac{\lambda_i}{x} \right).$$
If $x$ is greater than the spectral radius, then we can take the logarithm and expand it into a power series:
\begin{align*}
\log P_G(x) &= n \log x + \sum_{i=1}^n \log \left( 1 - \frac{\lambda_i}{x} \right) = n \log x - \sum_{i=1}^n \sum_{k=1}^{\infty} \frac{1}{k} \cdot \frac{\lambda_i^k}{x^k} \\
&= n \log x - \sum_{k=1}^{\infty} \frac{M_k(G)}{k x^k}.
\end{align*}
This formula, together with our main result, implies the following statement:
\begin{cor}\label{cor:charpoly}
For any tree $T$ with degree sequence $D$ and any $x > \rho(G(D))$, the inequality
$$P_T(x) \geq P_{G(D)}(x)$$
holds, with equality only if $T$ is isomorphic to $G(D)$.
\end{cor}
\section{Trees with different degree sequences}
\label{Sec:Diff}
In this section, we compare greedy trees with different degree sequence, in a similar way as it was done in \cite{biyikoglu2008graphs,Andriantiana2012,andriantiana2013greedy,zhang12}. This allows us to determine the maximal spectral moments of trees with different restrictions, e.g. given maximum degree or number of leaves.
To this end, we use a transformation on level greedy trees, where branches are moved between vertices at the same level. We study the
effect of such a transformation on the number of closed walks of given length. Unlike the procedure in the proof of Theorem~\ref{Th:Main}, the transformation that we consider in the following lemma does not preserve the degree sequence.
For any vertex $v$ in a rooted tree $T$, we denote by $T_v$ the rooted tree spanned by
$v$ and all its descendants, where $v$ is chosen to be the root.
\begin{lem}
\label{Lem:Snake}
Let $D=((i_{1,1}),(i_{2,1},\dots,i_{2,k_2}),\dots,(i_{n,1},\dots,i_{n,k_n}))$ be a leveled
degree sequence of a (vertex) rooted tree. For some $i$ and $j$ with $1< i < L(D)$ and $1<j\leq k_i$, let $B$ be a branch of $g^i_j$ in the level greedy tree $G = G(D)$ which does not contain the root. Choose the neighbor of $g_j^i$ in $B$ to be the
root of $B$. Let $T=G-g^i_j\ro(B)+g^i_{j'}\ro(B)$ for some $j'<j$ (see Figure~\ref{Fig:moving}). Then we have
$$C(k;T)\geq C(k;G)$$ for any nonnegative integer $k$. For even $k \geq 4$, the inequality is strict.
\end{lem}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1]
\node[fill=black,label=above right:{$x' = g^i_{j'}$},circle,inner sep=1pt] (v11) at (1.3,1.5) {};
\node[fill=black,label=above left:{$x = g^i_j$},circle,inner sep=1pt] (v21) at (4.7,1.5) {};
\draw[dashed] (0,0)--(6,0)--(3,5)--(0,0);
\draw (v21) -- (4.2,1);
\draw[dashed] (4.2,1)--(4.7,0.2)--(3.7,0.2)--(4.2,1);
\node at (4.2,0.5) {$B$};
\node at (4.7,1.1) {$e$};
\node[fill=black,label=above right:{$x' = g^i_{j'}$},circle,inner sep=1pt] (v12) at (9.3,1.5) {};
\node[fill=black,label=above left:{$x = g^i_j$},circle,inner sep=1pt] (v22) at (12.7,1.5) {};
\draw[dashed] (8,0)--(14,0)--(11,5)--(8,0);
\draw (v12) -- (9.8,1);
\draw[dashed] (9.8,1)--(10.3,0.2)--(9.3,0.2)--(9.8,1);
\node at (9.8,0.5) {$B$};
\node at (9.3,1.1) {$e'$};
\end{tikzpicture}
\caption{Moving a branch: the level greedy tree $G$ (left) and the resulting tree $T$ (right).}
\label{Fig:moving}
\end{figure}
\begin{proof}
We use the same labels for vertices in $T$ as in $G$. For notational convenience, set
$x= g^i_j$, $x' = g^i_{j'}$, $e=g^i_j\ro(B)$ and $e'=g^i_{j'}\ro(B).$
It is clear that $C(k;T)-C^{e'}(k;T) = C(k;G)-C^{e}(k;G)$
because $T-e'=G-e$. Thus it suffices to prove
\begin{equation}\label{eq:through_e}
C^{e'}(k;T) \geq C^{e}(k;G).
\end{equation}
Let $v=g^{i'}_l$ be the closest common
ancestor of $x = g^i_j$ and $x' = g^i_{j'}$ in $G$, and let
$u = g^{i'+1}_{h}$ and $u' = g^{i'+1}_{h'}$ be the neighbors of $v$ in the branches containing $g^i_{j}$ and $g^i_{j'}$ respectively.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1]
\node[fill=black,label=right:{$v$},circle,inner sep=1pt] (u) at (0,1) {};
\node[fill=black,label=right:{$u'$},circle,inner sep=1pt] (v1) at (-1,0) {};
\draw[dashed] (v1)--(-1,-4)--(-0.25,-4)--(v1);
\node[fill=black,circle,inner sep=1pt] (v11) at (-1.5,-1.5) {};
\draw[dashed] (v11)--(-1.85,-4)--(-1.1,-4)--(v11);
\node[fill=black,label=right:{$x'$},circle,inner sep=1pt] (v12) at (-2.75,-3) {};
\node[fill=white,label=right:{$C'_{i-i'}$},circle,inner sep=0pt] () at (-3.35,-3.75) {};
\node[fill=white,label=right:{$C'_2$},circle,inner sep=0pt] () at (-1.95,-3.75) {};
\node[fill=white,label=right:{$C'_1$},circle,inner sep=0pt] () at (-1.15,-3.75) {};
\draw[dashed] (v12)--(-3.5,-4)--(-2,-4)--(v12);
\node[fill=black,circle,inner sep=0.5pt] () at (-2.55,-2.7) {};
\node[fill=black,circle,inner sep=0.5pt] () at (-2.17,-2.22) {};
\node[fill=black,circle,inner sep=0.5pt] () at (-1.8,-1.85) {};
\node[fill=black,label=right:{$u$},circle,inner sep=1pt] (v2) at (1,0) {};
\draw[dashed] (v2)--(1,-4)--(0.25,-4)--(v2);
\node[fill=black,circle,inner sep=1pt] (v21) at (1.5,-1.5) {};
\draw[dashed] (v21)--(1.85,-4)--(1.1,-4)--(v21);
\node[fill=black,label=left:{$x$},circle,inner sep=1pt] (v22) at (2.75,-3) {};
\node[fill=white,label=left:{$C_{i-i'}$},circle,inner sep=0pt] () at (3.35,-3.75) {};
\node[fill=white,label=left:{$C_2$},circle,inner sep=0pt] () at (1.95,-3.75) {};
\node[fill=white,label=left:{$C_1$},circle,inner sep=0pt] () at (1.15,-3.75) {};
\node[fill=white,label=above:{\large $G(D)$},circle,inner sep=0pt] () at (5,-3.75) {};
\draw[dashed] (v22)--(3.5,-4)--(2,-4)--(v22);
\node[fill=black,circle,inner sep=0.5pt] () at (2.55,-2.7) {};
\node[fill=black,circle,inner sep=0.5pt] () at (2.17,-2.22) {};
\node[fill=black,circle,inner sep=0.5pt] () at (1.8,-1.85) {};
\draw (v1)--(u)--(v2);
\draw (v21)--(v2);
\draw (v1)--(v11);
\draw (u)--(0.5,1.5);
\draw[dashed] (1,2.5)--(-5,-4.1)--(7,-4.1)--(1,2.5);
\end{tikzpicture}
\caption{Decomposition of $G_v$ in the proof of Lemma~\ref{Lem:Snake}}
\label{Fig:isom_VertRoot}
\end{figure}
Since $G$ is level greedy, if we decompose $G_{v}$ as in Figure~\ref{Fig:isom_VertRoot},
then there is an isomorphism preserving roots between $C_r$ and a subgraph of $C'_r$ for any $r\in\{1,2,\dots,i-i'\}$.
Therefore one can find an injective homomorphism, say
$f:V\big(G_{u}\big)\longrightarrow V\big(T_{u'}\big)$, which satisfies
$f(u)=u'$, $f(x)=x'$ and $f(e)=e'$.
The map
\begin{align*}
F:\mathcal{C}^{e}(k;G)-\mathcal{C}^{v,e}(k;G) &\longrightarrow \mathcal{C}^{e'}(k;T)-\mathcal{C}^{v,e'}(k;T)\\
w_1\dots w_{k+1} &\longmapsto f(w_1)\dots f(w_{k+1})
\end{align*}
is injective because $f$ is injective. We also define a map
$$F':\mathcal{C}^{v}(k;G) \longrightarrow \mathcal{C}^{v}(k;T)$$
in a recursive way. Let $W=w_1\dots w_{k+1}\in \mathcal{C}^{v}(k;G),$
and let $m$ and $M$ be, respectively, the smallest and largest integers such that
$w_m=w_M=v$ and $1<m\leq M<k+1$, if there exist such integers. Then we define:
\begin{itemize}
\item If $v\notin \{w_2,\dots,w_k\}$ (and hence $w_1=w_{k+1}=v$)
and $w_sw_{s+1}\neq e$ for any $s=1,\dots, k,$ then $F'(W)=w_1\dots w_{k+1}.$
\item If $v\notin \{w_2,\dots,w_k\}$ and $w_sw_{s+1}= e$ for some $s\in\{1,\dots, k\}$, then
$$F'(W)=w_1f(w_2)\dots f(w_k)w_{k+1}.$$
\item Otherwise we set
$F'(W)=\phi(w_1\dots w_{m-1})F'(w_m\dots w_M)\phi(w_{M+1}\dots w_{k+1}),$
where $\phi(w_1\dots w_{m-1})=f(w_1)\dots f(w_{m-1})$ if $w_sw_{s+1}= e$ for some $s\in\{1,\dots, m-2\}$,
and $\phi(w_1\dots w_{m-1})=w_1\dots w_{m-1}$ otherwise.
\end{itemize}
In words, we break a walk into pieces separated by visits to vertex $v$. Each piece is either kept the same (if it does not contain $e$) or replaced by its image under the injection $f$ if it contains $e$. Since the decomposition is unique and $f$ is injective, the so constructed map $F'$ is also an injection, and so is its restriction to $\mathcal{C}^{v,e}(k;G)$. This proves inequality~\eqref{eq:through_e} and thus the main inequality.
For even $k \geq 4$, the inequality is strict, since $F$ is not surjective. The degree of $x$ in
$G$ is strictly less than the degree of $x'$ in $T$ by construction. Hence, there is an edge $e''$
incident to $x'$ that does not have a preimage under $F$, and so is any walk starting from $e'$ and uses $e''$.
There is such a closed walk for arbitrary even length larger than $4$.
\end{proof}
\begin{lem}
\label{Lem:Snake_e}
Let $D=((i_{1,1},i_{1,2}),(i_{2,1},\dots,i_{2,k_2}),\dots,(i_{n,1},\dots,i_{n,k_n}))$ be a leveled
degree sequence of an edge-rooted tree. For some $i$ and $j$ with $1\leq i < L(D)$ and $1<j\leq k_i$, let $B$ be a
branch of $g^i_j$ in the level greedy tree $G = G(D)$ which does not contain the root. Take the neighbor of $g^i_j$ in $B$ as root of $B$.
Let $T=G-g^i_j\ro(B)+g^i_{j'}\ro(B)$ for some $j'<j$. Then we have
$$
C(k;T)\geq C(k;G).
$$
for any nonnegative integer $k$. For even $k \geq 4$, the inequality is strict.
\end{lem}
\begin{proof}
Again, we keep the labels of vertices of $G$ in $T$. For simplicity, we write $x = g^i_j$, $x' = g^i_{j'}$, $e=g^i_j\ro(B),$ $e'=g^i_{j'}\ro(B)$, $r=\ro(G)$ and $r'=\ro(T).$
Let $G_1$ and $G_2$ be the components of $G-r$, and $T_1$ and $T_2$ those of $T-r'$, such that $|V(G_1)|\geq |V(G_2)|$ and $|V(T_1)|\geq |V(T_2)|$.
If $x$ and $x'$ are both vertices of the same component $G_m$, then the proof is exactly the same as that of Lemma~\ref{Lem:Snake}. So from now on, we assume that $x\in V(G_2)$ and $x'\in V(G_1)$.
If $G$ is decomposed as in Figure~\ref{Fig:isom}, then $C_r$ has a copy preserving levels
in $C'_r$ for any $1\leq r \leq i$. Because of this fact, we know that one can find a level
preserving injective homomorphism, say $f$, between $G_2$ and
$T_1$ (which has $G_1$ as a subgraph) which satisfies $f(g^1_2)=g^1_1$, $f(x)=x'$ and $f(e)=e'$.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1]
\node[fill=black,label=above:{$g^1_1$},circle,inner sep=1pt] (v1) at (-1,0) {};
\draw[dashed] (v1)--(-1,-4)--(-0.25,-4)--(v1);
\node[fill=black,circle,inner sep=1pt] (v11) at (-1.5,-1.5) {};
\draw[dashed] (v11)--(-1.85,-4)--(-1.1,-4)--(v11);
\node[fill=black,label=right:{$x'$},circle,inner sep=1pt] (v12) at (-2.75,-3) {};
\node[fill=white,label=right:{$C'_i$},circle,inner sep=0pt] () at (-3.1,-3.75) {};
\node[fill=white,label=right:{$C'_2$},circle,inner sep=0pt] () at (-1.95,-3.75) {};
\node[fill=white,label=right:{$C'_1$},circle,inner sep=0pt] () at (-1.15,-3.75) {};
\draw[dashed] (v12)--(-3.5,-4)--(-2,-4)--(v12);
\node[fill=black,circle,inner sep=0.5pt] () at (-2.55,-2.7) {};
\node[fill=black,circle,inner sep=0.5pt] () at (-2.17,-2.22) {};
\node[fill=black,circle,inner sep=0.5pt] () at (-1.8,-1.85) {};
\node[fill=black,label=above:{$g^1_2$},circle,inner sep=1pt] (v2) at (1,0) {};
\draw[dashed] (v2)--(1,-4)--(0.25,-4)--(v2);
\node[fill=black,circle,inner sep=1pt] (v21) at (1.5,-1.5) {};
\draw[dashed] (v21)--(1.85,-4)--(1.1,-4)--(v21);
\node[fill=black,label=left:{$x$},circle,inner sep=1pt] (v22) at (2.75,-3) {};
\node[fill=white,label=left:{$C_i$},circle,inner sep=0pt] () at (3.1,-3.75) {};
\node[fill=white,label=left:{$C_2$},circle,inner sep=0pt] () at (1.95,-3.75) {};
\node[fill=white,label=left:{$C_1$},circle,inner sep=0pt] () at (1.15,-3.75) {};
\draw[dashed] (v22)--(3.5,-4)--(2,-4)--(v22);
\node[fill=black,circle,inner sep=0.5pt] () at (2.55,-2.7) {};
\node[fill=black,circle,inner sep=0.5pt] () at (2.17,-2.22) {};
\node[fill=black,circle,inner sep=0.5pt] () at (1.8,-1.85) {};
\draw (v21)--(v2)--(v1)--(v11);
\end{tikzpicture}
\caption{Decomposition of $G$ in the proof of Lemma~\ref{Lem:Snake_e}}
\label{Fig:isom}
\end{figure}
Since we deal with closed walks, we are only interested in even $k=2l.$
We know that
$$
C(2l;T) - C^{r'}(2l;T)-[C(2l;G) - C^{r}(2l;G)]=C^{e'}(2l;T_1) - C^{e}(2l;G_2)
$$
is nonnegative: as $f$ is injective, so is the map
\begin{align*}
F:\mathcal{C}^{e}(2l;G_2)&\longrightarrow \mathcal{C}^{e'}(2l;T_1)\\
w_1\dots w_{k+1}&\longmapsto f(w_1)\dots f(w_{k+1}).
\end{align*}
Since $f$ is level preserving, we can even choose an arbitrary level sequence $S$ of walks
without two consecutive $1$s and still have
\begin{align}
C^{e'}(S;T_1)&\geq C^{e}(S;G_2), \nonumber \\
C(S,(T_1-B)\cup T_2)&= C(S;G_1\cup (G_2-B)), \label{eq:ls-ineq}
\end{align}
and hence $C(S;T_1\cup T_2)\geq C(S;G_1\cup G_2).$
Now we are left to show that
\begin{align}
\label{Eq:Sec}
C^{r'}(2l;T)-C^{r}(2l;G)=C^{e',r'}(2l;T)-C^{e,r}(2l;G)\geq 0
\end{align}
for any integer $l\geq 1$. Before that let us first show that
\begin{equation}
\label{Eq:Inter}
C^{e'}_{r'}(2l;T)\geq C^{e}_{r}(2l;G)
\end{equation}
for any positive integer $l$. Note the subtle difference between $C^{e,r}(2l;G)$ and $C^{e}_{r}(2l;G)$ here: the former counts walks that pass through $r$ \emph{at some stage}, while the latter counts walks that start with $r$. We reason by induction on $l$. For $l=1$ we have
$
C^{e'}_{r'}(2;T)=C^{e}_{r}(2;G)=0.
$
Assume that \eqref{Eq:Inter} holds whenever $l\leq m$ for some $m\geq 1.$ Since
$
C_{r'}(2l;T)-C^{e'}_{r'}(2l;T)=C_{r}(2l;G)-C^{e}_{r}(2l;G)
$
for all $l$, the induction hypothesis also implies that
$
C_{r'}(2l;T) \geq C_{r}(2l;G)
$
for all $l\leq m.$
Consider now the case where $l=m+1$. Let $\mathcal{C}^{e'}_{r'}(2l;T)=P_1(l)\cup Q_1(l)\cup R_1(l)$ and
$\mathcal{C}^{e}_{r}(2l;G)=P_2(l)\cup Q_2(l)\cup R_2(l)$, where the $P_i(l)$'s contain
walks whose level sequences start with $1,1,1,1$, the $Q_i(l)$'s contain walks whose level sequences start with $1,1,1,2$, and the level sequences of the elements of the $R_i(l)$'s
start with $1,1,2$. The induction hypothesis implies
\begin{align*}
|P_1(l)|
&=C^{e'}_{r'}(2(l-1);T) \geq C^{e}_{r}(2(l-1);G)=|P_2(l)|.
\end{align*}
It is easy to check that $|Q_1(1)|=|Q_2(1)|=0,$ $|Q_1(2)|=|Q_2(2)|\in \{0,1\}$. For $l\geq 3$, we define for any $j$ with $2 \leq j \leq 2l-3$ the
subset $Q_i^j(l)$ of $Q_i(l)$ whose elements have level sequence $(1,1,1,i_1,\dots,i_{2l-2})$,
where $i_j=i_{j+1}=1$ and $(i_s,i_{s+1})\neq(1,1)$ for $s = 1,\ldots,j-1$. Finally,
$$Q_i^{2l-2}(l)=Q_i(l)-\bigcup_{j=2}^{2l-3} Q_i^j(l)$$
is the subset of $Q_i(l)$ whose elements
have level sequence $(1,1,1,i_1,\dots,i_{2l-2})$, where $i_1=2$ and $(1,1)\neq (i_s,i_{s+1})$ for
$s=1,\dots,2l-3.$ Set
\begin{align*}
\mathbb{S}^1_j=\{(1,i_1,\dots,i_{j-1},1): i_1 = i_{j-1} = 2, (1,1)\neq(i_s,i_{s+1})\text{ for } s\in\{1,\dots,j-2\} \}.
\end{align*}
Now we decompose walks in $Q_1^j$ and $Q_2^j$: any such walk consists of two steps along the edge root (forwards and backwards), then continues to higher levels and returns to the first level after $j$ steps (possibly earlier as well, but without ever using the edge root). We call this part $U_1$; its level sequence lies in $\mathbb{S}^1_j$. Thereafter, the walk continues for another $2l-j-2$ steps, starting with the edge root; this part is called $U_2$. Since we know that a walk in $Q_1^j$ has to pass through $e'$, we have the following possibilities:
\begin{itemize}
\item The walk $U_1$ uses $e'$ (which means that it lies entirely in $T_1$), the walk $U_2$ is arbitrary.
\item The walk $U_1$ does not use $e'$, but stays in $T_1$ (thus it lies in $T_1 - B$), the walk $U_2$ contains $e'$.
\item The walk $U_1$ lies in $T_2$, thus it does not use $e'$. Then the walk $U_2$ has to contain $e'$.
\end{itemize}
For $Q_2^j$, there are three analogous possibilities. Making use of this decomposition, Lemma~\ref{Lem:Adj_uv},~\eqref{eq:ls-ineq} and the induction hypothesis, we obtain
\begin{align*}
&|Q_1^j(l)|\\
&=\sum_{S\in\mathbb{S}^1_j} C^{e'}(S;T_1) C_{\ro(T_1),\ro(T_2)}(2l-j-2;T)+\sum_{S\in\mathbb{S}^1_j}C(S;T_1-B) C^{e'}_{\ro(T_1),\ro(T_2)}(2l-j-2;T)\\
&\qquad+\sum_{S\in\mathbb{S}^1_j} C(S;T_2) C^{e'}_{\ro(T_2),\ro(T_1)}(2l-j-2;T)\\
&=\sum_{S\in\mathbb{S}^1_j} C^{e'}(S;T_1) \cdot \frac{1}{2} C_{r'}(2l-j-2;T)+\sum_{S\in\mathbb{S}^1_j}\left(C(S;T_1-B)+C(S;T_2)\right) \cdot \frac{1}{2} C^{e'}_{r'}(2l-j-2;T)\\
&\geq\sum_{S\in\mathbb{S}^1_j} C^{e}(S;G_2) \cdot \frac{1}{2} C_{r}(2l-j-2;G)+\sum_{S\in\mathbb{S}^1_j}\left(C(S;G_1)+C(S;G_2-B)\right) \cdot \frac{1}{2} C^{e}_{r}(2l-j-2;G)\\
&=|Q_2^j(l)|
\end{align*}
for all $j$ such that $2 \leq j \leq 2l-3$. For $j = 2l-2$, walk $U_2$ is empty, so we have
$$
|Q_1^{2l-2}(l)| =\sum_{S\in\mathbb{S}^1_j} C^{e'}(S;T_1) \geq \sum_{S\in\mathbb{S}^1_j} C^{e}(S;G_2) = |Q_2^{2l-2}(l)|.
$$
We conclude with the third subclass of walks whose level sequences start with $1,1,2$.
For any $j$ with $2 \leq j \leq 2l-2$, let $R_i^j(l)$ be the subset of $R_i(l)$ whose elements have level sequence $(1,1,i_1,\dots,i_{2l-1})$,
where $i_j=1$ and $1\notin\{i_1,\dots,i_{j-1}\}$. The case that $j=2l-1$ is not interesting since it does
not correspond to any closed walk. We decompose $R_1^j(l)$ and $R_2^j(l)$ in a similar way as we decomposed $Q_1^j(l)$ and $Q_2^j(l)$. Define
$$\mathbb{S}^2_j=\{(1,i_1,\dots,i_{j-1},1): 1\notin \{i_1,\dots,i_{j-1}\} \}.$$
A walk in $R_1^j(l)$ (or $R_2^j(l)$) consists of a step along the edge root, then moves to higher levels and only returns to the first level after $j$ steps. This part of $j$ steps has a level sequence in $\mathbb{S}^2_j$, the rest forms a closed walk starting with the edge root. Dividing into three cases again, depending on which part contains $e'$ ($e$, respectively), we obtain
\begin{align*}
|R_1^j(l)|
&=\sum_{S\in\mathbb{S}^2_j} C^{e'}(S;T_1) C_{\ro(T_2),\ro(T_1)}(2l-j;T)+\sum_{S\in\mathbb{S}^2_j} C(S;T_1-B) C^{e'}_{\ro(T_2),\ro(T_1)}(2l-j;T)\\
&\qquad+\sum_{S\in\mathbb{S}^2_j} C(S;T_2) C^{e'}_{\ro(T_1),\ro(T_2)}(2l-j;T)\\
&=\sum_{S\in\mathbb{S}^2_j} C^{e'}(S;T_1) \cdot \frac{1}{2} C_{r'}(2l-j;T)+\sum_{S\in\mathbb{S}^2_j}\left(C(S;T_1-B)+C(S;T_2)\right)\cdot \frac{1}{2}C^{e'}_{r'}(2l-j;T)\\
&\geq\sum_{S\in\mathbb{S}^2_j}C^{e}(S;G_2) \cdot \frac{1}{2} C_{r}(2l-j;G)+\sum_{S\in\mathbb{S}^2_j}\left(C(S;G_1)+C(S;G_2-B)\right) \cdot \frac{1}{2} C^{e}_{r}(2l-j;G)\\
&=|R_2^j(l)|.
\end{align*}
This completes the proof of \eqref{Eq:Inter}. We now proceed to the proof of \eqref{Eq:Sec}, making use of a similar argument as in Lemma~\ref{Lem:Closed_Pass}.
Any element, say $W$, in $\mathcal{C}^{e',r'}(2l;T)$ or $\mathcal{C}^{e,r}(2l;G)$
has a unique decomposition
\begin{equation}
\label{Eq:UniqDec}
W=W_1W_2W_3,
\end{equation}
where $W_2$ is a closed walk that starts with the edge root and is chosen to have maximal length and
$W'=W_1W_3$ forms a closed walk which never uses the edge root, but passes at least once through one of its ends (unless it is empty). The decomposition \eqref{Eq:UniqDec} is unique (as it was explained in the proof of Lemma~\ref{Lem:Closed_Pass}). Now let $$\mathbb{S}^3_j=\{(i_1,\dots,i_{j+1}): i_s=1\text{ for some }1\leq s \leq j+1\}.$$
The walk $W'$ has a level sequence in $\mathbb{S}^3_j$ for some $j$. Again, there are three possibilities for a walk in $\mathcal{C}^{e',r'}(2l;T)$:
\begin{itemize}
\item The walk $W'$ contains $e'$, and thus lies entirely in $T_1$, and $W_2$ is arbitrary.
\item The walk $W'$ does not contain $e'$, but still lies in $T_1$ (thus entirely in $T_1 - B$), and $W_2$ uses $e'$.
\item The walk $W'$ lies in $T_2$, thus does not use $e'$. Then $W_2$ has to use $e'$.
\end{itemize}
There are three analogous possibilities for $\mathcal{C}^{e,r}(2l;G)$. We obtain
\begin{align*}
&C^{e',r'}(2l;T) \\
&=\sum_{j=0}^{2l-2}\sum_{S\in\mathbb{S}^3_j} C^{e'}(S;T_1) C_{\ro(T_1),\ro(T_2)}(2l-j;T)+C(S;T_1-B) C^{e'}_{\ro(T_1),\ro(T_2)}(2l-j;T)\\
&\qquad+C(S;T_2) C^{e'}_{\ro(T_2),\ro(T_1)}(2l-j;T)\\
&=\sum_{j=0}^{2l-2}\sum_{S\in\mathbb{S}^3_j} C^{e'}(S;T_1) \cdot \frac{1}{2} C_{r'}(2l-j;T)+\left(C(S;T_1-B)+ C(S;T_2)\right) \cdot \frac{1}{2} C^{e'}_{r'}(2l-j;T)\\
&\geq\sum_{j=0}^{2l-2}\sum_{S\in\mathbb{S}^3_j} C^{e}(S;G_2) \cdot \frac{1}{2} C_{r}(2l-j;G)+\left(C(S;G_1)+ C(S;G_2-B)\right) \cdot \frac{1}{2} C^{e}_{r}(2l-j;G)\\
&= C^{e,r'}(2l;G).
\end{align*}
This concludes the proof of~\eqref{Eq:Sec} and thus the theorem. As in the previous lemma, the inequality is strict for even $k \geq 4$ since the map $F$ is not surjective.
\end{proof}
Given two degree sequences $B\preccurlyeq D$ of trees, by iteratively transferring branches, we can transform
$G(B)$ to become an element of $\mathbb{T}_D$. As seen in the proof of the next theorem, it turns out that
it is always enough to only use transfers of the type described in the two Lemmas~\ref{Lem:Snake}
and \ref{Lem:Snake_e} to obtain an element of $\mathbb{T}_D$ from $G(B)$, showing that $G(D)$ has more closed walks of any length than $G(B)$. This parallels analogous results for e.g. the number of subtrees \cite{zhang12} or the spectral radius \cite{biyikoglu2008graphs}.
\begin{thm}
\label{Th:comp}
Let $D=(d_1,\dots,d_n)$ and $B=(b_1,\dots,b_n)$ be degree sequences of trees of the same order
such that $B\preccurlyeq D$. Then for any integer $k\geq 0$ we have
$$
C(k;G(B)) \leq C(k;G(D)).
$$
If $B \neq D$ and $k$ is even and $\geq 4$, then the inequality is strict.
\end{thm}
\begin{proof}
The statement is obvious for $B = D$. From now, we assume that there exists some $i_0$ such that
$b_{i_0}\neq d_{i_0}$. Since
\begin{equation}
\label{Eq:Mjoo}
\sum_{i=1}^nb_i=\sum_{i=1}^nd_i,
\end{equation}
we know that the set $\{i : d_i\neq b_i\}$ must have at least two elements. Let
$l=\min \{i : d_i\neq b_i\}$ and $m=\max \{i : d_i\neq b_i\}.$ We must have $b_{l}< d_{l}$,
$b_{m}> d_m$ and hence $b_{l-1}=d_{l-1}\geq d_l\geq b_l+1$ and $b_{m+1}=d_{m+1}\leq d_m\leq b_m-1$.
Therefore, $B_1 = (b_1,\dots,b_{l-1},b_l+1,b_{l+1},\dots,b_{m-1},b_m-1,b_{m+1},\dots,b_n)$
is a valid degree sequence. It is easy to see that $B\preccurlyeq B_1.$ Consider two vertices $u$
and $v$ in the greedy tree $G(B)$ such that $\deg u=b_l$ and $\deg v=b_m$.
\medskip
\noindent {\bf Case 1:} The length of the path in $G(B)$ joining $u$ and $v$ is even. Let $w$ be the middle vertex of this path. Consider $G(B)$ as a level greedy tree whose root is $w$. Then $u$ and $v$ are on the same level, say level $h$. We have $u = g_i^h$ and $v = g_j^h$ for some $i < j$.
Let $w = g_r^{h+1}$ be a child of $v = g_j^h$,
and let $H = G(B)_{w}$ be the branch rooted at $w$.
Consider $T=G(B)-vw+uw$; the degree sequence of $T$ is $B_1$. By Theorem~\ref{Thm:Main_ver_root}
and Lemma~\ref{Lem:Snake}, it follows that
$$C(k;G(B_1)) \geq C(k;T) \geq C(k;G(B))$$
for all $k \geq 0$.
\medskip
\noindent {\bf Case 2:} The length of the path in $G(B)$ joining $u$ and $v$ is odd. The
argument is analogous to the previous case: we choose the middle edge of the path as root and then
we use Theorem~\ref{Th:Main_closed_wa_edge} and Lemma~\ref{Lem:Snake_e} instead of
Theorem~\ref{Thm:Main_ver_root} and Lemma~\ref{Lem:Snake}.
\medskip
In either case, we have
$$C(k;G(B_1)) \geq C(k;G(B))$$
for all $k \geq 0$. We repeat this process to obtain a sequence of degree sequences $B_0 = B, B_1,B_2,\ldots,B_r = D$ such that
$
B = B_0 \preccurlyeq B_1 \preccurlyeq \dots \preccurlyeq B_r = D
$
and
$$
C(k;G(B)) = C(k;G(B_0)) \leq C(k;G(B_1))\leq \dots \leq C(k;G(B_r)) = C(k;G(D))
$$
for all $k \geq 0$, which proves the theorem.
\end{proof}
Conjecture~\ref{Conj:1} follows as corollary of the two Theorems~\ref{Th:Main} and \ref{Th:comp}:
The degree sequence of the $n$-vertex Volkmann tree, which is of the form
$(\Delta,\dots,\Delta,r,1,\dots,1)$ for some $1\leq r<\Delta$, majorizes all possible degree
sequence of $n$-vertex trees with maximum degree $\Delta$.
More results can be obtained by similar arguments in the same way as Corollaries 5.1 -- 5.5 of
\cite{zhang12} and Corollaries 29 -- 32 of \cite{Andriantiana2012} are obtained. Let us state some more of these corollaries, which also recover some results that can be found in \cite{zhang2011estrada,du2011estrada2}:
\begin{cor}
For any $n$-vertex tree $T$ and for any $k\geq 0$, $$\M_{k}(S_n)\geq \M_k(T),$$ where $S_n$
is the star with $n$ vertices, whose degree sequence is $(n-1,1,\dots,1)$.
\end{cor}
\begin{cor}
Among trees $T$ of order $n$ with $s$ leaves, $\M_k(T)$
is maximized by the greedy tree $G(s,2,2,\ldots,2,1,1,\ldots,1)$ (the number of $2$s is $n-s-1$,
the number of $1$s is $s$) for any $k \geq 0$.
\end{cor}
\begin{cor}
Among trees $T$ of order $n$ with independence number $\alpha \geq n/2$ and among all trees $T$ with matching number
$n-\alpha\leq n/2$, $\M_k(T)$ is maximized by the greedy tree $G(\alpha,2,2,\ldots,2,1,1,\ldots,1)$
(the number of $2$s is $n-\alpha-1$, the number of $1$s is $\alpha$) for any $k \geq 0$.
\end{cor}
$\M_k$ in each of the above corollaries can of course be replaced by $\EE$ or more generally $\E_f$ for any $f$ with nonnegative coefficients in \eqref{Eq:f}. If infinitely many even-indexed coefficients are strictly positive (e.g., for $\EE$), then we even have strict inequality. Moreover, corollaries analogous to Corollary~\ref{cor:specrad} and Corollary~\ref{cor:charpoly} for the spectral radius and the values of the characteristic polynomial also follow easily.
\bibliographystyle{abbrv}
|
1,314,259,996,688 | arxiv | \section{Introduction}
The interplay between physics at hadron colliders and that at
$e^{+}e^{-}$ machines has traditionally been of great significance in
furthering our understanding of elementary particles and their interactions.
One can for example point to the specific case of the discovery of the $Z$ boson at a hadron collider \cite{Z} which was then followed by
high precision phenomenology at LEP which helped to establish firmly the standard model of particle physics, the current theory of elementary particles beyond which any discoveries are still to be made.
This tradition is set to continue with the strong expectation that the
LHC will lead to the discovery of the Higgs boson or help to clarify the Higgs sector as well as enabling the discovery of physics beyond the standard model. The extremely high energy hadronic collisions at the LHC, which make it a
powerful discovery machine, however come with a price which takes the form of a more complicated initial state (protons rather than elementary particles) and complications concerning non-perturbative effects such as beam remnant interactions (the underlying event) and pile-up which threaten to limit the
theoretical precision that one may be able to obtain. The most precise
determination of the parameters of the new physics such as masses and couplings would probably require a cleaner environment such as a high energy
$e^{+}e^{-}$ future linear collider.
Nevertheless as the Tevatron experience has to an extent confirmed,
calculations in perturbative QCD will have a strong role to play in the physics program of the LHC, particularly with regards to estimating accurately backgrounds to new physics. To this end significant effort has been devoted in the past years to develop QCD calculations specifically for important LHC processes in the discovery context. Moreover given the vast scale hierarchy inherent in high energy hadron collider physics (with scales ranging from the TeV range centre-of--mass energy $\sqrt{s}$, through typical jet transverse momenta $p_T$, the masses of electroweak scale particles down to the few GeV scales associated to non-perturbative physics) it is clear that techniques involving summation of
large logarithms in scale ratios would also be important in maximising the theoretical accuracy one may be able to achieve. The introduction of new and faster infrared and collinear (IRC) safe jet algorithms and a systematic
understanding of perturbative and non-perturbative properties of jets and jet substructure is also a rapidly developing and vital part of the current and future LHC physics program.
At the same time, as should be clear from the preceding discussion,
furthering the precision of QCD calculations for $e^{+}e^{-}$ annihilation
remains of continued importance for future phenomenology as well as remaining a simpler learning and testing ground for QCD practitioners. In this context
the development of next-to--next to leading order (NNLO) predictions and taking resummed computations from the state of the art next-to--leading logarithmic (NLL) level through to NNLL accuracy as well as possibly improving the current theoretical understanding of non-perturbative effects such as hadronisation
corrections will all play an important role.
In what follows below we present a brief summary of what we perceive to be some of the main developments and recent progress in QCD calculations for both
hadron colliders and $e^{+}e^{-}$ machines. It is impossible due to page
limitations to adequately cover all the relevant progress that has been made in the past few years and thus the selection of topics/references below is far
from complete. We shall aim to discuss briefly the progress in fixed-order
perturbative computations as well as all-order resummations both in the hadron collider and the $e^{+}e^{-}$ context, mention the status of $\alpha_s$ measurements and discuss progress in the definition and understanding of jets and their properties in and beyond QCD perturbation theory.
\section{QCD at fixed order}
Observables that have the property of infrared and collinear (IRC) safety can be calculated as an expansion in the strong coupling $\alpha_s$ using perturbative techniques based on the evaluation of Feynman graphs. By an IRC safe observable one essentially means the following: Let ${\mathcal{O}}_n \equiv {\mathcal{O}}(p_1,p_2,\cdots p_n)$ denote the value of the observable $\mathcal{O}$ due
to a configuration involving $n$ partons with momenta $p_1,p_2 \cdots p_n$.
Now consider adding an extra parton with momentum $p_{n+1}$. In the soft limit that the energy $E_{n+1} \to 0$ (with $E_1, \cdots ,E_n$ held finite) or the limit that $\vec{p}_{n+1} \to \vec{p}_i$ where $i=1,\cdots,n$ i.e the limit in which the three-momentum $\vec{p}_{n+1}$ is parallel to any of the three-momenta $\vec{p}_i$ (collinear limit) IRC safety implies that independent of $n$, $\mathcal{O}_{n+1} \to \mathcal{O}_n$. IRC safety ensures that real-virtual cancellation of divergences occurs and hence that finite results are obtained
in perturbation theory.
For a simple observable of the above kind, $V$, involving a single hard scale $Q^2$, we can then write the perturbation expansion as
\begin{equation}
V = \sum_{n=0}^{\infty} C_n \left(\frac{Q^2}{\mu^2} \right) \alpha_s^n(\mu^2),
\end{equation}
where $C_n$ are perturbatively calculable coefficients, $Q$ is the hard scale of the process and $\mu$ an arbitrary renormalisation scale, which however should be chosen to be of order $Q$ to avoid large logarithms in $Q^2/\mu^2$. The dependence on $\mu$ would in fact cancel if one were able to compute the observable to all orders exactly but in practice one is able to evaluate only a few terms in the above sum. The residual $\mu$ dependence in a calculation truncated at $n^{\mathrm{th}}$ order in $\alpha_s$ is of the order of uncalculated ${\mathcal{O}}\left(\alpha_s^{n+1}\right)$ terms. Thus scale dependence is usually
taken as a measure of the influence of uncalculated higher orders and hence the theoretical accuracy of a given prediction. \footnote{One should be aware that the scale dependence may in cases not be a reliable estimate of the true size of higher orders. For example if new hard scattering channels open up in higher perturbative orders varying scales in a lower order contribution cannot be expected to estimate the size of such new contributions.}
Generally speaking leading order (LO) calculations are too crude to be considered reliable estimates for most collider observables. NLO calculations on the
other hand may be expected to be correct, broadly speaking, to within order 10 percent while NNLO calculations represent high precision and as a rule of thumb ought to be accurate to within a few percent or so.
\footnote{There are exceptions to these broad statements which for instance only apply to observables not afflicted by multiple disparate hard scales. For
explicit counter examples see for instance Ref.~\cite{giant}.}
\begin{figure}
\includegraphics[width=0.6\textwidth]{LHC_Z_Mz_edit03}
\caption{An illustration of the scale uncertainty reduction with the order of
the perturbative estimate for the case of the rapidity ($Y$) distribution for
inclusive $Z$ production at the LHC. Figure taken from Ref.~\cite{Babis}.}
\label{fig:babz}
\end{figure}
An illustration of this is provided in Fig.~ 1 where one notes the progressive reduction in scale uncertainty with the increasing order of the perturbative evaluation for the case of the inclusive $Z$ rapidity distribution for the LHC.
For reliable estimates of backgrounds to LHC processes with new physics it
would thus appear that at least NLO accuracy is a must. For up to the production of three jets at hadron colliders NLO calculations encoded in the program
NLOjet++ have been available for some time \cite{ref:Nagy}.
However many of the relevant discovery processes involve high multiplicity final states with similar backgrounds involving e.g multiple hard final state jets for which it is much less straightforward to obtain NLO estimates. At present the current state of the art for NLO computations at hadron colliders is for $2 \to 4$ processes such as a $t \bar{t} b \bar{b}$ final state relevant for
Higgs production and decay in association with a $t \bar{t}$ pair \cite{Shitmeir1,Shitmeir2}. Similarly NLO calculations to $W+3j$ \cite{Shitmeir3,Shitmeir4} and $Z^0+3j$ \cite{Shitmeir5} have been recently computed. A significant development in the computation of NLO corrections has been the advent of unitarity based calculational methods alongside traditional Feynman-diagram techniques. A pedagogical review and further references can be found in Ref.~\cite{GZ}. The
automation of NLO computations is also an important step towards the calculation of several different collider processes. The automation of both real radiation terms \cite{SHERPA,MadGraph,Phegas} and virtual corrections \cite{Golem,BlackHat,CutTools,Rocket,Samurai} has been achieved in the past few years, for NLO corrections.
As far as NNLO calculations are concerned only a few processes are known to
such accuracy. For instance for hadron collisons fully exclusive NNLO corrections to vector boson production have been computed \cite{Catani,Petriello}
while for the case of $e^{+}e^{-}$ annihilation similar calculations have been performed using the method of antenna subtraction for the case of $e^{+}e^{-} \to 3j$ which has enabled a more accurate determination of $\alpha_s$ from data on LEP event shape variables \cite{Glover,thefreak,Bethke}.
Having briefly summarised the state of the art for QCD calculations at fixed-order we shall turn our attention to those observables where the involvement of more than one perturbative scale forces us to go beyond fixed-order
perturbation theory using resummation methods.
\section{QCD beyond fixed-order}
As mentioned above there exist several observables of phenomenological interest where multiple scales (typically the process hard scale and other scales introduced due to observable definition) play an important role. For such
observables, the classic examples of which remain event or jet shape variable
distributions \cite{Dassal}, one has to consider the role of large logarithms
in scale ratios and examine the possibility to resum these to all orders at a
given logarithmic accuracy.
To be more explicit consider the distribution in some shape variable $\tau$ in say $e^{+}e^{-}$ annihilation:
\begin{equation}
\frac{1}{\sigma} \frac{d\sigma}{d\tau} \sim \sum_n \frac{1}{\tau} \alpha_s^n \ln^{2n-1} \frac{1}{\tau}+\cdots
\end{equation}
The above behavior reflects the double-logarithmic enhancement of the shape
cross-section due to soft and collinear gluon emissions while the ellipsis denote less singular terms some of which also need to be accounted for for phenomenological purposes. This result is clearly divergent and
unphysical at small $\tau$ which reflects the inadequacy of fixed-order
predictions in that region and hence the need for resummation.
On resummation, for those variables that have the property of exponentiation
\cite{CTTW} one can write a result of the form
\begin{equation}
\frac{1}{\sigma} \frac{d\sigma}{d\tau} \sim \frac{d}{d\tau} e^{-C_F \alpha_s \ln^2 \frac{1}{\tau}}+\cdots
\end{equation}
which generalises with account of running coupling and less singular terms into the form ($L \equiv \ln 1/\tau$):
\begin{equation}
\frac{1}{\sigma}\frac{d\sigma}{d\tau}\sim \frac{d}{d\tau} \exp \left [Lg_1(\alpha_s L) +g_2(\alpha_s L) +\alpha_s g_3 (\alpha_s L) +\cdots \right].
\end{equation}
In the above result the leading and next-to--leading logarithmic (NLL) terms are represented by the functions $g_1$ and $g_2$. The current state-of--the art
for most observables at any collider process is NLL accuracy in the resummed exponent. The NNLL function $g_3$ is known only for some select variables
amongst which are the thrust and heavy jet-mass distribution in $e^{+}e^{-}$ annihilation (in fact computed in the framework of soft-collinear effective theory to $\mathrm{N^3 LL}$ accuracy \cite{BecNeu}) and for hadron collisions the Drell-Yan and Higgs transverse momentum ($Q_T$) distribution (see for instance
Ref.~\cite{CatNLL} and references therein). Most recently for the Drell-Yan case results have also been obtained including NNLL accuracy for the new $a_T$ and $\phi^*$ variables measured by the D0 collaboration \cite{d0} which broadly speaking are in good agreement with the data even without inclusion of non-perturbative effects \cite{at,DasMarBan}.
For $e^{+}e^{-}$ annihilation the role of resummation in
ensuring precision phenomenology has been clear for a long time \cite{CTTW}.
\begin{figure}
\label{fig:scet}
\includegraphics[width=0.4 \textwidth]{scetvspyth92G}
\includegraphics[width=0.4 \textwidth]{scetvspyth1T.eps}
\caption{Figure illustrating the comparison between various levels of resummation and results from PYTHIA for the thrust distribution in $e^{+}e^{-}$
annihilation for $Q=91.2$ GeV (left) and $Q=1$ TeV (right). Data from LEP are also shown in the former case. Figure taken from Ref.~\cite{BecNeu}}
\end{figure}
Consider as a recent example the comparison between various levels of resummation, event generators and $e^{+}e^{-}$ event shape data depicted in Fig.~2.
At the $Z$ peak it appears that there is excellent agreement between PYTHIA (at hadron level) and data. Moreover the PYTHIA (parton level) result appears rather closer to the $\mathrm{N^3LL}$ ($4^{th}$ order) curve than to the LL result which is where one may expect it to be. That this is an effect which arises
due to uncontrolled sub-leading terms and the tuning procedure inherent in
PYTHIA is revealed by going to $Q=1$ TeV, where for example subleading effects would be inconsequential, PYTHIA is much closer to the LL rather than the $\mathrm{N^3LL}$ result. It has hence been observed in Ref.~\cite{BecNeu} that using LL MC generators may potentially lead to a significant underestimate of certain QCD backgrounds at a future ILC (at about the few tens of percent level).
\begin{figure}
\label{fig:cae}
\includegraphics[width =0.8 \textwidth]{herwig2ymaxpartons-LHC_200-6bvar.eps}
\caption{Resummed predictions for global hadron collider event shapes compared to results from HERWIG. Figure taken from Ref.~\cite{BSZ}}
\end{figure}
While accurate resummed predictions have been an important requirement in say the determination of $\alpha_s$ from LEP event shapes, they are also in principle of great value for jet production in hadron collisions in terms of improving perturbative accuracy. However the more complex hadronic environment at a hadron collider makes all-order resummation a rather delicate affair. For instance care has to be taken in constructing observables such as event shapes to avoid contamination from beam remnants by constructing suitably central event shapes which then have the property of being non-global \cite{DassalNG1,DassalNG2}. Since the non-global single logarithms cannot be computed beyond the large $N_c$ limit, in order to ensure full NLL accuracy for observables such as event
shapes in hadronic dijet production, the observables have to be further
modified in such a way so as to ensure globalness, such as those variables
studied in Ref.~\cite{BSZ}. A yet more troublesome issue is the contamination as a result of effects such as pile-up which can potentially override the eventual accuracy which can be achieved via theoretical methods such as resummation. It is thus desirable to seek variables that are less prone to such effects in order to test resummed calculations hadron collider observables. Predictions for several hadron collider event shape variables as reported in Ref.~\cite{BSZ} are shown in Fig.~3. In some cases some discrepancy with corresponding results from leading-log and leading colour event generators such as HERWIG can also be noted. For more detailed comments on the role of tuning and the shower parameters in such comparisons we refer the reader to the comments in Ref.~\cite{BSZ}. Detailed phenomenological studies for hadron collider event shape variables are currently in progress \cite{BSZ,Aaltonen}.
Aside from a limited number of global event shapes and observables such as suitably defined dijet azimuthal correlations \cite{BanDasDel} as well as Drell-Yan $Q_T$ spectra, one may try to study via resummation other observables involving for instance jet-definition and the application of a jet algorithm. As an
example of this one can point to the case of jet masses and shapes for high $p_T$ jets at the LHC which are relevant in identifying the origin of a jet as being initiated by a QCD process (quark or gluon jet) or say by the decay of a boosted heavy particle. The QCD jet mass distribution for example would receive logarithmic enhancements $\sim \alpha_s \ln^2 \frac{R P_t}{M_j}$ where $P_t$ is the transverse momentum and $M_j$ the jet mass, with $R$ being the jet radius. Since at the LHC we will encounter jets with $P_t$ in the TeV range, the role of such logarithmic terms can be expected to be substantial even up to jet masses near the electroweak scale. The resummation of such logarithms while being
immensely desirable from the standpoint of perturbative accuracy however has
complex issues mainly to do with the role of non-global logarithms and
jet algorithms and was recently discussed in Ref.~\cite{DasMarBanKam}. While a very high formal level of precision in such cases is essentially ruled out it
should still be possible to develop resummation formulae that capture the numerically dominant terms in the result to sufficient accuracy for phenomenological purposes.
We conclude this section by presenting in Fig.~4, the current status of $\alpha_s$ determinations taken from Ref.~\cite{bethke}. The 2009 value for the world average for $\alpha_s(M_z)$ was reported as $0.1184 \pm 0.0007$. The individual contributions from various QCD observables used for $\alpha_s$ extraction are also shown in Fig.~4.
\begin{figure}
\label{fig:asbet}
\includegraphics[width=0.5\textwidth]{asmz-09.eps}
\caption{Figure displaying the $\alpha_s$ values extracted from various QCD
studies alongwith the world average value (dashed vertical line) and error (yellow band). Taken from the comprehensive 2009 review \cite{bethke}.}
\end{figure}
\section{Progress in jet definiton and understanding jet properties}
Although the precise definition of QCD jets may appear a detail not necessarily directly relevant to progress of high order QCD calculations discussed in the major part of this review, it is in fact the case that such calculational developments need to be supported by suitable IRC safe jet definitions. In other words higher order perturbative estimates for jet cross-sections and differential distributions only make sense when an infrared and collinear safe jet algorithm is used in jet definition. Although in many cases of interest such as inclusive jet $p_T$ spectra the IRC unsafety of a given jet algorithms may only appear at a relatively high order, for several LHC processes involving large multiplicity of final state jets (say as backgrounds to a new physics process) the IRC unsafety may appear already at leading order invalidating any level of perturbative accuracy \cite{SalSoy}.
Likewise it is not meaningful to compute all-order resummed predictions for quantities that will diverge at any fixed order due to the algorithm in use. This requirement coupled with experimental and practical considerations (speed of the algorithm for high multiplicity hadronic final states) make the definition of jets a non-trivial task. Fortunately there now exist several different practically feasible options for IRC safe jet definitions defined either using sequential recombination \cite{k_t,C_A,anti} or based on the idea of cone jets \cite{SalSoy}. The recent fast progress in the field of jet physics are expertly reviewed in Ref.~\cite{Salam} to which we point the interested reader for further details.
As a by product of the rapid developments in jet physics there has also recently been tremendous interest in using a somewhat more sophisticated understanding of jets and their properties, gained via relatively simple analytical
calculations, as a chisel for improving the prominence of new physics signals at the LHC. For example the idea of the optimal value of jet radius $R$ to be employed in various searches for new physics at the LHC based on analytical
estimates of both perturbative radiation and non-perturbative hadronisation
corrections was suggested in Ref.~\cite{Dassalmag}.
Moreover ideas about jet substructure \cite{Sey,ButtSal} have contributed to an explosion in the production of tools which can be used to distinguish QCD
jets from those produced by the decays of massive particles in the highly
boosted regime where the decay products may be captured within a single jet.
For a detailed exposition of substructure techniques we refer the reader to Ref.~\cite{boost} and references therein.
To conclude we finish with a reminder that much of the progress in developing
QCD precision tools and the consequent improvement in understanding QCD effects whether in the context of hadron colliders or $e^{+}e^{-}$ machines should
ultimately yield benefits beyond the particular context within which it was initiated. For instance the need for developing theoretical methods to further the precision that can be obtained via perturbative techniques at the LHC should
in many cases ultimately have spin-offs that would pay dividend in the attainment of even higher precision at future linear colliders. There is thus much
reason to be optimistic in light of the fact that the pace of developments of
QCD tools continues to be rapid (and possibly even accelerated) stimulated in
large part by the advent of LHC data.
|
1,314,259,996,689 | arxiv |
\section{Introduction}
The search for \ensuremath{C\!P}\xspace violation (\ensuremath{C\!PV}\xspace) in charm decays provides
a sensitive probe of physics beyond the Standard Model (SM).
Owing to its suppression within the SM,
a significant observation of direct \ensuremath{C\!PV}\xspace in charm decays
would indicate the possible presence of new physics (NP) effects in the decay processes.
In a previous article~\cite{delAmoSanchez:2011zza},
we reported a precise measurement of the \ensuremath{C\!P}\xspace asymmetry in the \ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode,
where the measured asymmetry was found to be consistent with the
value expected from indirect \ensuremath{C\!PV}\xspace in the \ensuremath{K^0}\xspace system.
The LHCb and CDF Collaborations have recently reported evidence for \ensuremath{C\!PV}\xspace in
charm decays by measuring the difference of \ensuremath{C\!P}\xspace asymmetries in the
$\ensuremath{D^0}\xspace \to \ensuremath{K^+}\xspace \ensuremath{K^-}\xspace$ and $\ensuremath{D^0}\xspace \to \ensuremath{\pi^+}\xspace \ensuremath{\pi^-}\xspace$ channels~\cite{Aaij:2011in,Collaboration:2012qw},
which is mainly sensitive to direct \ensuremath{C\!PV}\xspace.
The size of the world average direct \ensuremath{C\!P}\xspace asymmetry difference,
$(-6.56\pm 1.54)\times 10^{-3}$~\cite{Amhis:2012bh}, suggests
either a significant enhancement of SM penguin amplitudes
or of NP amplitudes (or both) in charm decays~\cite{Isidori:2011qw_etal}.
Improved measurements of the \ensuremath{C\!P}\xspace asymmetries in the individual two-body modes,
along with measurements in other channels, are needed to
determine the nature of the contributing amplitudes.
We present herein measurements of the decay rate \ensuremath{C\!P}\xspace asymmetry, $A_{\ensuremath{C\!P}\xspace}$,
defined as
\begin{equation}
A_{\ensuremath{C\!P}\xspace}=\frac{\Gamma(D^+_{(s)}\to f)-\Gamma(D_{(s)}^-\to \overline{f})}
{\Gamma(D^+_{(s)}\to f)+\Gamma(D^-_{(s)}\to \overline{f})},
\end{equation}
in the decay modes \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, and \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace.
Previous measurements of $A_{\ensuremath{C\!P}\xspace}$ in these channels have been reported by
the CLEO-c~\cite{:2007zt} and Belle Collaborations~\cite{Ko:2010ng}.
As for the $A_{\ensuremath{C\!P}\xspace}$ measurement in \ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace,
we expect an $A_{\ensuremath{C\!P}\xspace}$ asymmetry of $(\pm0.332\pm 0.006)\%$~\cite{Nakamura:2010zzi}
resulting from \ensuremath{C\!PV}\xspace in $\ensuremath{K^0}\xspace-\ensuremath{\Kbar^0}\xspace$ mixing~\cite{Lipkin:1999qz}.
The sign of the $\ensuremath{K^0}\xspace$-induced asymmetry is positive (negative) if a \ensuremath{K^0}\xspace (\ensuremath{\Kbar^0}\xspace)
is present in the corresponding tree-level Feynman diagram.
Because it is identified by its $\pi^+\pi^-$ decay, the intermediate state
is a coherent mix of \KS and \KL amplitudes. It has been shown in Ref.~\cite{Azimov:1981ss} that the \ensuremath{\KS \kern -0.2em - \kern -0.2em \KL}\xspace
interference term gives rise to a measured \ensuremath{C\!P}\xspace asymmetry that depends on the range in proper
time over which the decay rates are integrated, and on the efficiency for the reconstruction of the
intermediate state as a function of its proper flight time.
For this analysis we employ a technique similar to that used
for our measurement of \ensuremath{C\!PV}\xspace in the \ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode~\cite{delAmoSanchez:2011zza}.
As a result, reference to our previous publication
is given for the description of some of the analysis details.
\section{The \babar detector and event selection}
The data used for these measurements were recorded at or near the
$\Y4S$ resonance by the \babar detector at the PEP-II\xspace storage rings,
and correspond to an integrated luminosity of $469\,\ensuremath{\mbox{\,fb}^{-1}}\xspace$.
Charged particles are detected, and their momenta measured, by a combination of a silicon
vertex tracker, consisting of 5 layers of double-sided
detectors, and a 40-layer central drift chamber, both operating
in a 1.5 T axial magnetic field.
Charged-particle identification is
provided by specific ionization energy loss measurements in the tracking system, and by the
measured Cherenkov angle from an internally reflecting ring-imaging
Cherenkov detector covering the central region of the detector.
Electrons are detected by a CsI(Tl) electromagnetic calorimeter.
The \babar detector, and the coordinate system used throughout,
are described in detail in Refs.~\cite{Aubert:2001tu,Menges:2006xk}.
We validate the analysis procedure using
Monte Carlo (MC) simulation based on Geant4~\cite{Agostinelli:2002hh}.
The MC samples include $\ensuremath{e^+e^-}\xspace\to\ensuremath{q\overline q}\xspace\;(q=u,d,s,c)$ events, simulated with
JETSET~\cite{Sjostrand:2006za} and \BB decays simulated with the EvtGen
generator~\cite{Lange:2001uf}.
To avoid potential bias in the measurements
we finalize the event selection for each channel,
as well as the procedures for efficiency correction, fitting,
and the determination of the systematic uncertainties and possible biases in the
measurements, prior to extracting the value of $A_{\ensuremath{C\!P}\xspace}$ from the data.
Signal candidates are reconstructed by combining a $\KS$
candidate, reconstructed in the decay mode $\KS\to\pi^+\pi^-$,
with a charged pion or kaon candidate.
A \KS candidate is reconstructed from two oppositely charged
tracks with an invariant mass within
a $\pm 10\,\ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c^2}}\xspace$ interval centered on the nominal \KS mass~\cite{Nakamura:2010zzi},
which is approximately $\pm 2.5\,\sigma$ in the measured
\KS mass resolution. The $\chi^2$-probability of
the $\pi^+\pi^-$ vertex fit must be greater than $0.1\,\%$.
Motivated by MC studies, we require the measured flight length
of the \KS candidate to be at least three times greater than its uncertainty,
to reduce combinatorial background.
A reconstructed charged-particle track that has $p_T\ge 400\,\ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c}}\xspace$
is selected as a pion or kaon candidate, where $p_T$ is the magnitude
of the momentum in the plane perpendicular to the $z$ axis (transverse plane).
In our measurement, we require that a pion candidate not be identified
as a kaon, a proton, or an electron, and that a kaon candidate
be identified as a kaon, and not as a pion, a proton, or an electron.
Identification efficiencies and misidentification rates for
electron, pions, kaons, and protons with 2 \ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace momentum in the laboratory frame
are reported in Table \ref{tab:pideff}.
\begin{table}[tb]
\caption{Identification efficiencies and misidentification rates for
electron, pions, kaons, and protons with 2 \ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace momentum in the laboratory frame.
The values for kaons on the third row refers to the identification criterion
used to reject kaons from the pion sample,
while the values on the fourth row to the criterion used in the kaon selection.}
\begin{center}
\begin{tabular}{|l|c|c|c|}\hline
\multirow{2}{*}{Particle} & \multirow{2}{*}{Efficiency[\%]} &
\multicolumn{2}{|c|}{Misid. rate [\%]}\\
\cline{3-4}
& & \ensuremath{\pi^\pm}\xspace & \ensuremath{K^\pm}\xspace \\
\hline
\ensuremath{e^\pm}\xspace & 91 & 0.04 & $<0.2$ \\
\ensuremath{\pi^\pm}\xspace & 88 & $-$ &1 \\
\ensuremath{K^\pm}\xspace (applied to \ensuremath{\pi^\pm}\xspace) & 91 & 1 & $-$ \\
\ensuremath{K^\pm}\xspace (applied to \ensuremath{K^\pm}\xspace) & 99 & 8 & $-$ \\
\ensuremath{p^\pm}\xspace & 80 & 0.2 & 0.2 \\
\hline
\end{tabular}
\label{tab:pideff}
\end{center}
\end{table}
The criteria used to select pion or kaon candidates are very effective in reducing
the charge asymmetry from track reconstruction and identification, as inferred from
studying the data control samples described below.
A vertex fit to the whole decay chain, constraining the $\ensuremath{D_{(s)}}\xspace^\pm$ production vertex to
be within the \ensuremath{e^+e^-}\xspace interaction region, is then performed~\cite{Hulsbergen:2005pu}.
We retain only $\ensuremath{D_{(s)}}\xspace^\pm$ candidates having a $\chi^2$-probability for this fit
greater than 0.1\%, and an invariant mass $m(\KS h), h = \pi, K, $ within a $\pm65\ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c^2}}\xspace$
interval centered on the nominal $\ensuremath{D_{(s)}}\xspace^\pm$ mass~\cite{Nakamura:2010zzi},
which is approximately equivalent to $\pm 8\,\sigma$ in the measured
$\ensuremath{D_{(s)}}\xspace^\pm$ mass resolution.
We require further that the magnitude of the $D^\pm_{s}$ candidate momentum in the
$\ensuremath{e^+e^-}\xspace$ center-of-mass (CM) system, $p^*$,
be between 2.6 and 5.0 \ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace, in order to suppress combinatorial background from \BB events.
For the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace mode, the MC simulated sample shows that
retaining candidates with $p^*$ between 2.0 and 5.0 \ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace
allows signal candidates from $B$-meson decays,
without introducing an excessive amount of combinatorial background.
Assuming that \CPT is conserved, there is no contribution to $A_{\ensuremath{C\!P}\xspace}$ from \ensuremath{C\!P}\xspace violation
in $B$ meson decays from Standard Model processes.
Additional background rejection is obtained by requiring that the impact parameter
of the $\ensuremath{D_{(s)}}\xspace^\pm$ candidate with respect to the beam-spot~\cite{Aubert:2001tu},
projected onto the transverse plane,
be less than 0.3 cm, and that the $\ensuremath{D_{(s)}}\xspace^\pm$ proper decay time, $t_{xy}$,
be between $-15$ and $35$ ps.
The decay time is measured using $L_{xy}$, defined as the distance of
the $\ensuremath{D_{(s)}}\xspace^\pm$ decay vertex from the beam-spot
projected onto the transverse plane.
In order to further optimize the sensitivity of the $A_{\ensuremath{C\!P}\xspace}$ measurements,
we construct a multivariate algorithm, based on seven
discriminating variables for each $\ensuremath{D_{(s)}}\xspace^\pm$ candidate:
$t_{xy}$, $L_{xy}$,
$p^*$, the momentum magnitude and component in the transverse plane
for the \KS candidate, and also for the pion or kaon candidate.
For the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace and \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace modes the multivariate algorithm with the best performance
is a Boosted Decision Tree~\cite{Speckmayer:2010zz},
while for the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode the best algorithm is a Projective Likelihood method~\cite{Speckmayer:2010zz}.
The final selection criteria, based on the outputs of the multivariate selectors,
are optimized using truth-matched signal and
background candidates from the MC sample.
For the optimization, we maximize the $S/\sqrt{S+B}$ ratio, where
$S$ and $B$ are the numbers of signal and background candidates
with invariant mass within $\pm 30 \ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c^2}}\xspace$ of the nominal $\ensuremath{D_{(s)}}\xspace^\pm$ mass,
which is approximately $\pm 3\,\sigma$ in the measured mass resolution.
\section{Signal yield and asymmetry extraction}
For each mode the signal yield is extracted using
a binned maximum likelihood (ML) fit to the distribution
of the invariant mass $m(\KS h)$ for the
selected $\ensuremath{D_{(s)}}\xspace^\pm$ candidates.
The total probability density function (PDF)
is the sum of signal and background components.
The signal PDF is modeled as a sum of two Gaussian functions
for the \ensuremath{\Dps^\pm \to \KS \ensuremath{K^\pm}\xspace}\xspace modes, and as a single Gaussian function for the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode.
The background PDF is taken as the sum of two components: a distribution describing
the invariant mass of mis-reconstructed charm meson decays,
and a combinatorial background modeling the mass distribution from other sources.
For the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace (\ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace) mode the charm background is mainly from the tail of the invariant mass
distribution for \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace (\ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace) candidates.
For the \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace mode, the mis-reconstructed charm background originates mainly
from \ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace decays for which the \ensuremath{\pi^\pm}\xspace is misidentified as a \ensuremath{K^\pm}\xspace.
Assigning the wrong mass to the pion shifts the reconstructed invariant mass, and the resulting
distribution is a broad peak with mean value close to the $D_{s}^\pm$ mass.
For each mode, the invariant mass distribution due to charm background is modeled
using a histogram PDF obtained from a MC sample of simulated charm background decays.
The combinatorial background is described by a first(second)-order polynomial for the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode
(\ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace and \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace modes).
The fits to the $m(\KS h)$ distributions yield
$(159.4 \pm 0.8) \times 10^3$ \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace decays,
$(288.2 \pm 1.1) \times 10^3$ \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace decays, and
$(14.33 \pm 0.31) \times 10^3$ \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace decays.
The data and the fit results are shown in Fig.~\ref{fig1}.
All of the PDF parameters are extracted from fits to the data.
\begin{figure*}[tb]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.33\textwidth]{data_DToKsK} &
\includegraphics[width=0.33\textwidth]{data_DsToKsK} &
\includegraphics[width=0.33\textwidth]{data_DsToKspi} \\
\end{tabular}
\vspace{-0.3cm}
\caption{
Invariant mass distribution for (a) \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, (b) \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, and (c) \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace
candidates (points with error bars).
The solid curve shows the result of the fit to the data.
The dashed curve represents the sum of all background contributions,
while the dotted curve indicates combinatorial background only.}
\label{fig1}
\vspace{-0.7cm}
\end{center}
\end{figure*}
For each channel, we determine $A_{\ensuremath{C\!P}\xspace}$ by measuring the signal yield asymmetry $A$ defined as:
\begin{equation}
A=\frac{N_{\ensuremath{D_{(s)}}\xspace^+}-N_{\ensuremath{D_{(s)}}\xspace^-}}{N_{\ensuremath{D_{(s)}}\xspace^+}+N_{\ensuremath{D_{(s)}}\xspace^-}},
\end{equation}
where $N_{\ensuremath{D_{(s)}}\xspace^+}$($N_{\ensuremath{D_{(s)}}\xspace^-}$) is the number of $\ensuremath{D_{(s)}}\xspace^+$($\ensuremath{D_{(s)}}\xspace^-$) decays determined from the fit
to the invariant mass distribution.
The asymmetry $A$ contains two contributions in addition to $A_{\ensuremath{C\!P}\xspace}$, namely
the forward-backward (FB) asymmetry ($A_{{\scriptscriptstyle \rm FB}\xspace}$), and a detector-induced component.
We measure $A_{{\scriptscriptstyle \rm FB}\xspace}$ together with $A_{\ensuremath{C\!P}\xspace}$ using the selected dataset,
while we correct the data for the detector-induced component
using coefficients derived from a control sample.
\section{Correction of detector-related asymmetries}
We use a data-driven method, described in detail in Ref.~\cite{delAmoSanchez:2011zza},
to determine the charge asymmetry in track reconstruction as a function of the magnitude of
the track momentum and its polar angle in the laboratory frame.
The method exploits the fact that $\Y4S \to \BB$
events provide a sample evenly populated with positive and negative tracks,
free of any physics-induced asymmetries.
The off-resonance momentum distribution is subtracted from the on-resonance one,
to remove any contribution from continuum, for which there is a FB asymmetry in the CM frame.
This sample is used to compute the detector-related asymmetries in the reconstruction of charged-particle tracks.
Starting from a sample of $50.6\,\ensuremath{\mbox{\,fb}^{-1}}\xspace$ of data collected at the \Y4S resonance and
an off-resonance data sample of $44.8\,\ensuremath{\mbox{\,fb}^{-1}}\xspace$, we obtain a large sample of charged-particle tracks,
and apply the same charged pion or kaon track selection criteria
used in the reconstruction of the \ensuremath{\Dps^\pm \to \KS \ensuremath{K^\pm}\xspace}\xspace and \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace modes.
Then, after subtracting the off-resonance contribution from the on-resonance sample,
we obtain a sample of more than 120 million pion candidates, and 40 million kaon candidates,
originating from \Y4S decays.
We use the full off-resonance sample and an equivalent luminosity for
the on-resonance sample, because, due to the subtraction procedure, including additional
data in the on-resonance sample does not improve the statistical error
on the correction ratios mentioned below.
These candidates are then used to compute the efficiency ratios
for positive and negative pions and kaons.
The ratio values and their statistical errors for pions and kaons are shown
in Fig.~\ref{fig4} and Fig.~\ref{fig5}, respectively.
For the $\ensuremath{D_{(s)}}\xspace^{-} \to \KS \ensuremath{K^-}\xspace$ ($D_s^{-} \to \KS \ensuremath{\pi^-}\xspace$) modes,
the $\ensuremath{D_{(s)}}\xspace^{-}$ ($D_s^{-}$) yields, in intervals of kaon (pion) momentum and
cosine of its polar angle, $\cos\theta$, are weighted with the kaon (pion) efficiency ratios to
correct for the detection efficiency differences between \ensuremath{K^+}\xspace and \ensuremath{K^-}\xspace (\ensuremath{\pi^+}\xspace and \ensuremath{\pi^-}\xspace).
Momentum and cosine of its polar angle intervals are not uniform in order to have
similar statistics, and therefore similar correction uncertainty, in each interval.
Interval sizes vary from (0.05 \ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace, 0.06) to (4.4 \ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace, 0.96),
where the first number is the momentum interval, and the second its cosine of polar angle interval.
The largest correction is approximately 1\% for pions and 2\% for kaons.
After correcting the data for the detector-induced component only $A_{{\scriptscriptstyle \rm FB}\xspace}$ and $A_{\ensuremath{C\!P}\xspace}$
contribute to the measured asymmetry $A$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.47\textwidth]{plot_Ratio_pion}
\vspace{-0.3cm}
\caption{(top) The ratio between the detection efficiency
for $\pi^+$ and $\pi^-$, and (bottom) the
corresponding statistical errors.
The values are computed using the numbers of $\pi^+$ and $\pi^-$
tracks in the selected control sample.}
\label{fig4}
\vspace{-0.7cm}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.47\textwidth]{plot_Ratio_kaon}
\vspace{-0.3cm}
\caption{(top) The ratio between the detection efficiency
for $K^+$ and $K^-$, and (bottom) the
corresponding statistical errors.
The values are computed using the numbers of $K^+$ and $K^-$
tracks in the selected control sample.}
\label{fig5}
\vspace{-0.7cm}
\end{center}
\end{figure}
\section{Extraction of $A_{\ensuremath{C\!P}\xspace}$ and $A_{{\scriptscriptstyle \rm FB}\xspace}$}
Neglecting higher-order terms that contain $A_{\ensuremath{C\!P}\xspace}$ and $A_{{\scriptscriptstyle \rm FB}\xspace}$,
the resulting asymmetry can be expressed simply as the sum of the two.
Given that $A_{{\scriptscriptstyle \rm FB}\xspace}$ is an odd function of $\cos\theta^*_D$,
where $\theta^*_D$ is the polar angle of the $\ensuremath{D_{(s)}}\xspace^\pm$
candidate momentum in the CM frame,
$A_{\ensuremath{C\!P}\xspace}$ and $A_{{\scriptscriptstyle \rm FB}\xspace}$ can be written as a function of $|\cos\theta^*_D|$ as follows:
\begin{align}
A_{\ensuremath{C\!P}\xspace}(|\cos\theta^*_D|) &= \frac{A(+|\cos\theta^*_D|) + A(-|\cos\theta^*_D|)}{2}\\
\intertext{and}
A_{{\scriptscriptstyle \rm FB}\xspace}(|\cos\theta^*_D|) &= \frac{A(+|\cos\theta^*_D|) - A(-|\cos\theta^*_D|)}{2},
\label{eq:AcpAfb_intro}
\end{align}
where $A(+|\cos\theta^*_D|)$ ($A(-|\cos\theta^*_D|)$) is the measured asymmetry
for the $\ensuremath{D_{(s)}}\xspace^\pm$ candidates in a positive (negative) $\cos\theta^*_D$ interval.
A simultaneous ML fit to the $\ensuremath{D_{(s)}}\xspace^+$ and $\ensuremath{D_{(s)}}\xspace^-$ invariant mass distributions
is carried out to extract the signal yield asymmetry in each of
ten equally spaced $\cos\theta^*_D$ intervals, starting with interval 1 at $[-1.0,-0.8]$.
The PDF model that describes the distribution in each sub-sample
is the same as that used in the fit to the full sample,
but the following parameters are allowed to float separately
in each sub-sample (referred to as split parameters):
the yields for signal, charm background and combinatorial candidates;
the asymmetries for signal and combinatorial candidates;
the width, and the fraction of the Gaussian function with the larger contribution to the signal PDF;
and the first-order coefficient of the polynomial that models the combinatorial background.
For the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace mode the yields for the charm background candidates in intervals 1, 2, and 3
were fixed to 0 to obtain a fully convergent fit.
Since interval 10 contains the smallest number of candidates,
we use a single Gaussian function to model the signal PDF for the \ensuremath{\Dps^\pm \to \KS \ensuremath{K^\pm}\xspace}\xspace modes.
For the \ensuremath{C\!P}\xspace asymmetry of charm background candidates
we use the same floating parameters as for the signal
candidates, because the largest source of \ensuremath{C\!P}\xspace asymmetry for both samples is due to \ensuremath{C\!PV}\xspace in
\ensuremath{\Kz \kern -0.2em - \kern -0.2em \Kzb}\xspace mixing. For the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode, where the primary charm background channel, \ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace,
has the same magnitude but opposite-sign asymmetry due to \ensuremath{\Kz \kern -0.2em - \kern -0.2em \Kzb}\xspace mixing, we use a separate
parameter for the asymmetry of the charm background candidates.
To achieve a more stable fit,
if the fit results for a split parameter are statistically compatible
between two or more sub-samples,
the parameter is forced to have the same floating value among those sub-samples only.
For the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode the width of the first Gaussian function for the signal PDF is set to the same
floating value in intervals 1, 2, 3, and 4.
The first-order coefficient of the polynomial describing the combinatorial background
is set to the same floating value in intervals 4 to 8 (\ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace), in intervals 4 to 8 (\ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace), and
in intervals 2 to 7 (\ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace).
The final fit contains 70, 80, and 64 free parameters
for the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, and \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace modes, respectively.
The $A_{\ensuremath{C\!P}\xspace}$ and $A_{{\scriptscriptstyle \rm FB}\xspace}$ values for the five $|\cos\theta^*_D|$ bins are shown
in Fig.~\ref{fig6} for the three decay modes.
The weighted average of the five $A_{\ensuremath{C\!P}\xspace}$ values is
$(0.16 \pm 0.36)\%$ for the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace mode,
$(0.00 \pm 0.23)\%$ for \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace,
and $(0.6 \pm 2.0)\%$ for \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace,
where the errors are statistical only.
\begin{figure*}[tb] \begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.33\textwidth,clip=true]{data_DToKsK_acp} &
\includegraphics[width=0.33\textwidth,clip=true]{data_DsToKsK_acp} &
\includegraphics[width=0.33\textwidth,clip=true]{data_DsToKspi_acp} \\
\includegraphics[width=0.33\textwidth,clip=true]{data_DToKsK_afb} &
\includegraphics[width=0.33\textwidth,clip=true]{data_DsToKsK_afb} &
\includegraphics[width=0.33\textwidth,clip=true]{data_DsToKspi_afb} \\
\end{tabular}
\vspace{-0.3cm} \caption{
\ensuremath{C\!P}\xspace asymmetry, $A_{\ensuremath{C\!P}\xspace}$, for (a) \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, (b) \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, and (c) \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace
as a function of $|\cos\theta^*_D|$ in the data sample.
The solid line represents the central value of $A_{\ensuremath{C\!P}\xspace}$
and the gray band is the $\pm1\,\sigma$ interval, both
obtained from a $\chi^2$-minimization
assuming no dependence on $|\cos\theta^*_D|$.
The corresponding forward-backward asymmetries, $A_{{\scriptscriptstyle \rm FB}\xspace}$, are shown in (d), (e), and (f).
}
\label{fig6} \vspace{-0.7cm} \end{center} \end{figure*}
We perform two tests to validate the analysis procedure for each channel.
The first involves generating 5000 toy MC experiments with a statistics equal to
data using the PDF and the parameters obtained from the fit to data.
After extracting $A_{\ensuremath{C\!P}\xspace}$ from each experiment,
for the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace and \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace modes,
we deduce from the mean of the $A_{\ensuremath{C\!P}\xspace}$ pull distributions the presence of a small bias
in the fitted value of each fit parameter
(the means are $-0.036 \pm 0.014$ and $+0.041 \pm 0.014$, respectively).
To account for this effect we apply a correction to the final values equal to
$+0.013\%$ for the \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace mode, and $-0.01\%$ for the \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace mode.
The $A_{\ensuremath{C\!P}\xspace}$ pull distributions show that the fit provides an accurate estimate
of the statistical error for all the modes.
The second test involves fitting a large number of MC events from the
full \babar detector simulation.
We measure $A_{\ensuremath{C\!P}\xspace}$ from this MC sample to be consistent with the generated value of zero.
\section{Systematics}
The main sources of systematic uncertainty are listed in Table~\ref{tab_syst}
for each decay mode, together with the overall uncertainties.
\begin{table*}[tb]
\caption{Summary of the systematic uncertainty contributions
for the $A_{\ensuremath{C\!P}\xspace}$ measurement in each mode.
The values are absolute uncertainties, even though given as percentages.
The total value corresponds to the sum in quadrature of the individual contributions.}
\begin{center}
\begin{tabular}{|l|c|c|c|}\hline
Systematic uncertainty & \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace & \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace & \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace \\
\hline
Efficiency of PID selectors & 0.05\% & 0.05\% & 0.05\% \\
Statistics of the control sample & 0.23\% & 0.23\% & 0.06\% \\
Misidentified tracks in the control sample & 0.01\% & 0.01\% & 0.01\% \\
$\cos\theta^*_D$ interval size & 0.04\% & 0.02\% & 0.27\% \\
\ensuremath{\Kz \kern -0.2em - \kern -0.2em \Kzb}\xspace regeneration & 0.05\% & 0.05\% & 0.06\% \\
\ensuremath{\KS \kern -0.2em - \kern -0.2em \KL}\xspace interference & 0.015\% & 0.014\% & 0.008\% \\
\hline
Total & 0.25\% & 0.24\% & 0.29\% \\
\hline
\end{tabular}
\label{tab_syst}
\end{center}
\end{table*}
The primary sources of systematic uncertainty are
the detection efficiency ratios used to weight the $\ensuremath{D_{(s)}}\xspace^-$ yields, and
the contributions from mis-identified particles in the data control sample used
to determine the charge asymmetry in track reconstruction efficiency.
The technique used to remove the charge asymmetry due to detector-induced effects produces a
small systematic uncertainty in the measurement of $A_{\ensuremath{C\!P}\xspace}$ due to the statistical error in the
relative efficiency estimation. This systematic uncertainty depends only on the type of
charged particle (pion or kaon) in the final state, and not on the initial state. To
estimate the systematic uncertainty on $A_{\ensuremath{C\!P}\xspace}$ resulting from this source, the relative charged-particle
efficiency in each interval of momentum and $\cos\theta$ is randomly drawn from a Gaussian
distribution whose mean is the nominal relative efficiency in that interval,
and where the root-mean-squared (r.m.s.) deviation is the
corresponding statistical error. For each mode, we generate 500 such charged-particle
relative-efficiency distributions, and use them to obtain 500 $A_{\ensuremath{C\!P}\xspace}$ values, following the
procedure described earlier to determine the nominal value of $A_{\ensuremath{C\!P}\xspace}$.
The r.m.s.~deviation of these 500 values from the nominal $A_{\ensuremath{C\!P}\xspace}$ is taken to be the systematic uncertainty. For
the \ensuremath{\Dps^\pm \to \KS \ensuremath{K^\pm}\xspace}\xspace modes, the estimated systematic uncertainty is 0.23\%. For the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace
mode, we assign the same systematic uncertainty, 0.06\%, as that estimated for the
\ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode in Ref.~\cite{delAmoSanchez:2011zza}.
The small fraction of misidentified particles in the generic track sample
can introduce small biases in the estimation of the efficiencies,
and subsequently in the $A_{\ensuremath{C\!P}\xspace}$ measurements.
Because of the good agreement between data and MC samples, we can use the simulated MC candidates
to measure the shift in the $A_{\ensuremath{C\!P}\xspace}$ value from the fit when the corrections are applied,
and when they are not.
Again, this contribution depends only on the type of the charged-particle track.
Hence, for the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode, we assume the same shift obtained in Ref.~\cite{delAmoSanchez:2011zza}, namely +0.05\%.
By fitting the \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace MC sample when the corrections are applied, and again when not,
we obtain a shift of +0.05\% and we assume this for both the \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace and \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace modes.
For all the modes, we shift the measured $A_{\ensuremath{C\!P}\xspace}$ by this correction value
and then, conservatively, include the magnitude of this shift as a contribution to the systematic uncertainty.
Using MC simulation, we evaluate an additional systematic uncertainty
of $\pm0.01\%$ due to a possible charge asymmetry present in the control sample
before applying the selection criteria.
Another source of systematic uncertainty is due to the choice of
the $\cos\theta^*_D$ interval-size in the simultaneous ML fit.
The systematic uncertainty is taken to be the largest absolute difference
between the nominal $A_{\ensuremath{C\!P}\xspace}$ extracted using ten $\cos\theta^*_D$ intervals
and that obtained when the fit is performed using either 8 or 12 intervals in $\cos\theta^*_D$.
This is the dominant source of systematic uncertainty for the \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode,
as shown in Table~\ref{tab_syst}.
We also consider a possible systematic uncertainty due to the regeneration
of neutral kaons in the material of the detector.
The \ensuremath{K^0}\xspace and \ensuremath{\Kbar^0}\xspace mesons produced in the decay processes can interact with the
material in the tracking volume before they decay.
Following a method similar to that described in Ref.~\cite{Ko:2010mk},
we compute the probability for a \ensuremath{K^0}\xspace or a \ensuremath{\Kbar^0}\xspace meson to interact
inside the \babar tracking system, and
estimate systematic uncertainties of $0.05\%$ (\ensuremath{\Dps^\pm \to \KS \ensuremath{K^\pm}\xspace}\xspace) and $0.06\%$ (\ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace).
\begin{table*}[tbp]
\caption{Summary of the $A_{\ensuremath{C\!P}\xspace}$ measurements.
Where reported, the first uncertainty is statistical, and the second is systematic.}
\begin{center}
\begin{tabular}{|l|c|c|c|}\hline
& \ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace & \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace & \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace \\
\hline
$A_{\ensuremath{C\!P}\xspace}$ value from the fit & $(+0.155 \pm 0.360)\%$ & $(0.00 \pm 0.23)\%$ & $(+0.6 \pm 2.0)\%$ \\
\hline
Correction for the bias from toy MC experiments & $+0.013\%$ & $-0.01\%$ & $-$ \\
Correction for the bias in the PID selectors & $-0.05\%$ & $-0.05\%$ & $-0.05\%$ \\
Correction for the \ensuremath{\KS \kern -0.2em - \kern -0.2em \KL}\xspace interference ($\Delta A_{\ensuremath{C\!P}\xspace}$) & $+0.015\%$ & $+0.014\%$ & $-0.008\%$ \\
\hline
$A_{\ensuremath{C\!P}\xspace}$ final value & $(+0.13 \pm 0.36 \pm 0.25)\%$ & $(-0.05 \pm 0.23 \pm 0.24)\%$ & $(+0.6 \pm 2.0 \pm 0.3)\%$\\
\hline
$A_{\ensuremath{C\!P}\xspace}$ contribution from \ensuremath{\Kz \kern -0.2em - \kern -0.2em \Kzb}\xspace mixing
& $(-0.332 \pm 0.006)\%$ & $(-0.332 \pm 0.006)\%$ & $(+0.332 \pm 0.006)\%$ \\
\hline
$A_{\ensuremath{C\!P}\xspace}$ final value (charm only) & $(+0.46 \pm 0.36 \pm 0.25)\%$ & $(+0.28 \pm 0.23 \pm 0.24)\%$ & $(+0.3 \pm 2.0 \pm 0.3)\%$\\
\hline
\end{tabular}
\label{tab_final}
\end{center}
\end{table*}
Although the intermediate state is labelled as a \KS, we apply a correction term to the
measured $A_{\ensuremath{C\!P}\xspace}$ to include the effect of \ensuremath{\KS \kern -0.2em - \kern -0.2em \KL}\xspace interference in the intermediate state~\cite{Grossman:2011zk}.
This correction term depends on the proper time range over which decay distributions are integrated, and
on the efficiency of the reconstruction of the $\ensuremath{\pi^+}\xspace\ensuremath{\pi^-}\xspace$ final state as a function of proper time.
We compute the reconstruction efficiency distribution as a function of
proper time using MC truth-matched \KS decays after the full selection.
Following the method in Ref.~\cite{Grossman:2011zk} we estimate
the asymmetry-correction term $\Delta A_{\ensuremath{C\!P}\xspace}$ defined as:
\begin{equation}
\Delta A_{\ensuremath{C\!P}\xspace} = A_{\ensuremath{C\!P}\xspace}^{\textrm{corr}} - A_{\ensuremath{C\!P}\xspace}^{\textrm{fit}},
\end{equation}
where $A_{\ensuremath{C\!P}\xspace}^{\textrm{fit}}$ is the value obtained from the fit and
$A_{\ensuremath{C\!P}\xspace}^{\textrm{corr}}$ is the corrected value.
The correction terms are reported in Table~\ref{tab_final}
and, to be conservative, we include their absolute values
as contributions to the systematic uncertainty estimates.
We also estimate the correction factor for the \ensuremath{D^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace mode
using the \KS reconstruction efficiency distribution after the selection detailed
in Ref.~\cite{delAmoSanchez:2011zza}, and obtain the value +0.002\%.
All these corrections are rather small, even compared to those estimated in a
similar analysis~\cite{BABAR:2011aa}.
The smaller values of the corrections in the present analysis are due to the
improved efficiency for \KS mesons with short decay times that we obtain
by applying the requirement on the decay length divided by its uncertainty,
rather than on the decay length alone.
\section{Conclusion}
In conclusion, we measure the direct \ensuremath{C\!P}\xspace asymmetry, $A_{\ensuremath{C\!P}\xspace}$, in the
\ensuremath{D^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, \ensuremath{D_s^{\pm}\to\KS\ensuremath{K^\pm}\xspace}\xspace, and \ensuremath{D_s^{\pm}\to\KS\ensuremath{\pi^\pm}\xspace}\xspace modes using approximately
159\,000, 288\,000, and 14\,000
signal candidates, respectively.
The measured $A_{\ensuremath{C\!P}\xspace}$ value for each mode is reported in Table~\ref{tab_final},
where the first errors are statistical and the second are systematic.
In the last row of the table, we also report the $A_{\ensuremath{C\!P}\xspace}$ values after
subtracting the expected $A_{\ensuremath{C\!P}\xspace}$ contribution for each mode due to \ensuremath{\Kz \kern -0.2em - \kern -0.2em \Kzb}\xspace mixing.
The results are consistent with zero, and with the SM prediction,
within one standard deviation.
\section{Acknowledgements}
We are grateful for the
extraordinary contributions of our PEP-II\xspace colleagues in
achieving the excellent luminosity and machine conditions
that have made this work possible.
The success of this project also relies critically on the
expertise and dedication of the computing organizations that
support \babar.
The collaborating institutions wish to thank
SLAC for its support and the kind hospitality extended to them.
This work is supported by the
US Department of Energy
and National Science Foundation, the
Natural Sciences and Engineering Research Council (Canada),
the Commissariat \`a l'Energie Atomique and
Institut National de Physique Nucl\'eaire et de Physique des Particules
(France), the
Bundesministerium f\"ur Bildung und Forschung and
Deutsche Forschungsgemeinschaft
(Germany), the
Istituto Nazionale di Fisica Nucleare (Italy),
the Foundation for Fundamental Research on Matter (The Netherlands),
the Research Council of Norway, the
Ministry of Education and Science of the Russian Federation,
Ministerio de Ciencia e Innovaci\'on (Spain), and the
Science and Technology Facilities Council (United Kingdom).
Individuals have received support from
the Marie-Curie IEF program (European Union) and the A. P. Sloan Foundation (USA).
|
1,314,259,996,690 | arxiv | \section{Introduction}
Tile self-assembly is an algorithmically rich model of ``programmable crystal growth.''
Well-designed molecules (square-like ``tiles'') with specific binding sites can deterministically form a single target shape even subject to the chaotic nature of molecules floating in a well-mixed chemical soup.
Such tiles was experimentally implemented as DNA double-crossover molecules in 1998 \cite{WiLiWeSe1998}, and the last decade saw the drastic improvement of the reliability of DNA tile self-assembly; see, e.g., \cite{BaScRoWi2009}.
Shape-building is one primary goal of self-assembly; pattern-painting is another.
Based on the abstract Tile Assembly Model (aTAM) introduced by Winfree \cite{Winfree_PhDthesis}, Ma and Lombardi have first shed light on this problem and formalized it in the name of {\em patterned self-assembly tile set synthesis} ({\pats}) problem in \cite{MaLombardi2008,MaLombardi2009}.
In this framework, an optimization problem of interest aims at minimizing the number of tile types necessary for a rectilinear TAS (RTAS) to uniquely assemble a given rectangular pattern.
An exhaustive partition-search algorithm \cite{GoosOrponenDNA16} as well as a randomized search algorithm \cite{LempiainenCzeizlerOrponenDNA17} have been proposed for this problem.
{\pats} was recently proved to be \NP-hard\footnote{
This problem had been claimed \NP-hard even with $d=2$ \cite{MaLombardi2009}, but an error was found as being pointed out in \cite{CzeizlerPopa2012}.
}.
Nevertheless, it is not until being investigated under the restriction of the number of available colors that {\pats} gets practically meaningful, as summarized in DNA 2012 as: ``{\it any given logic circuit can be formulated as a colored rectangular pattern with tiles, using only a constant number of colors}.''
We will hence propose a variant of {\pats} parameterized by the size of palette $c$ and propose {\em $c$-{\pats}}.
The main contribution of this paper is the proof of the \NP-hardness of $c$-{\pats} for $c = \nc$.
Various techniques that have been invented for combinatorial optimization in shape assembly (see e.g.,~\cite{AdChGoHu2001,AdChGoHuKeEsRo2002,BrChDoKaSe2011,ChenDotySeki2011}) are useful but not sufficient in the proof because we now encounter a new challenge intrinsic to the optimization in pattern assembly.
That is the combinatorial explosion in the number of possible ways to color tile types.
This is observed even in assembling a two-colored pattern by TASs.
Two tile types are trivially necessary for that (there is no chameleon tile type).
If a TAS can use only 2 tile types, then there is no choice but to draw the types by distinct colors.
By contrast, once more tile types become available, TAS designers cannot do without considering how many tile types to be painted by a color\footnote{
It is not fair to say only due to this explosion that combinatorial optimization is more challenging for (multicolor) pattern assembly than shape building (monotone pattern assembly); colors often provide a visual clue to distinguish types of tiles.
}.
Since there is a pattern whose tile complexity is greater than the number of colors it contains, this combinatorial explosion must be addressed in some way.
In Section~\ref{subsec:basics}, we will propose subpatterns that are embedded into a bigger pattern $P$ and cooperatively force RTASs that uniquely assemble $P$ to paint at least certain number of their tile types with a specific color.
Being combined with an upperbound on the number of tile types available for the RTASs, which is given in $c$-{\pats}, the subpatterns prove their worth of providing the RTASs with the precise number of tile types to be drawn by each color.
\section{Preliminaries}
In this section, we mainly recall the abstract Tile Assembly Model (aTAM) proposed by Winfree \cite{Winfree_PhDthesis} and the problem {\sc Pattern self-Assembly Tile set Synthesis} (\pats) proposed by Ma and Lombardi \cite{MaLombardi2008}.
A variant of {\pats} will be the main focus of this paper.
\subsection{Abstract Tile Assembly Model (aTAM)}
\label{subsec:def_aTAM}
Let $\Sigma$ be an alphabet, and by $\Sigma^*$, we denote the set of finite strings over $\Sigma$.
By $\mathbb{Z}$ and $\mathbb{N}$, we denote the set of integers and the set of positive integers, respectively, and let $\mathbb{N}_0 = \mathbb{N} \cup \{0\}$.
In aTAM, $\mathbb{Z}^2$ is especially considered either as the two-dimensional integer lattice or as the set of all points on it.
Given a set of points $A \subseteq \mathbb{Z}^2$ on the integer lattice, the {\it full grid graph} of $A$ is the undirected graph $G^{\rm f}_A = (V, E)$, where $V = A$ and for all $u, v \in V$, there is an edge between $u$ and $v$ if and only if $||u-v||_2 = 1$, where $|| \cdot ||_2$ is the Manhattan distance, that is, $u$ and $v$ are adjacent points.
Let $\north, \west, \south, \east$ stand for the respective directions north, west, south, and east, and be also interpreted as the respective unit vectors $(0, 1), (-1, 0), (0, -1), (1, 0)$.
\begin{figure}[tb]
\begin{center}
\begin{minipage}{0.75\textwidth}
\includegraphics[scale=0.45]{half-adder.eps}
\end{minipage}
\begin{minipage}{0.2\textwidth}
{\footnotesize
\begin{tabular}{|cc|cc|}
\hline
A & B & S & C \\
\hline
0 & 0 & 0 & 0 \\
\hline
0 & 1 & 1 & 0 \\
\hline
1 & 0 & 1 & 0 \\
\hline
1 & 1 & 0 & 1 \\
\hline
\end{tabular}
}
\end{minipage}
\end{center}
\caption{
Four tile types (two blues, two oranges) implement together the half-adder, with two inputs A, B from the west and south, the output S to the north, and the carryout C to the east.
Just for reference, the truth table of half-adder is also presented.
}
\label{fig:half-adder}
\end{figure}
A {\it tile type} $t$ is a quadruple $t \in \Sigma^* \times \Sigma^* \times \Sigma^* \times \Sigma^*$, and is regarded as a unit square with four sides listed in the counter-clockwise order starting at the north (\north), each having a {\it glue label} (a.k.a., {\it glue}) taken from $\Sigma^*$; for instance, the second rightmost (orange) tile type in Figure~\ref{fig:half-adder} is represented as $(1, 1, 0, 0)$.
For each direction $d \in \{\north, \west, \south, \east\}$, let $t(d)$ be the glue label at the $d$ side of $t$.
Let $T$ be a {\it finite} set of tile types, and let us denote the (finite) set of all glues of tile types in $T$ by $\Lambda(T) \subseteq \Sigma^*$.
An {\it assembly} (a.k.a., {\it supertile}) is a positioning of tiles of types in $T$ on (part of) the integer lattice $\mathbb{Z}^2$.
It does not have to be a tessellation.
Hence, we can say that an assembly is a partial function $\mathbb{Z}^2 \dashrightarrow T$.
Given two assemblies $\alpha, \beta: \mathbb{Z}^2 \dashrightarrow T$, $\alpha$ is a {\it sub-assembly} of $\beta$, written as $\alpha \sqsubseteq \beta$, if $\dom(\alpha) \subseteq \dom(\beta)$ and for every point $p \in \dom(\alpha)$, $\alpha(p) = \beta(p)$, where $\dom$ denotes the domain of the function.
The aTAM models dynamics in the growth of assemblies based on the interaction among its basic building blocks, tiles.
A {\it strength function $g: \Lambda(T) \to \mathbb{N}_0$} endows tiles with an ability to interact with its neighboring tiles by assigning the strength $g(\ell)$ to the {\it matching} label $\ell$ of their abutting edges.
If the labels do not match or $g(\ell) = 0$, these tiles do not interact; otherwise, they do.
Tile interactions according to $g$ have an assembly $\alpha$ induce a {\it binding graph}, which is a grid graph whose vertices are $\dom(\alpha)$ and for two neighboring positions $p_1, p_2 \in \dom(\alpha)$, there is an edge between $p_1$ and $p_2$ on this graph if and only if the tiles $\alpha(p_1)$ and $\alpha(p_2)$ interact.
On this graph, an edge between vertices means that the corresponding tiles interact, and hence, their abutting edges share the same label $\ell$.
Thus, we can consider that the edge is labeled with $\ell$ and $g$ gives it the weight $g(\ell)$.
The assembly is {\it $\tau$-stable (with respect to $g$)} if every cut of its binding graph has strength at least $\tau$.
That is, the assembly is $\tau$-stable if at least energy $\tau$ is required to separate it into two parts.
A {\it (seeded) tile assembly system} (TAS) is a quadruple $\tas = (T, \sigma, g, \tau)$, where $T$ and $g$ are as stated above, $\tau \ge 1$ is an integer parameter called {\it temperature}, and $\sigma$ is a finite $\tau$-stable {\it seed assembly} consisting of tile types that are NOT included in $T$.
$\tas$ is provided with inexhaustible supply of copies of each tile type in $T$, each copy being referred to as a {\it tile}.
Given two $\tau$-stable assemblies $\alpha, \beta$, we write $\alpha \to_1^{\tas} \beta$ if $\alpha \sqsubseteq \beta$, $\dom(\beta) \setminus \dom(\alpha) = \{p\}$ for some position $p \in \mathbb{Z}^2$, and $\beta(p) \in T$.
Intuitively, this means that $\alpha$ can grow into $\beta$ by the addition of a single tile in $T$ at the position $p$.
Since $\beta$ is required to be $\tau$-stable, the new tile binds to $\alpha$ with strength at least $\tau$.
In this case, we say that {\it $\alpha$ $\tas$-produces $\beta$ in one step}.
A sequence of $\tau$-stable assemblies $\alpha_0, \alpha_1, \ldots, \alpha_k$ is a {\it $\tas$-assembly sequence} if for all $1 \le i \le k$, $\alpha_{i-1} \to_1^{\tas} \alpha_i$ holds.
We write $\alpha \to^\tas \beta$ and say {\it $\alpha$ $\tas$-produces $\beta$} (in 0 or more steps) if there is a $\tas$-assembly sequence $\alpha_0, \alpha_1, \ldots, \alpha_k$ of length $k = |\dom(\beta) \setminus \dom(\alpha)|$ with $\alpha_0 = \alpha$ and $\alpha_k = \beta$\footnote{
This definition of producibility is justified by our limited focus only onto the finite assemblies in this paper; for the infinite assembly, it is not appropriate; see \cite{BrChDoKaSe2011} for instance.
}.
An assembly $\alpha$ is {\it $\tas$-producible} or {\it producible by $\tas$} if $\sigma \to^\tas \alpha$.
A $\tau$-stable assembly $\alpha$ is {\it ($\tas$-)terminal} if for any $\tau$-stable assembly $\beta$, $\alpha \to^\tas \beta$ implies $\alpha = \beta$.
Let $\mathcal{A}[\tas]$ be the set of assemblies producible by $\tas$, and let $\mathcal{A}_\Box[\tas] \subseteq \mathcal{A}[\tas]$ be the set of terminal assemblies that are producible by $\tas$.
A TAS $\tas$ is {\it directed} if for each $\alpha, \beta \in \mathcal{A}[\tas]$, there exists $\gamma \in \mathcal{A}[\tas]$ such that $\alpha \to^\tas \gamma$ and $\beta \to^\tas \gamma$.
One can easily verify that $\tas$ is directed if and only if $|\mathcal{A}_\Box[\tas]|=1$, and that $\tas$ is {\em not} directed if and only if there exist $\alpha, \beta \in \mathcal{A}[\tas]$ and a position $p \in \dom(\alpha) \cap \dom(\beta)$ such that $\alpha(p) \neq \beta(p)$.
Throughout this paper, any TAS $\tas = (T, \sigma, g, \tau)$ is assumed to be free from useless tile types in the sense that for any tile type $t \in T$, there are a $\tas$-producible assembly $\alpha \in \mathcal{A}[\tas]$ and a position $p \in \dom(\alpha)$ such that $\alpha(p) = t$.
\subsection{Rectangular patterns and rectilinear TASs}
\label{subsec:RTAS}
For $w, h \ge 1$, a {\it (rectangular) pattern (of width $w$ and height $h$)} is a function from the rectangular domain $\{(x, y) \mid x \in \{0, 1, \ldots, w\}, y \in \{0, 1, \ldots, h\}\}$ to $\mathbb{N}$.
We call the image of a pattern $P$ the {\em color set of $P$}, and denote it by $\color(P)$.
That is, any color in $\color(P)$ occurs at least once on $P$.
When $|\color(P)| \le k$, we say that $P$ is {\it $k$-colored}.
The {\it rectilinear TAS} (RTAS) is a variant of {\em temperature-2} TAS that is specialized for the rectangular pattern assembly.
An RTAS is a quadruple $\tas = (T, \sigma, g, 2)$, where
\begin{enumerate}
\item $T$ and $g$ are as defined for TASs;
\item $\sigma$ is a 2-stable seed of L-shape, whose domain is $\{(x, 0) \mid 0 \le x \le w\} \cup \{(0, y) \mid 0 \le y \le h\}$;
\item for any $t \in T$ and $d \in \{\north, \west, \south, \east\}$, $g(t(d)) = 1$.
\end{enumerate}
The statement 3 characterizes the rectilinear manner of assembly.
The temperature being 2, the statement implies that any tile attachment needs the cooperation between the west glue and south glue, both of which are of strength~1.
As such, at the initial time point of assembly process, for example, the sole position at which a tile in $T$ can attach is $(1, 1)$.
Then it is a routine to verify that the assembly process proceeds from south-west to north-east {\it rectilinearly}.
We say that $\tas$ {\it uniquely self-assembles a $k$-colored pattern $P$} if there exists a coloring function $f$ from $T \cup T_\sigma$ to the color set $C = \{1, 2, \ldots, k\}$ such that for any terminal assembly $\alpha \in \mathcal{A}_\Box[\tas]$, $f(\alpha) = P$.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.3]{SA_counter.eps}
\end{center}
\caption{
From the L-shape seed, an RTAS $\tas_{\rm bc}$ uniquely self-assembles the infinite binary counter using the four tile types that implement the half-adder.
}
\label{fig:SA_counter}
\end{figure}
As an example, an RTAS $\tas_{\rm bc}$ that uniquely self-assembles the infinite binary counter is shown in Figures~\ref{fig:half-adder} and~\ref{fig:SA_counter}.
Two blue tile types and two orange ones in Figure~\ref{fig:half-adder} represent all four possiblities of 2-bit inputs {\tt 00}, {\tt 01}, {\tt 10}, {\tt 11} as their west and south glues, output the sum of the inputs to the north, and carry out to the east.
In this way, they implement the half-adder together.
$\tas_{\rm bc}$ fills the first quadrant defined by its L-shape seed by these four tile types in the {\em directed} manner (Figure~\ref{fig:SA_counter}).
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.45]{OR_gate.eps}
\end{center}
\caption{
Four tile types implement together the {\tt OR}-gate, with two inputs A, B and one output S.
}
\label{fig:or-gate}
\end{figure}
The L-shape seed of $\tas_{\rm bc}$ provides the sequence of 1-glues to the east and that of 0-glues to the north.
With different glue sequences on the seed, tiles of $\tas_{\rm bc}$ uniquely yield other blue-orange patterns.
Thus, we can say that the tile type set of $\tas_{\rm bc}$ is a machanism to convert a glue sequence on the seed given as input to a rectangular pattern.
Furthermore, this example suggests that not only the half-adder but any combinatorial logics with at most two inputs and at most two outputs can be thus implemented using at most four tile types.
Figure~\ref{fig:or-gate} presents such an implementation of {\tt OR}-gate, for instance.
In Section~\ref{sec:reduction}, we shall design a set of tile types that evaluates a {3\sat} instance according to a given Boolean value assignment.
\subsection{PATS: pattern self-assembly tile set synthesis problem}
Now, we introduce the pattern self-assembly tile set synthesis problem ({\pats}) originally proposed by Ma and Lombardi \cite{MaLombardi2008}.
{\pats} aims at computing the minimum size RTAS that uniquely self-assembles a given rectangular pattern, where the size of an RTAS is measured by the cardinality of its tile type set.
\begin{definition}[{\scshape Pattern self-Assembly Tile set Synthesis (Pats)}]
\ \\
\begin{tabular}{ll}
{\scshape Given}: & a pattern $P$ \\
{\scshape Find}: & a smallest directed RTAS that uniquely self-assembles $P$.
\end{tabular}
\end{definition}
\noindent
Note that the solution to {\pats} is required to be directed here, while it was not so in its original definition.
This, however, does not change the problem, as being observed in \cite{GoosOrponenDNA16}.
{\pats} is an \NP-hard problem \cite{CzeizlerPopa2012}.
By parameterizing it by the number of maximum colors $c$ used to draw patterns, we propose the following more practically meaningful variant of {\pats}.
\begin{definition}[{\scshape $c$-colored Pattern self-Assembly Tile set Synthesis ($c$-Pats)}]
\ \\
\begin{tabular}{ll}
{\scshape Given}: & a $c$-colored pattern $P$; \\
{\scshape Find}: & a smallest directed RTAS that uniquely self-assembles $P$.
\end{tabular}
\end{definition}
We will find out a constant $c$ that makes {$c$-\pats} \NP-hard, through a polynomial-time reduction of {3\sat} to the following decision variant of the problem:
\vspace*{2mm}
\begin{tabular}{ll}
{\scshape Given}: & a $c$-colored pattern $P$ and an integer $n$; \\
{\scshape Output}: & {\tt YES} if there exists a directed RTAS with at most $n$ tile types \\
& that uniquely self-assembles $P$.
\end{tabular}
\subsection{Basic Combinatorial Results}
\label{subsec:basics}
Before proceeding to the main result, let us present several basic results on directed RTASs, which we will be used in Section~\ref{sec:reduction}.
Let us begin with the most important property which characterize the directedness property by the west and south glues of their tile types.
\begin{proposition}\label{prop:directed_RTAS_characterization}
An RTAS is directed if and only if it contains no distinct tile types $t_1, t_2$ with $t_1(\west) = t_2(\west)$ and $t_1(\south) = t_2(\south)$\footnote{
This proposition is true as long as all tile types of $\tas$ appear on some assembly producible by $\tas$.
This is an assumption we make in this paper, as declared at the end of Section~\ref{subsec:def_aTAM}.
}.
\end{proposition}
This proposition enables us to design simple patterns that, being embedded into another pattern $P$ as a subpattern, necessitates {\em at least} 2 tile types in order for {\em any} directed RTAS to uniquely self-assemble $P$.
One of such patterns can be found in the binary counter pattern $P_{\rm bc}$ in Figure~\ref{fig:SA_counter}.
At the orange position $P_{\rm bc}(2, 2)$ and blue position $P_{\rm bc}(4, 2)$, an RTAS $\tas$ must put tiles of distinct types $t_1, t_2$.
Moreover, in order for $\tas$ to be directed, $t_1$ needs to disagree to $t_2$ with respect to either west glue or south glue (Proposition~\ref{prop:directed_RTAS_characterization}).
This implies that $\tas$ has two distinct blue tile types because the west and south neighbors of these positions are all blue.
This observation is formally described as follows.
\begin{lemma}\label{lem:at_least_2-1}
Let $\tas$ be a directed RTAS that uniquely self-assembles a pattern $P$.
For a color $i$ and positions $(x_1, y_1), (x_2, y_2)$, if $P(x_1-1, y_1) = P(x_1, y_1-1) = P(x_2-1, y_2) = P(x_2, y_2-1)=i$ but $P(x_1, y_1) \neq P(x_2, y_2)$, then $\tas$ has at least two tile types of color $i$.
\end{lemma}
\begin{figure}[tb]
\begin{minipage}{0.6\textwidth}
\begin{center}
\includegraphics[scale=0.4]{LB2.eps}
\end{center}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\includegraphics[scale=0.4]{glue_allocation.eps}
\end{minipage}
\caption{
(Left) If this blue-orange binary-color subpattern appears on a pattern $P$, a directed RTAS needs at least 2 blue tile types and at least 2 orange tile types in order to uniquely self-assemble $P$;
(Middle) This pattern also asks a directed RTAS to prepare 2 blue tile types.
(Right)
}
\label{fig:LB2}
\end{figure}
Lemma~\ref{lem:at_least_2-1} immediately leads us to the next lemma.
\begin{lemma}\label{lem:at_least_2-substitutor}
If a directed RTAS uniquely self-assembles a pattern on which the pattern shown in Figure~\ref{fig:LB2} (Left) appears, then it has at least 2 blue tile types and at least 2 orange tile types.
\end{lemma}
Another pattern of interest is shown in Figure~\ref{fig:LB2} (Middle).
With tiles of type $t$ at all the four blue positions, $t$ would satisfy $t(\west) = t(\east)$ and $t(\north) = t(\south)$ and hence a blue tile (of this type) would fill the orange position.
This observation is formally described as follows.
\begin{lemma}\label{lem:at_least_2-2}
Let $\tas$ be a directed RTAS that uniquely self-assembles a pattern $P$.
For a color $i$ and a position $(x, y)$, if $P(x-2, y) = P(x-1, y) = P(x, y-1) = P(x, y-2-1)=i$ but $P(x, y) \neq i$, then $\tas$ has at least two tile types of color $i$.
\end{lemma}
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.6]{LB2_multiple.eps}
\end{center}
\caption{
Two subpatterns that cooperatively force at least 2 colors to be used to draw at least 2 tile types.
To the right, a set of 9 tile types that uniquely self-assemble them is presented, which is the sole minimum one.
}
\label{fig:LB2_multiple}
\end{figure}
Next, we propose a mosaic pattern {\em parameterized by an integer $k$} that forces $k$ number of different colors to be used to draw at least 2 tile types.
For $k = 2$, this pattern can be found in Figure~\ref{fig:LB2_multiple}.
It contains 7 colors: white, blue, orange, lined black, lined white, {\tt A}, and {\tt B}.
On the assumption that the gray part be the seed, where, by definition, any information can be encoded, how many tile types are necessary and sufficient for a directed RTAS $\tas$ to uniquely self-assemble them.
Since ${\tt A} \neq {\tt B}$, the two tile types $t_A, t_B$ need to satisfy either $t_A(\west) \neq t_B(\west)$ or $t_A(\south) \neq t_B(\south)$.
In the former case, as shown in Figure~\ref{fig:LB2_multiple}, two lined black types and two lined white ones can carry the one bit information (to be {\tt A} or not to be), and one white, blue, and orange are enough.
On the other hand, in the latter case, we need to deliver the 1-bit information through the 3-colors (white, blue, orange) mosaic pattern and no matter how it is delivered, the path encounters all of these 3 colors.
If it were not for two blue tile types, the assembly process would lose the information once it hits the diagonal blue stripe, and this argument is valid for white and orange.
Thus in this case we need extra 3 tile types in contrast to the need for 2 tile types in the former case.
So, for instance, with only 9 tile types, the latter propagation strategy cannot be employed.
Let us note that this parameterized mosaic pattern is ``stretchable'' by duplicating each column arbitrary many times.
We have seen several patterns that forces an RTAS to draw 2 tile types by a specific color.
It is a challenging issue to design a pattern {\em without introducing many auxiliary (wasteful) colors} that forces more than 2 types to be drawn by a specific color.
Another issue of significance is to design a pattern that allocates glues on tile types in an intended manner.
In Figure~\ref{fig:LB2} (Right), you see two $2 \times 2$ square patterns, which consist of three colors (gray, blue, orange).
If a directed RTAS uniquely self-assembles a pattern including these square patterns, and moreover, using {\em only one} tile type of each of these three\footnote{In fact, it is fine that gray is equal to either blue or orange as long as blue is distinct from orange in order to reach the coming conclusion.} colors.
Let them be $t_g, t_b, t_o$.
Then $t_g(\east) = t_b(\west) = t_o(\west)$.
Thus, $t_b(\south) \neq t_o(\south)$ must hold (Proposition~\ref{prop:directed_RTAS_characterization}).
This can be strengthened as $t_b(\north) = t_b(\south) \neq t_o(\south) = t_o(\north)$ because blue and orange positions are in tandem.
\section{Polynomial-Time Reduction of 3SAT to \nc-Colored PATS}
\label{sec:reduction}
The following is the main theorem of this paper.
\begin{theorem}\label{thm:cpats_NPhard}
{\nc-\pats} is \NP-hard.
\end{theorem}
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.7]{blueprint.eps}
\end{center}
\caption{The pattern $P(\phi)$ to which a {3\sat} instance $\phi$ is reduced.}
\label{fig:blueprint}
\end{figure}
Our proof takes the classic approach: a polynomial-time many-one reduction from {3\sat} to the decision variant of {$c$-\pats}.
An instance of {3\sat} is a formula $\phi$ that is a conjunction of clauses consisting of exactly three literals (a variable or its negation); the $m$ variables of $\phi$ are indexed as $v_1$, $v_2$, and so on.
We will propose a set $T_{\rm 3SAT}$ with ${\nc}+24$ tile types and a pattern $P(\phi)$ to which a given {3\sat} instance $\phi$ is reduced such that
\begin{enumerate}
\item any RTAS with this tile type set is directed due to Proposition~\ref{prop:directed_RTAS_characterization};
\item if $\phi$ is satisfiable, a directed RTAS uniquely self-assembles $P(\phi)$ using $T_{\rm 3SAT}$;
\item if a directed RTAS uniquely self-assembles $P(\phi)$ using {\nc}+24 tile types, then its tile type set is isomorphic to $T_{\rm 3SAT}$ (up to glue label renaming) and (hence) $\phi$ is satisfiable.
\end{enumerate}
As shown in Figure~\ref{fig:blueprint}, $P(\phi)$ consists of four subpatterns: {3\sat} evaluator pattern $P_{\rm eval}(\phi)$ and three gadget patterns $P_A, P_B, P_D$.
The {3\sat} evaluator pattern is the main pattern and the other three gadget patterns play an auxiliary role in the proof of the third statement above.
\subsection{3SAT Evaluator Pattern}
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.4]{3sat_evaluator.eps}
\end{center}
\caption{
A digital circuit to evaluate a specific {3\sat} instance $(v_4 \vee v_3 \vee v_2) \wedge (v_4 \vee \neg v_3 \vee v_1)$ according to a given assignment of Boolean values.
Literals (white stripes) are evaluated at substitutors (blue squares), and the evaluation (black stripes with a white arrow) is transmitted to the north for the evaluation of clause they belong to.
It should be clear that the circuit design principle works for arbitrary number of variables or clauses.
}
\label{fig:3sat_evaluator}
\end{figure}
Let us begin with the {3\sat} evaluator pattern.
It is degisned based on a digital circuit which we call a {\it {3\sat} evaluator}.
Just as the name suggests, it evaluates a given {3\sat} formula $\phi$ according to an assignment given as input.
It must be noted first that this circuit is designed to be planar (no wire goes over the others) and rectilinear (signals always transmit from south-west to north-east) so that it can be easily transformed into $P(\phi)$.
Figure~\ref{fig:3sat_evaluator} illustrates the {3\sat} evaluator.
As input, it takes from the left an assignment $\vector{b} = (b_1, \ldots, b_m)$ according to which $\phi$ is evaluated, where $b_1, \ldots, b_m \in \{0, 1\}$ (0:false, 1:true).
It is provided with $m$ horizontal wires, which transmit the input to the right along with the index $i$ of the variable they represent.
It is also provided with vertical wires, which represent literals in $\phi$.
We refer to the horizontal wire for variable $v_i$ simply as the {\it variable wire $v_i$}, and a vertical wire for literal $v_j$ (resp.~$\neg v_j$) as a {\it literal wire $v_j$ (resp.~$\neg v_j$)}.
Any variable wire is connected with any literal wire at their intersection by a device called {\it substitutors}.
At such an intersection between the variable wire $v_i$ and a literal wire $v_j$ or $\neg v_j$, the substitutor compares their indices, and if $i = j$, it substitutes the Boolean value $b_i$ into the literal by an {\xnor} gate and transmits the result to the north; otherwise, it does nothing but merely ``lets one go over the other.''
For each clause, its three literals thus evaluated are conjugated by three {\tt OR} gates, which determines whether the clause is satisfied or not, and the evaluation is output at the top as LED signal (red:unsatisfied, green:satisfied; see Figure~\ref{fig:3sat_evaluator}, in which Clause 1 is evaluated to be satisfied, whereas Clause 2 is not).
\begin{figure}[p]
\begin{center}
\includegraphics[scale=0.75]{main_tileset.eps}
\end{center}
\caption{
The 51 tile types (of 27 colors) for the simulation of {3\sat} evaluator by a directed RTAS.
}
\label{fig:tileset}
\end{figure}
\begin{figure}[p]
\begin{center}
\includegraphics[scale=0.45]{main_idea.eps}
\end{center}
\caption{Using tiles in the set $T_{\rm 3SAT}$, this pattern uniquely self-assembles from the L-shape seed that encodes a {3\sat} instance with 3 variables $v_1, v_2, v_3$ and a clause $\{v_1, \neg v_2, v_3\}$ with the assignment $(1, 1, 1)$.}
\label{fig:main_idea}
\end{figure}
The pattern $P_{\rm eval}(\phi)$ is a slight modification of a pattern $P_{\rm eval}(\phi, \vector{b})$ of {3\sat} evaluator's circuit layout.
Now we will propose a tile type set $T_{\rm eval}$ (Figure~\ref{fig:tileset}) with which a directed RTAS simulates the {3\sat} evaluator and uniquely self-assembles $P_{\rm eval}(\phi, \vector{b})$.
This consist of 51 tile types of 27 colors: 1 type of each of 9 colors $c_{1, 1}, c_{1, 2}, \ldots, c_{1, 9}$, 2 types of each of 15 colors $c_{2, 1}, \ldots, c_{2, 15}$, and 4 types of each of 3 colors $c_{4, 1}, c_{4, 2}, c_{4, 3}$.
The assignment $\vector{b} = (b_1, \ldots, b_m)$ and instance $\phi$ are encoded on the L-shape seed.
Specifically, the assignment $\vector{b}$ is encoded as the following glue sequence on the vertical bar of the seed:
\begin{equation}\label{eq:assign_encoding}
0 {\tt e} ({\tt bg})^2 \ b_1 {\tt v}^2 \ ({\tt bg})^2 \ b_2 {\tt v}^4 \ ({\tt bg})^2 \cdots ({\tt bg})^2 \ b_m {\tt v}^{2m} \ ({\tt bg})^2 @,
\end{equation}
where $v_i$ is encoded with its assigned value $b_i$ as $b_i {\tt v}^{2i}$ and @ indicates the origin (0, 0), and ${\tt bg}, {\tt v}, {\tt e}$ are glues that stand for {\it background}, {\it variable}, and {\it evaluation}.
On the other hand, the literal $v_j$ and its negation are encoded as ${\tt v}^{2j} 1$ and ${\tt v}^{2j} 0$, respectively (1:positive, 0:negative).
Then we encode a clause $c = \{v_i, \neg v_j, v_k\}$ as:
\begin{equation}\label{eq:literal_encoding}
gs(c) = ({\tt bg})^2 \ {\tt l}^{2i} 1 \ ({\tt bg})^2 \ {\tt l}^{2j} 0 \ ({\tt bg})^2 \ {\tt l}^{2k} 1 \ ({\tt bg})^2 {\tt e},
\end{equation}
where {\tt l} represents the {\it literal}.
A glue sequence on the seed's horizontal bar that encodes the clauses $c_1, c_2, \ldots$ of $\phi$ is obtained by catenating $g(c_1)$, $g(c_2)$, and so on.
Concatenating it further with the glue sequence in \eqref{eq:assign_encoding} amounts to the L-shape seed, from which, the pattern $P_{\rm eval}(\phi, \vector{b})$ self-assembles (see Figure~\ref{fig:main_idea} and compare it with the {3\sat} evaluator).
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{substitutors_v2.eps}
\end{center}
\caption{
Patterns occurring at the intersection of the literal wire $v_2$
(Left) with the variable wire $v_3$,
(Right bottom) with the wire for matching variable $v_2$, or
(Right, middle) with the variable wire $v_1$.
(Right, top) Its intersection with the thinnest wire, which does not encode any variable.
}
\label{fig:substitutors1}
\end{figure}
On $P_{\rm eval}(\phi, \vector{b})$, any literal wire encounters the $m$ variable wires.
At the intersection of a literal wire $v_j$ with the variable wire $v_i$, tiles in $T_{\rm eval}$ self-assembles a pattern that visualizes the mechanism of substitutor in the {3\sat} evaluator.
How this assembly proceeds may be understood more easily visually than with words, so see Figure~\ref{fig:substitutors1}.
When the wires meet, the {\tt check-start} tile attachs and triggers the assembly of diagonal zig-zag snake.
When it hits the literal wire, one of the 2 {\tt check} tile attachs, and if to its north, the variable wire $v_i$ is waiting with the input $b_i$ (this happens if and only if $i = j$), one of the 4 {\xnor} tiles selectively attachs, substitutes $b_i$ into the literal, and outputs the result to the north (1:true, 0:false; note that before and after the substitution, the signal 1/0 through a literal wire is interpreted in different ways).
Figure~\ref{fig:substitutors1} shows how the literal wire goes over the variable wire without being substituted in other two cases when $i > j$ (Left) and $i < j$ (Right, middle).
Due to the encoding \eqref{eq:assign_encoding}, at of the substitution, the wire $v_j$ has already crossed the variable wires $v_m, v_{m-1}, \ldots, v_{j+1}$, which are thicker than itself, and it is going to transmit the substituted value while crossing the remaining variable wires $v_{j-1}, \ldots, v_1$ in this order, which are thinner.
With the property of $T_{\rm eval}$ that when a literal wire crosses a thicker variable wire, some tile types {\em visually} tell which of 1/0 the wire carries, while its crossing a thinner one leaves no such visual clue, the encounter order means that from $P_{\rm eval}(\phi, \vector{b})$, we cannot get any further clue of $\vector{b}$ than whether clauses are satisfied or not.
This cover-up plays a critical role in the proof of \NP-hardness.
The evaluation of each clause by 4 tile types of {\tt OR} color should be straightforward from Figure~\ref{fig:main_idea} (see its top).
The {\em only} positions on $P(\phi, \vector{b})$ whose color changes depending on the encoded assignment $\vector{b}$ are the LED positions.
Drawing all these positions by green (satisfied) yields the pattern $P_{\rm eval}(\phi)$.
By this construction, it is obvious that for a satisfiable $\phi$, a directed RTAS uniquely self-assembles $P_{\rm eval}(\phi)$, and actually the whole pattern $P(\phi)$.
Although for an unsatisfiable $\phi$, $P(\phi)$ cannot be uniquely self-assembled by $T_{\rm 3SAT}$, the possibility for another set of ${\nc}+24$ tile types cannot be ruled out.
That is when the gadget patterns come into play.
\subsection{Gadget Patterns}
Let us imagine that you are given {\nc}+24 tile types, all of which are uncolored, and asked to draw them so as for them to uniquely self-assemble $P(\phi)$ in a directed manner.
It goes without saying that we must draw at least one tile type by each of {\nc} colors on $P(\phi)$, and 24 tile types are left uncolored.
Applying Lemmas~\ref{lem:at_least_2-1}, \ref{lem:at_least_2-substitutor}, and \ref{lem:at_least_2-2} to $P(\phi)$ implies that we need at least 2 tile types of the 6 colors $c_{2, 1}, \ldots, c_{2, 4}$, $c_{2, 5}$ (white), and $c_{2, 6}$ (black) (see Figure~\ref{fig:tileset}).
Now 18 extra tile types remain uncolored.
The arguments up to now should be straightforward.
The role of the three gadget patterns is to cooperatively force us to draw them in the same way as $T_{\rm 3SAT}$, and furthermore, allocate glues in the isomorphic manner (see Figure~\ref{fig:tileset}).
More specifically,
\begin{description}
\item[$P_A$:] Due to this, at least 7 of uncolored tile types are to be drawn by colors included in this pattern;
\item[$P_B$:] Among the remaining (at most 11) uncolored tile types, at least 8 of them are to be drawn;
\item[$P_D$:]
This draws the remaining (at most 3) uncolored tile types.
The more important role is to allocate glues onto the 12 tile types of color $c_{4, 1}, c_{4, 2}, c_{4, 3}$ as shown in Figure~\ref{fig:tileset}, and hence, implement the {\tt OR} gate, wire crossing, and {\tt XNOR} gate.
\end{description}
At this point, the only possible coloring is to draw 4 tile types by each of $c_{4, 1}, c_{4, 2}, c_{4, 3}$ and 2 tile types by each of 15 colors $c_{2, 1}, \ldots, c_{2, 15}$ (see Figure~\ref{fig:tileset}).
Now we explain each gadget.
They are designed based on the parameterized mosaic pattern introduced in Section~\ref{subsec:basics}.
\subsubsection{Gadget Pattern $P_A$}
\begin{figure}[p]
\begin{center}
\includegraphics[scale=0.6]{gadget_A.eps}
\end{center}
\caption{The gadget pattern $P_A$.}
\label{fig:gadget_A}
\end{figure}
The gadget pattern $P_A$ is shown in Figure~\ref{fig:gadget_A}.
This pattern employs extra 14 auxiliary colors A1-A11 and AB1-AB3 (note that these auxiliary colors will appear in $P_B$ or $P_D$, and their auxiliary colors do not appear in the other, either).
The colors A1-A8 are used to make the mosaic pattern (in Figure~\ref{fig:gadget_A}, the mosaic is not described for the sake of clarity), which is stretched as being explained before such that to the right of $P_A$, the next gadget pattern $P_B$ can be assembled properly.
A9-A11 seal this pattern from the top so that the variable wires $v_1, \ldots, v_m$ can carry the assignment $\vector{b}$ to the {3\sat} evaluator pattern $P_{\rm eval}(\phi)$ without trouble (see, at the top of Figure~\ref{fig:gadget_A}, we can see the lower end of the variable wire $v_m$.
AB1-AB3 provides the glue $b$'s to the east for the gadget pattern $P_B$.
Due to the previously-mentioned property of mosaic pattern, we need to use at least 7 colors to draw tile types that have been uncolored, and unless the 7 colors are chosen to be $c_{2, 12}, c_{2, 13}, c_{2, 14}, c_{2, 15}, c_{4, 1}, c_{4, 2}, c_{4, 3}$, one more uncolored tile type would be drawn, which is not acceptable because for $P_B, P_D$, we have to set aside 11 uncolored tile types.
Even with the preferable (only one) choice, we further have to allocate glues to them so that 1-bit information (0/1) can be transmitted vertically because otherwise the information cannot help but penetrate the mosaic, which would cost at least one more uncolored tile type.
It is not the case that this would allocate glues to the tile types of these 7 colors in a way isomorphic to those in Figure~\ref{fig:tileset}.
The issue of glue allocation should be discussed after explaining the other two gadget patterns and we are convinced of the fact that the ${\nc}+24$ tile types available in total are drawn as specified in Figure~\ref{fig:tileset} (all the auxiliary colors introduced for $P_A, P_B, P_D$ are used to draw only one tile type).
\subsubsection{Gadget Pattern $P_B$}
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.3]{gadget_B.eps}
\end{center}
\caption{The gadget pattern $P_B$.}
\label{fig:gadget_B}
\end{figure}
The gadget pattern $P_B$ is a counterpart of $P_A$ for 8 colors $c_{2, 7}, \ldots, c_{2, 11}, c_{4, 1}, c_{4, 2}, c_{4, 3}$.
Hence, we mention only the fact that unless we use these colors to draw extra 8 tile types and allocate glues such that 0/1 signal can transmit, but rather horizontally, we would waste too many uncolored tile types.
\subsubsection{Gadget Pattern $P_D$}
\begin{figure}[p]
\begin{center}
\includegraphics[scale=0.6]{gadget_D.eps}
\end{center}
\caption{The gadget pattern $P_D$.}
\label{fig:gadget_D}
\end{figure}
The gadget pattern $P_D$ is shown in Figure~\ref{fig:gadget_D}.
This is actually a modification of the mosaic pattern using 7 colors D1-D7.
In addition, one color C-bg is found between the border between $P_D$ and the {3\sat} evaluator pattern $P(\phi)$.
Thus, $P_D$ introduces 8 new colors, and now all {\nc} colors have been introduced.
We claim that this pattern needs extra 3 uncolored tile types, and they must be drawn with the colors $c_{4, 1}, c_{4, 2}, c_{4, 3}$.
Hence, the other gadget patterns must be assembled in the intended way (with minimum consumption of the uncolored tile types).
In summary, $c_{4, 1}, c_{4, 2}, c_{4, 3}$ each is used to draw 4 tile types (in total 12 tile types), $c_{2, 1}, \ldots, c_{2, 15}$ each is used to draw 2 tile types (in total 30 tile types), and the other 41 colors including the auxiliary ones appears on only one tile type.
\subsubsection{Glue Allocation}
Having completed the coloring, we briefly explain the reason why we must not only color them thus but also allocate glues to them in such a way as shown in Figure~\ref{fig:tileset}.
Actually, it suffices to verify this isomorphism for colors that are responsible for transmitting or processing 0/1 signals for {3\sat} evaluation.
It is helpful to observe that for a color that draws only one tile type, all tile types at positions of this color on $P(\phi)$ are identical.
First, see the sole tile types $t_1, t_0, t_{\rm eval}$ of the respective colors $c_{1, 1}$ ({\tt 1}), $c_{1, 2}$ ({\tt 0}), and $c_{1, 4}$ ({\tt c-eval l-eval}).
In the pattern $P(\phi)$, we find two $c_{1, 4}$-colored positions whose north neighbors are colored {\tt 1} and {\tt 0}, respectively.
Thus, $t_1(\south) = t_0(\south) = t_{\rm eval}(\north)$.
Then Proposition~\ref{prop:directed_RTAS_characterization} implies that $t_1(\west) \neq t_0(\west)$.
Let $t_1(\west) = 1$ and $t_0(\west) = 0$ with $1 \neq 0$.
Since there is a {\tt 1}-colored position whose west neighbor is colored {\tt 0}, $t_0(\east) = 1$, and likewise $t_1(\east) = 0$.
In the same manner, for the tile types $t_P, t_N$ with respective colors $c_{1, 7}, c_{1, 8}$, we have $t_P(\west) = t_N(\west)$ but $t_P(\north) = t_P(\south) \neq t_N(\north) = t_N(\south)$.
Now these four tile types $t_1, t_0, t_P, t_N$ can be employed to allocate glues to the other tile types.
For example, see the right half of the gadget pattern $P_B$ (Figure~\ref{fig:gadget_B}).
Two positions of each of the colors $c_{2, 9}, c_{2, 10}, c_{2, 11}$ are found and they are sandwiched by {\tt 1} positions and {\tt 0} positions.
Thus, for example, we can say that there are two $c_{2, 9}$ tile types one of whose east and west glues are both {\tt 1} and the other's are both {\tt 0}.
This guarantees that when an assignment signal (0/1) crosses the border between two {3\sat} clauses, the signal is not converted (though $P_B$ guarantees that they carry the 1-bit signal horizontally, it cannot rule out the possibility that these signals be flipping).
$P_B$ does not contain an analogous pattern for the colors $c_{2, 7}, c_{2, 8}$, nevertheless they are also transmitting signals.
This is because in $P(\phi)$ they always appear as a pair and hence no matter whether each of their 2 tile types converts a signal or not, it can deliver the signal correctly as long as the delivery distance is even.
With the help of these four tile types, the gadget pattern $P_D$ allocates glues so as to implement the {\tt OR}-gate, the signal intersection, and {\tt XNOR}-gate.
The one risk to be taken into consideration resides in the 4 tile types for the signal intersection.
If the north glue of the D1-colored tile type is 1 and that of the D4-colored one is 0, then this converts the signal vertically.
In order to render this conversion harmless, we encode the variable $v_i$ on the seed rather as $b_i {\tt v}^{2i} 0$; this produces a (meaningless) wire at the lower end of the variable wire $v_i$.
As a result, any signal certainly goes through the signal intersection {\em even} times.
\section*{Acknowledgements}
We gratefully acknowledge valuable comments and encouragement from Ho-Lin Chen, Eugen Czeizler, David Doty, Aleck Christopher Johnson, Natasha Jonoska, Ming-Yang Kao, Steffen Kopecki, Florence Linez, Pekka Orponen, Amir Hossein Simjour, and Damien Woods.
In particular, Ho-Lin Chen suggested encoding wires unary instead of binary, and David Doty and Damien Woods helped us to decrease the number of colors in the gadget patterns.
The current design of {\nc}-colored pattern for the reduction was done during the first author's visit at Natasha Jonoska's research group at the University of South Florida and his stay in Nantes, France with Florence Linez.
This research is in part financially supported by HIIT Pump Priming Project Grants 902184/T30606 to the first author.
\bibliographystyle{plain}
|
1,314,259,996,691 | arxiv | \section{Introduction}
About one third of all binary star systems are thought to be members
of larger multiple systems. Most of these are hierarchical triples, in
which the (inner) binary is orbited by a third body in a much wider
orbit (see Tokovinin 1997a,b for recent results and compilations).
Secular perturbations in triples result from the gravitational
interaction between the inner binary and the outer object, possibly
coupled to other processes such as stellar evolution, tidal effects,
or, for compact objects, general relativistic effects. In strongly hierarchical
triples, the two orbits never approach each other closely, and an
analytic, perturbative approach can be used to calculate the evolution
of the system. One particularly important perturbation is that of the
orbital eccentricities. As the two orbits torque each other and
exchange angular momentum, their eccentricities will undergo periodic
oscillations over secular timescales (i.e., very long compared to the
orbital periods). For non-coplanar systems, corresponding oscillations
occur in the orbital inclinations.
In contrast, according to canonical perturbation theory, there is no
secular change in the semimajor axes, since the
energy exchange between the two orbits averages out to zero over
long timescales (see, e.g., Heggie 1975).
For triple systems that begin their life near the stability limit,
the result of an eccentricity increase can be catastrophic, leading to
a collision between the two inner stars, if they started as a close pair,
or, more typically, to the disintegration of the triple. This
disintegration proceeds through a phase of chaotic evolution whose
outcome is the ejection of one of the three stars (typically the
least massive body) on an unbound trajectory, while the other two are left in a
more tightly bound binary. A striking example of this process was
revealed by the recent HST/NICMOS observations of the TMR-1 system in
the Taurus star-forming region (Terebey et al.\ 1998). The HST images
reveal a faint companion, most likely a giant planet or brown dwarf,
that appears to have been ejected from its parent protostellar binary
system. More indirect observational evidence is provided in the form
of binary systems with anomalously high space velocities. In
particular, the disintegration of short-lived triples formed in dense
star forming regions may lead to binary OB runaway stars with very
large peculiar velocities, such as HD 3950 (Gies \& Bolton 1986).
The stability of triple systems has been the subject of many
theoretical studies. Most recently, Eggleton \& Kiseleva (1995
and references therein)
performed numerical experiments and provided an empirical stability
criterion in terms of a critical ratio $Y_{\rm min}$ between the
periastron distance of the outer orbit to the apastron distance of
the inner orbit. For systems containing three nearly equal masses,
one finds $Y_{\rm min}\simeq3-6$ depending on initial phases, eccentricities
and inclinations.
Holman \& Wiegert (1999) study the stability of planets in
binary systems, both for planets orbiting close to one of the two
stars, and for planets orbiting outside the binary. All these
stability analyses are based on numerical integrations of the
three-body problem that are limited to $10^4-10^6$ periods of the outer
orbit. In some cases, however, the secular evolution timescale of the
triple can be much longer than this, and therefore systems that remain
stable for the duration of the numerical integration may in fact turn
out to be unstable over secular timescales. Analytic results such as
those derived here can therefore help determining more accurate
stability criteria, e.g., by integrating the secular evolution
equations and verifying that the stability ratio remains $>Y_{\rm
min}$ over the entire cycle of secular perturbations.
Hierarchical triple star systems can play an important role in the
dynamical evolution of dense star clusters containing primordial
binaries. The cores of globular clusters, for example, are thought to
contain a small but dynamically significant population of triple
systems formed through dynamical interactions between primordial
binaries (McMillan, Hut, \& Makino 1991). Both stable and unstable
triples can form easily through exchange and resonant interactions
between binaries. In direct $N$-body integrations of the cluster
dynamics, marginally stable or unstable triples can represent a
significant computational bottleneck, since they require very long
integrations of the orbital dynamics in order to resolve the outcome
of the interaction (see, e.g., Mikkola 1997).
Direct observational evidence for the dynamical production of triple
systems in globular clusters is provided by the millisecond pulsar
system PSR B1620$-$26 (Rasio, McMillan, \& Hut 1995; Ford et al.\ 2000).
This radio pulsar is a member of
a hierarchical triple system located in the core of the globular
cluster M4. The inner binary of the triple contains the $\simeq 1.4
M_{\odot}$ neutron star with a $\simeq 0.3 M_{\odot}$ white-dwarf
companion in a 191-day orbit (Lyne et al.\ 1988; McKenna \& Lyne 1988).
The triple nature of the system was first proposed by Backer (1993) in
order to explain the unusually high residual second and third pulse
frequency derivatives left over after subtracting a standard Keplerian
model for the pulsar binary. The pulsar has now been timed for eleven
years since its discovery (see Thorsett et al.\ 1999 for the most recent
update). These observations have not only confirmed the triple nature
of the sytem, but they have also provided tight constraints on the
mass and orbital parameters of the second companion. Theoretical
modeling of the latest timing data (now including five pulse frequency
derivatives) and preliminary measurements of the orbital perturbations
of the inner binary have further constrained the mass of the second
companion, and strongly suggest that it is a giant planet or a brown
dwarf of mass $\sim 0.01\,M_\odot$ at a distance of $\sim 50\,$AU
from the pulsar binary (Joshi \& Rasio 1997; Ford et al.\ 2000).
Our treatment of the secular perturbations in this paper is based on
classical celestial mechanics techniques and assumes that all three
bodies are unevolving point masses. Whenever the stellar evolution
time of one of the components becomes comparable to any of the orbital
perturbation timescales computed here, the evolution of the triple can
be affected significantly through mass losses, or mass transfer. For a
recent discussion of stellar evolution in triples, see Iben \& Tutukov
(1999), who study the production of Type~Ia supernovae from the mergers
of heavy white dwarfs inside hierarchical triples.
Mikkola \& Tanikawa (1998) have studied the episodic mass transfer triggered
by large eccentricity oscillations of the inner binary in the secular
evolution of the triple system CH Cygni.
Eggleton \& Verbunt (1989) have discussed the possible
relevance of triple star evolution for the formation of low-mass X-ray
binaries.
Our work focuses on triple systems containing
well-separated components, in which the orbital perturbation
timescales are short compared to any tidal dissipation time. For
triple systems containing a close inner binary, tidal dissipation in
the inner components provides a sink of energy and angular momentum
which can change substantially the character of the secular
perturbations. For a recent discussion of tidal dissipation in triple
systems containing a close inner binary, see the paper by Kiseleva,
Eggleton, \& Mikkola (1998). Bailyn \& Grindlay (1987) have discussed
the combined effects of tidal interaction and mass transfer for
compact X-ray binaries in hierarchical triples.
Mazeh \& Shaham (1979) were the first to point out that the combination of
tidal dissipation and secular eccentricity perturbations in triples
could sometimes lead to a substantial orbital shrinking of the
inner binary.
One possible additional perturbation effect that we do take into
account in this work is the general relativistic precession of the
inner orbit if the inner binary contains compact objects. An example
is provided by the PSR B1620$-$26 triple system, in which the inner
binary contains a neutron star and a white dwarf. When the precession
periods from general relativity and from Newtonian perturbations in
the triple become comparable, a type of resonant effect is possible
which leads to increased magnitudes for the orbital perturbations. A
similar resonant effect has been mentioned by S\"oderhjelm (1984) for
triples where the inner binary precesses under the influence of a
rotationally-induced quadrupole moment in one of the stars.
Our paper is organized as follows. In \S 2 we present a derivation of
the octupole-order secular perturbation equations and we
compare our results to those obtained in the quadrupole approximation
and in classical planetary perturbation theory. In \S 3
the analytic results are compared to direct numerical integrations, and
the effects of varying all relevant parameters are explored.
In \S 4 we discuss the effects of the general relativistic precession of the
inner orbit on the secular evolution of the triple, using
the PSR B1620$-$26 system as an example.
In \S 5 possible applications of our results to
other observed triple systems are briefly discussed.
\section{Analytic Secular Perturbation Theory}
In this section we present a simple analytic treatment of the
long-term, secular evolution of hierarchical triple systems using
time-independent Hamiltonian perturbation theory in which the
small parameter is the ratio of semimajor axes. We discuss essential
aspects of the lowest-order (quadrupole-level) approximation, which
has been widely used to study hierarchical stellar triples. Then we
extend the approximation to the octupole level and compare our results
with results from the quadrupole and other approximations to show that
the octupole-level equations derived here are valid for a far greater
range of parameters.
\subsection{Summary of Previous Work}
Our derivation of the octupole-level secular perturbation equations is
based on classical perturbation methods of celestial mechanics.
Studies of the long-term behavior of the solar system led Lagrange and
Laplace to the creation of the first classical perturbation theory.
Their approach is applicable to a small class of planetary
configurations with parameters similar to those of the solar system.
The lunar problem was successfully attacked in the end of the
last century by Delaunay who was the first to apply the method of
canonical transformations to long-term perturbations. This method
possesses much greater generality and was used to study a broad
spectrum of problems. Brown (1936) was the first to apply canonical
averaging to stellar triples, and he obtained the transformed
quadrupole Hamiltonian. Kozai (1962) made use of the quadrupole
approximation while studying the long-term motion of asteroids and
noted several important properties of this approximation. Harrington
(1968) obtained quadrupole-level expressions similar to Kozai's for
general hierarchical systems of three stars. S\"oderhjelm (1984)
derived octupole-level equations in the limit of low eccentricities
and inclinations. In particular, he demonstrated that the quadrupole
approximation fails in this regime because the octupole term in the
Hamiltonian becomes dominant. Finally, Marchal (1990) averaged the
octupole Hamiltonian keeping all terms up to third order in
$\alpha$ and some terms of order $\alpha^{7/2}$. His Hamiltonian
truncated at third order is identical to the one used in this paper.
In the process of completing this work, we became aware of related
ongoing work by other groups. In particular, Krymolowski \& Mazeh
(1999) have derived octupole-order perturbation equations following
the same method used here. They retain some additional
terms, of order $\alpha^{7/2}$, which were also partly included
in Marchal's (1990) Hamiltonian. Based on a few numerical integrations
that Krymolowski \& Mazeh (1999) provide for
a fairly strongly coupled system ($\alpha=0.1$), it appears that
these higher-order terms have a negligible effect on
the perturbations, although they can lead to slightly shorter
periods of eccentricity oscillations for systems with low relative
inclination. Eggleton (2000) has used a perturbation method based
on the variation of the Runge-Lenz vector (see Heggie \& Rasio 1996)
to derive an extension of Kozai's theory to octupole order.
Similar work has been done by Georgakarakos (2000), who
concentrates on systems where the inner orbit is nearly circular.
\subsection{Octupole Theory}
A hierarchical triple system
consists of a close binary ($m_0$ and $m_1$) and a third
body ($m_2$) moving around the inner binary on a much wider orbit.
To describe this structure it is convenient to
use Jacobi coordinates, which are defined as follows. The vector ${\bf r}_1$
represents the position of $m_1$ relative to $m_0$, and ${\bf r}_2$ is
the position of $m_2$ relative to the center of mass of the inner binary
(See Fig.\ 1).
This coordinate system naturally divides the motion
of the triple into two separate motions, and makes it possible to
write the Hamiltonian as a sum of two terms representing the two
decoupled motions and an infinite series representing the coupling
of the orbits. Let the subscripts 1 and 2 refer to the inner
and outer orbits, respectively. The coupling term is written as a
power series in the ratio of the semi-major axes
$\alpha\equiv{a_1}/{a_2}$, which serves as the small parameter
in our perturbation expansion. The complete Hamiltonian of the
three-body system is given by (Harrington 1968)
\begin{eqnarray}
{\cal F}=\frac{k^2 m_0 m_1}{2 a_1} + \frac{k^2 (m_0+m_1) m_2}{2a_2}+
\frac{k^2}{a_2}\, \sum_{j=2}^\infty \alpha^{j} M_j \left(
\frac{r_1}{a_1} \right)^j \left(\frac{a_2}{r_2}\right)^{j+1}
P_j(\cos \Phi) \label{eq:Ham1},
\end{eqnarray}
where $k^2$ is the gravitational constant, $P_j$'s are the Legendre
polynomials, $\Phi$ is the angle between ${\bf r}_1$ and ${\bf r}_2$, and
\begin{eqnarray}
M_j = m_0m_1m_2\frac{m_0^{j-1}-(-m_1)^{j-1}}{(m_0+m_1)^j} .
\end{eqnarray}
We shall deal with the expansion only up to third order in $\alpha$.
\begin{figure}
\plotone{fig1.ps}
\caption{Diagram illustrating the coordinate system used to describe the
hierarchical triple system. \label{fig:coord}}
\end{figure}
Let us define a set of canonical
variables, known as Delaunay's elements, that provide a particularly
convenient dynamical description of our three-body system. The angle
variables are chosen to be
\begin{eqnarray}
l_1,l_2 &=& \rm {mean \ anomalies} \\ g_1,g_2 &=& \rm {arguments \
of \ periastron} \\ h_1,h_2 &=& \rm{longitudes \ of \ ascending \
nodes} \label{eq:def1}
\end{eqnarray}
and their conjugate momenta
\begin{eqnarray}
L_1 = \frac{m_0m_1}{m_0+m_1}\sqrt{k^2(m_0+m_1)a_1} \qquad
L_2=\frac{m_2(m_0+m_1)}{m_0+m_1+m_2}\sqrt{k^2(m_0+m_1+m_2)a_2},
\end{eqnarray}
\begin{eqnarray}
G_1 = L_1 \sqrt{1-e_1^2} \qquad G_2=L_2 \sqrt{1-e_2^2},
\end{eqnarray}
\begin{eqnarray}
H_1 = G_1 \cos i_1 \qquad H_2=G_2 \cos i_2, \label{eq:def2}
\end{eqnarray}
where $e_1$, $e_2$ are the orbital eccentricities and $i_1$, $i_2$ are the
orbital inclinations.
The usual canonical relations represent the equations of motion:
\begin{eqnarray}
\frac{dL_j}{dt}=\frac{\partial {\cal F}}{\partial l_j} \qquad
\frac{dl_j}{dt}=-\frac{\partial {\cal F}}{\partial L_j}, \label{eq:canon1a}
\end{eqnarray}
\begin{eqnarray}
\frac{dG_j}{dt}=\frac{\partial {\cal F}}{\partial g_j} \qquad
\frac{dg_j}{dt}=-\frac{\partial {\cal F}}{\partial G_j} ,
\end{eqnarray}
\begin{eqnarray}
\frac{dH_j}{dt}=\frac{\partial {\cal F}}{\partial h_j} \qquad
\frac{dh_j}{dt}=-\frac{\partial {\cal F}}{\partial H_j}, \label{eq:canon1b}
\end{eqnarray}
where $j=1,2$.
Note that $H_2$ is the $z$-component of the angular momentum contributed by
the perturbing body, the $z$-axis being the direction of the total angular
momentum, perpendicular to the invariable plane of the system (See Fig.\ 2).
Equations~(\ref{eq:canon1a})--(\ref{eq:canon1b}) appear to
have six degrees of freedom, but they can be reduced to four by the theorem
of elimination of nodes (Jeffrys \& Moser 1966). The Hamiltonian contains
$h_1$ and
$h_2$ only in the combination $h_1-h_2$, and it is symmetric with respect to
the orientation of the line of nodes when the invariable plane is chosen
as a reference plane. In other words, the Hamiltonian is symmetric with
respect to rotations about the total angular momentum vector $\bf H$.
Thus, $H_1$ and $H_2$ enter the Hamiltonian only as $H_1+H_2=H$ and can
be eliminated from the Hamiltonian using the relations
\begin{eqnarray}
H_1=\frac{H^2+G_1^2-G_2^2}{2H} \\
H_2=\frac{H^2+G_2^2-G_1^2}{2H}
\end{eqnarray}
\begin{figure}
\plotone{fig2.ps}
\caption{Diagram illustrating the relationships between the canonical
variables and angular momenta. \label{fig:vectors} }
\end{figure}
Using the new canonical elements we can write the first four terms of
the Hamiltonian~(\ref{eq:Ham1}) as
\begin{eqnarray}
{\cal F}_{oct} &=& {\cal F}_0 +{\cal F}_1 +{\cal F}_2 +{\cal F}_3 \\ \label{eq:Ham2}
&=&\frac{\beta_0}{2L_1^2}+\frac{\beta_1}{2L_2^2}+8\beta_2
\left(\frac{L_1^4}{L_2^6}\right) \left(\frac{r_1}{a_1}\right)^2
\left(\frac{a_2}{r_2}\right)^3(3\cos^2\Phi-1) \\
& & + 2 \beta_3
\left(\frac{L_1^6}{L_2^8}\right) \left(\frac{r_1}{a_1}\right)^3
\left(\frac{a_2}{r_2}\right)^4(5\cos^3\Phi-3\cos\Phi), \nonumber
\end{eqnarray}
where the mass parameters are
\begin{eqnarray}
\beta_0 &=& k^4 \frac{(m_0m_1)^3}{m_0+m_1}, \\
\beta_1 &=& k^4 \frac{(m_0+m_1)^3m_2^3}{m_0+m_1+m_2}, \\
\beta_2 &=& \frac{k^4}{16}\frac{(m_0+m_1)^7}{(m_0+m_1+m_2)^3}
\frac{m_2^7} {(m_0m_1)^3},\\
\beta_3 &=& \frac{k^4}{4}\frac{(m_0+m_1)^9}{(m_0+m_1+m_2)^4}
\frac{m_2^9(m_0-m_1)}{(m_0m_1)^5}.
\end{eqnarray}
Each term in the series is labeled according to the degree of the Legendre
polynomial associated with it (same as the power of $r_1/r_2$). Following the
standard
nomenclature associated with multipole expansions we shall call the
Hamiltonian containing the first three terms (up to $j=2$) ``quadrupole'' and
the third-order Hamiltonian~(\ref{eq:Ham2}) ``octupole.''
The first two terms in expression~(\ref{eq:Ham2}) describe the unperturbed motion
of the inner and outer binaries, and the higher-order terms describe the coupling.
The quadrupole
Hamiltonian contains the perturbation of order $\alpha^2$, the octupole Hamiltonian
extends this to order $\alpha^3$.
The complete Hamiltonian~(\ref{eq:Ham1}) contains the full description
of the system. However, we are going to restrict our study to the
long-term, secular behavior of the system, by averaging over short-period
effects.
Even though equation~(\ref{eq:Ham2}) is already an approximation of the
full Hamiltonian, it contains information about short-period perturbations
that needs to be eliminated. In particular the angle
$\Phi$ depends on the mean anomalies. Further simplification is achieved
through a canonical transformation of variables,
called the von Zeipel transformation.
Its essence is to replace the Delaunay elements with a set of new canonical
coordinates and momenta that rid the Hamiltonian of the dependence
on $l_1$ and $l_2$. The perturbed action variables are still periodic
functions of the perturbed angle variables, but the former are no
longer linear functions of time. The goal is to find such a set of
action-angle variables that the perturbed Hamiltonian will be a
function only of the action variables.
In the end we can think of the Hamiltonian as describing the
interaction between two weighted elliptical rings instead of point masses
in orbits.
It is important to note that we do not
simply average the Hamiltonian with respect to these variables,
since this would destroy the canonical structure of the equations
of motion, but instead proceed in a more cautious and intricate manner.
We start by requiring that the new Hamiltonian be equal to the old one,
since changing variables does not change the energy, and
expanding both sides of the equality as Taylor series in $\alpha$.
Then we go order by order to identify the terms in the transformed
Hamiltonian, using the result of the previous order calculation in each step.
The theory behind the Von Zeipel method is very well presented and
illustrated by Goldstein (1980, section 11-5) and Hagihara (1972).
Additionally, Harrington (1968, 1969)
has applied this method to the quadrupole Hamiltonian. We
followed exactly the same prescription but all the way to third order.
Here we present only the results of the von Zeipel averaging
procedure, omitting the laborious algebraic details.
Let us define the following convenient quantities
\begin{eqnarray}
\theta=\cos i = \frac{H^2-G_1^2-G_2^2}{2 G_1 G_2}, \label{eq:cosi}
\end{eqnarray}
where $\bf H= \bf G_1 + \bf G_2$ and $H=|\bf H|$ is given by
initial conditions and $i=i_1-i_2$ is the mutual inclination.
The angle $\varphi$ between the directions of periastron is given by
\begin{eqnarray}
\cos \varphi= -\cos g_1 \cos g_2-\theta \sin g_1 \sin g_2.
\end{eqnarray}
The doubly-averaged Hamiltonian is given by
\begin{eqnarray}
\bar {\cal F}_{oct} &=& C_2 \cdot \{(2+3e_1^2)(3\theta^2-1)+
15e_1^2(1-\theta^2)\cos 2g_1 \} \nonumber\\
& & +C_3 \cdot e_1 e_2 \Big\{A \cos \varphi + 10\theta(1-\theta^2)(1-e_1^2)
\sin g_1 \sin g_2 \Big\}, \label{eq:Ham3}
\end{eqnarray}
where
\begin{eqnarray}
C_2=\frac{k^4}{16}\frac{(m_0+m_1)^7}{(m_0+m_1+m_2)^3}
\frac{m_2^7}{(m_0m_1)^3}
\frac{L_1^4}{L_2^3G_2^3}, \label{eq:c2}
\end{eqnarray}
\begin{eqnarray}
C_3= \frac{15}{16}\frac{k^4}{4}\frac{(m_0+m_1)^9}{(m_0+m_1+m_2)^4}
\frac{m_2^9(m_0-m_1)}{(m_0m_1)^5}
\frac{L_1^6}{L_2^3G_2^5}, \label{eq:c3}
\end{eqnarray}
\begin{eqnarray} B=2+5e_1^2-7e_1^2\cos 2g_1,\end{eqnarray}
\begin{eqnarray} A=4+3e_1^2-\frac{5}{2}(1-\theta^2)B.\end{eqnarray}
Note that the Hamiltonian~(\ref{eq:Ham3}) does not contain any
dependence on $l_1$ or $l_2$ because these variables have been
integrated out as a result of the canonical transformation.
The actual differences between the
original and the transformed variables are small (of order $\alpha$ or
smaller) and periodic. Variables
appearing in equation~(\ref{eq:Ham3}) are only approximations of the old
variables defined in equations~(\ref{eq:def1})--(\ref{eq:def2}),
and they can be thought of
as the averages of the old variables. This Hamiltonian is equivalent to
the Hamiltonian given by Marchal (1990; see his eqs.~[252]--[255]).
The absence of $l_1$ and $l_2$ from the transformed Hamiltonian implies
that $L_1$ and $L_2$ are constants of the motion, which in turns implies that,
in our approximation, the transformed semi-major axes $a_1$ and $a_2$ are
constant. Thus, the only
secularly changing parameters in this model are $e_1$, $e_2$, $g_1$,
$g_2$, and $i$, which is coupled to $e_1$ and $e_2$ by
relation~(\ref{eq:cosi}).
The equations of motion are derived from the Hamiltonian~(\ref{eq:Ham3})
using the canonical relations
\begin{eqnarray} \frac{de_i}{dt}=\frac{\partial e_i}{\partial G_i}
\frac{\partial \bar {\cal F}_{oct}}{\partial g_i}, \label{eq:canon2}
\end{eqnarray}
and
\begin{eqnarray}
\frac{dg_i}{dt}=-\frac{\partial \bar {\cal F}_{oct}}{\partial G_i}.
\end{eqnarray}
After regrouping terms, the octupole-level secular perturbation equations
follow:
\begin{eqnarray}
\frac{dg_1}{dt} &=& C_2 \cdot 6 \left\{\frac{1}{G_1}\left[4 \theta^2+(5
\cos 2g_1
-1)(1-e_1^2-\theta^2)\right]+ \frac{\theta}{G_2} \left[2+e_1^2(3-5
\cos2g_1)\right]
\right\} \nonumber \\ & & -C_3\cdot e_2 \Bigg\{e_1\left(
\frac{1}{G_2}+ \frac{\theta}{G_1}\right) \Big[\sin g_1 \sin g_2 \left\{
A+10(3\theta^2-1)(1-e_1^2)\right\}
-5\theta B \cos \varphi \Big] \nonumber \\& &
- \frac{1-e_1^2}{e_1G_1}\Big[\sin g_1 \sin g_2 \cdot 10 \theta
(1-\theta^2)(1-3e_1^2) + \cos \varphi (3A-10\theta^2+2)
\Big]\Bigg\}, \label{eq:oct1a}
\end{eqnarray}
\begin{eqnarray}
\frac{de_1}{dt} &=& C_2 \cdot \frac{1-e_1^2}{G_1} \Big\{
30e_1 (1-\theta^2)\sin 2g_1\Big\} + C_3 \cdot e_2\frac{1-e_1^2}{G_1}
\Big\{35 \cos \varphi (1-\theta^2)e_1^2\sin 2g_1 \nonumber \\
& & - 10\theta(1-e_1^2)(1-\theta^2) \cos g_1 \sin g_2 -
A(\sin g_1 \cos g_2-\theta \cos g_1 \sin g_2)\Big\},
\end{eqnarray}
\begin{eqnarray}
\frac{dg_2}{dt} &=& C_2 \cdot 3\left\{
\frac{2\theta}{G_1}\left[2+e_1^2(3-5\cos 2g_1)
\right] + \frac{1}{G_2} \left[4+6e_1^2+
(5\theta^2-3)(2+e_1^2(3-5\cos 2g_1))\right]\right\} \nonumber \\
& & + C_3\cdot e_1 \Bigg\{\sin g_1 \sin g_2 \left[\frac{4e_2^2+1}{e_2G_2}
10 \theta (1-\theta^2)(1-e_1^2)- e_2 \left(\frac{1}{G_1}+
\frac{\theta}{G_2}\right)(A+10(3\theta^2-1)(1-e_1^2))\right] \nonumber \\
& & +\cos \varphi \left[ 5B\theta e_2\left(\frac{1}{G_1}+
\frac{\theta}{G_2}\right)+\frac{4e_2^2+1}
{e_2G_2}A \right]\Bigg\},
\end{eqnarray}
\begin{eqnarray}
\frac{de_2}{dt} &=& -C_3 \cdot e_1 \frac{1-e_2^2}{G_2} \Big\{
10\theta(1-\theta^2)(1-e_1^2)\sin g_1 \cos g_2 + A(\cos g_1 \sin g_2-
\theta \sin g_1 \cos g_2) \Big\}. \label{eq:oct1b}
\end{eqnarray}
This system of coupled nonlinear differential equations describes
the octupole-level behavior of hierarchical triples and can be easily
integrated numerically to determine the secular evolution of a
hierarchical triple for any initial configuration. We found that for
calculations with small eccentricities it is better to use the
transformed set of variables $e_1 \sin g_1$, $e_1 \cos g_1$, $e_2 \sin
g_2$, and $e_2 \cos g_2$. These equations can be integrated
numerically much more rapidly than the full equations of motion.
\subsection{Comparison with Quadrupole-Level Results}
The quadrupole results of Kozai (1962) and Harrington (1969) have
found numerous applications in recent studies of planetary, pulsar,
and stellar systems. Mazeh \& Shaham (1979) calculated the
quadrupole-level, long-term periodic behavior of triples and used
their results to study high-amplitude eccentricity modulations of
systems with large initial inclination. Holman et al.\ (1997) used
the quadrupole-level theory of Kozai to analyze the dynamical
evolution of the planet in the binary star 16 Cygni. Rasio, Ford, \&
Kozinsky (1997) used the same theory to study long-term eccentricity
perturbations in the PSR B1620$-$26 pulsar system.
The quadrupole Hamiltonian can be obtained from expression~(\ref{eq:Ham3}) by
dropping the $C_3$ term:
\begin{eqnarray}
{\cal F}_q=C_2 \cdot \{(2+3e_1^2)(3\theta^2-1)+
15e_1^2(1-\theta^2)\cos 2g_1 \}.
\end{eqnarray}
Note that ${\cal F}_q$ is independent of $g_2$, meaning that, in the quadrupole
approximation, $G_2$ is a constant of the motion, and so is $e_2$ by
equation~(\ref{eq:canon2}). Therefore, to quadrupole order in secular
perturbation theory, there is no variation in the eccentricity of the
outer orbit. This is a well-known result (see, e.g., Marchal 1990,
Sec.~10.2.3). Now there is only one degree of freedom left, so the
evolution of the inner eccentricity is described by
\begin{eqnarray}
\frac{dg_1}{dt} &=& C_2 \cdot 6 \left\{\frac{1}{G_1}\left[4 \theta^2+(5
\cos 2g_1
-1)(1-e_1^2-\theta^2)\right]+ \frac{\theta}{G_2} \left[2+e_1^2(3-5
\cos2g_1)\right]
\right\}, \label{eq:quad1a}
\end{eqnarray}
with
\begin{eqnarray}
\frac{de_1}{dt} &=& C_2 \cdot \frac{1-e_1^2}{G_1} \Big\{
30e_1 (1-\theta^2)\sin 2g_1\Big\}. \label{eq:quad1b}
\end{eqnarray}
The advantage of using the quadrupole approximation is that it can
describe the secular behavior of systems with high relative
inclination and a wide range of initial eccentricities, regimes not
covered by the classical planetary perturbation theory. As in the
classical theory, the quadrupole-level perturbation equations
(\ref{eq:quad1a})--(\ref{eq:quad1b}) can be solved exactly for the
period and amplitude of oscillation (Kozai 1962). The period of
eccentricity oscillations is given approximately by
\begin{eqnarray}
P_e \simeq P_{1} \left(\frac{m_0+m_1}{m_2}\right)
\left(\frac{a_2}{a_1}\right)^3 \left( 1- e_2^2 \right)^{3/2},
\end{eqnarray}
where $P_1$ is the orbital period of the inner binary
(Mazeh \& Shaham 1978).
This expression should be multiplied by a coefficient of order unity
which can be obtained using
Weierstrass's zeta function as shown by Kozai (1962).
The secular evolution can be visualized with the help of phase-space
diagrams of $e_1$ vs $\cos g_1$. An example is provided in Figure~3.
Each contour corresponds to an initial condition with a certain value
of the total angular momentum $H$. Since $G_2$ is fixed, $e_1$ is
coupled to $i$ through equation~(\ref{eq:cosi}), so the relative
inclination oscillates with the same period as $e_1$. The up-down
symmetry forces similar behavior in $g_1$-$e_1$ phase space for both
$g_1 \in [-\pi,0]$ and $g_1 \in [0,\pi]$. One obvious feature is the
existence of two regimes: libration and circulation. The libration
island generally appears when the initial inclination is greater than
some critical value, which for most systems is around
$i_{\rm crit}\simeq 40^{\circ}$.
Kozai (1962) calculated that for $\alpha \le 0.10$,
$\,38.960^{\circ} \le i_{\rm crit} \le 39.231^{\circ}$. From the shape
of the large libration island we see that $e_1$ can grow from a very
small initial value to a very large maximum. Holman et al.\ (1997)
approximate the maximum inner eccentricity as
\begin{eqnarray}
e_{1, {\rm max}} \simeq \left(1-(5/3) \cos^2 i_o\right)^{1/2},
\end{eqnarray}
where $i_o$ is the initial relative inclination.
Libration can occur for low inclinations as well, but this does
not lead to large eccentricity oscillations.
Several erroneous features of the quadrupole approximation are worth
noting. For an initial condition with $e_1=0$, no evolution occurs at
all. This is especially significant in a case where there is no
libration island, since in that case the eccentricity perturbation
would appear to
approach zero continuously as the initial eccentricity is decreased.
Similarly, in the coplanar case ($\theta=1$), the theory predicts no
evolution of eccentricity. From the classical planetary
perturbation theory (\S 2.4), which assumes low eccentricities and
inclinations, we know that this is not correct. These features also
contradict the octupole-level results (\S 2.2), as well as the results
of direct numerical integrations (\S 3.2). We conclude that the
quadrupole approximation fails for low inclination and for low inner
eccentricity. However, it remains qualitatively applicable in the
high-inclination regime.
\begin{figure}
\vspace{6.5in}
\special{psfile="fig3.ps" voffset=0 hoffset=0 vscale=70 hscale=70 angle=0}
\vspace{-1in}
\caption{Phase space trajectories obtained in the quadrupole
approximation for a system with $m_2/m_1 = 10^{-3}$, $m_3/m_1 =
1$, $\alpha^{-1} = 100$, $e_2 = 0.9$, and initial values of $e_1$ ranging
from $0.02$ to $0.9$. The libration contours were obtained by setting
the initial value of $g_1=90^{\circ}$.\label{fig:phase1}}
\end{figure}
\begin{figure}
\vspace{6.5in}
\special{psfile="fig4.ps" voffset=0 hoffset=0 vscale=70 hscale=70 angle=0}
\vspace{-1in}
\caption{Phase-space trajectory of the same system as in Figure~3 with initial
$e_1=0.02$, but with the octupole terms included in the integration.
(Note that this corresponds to a single trajectory in Fig.~3.)
\label{fig:phase2}}
\end{figure}
The octupole-level theory has more degrees of freedom and covers most
regimes of hierarchical triple configurations. The perturbation
equations~(\ref{eq:oct1a})--(\ref{eq:oct1b}) indicate that there are no
additional conserved quantities apart from the obvious ones (total
angular momentum and total energy). In contrast to the quadrupole theory, the
quantities $e_2$ and $g_2$ now vary with time and the behavior is much
harder to visualize. We can notice striking qualitative
differences between the two theories by looking at phase-space
diagrams. An example is provided in Figure~\ref{fig:phase2}. Trajectories
are no longer closed, and transitions between libration and
circulation occur, since the angular momentum of the outer orbit now
evolves with time. Thus, we now have more
than one frequency in the secular oscillations.
Figure~\ref{fig:bad}
demonstrates that a system can have a qualitatively different behavior
from what is expected in the quadrupole approximation. While the
quadrupole theory predicts periodic variations of constant amplitude,
according to the octupole equations (and in agreement with the results of
a direct numerical integration) the
amplitude grows very close to unity. This leads to a very small periastron
distance and the possibility of a tidal interaction or collision between
the two inner stars.
Thus, one must exercise great caution when modeling systems using
the quadrupole approximation. Ignoring octupole-level terms can
lead to completely invalid results.
\begin{figure}
\plotone{fig5.ps}
\caption{These results illustrate the potential danger of
using the quadrupole approximation.
The inner eccentricity is shown as a function of time for
a system with $m_1/m_0 = 10^{-3}$, $m_2/m_1 = 1$,
$\alpha^{-1} = 100$, initial eccentricities $e_1 = 0.05$
and $e_2 = 0.9$, and an initial relative inclination $i=70^{\circ}$.
Time is given in years assuming $a_1=1\,$AU and $m_0=1\,M_\odot$.
The solid line
is from the integration of the octupole-level perturbation equations, while
the dashed line is from a direct numerical integration of the three-body
system (see \S 3.1). In the quadrupole approximation all oscillations would
have the same amplitude as the first shown here. Notice how the eccentricity
oscillations in fact increase in amplitude, making the inner
periastron separation quite small. At some point other effects such as
general relativistic precession and tidal dissipation (not to mention
a collision between the two inner stars) could become significant.
\label{fig:bad} }
\end{figure}
\subsection{Comparison with Classical Planetary Perturbation Theory}
For application to many problems in the context of the Solar System, a
classical perturbation theory was developed many years ago that
applies to low-eccentricity, low-inclination orbits of planets around
a central star (one dominant mass). This theory does {\it not\/}
assume that the ratio of semimajor axes is small, and therefore it
provides results valid to all orders in $\alpha$. A detailed account
of the planetary theory can be found in Brouwer \& Clemence (1961,
Chap.\ 16). For an excellent pedagogic summary, see Dermott \&
Nicholson (1986). Rasio (1994, 1995) used the classical theory to
study the eccentricity perturbations in the PSR B1620-26 triple system
in the limit of coplanar orbits, and derived simple approximate
expressions for the period and amplitude of eccentricity oscillations
in various limits.
Here we will only consider the variations of the
eccentricities, since the results for inclinations are very
similar.
It turns out that the inclination evolution is decoupled from the
eccentricity evolution, and so the two can be solved separately.
Because eccentricities are very small and can vanish, it is
better to use the variables
\begin{eqnarray} h_1=e_1 \sin g_1 \qquad h_2=e_2 \sin g_2 \label{eq:h} \end{eqnarray}
\begin{eqnarray} k_1=e_1 \cos g_1 \qquad k_2=e_2 \cos g_2. \label{eq:k} \end{eqnarray}
Since angular
momentum is conserved and the mutual inclination stays
constant to first order, the two eccentricities vary $90^{\circ}$ out
of phase. The linear
system of equations describing the secular evolution of eccentricities
in planetary perturbation theory is
\begin{eqnarray} \frac{dh_1}{dt}=+A_{11}k_1-A_{12}k_2 \end{eqnarray}
\begin{eqnarray} \frac{dk_1}{dt}=-A_{11}h_1+A_{12}h_2 \end{eqnarray}
\begin{eqnarray} \frac{dh_2}{dt}=-A_{21}k_1+A_{22}k_2 \end{eqnarray}
\begin{eqnarray} \frac{dk_2}{dt}=+A_{21}h_1-A_{22}h_2, \end{eqnarray}
where the $A$'s are defined in terms of Laplace coefficients, which we
truncate at third order in $\alpha$ to obtain
\begin{eqnarray} A_{11}=\frac{3}{4}k\frac{\tilde{a}_1^{3/2}}{\tilde{a}_2^3}\frac{m_2}{\sqrt{m_0+m_1}},
\qquad
A_{12}=\frac{15}{16}k\frac{\tilde{a}_1^{5/2}}{\tilde{a}_2^4}\frac{m_2}{\sqrt{m_0+m_1}}, \end{eqnarray}
\begin{eqnarray} A_{21}=\frac{15}{16}k\frac{\tilde{a}_1^3}{\tilde{a}_2^{9/2}}\frac{m_1}{\sqrt{m_0+m_2}},
\qquad
A_{22}=\frac{3}{4}k\frac{\tilde{a}_1^2}{\tilde{a}_2^{7/2}}\frac{m_1}{\sqrt{m_0+m_2}}, \end{eqnarray}
where $\tilde{a}_1$ and $\tilde{a}_2$ are the averaged semimajor axes measured
from $m_0$ to $m_1$ and $m_2$, respectively.
Now we rewrite the system in terms of familiar quantities to find
\begin{eqnarray} \frac{de_1}{dt}=A_{12}e_2\sin g, \label{eq:de1dtcl} \end{eqnarray}
\begin{eqnarray} \frac{de_2}{dt}=-A_{21}e_1\sin g, \end{eqnarray}
\begin{eqnarray} \frac{dg}{dt}=A_{22}-A_{11}+A_{12}\frac{e_2}{e_1}\cos g-
A_{21}\frac{e_1}{e_2} \cos g, \end{eqnarray}
where $g=g_2-g_1$. Indeed the use of this variable is convenient, since for
coplanar orbits there is no well-defined line of nodes, and only the
relative longitudes of perihelia are important.
Upon expanding the octupole equations~(\ref{eq:oct1a})--(\ref{eq:oct1b})
to first order
in $e_1$ and $e_2$ we obtain an identical linear system
of differential equations, but with the $A$'s replaced by
\begin{eqnarray}
B_{11}&=&\frac{12C_2}{L_1}= \frac{3}{4}k\frac{a_1^{3/2}}{a_2^3}
\frac{m_2}{\sqrt{m_0+m_1}}, \\
B_{12}&=&\frac{4C_3}{L1} \ = \frac{15}{16}k\frac{a_1^{5/2}}{a_2^4}
\frac{m_2(m_0-m_1)}{(m_0+m_1)^{3/2}}, \\
B_{21}&=&\frac{4C_3}{L_2}=\frac{15}{16}k\frac{a_1^3}{a_2^{9/2}}
\frac{m_0m_1(m_0-m_1)\sqrt{m_0+m_1+m_2}}{(m_0+m_2)^3}, \\
B_{22}&=&\frac{12C_2}{L_2}=\frac{3}{4}k\frac{a_1^2}{a_2^{7/2}}
\frac{m_0m_1\sqrt{m_0+m_1+m_2}}{(m_0+m_1)^2}.
\end{eqnarray}
It is easy to see that, in the
limit where $m_0 \gg m_1$ and $m_0 \gg m_2$, the two sets of
equations coincide, as they should. This establishes the accuracy
of our analytic results in this limit.
In general, the two sets of equations differ in the dependence of
the coefficients on the masses. Although the two theories use
different coordinate systems (Jacobi vs heliocentric), this alone
does not explain the difference. Instead, one must remember
that the classical theory was derived from Lagrange's planetary
equations (see Brouwer \& Clemence 1961), which assume that the
disturbing functions (proportional to $m_1$ and $m_2$) are
small. Therefore the approximation is valid only if $m_0 \gg m_1,m_2$.
This can also be seen by considering the
$m_0=m_1$ case, for which Heggie \& Rasio (1996, App.~B) proved that
the variation of $e_1$ vanishes to all orders in $\alpha$ if the
initial $e_1=0$. In contrast, the classical planetary perturbation
equation~(\ref{eq:de1dtcl})
would predict a nonzero perturbation of $e_1$ for this case
(since $A_{12}\ne 0$, while our coefficient $B_{12}=0$ for $m_0=m_1$).
Our octupole-level analytic results do not depend on any assumption
about the three masses, as long as the system can be modeled as a
hierarchical triple.
The octupole equations predict constant eccentricities in the case
where $m_0=m_1$ and $i=0$. This happens because the odd-power
coefficients are proportional to $m_0-m_1$ in the Hamiltonian~(\ref{eq:Ham1}),
so the leading terms vanish. There is no reason to
expect the octupole approximation to work for this very particular case.
However, in \S 3.6 we show explicitly by comparison with direct
numerical integrations that the mass-dependences of our
equations~(\ref{eq:oct1a})--(\ref{eq:oct1b}) are valid for
wide ranges of both mass ratios.
\section{Comparison with Direct Numerical Integrations}
We have performed extensive numerical integrations of hierarchical
triple systems using both our octupole-order secular perturbation
equations (hereafter OSPE) and direct three-body integrations. In
this section we present a sample of results that establish the
validity and accuracy of our analytic results, and at the same time
illustrate the dependences of the perturbations on different
parameters.
\subsection{Numerical Methods}
For the numerical integration of the OSPE we change variables from
($e_1$, $g_1$, $e_2$, $g_2$) to ($e_1 \sin g_1$, $e_1 \cos g_1$, $e_2
\sin g_2$, $e_2 \cos g_2$) to remove singularities associated with the
longitude of pericenter for circular orbits. We perform the numerical
integration of the OSPE using the
Burlisch-Stoer integrator of Press et al.\ (1992). Energy and angular
momentum are automatically conserved, since the semimajor axes are
considered constant and the relative inclination of the orbits varies
to conserve angular momentum. We present results obtained using an
accuracy parameter ${\rm EPS}=10^{-8}$. We found that reducing the
integration step sizes did not make a significant numerical difference
for several test systems.
We have compared the results of the OSPE integrations with
direct three-body integrations using a fixed timestep
mixed-variable symplectic (hereafter MVS) integrator (Wisdom \& Holman 1991)
available in the software package SWIFT (Levison \& Duncan 1991). For
most integrations, we used a timestep of $P_1/40$, where $P_1$ is the
orbital period of the inner binary. Energy and angular momentum were
typically conserved to one part in $10^6$ and $10^{12}$, respectively.
For some high-eccentricity systems
we reduced the timestep to a value as small as
$P_1/600$. This MVS integrator was designed for systems in which
$m_0$ is much larger than both $m_1$ and $m_2$. For systems with $m_1/m_0
\mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.1$ we made use of a newer MVS integrator modified to accommodate
arbitrary mass ratios, kindly provided to us by Jack Wisdom and Matt Holman.
In a number of test calculations with this newer integrator, we varied the
timestep systematically to verify that our results are not affected
by numerical errors.
To complement integrations using the MVS integrator, we also used
SWIFT's Burlisch-Stoer (BS) integrator in a few test runs. This integrator
is valid for arbitrarily strong interactions between any pair of bodies, but is
not well suited for very long integrations. We have
only performed a small number of these tests, since the BS
integrations require a much longer computer time. Energy and angular
momentum in BS integrations were typically conserved to one part in
$10^5$. Most BS integrations were stopped after one full oscillation of
$e_1$, which we used to determine the ``maximum eccentricity
perturbation'' of the inner orbit, although, as pointed out in \S 2.3,
the true long-term secular evolution of the eccentricity will not, in
general, be strictly periodic. A typical BS run lasting for $\sim 10^5
P_2$ took about 400 CPU hours on a MIPS R10000 processor, while the
same run using the MVS integrator would only take about 2 CPU hours.
The numerical integrations all started with initial values of the
inner and outer arguments of pericenter of $0^{\circ}$ and
$180^{\circ}$, respectively. This choice leads
to the maximum eccentricity induced in the inner binary in both the
planetary limit (see \S 2.4) and the quadrupole approximation
(see \S 2.3). We have performed additional integrations to
verify that the remaining angles (longitudes of ascending node and
initial anomalies) do not significantly affect the secular evolution
of the system. For large inclinations, the initial argument of pericenter of
the inner
binary is important in determining whether the system will undergo circulation or
libration, if the inner orbit has a significant initial eccentricity (see Fig.~3).
However, if the inner orbit is nearly circular initially, then the initial
values of the angles are of little importance
since the inner orbit can switch from circulation to libration and vice versa.
For coplanar orbits, the magnitude of the angular momentum of the inner
orbit increases when
$g=g_2-g_1 <0$ and decreases when $g>0$. For small, nonzero inclinations this
can still serve as a guide when considering the effect of varying the angles.
\subsection{Eccentricity Oscillations}
\begin{figure}
\plotone{fig6.ps}
\caption[Maximum Eccentricity versus Inclination]
{Maximum eccentricity of the inner orbit after a single oscillation, as
a function of the relative inclination. Here $m_1/m_0 = 10^{-3}$,
$m_2/m_0 = 0.01$, $\alpha^{-1} = 100$, $e_2 = 0.05$, and the initial $e_1 =
10^{-5}$. The squares are from MVS integrations, and the
double dashes on either side are from OSPE integrations with varying
initial longitude of
periastron. The horizontal line indicates the amplitude of the
eccentricity oscillations calculated analytically in the planetary
theory (\S 2.4). The solid curve indicates the amplitude of
eccentricity oscillations calculated analytically according to the
quadrupole-level theory for $i\mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 40^{\circ}$.}
\end{figure}
First, we investigate the dependence of the maximum eccentricity
perturbation of the inner orbit on the initial relative
inclination (see Fig.\ 6). We see that, as expected (Sec.~2.3), for
small inclinations, $i\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 40^{\circ}$, the perturbations are dominated
by the octupole term, while for higher inclinations the quadrupole-level
perturbations dominate. In {\it both\/} regimes the OSPE results match
the direct numerical integrations very well. Near the transition,
numerical integrations (both OSPE and MVS) show a beat-like pattern of
eccentricity oscillations suggesting an interference between the
quadrupole and octupole terms.
Note that the results of Figure~6 are for a system with $m_1 \ll m_0$
and $m_2 \ll m_0$, for which the analytic results from the classical
planetary perturbation theory (Sec.~2.4) can be applied for small
eccentricities and inclinations. We see that the agreement with both MVS
and OSPE integrations is excellent for $i\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 30^{\circ}$.
We now discuss in some more detail the evolution of systems in the
low- and high-inclination regimes.
\subsubsection{Large-inclination Regime}
Figure~7 illustrates the evolution of
$e_1$, $g_1$, $e_2$, $g_2$, and $i$ obtained from a numerical
MVS integration for a typical system with large relative inclination.
For large inclination, the secular quadrupole-level perturbations
dominate the evolution. In the quadrupole approximation the inner
eccentricity undergoes periodic oscillations, while the outer
eccentricity remains constant. Indeed, we see in Figure~7 that
$e_1$ undergoes approximately periodic oscillations of large amplitude
(with corresponding oscillations in $i$),
while $e_2$ remains approximately constant. The small-amplitude
(about 10\%) fluctuations in $e_2$ are due mainly to the smaller,
octupole-level perturbations.
Deviations from strict periodicity in the variation of $e_1$ and $i$
are also caused by octupole-level perturbations.
The period of a quadrupole eccentricity
oscillation is a function of the mass ratios and the outer
eccentricity (see \S 2.3). Our numerical integrations of
the OSPE reveal that the most significant corrections to this period
come from the variable time spent at low eccentricities.
Equation~(\ref{eq:quad1b}) implies that the time derivative of $e_1$
is small when the inner orbit has a small eccentricity. Then the
octupole (and higher-order) perturbations can become important,
causing significant variation in the time a system will spend with a
small inner eccentricity. This effect can be seen clearly in Figure~7.
The OSPE do not correctly describe the evolution of a
systems starting with $e_1=0$. However for any system with
arbitrarily small but nonzero $e_1$, the inner orbit can switch back
and forth between libration and circulation in the $e_1$-$g_1$ plane,
achieving the full range of eccentricities. In our MVS integrations
the short-period variations (averaged out in secular perturbation theory)
provide the necessary perturbations to allow
for the full eccentricity oscillations, even if the initial $e_1=0$.
\begin{figure}
\plotone{fig7.ps}
\caption[Secular Evolution of a System with Large Relative Inclination]
{Typical evolution of the eccentricities, longitudes of periastron, and
relative inclination for a
system in the high-inclination regime. Here $m_1 / m_0 = 10^{-3}$,
$m_2/m_0 = 0.01$, $\alpha^{-1} = 100$, the initial inclination
$i=60^{\circ}$, and the initial eccentricities $e_1 = 10^{-5}$
and $e_2 = 0.05$.
Time is given in years assuming $a_1=1\,$AU and $m_0=1\,M_\odot$.
These results were obtained using numerical MVS integrations.}
\end{figure}
\subsubsection{Small-inclination Regime}
For small inclinations ($i\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 40^{\circ}$), the secular octupole-level
perturbations dominate and both $e_1$ and $e_2$ typically undergo
very small-amplitude fluctuations, as does the relative inclination
(Fig.~8). In the octupole approximation angular
momentum is periodically transferred from one orbit to the other. In
this regime, the special case of initially circular orbits is stable
to eccentricity oscillations. Short-period perturbations will still
cause small fluctuations in both eccentricities, but since the
inclination does not undergo large oscillations, the angular momentum
transferred from one orbit to another is limited by angular momentum
conservation, preventing large eccentricities from developing in
either orbit.
\begin{figure}
\plotone{fig8.ps}
\caption[Secular Evolution of a System with Small Relative Inclination]
{Typical evolution of a system in the low-inclination regime.
All parameters are as in Fig.~7, except that the initial inclination
$i=15^{\circ}$. These results were obtained using numerical MVS
integrations.}
\end{figure}
For inclinations approaching $\sim 40^{\circ}$, the octupole-level
interaction still leads to a
noticeable amplitude oscillation superimposed onto
the quadrupole-level result. For a small range of inclinations the
two eccentricity oscillations can become comparable leading to
a secular evolution with a period and amplitude larger than either
of the two oscillations in isolation.
\subsection{Dependence on the Ratio of Semimajor Axes}
Numerical results illustrating the dependence of the maximum
eccentricity perturbation of the inner orbit on the ratio of semimajor
axes are shown in Figure~9. Some small deviations between the OSPE and MVS
results appear for small $a_2/a_1$, where higher-order secular
perturbations may be significant.
As predicted by the quadrupole-level approximation, the
amplitude of the eccentricity oscillations becomes independent of
$\alpha$ for high inclinations.
\begin{figure}
\plotone{fig9.ps}
\caption[Maximum Eccentricity versus the Ratio of Semi-major Axes]
{Maximum eccentricity $e_1$ as a function of the ratio of semimajor
axes $\alpha^{-1}=a_2/a_1$. The integrations are for a system with $m_1/m_0 =
10^{-3}$, $m_2/m_0 = 0.01$, and initial eccentricities
$e_1=10^{-5}$ and $e_2 = 0.05$. The various symbols (lines) are from MVS (OSPE)
integrations of systems with various relative inclinations:
$0^{\circ}$ (open triangles, solid line), $15^{\circ}$ (open
squares, dotted line -- here quasi-indistinguishable from the solid
line, cf.\ subsequent figures), $30^{\circ}$ (open circles, short-dashed
line), $45^{\circ}$ (solid triangles, long-dashed line), $60^{\circ}$
(solid squares, short-dash-dotted line), $75^{\circ}$ (solid circles,
long-dash-dotted line), and $89^{\circ}$ (stars, short-dash-long-dashed
line). }
\end{figure}
\subsection{Dependence on the Initial Eccentricity}
Figure 10 shows the dependence of the maximum inner eccentricity on
its initial value. For low inclinations increasing the inner eccentricity
nearly adds to the
maximum induced eccentricity. In the high-inclination regime
increasing the initial inner eccentricity does not affect the maximum
inner eccentricity significantly until the two become comparable.
\begin{figure}
\plotone{fig10.ps}
\caption[Maximum Eccentricity versus the Inner Eccentricity]
{Maximum change in $e_1$ as a function of its initial value.
These integrations are for $m_1/m_0 = 10^{-3}$, $m_2/m_0
= 0.01$, $e_2 = 0.05$, and $\alpha^{-1}=100$. The symbols and lines
are as in Fig.~9.}
\end{figure}
\subsection{Dependence on the Outer Eccentricity}
Figure~11 shows the effect of varying the outer eccentricity. The
OSPE and MVS integrations agree precisely for moderate eccentricities,
but show discrepancies for both very large and very small $e_2$.
For low $i$ and $e_2 \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10^{-2}$, short-period eccentricity variations
become important. These are not included in the OSPE since they were
averaged out of the Hamiltonian. Formally, $i=e_1=e_2=0$ is a fixed point,
since it implies $de_1/dt$=0 in the OSPE. For low inclinations
and eccentricities, the short-period eccentricity oscillations determine
the maximum eccentricity, since this fixed point is stable to the
small-amplitude short-period perturbations. However, for large
relative inclinations, the initial condition $e_1 \simeq e_2\simeq 0$ will lead
to large amplitude oscillations as discussed in \S3.2.1.
Thus, the small-amplitude, short-period oscillations not included in
the OSPE allow the system to explore the full range of allowed
eccentricities.
For large $e_2$, some discrepancies may be
caused by inaccuracies in the MVS integrations: the fixed timestep
implies that periastron passages may not be fully resolved. We
have performed additional MVS integrations with a smaller timestep
(shown in Fig.~11) and a smaller number of BS integrations to verify
that most of the discrepancy is indeed caused by inaccuracies in the
MVS integrator and not the OSPE. However, for sufficiently large
$e_2$, the disagreement remains. As periastron passages begin to resemble
close dynamical encounters, the averaging over orbits becomes
invalid, and the OSPE are no longer applicable. In this limit where
the outer orbit is nearly parabolic, it may be better to treat each
periastron passage as a separate encounter. The results of Heggie \&
Rasio (1996) may be used to calculate analytically the eccentricity
perturbation of the inner binary after each encounter.
\begin{figure}
\plotone{fig11.ps}
\caption[Maximum Eccentricity versus the Outer Eccentricity]
{Maximum $e_1$ as a function of the
initial outer eccentricity, $e_2$, for a system with
$m_1/m_0 = 10^{-3}$, $m_2/m_0
= 0.01$, and initial $e_1=10^{-5}$. For $e_2 < 0.6$ we used $\alpha^{-1}=100$
as in previous figures, while
for $e_2 > 0.6$ we increased the value to $\alpha^{-1} = 300$
(to avoid close interactions with $m_2$). The symbols and line styles are
as in Fig.~9.}
\end{figure}
\subsection{Dependence on the Mass Ratios}
First, we investigate the dependence of the maximum
induced $e_1$ on $m_2$ (Fig.\ 12).
The MVS and OSPE integrations are in excellent agreement,
except for very large $m_2$. For sufficiently large
$m_2$, the binding energy of $m_1$ to $m_2$ becomes comparable to its
binding energy to $m_0$, and the inner orbit deviates significantly from
a Keplerian orbit, making the basic assumption of a hierarchical triple
invalid. As discussed in \S 3, the MVS integrator was not
designed for large $m_2/m_0$. However, we have performed a number of
test integrations, both BS and MVS (with a smaller timestep), and found
that the MVS integrations are generally accurate even for
$m_2 \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} m_0$, provided that the system is stable.
\begin{figure}
\plotone{fig12.ps}
\caption[Maximum Eccentricity versus the Outer Mass]
{Maximum $e_1$ as a function of the mass
of the outer body, $m_2$, for a system with
$m_1/m_0 = 10^{-3}$, $\alpha^{-1} = 100$, and initial eccentricities
$e_1 = 10^{-5}$ and $e_2 = 0.05$. The symbols and line styles are as in Fig.~9.}
\end{figure}
Next, we explore the effects of varying the ratio $m_1/(m_0+m_1)$
(Fig.\ 13). The agreement between MVS and OSPE results is very good, even
when $m_1\simeq m_0$.
In the OSPE, the octupole-level
perturbations vanish when $m_2/m_1 = 1$, removing the dominant term of the
expansion for low inclinations. Therefore we did not expect the OSPE
to properly model the systems with low inclinations.
Using the modified MVS integrator of Wisdom
and Holman (see \S3.1) for $m_1/m_0 \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.1$,
we find surprisingly good agreement between
the OSPE and MVS results for both low and high inclinations. In
particular the OSPE and MVS integrations agree on the maximum
induced eccentricity in the equal-mass case, $m_1= m_0$, which is an
important case for binary stars. Additionally, the OSPE and
MVS results show similar peaks in the maximum induced eccentricity around the
resonance between the quadrupole and octupole terms. Both MVS and limited
BS integrations also indicate that the vanishing of the
induced eccentricity for low-inclination systems when $m_0 = m_1$ is real.
(Unfortunately,
these systems are very time-consuming to integrate numerically with a
BS integrator, prohibiting us from doing a more thorough investigation.)
For example, for the initial
conditions $m_0=m_1=m_2$, $e_1=e_2=0$, $\alpha^{-1}=100$, and $i=0$, we
observed only short-term eccentricity fluctuations of magnitude $\sim
10^{-12}$. Thus, we conclude that the OSPE results are accurate for all values
of the inner mass ratio $m_1/(m_0+m_1)$.
\begin{figure}
\plotone{fig13.ps}
\caption[Maximum Eccentricity versus the Inner Mass]
{Maximum $e_1$ as a function of the mass ratio
of the inner binary, $m_1/(m_0+m_1)$, for a system with
the same parameters as in Fig.~12.
The symbols and line styles are as in Fig.~9.}
\end{figure}
\subsection{Summary and Discussion}
We have performed a large number of numerical integrations (including
many not shown here) to establish the validity of our analytic results
for a broad range of triple configurations.
The only significant difference we observed was in the regime where $e_1\simeq
0$. In that regime the system will chaotically choose circulation or libration
about an island in the $(e_1, g_1)$ phase space. Since $e_1=0$
creates a singularity in the OSPE, we circumvented this problem by
starting runs with $e_1=10^{-5}$. While varying the timestep affected
when $m_1$ chose to librate or circulate, it did not create any
significant difference in the ratio of circulation to libration time.
We conclude that the OSPE provide an accurate description of the
secular evolution of hierarchical triple systems (containing
unevolving point masses and in Newtonian gravity) for nearly all
inclinations, initial eccentricities, and mass ratios. The OSPE may
be used for small $e_1$, provided that $e_1 \ne 0$, since this can be
unstable to large oscillations. When secular perturbations are
sufficiently small, short-period perturbations may provide the larger
contribution to the eccentricity oscillations. The OSPE are not
applicable when $m_0/m_2 < \alpha \equiv a_1/a_2 $, since the inner orbit
is then no longer nearly-Keplerian. The OSPE also break down
whenever $ a_2 (1-e_2) / a_1 (1+e_1) \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 3-5$, since the triple system
is then likely unstable and its evolution will not be dominated by secular effects.
Similarly, the OSPE should not
be applied when $a_1 (1-e_1) \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} R_0$, where $R_0$ is the radius of
the larger of the two inner stars, since the tidal interaction with
that star would then be important. One should also be careful
whenever $e_1\simeq 1$, since a small fractional error in $e_1$ can lead
to a significant change in $r_{p,1}\equiv a_1 (1-e_1)$ which is important
in differentiating purely gravitational interactions from a strongly dissipative
tidal encounter or collision between the inner components.
\section{Resonant Perturbations: The Case of PSR B1620$-$26}
\subsection{Introduction}
Hierarchical triple systems can be affected by many different types of
perturbations acting on secular timescales. In general, during a
given phase in the evolution of a triple, only one type of
perturbation will be important. However, it is possible that, in some
cases, two perturbation mechanisms with different physical origins may
be acting simultaneously and combine in a nontrivial manner. In
particular, whenever two perturbations are acting on comparable
timescales, the possibility exists that they will reinforce each other
in a nonlinear way, leading to a kind of resonant amplification. This
is not to be confused with orbital resonances, which can lead to
nonlinear perturbations of two tightly coupled
Keplerian orbits when the ratio of
orbital periods is close to a ratio of small integers
(see, e.g., Peale 1976)
Perturbation effects coming from the stellar evolution of the
components or from tidal dissipation in the inner binary were
mentioned briefly in \S1 and will not be discussed extensively in this
paper.
Instead, we consider the case where the inner binary contains
compact objects and its orbit is affected by general relativistic
corrections on a timescale comparable to that of the Newtonian secular
perturbations calculated in \S 2. Rather than basing our discussion on
hypothetical cases, we concentrate on the real example provided by the
PSR B1620$-$26 system.
\subsection{The PSR B1620$-$26 Triple System}
PSR B1620$-$26 is a millisecond radio pulsar in a triple system,
located near the core of the globular cluster M4. The inner binary
consists of a $\simeq 1.4\,M_{\odot}$ neutron star with a $\simeq
0.3\,M_{\odot}$ white-dwarf companion in a 191-day orbit with an
eccentricity of $0.025$. The mass and orbital parameters of the third
body are less certain, since the duration of the radio observations
covers only a small fraction of the outer period. However, from the
modeling of the pulse frequency derivatives as well as short-term
orbital perturbation effects it appears that the second companion is
most likely a low-mass object ($m_2\simeq0.01\,M_\odot$) in a wide
orbit of semimajor axis $a_2\simeq50\,$AU (orbital period
$P_2\simeq300\,$yr) and eccentricity $e_2\simeq 0.45$ (Joshi \& Rasio
1997; Ford et al.\ 2000). The eccentricity of the inner binary,
although small, is several orders of magnitude larger than expected
for a binary millisecond pulsar of this type, raising
the possibility that it may have been produced by long-term secular
perturbations in the triple.
An analysis based on the classical planetary theory (i.e., for small
relative inclination ignoring general relativistic precession) shows that
a second companion of stellar mass
would be necessary to induce an eccentricity as large as 0.025 in the
inner binary (Rasio 1994, 1995). Such a large mass for the second
companion has now been ruled out by recent pulsar timing data, and by
the absence of an optical counterpart for the system (Ford et al.\
2000).
It is reasonable to assume
that the relative inclination is large, since the location of
the system near the core of a dense globular cluster suggests that the
triple was formed through a dynamical interaction between binaries
(Rasio, McMillan, \& Hut 1995; Ford et al.\ 2000).
For a sufficiently large relative inclination, we have seen (Fig.~6)
that it should always be possible to induce an arbitrarily
large eccentricity in the inner binary. Therefore, this would seem to
provide a natural explanation for the anomalously high eccentricity of
the binary pulsar in the PSR B1620$-$26 system (Rasio, Ford, \&
Kozinsky 1997). However, there are two additional conditions that
must be satisfied for this explanation to hold.
First, the timescale for reaching a high eccentricity must be shorter
than the lifetime of the triple system. In this case the lifetime of
the triple is determined by the timescale for encounters with passing
stars in the cluster, since any such encounter is likely to disrupt
the orbit of the (very weakly bound) second companion. As discussed in
detail by Ford et al.\ (2000), this is unlikely to be the case in
the high-inclination regime of secular perturbations, given the
parameters of PSR B1620$-$26 and its location near the core of M4
(or inside -- it is seen just inside the edge of the core in projection).
Second, the secular perturbation of the inner binary pulsar by its
distant second companion must be the dominant source of orbital
perturbation. Additional perturbations that alter the longitude of
periastron of the inner binary can indirectly affect the evolution of
its eccentricity. For a binary pulsar, general relativity contributes a significant
orbital perturbation. If the
additional precession of periastron induced by general relativity is
much slower than the precession due to the Newtonian secular
perturbations, then the eccentricity oscillations should not be
significantly affected. However, if the additional precession is
faster than the secular perturbations, then eccentricity oscillations
may be severely damped (Holman et al.\ 1997; Lin et al.\ 1998).
In addition, if the two precession periods are comparable, then a type of
resonance could occur, leading to a significant increase in the
eccentricity perturbation.
\subsection{Secular Evolution of the Eccentricity}
We have used the OSPE to study the secular evolution of the inner
binary eccentricity in the PSR B1620$-$26 system. We integrate the
system using the variables $h_1,\,h_2,\,k_1,\,k_2$ (eqs.\ \ref{eq:h} \&
\ref{eq:k}), which makes it easy to incorporate the first-order post-Newtonian
correction. We restrict our attention to the one-parameter family of
orbital solutions calculated by Ford et al.\ (2000), based on the
modeling of the four pulse frequency derivatives measured by
Thorsett et al.\ (1999). For each solution, the
maximum induced eccentricity of the inner orbit depends only on the
(unknown) relative inclination of the two orbits. In Figure~14, we
show this maximum induced eccentricity as a function of the second
companion's semimajor axis for several inclinations.
\begin{figure}
\plotone{fig14.ps}
\caption[Maximum Eccentricity versus Mass for a System with General
Relativistic Precession]
{The top panel compares several timescales in the PSR B1620$-$26
binary pulsar as a function of the semimajor axis, $a_2$, of its
second companion: $P_{\rm GR}$ is the general relativistic precession
period of the inner binary; $P_{\rm High~i}$ and $P_{\rm Low~i}$ are
the periods of the eccentricity oscillations in the high and low
relative inclination regimes, respectively; $\tau_{\rm c}$ and
$\tau_{\rm hm}$ are the expected lifetimes of the triple in the
core of M4 and at the half-mass radius, respectively. The bottom panel shows
the maximum induced eccentricity of the inner binary for several
different values of the (unknown) relative inclination. The peaks
correspond to a resonance between the general relativistic precession
of the inner orbit and the Newtonian secular perturbation by the
second companion.}
\end{figure}
For most solutions and most values of the inclination, the maximum
induced eccentricity remains significantly smaller than the observed
value of 0.025. However, for low enough inclinations, there is a
narrow range of solutions (around $a_2\simeq 45$)
for which the observed value can be
reached. Most remarkably, these solutions are also the ones currently
preferred if one takes into account preliminary measurements of the
fifth pulse derivative and short-term orbital perturbation effects in
the theoretical modeling (see Ford et al.\ 2000). We also see from
Figure~14 that the maximum induced eccentricity has a peak where the
precession period due to the secular perturbation of the second
companion is comparable to the precession period due to general
relativity, as expected. As the inclination increases, the maximum
induced eccentricity (at the peak) becomes smaller and the peak moves
towards lower values of $a_2$. This pattern continues for
inclinations slightly larger the normal cutoff for the low-inclination
limit ($\simeq 40^{\circ}$). For relative inclinations $50^{\circ}
\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} i \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 70^{\circ}$ we do not find a peak in the maximum induced
eccentricity. For inclinations $\mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 75^{\circ}$ we again find a peak,
which becomes smaller and moves towards larger separations as the
relative inclination of the two orbits is increased.
The maximum inner eccentricity is also limited in this case by the
relatively short lifetime of the triple system in M4, $\sim
10^7-10^9\,$yr depending on whether the system resides inside or
outside the cluster core (Ford et al.\ 2000). For solutions near a
resonance, the inner eccentricity starts growing linearly at nearly
the same rate as it would without general relativistic perturbations.
However the period of the eccentricity oscillations can be many times
the period of the classical eccentricity oscillations. Although this
allows the eccentricity to grow to a larger value, the timescale for
this growth is also longer. For PSR B1620$-$26, Ford et al.\ (2000) show
that, near resonance, the inner binary achieves an eccentricity of
0.025 in a time comparable to the expected lifetime of the triple in
the core of M4. However, the location of the pulsar near the edge of the
core (in projection) suggests that it may in fact reside well
outside the cluster core, where its lifetime can be significantly
longer. In summary, the resonance between general relativistic precession and
Newtonian secular perturbations by the outer companion provides a possible
explanation for the inner binary's eccentricity.
\subsection{Comparison with Direct Numerical Integrations}
We have conducted a few long integrations with our MVS symplectic
integrator for model systems similar to PSR B1620$-$26, in order to
check the validity of the OSPE in the presence of a resonance
(Fig.~15). Although there is good overall agreement, we find that
both the amplitude and the width of the peak is slightly narrower in
the MVS integrations. Note that the values of the masses and
semimajor axes were decreased in order to speed up the direct
integrations and to satisfy the assumptions of the well-tested MVS
integrator provided in SWIFT (i.e., $m_1\ll m_0$ and $m_2\ll m_0$).
This results in smaller values for the ratio of semimajor axes,
implying less accurate results from the OSPE.
Nevertheless, the OSPE integrator performs well, even near a resonance
such as the one
produced by general relativistic precession in a system like
the PSR B1620$-$26 triple.
\begin{figure}
\plotone{fig15.ps}
\caption[GR Resonance Comparison]
{Maximum induced eccentricity $e_1$ as a function of the ratio of
semimajor axes, $\alpha^{-1}=a_2/a_1$. The different symbols show the
results of numerical integrations with and without the general
relativistic term, using both OSPE and MVS integrators: MVS integrations are shown by squares, OSPE integrations are shown by triangles, integrations which include general relativistic precession are indicated by empty symbols, integrations which ignore GR are shown with solid symbols. These results are for a system similar to PSR B1620$-$26, but with the masses and semimajor axes altered to facilitate the numerical integrations: $m_0=1.4\,M_\odot$, $m_1=5\times 10^{-3}\,M_\odot$,
$m_2=8\times 10^{-4}\,M_\odot$, $e_{1,{\rm init}}=10^{-4}$, $e_{2,{\rm
init}}=0.193$, $a_1= 2 \times 10^{-4}\,$AU, and $i=0^{\circ}$.}
\end{figure}
\section{Application to Other Observed Triples}
\subsection{The 16 Cygni Binary and its Planet}
The 16~Cyg system contains two solar-like main-sequence stars in a
wide orbit (separation $\sim 10^3\,$AU) and a low-mass companion
orbiting 16~Cyg~B in a 2.2-yr ($\sim1.7\,$AU) eccentric orbit ($e=0.67$).
The amplitude of the observed radial velocity variations indicate that
the low-mass companion has a mass $m\sin i \simeq 0.6 M_{\rm Jup}$,
suggesting that it is a giant planet (Cochran et al.\ 1997). However
the large orbital eccentricity is surprising for a planet.
Holman et al.\ (1997) and Mazeh et al.\ (1997) pointed out that the
secular perturbation by 16~Cyg~A could explain the large eccentricity
of the planetary orbit for a sufficiently large relative inclination.
In order for the quadrupole perturbations to be effective the precession of
the longitude of periastron must be dominated by the secular perturbations
of 16~Cyg~A. The general relativistic precession period ($\sim 7 \times
10^7\,$yr) can be significant in the eccentricity evolution of the planet
(Holman et al.\ 1997).
Similarly, if additional companions to 16~Cyg~B are found in larger
orbits (like those recently detected around Upsilon Andromedae; see Butler
et al.\ 1999), these would also induce a secular change in the longitude
of pericenter of the presently known planet. Thus, additional companions
could also affect the eccentricity generated by the influence of 16~Cyg~A.
Such an interaction could prevent the quadrupolar secular
perturbation by 16~Cyg~A from accumulating long enough to produce the
observed large eccentricity.
Hauser \& Marcy (1999) have combined radial velocity and astrometric
data to compute a one-parameter family of solutions, which they
tabulate as a function of the line-of-sight component $z$ of the
position vector of B relative to A. There is
also a possibility that 16~Cyg~A may have an M-dwarf companion which
would affect the orbital solutions for the 16~Cyg~AB binary and hence
the secular perturbation timescale (Hauser \& Marcy 1999). We have
used their orbital solutions for 16~Cyg~A (with no M dwarf companion)
to estimate the effects of secular perturbations in this system.
If we assume that the eccentricity of 16~Cyg~Bb is due to quadrupolar secular
perturbations, then both general relativity and any additional
planets around 16~Cyg~B could constrain the orbit of the 16~Cyg~AB
binary. In Figure~16 we compare timescales for eccentricity oscillations
induced by 16~Cyg~A, general relativistic precession,
as well as the eccentricity
oscillations induced by a hypothetical second planet around 16~Cyg~B.
While the period of eccentricity oscillations is shorter than $5\,$Gyr
(the approximate age of 16~Cyg~A\&B; see Ford, Rasio, \& Sills 1998, 2000)
for about $75\%$ of the
orbital solutions listed by Hauser \& Marcy (1999; we actually used
an extended version of their Table~4 kindly provided by H.~Hauser),
the period of the eccentricity oscillations is
shorter than the general relativistic precession period for only
about $25\%$ of their solutions (assuming $\sin i\simeq 1$; with
$\sin i \simeq 0.5$ this fraction increases to about $60\%$).
Given the large mass
ratio of 16~Cyg~A to 16~Cyg~Bb and the high eccentricity of the orbit
($>0.53$), the ratio $C_3/C_2$ (see Eq.~\ref{eq:c2} \& \ref{eq:c3}) can approach
unity. Thus, the octupole term can be very significant in the
dynamics of this system on sufficiently long timescales.
As an example, we use the $z=0$ solution of Hauser \& Marcy (1999) and
assume that the planet was initially on a nearly circular orbit
with an initial relative inclination of $60^{\circ}$. We find that
the period of eccentricity oscillations is then $\sim 3\times 10^7\,$yr ($\sim
4 \times 10^7\,$yr if general relativistic precession is not included;
$\sim 6 \times 10^7\,$yr if neither general relativistic precession nor the
octupole term is included) and the amplitude of the first eccentricity
oscillation is $\simeq 0.685$ ($\simeq 0.767$ without GR;
$\simeq 0.764$ without GR or octupole term). However, there is a longer term
oscillation with a period of $\sim7 \times 10^8\,$yr and an amplitude of
0.766 ($\simeq 0.774$ without GR) which is not present when the octupole term
is ignored. Thus, in this example, the primary effect of general
relativistic precession is to reduce the fraction of the time where the
planet has a very high eccentricity.
\begin{figure}
\plotone{fig16.ps}
\caption{This plot compares the timescales for precession of the
longitude of pericenter of the planet around 16~Cyg~B due to
the secular perturbations by 16~Cyg~A ($P_{\rm A, High-i}$ and
$P_{\rm A, Low-i}$ for high and low relative inclination regimes,
respectively), general relativity ($P_{\rm GR}$), and secular
perturbations by a hypothetical second planet ($P_{\rm Bc}$), assuming
that it is coplanar with 16~ Cyg~Bb and has a mass of $1\,M_{\rm Jup}$.
Note that $P_{\rm A, High-i}$, $P_{\rm A, Low-i}$ and $P_{\rm GR}$
are plotted as a function of the binary semimajor axis, while
$P_{\rm Bc}$ is shown as a function of the orbital radius of the
additional planet. $P_{\rm A, High-i}$ and $P_{\rm A, Low-i}$ were
calculated for the one-parameter family of orbital solutions given
by Hauser \& Marcy (1999), which do not extend below $a\simeq 900\,$AU.
$\tau_{\rm ms}$ indicates the age of 16~Cyg~B.}
\end{figure}
\subsection{The Protobinary System TMR-1}
HST/NICMOS observations of the TMR-1 system by Terebey et al.\ (1998)
reveal, in addition to the two protostars (of masses
$\sim0.5\,M_{\odot}$) with a projected separation of 42\,AU, a faint
third body (TMR-1C) that appears to have been recently ejected from
the system. The association of TMR1-C with the protobinary is
suggested by a long, narrow filament that seems to connect the
protobinary to the faint companion. Given the observed luminosity of
TMR1-C, and using evolutionary models for young, low-mass objects, the
estimated age of $\sim 3\times 10^5\,$yr for the system leads to a
mass estimate of $\sim 2-5 M_{\rm Jup}$. This suggests that the object
may be a planet that was formed in orbit around one of the two
protostars, and later ejected from the system (Terebey et al.\ 1998).
If the age were increased to $\sim10^{7}\,$yr, the mass would increase
to $\sim 15\,M_{\rm Jup}$, and TMR1-C could also have been a low-mass,
brown dwarf companion to one of the stars.
If TMR-1C is indeed a planet that was ejected from the binary system,
this may place significant constraints on planet formation theory.
Here we speculate about the process which may have led to the ejection of
a planet from the TMR-1 system. In the standard planet
formation theory, TMR-1C must have formed in a nearly
circular orbit around one of the protostars. Secular perturbations by
the other protostar may then have driven a gradual increase in the
eccentricity of the planet's orbit, gradually pushing the system
towards instability. Large apocentre distances render perturbations
by the other protostar increasingly important. The planet could then
enter the chaotic regime in which it can
switch into an orbit around the other protostar, possibly switching
between stars many times before finally being ejected from
the system. The expected velocity after such an ejection is in agreement
with the estimated velocity of TMR-1C (de la Fuente Marcos \& de la
Fuente Marcos 1998). One concern with this scenario is that the timescale
for ejection may be short compared to the timescale for planet formation.
Early in the evolution the protostellar disk will damp the planet's
eccentricity. However, as the planet becomes more massive, the
gravitational perturbation
by the other protostar becomes dominant.
In fact, after the protoplanetary core has formed, it may be
able to accrete more mass than in the standard scenario, since it is no longer
confined to accrete from a narrow feeding zone.
We have investigated systems similar to TMR-1,
but with the low mass companion in a nearly circular orbit around one of
the stars. We assume that the two protostars
have masses of $1\,M_{\odot}$ and
$0.5\,M_{\odot}$, with a binary semimajor axis of 50 AU and a
planet mass of $5\,M_{\rm Jup}$.
We estimate when the triple system will become unstable by combining our
models of the secular eccentricity evolution of the binary with the
stability criterion of Eggleton \& Kiseleva (1995). We calculate the
amplitude of secular eccentricity perturbations and see if the system would
violate the Eggleton \& Kiseleva (1995) instability criterion
($Y\equiv a_2 (1-e_2) / a_1 (1+e_1) <Y_{\rm min}$) when
the planet's orbit reaches its maximum eccentricity. For orbits with a large
relative inclination, planetary semimajor axes from 14 AU to 8 AU are
expected to become unstable as the relative inclination is varied from
$40^{\circ}$ to $85^{\circ}$.
For nearly coplanar orbits, planetary semimajor axes from 16 AU to 3 AU
will become unstable according to this criterion, as the outer eccentricity
is varied from $0$ to $0.8$. Thus, it seems plausible that a
protoplanet could begin to form near the critical semimajor axis and
eventually be ejected from the system after it has accreted a large amount
of gas. If we assume the initial semimajor axis of the planet to be 5 AU,
then we can solve for a critical binary eccentricity, which we find to be
0.65. The period of the eccentricity oscillations responsible for reducing
the instability parameter
from $\simeq 10$ to $\simeq 3$ is about $3 \times 10^4\,$yr.
In the coplanar regime we can also apply the stability criteria of
Holman \& Wiegert (1999)\footnote{They define the critical semimajor
axis as the largest orbital radius for
which planets of all initial longitudes of periastron survived for $10^4$
binary periods. This is different from the criterion obtained by
combining secular perturbation theory with the results of
Eggleton \& Kisseleva (1995). However both criteria provide an
estimate of when the triple system becomes unstable. It is
reassuring that both criteria yield similar results.}.
As the eccentricity of the TMR-1 binary increases
from $0$ to $0.8$, the critical semimajor axis decreases from
about $11\,$AU to $2\,$AU. This is precisely
the region where giant planets are expected to form. If we assume the
initial semimajor axis of the planet to be $5\,$AU, then we can again
solve for a
critical binary eccentricity, which we find to be 0.48. The two estimates
are in reasonable agreement, and both also agree with the results of
preliminary numerical simulations which we have performed for this system.
We have conducted Monte Carlo simulations to study the process of planet
ejection from protobinaries (Fig.~17). For systems
with large inclinations, the most common outcome for unstable systems is a
collision of the planet with its parent star. However, for
systems with a low relative inclination, the most common outcome was for the
planet to be ejected from the system. Furthermore, we found that in many
cases it can take up to $\sim 10^7\,$yr for the planet to be ejected.
Since this is longer than a typical planetary formation timescale, the
scenario proposed above appears reasonable.
\begin{figure}
\plotone{fig17.ps}
\caption{
Results of Monte Carlo simulations
for the dynamical evolution of possible progenitor systems of TMR-1.
The systems contain a
$5\,M_{\rm Jup}$ planet in an initially circular $10\,$AU orbit around a $1
M_{\odot}$ star with a $0.5\,M_{\odot}$ companion star in a $50\,$AU
orbit. All other orbital parameters (initial phases, longitudes of
pericenter, relative inclination, and binary eccentricity)
were assigned random values.
Triangles correspond to
cases in which the planet was ejected (escaped) and squares correspond
to cases in which the planet collided with one of the stars.
The time to ejection (triangles) or collision (squares) is
shown as a function of the initial stability parameter
$Y\equiv {a_2 \left(1-e_2\right)}/{a_1}$. }
\end{figure}
\subsection{Systems with Short-Period Inner Binaries}
Finally, we discuss briefly some observations and related theoretical work
on triple systems containing a short-period inner binary.
If the outer period is also relatively short,
it may be possible to observe the secular perturbations directly, since the
timescale for eccentricity modulations and orbital precession may become
comparable to the timescale of observations.
Unfortunately, in these systems,
other perturbation effects such as tidal dissipation are likely to affect
the secular evolution, making the theoretical analysis more difficult.
\subsubsection{HD 109648}
HD 109648 is a triple-lined spectroscopic triple probably composed of three
main-sequence stars all with masses $\sim1\,M_{\odot}$
(Jha et al.\ 1999). The
inner orbit has a short period, $\simeq 5.5\,$d, so tidal dissipation effects
are likely to be important. The small but significant eccentricity
($e_1=0.0119\pm 0.0014$) of the inner orbit has been attributed to the
perturbation by the outer companion (Jha et al.\ 1999). This system is strongly
coupled, with $\alpha\simeq 0.1$, so the timescale for eccentricity
modulations is short, $P_e\sim 15\,$yr. Thus, the available
observations, spanning over 8 years, may already have
detected changes in the inner eccentricity and longitude of pericenter.
Theoretical models by Jha et al.\ (1999) based on current data provide a
loose constraint on the relative inclination of the orbits:
$5.9^{\circ} \le i \le 54^{\circ}$ or $126^{\circ}\le i \le 174.1^{\circ}$.
Future observations are likely to produce tighter
constraints on this and other orbital parameters for the triple
system. However, Jha et al.\ (1999) speculate that additional variations
may also be caused by the presence of a fourth object in a much wider orbit.
\subsubsection{HD 284163}
HD 284163 is a triple system in the Hyades. The inner binary consists
of a $0.72\,M_{\odot}$ primary and a secondary with a minimum mass of
$0.33\,M_{\odot}$ in a 2.4-day orbit (Griffin \& Gunn 1981; Ford \&
Rasio 2000). The outer companion (of mass $\sim 0.5\,M_{\odot}$) has
a projected separation of $7.4\,$AU (Patience et al.\ 1998).
Theoretical and empirical evidence indicate that tidal dissipation in
the primary should have circularized the inner binary (Ford \& Rasio
2000). However, the radial velocity curves indicate a significant
eccentricity, $e_1=0.057\pm 0.005$ (Griffin \& Gunn 1981). The
secular perturbation by the outer companion is likely responsible for
inducing this observed eccentricity. At present, however, the outer
orbit is not well constrained, making further analysis difficult.
\subsubsection{$\beta$ Per}
This is another triple system with a short-period (2.87 d) inner
binary ($1.7\,M_{\odot}$ + $3.7\,M_{\odot}$) which is expected to have a
very nearly circular orbit on the basis of tidal dissipation theory, but has
a significantly larger observed eccentricity of $0.0653$. Secular perturbations by
the
outer companion (mass $1.7\,M_{\odot}$ in a 1.86-yr orbit) are likely responsible for
maintaining the inner binary's eccentricity (Kisseleva et al.\ 1998).
Kisseleva et al.\ (1998) suggest that the inner binary may have
originally been significantly wider. In their scenario, quadrupole
perturbations drive a eccentricity increase. As the eccentricity
increases, tidal dissipation becomes significant and removes energy
from the orbit. As the orbit shrinks, precession of the longitude of
periastron due to the stellar quadrupole moments and general relativity
increase, eventually suppressing the eccentricity perturbations.
The secular
decrease in the semimajor axis due to the coupling of quadrupole
perturbations and tidal dissipation is then halted near the presently
observed orbit.
In this system the ratio of semimajor axes is rather small, $\simeq40$. Thus the
octupole-level perturbations could play an important role in the secular
evolution. In particular, this could lead to significantly
larger eccentricities in the initial orbit if other effects have not
yet started to suppress the
perturbations. Thus, the range of initial conditions that could lead
to such an evolution can be much larger than would be expected by
considering quadrupole-level perturbations only.
\acknowledgments
We are grateful to M.~Holman and J.~Wisdom for providing us with a version
of their MVS integrator modified for systems with more than one massive body,
and to H.~Hauser for sending us an extended table of orbital solutions for
the 16~Cygni binary.
We thank S.~Jha for pointing out to us the recent theoretical
work of Krymolowski and Mazeh and for providing us with a draft of his
paper on HD~109648 in advance of publication.
Some of the numerical simulations mentioned in \S 5.2 for TMR-1 were
performed at MIT by J.~Madic.
F.A.R.\ thanks the Theory Division of the Harvard-Smithsonian
Center for Astrophysics for hospitality.
This work was supported in part by NSF Grant AST-9618116 and NASA ATP
Grant NAG5-8460.
E.B.F.\ was supported in part by the Orloff UROP Fund and the UROP
program at MIT.
F.A.R.\ was supported in part by an Alfred P.\ Sloan Research Fellowship.
This work was supported by the National Computational Science Alliance
under Grant AST980014N and utilized the SGI/Cray Origin2000 supercomputer
at Boston University and the Condor system at the University of Wisconsin.
\newpage
|
1,314,259,996,692 | arxiv | \section{The fundamental group of \texorpdfstring{$\Spin(\Pi)$}{Spin(Pi)}}
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\title{Fundamental groups of split real Kac--Moody groups and generalized real flag manifolds}
\author{Paula Harring \hskip 1em Ralf K{\"o}hl\\[2ex] With appendices \\ by Tobias Hartnick and Ralf K{\"o}hl and \\ by Julius Gr\"uning and Ralf K\"ohl}
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\maketitle
\abstract{We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac--Moody groups endowed with the Kac--Peterson topology. In analogy to the finite-dimensional situation, because of the Iwasawa decomposition $G = KAU_+$ the embedding $K \hookrightarrow G$ is a weak homotopy equivalence, in particular $\pi_1(G) = \pi_1(K)$. It thus suffices to determine $\pi_1(K)$ which we achieve by investigating the fundamental groups of generalized flag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized flag variety is a CW decomposition -- in particular, we cover the complete symmetrizable situation; furthermore, the results concerning only the structure of $\pi_1(K)$ actually also hold in the non-symmetrizable two-spherical case.}
\section{Introduction} \label{setting}
The structure of maximal compact subgroups in semisimple Lie groups was investigated by Cartan and, later, Mostow: In \cite{Mos}, Mostow gives a new proof of a Cartan's theorem stating that a connected semisimple Lie group $G$ is a topological product of a maximal compact subgroup $K$ and a Euclidean space, implying in particular that $G$ and $K$ have isomorphic fundamental groups.
Subsequent case-by-case analysis provided the isomorphism types of these maximal compact subgroups and their fundamental groups; tables can be found in \cite[p 518]{Helga} and \cite[94.33]{Salz}.
Starting in the 1940's, Dynkin diagrams, introduced in \cite{Dynkin}, have been used to describe the structure of simple Lie groups.
In this article, we present a uniform result which makes it possible to determine the fundamental group of any algebraically simply connected split real simple Lie group -- and, more generally, any algebraically simply-connected semisimple split real topological Kac--Moody group -- directly from its Dynkin diagram.
In \cite[Theorem 1]{Ti}, Tits associates with every generalized Cartan matrix $\CA$ and every commutative ring $k$ a group $\calG_k(\CA)$. Let $\Pi$ be the Dynkin diagram of $\CA$.
\begin{de}
\label{Kac--Moody group}
We set $G(\Pi) := [\calG_\RR(\CA),\calG_\RR(\CA)]$ and refer to this group as the \emph{algebraically simply-connected semisimple split real Kac--Moody group of type $\Pi$}.
\end{de}
Kac--Moody groups endowed with the Kac--Peterson topology have been studied extensively by the second author together with Gl\"ockner and Hartnick in \cite{GGH} and with Hartnick and Mars in \cite{HKM}. Our result is applicable to those Kac--Moody groups whose Bruhat decompositions are CW decompositions and for which the embedding $K \hookrightarrow G$ is a weak homotopy equivalence.
\medskip
In order to fix notations,
let $G=G(\Pi)$ be the algebraically simply-connected split real semisimple Kac--Moody group associated to an irreducible diagram $\Pi = (V,E)$ endowed with the Kac--Peterson topology (for definitions, see Section \ref{Kac--Moody-basics}).
Let $K=K(\Pi)$ be the so-called maximal compact subgroup of $G(\Pi)$, i.e., the subgroup fixed by the Cartan--Chevalley involution $\theta$ of $G(\Pi)$.
Given the Dynkin diagram $\Pi = (V,E)$ with a fixed labelling $\lambda:\{1, \dots, n\} =: I \to V $, we define a modified diagram $\Pia$ with vertex set $V$ and $\{i^\lambda,j^\lambda\} \in V \times V$ edge if and only if $\epsilon(i,j) = \epsilon(j,i) = -1$, where $\epsilon(i,j)$ denotes the parity of the corresponding Cartan matrix entry. To each connected component $\bar{\Pi}^{\mathrm{adm}}$ of $\Pia$ we then assign a colour as follows: Let $\bar{\Pi}^{\mathrm{adm}}$ be coloured red (denoted by $r$) if it contains a vertex $i^\lambda$ such that there exists a vertex $j^\lambda \in V$ satisfying $\epsilon(i,j) = 1$ and $ \epsilon(j,i) = -1$. Let $\bar{\Pi}^{\mathrm{adm}}$ be coloured green ($g$) if it consists only of an isolated vertex, and blue ($b$) else.
One can then read off the isomorphism type of $\pi_1(G(\Pi))$ from the coloured diagram $\Pia$ as specified in the following theorem.
\begin{thm*}
Let $\Pi$ be an irreducible Dynkin diagram such that the Bruhat decomposition of $G(\Pi)$ provides a CW decomposition (i.e., such that the conclusion of Proposition \ref{bruhat decomp is cw decomp} holds) and such that the embeding $K \hookrightarrow G(\Pi)$ is a weak homotopy equivalence (i.e., such that the conclusion of Theorem~\ref{FundamentalGroups2} holds). Let $n(g)$ and $n(b)$ be the number of connected components of $\Pia$ of colour $g$ and $b$, respectively. Then
\[\pi_1(G(\Pi)) \iso \ZZ^{n(g)} \times C_2^{n(b)}.\] In particular, this statement holds in the symmetrizable case.
\end{thm*}
\begin{ex*}[Isomorphism types of $\pi_1(G(\Pi))$ for the spherical Dynkin diagrams\footnote{Dynkin diagram LaTeX styles kindly provided by Max Horn at \cite{Ho}}.]
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\[
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\end{tikzpicture}& \pi_1(\Spin(n,n+1)) \iso \begin{cases} \ZZ &\mathrm{if }\;n\leq 2,\\C_2&\mathrm{if }\; n>2.\end{cases}\Strut \\ \hline \Strut
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\pi_1(G_{2,2}) \iso C_2 \\ \hline \end{array} \]
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\hline \rule{0pt}{\normalbaselineskip}
\Pi & \Pia \;\mathrm{coloured\;by}\;\gamma & \pi_1(G(\Pi))\\
\hline \Strut
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\end{tikzpicture}& \pi_1(E_{10}) \iso C_2 \Strut \\ \hline \Strut
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\end{tikzpicture}& \pi_1(G(X)) \iso \ZZ^2 \times C_2^2 \\ \hline
\end{array}
\]
\end{ex*}
While in the classical case, one has a topological Iwasawa decomposition $G = K\times A \times U_+$ with $A$ and $U_+$ contractible, implying $\pi_1(K) \iso \pi_1(G)$, it is currently unknown whether the corresponding Iwasawa decomposition in the general Kac--Moody case is also topological. However, using a fibration result by Palais, in the appendix Hartnick and the second author prove that isomorphism of fundamental groups still holds in the general symmetrizable case, therefore reducing the problem to the computation of $\pi_1(K)$.
In \cite[Section~16]{GHKW}, the group ${\Spin(\Pi,\kappa)}$ -- where $\kappa$ denotes a so-called \emph{admissible colouring} of the vertices of $\Pi$ -- is defined as the canonical universal enveloping group of a $\Spin(2)$-amalgam $\mathcal{A}(\Pi, \Spin(2)) = \{\tilde{G}_{ij}, \tilde{\oldphi}_{ij}^i \mid i \neq j \in I\}$ where the isomorphism type of $\tilde{G}_{ij}$ depends on the $(i,j)$- and $(j,i)$-entries of the Cartan matrix of $\Pi$ as well as the values of $\kappa$ on the corresponding vertices.
It is shown in \cite[Section~17]{GHKW} that there exists a finite central extension $\Spin(\Pi, \kappa) \to K(\Pi)$ which implies that the subspace topology on $K(\Pi)$ defines a unique topology on $\Spin(\Pi, \kappa)$ that turns the extension into a covering map.
The resulting group topology on $\Spin(\Pi, \kappa)$ is called the {\em Kac--Peterson topology} on $\Spin(\Pi, \kappa)$.
In the simply-laced case, there is a unique non-trivial admissible colouring $\kappa$ and the corresponding group $\Spin(\Pi):= \Spin(\Pi,\kappa)$ double-covers $K$ as shown in \cite{GHKW}. We prove here that $\Spin(\Pi)$ is simply connected which then implies that $\pi_1(K) \iso C_2$.
The strategy of proof in the simply-laced case is to study fibre bundles of the form $$\Spin(3) \to \Spin(\Pi) \to \Spin(\Pi)/\Spin(3),$$ which yield exact sequences of the form \[ \{1\} = \pi_1(\Spin(3)) \rightarrow \pi_1(\Spin(\Pi)) \rightarrow \pi_1(\Spin(\Pi)/\Spin(3)).\]
One can use the building in order to prove that the fibre bundles\\ $\Spin(3) \to \Spin(\Pi)\to \Spin(\Pi)/\Spin(3)$ arising from the embedding of $\Spin(3)$ along a subdiagram of type $A_2$ indeed satisfy $\pi_1(\Spin(\Pi)/\Spin(3))=\{1\}$, which then establishes the simple-connectedness of $\Spin(\Pi)$.
A key to the proof both in the simply-laced and in the general case is the computation of the fundamental groups of \emph{generalized flag varieties} -- that is, spaces of the form $G/P_J$ for a parabolic subgroup $P_J$ of $G$ corresponding to an index subset $J \subseteq I$. We prove the following theorem:
\begin{thm*}
Let $\Pi$ be an irreducible Dynkin diagram such that the Bruhat decomposition of $G(\Pi)$ provides a CW decomposition (i.e., such that the conclusion of Proposition \ref{bruhat decomp is cw decomp} holds) and let $J \subseteq I$. Then a presentation of $\pi_1(G/P_J)$ is given by
\[ \left\langle x_i; \quad i \in I \mid x_ix_j^{\epsilon(i,j) } = x_jx_i,\quad x_k = 1; \quad i,j \in I, k \in J \right\rangle.\]
In particular, this statement holds in the two-spherical and in the symmetrizable case.
\end{thm*}
We refer to \cite{Wig} for the analog result in the finite-dimensional situation.
In order to determine $\pi_1(K)$ in the general case, we compute subgroups of $\pi_1(K)$ corresponding to the index sets of connected components of $\Pia$ using the above theorem and covering maps of type $K/K_J \to K/(K \cap T)K_J$. We then shown that $\pi_1(K)$ is a direct product of these subgroups.
In a very similar way, the fundamental group of $\Spin(\Pi,\kappa)$ is determined, establishing the following theorem:
\begin{thm*}
Let $\Pi$ be an irreducible Dynkin diagram such that the Bruhat decomposition of $G(\Pi)$ provides a CW decomposition (i.e., such that the conclusion of Proposition 3.5 holds).
Let $n(g)$ be the number of connected components of $\Pia$ of colour $g$. Let $n(b,\kappa)$ be the number of connected components of $\Pia$ on which $\kappa$ takes the value 1 and which have colour $b$. Then
\[\pi_1(\Spin(\Pi,\kappa)) \iso \ZZ^{n(g)} \times C_2^{n(b,\kappa)}.\] In particular, this statement holds in the two-spherical and in the symmetrizable case.
\end{thm*}
\medskip
\textbf{Acknowledgements.} The research leading to this article has been partially funded by DFG via the project KO 4323/11. The authors thank Julius Gr\"uning for various helpful remarks on an earlier version of this article.
\section{Split-real Kac--Moody groups}
\label{Split-real Kac--Moody groups}
\label{Kac--Moody-basics}
In \cite[\S1.3]{Kac2}, Kac associates with every generalized Cartan matrix $\CA = (a_{ij})_{1\leq i,j \leq n} \in \ZZ^{n\times n}$ a quadruple $(\g_\CC(\CA), \h_\CC(\CA), \Psi, \check{\Psi})$ of a complex Lie algebra $\g_\CC(\CA)$, an abelian subalgebra $\h_\CC(\CA)$ and linearly independent finite subsets $\Psi = \{\alpha_1, \dots, \alpha_n\} \subseteq \h_\CC(\CA)^*$ and $\check{\Psi} = \{\check{\alpha}_1, \dots, \check{\alpha}_n\} \subseteq \h_\CC(\CA)$ called \emph{simple roots} and \emph{simple coroots}, respectively, such that $\alpha_j(\check{\alpha}_i) = a_{ij}$. Associated with such a quadruple is a Lie algebra generating set $\{e_1, \dots, e_n,f_1,\dots,f_n\} \cup \h_\CC(\CA)$. The complex Lie algebra $\g_\CC(\CA)$ is called the \emph{complex Kac--Moody algebra} associated with $\CA$, and $\h_\CC(\CA)$ its \emph{standard Cartan subalgebra}.
Since $a_{ij} \in \RR$, one can analogously define a quadruple $(\g_\RR(\CA), \h_\RR(\CA), \Psi, \check{\Psi})$ where $\g_\RR(\CA)$ is a real Lie algebra that embeds naturally into $\g_\CC(\CA)$ as the real form given by the involution induced by complex conjugation. One refers to $\g_\RR(\CA)$ as the \emph{split real Kac--Moody algebra} associated with $\RR$ and to $\h_\RR(\CA)$ as its \emph{standard split Cartan subalgebra}.
Let $Q \subseteq \h_\RR(\CA)^*$ be the group generated by $\Psi$ and $Q_{\pm}$ the subsemigroups generated by $\pm \Psi$, respectively. For $k \in \{\CC, \RR\}$ and $\alpha \in \h_k(\CA)^*$ define the \emph{root space}
\[\g_{\alpha} := \{X \in \g_k(\CA) \mid \forall H \in \h_k(\CA)^*: [H,X] = \alpha(H)X\}. \] The \emph{set $\Delta$ of $\h_k(\CA)$ roots in $\g_k(\CA)$} is defined as $\Delta:=\{\alpha \in Q \setminus \{0\} \mid \g_\alpha^k \neq \{0\}\}$. One has the \emph{root space decomposition} \[\g_k(\CA) = \h_k(\CA) \oplus \bigoplus_{\alpha \in \Delta} \g_\alpha^k.\]
The set $\Delta$ decomposes as a disjoint union into the subsets $\Delta_{\pm}:= \Delta \cap Q_{\pm}$.
For $i = \{1,\dots, n\}$ define the \emph{fundamental root reflection} $\sigma_i \in \GL(\h_\RR(\CA)^*)$ by \[\sigma_i(\lambda):= \lambda -\lambda(\check{\alpha}_i)\alpha_i.\] Then the \emph{Weyl Group} of $\g_\RR(\CA)$ is defined as $W:= \langle \sigma_1, \dots, \sigma_n\rangle \leq \GL(\h_\RR(\CA)^*)$ and forms a Coxeter system together with the set of fundamental root reflections. Finally, define the \emph{set of real roots} $\Phi:= W.\Psi \subseteq \Delta$ and $\Phi^{\pm} := \Delta_{\pm} \cap \Phi$.
The construction in \cite{Ti} of $\calG_\RR(\CA)$ (see Definition \ref{Kac--Moody group}) provides a representation of $\calG_\RR(\CA)$ on $\g_\RR(\CA)$ by Lie algebra automorphisms, which is denoted by \[ \Ad: \calG_\RR(\CA) \to \Aut(\g_\RR(\CA)), \] and referred to as the \emph{adjoint representation} of $\calG_\RR(\CA)$. Since the subgroup $\Ad(G(\Pi))$ of $G(\Pi)$ under this representation preserves the commutator subalgebra $\g_\RR'(\CA)$, one obtains an adjoint representation \[ \Ad: G(\Pi) \to \Aut(\g_\RR'(\CA)) \] for $G(\Pi)$. The kernels of the adjoint representations of $\calG_\RR(\CA)$ and $G(\Pi)$ are given by the respective centres.
An element $X \in \g_\RR(\CA)$ is \emph{$\ad$-locally-finite} if for every element $Y \in \g_\RR(\CA)$ there exists an $\ad(X)$-invariant finite-dimensional subspace $W$ with $Y \in W$. As pointed out in \cite[p. 64]{Mar}, this implies that $\left.\ad(X)\right|_W$ is a (finite) matrix in some basis of $W$, so the exponential $\exp(\ad(X))$ can be defined in the ususal way.
By \cite[(KMG5), p.~545]{Ti} and the uniqueness properties of $G_\RR(\CA)$ established in \cite[Theorem 1]{Ti}, $\exp(\ad(X)) \in \Ad(G_\RR(\CA))$. Let $F_{\g_\RR(\CA)}$ and $F_{\g_\RR'(\CA)}$ be the subsets of $\ad$-locally-finite elements of the respective algebras.
The maps $\exp: F_{\g_\RR(\CA)} \to \Ad(\calG_\RR(\CA))$ and $\exp: F_{\g_\RR'(\CA)} \to \Ad(G(\Pi))$ given by $X\mapsto \exp(\ad(X))$ can be lifted to exponential functions $\exp: F_{\g_\RR(\CA)} \to \calG_\RR(\CA)$ and $\exp: F_{\g_\RR'(\CA)} \to G(\Pi)$.
For $X \in \h_\RR(\CA)\subseteq F_{\g_\RR(\CA)}$, one has
\begin{eqnarray}
\mathrm{ad}(X)(e_i) = [X,e_i] = \alpha_i(X) e_i,& \quad\quad & \mathrm{Ad}(\exp(X))(e_i) = e^{\alpha_i(X)}\cdot e_i,\label{AdRootSpaces} \\
\mathrm{ad}(X)(f_i) = [X,f_i] = -\alpha_i(X) f_i,& \quad\quad & \mathrm{Ad}(\exp(X))(e_i) = e^{-\alpha_i(X)}\cdot f_i,
\end{eqnarray}
cf.\ \cite[Section~6.1.6]{Kum}, \cite[(KMG5), p.~545]{Ti}.
The same constructions apply also to $\CC$ instead of $\RR$. Since $\h_\CC(\CA) \subseteq F_{\g_\CC(\CA)}$, one can define $T_\CC:= \exp(\h_\CC(\CA)
)$. Note that $\exp(\h_\RR(\CA) =: A_\RR \subsetneq T_{\RR} := T_\CC \cap \calG_\RR(\CA)$. In fact, $A_\RR$ is of index $2$ in $T_\RR$ and there is a unique Lie group topology on $T_\RR$ in which $T_\RR \iso (R^\times)^n$ and $A_\RR = T_\RR^\circ \iso (\RR_{>0})^n$. The centre of $\calG_\RR(\CA)$ is contained in $T_\RR$.
The intersection $T:= G(\Pi) \cap T_\CC$ is called the \emph{standard split maximal torus} of $G(\Pi)$; again, $A_\RR\cap T$ is of finite index in $T$ and $T$ contains the centre of $G(\Pi)$.
The Lie algebra $\mathfrak g_\RR({\mathbf A})$ admits a unique involution $\theta$ which maps $e_j$ to $f_j$ for all $j=1, \dots, r$ and acts as $-1$ on $\mathfrak h_\RR({\mathbf A})$. There exists a unique involutive automorphism $\theta: G_\RR(\CA) \to G_\RR(\CA)$ such that $\theta(\exp(X)) = \exp(\theta(X))$ for all $X \in F_{\mathfrak g_\RR({\mathbf A})}$, and this involutive automorphism is called the \emph{Cartan--Chevalley involution} of $G_\RR(\CA)$. We denote by $K_\RR(\CA) := G_\RR(\CA)^\theta \subset G_\RR(\CA)$ the fixed point subgroup of this involution and define $K(\Pi) := K_\RR(\CA) \cap G(\Pi)$.
Let $\alpha \in \Phi$ be a real root. Then $\g_\alpha^\RR$ is one-dimensional and consists of ad-locally-finite elements. One can therefore define the \emph{root group} $U_\alpha:= \exp(\g_\alpha^\RR) \subseteq \calG_\RR(\CA)$. Each root group $U_\alpha$ carries a unique Lie group topology such that $U_\alpha \iso \RR$ as topological groups.
Define the \emph{positive}, respectively \emph{negative maximal unipotent subgroup} $U^\pm$ of $\calG_\RR(\CA)$ as the group generated respectively by the positive and negative root groups. One has $U^\pm \subseteq G(\Pi)$. The groups $U^\pm$ are normalized by $T_\RR$ and intersect $T_\RR$ trivially. In particular, they intersect the centres of $\calG_\RR(\CA)$ and $G(\Pi)$ trivially and hence embed into both $\Ad(\calG_\RR(\CA))$ and $\Ad(G(\Pi))$.
If $\alpha \in \Phi^+$, then $-\alpha \in \Phi^-$ and the group $G_\alpha:= \langle U_\alpha, U_{-\alpha} \rangle \leq G(\Pi)$ is isomorphic to $\SL_2(\RR)$. The groups $G_\alpha$ with $\alpha \in \Phi^+$ are called the \emph{rank one subgroups} and the groups $G_1:=G_{\alpha_1}, \dots, G_n:=G_{\alpha_n}$ are called the \emph{fundamental rank one subgroups} of $G(\Pi)$.
One can show that the pair $((U_\alpha)_{\alpha \in \Phi}, T)$ defines an RGD system for $G(\Pi)$. For details concerning RGD systems, we refer the reader to \cite[Chapter~8]{AB}.
\begin{de+rem}
\label{definition kp}
The \emph{Kac--Peterson topology} on $\calG_\RR(\CA)$ equals the finest group topology on $\calG_\RR(\CA)$ such that the natural embeddings $(U_\alpha \hookrightarrow \calG_\RR(\CA))_{\alpha \in \Phi}$ and $T_\RR \hookrightarrow \calG_\RR(\CA)$ are continuous when $T_\RR$ and the root groups $U_\alpha$ are endowed with their Lie group topologies.
The Kac--Peterson topology is $k_\omega$ by \cite[Proposition 7.10]{HKM} and, in particular, Hausdorff. Moreover, for every $\alpha \in \Phi^+$, it induces the unique connected Lie group topology on $G_\alpha$ and on $T_\RR$ by \cite[Corollary 7.16]{HKM}
\end{de+rem}
For more details on the Kac--Peterson topology, see \cite[Chapter~7]{HKM}.
\begin{nota}
Throughout this paper, let $G:= G(\Pi)$ be the algebraically simply connected centered split real Kac--Moody group associated to an irreducible generalized Dynkin diagram $\Pi = (V,E)$ with (bijective) labelling $\lambda:\{1, \dots, n\} =: I \to V $. Let $K := K(\Pi)$ be the maximal compact subgroup of $G$, i.e., the subgroup fixed by the Cartan--Chevalley involution $\theta$.
The groups $G$ and $K$ are always endowed with the subspace topologies induced by the Kac--Peterson topology on $G_\RR(\CA)$ and $G/B$ with the quotient topology.
Denote by $B:= B_+$ the positive Borel subgroup of the twin $BN$-pair of $G$, by $T$ the standard split maximal torus and by $W$ the Weyl group of $G$ with generating set $S = \{\sigma_i\}_{i \in I}$. For each $\sigma_i \in S$, take $s_i \in G$ to be a fixed representative for $\sigma_i$. By \cite[Corollary 1.7]{DMGH}, one has an Iwasawa decomposition $G = KB$.
Unless specified more explicitly, the symbol $J$ will always denote an arbitrary subset of the index set $I$, the symbol $\Pi_J$ the subdiagram of $\Pi$ corresponding to $J$, the symbol $G_J$ the subgroup $G(\Pi_J)$ of $G$, and the symbols $K_J$ and $B_J$ the intersections $G_J \cap K$ and $G_J \cap B$, respectively. This is consistent with the notation for the fundamental rank one subgroups: One has $G(\Pi_i) = G_i = G_{\alpha_i}$.
\end{nota}
\begin{rem}
\label{homeos alpha beta}
Due to the structure theory of RGD systems (cf.\ \cite[Chapter~8]{AB}, most notably the fact that restricting an RGD system to a subdiagram again yields an RGD system), for each fundamental rank one subgroup $G_{i}$ there exists an (abstract) isomorphism $\gamma_{i}: \SL(2,\RR) \to G_{i}$ with the following properties: Let $B_{\SL(2,\RR)}$ be the group of upper triangular matrices in $\SL(2,\mathbb{R})$ and let $U_{\pm \beta}$ denote the canonical root subgroups of $\SL(2,\RR)$. Then
\begin{itemize}
\item $\gamma_{i}(U_{\pm \beta}) = U_{\pm \alpha_{i}}$.
\item $\gamma_{i}(B_{\SL(2,\RR)}) = B_{i}$.
\item For each $x \in \SL(2,\RR)$, $\gamma_{i}((x^t)^{-1}) = (\gamma_{i}(x))^{\theta}$, and hence
\item $\gamma_{i}(\SO(2,\RR)) = K_i$.
\end{itemize}
By \cite[Corollary~7.16]{HKM}, the restriction of the Kac--Peterson topology to any spherical subgroup $H$ of $G$ coincides with its Lie topology. That is, the groups $G_{i}$ inherit their Lie group topology from the topological Kac--Moody group $G$.
By the classical theory of Lie groups this yields the existence of a diffeomorphism $\gamma_i$ with the desired properties; in particular, $\gamma_{i}$ is open.
\end{rem}
\begin{de}
\label{basics}
Let
\begin{eqnarray*}
\delta: G/B \times G/B & \to & W \\ \delta(gB,hB) = w & \iff & g^{-1}h \in BwB
\end{eqnarray*}
be the Weyl distance function on $G/B$, and let $l_S$ be the length function that associates to each element the (unique) length of a corresponding reduced expression in $S$. Let $\leq$ be the strong Bruhat order on $W$: Recall that for $w_1, w_2 \in W$ one has $w_1 \leq w_2$ if there exist reduced expressions $s_{i_1}\dots s_{i_{l_S(w_1)}}$ of $w_1$ and $s_{j_1}\dots s_{j_{l_S(w_2)}}$ such that the former is a (not necessarily consecutive) substring of the latter.
For $w \in W$ and a chamber $gB \in G/B$ define $$C_w(gB) := \{hB \in G/B\mid \delta(gB, hB) = w\},$$ $$C_{\leq w}(gB) := \bigcup_{v \leq w}C_v(gB)$$ and $$C_{<w}(gB) :=C_{\leq w}(gB) \setminus C_w(gB).$$ In particular, one has $C_w(B)= BwB/B$ and $C_{\leq \sigma}(B) = B \langle s \rangle B/B$ for $\sigma \in S$ with representative $s \in \tilde{W}\subseteq G$ in the extended Weyl group $\tilde{W}$. A set $C_{\leq \sigma}(gB)$ is called a \emph{$\sigma$-panel}.
Moreover, for a subset $\{\sigma_i\}_{i \in J} \subseteq S$ with representatives $\{s_i\}_{i \in J} \subseteq \tilde{W}$ define $P_J$ to be the standard parabolic subgroup corresponding to the index set $J$, that is, $P_J := B\langle \{s_i\}_{i \in J} \rangle B$.
\end{de}
Throughout this paper, $C_w(gB)$ and $C_{\leq w}(gB)$ will always be endowed with the subspace topologies induced by $G/B$.
\begin{lem}
\label{P_i = G_iB}
Let $\sigma_i \neq \sigma_j \in S$.
Then the following hold:
\begin{enumerate}
\item $P_i = G_i B$ $= K_i B$. In particular, $C_{\leq \sigma_i}(B) = K_iB/B$.
\item $Bs_i s_jB = Bs_iB Bs_jB$. In particular, $C_{\leq \sigma_i \sigma_j}(B) = K_iBK_jB/B$
\end{enumerate}
\begin{proof} Assertions (a) and (b) follow from \cite[Remark~8.51]{AB} and \cite[Remark (2) after Theorem~6.56]{AB}, respectively, and the Iwasawa decomposition $G_i = K_iB_i$.
\end{proof}
\end{lem}
\section{The fundamental group of \texorpdfstring{$G/P_J$}{G/PJ}}
Throughout this section, let $J \subseteq I$, let $W_J$ be the subgroup of $W$ generated by $\{\sigma_i\}_{i \in J}$, and let $W^J \subseteq W$ be a set of representatives of the cosets in $W/W_J$ that have minimal length. The space $G/P_J$ is called \emph{generalized flag variety}.
\begin{lem}[(Bruhat decomposition)]
\label{bruhat}
One has $G/P_J = \bigsqcup_{w\in W^J}BwP_J/P_J$.
\end{lem}
\begin{proof}
By \cite[Theorem~6.56, Remark (1)]{AB}, the group $G$ has a Bruhat decomposition $$G = \bigsqcup_{w \in W} BwB = \bigcup_{w \in W} BwBW_JB = \bigcup_{w \in W} BwP_J.$$
Since double cosets partition $G$, one has $w_1 W_J = w_2 W_J$ if and only if $Bw_1 P_J = B w_2 P_J$ for $w_1, w_2 \in W$. This yields the desired disjoint decomposition of $G/P_J$.
\end{proof}
\begin{lem}
\label{quotient basics}
Let $G$ be a topological group and $H_1 \leq H_2$ subgroups of $G$ and endow $G/H_i$ with the quotient topology. Then the following hold:
\begin{enumerate}
\item The projection map $\pi: G \to G/H_1$ is continuous and open.
\item The canonical map $\psi: G/H_1 \to G/H_2$ is continuous and open.
\end{enumerate}
\begin{proof}
(a): Let $U \subseteq G$ open. Since $\pi$ is a quotient map, it suffices to show that $\pi^{-1}(\pi(U))$ is open. But this is true since $\pi^{-1}(\pi(U)) = UH_1 = \bigcup_{h \in H_1}Uh$ is a union of translates of open sets.
(b): This follows directly from (a) and the commutative diagram
\[\begin{tikzcd}
G \arrow[rd, "\phi"] \arrow[d, "\pi"] \\
G/H_1 \arrow[r, "\psi"] & G/H_2
\end{tikzcd}.\qedhere\]
\end{proof}
\end{lem}
\begin{de+rem}
\label{definition psi_w}
For $w \in W$, define the following restrictions of the canonical map $\psi: G/B \to G/P_J$:\begin{itemize}
\item $\psi_w: BwB/B \to BwP_J/P_J$,
\item $\psi_{\bar{w}}: \bigcup_{x \leq w} BxB/B \to \bigcup_{x \leq w} BxP_J/P_J$.
\end{itemize}
Since $\psi$ is continuous, the same holds for the two restrictions. The space $\bigcup_{x \leq w} BxB/B$ is compact by \cite[Corollary~3.10]{HKM} and so $\psi_{\bar{w}}$ is a quotient map.
\end{de+rem}
\begin{lem}
\label{hom between BwB/B and BwP/P}
Let $G$ be two-spherical or symmetrizable and let $w \in W^J$. Then the canonical map $\psi_w$ is a homeomorphism.
\end{lem}
\begin{proof}
By Remark \ref{definition psi_w}, $\psi_{\bar{w}}$ is a quotient map.
One has $\psi_{\bar{w}}^{-1}(BwP_J/P_J) = BwB/B$: Let $x \leq w$ such that $BxP_J/P_J = BwP_J/P_J$. Then $x\in BwP_J = BwW_JB$ where the equality holds since by definition of $W^J$ one has $l(ww')= l(w)+l(w')$ for all $w' \in W_J$ which implies $Bww'B = BwBw'B$. The Bruhat decomposition of $G$ yields $x \in wW_J$ and hence, $l(x) \geq l(w)$. This implies $x = w$.
Now, since $BwB/B$ is open in its closure $\bigcup_{x \leq w} BxB/B$ in $G/B$ (see \cite[Proposition~5.9]{HKM} plus Corollary~\ref{topstrong}), the preceding observations yield that $\psi_w$ is an injective quotient map and therefore a homeomorphism.
\end{proof}
\begin{lem}
\label{panels are spheres}
Let $\sigma_i \in S$. Then the panels satisfy $C_{\leq \sigma_i}(B) \hom S^1(\RR)$.
\end{lem}
\begin{proof}
The panel $C_{\leq \sigma_i}(B)$ is a subbuilding of $G/B$ corresponding to the RGD system $\{G_i, U_{\alpha_{i}}, U_{-\alpha_{i}}, T \cap G_i \}$. By Remark~\ref{homeos alpha beta} one has $G_i \iso \SL(2,\RR)$, $T \cap G_i \iso T_{\SL(2,\RR)}$ and $U_{\pm\alpha_{i}} \iso U_{\pm\alpha}$ where $T_{\SL(2,\RR)}$ denotes the subgroup of diagonal matrices and $U_{\pm\alpha}$ denote the canonical root subgroups of $\SL(2,\RR)$. This implies that $C_{\leq \sigma_i}(B)$ is homeomorphic to the building $\SL(2,\RR)/B_{\SL(2,\RR)} \hom \PP_1(\RR) \hom S^1(\RR)$.
\end{proof}
\begin{de}
Following \cite[Chapter 8]{Rot}, a \emph{CW complex} is an ordered triple $(X, E, \chi)$, where $X$ is a Hausdorff
space, $E$ is a family of cells in $X$, and $\chi = \{\chi_e\mid e\in E\}$ is a family of maps, such
that
\begin{enumerate}
\item $X = \bigsqcup_{e\in E}{E}$.
\item For $k \in \NN$, let $X^{(k)}\subseteq X$ be the union of all cells of dimension $\leq k$. Then for each $(k+1)$-cell $e \in E$, the map $\chi_e:(D^{k+1}, S^{k}) \to (e\cup X^{(k)}, X^{(k)})$, is a \emph{relative
homeomorphism}, i.e., it is a continuous map and its restriction $D^k \setminus S^{k-1} \to e$ is a homeomorphism.
\item If $e \in E$, then its closure $\cl{e}$ is contained in a finite union of cells in $E$.
\item $X$ has the weak topology determined by $\{\cl{e} \mid e \in E\}$, i.e., a subset $A$ of $X$ is closed if and only if $A \cap \cl{e}$ is closed in $\cl{e}$ for each $e \in E$.
\end{enumerate}
For $k \in \NN$, let $\Lambda_k$ be an index set for the $k$-dimensional cells, so that $X^{(k)} \setminus X^{(k-1)} = \bigsqcup_{\lambda \in \Lambda_k} e_\lambda$ and set $\chi_\lambda:= \chi_{e_\lambda}$. This map is called the \emph{characteristic map} of $e_\lambda$.
\end{de}
\begin{prop}
\label{bruhat decomp is cw decomp}
Let $G$ be two-spherical or symmetrizable. Then for each $w \in W$, the set $C_w(B) = BwB/B$ is a cell of dimension $l(w)$ that is open in its compact closure $C_{\leq w}(B)$ in $G/B$. For each subset $J \subseteq I$, the Bruhat decomposition $G/P_J = \bigsqcup_{w\in W^J}BwP_J/P_J$ is a CW decomposition.
\end{prop}
\begin{proof}
The first statement is immediate by \cite[Corollary~3.10 and Proposition~5.9]{HKM} plus Corollary~\ref{topstrong}, see also \cite[p. 170, 171]{Kra}. Furthermore, \cite[Proposition~5.9]{HKM} combined with Corollary~\ref{topstrong} states that the Bruhat decomposition of $G/B$ is a CW decomposition. By Lemma \ref{hom between BwB/B and BwP/P}, $G/P_J$ is composed of cells that are homeomorphic to cells in $G/B$, so composing the characteristic maps of the latter cells with the canonical map $\psi: G/B \to G/P_J$ yields characteristic maps for the cells in $G/P_J$.
For the closure-finiteness, let $BwP_J/P_J$ be a cell in $G/P_J$. Since $\psi$ is continuous and restricts to a homeomorphism $BwB/B \to BwP_J/P_J$, it maps
$\cl{BwB/B}$ surjectively onto $\cl{BwP_J/P_J}$. Now, $\cl BwB/B = \bigcup_{x \leq w} BxB/B$, which implies that
$$\cl{BwP_J/P_J} = \bigcup_{x \leq w} BxP_J/P_J = \bigcup_{\substack{x\leq w \\ x \in W^J}} BxP_J/P_J,$$ where the last equality holds since $W_J \subseteq P_J$. This proves that $\cl{BwP_J/P_J}$ is contained in a finite union of cells.
It remains to show that $G/P_J$ has the weak topology determined by the cell closures.
For $w\in W$ and a minimal-length representative $\tilde{w}\in W^J$ of $wW_J$, one has $BwP_J/P_J = B\tilde{w}P_J/P_J$. Let $e_w:= BwP_J/P_J = B\tilde{w}P_J/P_J$ and $e_w':= BwB/B$. Let $\bar{e}_w = \cl{e_w} = \bigcup_{x \leq \tilde{w}} BxP_J/P_J$ and $\bar{e}_w':= \cl{e'_w} = \bigcup_{x \leq w} BxB/B$.
Let $A$ be a closed subset of $G/P_J$ and let $e_w$, $w \in W^J$, be an arbitrary cell. Then $\psi^{-1}(A)$ is closed in $G/B$ since $\psi$ is continuous, so $\psi^{-1}(A) \cap \bar{e}_w'$ is closed in $\bar{e}_w'$ since $G/B$ is a CW complex. Now, \[ \psi^{-1}(A) \cap \bar{e}_w' = \psi^{-1}(A) \cap \psi^{-1}(\bar{e}_w) = \psi^{-1}(A \cap \bar{e}_w) = \psi_{\bar{w}}^{-1}(A \cap \bar{e}).\] Since $\psi_{\bar{w}}$ is a quotient map by Remark \ref{definition psi_w}, this implies that $A \cap \bar{e}_w$ is closed in $\bar{e}_w$.
Now, let $A$ be a subset of $G/P_J$ such that $A \cap \bar{e}_w$ is closed in $\bar{e}_w$ for all $w \in W^J$. Since for each $w \in W$ one has $e_w = e_{\tilde{w}}$ for any minimal-length representative $\tilde{w} \in W^J$ of $wW_J$, in fact $A \cap \bar{e}_w$ is closed in $\bar{e}_w$ for all $w \in W$. Therefore $\psi_{\bar{w}}^{-1}(A \cap \bar{e}_w)$ is closed in $\bar{e}_w'$ for all $w \in W$. Since $\psi_{\bar{w}}^{-1}(A \cap \bar{e}_w) = \psi^{-1}(A) \cap \bar{e}_w'$, the fact that $G/B$ is a CW complex implies that $\psi^{-1}(A)$ is closed in $G/B$. Since $\psi$ is open by Lemma \ref{quotient basics}, it follows that $A$ is closed in $G/P_J$. This proves that $G/P_J$ is a CW complex.
\end{proof}
\begin{nota}
Define $R: [0,1] \to \SO(2,\RR), s \mapsto \begin{pmatrix} \cos(s\pi) & -\sin(s\pi) \\ \sin(s\pi) & \cos(s\pi) \end{pmatrix}.$
\end{nota}
\begin{lem}
\label{the map R}
$R$ induces a continuous, surjective map $\tilde{R}: [0,1] \to \SL(2,\RR)/B_{\SL(2,\RR)}$ which maps the interior $(0, 1)$ homeomorphically onto its image and maps the boundary $\{0, 1\}$ surjectively onto its image.
\end{lem}
\begin{proof}
Let $\{x_0\}:= \left\langle \begin{pmatrix} 1 & 0 \end{pmatrix}^\intercal \right\rangle \in \PP^1$ where $\PP^1$ denotes the real projective line, modelled as the subset of one-dimensional subspaces of $\RR^2$. Since each one-dimensional subspace in $\PP^1 \setminus \{x_0\}$ contains exactly one element in the upper half circle $R([0,1]) \cdot \begin{pmatrix} 1 & 0 \end{pmatrix}^\intercal$ while $x_0$ contains the two boundary points corresponding to $R(0)$ and $R(1)$, one has a surjection from $[0,1]$ onto $\PP^1$ given by $t \mapsto \left\langle R(t) \cdot \begin{pmatrix} 1 & 0 \end{pmatrix}^\intercal \right\rangle$ which maps $(0,1)$ bijectively onto $\PP^1 \setminus \{x_0\}$. Since $\SL(2,\RR)$ acts transitively on the real projective line $\PP^1$ with $B_{\SL(2,\RR)}$ being the stabilizer of $x_0:= \left\langle \begin{pmatrix} 1 & 0 \end{pmatrix}^\intercal \right\rangle$, one has a bijective correspondence $gB \mapsto g x_0$ between $\SL(2,\RR)/B_{\SL(2,\RR)}$ and $\PP^1$. This yields the desired surjectivity and bijectivity properties of $\tilde{R}$. Continuity is clear, as well as the fact that the restriction to the interior is a homeomorphism.
\end{proof}
The following Lemma is a consequence of \cite[Ch. 7, Thm 2.1]{Mas}.
\begin{lem}
\label{presentation cw}
Let $X$ be a $CW$ complex with only one $0$-cell $x_0$. For each $\lambda \in \Lambda_2$, let $f_\lambda:[0,1] \to S^1$ be a loop whose homotopy class generates $\pi_1(S^1)$ and whose image $\gamma_\lambda:= \chi_\lambda \circ f_\lambda$ under $\chi_\lambda$ is a loop in $X^{(1)}$ starting at $x_0$. Then \[ \left \langle [\chi_\mu], \quad \mu \in \Lambda_1 \mid [\gamma_\lambda], \quad \lambda \in \Lambda_2 \right \rangle\] is a presentation of $\pi_1(X,x_0)$, where the brackets denote the respective homotopy classes in $X^{(1)}$.
\end{lem}
\begin{de}
Let $D^1 = [0,1]$ be the one-dimensional unit disc and note that $D^2 \hom D^1 \times D^1$. For $i,j \in I$ let $\gamma_i, \gamma_j$ be as in Remark \ref{homeos alpha beta}. Let $p:G \to G/B$ be the canonical projection. Define $\chi_i: D^1 \to G/B$ and $\chi_{(i,j)}: D^1 \times D^1 \to G/B$ by \begin{itemize}
\item $\chi_i(s) := p(\gamma_i(R(s))) = \gamma_i(R(s))\cdot B$,
\item $\chi_{(i,j)}(s,t):= p(\gamma_i(R(s))\gamma_j(R(t))) = \gamma_i(R(s))\gamma_j(R(t))\cdot B$.
\end{itemize}
\end{de}
The following Lemma was inspired by \cite[Ch. 10, second Proposition of 6.8]{Pro}, see also \cite[\S2.6, p.198]{Kac}.
\begin{lem}
\label{char maps}
Let $G$ be two-spherical or symmetrizable. Then the maps defined above are characteristic maps for the following cells:
\begin{enumerate}
\item $\chi_i$ for $C_{{\sigma}_i}(B) = B {s}_i B /B $,
\item $\chi_{(i,j)}$ for $C_{{\sigma}_i {\sigma}_j}(B) = B {s}_i {s}_j B /B$.
\end{enumerate}
\begin{proof}
(a): One has to show that $\chi_i([0,1]) \subseteq C_{\leq \sigma_i}(B)$ and that $\chi_i$ is a continuous map which maps $(0,1)$ homeomorphically to $C_{\sigma_i}(B)$. The first assertion is clear, since by Lemma \ref{P_i = G_iB} one has $C_{\leq \sigma_i} = G_iB/B$.
By Lemma \ref{P_i = G_iB}, one has $C_{\sigma_i}(B) = \{kB \mid k \in K_i \setminus (K_i \cap B)\}$. Let $k \in K_i \setminus (K_i \cap B)$. Then $\gamma_i^{-1}(p^{-1} (kB)) = \gamma_i^{-1}(k)\cdot B_{\SL(2,\RR)} \in \SL(2,\RR)/B_{\SL(2,\RR)} \setminus B_{\SL(2,\RR)}$. By Lemma \ref{the map R}, there exists a unique $s \in (0,1)$ satisfying $R(s)B_{\SL(2,\RR)} = \gamma_i^{-1}(k) B_{\SL(2,\RR)}$. Hence, $s$ is the unique preimage of $kB$ under $\chi_i$. This yields the desired bijectivity property. The continuity properties are clear.
(b): Since by Lemma \ref{P_i = G_iB} (c) one has $C_{\leq \sigma_i \sigma_j}(B) = K_iBK_jB/B$, it is clear that $\chi_{(i,j)}([0,1]\times [0,1]) \subseteq C_{\leq {\sigma}_i {\sigma}_j}(B)$. For the injectivity of the restriction, let $(s,t), (\tilde{s}, \tilde{t}) \in (0,1)^2$ such that $\chi_{(i,j)}(s,t) = \chi_{(i,j)}(\tilde{s}, \tilde{t})$. Then \begin{align*}
\gamma_i(R(s)) \gamma_j(R(t))B & = \gamma_i(R(\tilde{s})) \gamma_j(R(\tilde{t}))B\\
\iff (\gamma_i(R(\tilde{s})))^{-1} \gamma_i(R(s)) \gamma_j(R(t))B & = \gamma_j(R(\tilde{t}))B \in C_{\sigma_j}(B). \end{align*}
This implies $R(\tilde{s})^{-1} R(s) \in B_{\SO(2, \RR)}$, since otherwise the left expression is in $C_{\sigma_i\sigma_j}(B)$, contradicting $C_{\sigma_i \sigma_j}(B) \cap C_{\sigma_j}(B) = \emptyset$. Since $s, \tilde{s} \in (0,1)$, one obtains $\tilde{s} = s$. It follows that $\chi_j(t) = \chi_j(\tilde{t})$, hence $t = \tilde{t}$ by (a).
For the surjectivity, note that by Lemma \ref{P_i = G_iB} (c), one has $C_{\sigma_i\sigma_j}(B) = Bs_is_jB/B = Bs_iBBs_jB/B$. Let $x_i x_jB$ be an arbitrary element of $C_{\sigma_i\sigma_j}(B)$ with $x_i = b_1 s_i b_2 \in B s_i B$ and $ x_j \in B s_j B$. By (a), there exists an $s \in (0,1)$ with $\gamma_i(R(s))B = b_1 s_iB \in C_{\sigma_i}(B)$. Hence, there exists a $b \in B$ with $(\gamma_i(R(s))b = b_1 s_i b_2 = x_i$. Again by (a), there exists a $t \in (0,1)$ with $\gamma_j(R(t))B = bx_jB \in C_{\sigma_j}(B)$. This yields \begin{align*}
\chi_{i,j}(s,t) & = \gamma_i(R(s)) \cdot \gamma_j(R(t)) B\\
& = x_i b^{-1} \cdot bx_j B\\
=x_i x_j B.
\end{align*}
This proves that $\chi_{i,j}$ maps $(0,1) \times (0,1)$ bijectively to $C_{\sigma_i\sigma_j}(B)$ The continuity properties are clear.
\end{proof}
\end{lem}
\begin{nota}
\label{epsilon}
For $i,j \in I$, let $\epsilon(i,j):= (-1)^{\langle\check{\alpha}_i, \alpha_j\rangle},$ where $\langle\check{\alpha}_i, \alpha_j\rangle$ is the $(i,j)$-entry of the Cartan matrix of $\Pi$.
\end{nota}
\begin{lem}[{\cite[Remark 15.4(1)]{GHKW}}]
\label{vertauschen}
Let $e_i:= \gamma_i(-I) \in G_i$ with $\gamma_i$ as in Remark \ref{homeos alpha beta} and $k_j \in K_j$. Then $e_i k_j e_i = k_j^{\epsilon(i,j)}$.
\end{lem}
\begin{thm}
\label{fundamental group}
If the Bruhat decomposition satisfies the conclusion of Proposition \ref{bruhat decomp is cw decomp}, then a presentation of $\pi_1(G/P_J)$ is given by
\[ \left\langle x_i; \quad i \in I \mid x_ix_j^{\epsilon(i,j) } = x_jx_i,\quad x_k = 1; \quad i,j \in I, k \in J \right\rangle.\]
In particular, this statement holds in the two-spherical and the symmetrizable case.
\begin{proof}
By Lemma \ref{hom between BwB/B and BwP/P} and Proposition \ref{bruhat decomp is cw decomp}, the Bruhat decomposition $G/P_J = \bigsqcup_{w\in W^J}BwP_J/P_J$ is a $CW$ decomposition where each cell $BwP_J/P_J$ has dimension $l(w)$. For each 1-cell $Bs_iP_J/P_J$ and 2-cell $Bs_is_jP_J/P_J$, the compositions $\tilde{\chi}_i := \psi_{s_i} \circ \chi_i$ and $\tilde{\chi}_{(i,j)}:= \psi_{s_is_j} \circ \chi_{(i,j)}$ are, respectively, characteristic maps ($\psi_{s_i}$ and $\psi_{s_is_j}$ denoting the canonical homeomorphisms from Lemma \ref{hom between BwB/B and BwP/P}).
Lemma \ref{presentation cw} gives a presentation of $\pi_1(G/P_J)$. The generating elements are given by the homotopy classes $x_i:=[\tilde{\chi}_i]$ of the characteristic maps of the 1-cells -- namely, the cells $B s_i P_J/P_J$ where $i \in I \setminus J$. For the homotopy classes $x_k$ with $k \in J$, note that $\gamma_k(R(t)) \in G_k \subseteq P_J$, and so $\tilde{\chi}_k(t) = \gamma_k(r(t)) \cdot P_J = P_J$ which implies $x_k = [\tilde{\chi}_k] = 1_{\pi_1(G/P_J)}$. This yields the desired generating set as well as the trivial relation $x_k = 1$ for $i \in J$.
To obtain the set of relators, for $k = 1, \dots, 4$ let $\phi_k: [0,1] \to [0,1] \times [0,1]$ where \begin{align*}
\phi_1(t) &= (t,0),\\
\phi_2(t) &= (1,t),\\
\phi_3(t) &= (1-t,1),\\
\phi_4(t) &= (0,1-t).
\end{align*}
Then the concatenation $\phi:= \phi_1 * \phi_2 * \phi_3 * \phi_4$ is a loop in the relative boundary $\partial([0,1] \times [0,1]) \simeq S^1$ which generates its fundamental group. Moreover, for each characteristic map $\tilde{\chi}_{(i,j)}$ of a 2-cell, one has $\tilde{\chi}_{(i,j)}(\phi(0)) = \tilde{\chi}_{(i,j)}((0,0)) = \psi_{s_is_j}({\chi}_{(i,j)}(0,0)) = \psi_{s_is_j}(B) = P_J$ where $P_J$ is the unique 0-cell of the CW complex. Therefore, Lemma \ref{presentation cw} implies that the set of relators is given by $\{[\tilde{\chi}_{(i,j)} \circ \phi] \mid \sigma_i\sigma_j \in W^J, l(\sigma_i\sigma_j) = 2\}$. Now,
\begin{align*}
[\tilde{\chi}_{(i,j)} \circ \phi] & = [\tilde{\chi}_{(i,j)} \circ \phi_1] \cdot [\tilde{\chi}_{(i,j)} \circ \phi_2] \cdot [\tilde{\chi}_{(i,j)} \circ \phi_3]\cdot [\tilde{\chi}_{(i,j)} \circ \phi_4], \end{align*}
where
$\tilde{\chi}_{(i,j)} (s,t) = \alpha_i(R(s))\alpha_j(R(t))\cdot P_J$ with $R(0) = I_{\SO(2,\RR)}, R(1) = -I_{\SO(2,\RR)} \in B_{\SO(2,\RR)}$ which implies
\begin{align*}
[\tilde{\chi}_{(i,j)} \circ \phi_1] &= x_i, \\
[\tilde{\chi}_{(i,j)} \circ \phi_3] &= x_i^{-1},\\
[\tilde{\chi}_{(i,j)} \circ \phi_4] &= x_j^{-1}.
\end{align*}
Moreover, \begin{align*}
(\tilde{\chi}_{(i,j)} \circ \phi_2)(t) &= \alpha_i(-I)\alpha_j(R(t))\cdot P_J\\
& = \alpha_i(-I)\alpha_j(R(t)) \alpha_i(-I) \cdot P_J, \quad \text{ since }\alpha_i(-I)\in P_J\\
& = \alpha_j(R(t))^{\epsilon(i,j)}\cdot P_J \quad \text{ by Lemma \ref{vertauschen}}.
\end{align*}
Since $R(t)^{-1} = R(1-t)$, this yields $[\tilde{\chi}_{(i,j)} \circ \phi_2] = x_j^{\epsilon(i,j)}$. One therefore obtains $[\tilde{\chi}_{(i,j)} \circ \phi] = x_i \cdot x_j^{\epsilon(i,j)} \cdot x_i^{-1} \cdot x_j^{-1}$. This proves the assertion.
\end{proof}
\end{thm}
\begin{lem}
\label{cardinality fundamental group}
Let $\Pi$ be simply-laced and $J \neq \emptyset$. Then $\pi_1(G/P_J) \iso {C_2}^{n-|J|}$.
\begin{proof}
For each generator $x_h$ in the presentation of Theorem~\ref{fundamental group}, one has $x_h^2 = 1$: Recall that $\lambda$ denotes the labelling map $I \to V$ of the vertex set of $\Pi$. Since $\Pi$ is connected, one has a minimal path $(i_1, \dots, i_m = h)^\lambda$ in $\Pi$ such that $i_1 \in J$. If $m = 1$, one has $x_h = 1$ by the presentation above. Let $x_{i_1}, \dots, x_{i_{m-1}}$ have order $\leq 2$. Since $\Pi$ is simply-laced, $\epsilon(m-1,h) = -1 = \epsilon (h,m-1)$ which implies $x_h x_{i_{m-1}}^{-1} x_h^{-1} x_{i_{m-1}}^{-1} = 1$ and $x_{i_{m-1}} x_h^{-1} x_{i_{m-1}}^{-1} x_h^{-1} = 1$. Multiplying these expressions yields $x_h^2 = 1$.
Since each generator has order $\leq 2$, the relations show that the group is abelian. One concludes that $\pi_1(G/P_J) \cong {C_2}^{n-|J|}$.
\end{proof}
\end{lem}
\section{The fundamental groups of \texorpdfstring{$G(\Pi)$}{K(Pi)} and \texorpdfstring{$\Spin(\Pi,\kappa)$}{Spin(Pi,kappa)}}
\label{spin_simply_con}
\begin{lem}
\label{homeo K/(K cap P_J) to G/P_J}
The canonical map $\psi: K/(K \cap P_J) \to G/P_J$ is a homeomorphism.
\end{lem}
\begin{proof}
Bijectivity follows from the product formula for subgroups since $G = KP_J$.
By Lemma \ref{quotient basics}, the map $\tilde{\psi}: G/(K \cap P_J) \to G/P_J$ is continuous, so the same holds for its bijective restriction $\psi: K/(K \cap P_J) \to G/P_J$.
In order to show that $\psi$ is closed, let $P:= P_J$ and let $\tilde{P} := P_J \cap K$. Consider the commutative diagram \[\begin{tikzcd}
K/\tilde{P} \arrow[rd, "\psi"] \arrow[d, "\iota"] \\
G/\tilde{P} \arrow[r, "\phi"] & G/P
\end{tikzcd}\] where $\iota$ denotes the canonical embedding and $\phi$ denotes the canonical map from $G/\tilde{P}$ to $G/P$. Since $K$ is closed in $G$ by \cite[Section~3F]{FHHK}, the map $\iota$ is closed. By Lemma~\ref{quotient basics}, $\phi$ is open.
Let $X\tilde{P} \subseteq K/\tilde{P}$ be a closed subset of $K/\tilde{P}$ and
suppose that $\psi(X\tilde{P}) = X P$ is not closed in $G/P$. Then the complement $\comp_{G/P}(XP)$ is not open in $G/P$, hence the complement $\comp_{G/\tilde{P}}(\phi^{-1} (XP)) = \phi^{-1} (\comp_{G/P}(XP))$ is not open in $G/\tilde{P}$. Therefore, $\phi^{-1} (XP)$ is not closed in $G/\tilde{P}$. This yields that $X\tilde{P} = \psi^{-1}(XP) = \iota^{-1}(\phi^{-1}(XP))$ is not closed in $K/\tilde{P}$, a contradiction.
\end{proof}
\begin{cor}
\label{homeo G/P_J to K/(K cap T) K_J}
There exists a homeomorphism $G/P_J \to K/(K \cap T) K_J$.
\end{cor}
\begin{proof}
Since $P_J = G_JB$ and $\theta(P_J) \cap P_J = G_J T$, one has $P_J \cap K = K_J (K \cap T)$. Furthermore, $G_J$ is normal in $G_J T$ which implies $K_J(K \cap T) = (K \cap T) K_J$. The claim now follows from Lemma~\ref{homeo K/(K cap P_J) to G/P_J}.
\end{proof}
\begin{lem}
\label{covering basics}
Let $\phi: X \to Y$ be a continuous, open, surjective map between Hausdorff topological spaces. If all fibers are finite and of constant cardinality, then $\phi$ is a covering map.
\begin{proof}
Let $y \in Y$ and let $\phi^{-1}(y) = \{x_1, \dots, x_k\} \subseteq X$. Since $X$ is Hausdorff, for $i = 1, \dots, k$ there exist neighborhoods $U_i$ of $x_i$ with $\bigcap_{i = 1}^k U_i = \emptyset$. Let $V:= \bigcap_{i = 1}^k \phi(U_i)$. Then $V$ is open since $\phi$ is open and $V \neq \emptyset$ since $y \in V$. The preimage $\phi^{-1}(V)$ is a disjoint union of open sets $\tilde{U}_i:= \phi^{-1}(V) \cap U_i$ and each $\tilde{U}_i$ is mapped bijectively to $V$: Surjectivity is clear; for the injectivity let $y' \in V$. Then each $\tilde{U}_i$ contains a preimage of $y'$. Since all fibers have constant cardinality $k$, it follows that $|\tilde{U}_i \cap \phi^{-1}(y')| = 1$. This proves the assertion.
\end{proof}
\end{lem}
\begin{lem}
\label{degree of K to K/T covering}
The canonical map $\psi: K/K_J \to K/(K \cap T)K_J$ is a covering map of degree $2^{n-|J|}$.
\end{lem}
\begin{proof}
By Lemma \ref{quotient basics}, $\psi$ is continuous, open and surjective.
By \cite[Lemma~3.20 and the discussion after Prop 3.8]{FHHK}, the group $\tilde{T}:=(K \cap T)$ has order $2^n$.
Note that one has $T_J \cap T_{I\setminus J} = \{1\}$, since the Kac--Moody group $G$ being algebraically simply connected implies $T \iso T_J \times T_{I \setminus J}$.
Now, for $k \in K$ one has $\psi^{-1} (k\tilde{T}K_J) = \{ktK_J \mid t \in \tilde{T}\}$, and since $T_J \cap T_{I\setminus J} = \{1\}$, one has $k t_i K_J \neq k t_j K_J$ for $t_i \neq t_j \in T \cap K_{I\setminus J}$. This yields $|\psi^{-1} (k\tilde{T}K_J)| = |\{ktK_J \mid t \in \tilde{T}\}| = |\{ktK_J \mid t \in T \cap K_{I\setminus J}\}| = | T \cap K_{I\setminus J} | = | T_{I\setminus J} \cap K_{I\setminus J} |= 2^{n-|J|}$. Lemma \ref{covering basics} now shows that $\psi$ is a covering map.\end{proof}
\label{general case}
\begin{de}[{\cite[Definition~16.2]{GHKW}}]
Let $\Pia$ be the graph on the vertex set $V$ with edge set \[\{ \{i,j\} \in V \times V \mid i \neq j \in I, \epsilon(i,j) = \epsilon(j,i) = -1 \},\]
where $\epsilon(i,j)$ denotes the parity of the corresponding Cartan matrix entry, as defined in Notation \ref{epsilon}.
An \emph{admissible colouring} of $\Pi$ is a map $\kappa: V \to \{1,2\}$ such that
\begin{enumerate}
\item $\kappa(i^\lambda) = 1$ whenever there exists $j \in I \setminus\{i\}$ with $\epsilon(i,j) = 1$ and $ \epsilon(j,i) = -1$.
\item the restriction of $\kappa$ to any connected component of the graph $\Pia$ is a constant map.
\end{enumerate}
Define $c(\Pi,\kappa)$ to be the number of connected components of $\Pia$ on which $\kappa$ takes the value 2.
For a subgraph $\Pia_J$ of $\Pia$ that is a union of connected components of $\Pia$ let $\kappa_J$ be the corresponding restriction of $\kappa$.\end{de}
\begin{de}
\label{colouring}
Let the colouring $\gamma: V \to \{r,g,b\}$ of $\Pia$ be defined as follows:\begin{enumerate}
\item $\gamma(i^\lambda) = r$ whenever there exists $j \in I \setminus\{i\}$ with $\epsilon(i,j) = 1$ and $ \epsilon(j,i) = -1$.
\item $\gamma(i^\lambda) =g$ whenever for each $j \in I \setminus\{i\}$, one has $(\epsilon(i,j), \epsilon(j,i)) \in \{(1,1), (-1,1)\}$.
\item $\gamma(i^\lambda) = b$ whenever the connected component of $i^\lambda$ in $\Pia$ contains more than one vertex and case (a) applies to none of the vertices in this component.
\item The restriction of $\gamma$ to any connected component of the graph $\Pia$ is a constant map.
\end{enumerate}
\end{de}
We refer to the introduction for a discussion of various examples.
\begin{de+rem}
\label{Spin remark}
Recall from the introduction that in \cite[Definition~16.16]{GHKW}, the \emph{spin group $\Spin(\Pi,\kappa)$ with respect to $\Pi$ and $\kappa$} is defined as the universal enveloping group of a particular $\Spin(2)$-amalgam $\{\tilde{G}_{ij}, \tilde{\oldphi}_{ij}^i\mid i \neq j \in I \}$ where the isomorphism type of $\tilde{G}_{ij}$ depends on the $(i,j)$- and $(j,i)$-entries of the Cartan matrix of $\Pi$ as well as the values of $\kappa$ on the corresponding vertices. The group $K(\Pi)$ can be regarded as (being uniquely isomorphic to) the universal enveloping group of an $\SO(2,\RR)$-amalgam $\{G_{ij}, \oldphi_{ij}^i\mid i \neq j \in I \}$ where each $\tilde{G}_{ij}$ covers $G_{ij}$ via an epimorphism $\alpha_{ij}$.
By \cite[Lemma 16.18]{GHKW} there exists a canonical central extension $\rho_{\Pi,\kappa}: \Spin(\Pi, \kappa) \to K(\Pi)$ that makes the following diagram commute for all $i \neq j \in I$:
\[\begin{tikzcd}
\tilde{G}_{ij} \arrow[r, "\tilde{\tau}_{ij}"] \arrow[d, " \alpha_{ij}"] & \Spin(\Pi,\kappa) \arrow[d, "\rho_{\Pi,\kappa}"] \\
G_{ij} \arrow[r, "{\tau}_{ij}"] & K(\Pi)
\end{tikzcd}\]
Here, $\tilde{\tau}_{ij}$ and ${\tau}_{ij}$ denote the respective canonical maps into the universal enveloping groups.
By \cite[Proposition~3.9]{GHKW}, one has \[\ker(\rho_{\Pi,\kappa}) = \langle \tilde{\tau}_{ij}(\ker(\alpha_{ij})) \mid i \neq j \in I \rangle_{\Spin(\Pi,\kappa)}.\]
Each connected component of $\Pia$ that admits a vertex $i^\lambda$ with $\kappa(i^\lambda) = 2$ contributes a factor $2$ to the order of $\ker(\rho_{\Pi,\kappa})$ so that $\Spin(\Pi,\kappa)$ is a $2^{c(\Pi,\kappa)}$-fold central extension of $K(\Pi)$.
In particular, this implies that the subspace topology on $K(\Pi)$ defines a unique topology on $\Spin(\Pi)$ that turns the extension into a covering map. The resulting group topology on $\Spin(\Pi, \kappa)$ is called the {\em Kac--Peterson topology} on $\Spin(\Pi, \kappa)$.
In the case of a simply-laced diagram $\Pi$, the only admissible colourings are the trivial colouring and the constant colouring $\kappa: V \to \{2\}$ and we define the \emph{spin group $\Spin(\Pi)$ with respect to $\Pi$} as $\Spin(\Pi) := \Spin(\Pi, \kappa)$.
\end{de+rem}
Before turning to the general case, we will first consider the simply laced case and formulate and prove the corresponding simplified versions of the main theorems.
\begin{lem}
\label{homeo Spin(Pi)/Spin(Pi_{ij}) and K/K_{ij}}
Let $\Pi$ be simply laced and let $\{i,j\} \subseteq I$ be the index set of an $A_2$-subdiagram of $\Pi$. Then the spaces $\Spin(\Pi)/\Spin(\Pi_{ij})$ and $K/K_{ij}$ are homeomorphic.
\begin{proof}
From \cite{GHKW} (exact references below) it follows that the kernel of the covering map $\Spin(\Pi) \to K$ coincides with the kernel of the covering map $\Spin(\Pi_{ij}) \to K_{ij}$ and is equal to the group $Z:= \{\pm 1_{\Spin(\Pi)}\}$ (for the definition of $-1_{\Spin(\Pi)}$, see below).
This is a consequence of the following facts regarding an irreducible simply-laced diagram $\Pi$ (all referring to \cite{GHKW}):
\begin{itemize}
\item There is an epimorphism $\Spin(2) \to \SO(2,\RR)$ with kernel $\{\pm 1_{\Spin(2)}\}$ (see [Theorem~6.8]).
\item In $\Spin(\Pi)$, all elements $\tilde{\tau}_{ij}(\tilde{\oldphi}_{ij}^i(-1_{\Spin(2)}))$ coincide
(see [Lemma~11.7]).
\item Let $-1_{\Spin(\Pi)}:= \tilde{\tau}_{ij}(\tilde{\oldphi}_{ij}^i(-1_{\Spin(2)}))$ for an arbitrary pair $i \neq j \in I$.
Then $1_{\Spin(\Pi)} \neq -1_{\Spin(\Pi)}$ (see [Corollary~11.16]).
\item $\Spin(\Pi)$ is a 2-fold central extension of $K(\Pi)$ (see [Theorem~11.17]).
\end{itemize}
Hence, the 2-fold covering map $\tilde{\phi}: \Spin (\Pi) \to K(\Pi)$ induces a continuous bijective map $\phi: \Spin(\Pi) / \Spin(\Pi_{ij}) \to (\Spin(\Pi) / Z) / (\Spin(\Pi_{ij}) / Z) \to K / K_{ij}$. One has a commutative diagram \[\begin{tikzcd}
\Spin(\Pi) \arrow[r, "\tilde{\phi}"] \arrow[d, "\pi_1"] & K \arrow[d, "\pi_2"] \\
\Spin(\Pi)/\Spin(\Pi_{ij}) \arrow[r, "\phi"] & K/K_{ij}
\end{tikzcd}.\]
Since $\tilde{\phi}$ is open as a covering map and $\pi_2$ is open by Lemma \ref{quotient basics}, it follows that $\phi$ is a homeomorphism.
\end{proof}
\end{lem}
\begin{cor}
\label{K/K_J is simply connected.}
Let $\Pi$ be simply laced. Then $K/K_J$ is simply connected.
\end{cor}
\begin{proof}
$K/K_J$ is connected since $K$ is generated by connected groups isomorphic to $\SO(2,\RR)$. Hence by Lemma \ref {degree of K to K/T covering} it is a non-trivial cover of $K/(K \cap T)K_J$ of degree $2^{n-|J|}$. The claim now follows from Corollary~\ref{cardinality fundamental group} and Corollary~\ref{homeo G/P_J to K/(K cap T) K_J}.
\end{proof}
The following proposition provides our main result in the simply laced case.
\begin{prop} \label{simplyconnected}
Let $\Pi$ be irreducible and simply laced. Then $\Spin(\Pi)$ is simply connected with respect to the Kac--Peterson topology. In particular, $\pi_1(G) \iso C_2$.
\begin{proof}
By \cite[4.2.4]{Hus}, for a closed subgroup $H$ of a topological group $G$, the projection $p: G \to G/H$ is a principal $H$-bundle. By Lemma \ref{Palais},
this bundle is locally trivial if $H$ is a (closed) Lie group (note that, by \cite[Theorem~5.11]{HR}, every locally compact subgroup of a topological group is closed). Since locally trivial bundles admit local cross sections, \cite[Corollary in Section~7.4]{Ste1} implies that, if $H$ is a closed Lie group, then $p: G \to G/H$ is a fibre bundle with fibre $H$. This yields a \lila{locally trivial} fibre bundle
\[\begin{tikzcd}
\Spin(\Pi_{ij}) \arrow[r] & \Spin(\Pi) \arrow[r] & \Spin(\Pi)/\Spin(\Pi_{ij}).
\end{tikzcd}\]
By \cite[Chapter~4]{Hatcher}, this yields the homotopy long exact sequence
\begin{eqnarray}
\pi_4(\Spin(\Pi)/\Spin(\Pi_{ij})) & \to & \pi_3(\Spin(\Pi_{ij})) \to \pi_3(\Spin(\Pi)) \to
\pi_3(\Spin(\Pi)/\Spin(\Pi_{ij})) \notag \\ & \to & \pi_2(\Spin(\Pi_{ij})) \to \pi_2(\Spin(\Pi)) \to
\pi_2(\Spin(\Pi)/\Spin(\Pi_{ij})) \notag \\ & \to & \pi_1(\Spin(\Pi_{ij})) \rightarrow \pi_1(\Spin(\Pi)) \rightarrow \pi_1(\Spin(\Pi)/\Spin(\Pi_{ij})) \notag \\ \label{homotopylong}
\end{eqnarray}
from which one extracts the exact sequence
\[ \{1\} = \pi_1(\Spin(\Pi_{ij})) \rightarrow \pi_1(\Spin(\Pi)) \rightarrow \pi_1(\Spin(\Pi)/\Spin(\Pi_{ij})). \]
By Corollary~\ref{K/K_J is simply connected.} and Lemma~\ref{homeo Spin(Pi)/Spin(Pi_{ij}) and K/K_{ij}} one has $ \pi_1(\Spin(\Pi)/\Spin(\Pi_{ij})) \iso \pi_1(K/K_{ij}) = \{1\}$ and so by exactness $\pi_1(\Spin(\Pi)) = \{ 1 \}$.
The second assertion follows from the fact that $\pi_1(G) \iso \pi_1(K)$ by Corollary \ref{FundamentalGroups2} and the fact that $\Spin(\Pi)$ is a 2-fold central extension of $K$ by \cite[Theorem~11.17]{GHKW}.
\end{proof}
\end{prop}
We will now return to the case of a general irreducible Dynkin diagram $\Pi$.
\begin{nota}
\label{HJ}
For a subset $J \subseteq I$ let
\[ H_J:= \left\langle x_i; \quad i \in J \mid x_ix_j^{\epsilon(i,j) } = x_jx_i,; \quad i,j \in J\right\rangle.\]
\end{nota}
\begin{lem}
\label{connected component subgroups}
Let $J \subseteq I$ be the index set of a connected component $\Pia_J$ of $\Pia$. Then the following hold:
\begin{enumerate}
\item If $\Pia_J$ has colour $r$, then $H_J \iso C_2^{|J|}$.
\item If $\Pia_J$ has colour $g$, then $|J| = 1$ and $H_J \iso \ZZ$.
\item If $\Pia_J$ has colour $b$, then $|H_J| = 2^{|J|+1}$.
\end{enumerate}
\begin{proof}
(a): If $\Pia_J$ has colour $r$, then there exist $i \in J, j \in I \setminus\{i\}$ with $\epsilon(i,j) =1$ and $\epsilon(j,i) = -1$. This implies $x_ix_j = x_j x_i$ and $x_j x_i^{-1} = x_i x_j$ which yields $x_i^2 = 1$. Now, if $\{i^\lambda, k^\lambda\}$ is an edge in $\Pia$, then $x_i x_k^{-1}x_i^{-1} x_k^{-1} = 1 = x_k x_i^{-1} x_k^{-1} x_i^{-1}$. Multiplying these expressions shows that $x_i^2 = 1$ implies $x_k^2 = 1$. Since $\Pia_J$ is connected, this yields $x_k^2 = 1$ for each $k \in J$. Commutativity then follows from the relations of $H_J$.
(b): By definition, two nodes $i^\lambda$ and $j^\lambda$ in $\Pia$ are connected if and only if $\epsilon(i,j) = \epsilon(j,i) = -1$, and by definition a node $i^\lambda$ has color $g$ if and only if for each node $j^\lambda$ this is not the case -- namely, one of $\epsilon(i,j)$ and $\epsilon(j,i)$ is 1. Nodes of colour $g$ are therefore isolated in $\Pia$.
(c): Let $\Pi_J^{\mathrm{sl}}$ be the simply laced Dynkin diagram with vertex set $J^\lambda$ and edge set $\{ \{i,j\} \in J \times J \mid \{i,j\} \text{ edge in }\Pia\}$.
Let $\tilde{T}:= K(\Pi_J^{\mathrm{sl}}) \cap T(\Pi_J^{\mathrm{sl}})$ where $T(\Pi_J^{\mathrm{sl}})$ denotes the standard maximal torus of $G(\Pi_J^{\mathrm{sl}})$. Then by Lemma \ref{degree of K to K/T covering} and Proposition \ref{simplyconnected}, $\Spin(\Pi_J^{\mathrm{sl}}) \to K(\Pi_J^{\mathrm{sl}}) \to K(\Pi_J^{\mathrm{sl}})/\tilde{T}$ is a universal covering map where $K(\Pi_J^{\mathrm{sl}}) \to K(\Pi_J^{\mathrm{sl}})/\tilde{T}$ has degree $2^{|J|}$ and $\Spin(\Pi_J^{\mathrm{sl}}) \to K(\Pi_J^{\mathrm{sl}})$ has degree $2$ according to \cite[Theorem~11.17]{GHKW}. Since $\pi_1(K(\Pi_J^{\mathrm{sl}})/\tilde{T}) \iso H_J$ by Theorem \ref{fundamental group}, this implies $|H_J| = 2^{|J|+1}$.
\end{proof}
\end{lem}
\begin{prop}
\label{fundamental group as product}
Let $J_1 \sqcup \dots \sqcup J_k = I$ be the index sets of the connected components of $\Pia$. If the Bruhat decomposition satisfies the conclusion of Proposition \ref{bruhat decomp is cw decomp}, then \[\pi_1(G/B) \iso H_{J_1} \times \dots \times H_{J_K}. \]
\begin{proof}
By Theorem \ref{fundamental group}, $\pi_1(G/B) \iso H_I$ where \[H_I = \left\langle x_i; \quad i \in I \mid x_ix_j^{\epsilon(i,j) } = x_jx_i,; \quad i,j \in I\right\rangle\] as defined in \ref{HJ}. For $J \subseteq I$, let
\begin{equation}
\label{R_J}
R_J:= \{x_ix_j^{\epsilon(i,j)}x_i^{-1} x_j^{-1} \mid i,j \in J\},
\end{equation} the set of relators of $H_J$. Let \[R^c:= \bigcup_{\substack{i^\lambda, j^\lambda \text{in different}\\\text{conn. components}}}\{x_ix_j x_i^{-1} x_j^{-1}\},\] the set of commutators of pairs of generators from different connected components. Then \[H_{J_1} \times \dots \times H_{J_k} \iso \left \langle x_i; \quad i \in I \mid \bigcup_{l=1}^k R_{J_l} \cup R^c \right\rangle =: H.\] Let $\pi_{H_I}$ and $\pi_{H}$ be the canonical homomorphisms from the free group $\langle x_i; \quad i \in I\rangle$ to $H_I$ and $H$, respectively. It suffices to show that $\bigcup_{l=1}^k R_{J_l} \cup R^c \subseteq \ker \pi_{H_I}$ and $R_{I} \subseteq \ker \pi_H$.
It is clear that a relator $x_ix_j^{\epsilon(i,j)}x_i^{-1} x_j^{-1} \in R_I$ with $i^\lambda$ and $j^\lambda$ in a common connected component is contained in $\bigcup_{l=1}^k R_{J_l} \subseteq \ker \pi_H$, so let $x_ix_j^{\epsilon(i,j)}x_i^{-1} x_j^{-1} \in R_I$ with $i^\lambda$ and $j^\lambda$ in different connected components. Then one has $(\epsilon(i,j), \epsilon(j,i)) \in \{ (1,1), (1,-1), (-1,1)\}$. If $\epsilon(i,j) = 1$, then $x_ix_j^{\epsilon(i,j)}x_i^{-1} x_j^{-1} \in R^c \subseteq \ker \pi_H$, so let $\epsilon(i,j) = -1$ and $\epsilon (j,i) = 1$. Then $j^\lambda$ is contained in a connected component $\Pia_{J_m}$ of colour $r$, and by Lemma \ref{connected component subgroups}, $\langle x_l; \quad l \in J_m \mid R_{J_m} \rangle = H_{J_m} \iso C_2^{|J_l|}$.
This implies that $x_j$ has order $2$ in ${H_{J_m}}$, hence $x_j^2 \in \langle \langle R_{J_m} \rangle \rangle_{\langle x_i; i \in I\rangle}$, the normal closure of $R_{J_m}$ in the free group.
Since $\langle \langle R_{J_m} \rangle \rangle_{\langle x_i; i \in I\rangle} \subseteq \ker \pi_H$, one obtains $x_j^2 \in \ker \pi_H$. Since $x_j x_i x_j^{-1} x_i^{-1} \in R^c \subseteq \ker \pi_H$ and $\epsilon(i,j) = -1$, one therefore has \[\pi_H(x_ix_j^{\epsilon(i,j)}x_i^{-1} x_j^{-1}) = \pi_H (x_j x_i x_j^{-1} x_i^{-1} \cdot x_ix_j^{\epsilon(i,j)}x_i^{-1} x_j^{-1} ) = 1_H.\]
Conversely, it is clear that $\bigcup_{l=1}^k R_{J_l} \subseteq R_{I} \subseteq \ker \pi_{H_I}$, so let $x_i x_j x_i^{-1} x_j^{-1} \in R^c$ with $i^\lambda$ and $j^\lambda$ in different connected components. As above, we can assume that $\epsilon(i,j) = -1$ and $\epsilon(j,i) = 1$. Since $x_j x_i^{\epsilon(j,i)} x_j^{-1} x_i^{-1} \in \ker \pi_{H_I}$, this implies \[ \pi_{H_I}(x_i x_j x_i^{-1} x_j^{-1}) = \pi_{H_I}(x_j x_i^{\epsilon(j,i)} x_j^{-1} x_i^{-1} \cdot x_i x_j x_i^{-1} x_j^{-1}) = 1_{H_I}.\] This proves the assertion.
\end{proof}
\end{prop}
\begin{thm}
\label{fundamental group of K}
Let $\Pi$ be an irreducible Dynkin diagram such that $G(\Pi)$ satisfies the conclusions of Proposition \ref{bruhat decomp is cw decomp} and of Theorem \ref{FundamentalGroups2}. Let $n(g)$ and $n(b)$ be the number of connected components of $\Pia$ of colour $g$ and $b$, respectively. Then
\[\pi_1(G(\Pi)) \iso \ZZ^{n(g)} \times C_2^{n(b)}.\] In particular, this statement holds in the symmetrizable case.
\begin{proof}
By Theorem \ref{FundamentalGroups2}, $\pi_1(G) \iso \pi_1(K)$, so it suffices to prove that $\pi_1(K)$ is of the given isomorphism type; note that Theorem \ref{FundamentalGroups2} has only been established in the symmetrizable case.
Let $J \subseteq I$. The diagram \[\begin{tikzcd}
K \arrow[r, "\phi"] \arrow[d, "p"] & K/K_J \arrow[d, "q"] \\
K/(K \cap T)\arrow[r, "\psi"] & K/(K \cap T)K_J \end{tikzcd},\] with all maps being the respective canonical maps, commutes. Since the maps are continuous by Lemma \ref{quotient basics}, one obtains a commutative diagram of induced homomorphisms \[\begin{tikzcd}
\pi_1(K) \arrow[r, "\phi_*"] \arrow[d, "p_*"] & \pi_1(K/K_J) \arrow[d, "q_*"] \\
\pi_1(K/(K \cap T))\arrow[r, "\psi_*"]& \pi_1(K/(K \cap T)K_J) \end{tikzcd},\]
where $p_*$ and $q_*$ are injective. By Theorem \ref{fundamental group} and Lemma \ref{homeo K/(K cap P_J) to G/P_J}, $\pi_1(K/(K \cap T))$ and $\pi_1(K/(K \cap T)K_J)$ can be identified with $H_I = \langle x_i; \quad i \in I \mid R_I\rangle$ and $\langle x_i; \quad i \in I \mid R_I \cup \{x_J \mid j \in J\}\rangle$, respectively ($R_I$ as in (\ref{R_J}) in the above proof), where $\psi_*$ corresponds to the canonical homomorphism between these groups as the proof of Theorem \ref{fundamental group} shows.
For the index set $J_m$ of a connected component of $\Pia$, let $\bar{J}_m:= I \setminus J_m$. Then by Proposition \ref{fundamental group as product}, \[\left\langle x_i; \quad i \in I \mid R_I \cup \{x_J \mid j \in \bar{J}_m\}\right \rangle \iso \left(\prod_{i=1}^k{H_{J_i}}\middle/\prod_{\substack{i=1\\i \neq m}}^k{H_{J_i}}\right) \iso H_{J_m}.\] Summing up, one obtains a commutative diagram \[\begin{tikzcd}
\pi_1(K) \arrow[r, "\phi_*"] \arrow[d, "p_*"] & \pi_1(K/K_{\bar{J}_m}) \arrow[d, "q_*"] \\
\prod_{i=1}^k{H_{J_i}} \arrow[r, "\pi_m"]& H_{J_m} \end{tikzcd},\] having replaced $p_*$ and $q_*$ from above with the correponding monomorphisms.
By Lemma \ref{degree of K to K/T covering}, the covering $K/K_{\bar{J}_m} \to K/K_{\bar{J}_m}(K \cap T)$ has degree $2^{n-|\bar{J}_m|} = 2^{|J_m|}.$ This implies that $\tilde{H}_m:= q_*(\pi_1(K/K_{\bar{J}_m}))$ is a subgroup of $H_{J_m}$ of index $2^{|J_m|}$. The isomorphism type of $\tilde{H}_m$ is uniquely determined by this index and Lemma \ref{connected component subgroups}: One has \[\tilde{H}_m \iso \begin{cases} \{1\}, & \text{if }\Pia_{J_m}\text{ has colour }r,\\
2\ZZ \iso \ZZ, & \text{if }\Pia_{J_m}\text{ has colour }g,\\
C_2 & \text{if }\Pia_{J_m}\text{ has colour }b. \end{cases}\]
Again by Lemma \ref{degree of K to K/T covering}, the covering $K \to K/(K \cap T)$ has degree $2^n$, so $p_*(\pi_1(K))$ is a subgroup of index $2^n$ of $\prod_{i=1}^k{H_{J_i}}$. The commutative diagram above implies that $\pi_1(K) \iso p_*(\pi_1(K)) \subseteq \pi_m^{-1}(\tilde{H}_m)$. Since this holds for the index set of every connected component of $\Pia$, one has $p_*(\pi_1(K)) \subseteq \tilde{H}_1 \times \dots \times \tilde{H}_m$. But the latter is a subgroup of index $2^{|J_1|}\cdot \dots \cdot 2^{|J_m|} = 2^n$ of $\prod_{i=1}^k{H_{J_i}}$, so equality holds. This proves the assertion.
\end{proof}
\end{thm}
\begin{thm}
\label{fundamental group of Spin}
Let $\Pi$ be an irreducible Dynkin diagram such that $G(\Pi)$ satisfies the conclusion of Proposition \ref{bruhat decomp is cw decomp}. Let $n(g)$ be the number of connected components of $\Pia$ of colour $g$. Let $n(b,\kappa)$ be the number of connected components of $\Pia$ on which $\kappa$ takes the value 1 and which have colour $b$. Then
\[\pi_1(\Spin(\Pi,\kappa)) \iso \ZZ^{n(g)} \times C_2^{n(b,\kappa)}.\] In particular, this statement holds in the two-spherical and the symmetrizable case.
\begin{proof}
By \cite[Theorem~17.1]{GHKW}, the map $ \rho_{\Pi,\kappa}:\Spin(\Pi, \kappa) \to K$ is a $2^{c(\Pi, \kappa)}$-fold central extension. Let $J$ be the index set of a connected component of $\Pia$ and let $\bar{J}:= I \setminus \bar{J}$. Let $U_{\bar{J}} := \langle \tilde{G}_{ij} \mid i \neq j \in J \rangle_{\Spin(\Pi,\kappa)}$.
Since $\rho_{\Pi,\kappa}(U_{\bar{J}}) \subseteq K_J$, one has a continuous induced map $\rho_{\Pi,\kappa}^J: \Spin(\Pi,\kappa) / \Spin(\Pi_{\bar{J}},\kappa_{\bar{J}}) \to K/K_{\bar{J}}$ making the following diagram commute, where $\tilde{\phi}$ and $\phi$ denote the respective canonical maps:
\[\begin{tikzcd}
\Spin(\Pi, \kappa) \arrow[r, "\tilde{\phi}"] \arrow[d, " \rho_{\Pi,\kappa}"] & \Spin(\Pi,\kappa) /U_{\bar{J}} \arrow[d, "\rho_{\Pi,\kappa}^J"] \\
K \arrow[r, "\phi"] & K/K_{\bar{J}}
\end{tikzcd}\]
Each fiber of $\rho_{\Pi,\kappa}^J$ has cardinality \begin{align*}
|\{xU_{\bar{J}} \mid x \in \ker \rho_{\Pi,\kappa}\}| &= |\ker(\rho_{\Pi,\kappa}) /(U_{\bar{J}} \cap \ker(\rho_{\Pi,\kappa}))|\\
& = 2^{c(\Pi, \kappa) - c(\Pi_{\bar{J}},\kappa_{\bar{J}}) }\text{ by Remark \ref{Spin remark}}
\end{align*}
Since $\rho_{\Pi,\kappa}$ is open as a covering map and $\phi$ is open by Lemma \ref{quotient basics}, it follows from Lemma \ref{covering basics} that $\rho_{\Pi,\kappa}^J$ is a covering map.
From here the proof is analogous to the proof of Theorem \ref{fundamental group of K}, after extending the commutative diagram at the beginning of the latter proof:
\[\begin{tikzcd}
\Spin(\Pi, \kappa) \arrow[r, "\tilde{\phi}"] \arrow[d, " \rho_{\Pi,\kappa}"] & \Spin(\Pi,\kappa) / U_{\bar{J}} \arrow[d, "\rho_{\Pi,\kappa}^J"] \\
K \arrow[r, "\phi"] \arrow[d, "p"] & K/K_J \arrow[d, "q"] \\
K/(K \cap T)\arrow[r, "\psi"] & K/(K \cap T)K_J \end{tikzcd}.\]
One obtains that $\pi_1(\Spin(\Pi, \kappa)) \iso \prod_{i = 1}^k H'_{J_i}$ where each $H'_{J_m}$ is a subgroup of index $2^{c(\Pi, \kappa) - c(\Pi_{\bar{J}_m},\kappa_{\bar{J}_m})}$ of \[\tilde{H}_m \iso \begin{cases} \{1\}, & \text{if }\Pia_{J_m}\text{ has colour }r,\\
2\ZZ \iso \ZZ, & \text{if }\Pia_{J_m}\text{ has colour }g,\\
C_2 & \text{if }\Pia_{J_m}\text{ has colour }b. \end{cases}\] Since $\Pia_{\bar{J}_m}$ is the union of all connected components except $\Pia_{J_m}$, one has $c(\Pi, \kappa) - c(\Pi_{\bar{J}_m},\kappa_{\bar{J}_m}) \in \{0,1\}$, depending on whether $\kappa$ is constant $1$ or $2$ on $\Pia_{J_m}$. This implies \[{H}'_m \iso \begin{cases} \{1\}, & \text{if }\Pia_{J_m}\text{ has colour }r,\\
\ZZ, & \text{if }\Pia_{J_m}\text{ has colour }g,\\
C_2 & \text{if }\Pia_{J_m}\text{ has colour }b\text{ and }\kappa\equiv 1\text{ on }\Pia_{J_m},\\
\{1\}, & \text{if }\Pia_{J_m}\text{ has colour }b\text{ and }\kappa\equiv 2\text{ on }\Pia_{J_m}.
\end{cases}\]
This proves the assertion.
\end{proof}
\end{thm}
Now all theorems from the introduction have been proved.
|
1,314,259,996,693 | arxiv | \section{Introduction}
Speech is one of the most natural ways of human communication, and the development of systems capable of transcribing and understanding speech automatically have shown great importance and applicability in various scenarios, as in personal assistants, tools for customer attendance and other products \cite{karpagavalli2016review}. These products are built using systems known as Automatic Speech Recognition (ASR)s.
Despite its
\section{Introduction}
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\bibliographystyle{splncs04}
\section{Introduction}
Speech is one of the most natural ways of human communication, and the development of systems, known as Automatic Speech Recognition (ASR) Systems, capable of transcribing speech automatically have shown great importance and applicability in various scenarios, as in personal assistants, tools for customer attendance and other products \cite{karpagavalli2016review,goodfellow,yu2016automatic}.
The task to transcribe speech can be understood as a mapping of an acoustic signal containing speech to a corresponding sequence of symbols intended by the speaker \cite{goodfellow}. The research in the field started with the recognition of spoken digits \cite{davis1952automatic} and has shown great advances recently with the use of end-to-end deep learning models, specially for the English language.
ASR systems can be built using different learning approaches. The most common is probably the supervised version, where a dataset containing speech data with its respective labels is used to train a model with the objective to predict transcriptions of a given audio. ASR can be trained using unsupervised techniques as well, and this can be an option where there is a lack of available data. Usually supervised learning requires aligned labels with the respect to each speech audio at different levels, such as phones, words or sentences. Unsupervised learning can use unaligned text and speech as part of the training process.
Besides these two approaches, there are another options. In special, self-supervised learning, has gaining interest. This technique usually consists of using the speech audio for training a model that is able to learn representations from the samples themselves. These representations can be learned by predicting or reconstructing missing and masked parts of each sample or by solving a contrastive task, where the model needs to differentiate distinct samples from false examples \cite{baevski2019vq}.
Despite its progress in the area, the development of robust ASR models for languages other than English can still be considered a difficult task, mainly because state-of-the-art (SOTA) models usually needs many hours of annotated speech for training to achieve good results \cite{amodei2016deep,quintanilha2020open}. This can be a challenge for some languages, such as Brazilian Portuguese, that has just a fraction of open resources available, if compared to the English language \cite{neto2011free,neto2008spoltech}.
To overcome these challenges, one could manually transcribe the amount of data needed to produce robust models, which is an expensive and time-consuming process. Another option is the use of modern techniques such as self-supervised learning, which has shown great results in the development of models where there is a lack of available labeled data. Self supervised learning allows the learning of representations for posterior use in a fine-tuning process using less labeled data in a supervised task \cite{baevski2020Wav2vec}.
Wav2vec 2.0 \cite{baevski2020Wav2vec} is a modern deep learning architecture that employs self supervised learning for speech representation learning through speech masking and contrastive task solving during pre-training, followed by a supervised fine-tuning on transcribed speech. The authors shows that Wav2vec 2.0 allows the training of a robust ASR model in English using only 10 minutes of labeled data, which outperformed several SOTA models. This demonstrates a good alternative for languages with few resources available.
Although Wav2vec 2.0 achieves good results in English with few labeled data, it is not clear if the same results could be obtained when using the same pre-trained model in other languages.
In other words, it is possible that monolingual training could led to poor results when using this model for a language different than the original. Fortunately, \cite{conneau2020unsupervised} pre-trained a Wav2vec 2.0 model using 53k hours of speech audio of 53 different languages, including Brazilian Portuguese. Besides masking and contrastive learning, this model jointly learned quantized representations across multiple languages, producing a cross-lingual model that significantly outperforms monolingual pre-training and enables a single multilingual speech recognition model which is competitive to the individual ones.
This work uses an XLSR-53 model pre-trained in many languages to build an ASR for Brazilian Portuguese, using only open available data. The presented model is public available\footnote{\url{https://huggingface.co/lgris/wav2vec2-large-xlsr-open-brazilian-portuguese}}.
This work is organized as follows: Section \ref{sec:wav2vec} explains the Wav2vec series architectures, Section \ref{sec:asr-br} provide details about the best available open model for ASR in Brazilian Portuguese. Then, Section \ref{sec:methods} discusses the proposed method and Section \ref{sec:results} demonstrates the obtained results. Sections \ref{sec:comparison-br} and \ref{sec:comparison-en} discusses some comparisons with the best open Brazilian Portuguese ASR model and the original Wav2vec 2.0 models in English, respectively. Finally, Section \ref{sec:conc} presents some conclusions.
\section{Wav2vec}\label{sec:wav2vec}
Wav2vec \cite{schneider2019wav2vec} is a convolutional neural network that uses unsupervised learning to create speech representations from raw audio speech. After training, this representations can be fed into ASR models as features in a supervised way. The topology is composed by two stacked neural networks. The encoder is a function $f: X \longmapsto Z$, which embeds the raw audio $X$ with $x_i$ frames in latent representations $Z$. Whereas the context network is a function $g: Z \longmapsto C$ that combines multiple time-steps to obtain contextual representations $C$. The model is trained using a loss function $\mathcal{L}_{k}=-\sum_{i=1}^{T-k}\left(\log \sigma\left(\mathbf{z}_{i+k}^{\top} h_{k}\left(\mathbf{c}_{i}\right)\right)+\lambda_{\tilde{\mathbf{z}} \sim p_{n}} \mathbb{E}\left[\log \sigma\left(-\tilde{\mathbf{z}}^{\top} h_{k}\left(\mathbf{c}_{i}\right)\right)\right]\right)$, where $h_k$ is an affine transformation $h_{k}\left(\mathbf{c}_{i}\right)=W_{k} \mathbf{c}_{i}+\mathbf{b}_{k}$. The loss function has the objective to distinguish a sample $z_{i+k}$ that is $k$ time-steps in the future ($T$ \textit{timesteps} in total), from $\tilde{z}$ false examples.
Wav2vec classic demonstrated promising results using few labelled data, but the Vq-wav2vec\cite{baevski2019vq} architecture achieved better results. Vq-wav2vec is an improvement of Wav2vec classic that uses a quantization module that learns vector quantized (VQ) representations. This architecture incorporates some aspects of self-supervised learning and motivated the next topology of the Wav2vec series, the Wav2vec 2.0.
Wav2vec 2.0 is the most recent architecture of the Wav2vec series, and is based on concepts of Wav2vec classic and Vq-wav2vec. In Wav2vec 2.0, the learning is divided into two main steps called pre-training and fine-tuning. During pre-training, the model solves a contrastive task to differentiate true quantized latent representations from a set of distractors, which allows the model to learn audio representations from speech. After pre-training, a projection is added to the last layer of the model containing the respective vocabulary and the model is trained in a supervised way using aligned labeled speech.
Wav2vec 2.0 is an end-to-end model inspired on the previous works from \cite{schneider2019wav2vec} and \cite{baevski2019vq}. As in Vq-wav2vec, the model is pre-trained in a self-supervised way using the raw audio to learn discrete representations of speech, but it has an simpler architecture when compared to the Vq-wav2vec. The Wav2vec 2.0 model is based on the idea of taking discrete speech representations directly as input to the Transformer (called Context Network), solving the ASR as an end-to-end task \cite{baevski2020Wav2vec}, whereas in Vq-wav2vec a BERT\cite{devlin2018bert} model was trained using discrete representations, and then used as input to a separated acoustic model.
The architecture of the Wav2vec 2.0 is presented in Figure \ref{fig:arquitetura-wav2vec2}. Similarly to original model, Wav2vec 2.0 architecture is composed by a multi-layer convolutional encoder $f: X \longmapsto Z$ which maps raw speech $X$ in latent speech representations $z_1, \ldots, z_T$ in $T$ time-steps. The convolutional blocks of the encoder consists of causal convolutions followed by layer normalization \cite{ba2016layer} and the GELU activation function \cite{hendrycks2020gaussian}. The output of the encoder is then provided to the context network $g: Z \longmapsto C$ which maps the latent representations $Z$ in contextualized representations $c_1, \ldots, c_T$.
\begin{figure}[htp]
\centering
\includegraphics[width=.8\textwidth]{figures/wav2vec2.png}
\caption{Wav2vec 2.0 architecture. \cite{baevski2020Wav2vec}}
\label{fig:arquitetura-wav2vec2}
\end{figure}
The context network follows the Transformer architecture. In this network, the positional encoding is replaced by a convolutional layer with GELU activation function and layer normalization that acts as a relative positional embedding.
The output of the encoder $z$ is discretized by product quantization \cite{jegou2010product} to a finite set of speech representations $Q = q_1, \ldots, q_t$ by a quantization module $h: Z \longmapsto Q$ during the pre-training phase. The Wav2vec 2.0 also uses the Gumbel-Softmax activation function to allow choosing discrete codebook entries ($V$) in a fully differentiable way during training, which are stored in data structures called codebooks ($G$).
In a similar form to BERT, during pre-training, some time-steps are masked in the output of the encoder. The masking has the goal to allow the model to predict masked parts. The inputs of the quantization module are not masked.
During the pre-training phase, the model has the objective to learn the speech representations solving a contrastive task $\mathcal{L}_{m}$. This corresponds to the task of identifying a true quantized representation $q_t$ from a set of false examples in a masked time-step context. Besides the contrastive function, the loss function consists of an additional diversity loss $\mathcal{L}_{d}$ that acts as an entropy function, proportional to an hyper-parameter $\alpha$. This allows to control the choosing of the quantized codebook
representations in a more balanced way \cite{baevski2020Wav2vec}. The objective function during pre-training is presented in Equation \ref{eq:objetive-wav2vec2}. The contrastive and diversity functions are shown in equations \ref{eq:contrastive-wav2vec2} and \ref{eq:diversity-wav2vec2}, respectively. In Equation \ref{eq:objetive-wav2vec2}, $\operatorname{sim}$ is defined as the cosine similarity between context representations
and quantized latent speech representations and $k$ controls the contrastive loss temperature.
\begin{equation}
\mathcal{L} = \mathcal{L}_{m} + \alpha \mathcal{L}_{d} \\
\label{eq:objetive-wav2vec2}
\end{equation}
\begin{equation}
\mathcal{L}_{m}=-\log \frac{\exp \left(\operatorname{sim}\left(\mathbf{c}_{t}, \mathbf{q}_{t}\right) / \kappa\right)}{\sum_{\tilde{\mathbf{q}} \sim \mathbf{Q}_{t}} \exp \left(\operatorname{sim}\left(\mathbf{c}_{t}, \tilde{\mathbf{q}}\right) / \kappa\right)} \\
\label{eq:contrastive-wav2vec2}
\end{equation}
\begin{equation}
\mathcal{L}_{d}=\frac{1}{G V} \sum_{g=1}^{G}-H\left(\bar{p}_{g}\right)=\frac{1}{G V} \sum_{g=1}^{G} \sum_{v=1}^{V} \bar{p}_{g, v} \log \bar{p}_{g, v} \\
\label{eq:diversity-wav2vec2}
\end{equation}
After pre-training, a projection of $n$ targets is added to the output of the context network and the model is fine-tuned in a supervised task for speech recognition \cite{baevski2020Wav2vec}. The model is trained using the Connectionist Temporal Classification (CTC) \cite{graves2006connectionist} as loss function. SpecAugment \cite{Park_2019} is also used in this phase, promoting a better generalization to the ASR model.
The authors proposed a series of experiments using two versions of the model, BASE and LARGE (the LARGE has more parameters in the context network) and varying the datasets used to pre-train and fine-tune the model. \cite{baevski2020Wav2vec} showed that is possible to build ASR models even with few labelled data available. The experiments with 10 minutes of labeled data showed a Word Error Rate (WER) of 4.8/8.2\% in both the test sets of LibriSpeech, namely clean and other.
The Wav2vec 2.0 can be used in other languages besides English. \cite{conneau2020unsupervised} made available a model called XLSR-53, pre-trained in many languages. The XLSR-53 follows the LARGE architecture of Wav2vec 2.0 and is trained in a dataset containing 53k hours from the Common Voice, MLS (Multilingual LibriSpeech) \cite{Pratap_2020} and BABEL datasets. The results of XLSR-53 suggests that cross-lingual pre-training significantly outperforms monolingual pre-training.
\section{ASRs for Brazilian Portuguese}\label{sec:asr-br}
Despite the development of new technologies among ASR researches, there are still few work related to speech recognition for Brazilian Portuguese. One possible explanation is the limitation of open labeled data available and the lack of models capable of generalizing well in this condition. However, new methods to overcome this limitation has been created. Besides, new datasets are being made available free for use, for instance, MLS and CETUC \cite{alencar2008lsf}. These datasets and methods open new possibilities for the development of ASR models for Brazilian Portuguese.
In the context of open ASRs for Brazilian Poruguese, the works of \cite{quintanilha2017end} and \cite{quintanilha2020open} can be highlighted as important recent advances. \cite{quintanilha2017end} compiled a dataset in Portuguese composed by various freely available datasets (Sid, VoxForge, LapsBM) and the proprietaty dataset CSLU Spoltech \cite{schramm2006cslu}. The Spoltech dataset contains 8,080 sentences and 477 speakers, the Sid contains 5,777 sentences and 72 speakers, the VoxForge contains 4,090 sentences and more than 111 speakers and LapsBM contains 700 sentences and 35 speakers.
The end-to-end model proposed by \cite{quintanilha2017end} is based on a simple architecture containing Bidirectional LSTM\cite{hochreiter1997long} layers. The audios are pre-processed and the MFCC\cite{mohamed2014deep,SAHIDULLAH2012543} (Mel-frequency Cepstral Coefficients) are extracted to be fed as input to the model. The author obtained an WER of 25,13\% in the proposed test set.
More recently, \cite{quintanilha2020open} proposed a better version of the previous dataset and trained a topology based on the DeepSpeech 2 \cite{amodei2016deep}, containing two convolutional layers and five bidirectional recurrent layers. The authors merged the CETUC dataset with the previous one, which allows the training of deeper models, and trained KenLM\cite{heafield2011kenlm} based Language Models (LMs) for post-processing the transcriptions. \cite{quintanilha2020open} obtained 25,45\% in the proposed test set.
These works represents an important contribution to this paper. Many of the datasets and a language model used by \cite{quintanilha2020open} were also used in this work.
\section{Methods}\label{sec:methods}
This section discusses the proposed methods for the fine-tuning of the Wav2vec 2.0 model for Brazilian Portuguese.
We used various open datasets to train the final model. Preliminary experiments were conducted to verify the Wav2vec 2.0 capability of generalization in each dataset, which demonstrated promising results in all tested training sets. The final model was trained with a dataset containing all the available data in Brazilian Portuguese.
\subsection{Datasets}
The compiled dataset is very similar to the base presented by \cite{quintanilha2020open}. Besides the data used by these authors, the MLS and Common Voice (CV) was also added to compile the final dataset for training.
For each gathered dataset, we used all the audios in that dataset for training. The exception was Common Voice, in which we used only part of the audios for training. The reason for this is that two Common Voice subsets, namely test and development, were used as our testing and validation sets, respectively.
The sets of validation and test of Common Voice were selected because they are closer to real use cases and also because they simplify comparing our results with other works. These data were collect by volunteers in the Web and contains a wide variety of speakers and audio characteristics.
The majority of audios have short durations (between 1 to 10 secs). Only the MLS has audios with long durations. Figure \ref{fig:datasets-histograma} shows the histograms of durations of each dataset.
\begin{figure}[htp]
\centering
\subfigure[]{\includegraphics[width=0.45\textwidth]{figures/histo-cetuc.png}}
\subfigure[]{\includegraphics[width=0.45\textwidth]{figures/histo-cv.png}}
\subfigure[]{\includegraphics[width=0.45\textwidth]{figures/histo-lapsbm.png}}
\subfigure[]{\includegraphics[width=0.45\textwidth]{figures/histo-mls.png}}
\subfigure[]{\includegraphics[width=0.45\textwidth]{figures/histo-vox.png}}
\caption{Histograms showing durations in seconds for each datasets: (a) CETUC (b) Common Voice (c) LAPSBM (d) MLS (e) VOXFORGE.}
\label{fig:datasets-histograma}
\end{figure}
Regarding quantity of speakers, Common Voice is the dataset that contains the greatest amount: 1,120 in total. In contrast, MLS has only 62 speakers. It is expected that training in data with more variety of speakers led to best results if compared to less variety.
The MLS also presents a possible problem: the audio of this dataset is composed by book readings, and some texts of LibriVox have some speelings of other epochs, for instance, words with greek and roman influence (\textit{phono}, \textit{pharmacia}, \textit{theatro}, among others). Most of these words underwent changes with more recent spelling reforms for the Brazilian Portuguese, but some texts were not updated to the new rules. This may decrease the model performance slightly.
The datasets proposed are:
\begin{itemize}
\item CETUC \cite{alencar2008lsf}: contains approximately 145 hours of Brazilian Portuguese speech distributed among 50 male and 50 female speakers, each pronouncing approximately 1,000 phonetically balanced sentences selected from the CETEN-Folha\footnote{\url{https://www.linguateca.pt/cetenfolha/}} corpus.
\item LAPSBM 1.4\footnote{``Falabrasil -- UFPA'' (\url{https://github.com/falabrasil/gitlab-resources})} is a dataset used by the Fala Brasil group to benchmark ASR systems in Brazilian Portuguese. Contains 35 speakers (10 females), each one pronouncing 20 unique sentences, totalling 700 utterances in Brazilian Portuguese. The audios were recorded in 22.05 kHz without environment control;
\item VoxForge\footnote{\url{http://www.voxforge.org/}}: is a project with the goal to build open datasets for acoustic models. The corpus contains approximately 100 speakers and 4,130 utterances of Brazilian Portuguese, with sample rates varying from 16kHz to 44.1kHz.
\item Common Voice 6.1: is a project proposed by Mozilla Foundation with the goal to create a wide open dataset in different languages to train ASR models. In this project, volunteers donate and validate speech using the oficial site\footnote{\url{https://commonvoice.mozilla.org/pt}}. The set in Portuguese (mostly Brazilian variant) used in this work is the 6.1 version (pt\_63h\_2020-12-11) that contains about 50 validated hours and 1,120 unique speakers;
\item Multilingual LibriSpeech \cite{Pratap_2020}: a massive dataset available in many languages. The MLS is based on audiobook recordings in public domain like LibriVox\footnote{\url{https://librivox.org/}}. The dataset contains a total of 6k hours of transcribed data in many languages. The set in Portuguese used in this work\footnote{\url{http://www.openslr.org/94/}} (mostly Brazilian variant) has approximately 284 hours of speech, obtained from 55 audiobooks read by 62 speakers.
\end{itemize}
\section{Results}\label{sec:results}
The final compiled dataset has 325 hours of speech, approximately. This amount can be considered appropriate, considering Wav2vec 2.0 reaches optimal results in English for experiments based on 100 and 960 hours of audio. As presented in Figure \ref{fig:comp-bases}, the majority of the dataset is composed by the CETUC dataset, followed by MLS and Common Voice. The VoxForge and LapsBM datasets represented less than 3\% of the total. In special, LapsBM corresponds to less than 1\% of the final dataset. This means that this dataset might make little contribution to the model training. In future experiments, this set can be used as test set, it can be more interesting the along Common Voice. Another important aspect is the duration of the audios: the MLS dataset corresponds to approximately 50\% of the final dataset (Figure \ref{fig:comp-bases}(b)).
\begin{figure}[htp]
\centering
\subfigure[]{\includegraphics[width=.49\textwidth]{figures/composicao-base.png}}
\subfigure[]{\includegraphics[width=.49\textwidth]{figures/composicao-base-dur.png}}
\caption{Compiled Dataset Composition: (a) by audios, (b) by hours.}
\label{fig:comp-bases}
\end{figure}
Some preliminary experiments were conducted in a NVIDIA TITAN V 12GB. The final model was training in a NVIDIA TESLA V100 32GB. Fine-tuning parameters were defined using the same configurations of the 100-hour experiment proposed by the original authors and is available at the official fairseq\footnote{\url{https://github.com/pytorch/fairseq/tree/master/examples/wav2vec}} repository. Parameters include: a maximum update of 80k, a learning rate of $3 \times 10^{-5}$, a max quantity of tokens of 1,280,000 (which defines the batch size automatically) and a update frequency of 24 steps. The model trained for a total of 108 epochs in approximately 8 days.
The validation WER and the validation and training loss of the final experiment is presented in Figure \ref{fig:graficos-principal}. As showed in \ref{fig:graficos-principal}(a), the model training converged during the first epochs, as expected, right after the Transformer is unfreeze (after 10k updates, by default). It is important to note that the model present subtle improvements in the posterior updates, and also the learning rate depends on the total of defined updates.
\begin{figure}[htp]
\centering
\subfigure[]{\includegraphics[width=.45\textwidth]{figures/train-loss-principal.png}}
\subfigure[]{\includegraphics[width=.45\textwidth]{figures/val-loss-principal.png}}
\subfigure[]{\includegraphics[width=.45\textwidth]{figures/wer-principal.png}}
\caption{Model performance over steps: (a) training loss, (b) validation loss, (c) validation WER}
\label{fig:graficos-principal}
\end{figure}
It is also import to consider that the model could reach a local optimum. Considering the total duration of the dataset, it is possible that an experiment with 320k updates (as proposed by the original authors in a dataset of 1000 hours) can led to better results.
During the validation, the best model is saved considering the WER obtained in the validation set. The best model was then tested against the test set. Table~\ref{tab:final} presents the test results for the best trained model. The lowest WER obtained was 11.95\% in the Common Voice test set using the 5-gram language model trained by \cite{quintanilha2020open}. Without a language model, the WER was 12.69\%. Both results can be considered promising in the context of the development of ASR models given the SOTA for Brazilian Portuguese.
\begin{table}[htp]
\centering
\caption{Lowest WERs on Common Voice test set.}
\begin{tabular}{|l|l|}
\hline
\textbf{Test} & \textbf{WER} \\ \hline
Without LM & 12.69\% \\ \hline
Using LM 5-gram-char \cite{quintanilha2020open} & \textbf{11.95\%} \\ \hline
\end{tabular}
\label{tab:final}
\end{table}
\fred{Traduzi as colunas da tabela}
The obtained errors were more common less frequent words, foreign words and proper names. Table \ref{tab:exemplos-erros-modelo} presents some errors made by the model. Errors that does not change the meaning of the sentence also occur, for instance, the swapping of letters with similar pronunciation and omission of ``s'' on the word ends that denote plural forms . Another frequent error is the junction of two or more words in a unique word. Most of these errors can be fixed with a proper language model.
\fred{Traduzi as colunas da tabela}
\begin{table}[ht]
\centering
\caption{Incorrect transcriptions without language model. The incorrect words are highlighted.}
\label{tab:exemplos-erros-modelo}
\begin{tabular}{|p{6cm}|p{6cm}|}
\hline
\textbf{Text} & \textbf{Transcription } \\
\hline \hline
É comum os usuários confundirem software livre com software livre & É comum os \textbf{usuares} \textbf{confunder} em \textbf{softwerlivr} com \textbf{softwerlivre} \\
\hline \hline
Ele fez tanto ghostwriting que ele começa a se sentir como um fantasma também &
Ele fez tanto \textbf{golstraitn} que ele \textbf{começou} a se sentir como um fantasma também \\
\hline \hline
Arnold apresentou um gráfico mostrando quantas cegonhas ele havia contado nos últimos dez anos
&
Arnold apresentou um gráfico mostrando quantas \textbf{segonhas} ele havia contado nos últimos dez anos \\
\hline \hline
Mais cedo ou mais tarde eles descobrirão como ler esses hieróglifos &
Mais \textbf{sedo} ou mais tarde eles descobriram como \textbf{de} esses \textbf{ierogrôficos} \\
\hline \hline
Viver juntos compartilhar objetivos e ter um bom relacionamento &
\textbf{E ver} juntos \textbf{signafica} viver juntos ou \textbf{fartlhar} objetivos ter um bom \textbf{relacionamentoo} \\
\hline \hline
Da mesma forma uma patente pode impedir que concorrentes desenvolvam produtos similares &
Da mesma forma uma patente pode impedir que concorrentes \textbf{desenvolva} produtos similares \\
\hline \hline
Duas mulheres e uma menina levantam com troféus &
Duas mulheres e uma menina levantam com \textbf{trofés} \\
\hline \hline
Esse acrobata de circo deve ter um sistema vestibular bem treinado pensou o espectador &
Esse acrobata de \textbf{cirko} deve ter um sistema vestibular \textbf{bemtreinado} pensou o espectador \\
\hline \hline
Durante a exposição o tribunal pode fazer quaisquer perguntas ou esclarecimentos que considere apropriados &
Durante a exposição o tribunal pode fazer quaisquer perguntas ou esclarecimentos que considere \textbf{apropriado} \\
\hline
\end{tabular}
\end{table}
\section{Comparison with Brazilian Portuguese models}\label{sec:comparison-br}
The results from \cite{quintanilha2020open} are interesting, considering that the model training was made entirely in a end-to-end supervised form. However, the result of 25.45\% of WER is above of the WER of this work. Although the results are not directly comparable, since author's test set differ from this work test set, the obtained result suggests that our model achieves SOTA performance for the Brazilian Portuguese, considering public available models. The best model in this work obtained a WER of only 12.79\%, without language model, in the Common Voice test set, a challenge dataset that contains noise and low quality recordings.
In a general form, our results suggests that the unsupervised or self-supervised techniques can be more better alternatives for the development of ASR models for Brazilian Portuguese, specially while the pure supervised techniques need a great quantity of labeled data to generalize as well as the English ones.
We believe that our results were superior from the work of \cite{quintanilha2020open} for two main reasons. First, we used Wav2vec 2.0, a more modern neural architecture. Second, we trained over more data, since recently several public access dataset were released.
\section{Comparison with English Wav2vec models}\label{sec:comparison-en}
Various experiments of fine-tuning were made by \cite{baevski2020Wav2vec}. These experiments varied the architecture used and the problem domain (characters or phonemes transcription). Besides, the authors made different experiments varying the dataset used in both pre-training and fine-tuning phases.
Considering the work of \cite{baevski2020Wav2vec}, the experiments that are more similar to our work are the fine-tunings with 100 and 960 hours of audio using the LARGE Wav2vec topology pre-trained on either LibriVox or Librispeech. The best model was pre-trained using the LibriVox dataset and fine-tuned using 960h of speech, obtaining a WER of 2\% and 4\% on the LibriSpeech test clean and test other sets. However, for a better comparison, a test using the Common Voice dataset is more desirable.
Table \ref{tab:comparacao-final} compares the result obtained in this work with SOTA version of Wav2vec, specifically, two pre-trained versions on the English Common Voice test set. Table \ref{tab:comparacao-final} also shows the result of the test using the best obtained model by the authors. The results shows that the model of this work presents a superior performance if compared to the original model for English, against the Common Voice dataset. Besides the size and variety of speakers of the English version be considerable higher than the Portuguese one, there are other explanations for this result: one is that our model were trained over more noisy audios and has an advantage on Common Voice, which also presents noise; another hypotheses is the superior performance of the XLSR-53 model, that was trained using 53 different languages, whilst the original version was trained only using English speech.
Therefore, it is important to note the difference in the experiments of \cite{baevski2020Wav2vec}, which were fine-tuned using high quality audios. We believe relevant to include real data or perform some data augmentation for a better generalization in different scenarios.
\begin{table}[H]
\centering
\caption{Comparison with English Language Results using no Language Model. LARGE models were tested in English and XLSR-53 model (this work), were tested in Portuguese.}
\begin{tabular}{l|l|l|c}
\hline
\multicolumn{1}{c|}{\textbf{Topology}} & \multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Pre-trained \\ Dataset\end{tabular}}} & \multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Fine-tuning \\ Dataset\end{tabular}}} & \textbf{\begin{tabular}[c]{@{}c@{}}WER \\ (Common Voice)\end{tabular}} \\ \hline
\multirow{3}{*}{LARGE} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}LibriVox \\ (60k hours)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Libri-Light + \\ Librispeech (100h)\end{tabular} & 29.83 \\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Libri-Light + \\ Librispeech (960h)\end{tabular} & 23.00 \\ \cline{2-4}
& LibriSpeech (960h) & LibriSpeech (100h) & 37.82 \\ \hline
XLSR-53 (this work) & \begin{tabular}[c]{@{}l@{}}MLS, CommonVoice, \\ BABEL (56h)\end{tabular} & \begin{tabular}[c]{@{}l@{}}CETUC, MLS, CV, \\ VF, LapsBM (325h)\end{tabular} & 12.69 \\ \hline
\end{tabular}
\label{tab:comparacao-final}
\end{table}
\section{Conclusions}\label{sec:conc}
In these work we presented a model for Automatic Speech Recognition for the Brazilian Portuguese Language. According to our best knowledge, the model achieves state-of-the-art results among the open available models for the target language. Our model obtained WERs of 12.69\% and 11.95\% using no language model or a 5-gram-char language model, respectively. The results are competitive to both to English and Portuguese open models.
As future work, we plan to increase the model performance using new datasets available for Portuguese. We also plan to perform more detailed experiments using different language models and applying data-augmentation techniques such as noise insertion, to improve performance in non controlled environments.
\bibliographystyle{splncs04}
|
1,314,259,996,694 | arxiv | \section{Introduction}
\label{sec-intro}
One of the most interesting questions in high-energy nuclear physics is
that about the properties of the hot and dense medium created in
ultra-relativistic heavy-ion collisions. Finite-temperature lattice-QCD
(lQCD) calculations of strong\-ly-in\-ter\-ac\-ting matter predict a
phase transition from hadronic matter to a quark-gluon plasma (QGP) at a
critical temperature, $T_c \simeq 180 \;
\mathrm{MeV}$~\cite{Karsch:2007vw}. In the recent years the experimental
program at the Relativistic Heavy-Ion collider has resulted in
convincing evidence for the formation of such a hot and dense partonic
state~\cite{Arsene:2004fa,Back:2004je,Adams:2005dq,Adcox:2004mh}.
The heavy charm and bottom quarks are particularly valuable probes for
the properties of this medium since they are created in the primordial
hard collisions of the nucleons within the colliding nuclei. Thus, they
form a rather well defined initial state and interact with the hot and
dense fireball during its entire evolution. Recently, measurements of
the transverse-momentum distributions of ``non-photonic single
electrons'' ($e^{\pm}$), which originate mainly from the semi-leptonic
decays of open-charm and -bottom mesons, in $200 \; A\mathrm{GeV}$ Au-Au
collisions at RHIC have found a surprisingly large suppression at high
transverse momenta ($p_t$) (i.e., a small nuclear modification factor,
$R_{AA}$) and a large elliptic-flow parameter, $v_2$. Both findings
indicate that during the lifetime of the hot and dense fireball heavy
quarks come close to thermal equilibrium with the
medium~\cite{Adler:2005xv,Abelev:2006db,Adare:2006nq}.
The theoretical challenge is to understand the corresponding
thermalization times of heavy quarks from the underlying microscopic
scattering processes with the constituents of the QGP, in particular how
the heavy quarks, despite their large masses, $m_Q \gg T_c$, become part
of the collective flow of the fireball. In calculations of the pertinent
transport coefficients from perturbative QCD (pQCD), based on
gluon-brems\-strah\-lung energy loss, including elastic HQ scattering,
one has to artificially tune the coupling strength beyond the
applicability range of perturbation
theory~\cite{Armesto:2005mz,Wicks:2005gt}. It has also been shown that
the convergence of the perturbative series for the HQ diffusion
coefficient is quite poor~\cite{CaronHuot:2007gq}. Thus,
non-per\-tur\-ba\-tive approaches have to be used to explain the strong
HQ couplings necessary. One suggested mechanism is the formation of $D$-
and $B$-meson resonance excitations in the deconfined phase of QCD
matter~\cite{vanHees:2004gq,vanHees:2005wb} which has lead to a quite
satisfactory description of the $e^{\pm}$ data at RHIC.
This paper is organized as follows: In Sec.~\ref{sec-t-matrix} we use HQ
static potentials from lattice-QCD calculations at finite temperature in
a many-body Brueckner $T$-matrix approach to calculate elastic HQ
light-quark scattering-ma\-trix elements in the medium
\cite{Mannarelli:2005pz,vanHees:2007me}. We show that after inclusion of
a complete set of color channels, taking into account $l=0,1$ states in
the partial-wave expansion of the $T$-matrix, the resonance states,
conjectured in the earlier approaches, are confirmed by these
interactions, which are in principle free of tunable parameters. The
resulting elastic-scattering amplitudes are used in
Sec.~\ref{sec-langevin} to calculate drag and diffusion coefficients for
a Fokker-Planck
equation~\cite{Svetitsky:1987gq,vanHees:2004gq,Moore:2004tg}, describing
the rescattering of the heavy quarks within the hot and dense sQGP
fireball. In the next step we employ a relativistic Langevin simulation
to find the corresponding HQ $p_t$ distributions, using a
thermal-fireball parameterization, including elliptic flow for
non-central heavy-ion collisions. To confront these spectra with the
$e^{\pm}$ data from the PHENIX and STAR collaborations at RHIC, in
Sec.~\ref{sec-observables} we use a combined quark-coalescence and
fragmentation model to hadronize the heavy quarks to $D$ and $B$ mesons
which then are decayed semi-leptonically leading to the final $e^{\pm}$
spectra which can be directly confronted with recent data on nonphotonic
single electrons in $200 \; A \text{GeV}$ Au-Au collisions at RHIC. The
paper closes with brief conclusions and an outlook
(Sec.~\ref{sec-conclusions}).
\section{HQ scattering in the QGP}
\label{sec-t-matrix}
In this Sec. we calculate in-medium matrix elements for elastic
scattering of heavy quarks ($Q=c,b$) with light quarks $q=u,d,s$ in a
Brueckner-like many-body approach, assuming that a static heavy-quark
light-quark potential, $V(r)$, can be employed as the interaction
kernel. Such a model has been used in the vacuum to successfully
describe $D$-meson spectra and
decays~\cite{Godfrey:1985xj,Avila:1994vi}. Further, we assume that the
effective in-medium potential can be extracted from finite-temperature
lQCD calculations of the color-singlet free energy
$F_1(r,T)$~\cite{Kaczmarek:2003dp,Kaczmarek:2005gi} for a static
$\bar{Q} Q$ pair as the internal potential energy by the usual
thermodynamic
relation~\cite{Mannarelli:2005pz,Shuryak:2004tx,Wong:2004zr,Cabrera:2006wh},
\begin{equation}
\label{F-to-U}
U_1(r,T)=F_1(r,T)-T \frac{\partial F_1(r,T)}{\partial T}.
\end{equation}
For application as a scattering kernel in a $T$-matrix equation, the
potential as to vanish for $r \rightarrow \infty$. Thus we choose the
accordingly subtracted internal potential energy,
\begin{equation}
\label{U-to-V}
V_1(r,T)=U_1(r,T)-U_1(r \rightarrow \infty,T).
\end{equation}
In lQCD simulations one finds that $U_1(r \rightarrow \infty,T)$ is a
decreasing function with temperature which could be associated as a
contribution to the in-medium HQ mass, $m_Q(T)=m_0+U_1(r \rightarrow
\infty,T)/2$ where $m_0$ denotes the bare mass. However, this leads to
problems since close to $T_c$ the asymptotic value, $U_1(r \rightarrow
\infty,T)$, develops a pronounced peak structure. Thus, in this
calculation, we assume constant effective HQ masses, $m_c=1.5 \;
\mathrm{GeV}$ and $m_b=4.5 \; \mathrm{GeV}$.
We also consider the complete set of color channels for the $Q \bar{q}$
(singlet and octet) and $Q q$ (anti-triplet and sextet) systems, using
Casimir scaling as in leading-order pQCD,
\begin{equation}
\label{casimir-scaling}
V_{8}=-\frac{1}{8} V_1, \quad V_{\bar{3}}=\frac{1}{2} V_1, \quad
V_6=-\frac{1}{4} V_1,
\end{equation}
which is also justified by recent lQCD calculations of the finite-$T$ HQ
free energy~\cite{Nakamura:2005hk,Doring:2007uh}.
This approach is in principle parameter free in the choice of the
interactions, since their strength is taken from first-principle lQCD
simulations. However, there are considerable uncertainties in the
potentials (a) between different lattice calculations and (b) in the
extraction and parameterization of the corresponding free energies,
particularly their temperature dependence needed to subtract the entropy
term in Eq.~(\ref{F-to-U}). In addition, the very notion of an
``in-medium potential'' is not a unique concept~\cite{Brambilla:2008cx},
and its identification with the internal potential energy may be seen as
an upper limit in interaction strength. We use three different
parameterizations of $F_1$
\cite{Wong:2004zr,Shuryak:2004tx,Mannarelli:2005pz}:
\begin{description}
\item[{\qquad [Wo]}] of quenched lQCD~\cite{Kaczmarek:2003dp},
\item[{\qquad [SZ]}] of two-flavor
lQCD~\cite{Kaczmarek:2003ph}, and
\item[{\qquad [MR]}] of three-flavor
lQCD~\cite{Petreczky:2004pz}.
\end{description}
The resulting potentials from [Wo] and [SZ] are comparable to a
numerical extraction from three-flavor lQCD~\cite{Petreczky:2004pz},
while that from [MR] is deeper than the other two for $T \lesssim 1.6
T_c$, but falls off faster at higher temperatures. The resulting
uncertainty in the transport coefficients (see Sec.~\ref{sec-langevin})
amounts to up to $40 \%$.
\begin{figure}
\centerline{\includegraphics[width=0.9\linewidth]{brueckner.eps}}
\caption{(Color online) Diagrammatic representation of the Brueckner
many-body scheme for the coupled system of the $T$-matrix based on the
lQCD static internal potential energy as the interaction kernel and
the HQ self-energy.}
\label{fig.brueckner}
\end{figure}
To define the Brueckner-type many-body scheme the four-di\-men\-sio\-nal
(4D) Bethe-Salpeter (BS) ladder approximation, symbolized in
diagrammatical form by the upper panel of Fig.~\ref{fig.brueckner}, has
to be reduced to a 3D Lippmann-Schwinger (LS) equation, neglecting
antiparticle components in the quark propagators, in order to implement
the static potential from lQCD
via~Eqs.~(\ref{F-to-U}-\ref{casimir-scaling}). After this reduction the
LS equation in the color channel, $a \in \{1,\bar{3},6,8\}$
reads~\cite{Mannarelli:2005pz}\footnote{Here and in the following all
vertex and Green's functions are understood as the retarded real-time
quantities which can be derived as analytic continuations of the
corresponding imaginary-time (Matsubara) quantities of thermal quantum
field theory.}
\begin{equation}
\begin{split}
\label{LS}
T_a(E;\vec{q}',\vec{q}) = &V_a(\vec{q}',\vec{q})-\int \frac{\mathrm{d}^3
\vec{k}}{(2 \pi)^3} V_q(\vec{q}',\vec{k}) G_{qQ}(E;k) \\
& \times T_a(E;\vec{k},\vec{q})[1-f_F(\omega_k^q)-f_F(\omega_k^Q)]
\end{split}
\end{equation}
with the Fourier-transformed potentials,
\begin{equation}
\label{v-four}
V_a(\vec{q}',\vec{q})=\int \mathrm{d}^3 \vec{r} V_a(r) \exp[\mathrm{i}(\vec{q}-\vec{q}')\vec{r}].
\end{equation}
Further, $E$, $\vec{q}$ and $\vec{q}'$ denote the energy and incoming
and outgoing momenta in the center-of-mass (CM) frame, respectively.
\begin{equation}
\label{fd-dist}
f_F=\frac{1}{\exp(\omega/T)+1}
\end{equation}
is the Fermi-Dirac distribution. The quark-dispersion relations are
determined in quasi-particle approximation by
\begin{equation}
\omega_k^{q,Q}=\sqrt{k^2+m_{q,Q}^2},
\end{equation}
where for simplification we do not solve the fully self-consistent
scheme in Fig.~\ref{fig.brueckner} but use a fixed mass of $m_q=0.25 \;
\text{GeV}$, $m_c=1.5 \; \text{GeV}$, and $m_b=4.5 \; \text{GeV}$ for
the light, charm, and bottom quarks, respectively. Finally, the
two-particle-$qQ$ propagator in~(\ref{LS}) is given in terms of the
Thompson-reduction scheme~\cite{Thompson:1970wt}
\begin{equation}
\label{thompson-prop}
G_{qQ}(E;k)=\frac{1}{4} \frac{1}{E- (\omega_k^q +\mathrm{i}
\Sigma_I^q)-(\omega_k^Q +\mathrm{i} \Sigma_I^Q)}
\end{equation}
with a quasi-particle width for both light and heavy quarks of
$-2\Sigma_{I}^{q,Q}=0.2 \; \text{GeV}$.
The solution of the LS equation (\ref{LS}) is simplified by using a
partial-wave expansion of the potential and $T$-matrix,
\begin{equation}
\begin{split}
\label{part-wave}
V_a(\vec{q}',\vec{q})=4 \pi \sum_{l} (2l+1) V_{a,l}(q',q) P_l[\cos
\angle(\vec{q},\vec{q}')], \\
T_a(E;\vec{q}',\vec{q})=4 \pi \sum_{l} (2l+1) T_{a,l}(E;q',q) P_l[\cos
\angle(\vec{q},\vec{q}')],
\end{split}
\end{equation}
which leads to the 1D LS equations,
\begin{equation}
\begin{split}
\label{LS-1D}
T_{a,l}(E;q',q)=V_{a,l}&(q',q) + \frac{2}{\pi} \int \mathrm{d} k k^2
V_{a,l}(q',k) G_{Qq}(E;k) \\
&\times T_{a,l}(E;k,q)[1-f_F(\omega_k^Q)-f_F(\omega_k^q)],
\end{split}
\end{equation}
for the partial-wave components, $T_{a,l}$, of the $T$-matrix, which are
solved numerically with the matrix-inversion algorithm of Haftel and
Tabakin~\cite{Haftel1970}. We restrict ourselves to $S$ ($l=0$) and $P$
($l=1$) waves.
\begin{figure}
\centerline{\includegraphics[width=0.9\linewidth]{ImT-mat.eps}}
\caption{(Color online) Imaginary part of the $S$-wave in-medium $T$
matrix for $c\bar{q}$ and $cq$ scattering in the color-singlet and
-antitriplet channels based on the parameterization of the lQCD
potential energy by [Wo].}
\label{fig.ImT}
\end{figure}
As can be seen from Fig.~\ref{fig.ImT}, in the dominating attractive
color-singlet $Q\bar{q}$ and color-antitriplet $Q q$ channels, close to
the critical temperature, $T_c \simeq 180 \; \text{MeV}$, resonance
states above the threshold, $E_{\text{thr}}=m_Q+m_q$ are formed, similar
as conjectured in~\cite{vanHees:2004gq,vanHees:2005wb}. However, in this
full in-medium scheme the resonances melt at higher temperatures $T
\gtrsim 1.7 T_c$ and $T \gtrsim 1.4 T_c$, respectively. As we shall see
in the next section, contrary to the expectation from perturbative
calculations, using the [Wo] parameterization of the potential, this
leads even to \emph{decreasing} transport coefficients with increasing
temperature, i.e., the decreasing interaction strength of the potential
overcompensates the higher density of the medium.
The HQ self-energy, diagrammatically represented by the lower panel of
Fig.~\ref{fig.brueckner}, is then given by
\begin{equation}
\begin{split}
\label{self-en}
\Sigma_a^Q(\omega,p)=&\frac{d_{\text{SI}} d_a}{6} \int \frac{k^2 \mathrm{d} k \mathrm{d}
x}{4\pi^2} [f_F(\omega_k)+f_B(\omega+\omega_k)] \\
& \times T_a(E;\vec{p},\vec{k}),
\end{split}
\end{equation}
where $d_{\text{SI}}=4(2l+1) N_f$ denotes the spin-isospin degeneracy of
the light quarks (which is already implicitly assumed in the light-heavy
quark interaction of our approach, which is in line with the free
$D$-meson spectrum \cite{Abe:2003zm}) in the $l^{\text{th}}$ partial
wave and $d_a$ the color degeneracy in the corresponding channel. Here
we assume an effective number of light-quark flavors, $N_f=2.5$, to
account for the smaller strange-quark density. For the charm quarks,
cf. Fig.~\ref{fig.HQSE} the calculation leads to an in-medium width of
$\Gamma_c=-2 \im \Sigma_c \simeq 200 \;\text{MeV}$, which justifies our
simplifying assumption in the $T$-matrix calculation above.
\begin{figure}
\centerline{\includegraphics[width=0.9\linewidth]{SigmaQ.eps}}
\caption{(Color online) Real and imaginary parts of the $c$-quark
self-energy as a function of three-momentum at different
temperatures.}
\label{fig.HQSE}
\end{figure}
\section{HQ transport in the QGP}
\label{sec-langevin}
To evaluate the motion of the heavy quarks in the hot and dense
fireball, consisting of light quarks and gluons, we employ a Langevin
simulation of the Fokker-Planck equation,
\begin{equation}
\label{FP}
\frac{\partial f_Q}{\partial t}=\frac{\partial}{\partial p_i} (p_i \gamma
f_Q) + \frac{\partial^2}{p_i p_j} (B_{ij} f_Q).
\end{equation}
The drag or friction coefficient, $\gamma$, and diffusion coefficients,
\begin{equation}
\label{diff}
B_{ij}=B_0 \frac{p_i p_j}{p^2} + B_1 \left (1-\frac{p_i p_j}{p^2} \right ),
\end{equation}
are calculated from the invariant scat\-te\-ring-ma\-trix
elements~\cite{Svetitsky:1987gq}. Taking into account elastic scattering
of the heavy quark with a light quark or antiquark the latter given in
terms of the above calculated $T$-matrix by
\begin{equation}
\begin{split}
\label{Msq}
\sum |\mathcal{M}|^2=\frac{64\pi}{s^2} (s-m_q^2+m_Q^2)^2(s-m_Q^2+m_q^2)^2 \\
\times N_f\sum_{a} d_a (|T_{a,l=0}(s)|^2 +3 |T_{a,l=1} (s)
\cos(\theta_{\text{cm}})|^2).
\end{split}
\end{equation}
We define the averaging operator
\begin{equation}
\begin{split}
\label{erw}
\erw{X(\vec{p}{\,})} &= \frac{1}{2 E_p} \int \frac{\mathrm{d}^3 \vec{q}}{(2
\pi)^3 2E_q} \int \frac{\mathrm{d}^3 \vec{q}{\,}'}{(2 \pi)^3 2E_{q'}} \\
& \quad \int \frac{\mathrm{d}^3
\vec{p}{\,}'}{(2\pi)^3 2E_{p'}} \frac{1}{\gamma_c} \sum |\mathcal{M}|^2 \\
& (2 \pi)^4 \delta^{(4)}(p+q-p'-q') f_q(\vec{q}) X(\vec{p}{\,}') \ ,
\end{split}
\end{equation}
for a heavy-quark observable, $X$, over the elastic scatterings per unit
time of the heavy quark with momentum $\vec{p}$ with a light quark of
momentum $\vec{q}$, changing their momenta to $\vec{p}'$ and $\vec{q}'$.
Here, $f_q$ is the (thermal) distribution of the light quarks in the
me\-dium. Then we can calculate the transport coefficients as
\begin{align}
\gamma(|\vec{p}|) &= \erw{1} -\frac{\erw{\vec{p} \cdot \vec{p}{\,}'}}{\vec{p}^2},
\label{A}
\\
B_0(|\vec{p}|) &= \frac{1}{4} \left [
\erw{\vec{p}{\,}'{}^2}-\frac{\erw{(\vec{p} \cdot \vec{p}{\,}')^2}}{\vec{p}^2}
\right],
\label{B0}
\\
B_1(|\vec{p}|) &= \frac{1}{2} \left[
\frac{\erw{(\vec{p} \cdot \vec{p}{\,}')^2}}{\vec{p}^2}
-2 \erw{\vec{p}{\,}' \cdot \vec{p}} + \vec{p}^2 \erw{1} \right].
\label{B1}
\end{align}
However, it turns out that, in order to guarantee the pro\-per equilibrium
limit of the heavy quarks with the me\-dium, we have to enforce Einstein's
fluctuation-dissipation relation for the longitudinal diffusion
coefficient, which is used here in its relativistic
form~\cite{Moore:2004tg,vanHees:2005wb},
\begin{equation}
\label{fluc-diss}
B_1=E T \gamma.
\end{equation}
The nonperturbative HQ light-quark scat\-te\-ring-ma\-trix ele\-ments are
supplemented by the corresponding perturbative elastic HQ
gluon-scattering ones~\cite{Combridge:1978kx}. The $t$-channel
singularity is regulated by a gluon-Debye screening mass of $m_g=g T$
with a strong coupling constant $g=\sqrt{4 \pi \alpha_s}$, using
$\alpha_s=0.4$.
\begin{figure}
\centerline{\includegraphics[width=0.9\linewidth]{fp-coeffs-allT.eps}}
\caption{(Color online) The drag coefficient, $\gamma$, as a function of
HQ momentum, calculated via (\ref{A}) with scattering-matrix elements
from the non-perturbative $T$-matrix calculation (using the
parameterization of the lQCD internal potential energies by [Wo])
compared to a LO perturbative calculation based on matrix elements
from~\cite{Combridge:1978kx}.}
\label{fig.frict-coeff}
\end{figure}
As shown in Fig.~\ref{fig.frict-coeff}, close to $T_c$ the equilibration
times of $\tau_{\text{eq}}=1/\gamma \simeq 7 \; \text{fm}/c$ for charm
quarks are a factor of $\sim 4$ larger than the values from a
corresponding pQCD calculation, reminiscent to the results based on the
model, assuming the survival of $D$-meson like resonance states above
$T_c$~\cite{vanHees:2004gq,vanHees:2005wb}. In contrast to this and
other calculations of the HQ transport coefficients, here the drag
coefficients \emph{decrease} with increasing temperature because of the
``melting'' of the dynamically generated resonances at increasing
temperatures due to the diminishing interaction strength from the lQCD
potentials.
To solve the Fokker-Planck equation~(\ref{FP}) under conditions of the
sQGP medium produced in heavy-ion collisions, we use an isentropically
expanding thermal fireball model, assuming an ideal-gas equation of
state of $N_f=2.5$ effective massless light-quark flavors and
gluons. The total entropy is fixed by particle multiplicities at
chemical freeze-out which we assume to occur at the critical
temperature, $T_c = 180 \; \text{MeV}$. For semi-central collisions the
fireball is chosen to be of elliptic-cylindrical shape with isobars
given by confocal ellipses with a perpendicular radial-flow field,
scaling linearly with the distance from the center as seen in
hydrodynamic calculations~\cite{Kolb:2000sd} to which also the (average)
radial flow velocity and ellipticity, $v_2$, is fixed. To compare to
``minimum-bias'' data on $e^{\pm}$ spectra in $200 \; A \text{GeV}$
Au-Au reactions at RHIC we simulate collisions with an impact parameter
of $b=7 \; \text{fm}$, implying an initial spatial eccentricity of about
$0.6$. Using a QGP-formation time of $0.33 \; \text{fm}/c$ leads to an
initial temperature of $340 \;\text{MeV}$. The evolution stops after the
fireball has undergone a mixed QGP-ha\-dro\-nic phase after about $5 \;
\text{fm}/c$, at which the radial velocity of the fireball has reached
about a radial velocity of about $v_{\perp}=0.5 c$ at the surface and an
ellipticity of $v_2 =5.5\%$.
Given this description of the medium, (\ref{FP}) is solved with help of
an equivalent relativistic Langevin simulation which is defined by the
stochastic equation of motion for a heavy quark at position, $\vec{x}$,
and momentum $\vec{p}$:
\begin{equation}
\label{langevin}
\delta \vec{x}=\frac{p}{E} \delta t, \quad \delta
\vec{p}=-\gamma(t,\vec{p}) \vec{p} \delta t + \delta
\vec{W}(t,\vec{p}+\delta \vec{p}),
\end{equation}
where $\delta \vec{W}$ is a stochastic force distributed normally,
\begin{equation}
\label{rand-force}
P(\delta \vec{W}) \propto \exp \left [-\frac{(B^{-1})_{jk} \delta W^{j}
\delta W^{k}}{4 \delta t} \right ],
\end{equation}
$B^{-1}$ denoting the inverse of the diffusion-coefficient matrix
(\ref{diff}). Note that in (\ref{langevin}) and (\ref{rand-force}) the
diffusion coefficients are to be evaluated at the updated momenta
$\vec{p}+\delta \vec{p}$. This H{\"a}nggi-Klimontovich realization of
the stochastic process together with the dis\-si\-pa\-tion-fluctuation
relation~(\ref{fluc-diss}) ensures the correct equilibrium limit in the
long-time regime $t \gg 1/\gamma$~\cite{Haenggi:2005}. After evaluation
of the time step (\ref{langevin}) the resulting momenta are Lorentz
boosted to the laboratory frame.
The initial condition for (\ref{langevin}) is given by the phase-space
distribution of the heavy quarks. The spatial distribution is determined
with a Glauber model for heavy-quark production. The initial $p_t$
spectrum is determined from data on $p$-$p$ and d-Au collisions at RHIC
as follows: The $c$-quark spectra are taken from a modified PYTHIA
calculation to fit $D$ and $D^*$ spectra in d-Au
collisions~\cite{Adams:2004fc}, assuming $\delta$-function
fragmentation. After decaying this spectrum to single $e^{\pm}$ they
saturate corresponding data from $p$-$p$- and
d-Au~\cite{Adler:2005xv,Tai:2004bf} collisions up to $p_t \simeq 3.5 \;
\text{GeV}$. The missing yield at higher $p_t$ is assumed to be filled
with the corresponding contributions from $B$ mesons, leading to a
cross-section ratio of $\sigma_{b\bar{b}}/\sigma_{c\bar{c}} \simeq 5 \cdot
10^{-3}$ and a crossing of the $c$- and $b$-decay electron spectra at
$p_t \simeq 5\;\text{GeV}$.
\begin{figure}
\centerline{\includegraphics[width=0.9\linewidth]{quark-RAA-central.eps}}
\centerline{\includegraphics[width=0.9\linewidth]{quark-v2-semicentral.eps}}
\caption{(Color Online) Nuclear modification factor, $R_{AA}$, for
central (upper panel) and elliptic flow, $v_2$, for semicentral (lower
panel) $200 \; A\text{GeV}$ Au-Au collisions for charm quarks from the
Langevin simulation, using the $T$-matrix results for the transport
coefficients employing the parameterizations [Wo] and [SZ] for the
lQCD potentials, compared to a calculation based on pQCD and the
resonance-model interactions in~\cite{vanHees:2005wb}}
\label{fig.cquark-raa-v2}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.9\linewidth]{quark-RAA-bot-central.eps}}
\centerline{\includegraphics[width=0.9\linewidth]{quark-v2-bot-semicentral.eps}}
\caption{(Color online) The same as Fig.~\ref{fig.cquark-raa-v2} but for
bottom quarks}
\label{fig.bquark-raa-v2}
\end{figure}
In Figs.~\ref{fig.cquark-raa-v2} and \ref{fig.bquark-raa-v2} we show the
nuclear modification factor, $R_{AA}$, defined by
$R_{AA}=P_Q(t_{\text{fin}},p_t)/P_Q(0,p_t)$ (where $t_{\text{fin}}$
denotes the time at the end of the mixed QGP-hadronic phase) in central
and the elliptic flow,
\begin{equation}
v_2=\erw{(p_x^2-p_y^2)/p_t^2},
\end{equation}
in semicentral $200\;A\text{GeV}$ Au-Au collisions. We compare the
results from the $T$-matrix model with the parameterizations by [Wo] and
[SZ] of the lQCD potentials with the those using pQCD or the
resonance-model interactions of Ref.~\cite{vanHees:2005wb}. While for
charm quarks for the [Wo] potential the result for $R_{AA}$ is
comparable to the upper end of the uncertainty band of the
resonance-model calculation, the $v_2$ is slightly enhanced at low
$p_t$. The reason for this behavior is the decrease of the transport
coefficients with increasing $T$: While the suppression of the $p_t$
spectra at high $p_t$ is due to the evolution along the whole history of
the fireball, leading to comparable effects at the end of the mixed
phase, the anisotropic flow is mostly developed at the later stages and
thus can be transferred to the heavy quarks at the end of the evolution
efficiently, when the drag coefficient become larger due to the
dynamical formation of the resonance states. The $T$-matrix result with
the somewhat less attractive [SZ] potential leads to the usual ordering
of the coefficients (increasing with increasing temperature) and thus
shows weaker effects for both the $R_{AA}$ and $v_2$ than the result of
the resonance model. For $b$ quarks the $T$-matrix calculations yield
larger medium modifications of the $p_t$ spectra than the resonance
model which is due to the mass effect, leading to stronger binding
effects for the resonances in the $T$-matrix calculation. As to be
expected, the effects of pQCD-based transport coefficients on the HQ
spectra for both charm and bottom quarks is much weaker than the
non-perturbative ones via the resonance-scattering mechanism.
\section{Single-electron observables at RHIC}
\label{sec-observables}
The last step toward a comparison of the above described mo\-del for HQ
diffusion in the QGP with the single-electron $p_t$ data from RHIC is
the hadronization of the HQ spectra to $D$- and $B$-mesons and their
subsequent semileptonic decay to $e^{\pm}$. Here we use the
quark-coalescence model described
in~\cite{Greco:2003mm,Greco:2003vf}. In the recent years, the
coalescence of quarks in the hot and dense medium created in heavy-ion
collisions has been shown to provide a successful ha\-dro\-ni\-za\-tion
mechanism to explain phenomena such as the scaling of hadronic
elliptic-flow parameters, $v_2$ with the number of constituent quarks,
$v_{2,h}(p_t) = n_h v_{2,q}(p_t/n_h)$, where $n_h=2(3)$ for mesons
(hadrons) denotes the number of constituent quarks contained in the
hadron, $h$, and the large $p/\pi$ ratio in Au-Au compared to $p$-$p$
collisions~\cite{Greco:2003mm,Hwa:2002tu,Fries:2003kq}. Quark
coalescence is most efficient in the low-$p_t$ regime where most $c$ and
$b$ quarks combine into $D$ and $B$ mesons, respectively. To conserve
the total HQ number, we assume that the remaining heavy quarks hadronize
via ($\delta$-func\-tion) fragmentation.
\begin{figure}
\centering
\includegraphics[width=0.9 \linewidth]{raa-central-v2mb-wo-sz-HP08.eps}\\[0.3cm]
\includegraphics[width=0.9 \linewidth]{raa-central-v2mb-wo-D-B.eps}
\caption{(Color online) Upper panel: single-electron spectra from the
$T$-matrix calculation of HQ diffusion in the QGP based on the [Wo]
(solid line) and the [SZ] (dash-double-dotted line) parameterizations
of the lQCD static HQ potential in comparison to data from the
PHENIX~\cite{Abelev:2006db} and STAR~\cite{Abelev:2006db}
collaborations in $200 \; A \text{GeV}$ Au-Au collisions at RHIC. The
dashed line shows the result when only $\delta$-function fragmentation
is considered for hadronization. Lower panel: $R_{AA}$ and $v_2$, as
in the upper panel for the [Wo] parameterization for the electrons
from $D$ (dashed line) and $B$ mesons (dash-dotted line) separately.}
\label{fig.electrons}
\end{figure}
As shown in Fig.~\ref{fig.electrons} the Langevin simulation of the HQ
diffusion based on transport coefficients from the lQCD static
potentials, followed by the combined quark-coa\-le\-scen\-ce
fragmentation description of hadronization to $D$ and $B$ mesons and
their subsequent semileptonic decay, successfully accounts
simultaneously for both the nuclear modification factor, $R_{AA}$, and
the elliptic flow, $v_2$, of single electrons in $200 \; A\text{GeV}$
Au-Au collisions~\cite{Adare:2006nq,Abelev:2006db} at RHIC. The
uncertainty due to the two different parameterizations of the potentials
by [Wo] and [SZ] is not so large. However, the deviations from other
parameterizations are bigger and will be demonstrated
elsewhere~\cite{vanHees:2008xxx}. The effects from the ``momentum kick''
of the light quarks in quark coalescence, an enhancement of both,
$R_{AA}$ and $v_2$, is important for the quite good agreement of both
observables with the data. As can be seen from the lower panel in
Fig.~\ref{fig.electrons}, within our model the effects from the mixing
of the $B$-meson decay contribution to the $e^{\pm}$ spectra becomes
visible in the region of $p_t \simeq 2.5$-$3 \; \text{GeV}$. A closer
inspection of the time evolution of the $p_t$ spectra shows that the
suppression of high-$p_t$ heavy quarks occurs mostly in the beginning of
the time evolution, while the $v_2$ is built up later at temperatures
close to $T_c$ which is to be expected since the $v_2$ of the bulk
medium is fully developed at later stages only. This effect is also
pronounced for the [Wo] parameterization of the HQ potential since in
this case due to resonance formation the transport coefficients become
largest close to $T_c$.
\section{Conclusions and outlook}
\label{sec-conclusions}
We have used static potentials from finite-temperature lQCD calculations
within a Brueckner-type many-body calculation, complemented by pQCD
HQ-gluon elastic-scattering matrix elements, to assess drag and
diffusion coefficients for $c$ and $b$ quarks in the QGP in a
principally parameter free approach, however plagued with large
uncertainties in the determination of the relevant potential from
lattice data. The diffusion of heavy quarks in the QGP is calculated
with a Langevin simulation. The medium is parameterized as an expanding
thermal fireball (including an\-iso\-tro\-pic flow for semicentral
heavy-ion collisions) with an equation of state of a massless gas of
gluons and $N_f=2.5$ light-quark flavors. To confront this model with
data on non-photonic single electrons in $200 \; A\text{GeV}$ Au-Au
collisions at RHIC, we have used a combined coalescence-fragmentation
model to hadronize the heavy quarks to $D$ and $B$ mesons which
subsequently decay semi-leptonically. The resulting $p_t$ spectra agree
with recent data on the nuclear modification factor, $R_{AA}$ and
elliptic flow, $v_2$ quite well.
In a schematic estimate from the evaluated drag and diffusion
coefficients leading to this results, based on kinetic theory using
either pQCD for a weakly coupled plas\-ma
\cite{Israel:1970,Danielewicz:1984ww} or the strong-coupling limit
applying AdS/CFT
correspondence~\cite{Herzog:2006gh,CasalderreySolana:2006rq}, we find
values for the space-dif\-fu\-sion coefficient $2 \pi T D_s=4$-$6$ and a
viscosity to entropy-density ratio of $\eta/s=2$-$5/(4 \pi)$ (to be
compared to the conjectured AdS/CFT bound $(\eta/s)_{\mathrm{min}}=1/(4
\pi)$), indicating a strong\-ly coupled (liquid like) quark-gluon plasma
close to the phase transition~\cite{Rapp:2008qc}.
In future works detailed studies of the uncertainties in the potential
approach is necessary, in particular about the question whether a
static-potential approach is justified and which in-medium potential
(i.e., free or internal potential energy or combinations thereof) should
be used to describe the interactions of heavy quarks within the
sQGP. First steps in this direction have been made
in~\cite{Brambilla:2008cx} in an effective-field theory approach for
non-relativistic in-medium QCD-bound states. Also an inclusion of
inelastic processes like gluon bremsstrahlung which should become
effective at higher $p_t$ is mandatory for a complete picture of the
in-medium behavior of heavy quarks~\cite{Vitev:2007jj}.
Another step, which is quite natural, given that with\-in our model the
underlying microscopic effect of the strong coupling of the heavy quarks
to the medium is the formation of resonance states close to $T_c$, is to
substitute the quark-coalescence model with a transport-model based
resonance-recombination model for hadronization in the
QGP~\cite{Ravagli:2007xx} which obeys the conservation laws for energy
and momentum as well as the second law of thermodynamics. This model
shares with quark-coalescence hadronization the phenomenologically
successful feature of the scaling of $v_2$ with the hadrons' number of
constituent quarks. Additionally it leads to scaling with the transverse
kinetic energy well, provided the parton distribution and flow is
calculated with a dynamically consistent scheme as the Langevin
simulation discussed in the present paper~\cite{Ravagli:2008rt}.
\begin{acknowledgement}
This work has been supported by the U.S. National Science foundation
under CAREER grant PHY-0449489 (HvH,RR), and by the Ministerio de
Educaci{\'o}n y Ciencia under grant AYA 2005-08013-C03-02 (MM).
\end{acknowledgement}
|
1,314,259,996,695 | arxiv | \section{Introduction}
The Arctic is warming at least twice as fast as the global average \citep{Osborne2018}. This phenomenon of Arctic amplification in surface air temperature is closely connected to a dramatic multi-decade reduction in Northern sea ice. Indeed, since accurate satellite measurements began in 1978, the extent of Arctic summer sea ice has shrunk by about 40 percent, a loss in area comparable to the western continental United States. This drop in sea ice is one of the most conspicuous warning signs of ongoing climate change, but the reduction of sea ice will also play an important role in determining the pace of future global climate change and has wide-ranging implications for the polar region and the rest of the world.
At a regional level, diminishing sea ice alters polar ecosystems and habitats and introduces major economic opportunities and risks. For example, new deposits of natural gas, petroleum, and other natural resources will become accessible for extraction, emission, and possible spillage \citep{Petrick2017}. Also, reduced ice coverage facilitates tourism and the use of Arctic shipping lanes, which are shorter than traditional passages via the Suez or Panama Canals. These new routes reduce sailing times but increase Arctic environmental risks from, for example, discharges, spills, and soot deposits \citep{Bekkers2016}. Finally, melting sea ice will have geopolitical consequences for Arctic sea-lane control \citep{ebinger2009}.
Although these proximal Arctic effects are important, the far-reaching implications of diminished Arctic sea ice for regulating global climate and weather are even more consequential. Less sea ice and more open water diminishes the reflectivity (or albedo) of the Arctic region, so that, over time, a greater share of solar heat is absorbed by the earth, which leads to further Arctic amplification and increased temperatures worldwide \citep{Hudson2011}. These higher Arctic temperatures promote thawing and erosion of the polar permafrost, which can result in the release of large amounts of carbon dioxide and methane and provide a significant impetus to further global warming \citep{Tanskietal2019}. Increasing Arctic temperatures also hasten the melting of the Greenland ice sheet, further pushing up sea levels \citep{Trusel2018}. Finally, a warming Arctic and loss of sea ice cover can alter the global dynamics of ocean and air streams, and this effect already appears to be changing weather patterns at sub-polar latitudes \citep{PetoukhovandSemenov2010} and weakening thermohaline ocean circulation including the current that warms Europe \citep{LiuFedorov2019}.
In brief, the loss of Arctic sea ice is not only a stark signal of a changing climate, but it also plays an integral role in the timing and intensity of further global climate change. Not surprisingly then, the downward trend in Arctic sea ice has been the subject of hundreds of research studies. The forecasting literature alone is voluminous and impressive in both methodology and substance.\footnote{Recent research includes \cite{petty2017}, \cite{Onoetal2018}, \cite{Peng2018}, \cite{SerrezeAndMeier2019}, and \cite{Ionita2019}. Ongoing prediction research forums include the Sea Ice Prediction Network at \url{https://www.arcus.org/sipn} and the Polar Prediction Project at \url{https://www.polarprediction.net}.} Although the importance of accurate polar prediction is hard to overstate, substantial uncertainty still remains about the future evolution of sea ice. Indeed, obtaining a deeper understanding of Arctic sea ice loss has been called a ``grand challenge of climate science" \citep{Katssovetal2010}.
Much forward-looking sea ice analysis has been based on large-scale climate models, which represent of the fundamental physical, chemical, and biological drivers of the earth's climate. These models attempt to capture the dynamics of the oceans, atmosphere, and cryosphere at a high frequency and a granular level of geographic and spatial detail. Such structural physical models are invaluable for understanding climate variation, determining event and trend attribution, and assessing alternative scenarios. However, from a forecasting perspective, climate models have generally underestimated the amount of lost sea ice in recent decades (\citealp{Stroeve2007}; \citealp{Stroeve2012}; \citealp{Jahn2016}; and \citealp{Rosenblum2017}). In addition, long-range sea ice projections can differ widely across climate models \citep{StroeveNotz2015}.
Given the global significance of Arctic conditions and the progress yet to be made on specifying structural global climate models, purely \emph{statistical} projections of sea ice are an obvious complementary approach. As a practical matter -- across many disciplines -- parsimonious statistical representations often produce forecasts that are at least as accurate as detailed structural models. Specifically for forecasting Arctic sea ice, there is already some evidence that small-scale statistical models with no explicitly embedded physical science can have some success (\citealp{Guemasetal2016}; \citealp{Wangetal2016}). Therefore, we provide a statistical analysis of the long-run future evolution of Arctic sea ice. Our work is distinguished by its use of intrinsically stochastic ``unobserved components" models, with detailed attention to trend, seasonality, and serial correlation. Based on several decades of satellite data, we provide statistical forecasts of the future loss of Arctic sea ice.\footnote{Our analysis does not examine prediction skill in real-time repeated forecasting, as in \cite{Ionita2019}, but considers very long-range projections at a point in time, as in \cite{Guemasetal2016} .} Importantly, these forecasts provide probability assessments for a range of long-run outcomes and quantify both model parameter uncertainty and intrinsic uncertainty. Of particular interest are probability assessments of the timing of an ice-free Arctic, an outcome with vital economic and climate consequences (\citealp{Massonnet2012}; \citealp{Snape2014}; and \citealp{Jahn2016}). For this analysis, we also introduce a novel statistical modeling mechanism -- a shadow ice interpretation -- that allows us to readily account for the zero lower bound on the extent of Arctic sea ice in our model.
Our resulting distributional forecasts suggest an ice-free Arctic summer is more likely than not within two decades -- \emph{much} sooner than the projections from many large-scale climate models. In particular, we contrast our statistical forecasts with projections from the ensemble of model simulations conducted for the fifth Coupled Model Intercomparison Project (CMIP5) -- a highly-regarded central source for international global climate model projections. On average, these climate models envisage ice-free Arctic conditions close to the end of the century (assuming a range of business-as-usual carbon emissions paths). Thus, besides their relevance for environmental and economic planning, our probability assessments may also provide a useful benchmark for assessing or calibrating global climate models going forward.
We proceed as follows. In section \ref{linmod}, we introduce a linear statistical trend model and use it to produce long-range sea ice point forecasts. In section \ref{shadowice}, we introduce the ``shadow ice" concept to account for the zero-ice lower bound. In section \ref{quadmod}, we generalize to a nonlinear (quadratic) statistical model and to interval forecasts that incorporate several forms of uncertainty. In section \ref{GCM}, we compare our statistical model forecasts to global climate model forecasts with particular attention to hard versus soft landings at zero ice. In section \ref{prob}, we make probabilistic assessments of several sea ice scenarios. We conclude in section \ref{conclsec}.
\section{A Linear Statistical Model and Point Forecasts} \label{linmod}
Arctic sea ice has been continuously monitored since 1978 using satellite-based passive microwave sensing, which is unaffected by cloud cover or a lack of sunlight. For a polar region divided into a grid of individual cells, the satellite data provide cell-by-cell brightness readings, which can be converted into fractional ice surface coverage estimates for each cell. Sea ice extent, $SIE$ -- a very commonly-used measure of total ice area -- is the total area of all cells with at least 15 percent ice surface coverage. That is, $SIE$ rounds down cells with measured coverage of less than 15 percent to zero and rounds up cells that pass the 15 percent threshold to full coverage. The up-rounding in $SIE$ is effectively a useful bias correction, as melting pools on summer ice surfaces can be mistaken for ice-free open water. For this reason, we follow common practice and use monthly average $SIE$ data from November 1978 through October 2019 from the National Snow and Ice Data Center (NSIDC).\footnote{Another measure of Arctic ice is sea ice area ($SIA$), which adds together the measured fractions of ice-covered areas of all cells that pass the 15 percent threshold. For brevity, we report results only for $SIE$, but results for $SIA$ are qualitatively identical.} The NSIDC data use the NASA team algorithm to convert the satellite microwave brightness readings into measured ice coverage \citep{Fettereretal2017}.\footnote{We interpolate the missing December 1987 and January 1988 observations with fitted values from a regression on trend and monthly dummies estimated using the full data sample.}$^,$\footnote{For a comparison of algorithms, and evidence of strong performance of the NASA team algorithm, see \cite{IcePlus}.}
\begin{figure}[t]
\caption{Arctic Sea Ice Extent ($SIE$) and Fitted Linear Trend}
\begin{center}
\includegraphics[trim= 6mm 0mm 0mm 0mm, clip, scale=.13]{newgraph1.jpg}
\label{tsplot}
\end{center}
\begin{spacing}{1.0} \footnotesize \noindent Notes: We show monthly average Arctic sea ice extent ($SIE$) from November 1978 to October 2019 with a fitted linear trend. Each monthly observation is a dot, and September and March observations are colored red and blue, respectively.
\end{spacing}
\end{figure}
Figure \ref{tsplot} plots the time series of Arctic $SIE$ -- each monthly average observation is a dot -- with an overall estimated linear trend superimposed. The clear downward trend is accompanied by obvious seasonality. A more subtle feature is the possible time variation in the seasonal effects, which may be trending at different rates and possibly nonlinearly. These effects turn out to be of interest in a complete statistical representation of sea-ice dynamics.
\begin{figure}[t]
\caption{$SIE$: Linear Model Fits and Point Forecasts}
\label{forec_lin}
\begin{center}
\includegraphics[scale=0.11]{unrestrictedLinearSlide2.jpg}
\end{center}
\begin{spacing}{1.0} \footnotesize \noindent Notes: The lines are the twelve in-sample fitted trends and out-of-sample extrapolations for Arctic $SIE$ in each month based on the linear model (\ref{eq:SIE}). The trends for the months with maximal and minimal sea ice extent are colored -- March in blue and September in red -- and the blue and red dots are the March and September historical $SIE$ data. The out-of-sample period is shaded gray.
\end{spacing}
\end{figure}
A simple initial representation to capture this variation is a linear statistical model with twelve intercepts, one for each month, each of which may be differently trending, and potentially serially correlated stochastic shocks:
\begin{equation}
\label{eq:SIE}
SIE_t = \sum_{i=1}^{12} \delta_i D_{it} + \sum_{j=1}^{12} \gamma_j D_{it} {\cdot} TIME_t + \varepsilon_t
\end{equation}
$$
\varepsilon_t = \rho \varepsilon_{t-1} + v_t
$$
$$
v_t \sim iid (0, \sigma^2),
$$
where the $D_{i}$'s are monthly dummy variables ($D_{it}{=}1$ in month $i$ and 0 otherwise, $i{=}1, ..., 12$) and $TIME$ is a time dummy ($TIME_t {=} t$).\footnote{Earlier work in this tradition includes \cite{Peng2018}.}$^,$\footnote{Note that although we allow for seasonal and trending intercepts, we assume a constant disturbance variance across months and time. There are so many interesting directions that we have reserved volatility modeling for future work. For example, high-frequency (daily) $SIE$ data are available, from which one could accurately estimate, and then model, lower-frequency (monthly) realized $SIE$ volatility.} We estimate model (\ref{eq:SIE}) -- and other versions below -- by maximizing the Gaussian likelihood. Detailed regression results for model (\ref{eq:SIE}) are in column (6) of Table \ref{vvv1} in Appendix \ref{App1}.
Figure \ref{forec_lin} shows the resulting linear trends for all twelve months, highlighting March in blue and September in red.
All of the monthly trends slope downward -- an indication of a warming climate -- and are highly significant. The slopes of the linear trends also differ across months (\citealp{SerrezeAndMeier2019}; \citealp{Cavalierietal2012}). In particular, the seasonally low-ice months of July through October have notably steeper downward sloping trends than the seasonally high-ice months of December through May. The estimated September trend, for example, is \textit{twice} as steep as the March trend, and the difference is highly statistically significant. These linear trends are also extrapolated out of sample (shaded gray) through the end of the century. For example, September sea ice extent is projected to reach zero just after 2072.
Such linear point forecasts are a useful first step, but they can be improved by allowing for nonlinearity in the trends and by quantifying forecast uncertainty -- as described in section \ref{quadmod}. First, however, we elucidate a ``shadow ice" modeling approach that takes into account the fact that the measured amount of sea ice is bounded below by zero.
\section{A Shadow Ice Interpretation} \label{shadowice}
One consideration for downward trending statistical models for Arctic sea ice is that the measured amount of sea ice will always be non-negative. By contrast, extrapolations of simple trend models will eventually push into negative territory. There are various functional forms that can be used to model such bounded time series, and the appropriate representation depends very much on the details of the real-world phenomenon under examination.\footnote{One modeling approach is to rescale the bounded time series data to the real line using, say, a log-ratio transformation \citep{Wallis1987}. Alternatively, a time series can be modeled in the original bounded sample space using, for example, the beta autoregressive model of \cite{Rocha2008}. By contrast, some physical science analyses simply terminate model simulations when zero ice conditions are reached, as in \cite{Obryketal2019}.} Some bounds act like reflecting barriers, so the variable of interest spends very little time at the constraint. Other bounds are absorbing states, and once reached, they may be sustained for some time.
With positive amounts of sea ice, fluctuations in \textit{SIE} can serve as a rough approximation for changes in the amount of thermal energy in the Arctic; that is, hotter and colder ocean surface temperatures are reflected in less or more ice, respectively.\footnote{The amount of sea ice depends on a variety of factors including ocean and air temperature, ocean salinity, cloud cover, and wind, current, and wave action. However, several studies have identified ocean heat content as a major driver of sea ice coverage, notably, \cite{Arthun2012}, \cite{Schlichtholz2011}, and \cite{Selyuzhenok2020}.} However, this connection breaks down when the ice disappears: While \textit{SIE} is fixed at zero, the surface temperature of the Arctic ocean can continue to warm. Furthermore, the warmer the ocean becomes, the less likely there will be a quick return of sea ice, which is indicative of a partially-absorbing state.\footnote{See \cite{StroeveNotz2018} for a description of the phenomenon by which the ocean must release heat back into the atmosphere before sea ice can form again in winter.}
To account for this effect, the negative values of sea ice produced by a statistical model can be viewed as a rough expression of ocean temperature. Thus, we redefine the left-hand side variable of the unconstrained model as a \textit{shadow} surface ice extent, $SIE^*$. We view $SIE^*$ as a notional variable that equals measured surface ice when positive, but that may also go negative to represent further increases in ocean thermal energy more broadly. Formally, to translate negative model-based sea ice values into nonnegative sea ice observations, our shadow ice model modifies the unconstrained model \eqref{eq:SIE} to respect the zero lower bound for ice:
\begin{equation} \label{eq:Shadow_a}
SIE^*_t = \sum_{i=1}^{12} \delta_i D_{it} + \sum_{j=1}^{12} \gamma_j D_{jt} {\cdot} TIME_t + \varepsilon_t
\end{equation}
$$ \label{eq:Shadow_b}
\varepsilon_t = \rho \varepsilon_{t-1} + v_t
$$
$$ \label{eq:Shadow_c}
v_t \sim iid (0, \sigma^2)
$$
$$ \label{eq:Shadow_d}
SIE_t = \max(SIE^*_t, 0).
$$
That is, we now interpret our earlier unconstrained linear model of surface ice as a model of shadow ice, $SIE^*$, so that the observed extent of sea ice, \textit{SIE}, is the maximum of $SIE^*$ and zero.\footnote{A similar framework has been successfully applied in finance to model nominal interest rates near their zero lower bound \citep{CR2014, BR2016}.}
The shadow ice model respects the nonlinearity of observed ice at the zero lower bound but retains tractability. It also allows us to translate the long-range forecasts from downward trending models like model \eqref{eq:SIE} -- including distributional projections -- into observed data that are always non-negative. As a matter of physical interpretation, very negative values of shadow ice extent, $SIE^*$, represent environments in which the thermal content of the Arctic ocean is high enough that an immediate return to a positive \textit{SIE} is unlikely. This shadow ice structure provides an intuitive and simple approximation of the thermodynamics of the Arctic ocean transition between sea ice and open water and serves as a useful modeling tool for observed \textit{SIE} dynamics.\footnote{\cite{Wangetal2016} take a different approach by simply constraining the model by the lower bound (for sea ice concentration in their case), so negative predicted values are set to zero. In a dynamic model with lagged sea ice, this procedure will result in a representation of a physical bound that is much closer to a reflecting barrier.}
\section{A Quadratic Statistical Model and Interval Forecasts} \label{quadmod}
A downward linear trend is a common representation of the secular decline in Arctic sea ice, but linearity is not assured by the physical science. There are a variety of climate feedback mechanisms that could hasten or retard the pace of sea ice loss. The well-known ice albedo effect occurs as sea ice cover is reduced, and the resulting darker ocean surface absorbs more energy, which in turn further reduces sea ice \citep{Stroeve2012}. This feedback effect amplifies sea ice seasonality and may progressively steepen the downward trend in $SIE$ over time \citep{Schroder2014}. The geography of the perimeter of the Arctic Ocean, which is partially constrained by land that blocks expanding winter ice, becomes less relevant as sea ice shrinks, and relaxing this constraint may allow greater seasonal variation and a steeper downward sea ice trend \citep{SerrezeAndMeier2019}.\footnote{Currently, the southern limit of winter ice is bounded by land except in the Bering Sea, Sea of Okhotsk, East Greenland Sea, Barents Sea, and Baffin Bay.} However, there are offsetting negative feedback mechanisms -- associated, for example, with increased cloud cover -- that could slow the rate of sea ice loss over time \citep{IPCC2019}. Indeed, \cite{StroeveNotz2015} argue against trend amplification in favor of trend constancy (linearity) or trend attenuation. Moreover, as described in the next section, long-range $SIE$ projections from large-scale global climate models appear dominated by feedback mechanisms that slow the rate of September sea ice loss over time.\footnote{More extreme forms of nonlinearity -- such as discontinuous breaks, tipping points, and thresholds -- are possible but viewed as less likely \citep{StroeveNotz2015}. \cite {GoldsteinEtAl18} argue that Arctic sea ice is best modeled by step-like shifting means at fitted breakpoints. However, modified statistical information criteria that properly account for the implicit flexibility of such breakpoints -- following \cite{Hall2013} -- do not favor such a shifting mean models relative to a linear trend.}
The lack of a complete understanding of the drivers of Arctic sea ice recommends consideration of a flexible empirical $SIE$ specification, so we generalize from linear to quadratic trends:
\begin{equation} \label{eq:Shadow2}
SIE^*_t = \sum_{i=1}^{12} \delta_i D_{it} + \sum_{j=1}^{12} \gamma_j D_{jt} {\cdot} TIME_t + \sum_{k=1}^{12} \alpha_k D_{kt} {\cdot} TIME_t^2 + \varepsilon_t
\end{equation}
$$
\varepsilon_t = \rho \varepsilon_{t-1} + v_t
$$
$$
v_t \sim iid (0, \sigma^2)
$$
$$
SIE_t = \max(SIE^*_t, 0).
$$
We label model (\ref{eq:Shadow2}) as the ``general" quadratic model as no constraints are imposed on the twelve quadratic ($\alpha_k$) parameters. The linear model (\ref{eq:Shadow_a}) of course emerges under the constraint $\alpha_{1} {=} ... {=} \alpha_{12} {=} 0$.
\begin{figure}[t]
\caption{$SIE$: General Quadratic Model Fits and Point Forecasts}
\begin{center}
\includegraphics[scale=0.11]{unrestrictedQuadraticSlide3.jpg}
\label{forec_quad}
\end{center}
\begin{spacing}{1.0} \footnotesize \noindent Notes: The curves are the twelve in-sample fitted trends and out-of-sample forecasts for Arctic $SIE$ in each month from the general quadratic model (\ref{eq:Shadow2}). The months with maximal and minimal sea ice extent are colored -- March in blue and September in red -- and the blue and red dots are the March and September historical $SIE$ data. The out-of-sample period is shaded gray.
\end{spacing}
\end{figure}
Figure \ref{forec_quad} shows the $SIE$ estimation fits and forecasts from the general quadratic model, by month, with March and September again highlighted in blue and red, respectively. The trend curvatures for all months are strikingly similar: the trends for all months decrease \textit{at an increasing rate}. That is, the estimated $\alpha_k$ coefficients are negative for every month, indicating that $SIE$ is diminishing at an increasing rate. (Detailed estimation results of model (\ref{eq:Shadow2}) appear in column (1) of Table \ref{vvv1} in Appendix \ref{App1}.) The size of the estimated negative coefficients on the quadratic trend terms and their statistical significance vary by month. The most negative and significant $\alpha_k$ coefficients are in the seasonally low-ice ``summer"
months of August, September, and October, and these months show the greatest trend rates of decline.\footnote{Minimum ice coverage occurs at the end of summer in September because sea ice coverage continues to shrink for any above-freezing temperatures. For convenience, we refer to the low-ice months of August, September, October as ``summer" even though they span late summer through early autumn.} An $F$ test of the joint hypothesis that $\alpha_8{=} \alpha_9{=}\alpha_{10}{=}0$ produces a $p$-value of 0.00.\footnote{Because $TIME^2$ and $TIME$ are correlated, an insignificant $\alpha_k$ coefficient would not necessarily imply that nonlinearity is unimportant.} Relative to a linear trend model, the nonlinear trend model forecasts lower sea ice at long horizons. February-April $SIE$ point forecasts nevertheless remain well above zero through the century, but August-October point forecasts approach zero much more quickly. Indeed the quadratic September point forecast hits zero in 2045.
Table \ref{vvv} summarizes the results from investigation of the summer and non-summer differences in quadratic trend curvature using two standard model selection criteria: the Akaike information criterion (\textit{AIC}) and the Bayesian information criterion (\textit{BIC}). The \textit{AIC} and \textit{BIC} are estimates of out-of-sample forecasting performance (mean-squared error), formed by penalizing estimates of in-sample forecasting performance for degrees of freedom used in model fitting and differing only in the precise penalty applied \citep{Diebold2007}. The table reports these model selection criteria for six versions of the quadratic trend model (\ref{eq:Shadow2}) with various equality constraints imposed on the quadratic coefficients $\alpha_1, ..., \alpha_{12}$. Corresponding estimation results appear in Table \ref{vvv1} in Appendix \ref{App1}.
The models favored by \textit{AIC} and \textit{BIC} -- that is, those with smaller values -- are very similar and involve summer and non-summer restrictions. The \textit{AIC} selects a model with equal $\alpha_k$'s for the nine non-summer months (November-July) and unconstrained $\alpha_k$'s for the three summer months (August-October). The \textit{BIC}, which penalizes degrees of freedom more harshly, selects a slightly more constrained model, with the non-summer $\alpha_k$'s again constrained to be equal and the three summer $\alpha_k$'s \textit{also} constrained to be equal. Compared to the unconstrained version of equation (\ref{eq:Shadow2}) -- the general model -- both the \textit{AIC} and \textit{BIC} prefer specifications with some summer and non-summer equality constraints imposed, because their imposition saves degrees of freedom without substantially degrading fit. All told, the results of Table \ref{vvv} suggest a ``simplified" quadratic model, namely model (\ref{eq:Shadow2}) with both summer and non-summer quadratic coefficients constrained separately to equality: $\alpha_{8} {=} \alpha_{9} {=} \alpha_{10}$ and $\alpha_{11} {=} \alpha_{12} {=} \alpha_{1} {=} ... {=} \alpha_{7}$. The regression diagnostics reported in Appendix \ref{App1} suggest good performance of the simplified quadratic model: $R^2$ is almost perfect; the residual graph and Durbin-Watson statistic (DW) are consistent with random noise; and residual skewness, kurtosis, and density estimates are consistent with normality. All told, the model appears to do a fine job of capturing $SIE$ signal, reducing its complicated dynamics to Gaussian white noise.
\begin{table} [t]
\caption{Akaike and Bayes Information Criteria for Quadratic Coefficient Constraints}
\label{vvv}
\begin{center}
{
\begin{tabular}{ c c c c c c c}
\toprule
&(1)&(2)&(3)&(4)&(5)&(6)\\
& NONE & Seq & NSeq & Seq+NSeq & ALLeq & ALL0 \\
\midrule
\textit{AIC} & -0.0673 [3] & -0.0651 [4] & \textbf{-0.0913 [1]} & \textbf{-0.0877 [2]} & -0.0639 [5] & -0.0569 [6] \\
\textit{BIC} & \phantom{-}0.2569 [6] & \phantom{-}0.2421 [5] & \textbf{\phantom{-}0.1647 [2]} & \textbf{\phantom{-}0.1513 [1]} & \phantom{-}0.1665 [4] & \phantom{-}0.1649 [3]\\
\bottomrule
\end{tabular}
}
\end{center}
\begin{spacing}{1.0} \footnotesize \noindent Notes: We show \textit{AIC} and \textit{BIC} values for the quadratic model with various equality constraints imposed on the quadratic coefficients $\alpha_1, ..., \alpha_{12}$. ``NONE" denotes no constraints, which corresponds to the general quadratic model (\ref{eq:Shadow2}). ``Seq" (``Summer equal") denotes August-October equal ($\alpha_{8} {=} \alpha_{9} {=} \alpha_{10}$). ``NSeq" (``Non-Summer equal") denotes November-July equal ($\alpha_{11} {=} \alpha_{12} {=} \alpha_{1} {=} ... {=} \alpha_{7}$). ``Seq+NSeq" denotes summer months equal and non-summer months (separately) equal, which corresponds to the simplified quadratic model. ``ALLeq " denotes all months equal ($\alpha_{1} {=} ... {=} \alpha_{12}$). ``ALL0" denotes all months 0 ($\alpha_{1} {=} ... {=} \alpha_{12} {=} 0$), which corresponds to the linear model (\ref{eq:SIE}). Model ranks appear in brackets, where [1] denotes the best (smallest) criterion value. We show in boldface the best two models according to each criterion.
\end{spacing}
\end{table}
\begin{figure}[t]
\caption{$SIE$: Simplified Quadratic Model Fits, Point Forecasts, and Interval Forecasts}
\begin{center}
\includegraphics[scale=0.11]{confidenceIntervalsSlide5.jpg}
\label{forec_quads}
\end{center}
\begin{spacing}{1.0} \footnotesize{Notes: The curves are the twelve in-sample fitted trends and out-of-sample forecasts for Arctic $SIE$ in each month from the simplified quadratic model. The months with maximal and minimal sea ice extent are colored -- March in blue and September in red -- and the blue and red dots are the March and September historical $SIE$ data. The out-of-sample period is shaded gray. The March and September forecasts also have $\pm \, 2 \, s.e.$ blue and red dotted bands that account for parameter estimation uncertainty and intrinsic uncertainty error.}
\end{spacing}
\end{figure}
Figure \ref{forec_quads} shows the simplified quadratic model fits and forecasts. The simplified model point forecasts are very similar to those of the general quadratic model in Figure \ref{forec_quad}, with a zero-ice September also reached in 2045, but the rank ordering of the months in terms of $SIE$ is better preserved going forward. Going beyond these point forecasts, an important advantage of a formal statistical approach is that it can quantify the amount of future uncertainty. Figure \ref{forec_quads} supplements the simplified quadratic trend point forecasts with interval or probability density forecasts. When making long-horizon interval forecasts, it is crucial to account for parameter estimation error, because its deleterious effects grow with the forecast horizon. For example, although parameter estimation error may have small effects on 6-month-ahead intervals, it will be greatly compounded for 600-month-ahead intervals. An estimate of the time-$t$ standard deviation of the forecast error, which accounts for parameter estimation error, is $\delta_t {=} s \sqrt{1 + x_t' (X'X)^{-1} x_t}$,
where $s$ is the standard error of the regression, $x_t$ is a 36$\times$1 column vector of time-$t$ right-hand-side variables, $X$ is a $T$$\times$36 matrix whose columns contain the regression's right-hand-side variables over time, and $T$ is sample size.\footnote{See for example \cite{Johnston1972}, pp. 153-155, for derivation of this canonical result.} We use $\delta_t$ to produce the $\pm \, 2 \, \delta_t$ pointwise prediction intervals of Figure \ref{forec_quads}. Under normality of the shocks underlying the simplified quadratic model, the $\pm \, 2 \, \delta_t$ intervals are approximate 95 percent confidence intervals.\footnote{Normality of the $v$ shocks does not appear unreasonable, as the simplified quadratic model residuals have skewness and kurtosis of -0.17 and 3.73, respectively. We also obtained similar results without assuming normality via bootstrap simulation, which we discuss in section \ref{prob}.} The intervals widen rapidly; indeed the September interval starts to include zero before 2040.\footnote{By contrast, interval forecasts that fail to account for parameter estimation uncertainty quickly approach (by about 12 months ahead) the fixed-width interval $\pm \, 2 \, \delta$, where $\delta$ is the estimated unconditional standard deviation of the $AR(1)$ disturbance in equation (\ref{eq:Shadow2}), $\sqrt{\hat{\sigma}^2 / (1 {-} \hat{\rho}^2)} \approx 0.32$ and fail to widen with forecast horizon.}
\begin{figure}[t]
\caption{$SIE^*$: Simplified Quadratic Model Point and Interval Forecasts}
\begin{center}
\includegraphics[trim= 10mm 30mm 0mm 30mm, clip, scale=.13]{graph7s2.jpg}
\label{shadow}
\end{center}
\begin{spacing}{1.0} \footnotesize Notes: We show the forecast of Arctic shadow sea ice extent, $SIE^*$, for 2060-2062, based on the simplified quadratic model. The solid black line is the point forecast, the shaded area is the $\pm \, 2 \, s.e.$ band, and the red line denotes zero.
\end{spacing}
\end{figure}
Finally, in Figure \ref{shadow} we zoom in on the 2060-2062 (36-month) segment of the simplified quadratic model shadow ice forecast. The point forecast trends down and is seasonally below zero by then. (Indeed, as already discussed, the point forecast is seasonally below zero well before then.) Moreover, the $\pm \, 2 \, s.e.$ intervals widen over time, and entire \textit{intervals} are seasonally below zero by then. Hence it appears that, {with near certainty}, summer $SIE$ will vanish by 2060. This result is also clear from our earlier Figure \ref{forec_quads}, but Figure \ref{shadow} highlights it in a different and complementary way.
\section{Statistical versus Climate Model Projections} \label{GCM}
Of the many analyses of the long-term future evolution of Arctic ice, most have focused on projections from large-scale climate models. Such models are based on the underlying physical, chemical, and biological processes that govern the dynamics of weather and climate across ocean, air, ice, and land. The models fit an immense number of variables at a high temporal frequency and a granular spatial scale (e.g., a 30-minute time interval. a 100km worldwide horizontal resolution, and 40 vertical levels in the oceans and atmosphere). Dozens of scientific groups around the world have constructed and currently maintain such models. Occasionally, these groups conduct concurrent simulations as part of the Coupled Model Intercomparison Project (CMIP) that involve common sets of inputs including carbon emissions scenarios. The most recently completed iteration or phase of this project is the fifth one, denoted CMIP5 \citep{Taylor2012}. The CMIP5 model comparison study was the main source of climate projections included by the International Panel on Climate Change (IPCC) in its landmark Fifth Assessment Report.
Arctic $SIE$ is a key variable projected by climate models, and the models included in CMIP5 are generally judged to provide a better fit to Arctic sea ice than earlier CMIP iterations.\footnote{\cite{Stroeve2007} describes the poor sea ice fit of the CMIP3 models, while the somewhat improved fit of the CMIP5 models is noted by in \cite{Stroeve2012}.} Figure \ref{CModel} shows three projections for September $SIE$ constructed as averages across sets of CMIP5 global climate models. These multi-model mean projections are constructed under three different scenarios, or Representative Concentration Pathways (RCPs), for future greenhouse gas concentrations. The brown, yellow, and blue lines are averages of the climate models, respectively, under a high level of carbon emissions (RCP8.5), a medium level (RCP6.0, yellow), and a low level (RCP4.5, blue).\footnote{The climate model data are described in \cite{SerrezeAndMeier2019} and \cite{Stroeve2012} and were kindly provided by Andrew Barrett at the National Snow and Ice Data Center. The model sets averaged for each scenario are not identical, with the RCP4.5, RCP6.0, and RCP8.5 scenarios based on 26, 8, and 25 climate models, respectively.} The first two higher emissions scenarios are viewed as more likely business-as-usual outcomes and are the most relevant to compare to statistical projections that extend the historical sample of past data and assume a continuation of the world economy's current population and development trajectories.\footnote{In the RCP8.5 scenario, continuing increases in greenhouse gas emissions through the end of the century raise the 2100 global average temperature by about 4.0-6.0$^{\circ}$C above pre-industrial levels \citep{USGCRP2018}. In the RCP4.5 scenario, greenhouse gas emissions level off before mid-century, and the 2100 global average temperature is approximately 2.0-3.0$^{\circ}$C above pre-industrial levels. }
The solid red line in Figure \ref{CModel} shows the September trend from the simplified quadratic model estimated on the full sample from November 1978 to October 2019. The dotted lines bracketing this forecast provide an approximate 95 percent confidence interval. However, the CMIP5 climate model projections are only based on data through 2005 and do not include the past dozen or so years of sea ice observations. For comparability to these climate model projections, we re-estimated the simplified quadratic model using data from 1978 through 2005, and the September trend from this pre-2006 model is shown as the dashed red line. An interesting first result is that from the close conjunction of the two simplified model statistical projections (the solid and dashed red lines), the addition of data from 2006 to 2019 does not lead to a significant revision in the statistical trend model. The very modest differences between the pre-2006 estimated quadratic trend and the full-sample version is an indication of the stability and suitability of the simplified statistical model.
\begin{figure}[tb]
\caption{September $SIE$: Statistical and Climate Model Fits, Point Forecasts, and Interval Forecasts}
\begin{center}
\includegraphics[scale=0.12]{gcmSlide8.jpg}
\label{CModel}
\end{center}
\begin{spacing}{1.0} \footnotesize \noindent Notes: We show fitted values and forecasts of Arctic sea ice extent, $SIE$. The solid black line is the historical data for September Arctic sea ice extent from 1979 to 2019. Mean projections from CMIP5 climate models are shown assuming underlying pathways of a high level of emissions (RCP8.5, brown), medium emissions (RCP6.0, yellow), and lower emissions (RCP4.5, blue). The climate model projections start in 2006. The red dashed line is the fitted and projected simplified quadratic model estimated using data from 1979 to 2005. The red solid line is the fitted and projected simplified quadratic trend model estimated using the full sample from 1979 to 2019, and the red dotted lines are $\pm \, 2 \, s.e.$ intervals (approximate 95\% confidence intervals) around that projection accounting for parameter estimation uncertainty and intrinsic uncertainty.
\end{spacing}
\end{figure}
Comparing the statistical and climate model projections in Figure \ref{CModel} reveals two salient features. First, throughout the forecast period, the full-sample statistical model projection is significantly lower than any of the climate multi-model mean projections. Indeed, the climate model mean projections are well outside the 95 percent confidence intervals. Relative to the statistical model estimated on data before 2006 or the full-sample version, the climate model means are overestimating the 2019 $SIE$ trend by about 1.0 million km\textsuperscript{2}. This wedge is projected to increase dramatically over the next two decades. Indeed, the pre-2006 and full-sample statistical trend models project zero ice in 2042 and 2044, respectively, but none of the climate model mean projections reach a completely ice-free Arctic in this century. Notably, even assuming a high level of emissions (RCP8.5) -- which is a scenario with continuing increases in average global surface temperatures throughout this century -- the multi-model mean projection never reaches zero Arctic summer sea ice.
Second, in contrast to the statistical projection, the climate model mean projections show a decreasing rate of ice loss over time -- that is, a concave rather than convex structure. Specifically, the climate model projections all display a roughly linear decline for the first couple of forecast decades (through around 2040) and then start to level out. For the lower emissions RCP4.5 scenario, the leveling out and deceleration of sea ice partly reflects a slowdown in the pace of global temperature increases. This effect is not at work in the RCP8.5 scenario, which has global temperatures steadily climbing through 2100. However, very close to zero sea ice extent, the leveling out of $SIE$ in RCP8.5 appears to reflect the hypothesized difficulty of melting the thick sea ice clinging near northern coastlines -- notably in Greenland and Canada. Climate models generally assume that these coastal regions will retain landfast sea ice for a time even after the open Arctic Sea is free of ice. Therefore, a common definition of ``ice-free" or ``nearly ice-free" in the literature is a threshold of 1.0 million km\textsuperscript{2} rather than zero SIE \citep{WangOverland2009}. Still, even with this higher threshold, the three climate model mean projections only reach a nearly ice-free Arctic in 2068, 2089, and $>$2100 for successively lower emissions scenarios, respectively. By contrast, the pre-2006 statistical projection reaches the higher 1.0 million km\textsuperscript{2} threshold in 2037, and the full-sample statistical model reaches that level in 2039.
Some have argued that climate models generally do well in representing the large-scale evolution of Arctic sea ice \citep{StroeveNotz2015}, but a number of studies have noted that the CMIP5 global climate models overpredicted the amount of Arctic sea ice (\citealp{Massonnet2012}; \citealp{StroeveEtAl2012b}; \citealp{SerrezeAndMeier2019}). That overprediction continues, and its source is not well understood. One proposed correction to this overprediction has been to focus on the models that fit the historical $SIE$ data better according to certain metrics \citep{WangOverland2012}. However, there is no agreed upon model selection criterion. Also, \cite{Rosenblum2017} discount such model selection because models with more accurate sea ice readings also tend to overpredict global temperatures, so the selected climate models may be getting sea ice loss right for the wrong reason. Finally, it should be noted that from a statistical viewpoint, focusing on a simple average forecast from many models has been shown to be a robust prediction strategy (\citealp{DieboldAndShin2019}).
In some respects, the wide differences between the statistical and climate model projections are not too surprising. In climate models, the monthly observations on total Arctic SIE are a high-level output from complex, nonlinear, granular representations of the relevant underlying science. Obtaining good SIE predictions from these models requires correctly specifying a host of detailed subsidiary processes. Such a bottom-up modeling procedure has important advantages in structural interpretation and counterfactual scenario analysis. However, in a variety of disciplines, a bottom-up procedure, which carries the possibility that small misspecifications can accumulate and affect high-level aggregates, has not been found to improve prediction relative to a top-line procedure that directly models the object of interest (\citealp{Diebold2007}). Thus, based on broad previous experience, we believe that direct statistical projections of Arctic SIE are likely to be relatively accurate.
\section{Probability Assessments of an Ice-Free Arctic} \label{prob}
An advantage of a formal statistical model is its ability to make probability density forecasts for a range of possible reduced Arctic sea ice scenarios. Of particular interest are probability assessments of an ice-free or nearly ice-free Arctic, that is, the probability that $SIE_t$ equals zero or is less than or equal to some threshold $\gamma$, respectively. Formally, such event probabilities can be denoted as $P(SIE_t{\le} \gamma)$, which represents the probability that sea ice extent is less than or equal to $\gamma$ in month $t$.
We estimate these scenario probability distributions using the simplified quadratic model and a stochastic simulation procedure that accounts for parameter estimation uncertainty and allows for potentially non-Gaussian serially correlated stochastic shocks. From a given set of simulated paths, we estimate the event probabilities of interest as the proportion of simulated paths in which the event occurs out of the total number of paths.\footnote{We provide details on this simulation procedure in Appendix \ref{App2}. For further discussion of the econometrics of threshold event probabilities, see \cite{BR2016}.}
\begin{figure}[t]
\caption{Probability Distributions of First Ice-Free September and First Ice-Free Summer }
\begin{center}
\includegraphics[scale=0.131]{probdistslide10.jpg}
\label{densityS}
\end{center}
\begin{spacing}{1.0} \footnotesize{Notes: We display probability distributions from the simplified quadratic trend model for the date of the first effectively ice-free September (in red) and first ice-free summer (in black). The ``ice-free" threshold is defined as $SIE = 0$, 1, or 2 million km\textsuperscript{2}. The brown vertical line denotes the date that the mean CMIP5 climate model projection for RCP8.5 reaches $SIE \le $ 1 million km\textsuperscript{2}.}
\end{spacing}
\end{figure}
An event that has attracted much attention in the literature is the initial occurrence of an ice-free or nearly ice-free September. We calculate the probability for each September with date $t_0$ so that $SIE_{t_0}{\le} \gamma$ and $SIE_t {>} \gamma$ for all $t{<}t_0$. Specifically, for a given $\gamma$ and simulation $i$, we determine the year in which September $SIE_t$ first reaches $\gamma$, and then we cumulate across all simulations to build a distributional estimate. The red lines in Figure \ref{densityS} provide the resulting probability distributions for an initial ``ice-free" Arctic September for $\gamma{=}0$, $\gamma{=}1$, and $\gamma{=}2$, that is, for progressively more lenient definitions of ``ice-free." As noted above, the middle value of $\gamma$, which represents Arctic sea ice of less than $1$ million ${\rm km}^2$, is a popular benchmark in the literature \citep{WangOverland2012}. For that particular threshold, the statistical model produces a distribution centered at 2039.\footnote{Both the mean and median round to 2039 despite a slightly skewed distribution toward longer times.} Of particular interest is the probability distribution of dates taking into account model parameter uncertainty and stochastic shock uncertainty, and this distribution shows about a 60 percent probability of an effectively ice-free September Arctic occurring in the 2030s.
The distribution with $\gamma{=}1$ is bracketed on either side by distributions that use the higher and lower thresholds. The $\gamma{=}0$ distribution has a median date of 2044 and a 95 percent range from 2039 to 2053. The $\gamma{=}2$ distribution has a median date of 2033 and an earlier and slightly narrower 95 percent range from 20230 to 2039. The climate modeling literature has pointed to several factors that underpin the uncertainty in the timing of an initial September ice-free Arctic including natural climate variability, emissions path uncertainty, and uncertain sea ice dynamics \citep{SerrezeAndMeier2019}. These factors are at least partially accounted for in our analysis.
The multi-model mean CMIP5 climate projections described above are well outside the above-described distributional ranges. For a threshold of 1 million km\textsuperscript{2}, the mean projections reach a nearly ice-free Arctic in 2068 and 2089 under the RCP8.5 and RCP6.0 scenarios, respectively. The former date is denoted by a vertical brown bar in Figure \ref{densityS}. The range of dates across individual models is also extremely wide, stretching well over a century \citep{Jahn2016}. To narrow this range, researchers have omitted models with poor performance using a variety of sea ice metrics. With such model selection procedures, the range of dates for a first nearly ice-free September is narrowed greatly to a 20-year span that runs from the 2040s to 2060s (\citealp{Massonnet2012}; \citealp{ThackHall2019}).\footnote{Again, the caveats of \cite{Rosenblum2017} regarding such model selection are relevant.} Even such a carefully circumscribed span is roughly a decade later than the simplified quadratic trend model produces. Moreover, the climate model simulation exercises are not designed to yield formal error estimates or measures of uncertainty as the spread of climate model forecasts in an ensemble is insufficient to completely characterize forecast uncertainty.
Finally, we note the availability of density forecasts for a variety of richer joint scenarios of interest. As one example, \cite{Jahn2016} describe other definitions of ``ice-free" that involve, for example, 5-year running means. Alternatively, ``ice-free" may require no ice for several consecutive months, for example, to accommodate meaningful freight shipping, tourism, mining, and commercial fishing \citep{Aksenov2017}. For example, the strong Autumn demand for international freight shipping to satisfy year-end Western holiday consumer demand could make a multi-month ice-free Arctic shipping lane of interest. In this case, the probability distribution of the initial occurrence of an ``ice-free" summer -- a joint ice-free August, September, and October -- could be relevant. Figure \ref{densityS} shows this distribution in black assuming an ice-free threshold of $\gamma{=}1$. This density is notably shifted right -- and more right-skewed -- compared to the September scenarios.
\section{Concluding Remarks} \label{conclsec}
A rapidly warming Arctic is an ominous sign of the broader climate change caused by human activity, but declining Arctic sea ice also has an important influence on the pace and intensity of future climate change. Using statistical models, we have provided probabilistic projections of 21st-century Arctic sea ice that account for both intrinsic uncertainty and parameter estimation uncertainty. These projections indicate that summertime Arctic sea ice will quickly diminish and disappear -- with about a 60 percent probability of an effectively ice-free September Arctic occurring within two decades.
By contrast, the average projection from leading climate models implies an initial seasonal ice-free Arctic several decades later -- even assuming a business-as-usual emissions path. The slow and decreasing pace at which large-scale climate models reach an ice-free Arctic may be a serious shortcoming, and such conservative projections of sea ice loss could be a misleading guide for global climate policy.\footnote{Our statistical results are in line with the concerns of \cite{Stroeve2007} regarding the possibility that the slow projected decline in Arctic sea ice by climate models suggests they are underestimating the effects of greenhouse gases. More recently, \cite{Guarinoetal2020} also emphasize deficiencies in climate models' Arctic representations.}
There are numerous examples, across many disciplines, showing that parsimonious statistical representations can provide forecasts that are at least as accurate as the ones from detailed structural models.\footnote{A classic example from economics is \cite{Nelson1972}, who showed that simple statistical models forecast the economy as well as large-scale structural models based on economic theory.} However, rather than treat statistical models as just forecast competitors to structural climate models, there is likely to be scope to use them as complementary representations going forward. The mechanisms governing Arctic sea ice loss and connecting that loss to atmospheric, oceanic, and permafrost responses are not fully captured in climate models. Statistical models may be able to help assist in bridging such gaps until a more complete understanding is available \citep{Parkinson2010}.
In addition, statistical models may also be able to provide a benchmark for model performance that can be used for climate model evaluation, estimation/calibration, and selection. In particular, our statistical approach summarizes a key aspect of the data, namely its historical dynamics and the associated forecast path (call it $PATH_D$, where $D$ stands for ``Data"). A climate model with calibrated parameters $\theta$ produces a corresponding summary for the model (call it $PATH_M(\theta)$, where $M$ stands for ``Model"). So, at least conceptually, we can consider
\begin{enumerate}
\item \textit{evaluating} a climate model by examining a measure of divergence, $|| PATH_D {-} PATH_M(\theta) ||$
\item \textit{estimating} a climate model by finding $\hat{\theta} = argmin_{\theta} \, || PATH_D {-} PATH_M(\theta) ||$ (this is precisely indirect inference \citep{Gour1993}, using $PATH$ as the estimation window)
\item \textit{selecting} a climate model by finding the model with smallest $|| PATH_D {-} PATH_M(\hat{\theta}) ||$, adjusted for d.f.
\end{enumerate}
In practice such formal procedures are infeasible for massive global climate models, which attempt to address all aspects of the high-dimensional climate state, of which sea ice is but one small part. Hence, for example, the ``right" window for complete climate model estimation by indirect inference is much more complicated than just a sea ice path. Less formal procedures directly motivated by the above considerations can nevertheless be highly revealing, as in our key Figure 6.
\clearpage
|
1,314,259,996,696 | arxiv |
\section{Introduction}
Interactive robotic and autonomous applications require detecting objects as well as the actions being performed by them \cite{jiang2014modeling, rezazadegan2017action}. For example, an autonomous car not only needs to detect pedestrians but the action being performed by them to determine its own course of action. Action detection is an important component of the perception engine and it has received considerable progress in recent years due in part to deep-learning based architectures. The effective solutions repurpose deep learning based object detection architectures such as Faster RCNN and SSD and add temporal information to them \cite{peng2016multi, kalogeiton2017action, he2018generic, simonyan2014two, singh2017online, saha2017amtnet, singh2018tramnet, zhao2019dance}. Yet, despite such advances, spatio-temporal action detection in unconstrained environments remains a hard problem due to video-specific artifacts such as motion blurring and jitter. Particularly for moving platforms, a common cause for the above artifacts is due to relative motion between the actor and the camera. Previously, camera motion has only been explored in the context of the task of action classification \cite{wu2011action, jain2013better, wang2013action, avgerinakis2015moving} and not action detection. In this work, we take a step towards understanding the role of camera motion in action detection and computational steps to achieve robustness to it.
\begin{table*}[b]
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{|l|c|c|c|}
\hline
Method & Global Features & Feature Alignment & End Task\\
\hline
CoupleNet \cite{zhu2017couplenet} & Concat & - & Frame Object Detection\\
\hline
FGFA \cite{zhu2017flow} & - & Optical Flow Warping & Video Object Detection\\
\hline
STSN \cite{bertasius2018object} & - & Deformable Convolutions Single Scale & Video Object Detection\\
\hline
TramNET \cite{singh2018tramnet} & - & Proposal linking & Video Action Detection\\
\hline
RTPR \cite{li2018recurrent} & Concat & - & Video Action Detection\\
\hline
\textbf{This Work} & Weighted Avg. & Deformable Convolutions. Refinement using Pyramids & Video Action Detection\\
\hline
\end{tabular}}
\caption{Comparison with Prior Work}
\label{tab:algo-improvements}
\end{table*}
We focus on the task of spatio-temporal action detection in videos with high amounts of camera motion. To quantify the degree of camera motion we propose a novel ranking method that uses dense optical flow. Next, we observe that current action detection models fail in challenging videos with high amounts of camera motion. We hypothesize the reduction in accuracy is due to reduced amounts of overlap of features of the actor across frames and the absence of discriminative actor features. Inspired by camera motion compensated features for action classification \cite{wang2013action, jain2013better, avgerinakis2015moving}, we propose the addition of a fully learnable feature alignment module that predicts spatiotemporal offsets and uses deformable convolutions for compensation \cite{bertasius2018object}. The concept of deformable convolutions for feature alignment is motivated by the STSN which predicts offsets at a single scale but unlike the STSN we predict offsets at multiple scales and propose a novel offset refinement step by creating a pyramid of offsets. In addition to that, we propose the addition of contextual information \cite{zhu2017couplenet, li2018recurrent} but with a novel input-dependent weighted averaging strategy for fusion of local and global features.
We propose RADNet, a clip-based action detection architecture that takes in a stacked set of frames $f_t...f_{t+k}$ and generates feature maps at various scales (specifically 6 here) for all the frames within the clip. The feature maps at each scale and at each time step are aligned with respect to a reference frame within the clip using our multi-scale deformable convolution and refinement procedure. The aligned features are then fused with global scene features which are then used for prediction.
Once trained, our model can handle arbitrary camera motions, an idea that we verify by training on existing action detection datasets (UCF101-24 \cite{soomro2012ucf101}) and evaluating on videos with high camera motion. We create a custom dataset of real-world moving camera videos with spatio-temporal annotation of actions. These videos comprise our \textit{MOVE dataset}, which we will release. These videos contain a diverse array of actions in challenging scenarios with high degrees of camera motion. We verify the camera motion both qualitatively and quantitatively using our ranking method. Empirically, we observe that our model improves frame mAP and video mAP by 4.1\% and 17\% respectively on the MOVE dataset. The experiment validates that once trained on any action detection dataset, our model can be generalized to consider arbitrary camera motion.
In summary, our contributions are
\begin{itemize}
\itemsep0em
\item We construct a novel ranking method for quantifying degree of camera motion within a video and characterize the effect of camera motion on the accuracy of action detection.
\item We propose a feature alignment module that predicts offsets at multiple scales and refines them by creating a pyramid of offsets.
\item We propose a novel global feature fusion strategy that uses weighted averaging of local and global features.
\item We propose a combined model with feature alignment and global feature fusion that improves overall action detection accuracy.
\item We collect a dataset of challenging internet videos (MOVE dataset) with high degree of camera motions and annotate the actions within them and show the efficacy of our proposed network on it.
\end{itemize}
\section{Related Work} \label{sec:related}
\textbf{Action Detection} The effective approach in literature is to adapt object detection architectures such as RCNN and SSD and incorporate temporal information to the architecture \cite{gkioxari2015finding, weinzaepfel2015learning, peng2016multi, suman16bmvc, feichtenhofer2016convolutional, kalogeiton2017action, saha2017amtnet, singh2018tramnet, singh2018predicting, he2018generic, zhao2019dance}. The temporal information is added either by adding optical flow as an additional modality to create a two-stream framework \cite{gkioxari2015finding} or by processing a stack of frames \cite{kalogeiton2017action, saha2017amtnet, singh2018tramnet} or by using 3D CNNs as feature extractors \cite{hou2017end, gu2018ava, zhang2019structured}.
In fully supervised approaches, ROAD \cite{singh2017online} is a single frame approach using SSD networks trained with annotated actions. ACT \cite{kalogeiton2017action} and TRAMNet \cite{singh2018tramnet} are SSD-based architectures while AMTNet \cite{saha2017amtnet} and TPN \cite{he2018generic} are Faster R-CNN-based architectures that use multiple frames. In addition, multi-task formulations \cite{singh2018predicting, song2019tacnet}, recurrent classifiers \cite{li2018recurrent} and early/hybrid fusion of optical flow and RGB \cite{zhao2019dance, yang2019step} are used to improve the action detection performance. The tubelet or frame level detections from these models are linked together to form action tubes through various methods including EM/Viterbi \cite{suman16bmvc, singh2017online, kalogeiton2017action} tracking by detection \cite{weinzaepfel2015learning} or learning-based \cite{yang2019step}. This paper uses the ACT model \cite{kalogeiton2017action} for benchmarking on action detection in moving camera sequences as it is a simple and powerful baseline.
\textbf{Actions in Moving Camera Sequences} The role of camera motion has been studied for action classification \cite{avgerinakis2015moving, wu2011action, wang2013action}. These approaches separate out camera-induced and object-induced features such as superpixels \cite{avgerinakis2015moving}, dense trajectories
\cite{wang2013action} through homography \cite{avgerinakis2015moving, wang2013action} or low-rank optimization \cite{wu2011action}. Other approaches include using human detectors \cite{rezazadegan2017action} or a CRF \cite{tokmakov2017learning} to crop object boundaries and use the cropped features to perform classification. \textbf{However, Action detection in moving camera sequences has not received much attention in prior work}.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\linewidth]{fig/move_videos.png}
\caption{Some sample frames from the videos in the MOVE dataset. The videos contain diverse actions but with strong camera motion and jitter that leads to video artifacts such as motion blurring.}
\label{fig:move_videos}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\linewidth]{fig/stabilized_biking.png}
\caption{A video after image stabilization. The movement of the borders shows the level of compensation applied to stabilize the video}
\label{fig:move_video_stablized}
\end{figure*}
\textbf{Datasets} Action detection datasets contain both spatial and temporal labelling of actions. JHMDB-21 \cite{cheron2015p} contains 928 clips of 21 everyday actions such as \textit{catch, pick, walk, and wave}. UCFSports \cite{soomro2012ucf101} contains 150 videos of 10 sports-based actions such as \textit{diving, running, kicking}. UCF101-24 \cite{soomro2012ucf101} is a challenging dataset of 3207 untrimmed videos with 24 actions. The actions in UCF101-24 are also sports-centric i.e. \textit{diving, cricket, diving, biking}, etc. AVA \cite{gu2018ava} is a large-scale dataset of over 80 actions in 430 15-minute video clips. \textbf{The videos in these datasets are either captured from a static viewpoint or have very smooth camera motions}.
\vspace{-0.5em}
\subsection{Our contributions in Context of Prior Work}
A summary of the qualitative comparisons with prior work is given in Table \ref{tab:algo-improvements}.
\textbf{Feature Alignment in Video Data:} FGFA \cite{zhu2017flow} aligns features from multiple frames to a keyframe by using optical flow based warping. In STSN \cite{bertasius2018object} the optical flow warping step is replaced with deformable convolutions \cite{dai2017deformable}. DT\&T \cite{feichtenhofer2017detect} track the detected objects and use cross-correlated feature maps for detection. Non-local Neural Networks \cite{wang2018non} do non-local mean filtering over space and time for video action classification. Lei et al. use deformable temporal operations in which they predict temporal offsets for selecting feature maps for prediction \cite{lei2018temporal} but they do not predict spatial offsets.
Our automated feature alignment step is most similar to the STSN \cite{bertasius2018object} proposed for video object detection. The key difference is that we generate offsets at multiple scales. Additionally, we refine the offsets by upsampling the coarse offsets and adding to the finer-scale offsets similar to a pyramid. The pyramid approach is similar to FPN \cite{lin2017feature} but we create a pyramid of offsets and not features. An orthogonal approach is used by TramNET \cite{singh2018tramnet} that aligns the final predicted boxes. Unlike their approach we do the alignment step at the feature map level.
\textbf{Global Features:} Average fusion of scene and local features is used in CoupleNet \cite{zhu2017couplenet} and RTPR \cite{li2018recurrent} does concatenation of the local and global feature maps. We develop a weighted averaging scheme for fusion.
\section{Action Detection in Moving Camera Sequences}
We focus on the problem of action detection for moving camera sequences. First, we develop a camera motion detection model based on dense optical flow. The videos containing actions are then ranked with the camera motion model. A baseline action detector is used to benchmark the action detection metrics.
\subsection{Camera Motion Detection Model}
Our model relies on dense optical flow to model the global camera motion between frames (Figure \ref{fig:ranking-procedure}). The dense optical flow is computed using the Brox Method \cite{brox2004high}. The average magnitude of the optical flow is stored for each frame (since we do not care about the direction of the apparent motion). Next, regions containing human motions are masked out as they are not always in agreement with the camera motion and may even provide erroneous estimates \cite{wang2013action}. The masking is performed using the bounding box annotations. A moving difference of the average flow is taken and normalized by the length of the video to generate the ranking (Equations \ref{eq:flow} and \ref{eq:ranking}). Videos with a higher ranking are considered to have a higher degree of camera motion. We verify the ranking through visual inspection.
\vspace{-0.5em}
\begin{equation} \label{eq:flow}
\small flow(t) = \frac{1}{x*y}*\sum_{x} \sum_{y} ||I_{masked}(t,x,y)||
\end{equation}
\vspace{-0.5em}
\begin{equation} \label{eq:ranking}
\small rank = \frac{1}{nframes}*\sum_{i=1}^{nframes-1} |flow(i+1) - flow(i)|
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{fig/ranking_procedure.png}
\caption{Model for ranking videos by camera motion content}
\label{fig:ranking-procedure}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{fig/move_ranking.png}
\caption{Left: Histogram of videos ranked according to the degree of camera motion content. Right: Zoomed area shows count of videos with larger camera motion content and the black line shows frame AP at a threshold of 0.5 for the different clusters on the MOVE dataset with the baseline ACT ResNet50.}
\label{fig:move_rankings}
\vspace{-1em}
\end{figure}
We compared our ranking with a video stabilization algorithm that uses Shi-Tomasi Corner Features and Pyramid Lucas-Kanade flow \cite{vidstab2019}. The stabilization transforms are summed up and length normalized to generate the video ranking. Our ranking method gave a higher ranking to videos with jitter and irregular camera motions. The top ranking videos in the UCF-101-24 dataset with both rankings are shown in the supplementary material.
\begin{figure*}[t]
\centering
\includegraphics[width=0.95\linewidth]{fig/model_picture.png}
\caption{The RADNet action detection model tailored for detection in moving camera sequences. Feature alignment is done by predicting likely locations of the objects in different frames using an offset predictor and the global scene features are aggregated with the local actor features.}
\label{fig:full-model}
\end{figure*}
\subsection{MOVE: A Dataset of Moving Camera Sequences}
The camera motion ranking model shows that only a small subset of videos in the UCF101-24 dataset contains videos with high degree of camera motion. This motivates us to collect videos containing actions and with natural camera motions. We collected a dataset of moving camera videos from Youtube to empirically evaluate performance of action detection models on real-world sequences with natural camera motion. We chose videos containing activities that overlapped with the UCF-101-24 dataset and we collected 43 sequences with various actions and high degrees of camera motion. Some of these sequences are shown in Figure \ref{fig:move_videos}. A sample video is also shown with video stabilization applied to it in Figure \ref{fig:move_video_stablized} to show the degree of camera motion present. In these videos, actions were annotated spatially and temporally. Of the 24 actions present in UCF-24, we were able to collect videos containing 15 of those actions with clip lengths varying from 2 to 12 seconds. A comparison of the rankings for videos in UCF101-24 and the MOVE dataset in Figure \ref{fig:move_rankings} shows the stark difference in estimated motion for the videos in UCF101-24 and our dataset. Over 23 videos in our dataset rank higher than the highest ranked videos in UCF101-24. \textbf{Our objective is not to train on this data but to simply use for evaluation.}
\section{The RADNET Model}
The RADNet model for action detection builds upon the ACT \cite{kalogeiton2017action} framework for action detection. The network uses a stacked set of frames (clip/tubelet) to generate predictions. For each frame within the clip, feature maps (at multiple scales for one-stage detection) are generated by applying a convolutional backbone to the input frames and the generated feature maps (at the same scale) are combined together in the time dimension. Anchor cuboids are generated apriori for each location in the feature map by extending anchor boxes in the same spatial location over time. For each anchor cuboid, box regression and action classification is performed. The key difference is the addition of two layers. An automated feature alignment layer with multi-scale refinement spatially aligns features from multiple frames while a global feature fusion layer incorporates contextual information with local-actor specific features.
\subsection{Feature Alignment with Multi-Scale Refinement}
The feature alignment step aligns features spatially across the time dimension. Formally, given a stacked set of features $X_{stacked}$ corresponding to a stacked set of frames through timesteps $N-K$ to $N$, a prediction module $M$ makes class predictions and localizations for every point $(i,j)$ on that feature map (Equation \ref{eq: prediction}) for time step $N$.
\vspace{-0.5em}
\begin{equation}
\small
X_{stacked} = DepthConcat(X_{N-K}, X_{N-K+1}, ..., X_N)
\end{equation}
\vspace{-0.5em}
\begin{equation} \label{eq: prediction}
\small cls, loc (N,i,j) = M(X_{stacked})
\end{equation}
For a point on the feature map $(i,j)$ our objective is to transform $X_{N-K}$ through $X_N$ so that features relevant to $(i,j)$ are passed through (Equation \ref{eq: transform}). The transformation is achieved by tuning the receptive field of the filters at that location by adding an offset from the set of predicted offsets $\Delta P$.
\vspace{-0.5em}
\begin{equation} \label{eq: transform}
\small X(t,i,j) = \sum X(t, p + \Delta p(t,i,j) ), {t \in T}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{fig/coarse2finearch.png}
\caption{Refinement of predicted offsets at different scales}
\label{fig:coarse2finearch}
\end{figure}
The set of offsets $\Delta P$ is computed from the stacked set of feature maps $X_{stacked}$ for every point in the sampling grid $G$. Let $F$ be the function that predicts offsets for each time step for every spatial point in the feature map. We apply $F$ to get $\Delta P$ as shown in Equation \ref{eq:per-frame-offset}.
\begin{equation} \label{eq:per-frame-offset}
\small \Delta P = F(X_{stacked})
\end{equation}
$F$ can be any transformation such as optical flow between frames. For a fully end-to-end learning framework, $F$ needs to be differentiable. The STSN framework using deformable convolutions is adopted to model $F$. Briefly, a 2D convolution layer predicts offsets which are then used to offset the sampling locations for the convolution operator. However, convolution is a very local operation with limited receptive fields i.e. 3x3 and cannot model the global camera motion effectively. The solution is to create a pyramid representation of the offsets. The formulation for the pyramid is trivial \cite{bouguet2001pyramidal} and is adapted to RADNet as follows.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{fig/beforeafter_coarse2fine.png}
\caption{Before and after refinement of offsets at the finest scale}
\label{fig:coarse2fineresults}
\vspace{-1em}
\end{figure}
Let $O$ be the set of offsets with $o^{k}$ being the offsets computed at the $kth$ scale. In SSD, the feature maps at 6 scales are used for prediction thus there are 6 levels of offsets in $O$ with scales ranging from 38 x 38 to 1 x 1. The offsets at the coarsest scale (1 x 1) model the average camera motion component. They are then upsampled and added to the finer scales recursively. The up-sampled is performed using bilinear sampling. The block diagram for the refinement is shown in Figure \ref{fig:coarse2finearch}. A qualitative comparison before and after refinement is shown in Figure \ref{fig:coarse2fineresults}. The noisy local motion components are smoothed using the refinement procedure.
\subsection{Global and Local Feature Fusion}
To incorporate contextual information, we extract features corresponding to the whole scene and fuse them with the local actor specific features $f_{local}$. The global features $f_{global}$ are generated by average pooling the feature map at that scale. The global feature map is then aggregated with the local feature maps for prediction. Multiple fusion strategies for aggregation are implemented including concatenation, averaging and weighted averaging as shown in Figure \ref{fig:arch-globalfusion}.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{fig/fusion_aggregation.png}
\caption{Aggregation strategies for local features and global features}
\label{fig:arch-globalfusion}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{fig/results_move.png}
\caption{Results on the MOVE dataset after post-processing of detections. Only the action labelled for the video is shown for visual clarity. Top Row: With the ACT detector. Bottom: Our model.
In the volleyball video, the ACT detector does not detect the correct actor and has many false positives}
\label{fig:results-move}
\end{figure*}
\textbf{1. Concatenation} In this scheme, the local and global feature maps are concatenated along the channel dimension. The width of the next layer needs to be doubled to accommodate this.
\textbf{2. Average/Summation} The local and global feature maps are averaged or summed up along the channel dimension. The width of the next layer does not change in this scheme.
\textbf{3. Weighted Averaging} A novel weighted averaging scheme is implemented that dynamically weighs the global and local features for every input sample and for every input spatial location. The weighted averaging is implemented by using an embedding function to generate weights for the actor features and global features. Formally let the embedding function be $f_{embed}$ and let the actor features weight be $l^i$ and the global feature be $g^i$ for the $ith$ input sample. The embeddings for both are generated by applying $f_{embed}$ as given in Equation \ref{eq:embed1}.
\begin{equation} \label{eq:embed1}
\small l_{embed}^i, g_{embed}^i = f_{embed}(l^i), f_{embed}(g^i)
\end{equation}
The normalized weight embeddings $w_{normalized}^i$ are then generated by applying the softmax function over concatenation of $l_{embed}$ and $g_{embed}$ (Equation \ref{eq:embed2}).
\begin{equation} \label{eq:embed2}
\small w_{normalized}^i = softmax(concat(l_{embed}^i, g_{embed}^i))
\end{equation}
The $w_{normalized}$ is then used to do the weighted summation of the global and local features as given in Equation \ref{eq:embed4} to get the output feature map $p$ for prediction.
\begin{equation} \label{eq:embed4}
\small p^i = l^i * w^i_{normalized}[0] + g^i * w^i_{normalized}[1]
\end{equation}
We use a 3 layer DNN \cite{zhu2017flow} EmbedNet to implement the $f_{embed}$ function. The EmbedNet takes in the local features and the global features of dimension $(B,C,H,W)$. The output of EmbedNet is a one-channel feature map $(B,1,H,W)$. The local and global embeddings are then concatenated and the softmax is applied to get a normalized weighted feature map. The global and local weights are then multiplied with the global and local features respectively.
\section{Experimental Results and Analysis} \label{sec:experiments}
We evaluate the different configurations of RADNet on the MOVE Dataset. In addition, we also benchmark RADNet on existing action detection datasets JHMDB-21 and UCF101-24. For accuracy benchmarking, frame AP and video AP metrics are used. Frame AP evaluates correctness of detections only at the frame level while video AP checks for temporal consistency of detections for the duration of the video.
\textbf{Implementation Details :} We use the ResNet-50 backbone in all of our models. 3 extra residual blocks are added and feature maps from res3, res4, res5 and the extra layers are used. All images are resized to 300 x 300 for training and testing. The model is trained with a step learning schedule and a starting learning rate of 1e-3 for a total of 130000 iterations and a batch size of 8. The feature alignment layers are added to the output of all scales.
\begin{table}[!t
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{|c|l||c|c||c|c|}
\hline
\multirow{2}{*}{detector} & \multirow{2}{*}{method} & \multicolumn{2}{|c||}{JHMDB-21} & \multicolumn{2}{|c|}{UCF-101-24} \\
\cline{3-6}
& & 0.2 & 0.5:0.95 & 0.2 & 0.5:0.95\\ \hline \hline
\multirow{3}{*}{SSD}
& ROAD ~\cite{singh2017online} & 73.8 & 41.6 & 73.5 & 20.4 \\
& ACT~\cite{kalogeiton2017action} & 74.2 & 44.8 &
76.5 & 23.4 \\
& TramNet $\Delta$1~\cite{singh2018tramnet} & - & - & 79.0 & 23.9 \\
& AMTNet-L~\cite{singh2018tramnet} & - & - & \textbf{79.4} & 23.4 \\
& TACNET \cite{song2019tacnet} & 74.1 & 44.8 & 77.5 & 24.1 \\
& \textbf{ours} & \textbf{75.1} & \textbf{48.3} & 78.4 & \textbf{24.5} \\
\hline
\end{tabular}}
\caption{Comparison Results on video mAP of our method to the state of the art at various detection thresholds.
}
\label{table:sotavideoAP}
\vspace{-4mm}
\end{table}
\begin{table}[tbh]
\centering
\resizebox{0.95\linewidth}{!}{
\begin{tabular}{l|c|c|c|c}
\multirow{2}{*}{Network} & frameAP & \multicolumn{3}{c}{Video mAP} \\
\cline{2-5}
& 0.5 & 0.1 & 0.2 & 0.3 \\
\hline
ACT ResNet-50 & 27.7 & 57.8 & 52.3 & 48.7\\% & 35.5 & 17.5 \\
\hline
\multicolumn{5}{c}{\textbf{Feature Alignment}} \\
\hline
+STSN\cite{bertasius2018object} & 27.1 & 55.9 & 50.5 & 43.9\\% & 93.1 & 39.1 \\
+STSN+refinement & 31.1 & 67.2 & 63.9 & 57.7\\% & 93.1 & 39.1\\
\end{tabular}}
\caption{Comparison on the MOVE dataset with and without feature alignment \label{table:move-align}}
\end{table}
\begin{table}[tbh]
\centering
\resizebox{0.95\linewidth}{!}{
\begin{tabular}{l|c|c|c|c}
\multirow{2}{*}{Network} & frameAP & \multicolumn{3}{c}{Video mAP} \\
\cline{2-5}
& 0.5 & 0.1 & 0.2 & 0.3 \\
\hline
ACT ResNet-50 & 27.7 & 57.8 & 52.3 & 48.7\\% & 35.5 & 17.5 \\
\hline
\multicolumn{5}{c}{\textbf{Global Feature Fusion}}\\
\hline
Concat & 29.0 & 54.6 & 48.6 & 46.7\\
Avg & 29.4 & 59.9 & 56.4 & 45.0\\
Weighted Avg & 29.2 & 66.1 & 60.5 & 51.9\\
\end{tabular}}
\caption{Comparison on the MOVE dataset with and without global feature fusion \label{table:move-global}}
\end{table}
\begin{table}[tbh]
\centering
\resizebox{0.95\linewidth}{!}{
\begin{tabular}{ccc|c|c|c}
STSN\cite{bertasius2018object} & global & refinement & frame AP & Params & FLOPS \\
\hline
& & & 27.7 & 35.5 & 17.5\\
\checkmark & & & 27.1 & 93.1 & 39.1\\
\checkmark & & \checkmark & 31.1 & 93.1 & 39.1\\
& \checkmark & & 29.2 & 63.6 & 28.9\\
\checkmark & \checkmark & & 31.6 & 121.4 & 49.6\\
\checkmark & \checkmark & \checkmark & \textbf{31.8} & 121.4 & 49.6\\
\hline
\end{tabular}}
\caption{Ablation study on the MOVE Dataset \label{table:move-ablation}}
\end{table}
\begin{table}[tbh]
\centering
\resizebox{0.95\linewidth}{!}{
\begin{tabular}{l|c|c|c|c}
\multirow{2}{*}{Network} & frameAP & \multicolumn{3}{c}{Video mAP} \\
\cline{2-5}
& 0.5 & 0.1 & 0.2 & 0.3 \\
\hline
ACT ResNet-50 & 27.7 & 57.8 & 52.3 & 48.7\\% & 35.5 & 17.5 \\
RADNet & 31.8 & 74.9 & 70.8 & 65.7\\
\end{tabular}}
\caption{Comparison on the MOVE Dataset with RADNeT and prior work \label{table:move-prior}}
\end{table}
\subsection{Evaluation on Existing Datasets}
Evaluation of video mAP on the UCF101-24 and JHMDB-21 datasets (Table \ref{table:sotavideoAP}) is performed. For direct comparison, both RGB and optical flow modalities are used and we use the two-stream network formulation \cite{gkioxari2015finding}. Similar to prior work, we average the results for all splits for JHMDB and for UCF101 we report the results for the first split only. For the JHMDB dataset, we improve the video mAP at all thresholds. For UCF101-24, we improve the video mAP at at 0.5:0.95 by 0.4\%. The small increase in video mAP is attributed to the fact that the datasets contain few moving camera videos so our network does not improve the performance by much.
\subsection{Evaluation on MOVE Dataset}
Evaluation on the MOVE dataset is only performed on the RGB images and with the RGB-trained network.
\textbf{Feature Alignemnt} The effect of feature alignment is discussed here (Table \ref{table:move-align}). In the first experiment the STSN module without refinement is added to the network. The accuracy metrics are decreased. We attribute this to the inaccurate offsets being computed. With the addition of the refinement the frame AP is boosted by 3.4\% and the video AP is also improved by 9.0\% at a threshold of 0.3 This validates quantitatively the advantage of the refinement step. It also comes at a minimal cost with no addition in the parameters of the model.
\textbf{Global Fusion Strategies:} Next, the different global fusion strategies are evaluated in the network (Table \ref{table:move-global}). The weighted averaging shows a clear advantage over the concatenation and simple averaging with increase in video mAP (66.1\% vs. 57.8\%) at a threshold of 0.1 and 51.9\% vs 48.7\% at a threshold of 0.3.
\textbf{Ablation Studies:} Ablation studies were performed on the MOVE dataset with different configurations of the model (Table \ref{table:move-ablation}). It is interesting to note that the combined STSN and global feature fusion model improve the frame AP more than their individual contributions. Thus, their effects are complementary. The weighted averaging scheme is used for fusion in all the configurations. The highest frame AP and video AP are observed with the offset refinement and weighted averaging global feature fusion (31.8\%). For RADNet we use this configuration to compare with ACT and improve frame AP by 4.1\% and video AP by 17\% at a threshold of 0.3 (Table \ref{table:move-prior}).
\section{Conclusion}
In this paper, we have presented the problem of spatio-temporal action detection in moving camera sequences. We have collected and presented a test dataset containing moving camera sequences with annotated actions. We have shown a novel action detection architecture that is robust to the effect of camera motion in action detection. For future work, we intend to collect more videos for testing and possibly for training as well. In addition we will investigate the two-stream model for fusing optical flow input in the moving camera context. We hope that this will encourage further research into this domain.
\section*{Acknowledgements}
The research reported here was supported in part by the Defense
Advanced Research Projects Agency (DARPA) under
contract number HR0011-17-2-0045. The views and conclusions
contained herein are those of the authors and should not
be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA.
|
1,314,259,996,697 | arxiv | \section{Introduction}
Efficient estimation of rare event probabilities finds various applications in the performance evaluation/prediction of wireless communication systems operating over fading channels \cite{alouini}. In particular, the left-tail of the cumulative distribution function (CDF) of sums of nonnegative independent and identically distributed (i.i.d.) random variables is an example of a rare event probability that is of practical importance. More specifically, the outage probability at the output of equal gain combining (EGC) and maximum ratio combining (MRC) receivers can be expressed as the CDF of the sum of fading channel envelops (for EGC) and fading channel gains (for MRC) \cite{7328688}.
The accurate estimation of the left-tail of the CDF of sums of random variables requires the use of variance reduction techniques because the naive Monte Carlo sampler is computationally expensive \cite{opac-b1132466,rubino2009rare,opac-b1123521}. Moreover, the existing closed-form approximations \cite{8737752,8744610,4570452,1388730,4781943,4939219,4814351} fail to be accurate when the tail of the CDF is considered. The literature is rich in works in which variance reduction techniques were developed to efficiently estimate rare event probabilities corresponding to the left-tail of the CDF of sums of random variables, see \cite{asmussen2016exponential,7328688, 8472928,botev2019fast,gulisashvili2016,Nadhir_SLN,9014029} and the references therein. For instance, the authors in \cite{asmussen2016exponential} used exponential twisting, which is a popular importance sampling (IS) technique, to propose a logarithmically efficient estimator of the CDF of the sum of i.i.d. Log-normal random variables. The logarithmic efficiency is a popular property in rare event simulation used to ensure estimators' efficiency \cite{7328688}. Let $\hat{\alpha}$ be an unbiased estimator of $\alpha$, i.e., $\mathbb{E}[\hat{\alpha}]=\alpha$. We say that $\hat{\alpha}$ is logarithmically efficient if $\lim_{\alpha \rightarrow 0}\frac{\log(\mathbb{E}[\hat{\alpha}^2])}{\log(\alpha^2)}=1$. In \cite{gulisashvili2016}, the CDF of the sum of correlated Log-normal random variables was considered. The authors developed an IS estimator based on shifting the mean of the corresponding multivariate Gaussian distribution. Under mild assumptions, they proved that their proposed estimator is logarithmically efficient. Based on \cite{gulisashvili2016} and under the assumption that the left-tail sum distribution is determined by only one dominant component, the authors in \cite{Nadhir_SLN} combined IS with a control variate technique to construct an estimator with the asymptotically vanishing relative error property, which is the most desired property in the field of rare event simulations \cite{opac-b1132466}. In \cite{7328688}, two unified IS approaches were developed using the hazard rate twisting concept \cite{Juneja:2002:SHT:566392.566394,BenRached2016} to efficiently estimate the CDF of sums of independent random variables. The first estimator is shown to be logarithmically efficient, whereas the second achieves the bounded relative error property for i.i.d. sums of random variables and under the given assumption that was shown to hold for most of the practical distributions used to model the amplitude/power of fading channels. The bounded relative error is a stronger criterion than the logarithmic efficiency. We say that an unbiased estimator $\hat{\alpha}$ of $\alpha$ achieves the bounded relative error property if $\frac{\mathrm{var}[\hat{\alpha}]}{\alpha^2}$ is asymptotically bounded when $\alpha$ goes to $0$, see \cite{7328688}
The efficiency of the above mentioned estimators was studied when the number of summand $N$ was kept fixed. More specifically, recall that the objective is to efficiently estimate the probability that the sum of nonnegative i.i.d. random variables falls below a given threshold, i.e., $\mathbb{P}(\sum_{i=1}^{N}{X_i} \leq \gamma)$. A close look at the above mentioned estimators shows that the efficiency results were proved when the rarity parameter $\gamma$ decreases whereas $N$ is kept fixed. However, in most cases, the efficiency of the existing estimators is considerably affected when $N$ increases. This represents the main motivation of the present work. We aim to introduce a highly accurate estimator that efficiently estimate $\mathbb{P}(\sum_{i=1}^{N}{X_i} \leq \gamma)$ in the rare event regime when $N$ is large and/or $\gamma$ is small.
It is well-acknowledged that the exponential twisting technique compares favorably, in most cases, to existing estimators. It is the optimal IS probability density function (PDF) in the sense that it minimizes the Kullback-Leibler (KL) divergence with respect to the underlying PDF under certain constraints \cite{doi:10.1177/0037549707087713}. However, it has some limitations. First, it requires the knowledge of the moment generating function of $X_i$, $i=1,2,\cdots,N$. Second, sampling according to the new IS PDF is not straightforward and might be expensive. Moreover, the twisting parameter is not available in a closed-form expression and needs to be estimated numerically.
Motivated by the above limitations, we summarize the main contributions of the present work as follows:
\begin{itemize}
\item We propose an alternative IS estimator that approximately yields, for certain classes of distributions and in the rare event regime, at least the same efficiency as the one given by the estimator based on exponential twisting and at the same time does not introduce the above limitations.
\item The first class includes distributions whose PDFs vanish at zero polynomially. For this class of distributions, the Gamma IS PDF with appropriately chosen parameters retrieves approximately, in the regime of rare events corresponding to small values of $\gamma$ and/or large values of $N$, the same performances as the exponential twisting PDF.
\item The above result does not apply to the Log-normal setting as the corresponding PDF approaches zero faster than any polynomials. We show numerically that in this setting, the Gamma IS PDF with optimized parameters achieves a substantial amount of variance reduction compared to the one given by exponential twisting.
\item Numerical comparisons with some of the existing estimators validate that the proposed estimator can deliver highly accurate estimates with low computational cost in the rare event regime corresponding to large $N$ and/or small $\gamma.$
\end{itemize}
The paper is organized as follows. In section 2, we define the problem setting and motivate the work. In section 3, we introduce the exponential twisting approach and present its limitations. The main contribution of this work is presented in section 4, where we show that the Gamma IS PDF with optimized parameters retrieves approximately, for certain classes of distributions and in the rare event regime, at least the same performance as the exponential twisting technique. Finally, numerical experiments are shown in section 5 to compare the proposed estimator with various existing estimators.
\section{Problem Setting and Motivation}
Let $X_1,X_2,\cdots,X_N$ be i.i.d. nonnegative random variables with common PDF $f_X(\cdot)$ and CDF $F_{X}(\cdot)$. Let $\boldsymbol{x}=(x_1,\cdots,x_N)^{t}$ and $h_{\bold{X}}(\boldsymbol{x})=\prod_{i=1}^{N}{f_X(x_i)}$ be the joint PDF of the random vector $(X_1,\cdots,X_N)^{t}$. We consider the estimation of
\begin{align}\label{qoi}
\alpha(\gamma,N)=\mathbb{P}_{h_{\bold{X}}} \left (\sum_{i=1}^{N}{X_i} \leq \gamma \right ),
\end{align}
where $\mathbb{P}_{h_{\bold{X}}} (\cdot)$ is the probability under which the random vector $\bold{X}=(X_1,\cdots,X_N)^{t}$ is distributed according to $h_{\bold{X}}(\cdot)$, i.e., for any Borel measurable set $A$ in $\mathbb{R}^N$, we have $\mathbb{P}_{h_{\bold{X}}} \left ( \bold{X} \in A\right )=\int_{A} h_{\bold{X}}(\boldsymbol{x})d\boldsymbol{x}$.
As an application, the quantity of interest $\alpha(\gamma,N)$ could represent the outage probability at the output of EGC and MRC wireless receivers operating over fading channels. In fact, the instantaneous signal to noise ratio (SNR) at EGC or MRC diversity receivers is given as follows \cite{7328688}
\begin{align}\label{snr}
\gamma_{end}=\frac{E_s}{N_0\sqrt{N^{1-p+q}}} \left ( \sum_{i=1}^{N}{R_i^p}\right )^q,
\end{align}
where $N$ is the number of diversity branches, $\frac{E_s}{N_0}$ is the SNR per symbol at the transmitter, $R_i$, $i=1,2,...,N$, is the fading channel envelope and
\begin{align}
(p,q) = \begin{cases} (1,2) & \text{EGC}, \\ (2,1)& \text{MRC}.\end{cases}
\end{align}
The outage probability is defined as the probability that the SNR falls below a given threshold. Using (\ref{snr}), it can be easily shown that the outage probability at the output of EGC and MRC receivers can be expressed as the CDF of the sum of fading channel envelops (for EGC) and fading channel gains (for MRC), and hence can be expressed as in (\ref{qoi}).
We focus on the estimation of $\alpha(\gamma,N)$ when $N$ is large and/or $\gamma$ is small. Before delving into the core of the paper, we illustrate via a simple example that the efficiency of an IS estimator, that performs well when $\gamma$ decreases and $N$ is not sufficiently large, can deteriorate when we increase the values of $N$.
We first write the quantity of interest as
\begin{align}\label{transformation}
\nonumber \mathbb{P}_{h_{\bold{X}}}\left (\sum_{i=1}^{N}{X_i}\leq \gamma \right )&= \mathbb{P}_{h_{\bold{X}}}\left (\sum_{i=1}^{N}{X_i}\leq \gamma, X_i \leq \gamma \text{ }\forall i \right )\\
\nonumber &= \mathbb{P}_{h_{\boldsymbol{w}}} \left (\sum_{i=1}^{N}{w_i}\leq 1 \right ) \left (F_{X}(\gamma)\right )^N\\
&= \mathbb{E}_{h_{\boldsymbol{w}}} \left [ (F_X(\gamma))^N \textbf{1}_{(\sum_{i=1}^{N}{w_i} \leq 1)}\right ]=\mathbb{E}_{h_{\boldsymbol{w}}} \left[ \hat{\alpha}(\gamma,N)\right ],
\end{align}
where $w_i$ is equal in distribution to $\frac{X_i}{\gamma}$ conditional on the event $\{X_i \leq \gamma\}$, $i=1,2,.\cdots,N$, and $h_{\boldsymbol{w}}(\boldsymbol{w})=\prod_{i=1}^{N}{f_w(w_i)}$ with $f_{w}(\cdot)$ is the PDF of $w_i$, i.e., the conditional PDF of $\frac{X_i}{\gamma}$ given the event $\{ \frac{X_i}{\gamma} \leq 1\}$, and is given by $f_w(w)=\frac{\gamma f_X(\gamma w)}{F_X(\gamma)} \textbf{1}_{(w<\gamma)}$. Note that $\mathbb{E}_{h_{\boldsymbol{w}}}[\cdot]$ denotes the expectation under $h_{\boldsymbol{w}}(\cdot)$.
The estimator is then given by estimating the right-hand side term of (\ref{transformation}) by the naive Monte Carlo method
\begin{align*}
\hat{\alpha}_{M}(\gamma,N)=\frac{1}{M} \sum_{k=1}^{M}{(F_X(\gamma))^N \textbf{1}_{(\sum_{i=1}^{N}{w_i^{(k)}} \leq 1)}},
\end{align*}
where $(w_1^{(k)},\cdots,w_N^{(k)})$ , $k=1,\cdots,M$, are independent realizations sampled according to $h_{\boldsymbol{w}}(\cdot)$.
Note that this estimator can be understood as applying IS with IS PDF being the truncation of the underlying PDF over the hypercube $[0,\gamma]^N$. It can be easily proved that for fixed $N$, this estimator achieves the desired bounded relative error property with respect to the rarity parameter $\gamma$ for distributions that satisfy $f_{w}(x) \sim b x^{p}$ as $x$ approaches zero and for $p>-1$ and $b>0$, see \cite{8472928}. This property means that the squared coefficient of variation, defined as the ratio between the variance of an estimator and its squared mean, remains bounded as $\gamma \rightarrow 0$,
see \cite{opac-b1132466}. More precisely, when this property holds,
the number of required samples to meet a fixed accuracy requirement remains bounded independently of how small $\alpha(\gamma,N)$ is. The question now is what happens when $N$ is large.
Using the Chernoff bound, we obtain for all $\eta>0$
$$
\mathbb{P}_{h_{\boldsymbol{w}}}(\sum_{i=1}^{N}{w_i}\leq 1) \leq \exp \left ( \eta+N \log \left ( \mathbb{E}_{f_w}{}[\exp(-\eta w)]\right )\right ),
$$
where $\mathbb{E}_{f_w}[\cdot]$ denotes the expectation under $f_{w}(\cdot)$. The squared coefficient of variation of $\hat{\alpha}(\gamma,N)$ in (\ref{transformation}) is given by
\begin{align*}
\nonumber \text{SCV}(\hat{\alpha}(\gamma,N))=\frac{\mathrm{var}_{h_{\boldsymbol{w}}}[\hat{\alpha}(\gamma,N)]}{\alpha^2(\gamma,N)}&=\frac{\mathbb{P}_{h_{\boldsymbol{w}}}(\sum_{i=1}^{N}{w_i}\leq 1)(1-\mathbb{P}_{h_{\boldsymbol{w}}}(\sum_{i=1}^{N}{w_i}\leq 1))}{\left (\mathbb{P}_{h_{\boldsymbol{w}}}(\sum_{i=1}^{N}{w_i}\leq 1) \right )^2}\\
&=\frac{1-\mathbb{P}_{h_{\boldsymbol{w}}}(\sum_{i=1}^{N}{w_i}\leq 1)}{\mathbb{P}_{h_{\boldsymbol{w}}}(\sum_{i=1}^{N}{w_i}\leq 1)}
\end{align*}
In particular, when $\eta=1$, the squared coefficient of variation (which is asymptotically equal to $1/ \mathbb{P}_{h_{\boldsymbol{w}}}(\sum_{i=1}^{N}{w_i}\leq 1) $ in the regime of rare events) is lower bounded by $\exp \left ( -1-N \log \left ( \mathbb{E}_{f_w}{}[\exp(-w)]\right )\right )$.
This shows that the squared coefficient of variation increases at least exponentially, which proves that the efficiency of the estimator deteriorates when $N$ is large.
\section{Exponential Twisting}
In this section, we review the popular exponential twisting IS approach and enumerate its limitations in estimating the quantity of interest. When applicable, it is well-acknowledged that the exponential twisting technique is expected to produce a substantial amount of variance reduction and to compare favorably, in most cases, to other estimators \cite{asmussen2016exponential}. For distributions with light right tails and under the i.i.d. assumption, the estimator based on exponential twisting can be proved, under some regularity assumptions, to be logarithmically efficient when the probability of interest is either $\mathbb{P}_{h_{\bold{X}}}(\sum_{i=1}^{N}{X_i}>\gamma)$ and $\gamma \rightarrow +\infty$ or $\mathbb{P}_{h_{\bold{X}}}(\sum_{i=1}^{N}{X_i}>\gamma N)$ and $N \rightarrow +\infty$ \cite{opac-b1123521}. In the left tail setting, which is the region of interest in the present work, the exponential twisting was shown in \cite{asmussen2016exponential} to achieve the logarithmic efficiency property in the case of i.i.d. Log-normal random variables when the probability of interest is $\mathbb{P}_{h_{\bold{X}}}(\sum_{i=1}^{N}{X_i} <N\gamma)$ and either $N\rightarrow +\infty$ or $\gamma \rightarrow 0$.
In \cite{doi:10.1177/0037549707087713}, the exponential twisting technique was also shown to be optimal in the sense that it minimizes the KL divergence with respect to the underlying PDF under the constraint that the rare set $\{\boldsymbol{x} \in \mathbb{R}^{N}_{+}, \text{ such that } \sum_{i=1}^{N}{x_i} \leq \gamma \} $ is no longer rare. The IS PDF is selected to be the solution of the following optimization problem, see \cite{doi:10.1177/0037549707087713},
\begin{align}
\nonumber &\inf_{h^*_{\bold{X}} \geq 0} \int{h^*_{\bold{X}}(\boldsymbol{x})\log \left ( \frac{h^*_{\bold{X}}(\boldsymbol{x})}{h_{\bold{X}}(\boldsymbol{x})}\right )d\boldsymbol{x}}\\
&s.t \hspace{4mm} \int{h^*_{\bold{X}}(\boldsymbol{x})d\boldsymbol{x}}=1\\
\nonumber &\hspace{8mm} \mathbb{E}_{h^*_{\bold{X}}}\left [\sum_{i=1}^{N}{X_i} \right ]=\gamma\\
& \nonumber \hspace{8mm} h^*_{\bold{X}}(\boldsymbol{x}) \geq 0, \hspace{2mm} x_i \geq 0 \text{ for all } i \in {1,2,\cdots,N}.
\end{align}
The solution of this problem is given as (see \cite{doi:10.1177/0037549707087713} for a more general setting)
\begin{align}
h^*_{\bold{X}}(\boldsymbol{x})=\frac{h_{\bold{X}}(\boldsymbol{x}) \exp \left (\theta^* \sum_{i=1}^{N}{x_i}\right )}{\mathbb{E}_{h_{\bold{X}}} \left [\exp \left ( \theta^* \sum_{i=1}^{N}{X_i}\right )\right ]}, \text{ } \boldsymbol{x} \in \mathbb{R}^N_+
\end{align}
and $\theta^*$ solves
$$
\frac{\mathbb{E}_{h_{\bold{X}}}\left [\sum_{i=1}^{N}{X_i}\exp \left ( \theta^* \sum_{i=1}^{N}{X_i}\right )\right ]}{\mathbb{E}_{h_{\bold{X}}}\left [\exp \left ( \theta^* \sum_{i=1}^{N}{X_i}\right )\right ]}=\gamma.
$$
Hence, by writing $h_{\bold{X}}^*(\boldsymbol{x})=\prod_{i=1}^{N}{f_{\bold{X}}^*(x_i)}$, we clearly observe that the optimal density is given by exponentially twisting each univariate PDF $f_X(\cdot)$
$$
f_{X}^*(x)=\frac{f_X(x) \exp(\theta^* x)}{M(\theta^*)}, \hspace{2mm} x\geq 0,
$$
with $M(\theta)=E_{f_X}[\exp(\theta X)]$ and the optimal twisting parameter $\theta^*$ satisfies
$$
\frac{M^{'}(\theta^*)}{M(\theta^*)}=\frac{\gamma}{N}.
$$
Since the left-tail of sums of random variables is considered in this work, we have that $\theta^* \rightarrow -\infty$ as $\gamma \rightarrow 0$ and/or $N \rightarrow +\infty$ \cite{asmussen2016exponential}. Using the exponential twisting technique, the IS estimator of $\alpha(\gamma,N)$ using $M$ i.i.d. samples of $\bold{X}$ from $h_{\bold{X}}^*(\cdot)$ is given as follows
\begin{align*}
\hat{\alpha}_{\textrm{exp},M}(\gamma,N)= \frac{1}{M}\sum_{k=1}^{M}{\textbf{1}_{(\sum_{i=1}^{M}{X_i^{(k)}} \leq \gamma)} (M(\theta^*))^N \exp \left ( -\theta^* \sum_{i=1}^{N}{X_i^{(k)}}\right )}
\end{align*}
Observe, however, that the exponential twisting technique has some restrictive limitations. The main one is that sampling according to $f_X^*(\cdot)$ is not straightforward. One generally needs the use of an acceptance-rejection technique, the complexity of which can be dramatic when the probability of acceptance is relatively small. In such a case, the computational complexity of the algorithm can be huge and even worse than the naive Monte Carlo method. There are other less critical drawbacks. First, computations are much simpler if the moment generating function $M(\theta)$ is known in closed-form. Such a requirement does not hold in general. Also, the twisting parameter $\theta^*$ does not have, in general, a closed-form expression, and hence, it should be approximated numerically.
\section{Gamma Family as IS PDF}
The objective of this paper is to propose an alternative IS PDF that approximately yields, for certain classes of distributions that include most of the common distributions and in the rare event regime corresponding to large $N$ and/or small $\gamma$, at least the same performance as the exponential twisting technique and at the same time does not introduce serious limitations. We distinguish three scenarios depending on how the PDF $f_X(\cdot)$ approaches zero.
\subsection{$f_X(x) \sim b$ as $x$ goes to $0$ and $b>0$ is a constant}
Recall that the exponential twisting IS PDF satisfies
$$
f_X^*(x) \propto f_X(x) \exp(\theta^* x ), \hspace{2mm} x\geq 0,
$$
with $\theta^* \rightarrow -\infty$ as $\gamma \rightarrow 0$ and/or $N \rightarrow +\infty$. Therefore, as $f(x)\sim b$ and $b>0$, and by letting $\tilde{M}(\theta)=-\frac{1}{\theta}$, we instead consider the following IS PDF
$$
\tilde{f}_X(x)=\frac{\exp(\theta x)}{\tilde{M}(\theta)}, \hspace{2mm} x\geq 0.
$$
We choose $\theta$ to be equal to $\tilde{\theta }$ such that $\frac{\tilde{M}^{'}(\tilde{\theta})}{\tilde{M}(\tilde{\theta})}=\frac{\gamma}{N}$. Through simple computation, we obtain $\tilde{\theta}=-\frac{N}{\gamma}$. To conclude, when $f(x)\sim b$ and $b>0$, we propose an IS PDF given by the exponential distribution with rate $\frac{N}{\gamma}$.
\subsection{$f_X(x)=x^{p} g(x)$ with $g(x) \sim b$ as $x$ goes to $0$, $p>-1$, and $b>0$ is a constant}
Using the same methodology as in section 4.1, the IS PDF that we consider is
\begin{align}\label{gamma_is}
\tilde{f}_X(x)=\frac{x^{p}\exp(\theta x)}{\tilde{M}(\theta)}, \hspace{2mm} x\geq 0.
\end{align}
Therefore, the new PDF corresponds to the Gamma PDF with shape parameter $p+1$ and scale parameter $-1/\theta$. The normalizing constant is $\tilde{M}(\theta)=\frac{\Gamma(p+1)}{(-\theta)^{p+1}}$. Hence, the value $\theta$ is chosen to be equal to $\tilde{\theta}$ such that $\frac{\tilde{M}^{'}(\tilde{\theta})}{\tilde{M}(\tilde{\theta})}=\frac{\gamma}{N}$ and is given by
\begin{align}\label{tildetheta}
\tilde{\theta}=-\frac{N}{\gamma}(p+1).
\end{align}
Using the Gamma IS PDF in (\ref{gamma_is}), the proposed IS estimator of $\alpha(\gamma,N)$ using $M$ i.i.d. samples of $\bold{X}$ from $\tilde{h}_{\bold{X}}(\boldsymbol{x})=\prod_{i=1}^{N}{\tilde{f}_{X}(x_i)}$ is
\begin{align*}
\nonumber \hat{\alpha}_{is,M}(\gamma,N)&=\frac{1}{M}\sum_{k=1}^{M}{\textbf{1}_{(\sum_{i=1}^{N}{X_i^{(k)}} \leq \gamma)}\prod_{i=1}^{N}{\frac{f_{X}(X_i^{(k)})}{\tilde{f}_X(X_i^{(k)})}}}\\
&=\frac{1}{M}\sum_{k=1}^{M}{\textbf{1}_{(\sum_{i=1}^{N}{X_i^{(k)}} \leq \gamma)}(\tilde{M}(\tilde{\theta}))^N\prod_{i=1}^{N}{\frac{f_{X}(X_i^{(k)})\exp \left (-\tilde{\theta} X_{i}^{(k)}\right )}{(X_{i}^{(k)})^p }}}
\end{align*}
\begin{figure*}
\begin{minipage}[h]{0.9775\textwidth}
\small
\begin{center}
\begin{tabular}{|l|l|l|}
\multicolumn{3}{c}{Table I: Some PDF asymptotics around zero \footnote{ Functions $I_{\xi}(\cdot)$, and $K_{\xi}(\cdot)$ are respectively the modified Bessel functions of the first kind and order $\xi$ and the second kind and order $\xi$ \cite{gradshteyn2007}. }}\\
\multicolumn{3}{c}{}\\
\hline
\textbf{Distribution} & PDF & Proportional to \\
&& as $x \rightarrow 0$ \\
\hline
Exponential & $k \exp(-k x)$ & 1\\[1ex]
$k>0$ & & \\[1ex]
\hline
Gamma & $\frac{1}{\beta^k\Gamma(k)}x^{k-1}\exp(-\frac{x}{\beta})$ & $x^{k-1}$ \\[1ex]
$k,\beta>0$ & & \\[1ex]
\hline
Weibull & $\frac{k}{\lambda} (\frac{x}{\lambda})^{k-1} \exp(-(\frac{x}{\lambda})^k)$ & $x^{k-1}$ \\[1ex]
$k,\lambda>0$ & & \\[1ex]
\hline
Nakagami-m & $\frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1}\exp(-\frac{m}{\Omega}x^2)$ & $x^{2m-1}$ \\[1ex]
$m,\Omega>0$ & &\\[1ex]
\hline
Generalized Gamma & $\frac{p/a^d}{\Gamma(d/p)}x^{d-1}\exp (-(\frac{x}{a})^p)$ & $x^{d-1}$ \\[1ex]
$a,d,p>0$ & &\\[1ex]
\hline
Rice & $\frac{x}{\sigma^2} \exp(-\frac{x^2+\nu^2}{2\sigma^2})I_0((\frac{x\nu}{\sigma^2}))$ & $x$ \\[1ex]
$\sigma>0,\nu \geq 0>0$ & &\\[1ex]
\hline
Gamma-Gamma & $\frac{2(km)^{\frac{k+m}{2}}}{\Gamma(k)\Gamma(m)\Omega} (\frac{x}{\Omega})^{\frac{k+m}{2}-1} K_{k-m}\left ( 2\sqrt{\frac{km x}{\Omega}}\right )$ & $x^{k-1}$ \\[1ex]
$\Omega>0,m>k>0,m-k \notin \mathbb{N}$ & &\\[1ex]
\hline
$\kappa-\mu$ distribution & $\frac{2\mu(1+\kappa)^{\frac{\mu+1}{2}}x^{\mu}}{\Omega^{\frac{\mu+1}{2}}\kappa^{\frac{\mu-1}{2}}\exp(\mu \kappa)}\exp(-\frac{(1+\kappa)\mu x^2}{\Omega}) I_{\mu-1} \left ( 2\mu \sqrt{\frac{\kappa(\kappa+1)}{\Omega}}x\right )$ & $x^{2\mu-1}$ \\[1ex]
$\kappa,\mu>0$ & & \\[1ex]
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{figure*}
In Table I, we provide a non-exhaustive list of distributions that belong to section 4.2 (note that distributions in section 4.2 include those in section 4.1). These distributions are among the most used distributions to model the amplitudes and powers of wireless communications fading channels.
\begin{rem} \hspace{2mm}It is worth mentioning that for distributions satisfying $f_X(x)=x^{p} g(x)$ with $g(x) \sim b$ as $x$ goes to $0$, $p>-1$, and $b>0$ is a constant, the proposed approach with the Gamma IS PDF in (\ref{gamma_is}) with parameters $p$ and $\tilde{\theta}$ in (\ref{tildetheta}) achieves approximately, as $\gamma$ decreases to $0$ and/or $N$ increases, the same performance as the one given by the exponential twisting without introducing serious limitations.
Let $A_1$ and $A_2$ be the second moments of the proposed and the exponential twisting estimators, respectively. Then, the ratio between $A_1$ and $A_2$ has the following expression
\begin{align}\label{ratio}
\nonumber \frac{A_1}{A_2}&=\frac{\mathbb{E}_{\tilde{h}_{\boldsymbol{x}}} \left [\bold{1}_{(\sum_{i=1}^{N}{X_i} \leq \gamma)} \prod_{i=1}^{N}{\frac{f_X^2(X_i)}{\tilde{f}_{X}^{2}(X_i)}} \right ]}{\mathbb{E}_{h^*_{\boldsymbol{x}}} \left [\bold{1}_{(\sum_{i=1}^{N}{X_i} \leq \gamma)} \prod_{i=1}^{N}{\frac{f_X^2(X_i)}{(f_{X}^{*}(X_i))^2}}\right ]}\\
\nonumber &=\frac{(\tilde{M}(\tilde{\theta}))^{2N}\mathbb{E}_{\tilde{h}_{\boldsymbol{x}}} \left [\bold{1}_{(\sum_{i=1}^{N}{X_i} \leq \gamma)} \prod_{i=1}^{N}{g^2(X_i)} \exp \left (-2\tilde{\theta} \sum_{i=1}^{N}{X_i} \right )\right ]}{(M(\theta^*))^{2N}\mathbb{E}_{h^*_{\boldsymbol{x}}} \left [\bold{1}_{(\sum_{i=1}^{N}{X_i} \leq \gamma)} \exp \left (-2\theta^* \sum_{i=1}^{N}{X_i} \right )\right ]}\\
\nonumber &=\frac{(\tilde{M}(\tilde{\theta}))^N\mathbb{E}_{h_{\boldsymbol{x}}} \left [\bold{1}_{(\sum_{i=1}^{N}{X_i} \leq \gamma)} \prod_{i=1}^{N}{g(X_i)} \exp \left (-\tilde{\theta} \sum_{i=1}^{N}{X_i} \right )\right ]}{(M(\theta^*))^N\mathbb{E}_{h_{\boldsymbol{x}}} \left [\bold{1}_{(\sum_{i=1}^{N}{X_i} \leq \gamma)} \exp \left (-\theta^* \sum_{i=1}^{N}{X_i} \right )\right ]}\\
&= \frac{(\tilde{M}(\tilde{\theta}))^N \int_{(\sum_{i=1}^{N}{x_i} \leq \gamma)}{\prod_{i=1}^{N}{g(x_i)f_X(x_i)} \exp \left (-\tilde{\theta} \sum_{i=1}^{N}{x_i} \right )dx_1\cdots dx_N }}{(M(\theta^*))^N \int_{(\sum_{i=1}^{N}{x_i} \leq \gamma)}{\prod_{i=1}^{N}{f_X(x_i)} \exp \left (-\theta^* \sum_{i=1}^{N}{x_i} \right )dx_1\cdots dx_N }}.
\end{align}
First observe that $M(\theta)=\int_{0}^{\infty}{\exp(\theta x)x^p g(x)dx}$ is well-approximated by $b\tilde{M}(\theta)=b \int_{0}^{\infty}{\exp(\theta x)x^p dx}$ for sufficiently small negative values of $\theta$. Moreover, recall that $\theta^*$ and $\tilde{\theta}$ go to $-\infty$ as either $\gamma \rightarrow 0$ or $N \rightarrow \infty$,
and that $\theta^*$ and $\tilde{\theta}$ satisfy $\frac{M'(\theta^*)}{M(\theta^*)}=\frac{\gamma}{N}$ and $\frac{\tilde{M}'(\tilde{\theta})}{\tilde{M}(\tilde{\theta})}=\frac{\gamma}{N}$, respectively. Thus, as $\gamma \rightarrow 0$ and/or $N \rightarrow \infty$, we obtain that $\theta^*$ is well-approximated by $\tilde{\theta}$, and hence $M(\theta^*)$ is well-approximated by $b \tilde{M}(\tilde{\theta})$. Finally, using the latter two approximations and the fact that $g(x) \sim b$ as $x$ goes to $0$, we conclude from (\ref{ratio}) that $A_1$ is approximately equal to $A_2$ when $\gamma$ goes to $0$. For large values of $N$, the same conclusion can be deduced by observing that $\mathbb{E}_{f_{X}^*}[X_i]=\mathbb{E}_{\tilde{f}_X}[X_i]=\frac{\gamma}{N}$, $i=1,2,\cdots,N$. Thus, the random variables $X_1,X_2,\cdots,X_N$ take, when sampled according to the IS PDFs, sufficiently small values when $N$ is sufficiently large.
\end{rem}
\subsection{The Log-normal Case}
Distributions that do not approach 0 polynomially are much more difficult to handle and need to be tackled on a case-by-case basis. In this work, we consider the case of the sum of i.i.d. standard Log-normal random variables. The density decreases to $0$ at a faster rate than any polynomials and thus the Gamma distribution with fixed shape parameter will not recover the results given by the use of the exponential twisting technique. Note that in \cite{asmussen2016exponential}, the exponential twisting technique was applied to the sum of i.i.d. standard Log-normals by i) providing an unbiased estimator of the moment generating function, ii) approximating the value of $\theta$, and iii) using acceptance-rejection to sample from the IS PDF.
The main difficulty is that the PDF of the Log-normal distribution does not have a Taylor expansion at $x=0$. The first estimator we propose is based on truncating the support $[0,+\infty]$ and only working on $[a,+\infty]$ with $a=\delta\gamma/N$. This allows the use of a Taylor expansion at $x=a$. This procedure, however, introduces a bias that needs to be controlled. We show numerically that this estimator exhibits better performances than the one based on exponential twisting. Moreover, we observe that, in the regime of rare events, the proposed estimator achieves approximately the same performances as the Gamma IS PDF with shape parameter equal to $2$. This is the main motivation behind introducing a second estimator whose IS PDF is a Gamma PDF with optimized parameters. The numerical results show that the second estimator achieves substantial variance reduction with respect to the first estimator.
\subsubsection{Biased estimator}
We rewrite the quantity of interest as
\begin{align}\label{eq_ln}
\nonumber \mathbb{P}_{h_{\bold{X}}}\left ( \sum_{i=1}^{N}{X_i} \leq \gamma\right )&\approx \left (1-F_X(\frac{\delta \gamma}{N}) \right )^N\\
& \times \mathbb{P} _{h_{\bold{X}}}\left ( \sum_{i=1}^{N}{X_i}\leq \gamma \Big{|}X_i>\frac{\delta\gamma}{N}, \forall i\right ),
\end{align}
where $\delta$ is a fixed value belonging to $[0,1)$. The first factor on the right-hand side has a known closed-form expression. Let $\bar{f}_X(\cdot)$ be the PDF of $X_i |\{X_i >\frac{\delta \gamma}{N}\}$, $i=1,2,\cdots,N$, whose expression is given as follows:
$$
\bar{f}_X(x)=\frac{1}{x\sqrt{2 \pi}} \frac{\exp \left ( -\frac{(\log(x))^2}{2}\right )}{P(X_i>\frac{\delta \gamma}{N})}, \hspace{2mm} x\geq \frac{\delta \gamma}{N}.
$$
Next, we write the second factor on the right hand side of (\ref{eq_ln}) as follows:
$$
\mathbb{P}_{h_{\bold{X}}} \left ( \sum_{i=1}^{N}{X_i}\leq \gamma \Big{|}X_i>\frac{\delta\gamma}{N}, \forall i\right )= \mathbb{P}_{\bar{h}_{\bold{X}}} \left ( \sum_{i=1}^{N}{X_i} \leq \gamma\right ),
$$
with $\bar{h}_{\bold{X}}(\boldsymbol{x})=\prod_{i=1}^{N}{\bar{f}_{X}(x_i)}$. The exponential twisting IS PDF is then given by
$$
\bar{f}_X^*(x) \propto \bar{f}_X(x) \exp(\theta x), \text{ }x \geq \frac{\delta \gamma}{N}.
$$
Now, by using the Taylor expansion of $\bar{f}_{X}(\cdot)$ at the point $x=\delta\gamma/N$, we write
$$
\bar{f}_X(x)= \bar{f}_X(\frac{\delta \gamma}{N})+(x-\frac{\delta\gamma}{N})\bar{f}_X^{'}(\frac{\delta \gamma}{N})+\frac{(x-\frac{\delta \gamma}{N})^2}{2}\bar{f}_X^{''}(\xi_{x,\delta,N}),
$$
where $\xi_{x,\delta,N}$ is between $\frac{\delta \gamma}{N}$ and $x$.
Hence, the approximate exponential twisting IS PDF is given by
\begin{align}\label{change_LN}
\tilde{f}_X(x)=\frac{ \bar{f}_X\exp ( \theta x)+(x-\frac{\delta\gamma}{N})\bar{f}_X^{'} \exp(\theta x)}{\tilde M(\theta)},\hspace{2mm} x\geq \frac{\delta \gamma}{N},
\end{align}
with the notation $\bar f_X=\bar{f}_X(\frac{\delta \gamma}{N})$ and $\bar{f}_X^{'}=\bar{f}_X^{'}(\frac{\delta \gamma}{N})$. We assume that $\frac{\delta \gamma}{N}$ is strictly less than $\exp(-1)$ to ensure that $\bar{f}_X^{'}>0$. This assumption is not restrictive, as we are interested in the rare event regime corresponding to $N$ large and/or $\gamma$ small. Through a simple computation, we get
$$
\tilde{M}(\theta)=-\frac{\exp \left ( \theta \delta \gamma/N\right )}{\theta}\bar{f}_X+\frac{\exp \left ( \theta \delta \gamma/N\right )}{\theta^2}\bar{f}_X^{'}.
$$
The value of $\theta$ that solves $\frac{\tilde{M}^{'}(\theta)}{\tilde{M}(\theta)}=\frac{\gamma}{N}$ is given by
$$
\theta=-\frac{\bar f_X-c \bar{f}_X^{'}+\sqrt{(\bar f_X-c \bar{f}_X^{'})^2+8\bar{f}_X\bar{f}_X^{'}c}}{2c\bar f_X},
$$
with $c=\frac{\gamma}{N}(1-\delta)$. The remaining part is to sample from $\tilde{f}_X(\cdot)$. To do this, we write
$$
\tilde{f}_X(x)=-\frac{\bar{f}_X\exp(\theta \delta \gamma/N)}{\tilde{M}_X(\theta)\theta}\tilde{f}_1(x)+\frac{\bar{f}_X^{'}\exp(\theta \delta \gamma/N)}{\tilde{M}_X(\theta)\theta^2}\tilde{f}_2(x),
$$
where $\tilde{f}_1(x)=-\frac{\theta \exp(\theta x)}{\exp(\theta \delta \gamma/N)}$ and $\tilde{f}_2(x)=\frac{\theta^2(x-\delta\gamma/N)\exp(\theta x)}{\exp(\theta \delta \gamma/N)}$ are two valid PDFs for $x>\delta \gamma/N$.
The question that remains is related to controlling the bias through a proper choice of the parameter $\delta$. Let $\alpha_1(\gamma,N)=\left (1-F_X(\frac{\delta \gamma}{N}) \right )^N \mathbb{P}_{h_{\bold{X}}} \left ( \sum_{i=1}^{N}{X_i}\leq \gamma \Big{|}X_i>\frac{\delta\gamma}{N}, \forall i\right )$. Then, the global relative error can be upper bounded as follows:
\begin{align}\label{error_split}
\left |\frac{\alpha(\gamma,N)-\hat{\alpha}_{1,is,M}}{\alpha(\gamma,N)} \right | \leq \frac{\alpha(\gamma,N)-\alpha_1(\gamma,N)}{\alpha(\gamma,N)}+ \left |\frac{\alpha_1(\gamma,N)-\hat{\alpha}_{1,is,M}}{\alpha_1(\gamma,N)}\right |,
\end{align}
where $\hat{\alpha}_{1,is,M}$ is the IS estimator of $\alpha_1(\gamma,N)$ based on $M$ i.i.d. realizations sampled according to $\tilde{h}_{\bold{X}}(\boldsymbol{x})=\prod_{i=1}^{N}{\tilde{f}_X(x_i)}$ where the the PDF $\tilde{f}_X(\cdot)$ is given in (\ref{change_LN})
\begin{align*}
\hat{\alpha}_{1,is,M}(\gamma,N)=\frac{1}{M}\sum_{k=1}^{M}{\left (1-F_X(\frac{\delta \gamma}{N}) \right )^N\textbf{1}_{(\sum_{i=1}^{N}{X_i^{(k)}} \leq \gamma)}\prod_{i=1}^{N}{\frac{\bar{f}_{X}(X_i^{(k)})}{\tilde{f}_X(X_i^{(k)})}}}.
\end{align*}
The parameter $\delta$ is then chosen to control the bias term in (\ref{error_split}), that is the first term on the right-hand side of (\ref{error_split}). The second term on the right-hand side is the statistical relative error of estimating $\alpha_1(\gamma,N)$ by $\hat{\alpha}_{1,is,M}$. From the Central Limit Theorem (CLT), this error term is approximately proportional to the coefficient of variation of $\hat{\alpha}_{1,is,M}$.
To achieve a global relative error of order $\epsilon$, it is sufficient to bound the two error terms, i.e., the statistical relative error and the relative bias, by $\epsilon/2$. Hence, the value of $\delta$ is selected such that the following inequality holds
\begin{align}\label{bias}
0 \leq \frac{\alpha(\gamma,N)-\alpha_1(\gamma,N)}{\alpha(\gamma,N)}\leq \epsilon/2.
\end{align}
The following lemma provides the relation between $\delta$ and $\epsilon$ such that (\ref{bias}) is fulfilled.
\begin{lem}
\hspace{2mm}The following expression of $\delta(\epsilon,N,\gamma)$
\begin{align}
\delta(\epsilon,N,\gamma)=\frac{N}{\gamma} \exp \left ( \Phi^{-1} \left ( \frac{\epsilon}{2N} \frac{(\Phi(\log(\gamma/N)))^N}{(\Phi(\log(\gamma)))^{N-1}}\right )\right ),
\end{align}
where $\Phi(\cdot)$ is the CDF of the standard Normal distribution, ensures that (\ref{bias}) holds.
\end{lem}
\begin{proof}
We first write that
\begin{align}
\nonumber &\alpha(\gamma,N)-\alpha_1(\gamma,N)\\
\nonumber &=\mathbb{P}_{h_{\bold{X}}} \left ( \{\sum_{i=1}^{N}{X_i} \leq \gamma\} \cap \cup_{i=1}^{N} \{ X_i \leq \delta \gamma/N\}\right )\\
\nonumber & \leq \mathbb{P}_{h_{\bold{X}}}(\cup_{i=1}^{N} \{X_i \leq \delta \gamma/N\}\cap \cap_{i=1}^{N} \{ X_i \leq \gamma\})\\
\nonumber & \leq \sum_{i=1}^{N}\mathbb{P} _{h_{\bold{X}}}\left ( \{X_i \leq \delta \gamma/N\}\cap \cap_{j\neq i} \{ X_j \leq \gamma\}\right )\\
\nonumber &=N \mathbb{P} _{h_{\bold{X}}}\left (X_1 \leq \delta \gamma/N,X_2 \leq \gamma,\cdots, X_N \leq \gamma\right )\\
&=N \Phi(\log(\delta \gamma/N)) \left ( \Phi(\log(\gamma))\right )^{N-1}.
\end{align}
On the other hand, we have
$$
\alpha(\gamma,N) \geq \left ( \Phi(\log(\gamma/N))\right )^N.
$$
Therefore, we get
$$
\frac{\alpha(\gamma,N)-\alpha_1(\gamma,N)}{\alpha(\gamma,N)} \leq N\frac{\Phi \left (\log(\delta\gamma/N)\right ) \left ( \Phi(\log(\gamma))\right )^{N-1}}{\left (\Phi (\log(\gamma/N))\right )^{N}}.
$$
By equating the right-hand side of the above inequality with $\epsilon/2$, we obtain
$$
\delta(\epsilon,N,\gamma)=\frac{N}{\gamma} \exp \left ( \Phi^{-1} \left ( \frac{\epsilon}{2N} \frac{(\Phi(\log(\gamma/N)))^N}{(\Phi(\log(\gamma)))^{N-1}}\right )\right ),
$$
and hence the proof is concluded.
\end{proof}
\subsubsection{The Gamma family as an IS PDF}
When we consider a sufficiently small value of $\delta$ in the above analysis, we observe from the expression of the IS PDF in (\ref{change_LN}) that the proposed estimator with the IS PDF in (\ref{change_LN}) achieves approximately the same performance as the Gamma IS PDF with shape parameter equal to $2$. This suggests investigating whether the Gamma family can achieve further variance reduction with respect to the approach in the previous subsection. Note that the advantage of using the Gamma family as IS PDFs compared to the approach in the previous subsection is that the estimator is unbiased. Recall that the Gamma PDF is given by
\begin{align}
\tilde{f}_X(x)=\frac{x^{k-1}\exp(- x/\theta)}{\Gamma(k)\theta^k}, \text{ } x>0,
\end{align}
where $\theta>0$ and $k>0$ are the scale and shape parameters. The value of $\theta$ is chosen to be equal to $\theta=\frac{\gamma}{Nk}$ to ensure that the expected value of each of the $X_i$'s, $i=1,2,\cdots,N$, under the PDF $\tilde{f}_{X}(\cdot)$ is equal to $\frac{\gamma}{N}$. The likelihood ratio is then given by
\begin{align*}
\mathcal{L}(x_1,x_2,\cdots,x_N)=\frac{(\Gamma(k)\theta^k)^N\exp(\frac{\sum_{i=1}^{N}{x_i}}{\theta}-\frac{1}{2}\sum_{i=1}^{N}{(\log(x_i))^2})}{\prod_{i=1}^{N}{x_i^k}(\sqrt{2 \pi})^N}.
\end{align*}
The second moment of the IS estimator is bounded by
\begin{align*}
& \mathbb{E}_{\tilde{h}_{\bold{X}}} \left [\mathcal{L}^2(X_1,X_2,\cdots,X_N) \bold{1}_{(\sum_{i-1}^{N}{X_i} \leq \gamma)} \right ]\\
& \leq (\frac{\Gamma(k)\theta^k}{\sqrt{2\pi}})^{2N} \exp(\frac{2\gamma}{\theta})\\
& \times \mathbb{E}_{\tilde{h}_{\bold{X}}} \left [ \exp(-\sum_{i=1}^{N}{(\log(x_i))^2}-2k \sum_{i=1}^{N}{\log(x_i)}) \right ]\\
& \leq (\frac{\Gamma(k)(\frac{\gamma}{Nk})^k}{\sqrt{2\pi}})^{2N} \exp(2kN+k^2N).
\end{align*}
The last upper bound is found by maximizing the function $x \rightarrow -(\log(x))^2-2k \log(x)$ for $x>0$.
Next, using Stirling's formula for the gamma function
$\Gamma(k)= \sqrt{2\pi}k^{k-\frac{1}{2}} \exp(-k)(1+\mathcal{O}(\frac{1}{k}))$, we get
\begin{align*}
&\mathbb{E}_{\tilde{h}_{\bold{X}}} \left [\mathcal{L}^2(X_1,X_2,\cdots,X_N) \bold{1}_{(\sum_{i-1}^{N}{X_i} \leq \gamma)} \right ]\\
& \lesssim C k^{-N} \left ( \frac{\gamma}{N}\right )^{2Nk} \exp (k^2N)\\
&=C\exp(N(k^2-2k\log(N/\gamma)-\log(k)))
\end{align*}
where $C$ is a constant. Next, the value of $k$ is chosen such that it minimizes the above right-hand side term. The solution of this minimization problem is given as follows:
\begin{align}\label{optimal_k_value}
k^*=\frac{1}{2} \left (\log(\frac{N}{\gamma})+\sqrt{(\log(\frac{N}{\gamma}))^2+2} \right ).
\end{align}
Note that when $N$ is large and/or $\gamma$ is small, the value of $k^*$ satisfies $k^* \sim \log(\frac{N}{\gamma})$.
\section{Numerical results}
In this section, we show some selected numerical results to compare the performance of the proposed estimators compared to some of the existing estimators. We consider three scenarios depending on the distribution of $X_i$, $i=1,2,\cdots,N$: the Weibull, the Gamma-Gamma, and the Log-normal distributions. Note that the proposed approach is not restricted to these three distributions (see Table I for a non-exhaustive list of distributions that can be handled).
We recall that the squared coefficient of variation of an unbiased estimator $\hat{\alpha}(\gamma,N)$ of $\alpha(\gamma,N)$ has the following expression
\begin{align}
\text{SCV}(\hat{\alpha}(\gamma,N))=\frac{\mathrm{var} \left [ \hat{\alpha}(\gamma,N)\right ]}{\alpha^2(\gamma,N)}.
\end{align}
Note that, from the CLT, the number of required samples to meet $\epsilon$ statistical relative error with $95\%$ confidence is equal to $(1.96)^2 \text{SCV}(\hat{\alpha}(\gamma,N))/\epsilon^2$. Therefore, when we compare two estimators, the one with the smaller squared coefficient of variation exhibits better performance than the other.
\subsection{Weibull Case}
In this section, we assume that $X_i$, $i=1,2,\cdots,N$, are distributed according to the Weibull distribution whose PDF is given in Table I.
The comparison is made with respect to the second IS approach of \cite{7328688} that is based on using the hazard rate twisting (HRT). In Figure \ref{fig_weibull_1} and Figure \ref{fig_weibull_2}, we plot the squared coefficient of variations given by the HRT technique and the proposed approach for two different values of the shape parameter: $k=1.5$ and $k=0.5$, respectively.
The value of $\alpha(\gamma,N)$ ranges approximately from $10^{-20}$ to $10^{-6}$ (respectively from $10^{-16}$ to $10^{-6}$) using the system's parameters of Figure \ref{fig_weibull_1} (respectively of Figure \ref{fig_weibull_2}).
These figures show that the proposed approach clearly outperforms the one based on HRT.
For instance, when $k=1.5$, $\lambda=1$, $\gamma=0.5$, and $N=12$, the proposed approach is approximately $270$ times more efficient than the one based on HRT. More specifically, to meet the same accuracy, the number of samples needed by the approach based on HRT should be approximately $270$ times the number of samples needed by the proposed approach.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.45]{Coeff_variation_weibull_first_scenario_1}
\caption{Squared coefficient of variation as a function of $N$ where $X_i$ are i.i.d. Weibull random variables with rate $\lambda=1$, $k=1.5$, and $\gamma=0.5$.}
\label{fig_weibull_1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.45]{Coeff_variation_weibull_second_scenario_2}
\caption{Squared coefficient of variation as a function of $N$ where $X_i$ are i.i.d. Weibull random variables with rate $\lambda=1$, $k=0.5$, and $\gamma=0.01$.}
\label{fig_weibull_2}
\end{figure}
In the next experiment, we aim to compare the proposed approach with the HRT one when $N$ is fixed and $\gamma$ decreases.
In Figure \ref{fig_weibull_4}, we compare the efficiency of both approaches in terms of squared coefficient of variations plotted as a function of $\gamma$ for two scenarios depending on the value of $N$ ($N=8$ and $N=10$). In this case, the value of $\alpha(\gamma,N)$ ranges approximately from $10^{-16}$ to $10^{-6}$ for $N=8$ and from $10^{-22}$ to $10^{-8}$ for $N=10$.
We observe a clear outperformance of the proposed approach compared to the one based on using HRT for both values of $N$. While the HRT approach was proved in \cite{7328688} to achieve the bounded relative error property with respect to $\gamma$ and for a fixed value of $N$, it is clear from Figure \ref{fig_weibull_4} that the asymptotic bound increases substantially with respect to $N$, and hence the performance of the HRT approach is dramatically affected by increasing $N$. On the other hand, we observe that increasing the value of $N$ has a minor effect on the efficiency of the proposed approach, i.e., the squared coefficient of variation is approximately unchanged for both values of $N$ and for the considered range of $\gamma$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.50]{coeff_variation_vs_gamma_1}
\caption{Squared coefficient of variation as a function of $\gamma$ where $X_i$ are i.i.d. Weibull random variables with rate $\lambda=1$, $k=1.5$.}
\label{fig_weibull_4}
\end{figure}
This numerical observation suggests to conclude that the proposed approach satisfies the bounded relative error property with an asymptotic bound that increases with a very slow rate, compared to the one given by the HRT approach, as we increase $N$. For illustration, the proposed approach is approximately $18$ (respectively $64$) times more efficient than the HRT one when $N=8$ (respectively $N=10$) and $\gamma=0.2$. Note that the previous observations are valid independently of the value of $\alpha(\gamma,N)$ (see Figure \ref{fig_weibull_4}, where the squared coefficient of variation is approximately constant for a fixed value of $N$ and for the considered range of $\gamma$). This experiment and the numerical results in Figures \ref{fig_weibull_1} and \ref{fig_weibull_2} validate the ability of the proposed approach to deliver a very accurate and efficient estimate of $\alpha(\gamma,N)$ when $N$ increases and/or $\gamma$ decreases.
\subsection{Gamma-Gamma Case}
The Gamma-Gamma distribution is used for various challenging applications in wireless communications. For instance, it exhibited a good fit to experimental data and was used to model wireless radio-frequency channels \cite{EP} and to model atmospheric turbulences in free-space optical communication systems \cite{8238201}. The PDF of $X_i$ is given in Table I.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{Coeff_variation_GG_v1_1}
\caption{Squared coefficient of variation as a function of $N$ where $X_i$ are i.i.d. Gamma-Gamma random variables with $m=4$, $k=1.7$, $\Omega=1$, and $\gamma=0.5$.}
\label{fig_GG}
\end{figure}
In Figure~\ref{fig_GG}, we compare the proposed approach with the one in \cite{7835220} by plotting the corresponding squared coefficient of variations as a function of $N$ and for a fixed value of $\gamma$. Note that in \cite{7835220}, the proposed IS PDF is simply another Gamma-Gamma PDF with shifted mean. We call this method the IS-based mean-shifted approach. The range of the quantity of interest $\alpha(\gamma,N)$ is approximately from $10^{-18}$ to $10^{-5}$. We observe that the proposed estimator outperforms the one in \cite{7835220}. Also, we observe that the outperformance of the proposed estimator compared to the one based on mean shifting increases as we increase $N$. Moreover, we should note here that the cost per sample (in terms of CPU time) of the approach in \cite{7835220} is twice the cost of the proposed approach. This is because a Gamma-Gamma random variable is generated by the product of two independent Gamma random variables, see \cite{5425871}. For illustration, we observe from Figure~\ref{fig_GG} that when $N=12$, the proposed approach is approximately $2.5$ times (five times if we include the computing time in the comparison) more efficient than the one of \cite{7835220}.
\subsection{Log-normal Case}
The Log-normal distribution can be used to model several types of attenuation including shadowing \cite{580779}, and weak-to-moderate turbulence channels in free-space optical communications \cite{6966082}. The standard Log-normal PDF (the associated Gaussian random variable has zero mean and unit variance) is given by
$$
f_X(x)=\frac{1}{x\sqrt{2\pi}} \exp \left ( -\frac{(\log(x))^2}{2}\right ), \text{ }x>0.
$$
\begin{figure}[h!]
\centering
\includegraphics[scale=0.55]{coeff_variation_LN_res_3_1}
\caption{Squared coefficient of variation as a function of $N$ where $X_i$ are i.i.d. standard Log-normal random variables with $\gamma=0.5$, and $\epsilon=0.05$.}
\label{fig_ln_1}
\end{figure}
Figure \ref{fig_ln_1} shows the squared coefficient of variation given by the exponential twisting \cite{asmussen2016exponential}, and the two proposed approaches, i.e., the one based on the biased estimator and the other based on using the Gamma distribution as an IS PDF. The value of $\alpha(\gamma,N)$ ranges approximately from $10^{-20}$ to $10^{-2}$.
For the considered range of $N$, we observe that out of these three approaches, it is the one using the Gamma distribution as an IS PDF that outperforms the others. When $N=9$, it is approximately $30$ times more efficient than the one based on exponential twisting. In addition to the efficiency in terms of number of samples, it is worth recalling that the exponential twisting technique developed in \cite{asmussen2016exponential} is computationally expensive in terms of computing time compared to the proposed approaches. Moreover, Figure \ref{fig_ln_1} also shows that the approach based on the biased estimator achieves better performances than the one based on exponential twisting. It is important to mention here that, for the comparison to be fair, the required number of samples of the biased estimator should be multiplied by $4$. This follows from the error analysis in (\ref{error_split}), in which the statistical relative error should be bounded by $\epsilon/2$, where $\epsilon$ is the required global relative error.
In Figure \ref{fig_ln_2}, we plot the squared coefficient of variations given by the three approaches as a function of $\gamma$ and for two different values of $N$ ($N=8$ and $N=10$). The quantity of interest $\alpha(\gamma,N)$ ranges approximately from $10^{-15}$ to $10^{-6}$ for $N=8$ and from $10^{-21}$ to $10^{-9}$ for $N=10$.
We observe that the approach based on using the Gamma distribution as an IS PDF clearly asymptotically outperforms the two other approaches. For both values of $N$, the outperformance increases as we decrease $\gamma$. Moreover, the biased estimator exhibits better performances than the exponential twisting one for both values of $N$ and for the considered range of $\gamma $ values.
Furthermore, increasing $N$ has a considerable negative effect on the performances of the exponential twisting and the biased IS-based approaches. On the other hand, Figure \ref{fig_ln_2} shows that increasing $N$ does not largely effect the performance of the IS estimator based on the use of the Gamma distribution as an IS PDF. For illustration, the approach based on using the Gamma distribution as an IS PDF is approximately $15$ times (respectively $35$) more efficient that the exponential twisting one when $N=8$ (respectively $N=10$) and $\gamma=0.6$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{coeff_variation_LN_gamma_v1_1}
\caption{Squared coefficient of variation as a function of $\gamma$ where $X_i$ are i.i.d. standard Log-normal random variables with $\epsilon=0.05$.}
\label{fig_ln_2}
\end{figure}
It is important to mention that the outperformance of the estimator based on using the Gamma distribution as an IS PDF over the one based on using the biased estimator is expected. As it was mentioned above, the latter approach gives approximately the same performance as the Gamma distribution with shape parameter equal to $2$ while the former one uses the Gamma distribution as an IS PDF with an optimized shape parameter (the shape parameter was chosen to minimize an upper bound of the second moment of the proposed estimator, see the expression of $k^*$ in (\ref{optimal_k_value})).
All of the above comparisons have been carried out in terms of the number of sampled needed to meet a fixed accuracy requirement. In order to include the computing time in our comparison, we define the Work Normalized Relative Variance (WNRV) metric of an unbiased estimator $\hat{\alpha}(\gamma,N)$ of $\alpha(\gamma,N)$ as follows (see \cite{8472928}):
\begin{align}
\text{WNRV}(\hat{\alpha}(\gamma,N))=\frac{\text{SCV}(\hat{\alpha}(\gamma,N))}{M} \times \text{computing time in seconds}.
\end{align}
The computing time is the time in seconds needed to get an estimator of $\alpha(\gamma,N)$ using $M$ i.i.d. samples of $\hat{\alpha}(\gamma,N)$. When comparing two estimators, the one that exhibits less WNRV is more efficient than the other estimator. More precisely, an estimator is efficient in terms of WNRV than another estimator means that it achieves less relative error for a given computational budget, or equivalently it needs less computing time to achieve a fixed relative error. Using the same setting as in Figure \ref{fig_ln_2}, we plot in Figure \ref{fig_ln_3} the WNRV metric as a function of $\gamma$ for two scenarios depending on the value of $N$ ($N=8$ and $N=10$).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{wnrv_ln_1}
\caption{WNRV as a function of $\gamma$ where $X_i$ are i.i.d. standard Log-normal random variables with $\epsilon=0.05$.}
\label{fig_ln_3}
\end{figure}
We observe that as $\alpha(\gamma,N)$ is getting smaller, it is the approach based on using the Gamma PDF as an IS PDF that outperforms the two other approaches in terms of WNRV (the efficiency increases as the event becomes rarer). It is worth recalling that the WNRV of the approach based on biased PDF should be multiplied by $4$ in order for the analysis to be fair (this follows from the error analysis that was performed in section 4.3.1). Moreover, Figure \ref{fig_ln_3} shows that, in addition to reducing the variance, as shown in Figure~\ref{fig_ln_2}, the approach based on using the Gamma IS PDF also reduces the computing time compared to the one using the exponential twisting technique. To see that, for $N=10$ and $\gamma=0.6$, the approach based on using the Gamma IS PDF is approximately $35$ times (respectively $340$ times) more efficient than the one based on exponential twisting when using the squared coefficient of variation metric (respectively the WNRV metric). More specifically, the Gamma based IS approach approximately reduces the computing time by a factor of $10$ with respect to the exponential twisting approach.
\section{Conclusion}
We developed efficient importance sampling estimators to estimate the rare event probabilities corresponding to the left-tail of the cumulative distribution function of large sums of nonnegative independent and identically distributed random variables. The proposed estimators achieve asymptotically at least the same performance as the exponential twisting technique, in the regime of rare events and for certain classes of distributions that include most of the common distributions. The main conclusion is that the Gamma PDF with suitably chosen parameters achieves for most of the common distributions substantial variance reduction, and at the same time avoids the restrictive limitations of the exponential twisting technique. The numerical results validate the efficiency of the proposed approach in being able to accurately and efficiently estimate the quantity of interest in the rare event regime corresponding to large $N$ and/or small $\gamma$. One possible extension of the present work is to connect it to the works in \cite{BESKOS20171417,Jasra} by creating a sequence of approximate measures corresponding to increasing the values of $N$.
|
1,314,259,996,698 | arxiv | \section{Introduction}
One of the outstanding properties in the field of atomic physics is the ability to control interatomic interactions using magnetically tunable Feshbach resonances (FRs) \cite{Chin2010}. They allow to address key problems in several fields of physics. For example, in order to explore molecular physics, one can create deeply bound molecules via Feshbach association \cite{Regal2003,Herbig2003}, followed by stimulated Raman adiabatic passage \cite{Ni2008,Danzl2008,Lang2008}. Such molecules can be used for the study of molecular structure, ultracold chemistry, and precision tests of fundamental laws of nature \cite{Carr2009}. Another example for the use of FRs is the study of the BEC-BCS crossover regime \cite{Regal2004,Bartenstein2004,Zwierlein2004} and the transition from weak to strong interactions \cite{OHara2002,Donley2001} in atomic many body physics. The tunability of the two-body scattering length is applied for the creation of Efimov trimers \cite{Braaten2007} in order to investigate few-body physics.
For the study of the above mentioned phenomena, precise knowledge of the field-dependent scattering lengths is essential. This can be obtained via a straightforward numerical coupled channels calculation (CC), which often employs a large number of channels $N$. As the time for the matrix operation required to solve this problem is on the order of $N^{3}$ \cite{Croft2011}, such a calculation can be computationally expensive. However, sufficient insight can be gained by applying models that approximately describe the scattering properties, while reducing the computational effort enormously. Two such models have been proven as powerful alternatives.
One of these models is the asymptotic bound state model (ABM) \cite{Tiecke2010a,Wille2008}, which uses only the bound states close to the asymptote to describe observables like FRs and the scattering length, removing the computation of the spatial part of the Schr\"odinger equation and the continuum of scattering states. A second approach to calculate scattering observables is the multichannel quantum defect theory (MQDT) \cite{Greene1979,Greene1982}, which uses the separation of length and energy scales to facilitate the calculation.
Even though Feshbach resonances have been extensively reviewed in Ref.\cite{Chin2010}, the literature is currently lacking a detailed juxtaposition of the aforementioned models. The goal of the present paper is to fill this gap by comprehensively comparing the approaches of CC calculation, ABM, and MQDT and by providing quantitative results based on the example of the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ system.
The reason for choosing this specific atom combination is the special role it will exhibit for the investigation of the above mentioned applications of FRs. For example, with the largest permanent electric dipole moment among all alkali-atom combinations of 5.5 Debye \cite{Aymar2005,Deiglmayr2010}, LiCs molecules in their rovibrational ground state \cite{Deiglmayr2008} are a unique candidate for the study of dipolar quantum gases \cite{Pupillo2008a}. Additionally, the large mass ratio of $m_{Cs}/m_{Li}\approx 22$ results in a very favorable Efimov scaling factor of 4.88 \cite{DIncao2006}, thus enabling the observation of a series of Efimov resonances~\cite{Pires2014,Tung2014}. Moreover, the system is also an excellent candidate for the study of polaron physics \cite{Tempere2009,Cucchietti2006}, because one resonance overlaps with a zero crossing of the ${}^{133}\text{Cs }$ scattering length, which allows for a strong coupling of a ${}^{6}\text{Li }$ impurity to a noninteracting Cs BEC.
We have recently reported on the observation of 19 intraspecies ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ $s$- and $p$-wave FRs, which have been accurately assigned via a CC calculation \cite{Repp2013} with a root mean square (rms) deviation $\delta B^{rms}$ (for a definition see Eq.~\ref{eq:rms}) of 39 mG for the field positions of the observed resonances. An application of the crudest version of the ABM with six free fit parameters, similar to the one done in Ref.~\cite{Repp2013}, yields $\delta B^{rms}=877$~mG. However, leaving all six parameters as free parameters in the fit yields unphysical fit values because the parameters are significantly correlated. Therefore, we demonstrate how this fit can be improved
by minimizing the amount of free fit parameters and by including
magnetic dipole-dipole interaction, yielding a slightly increased $\delta B^{rms}=965$ mG but parameters that are physically consistent and are coming close to those derived in the CC analysis.
The ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ combination is a good system for the illustration of extensions to the ABM,
because its small reduced mass leads to a large spacing between vibrational states. Therefore, only the least bound states need to be included, which keeps the number of parameters low, and minimizes the computational effort. Other systems with higher reduced mass would require a larger number $n$ of bound states, which results in $2n+n^{2}$ fit parameters ($2n$ bound states in singlet and triplet potentials and $n^2$ respective overlap parameters). For example in Rb-Cs at least five vibrational levels have to be included. The required 35 parameter fit to the observed resonances is asking for an appropriate number of observations if no further theoretical input is available.
We additionally apply the dressed ABM, which includes the coupling of the bound molecular state to the scattering state of the incoming atoms \cite{Tiecke2010a}, to improve the agreement with experimental FR positions in the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$~system even further. The application of this model is not straightforward due to a subtlety in the ${}^{6}\text{Li} {}^{133}\text{Cs}$~triplet potential. A virtual state, which is close to the atomic threshold, is not resonant enough to dominate the scattering behavior in the open channel. Therefore, neither the limiting case where a bound state dominates \cite{Tiecke2010a}, nor the case where only the virtual state dictates the behavior \cite{Marcelis2004} is applicable. We will bridge this gap by demonstrating a phenomenological method that includes both effects, leading to a convincing description of the observed FRs with a rms deviation of 263 mG.
Unlike the ABM, the MQDT handles the spatial part of the scattering problem at large separation $R$ explicitly, and the formalism does not differentiate between dominating bound or virtual states. Thus, the latest version of the MQDT as described in Ref.~\cite{Ruzic2013} can be directly applied without extension, resulting in a rms deviation of 40 mG. Besides giving the results for the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$~case, we demonstrate how a frame transformation (FT) in a MQDT ansatz can be applied to a system where no accurate potentials and only experimental data for FR positions are available, in order to assign these resonances and predict other resonance positions. The rms deviation of the FT approximation for the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$~system becomes 48 mG.
This paper is organized as follows. In Sect.~\ref{sec:nutshell} we explain the basic approach and the underlying assumptions of the three models to the scattering problem. The results of CC calculation are given in Sect.~\ref{sec:channels}. Sect.~\ref{sec:ABM} demonstrates how the ABM can be stepwise extended to predict the position of the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ FRs more accurately. In Sect.~\ref{sec:MQDT} we discuss the results of the MQDT calculation and finally, in Sect.~\ref{sec:Conclusion} we provide the quantitative comparison of the models and summarize our results.
\section{Approaches to the scattering problem in a nutshell}
\label{sec:nutshell}
The scattering process of two colliding atoms can be described by the following Hamiltonian \cite{Stoof1988}:
\begin{equation}
\label{eq:Hamiltonian}
H=T+V+H_{\text{hf}}+H_{\text{Z}}+H_{\text{dd}},
\end{equation}
where $T=-\hbar^{2}\Delta^{2}/(2\mu)$ denotes the relative kinetic energy term, with reduced mass $\mu$, and $V$ denotes the potential energy curves. The hyperfine energy operator
\begin{equation}
\label{eq:Hyperfineterm}
H_{\text{hf}}=\sum_{\beta=A,B} \alpha_{\beta}(R) \vec{s}_{\beta}\cdot \vec{i}_{\beta}/\hbar^{2},
\end{equation}
contains the electronic and nuclear spin operators $ \vec{s}$ and $\vec{i}$, respectively, and the summation is performed over the two atoms A and B. In the limit of large separations, the functions $\alpha_{\beta}(R)$, which depend on the internuclear separation $R$, approach the atomic hyperfine constant $\mathrm{a}_\mathrm{hf}$. The Zeeman interaction is given by
\begin{equation}
\label{eq:ZeemanTerm}
H_{\text{Z}}=\sum_{\beta=A,B} (g_{s,\beta}s_{z,\beta}+g_{i,\beta}i_{z,\beta})\mu_{B}B/ \hbar,
\end{equation}
where $g_{s}$ ($g_{i}$) is the electron (nuclear) g-factor, with respect to the Bohr magneton $\mu_{B}$ (see Ref.~\cite{Arimondo1977}). $H_{\text{dd}}$ is the Hamiltonian describing direct magnetic spin-spin, as well as second-order spin-orbit interactions, which causes for example the observed splitting of $p$-wave resonances \cite{Repp2013}. It can be given in its effective form \cite{Strauss}:
\begin{equation}
\label{eq:Vdipole}
V_\mathrm{dip}(R)=\frac{2}{3} \lambda(R)(3S_Z^2-S^2),
\end{equation}
where $S_Z$ is the total electron spin $S$ projected onto the molecular axis. The function
\begin{equation}
\label{eq:lamda}
\lambda(R) = -\frac{3}{4}
\alpha^2\left(\frac{1}{R^3}+a_\mathrm{SO}
\exp{\left(-bR\right)}\right),
\end{equation}
is given in atomic units with $\alpha$ the universal fine structure constant. Because the parameters $b$ and $a_\mathrm{SO}$ for the assumed effective functional form of the second order spin-orbit interaction are not available in the literature, they become fitting parameters in the following discussion. For binary collisions of alkali atoms, the total spin $S=s_{A}+s_{B}$ can only be 0 or 1. Therefore, the interatomic interaction $V=P_{0}V_{0}+P_{1}V_{1}$ is projected onto the singlet ($V_{S=0}$) and triplet ($V_{S=1}$) components by the projection operators $P_{0}$ and $P_{1}$, respectively, and additionally contains a centrifugal term from the separation of $T$ in radial and angular motion. The manifold of different internal states connected to the Hamiltonian of Eq.~\ref{eq:Hamiltonian} defines a number of channels for a given space fixed projection M of the total angular momentum of the system. Unless otherwise stated, the coordinates connected to spin and angular momentum are characterized by use of an appropriate basis set like in Hund's coupling case (e) for an atom pair AB:
\begin{equation}
\label{channelsdefinition}
|\chi \rangle \equiv|(s_A,i_A)f_A,m_A;(s_B,i_B)f_B,m_B,l,M>,
\end{equation}
where the electron spin $s$ couples with the nuclear spin $i$ to the atomic angular momentum $f$ with its projection $m$ on the space fixed axis. $l$ is the quantum number of the overall rotation of the atom pair. The basis vectors in Eq.~(\ref{channelsdefinition}) can be interpreted in two ways, namely for the field-free case, where $f_A$ and $f_B$ are good quantum numbers or in a magnetic field where the pair is build up by the eigenvectors of the Breit-Rabi formula and $f_A$ and $f_B$ are approximate quantum numbers to label the corresponding eigenvector. The channel with the same spin state as the incoming atoms, for which we want to find the FRs, is called entrance channel. Those channels with an asymptotic ($R \rightarrow \infty$) energy larger than that of the entrance channel are called closed channels, all others are referred to as open channels.
In principle, it is impossible to solve the corresponding Schr\"odinger equation without any approximations due to the fact that an infinite number of coupled channel equations, from an infinite number of basis states, are involved. In the following, we will give a general description of three different models to overcome this difficulty in order to obtain an accurate description of resonance positions, using the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ \ system as an example.
\subsection{Coupled Channels Calculation}
\label{sec:NutshellCC}
The coupled channels calculation is a numerical approach to solve the Schr\"odinger equation resulting from the Hamiltonian of Eq.~\ref{eq:Hamiltonian}. For bound states, $R$ is represented on a grid and the resulting matrix is diagonalized, while for scattering solutions, the logarithmic derivative of the wave function is propagated in discrete steps with optimized step size from low $R$ to large $R$, from which the phase shift is determined by comparing with asymptotic wave functions. To calculate bound states, the wave functions at small separations $R_{in}$ and large separations $R_{out}$ (up to 10 000 $a_{\mathrm{0}}$ for the weakest bound levels, where $a_{\mathrm{0}}$ represents the Bohr radius) are set to zero as boundary conditions. This is equivalent to adding an infinitely high potential wall at $R_{in}$ and $R_{out}$, resulting in discretized continuum states, often referred to as box states. As this leads to shifts of the calculated resonance states, the size of the modeled box potential will be increased for achieving the desired accuracy.
Furthermore, in order to obtain a finite number of equations, the basis set is truncated, which is usually called close-coupling calculation. The attribute "close" refers to the fact that only states which are "close" in energy to each other, are retained. In the present approach the truncation is only in the space spanned by the rotational quantum number $l$ and naturally by using only the two molecular ground states X$^1\Sigma ^+$ and a$^3\Sigma ^+$. We span all spin channels allowed by given $s_A$ and $s_B$ as well as $i_A$ and $i_B$ and the chosen space fixed projection $M$ of the total molecular angular momentum. The coupling to higher electronic states exists but is weak and to some degree contained in $H_{dd}$. For collisions of alkali atoms in the ground state at ultracold temperatures, only a limited number of partial waves $l$ has to be included, owing to the small collision energy.
Performing the numerical procedure for a fine grid of magnetic fields yields the field dependent collisional properties, e.g. scattering lengths, collisional cross sections and collision rates. The procedure as we apply it, is specified in Refs.~ \cite{Marzok2009,Schuster2012}, and our results for ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ are provided in Sect.~\ref{sec:channels}.
\subsection{Asymptotic Bound State Model}
\label{sec:NutshellABM}
The ABM simplifies the calculation of the coupled Schr\"odinger equations by replacing the kinetic energy term and the interatomic potentials in Eq~(\ref{eq:Hamiltonian}) by their bound-state energies as adjustable parameters for describing the observed FRs, and neglecting the scattering continuum \cite{Tiecke2010a,Wille2008}. Therefore, neither accurate potentials, which are often not available, nor numerical integration of the spatial Schr\"odinger equation are needed. Solving the eigenvalue problem with the approximate Hamiltonian reduces to a simple matrix diagonalization of low dimension, which is the major benefit of the model. The ABM \cite{Tiecke2010a} has been introduced in Ref.~\cite{Wille2008} and builds upon a model by Moerdijk et al. \cite{Moerdijk1995}. Since then it has been extended to include various physical phenomena which has been applied to describe Feshbach resonances in many systems \cite{Wille2008,Li2008,Voigt2009,Deh2010,Knoop2011,Goosen2010,Tscherbul2010,Repp2013,Park2012,Goosen2011}. The ABM model is explained in detail in Ref. \cite{Tiecke2010a} and here we present a summary and describe various extensions to the model.
We begin by considering zero-energy collisions ($E_{kin}=0$) and restrict ourselves to $s$-wave collisions where $\left\langle H_\mathrm{dd}\right\rangle =0$. The model introduced by Moerdijk et al. \cite{Moerdijk1995} neglected coupling of the singlet and triplet states reducing the Hamiltonian (\ref{eq:Hamiltonian}) to: $H=\epsilon_{0,1}+H_\mathrm{hf}^+ + H_\mathrm{Z}$ where $\epsilon_{0,1}$ represent the singlet and triplet bound state energies and $H_\mathrm{hf}^+$ is the part of the hyperfine interaction which does not couple singlet and triplet states. This is a valid approximation for the special case that the spacing between the singlet and triplet energies is larger than the hyperfine energy.
In the ABM, the full hyperfine interaction $H=\epsilon_{0,1}+H_\mathrm{hf} + H_\mathrm{Z}$ is included, which generalizes the Moerdijk model to systems with arbitrary bound state energies, and the singlet-triplet coupling is characterized by the overlap integral $\zeta_{l}=\left\langle \Psi^{l}_{S=0}|\Psi^{l}_{S=1}\right\rangle $ of the singlet ($|\Psi^{l}_{S=0}\rangle $) and triplet ($|\Psi^{l}_{S=1}\rangle $) wavefunctions times the nondiagonal part of the Hamiltonian.
In the ABM the Hilbert space consists of only bound states and no scattering states. Therefore, the calculation includes only the basis states
\begin{equation}
|\sigma \rangle\equiv |S M_{S} m_{iA} m_{iB} v_{n,S} l>
\label{eq:NewABM_basis}
\end{equation} of pure electon spin states $S=0$ or $S=1$, which will be related to the respective channels (see Eq.~\eqref{channelsdefinition}) at a later stage for a pair of vibrational levels of the singlet and triplet state together. $M_{S},m_{iA}$ and $m_{iB}$ are the projections onto the space fixed axis of the operators $S,i_{A}$ and $i_{B}$, respectively, and $v_{n,S}$ is the $n$-th vibrational state in the $S=1$ or $S=0$ state.
The FRs are found at the magnetic fields for which an eigenstate exists at the energy of the incoming atom pair at that field. This condition corresponds to $E_{kin}=0$. Additionally, if $\left\langle H_\mathrm{dd}\right\rangle$ is small enough to be neglected, the Hamiltonian (\ref{eq:Hamiltonian}) is diagonal in the partial wave quantum number $l$. As a result, the only parameters needed for the calculation of the FRs in each partial wave $l$ are the energies of the bound states $\epsilon^{l}_{S}$ of the singlet ($S=0$) and triplet ($S=1$) potentials and their wavefunction overlap $\zeta_{l}$. In fact, only a small number of such states has to be taken into consideration, because the FRs usually arise from the least bound states close to the asymptote. The energies $\epsilon^{l}_{S}$ and the overlap parameters $\zeta_{l}$ are the free parameters of the ABM and are typically obtained by fitting to experimentally observed FRs.
The resulting Schr\"odinger equation can be written in the form of a $N \times N$ matrix, denoted by $\underline{M}_{ABM}$, where $N$ is determined by the number of spin channels and the number of selected vibrational states; N is on the order of a few tens. The diagonalization of this matrix for different fields provides the molecular energies as a function of the magnetic field. A comparison of this function to the energy sum of the two atoms yields the magnetic fields, at which the energies of bound-states and incoming free atoms are degenerate, thus marking the position of the FRs, as depicted in Fig.~\ref{fig:EnergyLevels}.
Close to a $s$-wave resonance, the molecular state --and therefore the resonance position-- is shifted due to coupling to the scattering states of the open channel. These states are continuum states and hence not included in the ABM model as described above. However, in some systems, the coupling has such a severe effect on the resonance position that it cannot be neglected, but it can be approximated by the coupling of the resonant molecular state to the least bound state of the open channel\cite{Tiecke2010a}, which requires assigning the bound-states of $\underline{M}_{ABM}$ to the scattering channels.
For this purpose, a rotation of the basis of $\underline{M}_{ABM}$ is performed: from the $|\sigma\rangle$ basis (constructed for a singlet and triplet vibrational level) to the basis formed by the eigenvectors of $H_\mathrm{hf}+H_\mathrm{Z}$ at the desired magnetic field (see Eq.~\eqref{channelsdefinition}). This can be ordered in the block matrix
\begin{equation}
\label{eq:ABMmatrix}
\underline{M'}_{ABM}=\left(\begin{array}{cc}
\mathcal{H}_{PP} & \mathcal{H}_{PQ} \\
\mathcal{H}_{QP} & \mathcal{H}_{QQ}
\end{array} \right),
\end{equation}
where the index $P$ ($Q$) stands for the spin states which are associated with an open (closed) channel and might include possible $l$ partial waves. A diagonalization of the submatrix $\mathcal{H}_{QQ}$ provides the bare molecular energies $\epsilon_{Q}$, which are the energies of the molecular state when no coupling to the open channel bound state occurs. Typically, only one of these states is the resonant state which causes the FR under consideration.
With the assumption that near a resonance the system can be described in a two channel picture, with one incoming, open channel and one resonant, closed channel, the total $S$-matrix of the scattering problem in the open channel can be written in the simple form of Eq.~(22) in Ref.~\cite{Tiecke2010a} at energy E with wave vector amplitude $|k|=(2\mu |E|)^{1/2}/\hbar$.
For the calculation of the Feshbach resonances, which are given by the poles of the scattering matrix, the complex energy shift $\mathcal{A}(E)$ locating the pole needs to be estimated. Depending on whether a bound state or a virtual state dominates the scattering behavior, different expressions have to be used for $\mathcal{A}(E)$.
E.g. for $^{40}$K-$^{40}$K collisions a real bound state of the open channel (with wavenumber $k_{p}=i\kappa_{bs}$ with $\kappa_{bs}>0$) occurs close to resonance resulting in a large positive background scattering length.
In this case $\mathcal{A}(E)$ is given by \cite{Tiecke2010a}
\begin{equation}
\label{eq:BScomplexShift}
\mathcal{A}(E)=\frac{\mu}{\hbar^{2}} \frac{-iA}{\kappa_{bs}(k-i\kappa_{bs})},
\end{equation}
where $\kappa_{bs}$ is the wavevector associated with the bare energy of the open channel $\epsilon_{bs}<0$, which is found on the diagonal of the submatrix $\mathcal{H}_{PP}$ in Eq.~\eqref{eq:ABMmatrix}. The coupling term $A$ is the square of the appropriate off-diagonal matrix element in $\mathcal{H}_{PQ}$ between the $P$-channel and the resonant $Q$-channel, after the $Q$ subspace has been diagonalized and $\underline{M'}_{ABM}$ has been transformed to the eigenvector of $Q$ space. This procedure allows for a prediction of the resonance width (imaginary part of $\mathcal{A}(E)$) and shift (real part of $\mathcal{A}(E)$) arising from coupling to the continuum without additional parameters. Using the $S$-matrix, the scattering properties around the resonance can be derived. In the present case we consider only the positions of Feshbach resonances; these will appear at $E=0$ and $k=0$ for a magnetic field where the bare molecular energy satisfies $\epsilon_Q=-(\mu/\hbar^2) A/ \kappa_{bs}^2=-A/2|\epsilon_{bs}|$.
A virtual state, which is also often referred to as an anti-bound state ($k_{p}=-i\kappa_{vs}$ and $\kappa_{vs}>0$ \cite{Marcelis2004}) results in a large negative background scattering length. The $^{6}$Li$-^{6}$Li \cite{Abraham1997} and $^{133}$Cs$-^{133}$Cs \cite{Leo2000} systems are excellent examples for a system with a dominating virtual state.
In this scenario, the complex energy shift is given by \cite{Marcelis2004}
\begin{equation}
\label{eq:VScomplexShift}
\mathcal{A}(E)=\frac{\mu}{\hbar^{2}} \frac{-iA_{vs}}{\kappa_{vs}(k+i\kappa_{vs})},
\end{equation}
where the coupling between virtual and bound state $A_{vs}$ enters as new parameter, while $\kappa_{vs}$ can be estimated from the van der Waals range $r_{0}$ via $a_{bg}=r_{0}-1/\kappa_{vs}$. To find the position of Feshbach resonances one has to look for magnetic fields where the binding energy of the bare molecular state $\epsilon_Q=+(\mu/\hbar^2) A_{vs}/\kappa_{vs}^2$.
To calculate the background scattering length of the desired open channel $a_{bg}$, one requires the singlet ($a_{S}$) and triplet ($a_{T}$) background scattering lengths, as well as a decomposition of the ABM matrix eigenstates into singlet and triplet components. $a_{S}$ and $a_{T}$ can be estimated via the accumulated phase method, which employs a numerical calculation of the singlet and triplet wave functions from the asymptotic form of the inter-atomic potential $V_{as}$, using only the van der Waals tail plus adding the centrifugal barrier and the bound state energies. This procedure is described in Refs. \cite{Tiecke2010a,Verhaar2009}.
Obtaining the poles of the $S$-matrix for a system in which the virtual state dominates the scattering behavior has been utilized in Ref. \cite{Park2012} to explain FRs in a NaK mixture using the ABM.
The ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ system, however, is in an intermediate regime, where both the bound state and the virtual state in the open channel are required to describe the FR positions. In Sect.~\ref{sec:ABM} we demonstrate an extension of the existing models, that starts from the virtual state description, but includes the coupling to the bound state in a phenomenological way.
\subsection{Multichannel Quantum Defect Theory}
The MQDT uses a separation of the solution to the Schr\"odinger equation into a long-range and a short-range part. It is based on a model by Seaton \cite{Seaton1983}, which was originally introduced to describe the properties of an electron in the field of an ion. However, it has been generalized in Refs.~\cite{Greene1979,Greene1982} and can now be applied to a variety of collisional partners, with all sorts of interaction potentials (see Ref.~\cite{Croft2011} and references therein). For example, it has been applied successfully to various neutral atom pairs \cite{Burke1998,Gao2005,Raoult2004,Mies2000,Julienne1989,Gao1998,Gao2001}, and can, in general, be used for all alkali atom combinations without adaptation. The most recent modification improves the model for an accurate description of higher partial waves \cite{Ruzic2013}.
The main benefit of the model stems from the separate treatment of the long-range part of the scattering problem, where the van der Waals interaction dominates over exchange interactions and higher order terms. It can be solved accurately using the Milne phase amplitude method (see Ref. \cite{Burke1998} and references therein). This results in a linearly independent pair of functions $(f^{0},g^{0})$, referred to as base pair, which are smooth and analytic functions of energy. In the short-range part, the coupled Schr\"odinger equation at energy $E$ is numerically integrated outwards to a radius $R_{lr}$ on the order of a few tens of atomic units (typically 30 $a_{0}$), beyond which the exchange interaction is negligible. At $R_{lr}$ it is then connected to the long-range part of the solution.
The calculation incorporates only those channels which have a non-negligible effect on the scattering behavior of the system by truncating the basis set of Eq.~\eqref{channelsdefinition} in the same manner as for the CC model. The solution is given by the square matrix $\underline{M}(R)$, which contains the independent solutions of each channel in its columns. Beyond $R_{lr}$, $\underline{M}(R)$ can be given as superposition of the base pair:
\begin{equation}
\label{eq:Mmatrix}
\underline{M}(R)=\underline{f}^{0}(R)-\underline{g}^{0} \underline{K}^{sr},
\end{equation}
where $\underline{f}^{0} \ \mathrm{and} \ \underline{g}^{0}$ are diagonal matrices which contain the base pair evaluated at the appropriate channel energies $\epsilon_{i}=E-E_{i}$. In this notation $E_{i}$ is the energy of the asymptote of channel $i$. The short-range reaction matrix $\underline{K}^{sr}$ contains all the system specific information for the scattering behavior at low energies. Besides the short-range reaction matrix, one needs four coefficients in order to construct the $S$-matrix, which delivers the physical observables. Detailed instructions on how to obtain these coefficients, which are often noted as $A$, $\mathcal{G}$, $\gamma$ and $\eta$, are given in Refs.~\cite{Ruzic2013,Burke1998,Burke1999}.
The next level of simplification of the MQDT is the assumption that $\underline{K}^{sr}$ depends only very weakly on energy. Thus it only needs to be calculated for a few energies, and can then be interpolated between these values. In the best case, a $\underline{K}^{sr}$ matrix which is only calculated for one energy (typically close to threshold) and at zero magnetic field can be utilized to describe the scattering properties over a wide range of energies and magnetic fields. However, for obtaining $\underline{K}^{sr}$, one still has to solve the coupled channel equations at short-range.
Nevertheless, the calculation can be facilitated further by using a FT approach. The general form of the FT theory as applied to ultracold collisions of two alkali atoms has been written in Refs.\cite{Burke1998, Burke1999, Gao2005}. The main simplification is to neglect the hyperfine interaction at short-range. This is justified by the fact that the exchange splitting is much larger than the hyperfine and Zeeman energy. In this case the atomic motion is described by a set of uncoupled channel equations, which can be solved numerically. Matching the solutions to the analytic base pair allows one to determine the short-range energy-analytic scattering information in terms of quantum defects $\mu^{sr}_{S}(\epsilon_{S})$ in the single-channel singlet-triplet basis (equivalently the singlet and triplet scattering lengths recast as quantum defects) in a diagonal short-range reaction matrix $K^{sr}_{\mathrm{diag}}=\mathrm{tan}(\pi \mu)$. An energy independent real orthogonal transformation turns this short-range single-channel scattering information into the final channel structure applicable at $R\rightarrow \infty$, namely the representation of hyperfine plus Zeeman atomic energy eigenstates. This procedure, delivers the real, symmetric, short-range reaction matrix $K^{sr}$ (or the corresponding smooth quantum defect matrix $\mu^{sr}$):
\begin{equation}
\label{eq:Frametransform}
K^{sr}_{ij}=\sum_{\alpha} U_{i,\alpha} \tan(\pi \mu_\alpha) \tilde{U}_{\alpha,j}.
\end{equation}
Here the tilde denotes the matrix transpose. The dissociation channel index $i$ represents the set of quantum numbers according to Eq.~(\ref{channelsdefinition}) for non-zero magnetic field needed to characterize the internal energies of the separating atoms as well as their relevant angular momentum couplings with each other and with the orbital angular momentum quantum number $l$ and its projection $m_l=M-m_A-m_B$.
As was stressed by Bo Gao in his ''angular momentum insensitive'' form of quantum defect theory for a van der Waals long-range potential, the $l$-dependence is known approximately \cite{Gao2001} as $\mu^{sr}_{S,l} \approx \mu^{sr}_S-l /4$~\cite{Ruzic2013}. When higher accuracy is needed, a small $l-$dependent correction $\alpha_l$ can be introduced to this equation, which leads to:
\begin{equation}
\label{eq:defects}
\mu^{sr}_{S,l} \approx \mu^{sr}_S-l /4+\alpha_l,
\end{equation}
where $\alpha_0 \equiv 0$ by definition.
The FT then simply approximates the real, orthogonal matrix that diagonalizes $K^{sr}$ as the angular momentum recoupling matrix that connects the short-range eigenstates with those appropriate at large $R$. Specifically, in the absence of any magnetic field, good quantum numbers of the atomic energy levels are given by Eq.~\eqref{channelsdefinition}. In the presence of an external magnetic field $B$ directed along the $z$-axis, $f_a$ and $f_b$ are no longer good quantum numbers but $m_{a},m_{b}$ are still conserved for the atoms at infinite separation. However one must diagonalize the atomic hyperfine plus Zeeman Hamiltonian to obtain a numerical eigenvector $\langle f_{A} m_{A},f_{B} m_{B} | m_{A} k_{A}, m_{B} k_{B} \rangle \equiv \langle i| j \rangle $ and the corresponding field-dependent channel energies, $E_{m_{A} k_{A},m_{B} k_{B}}(B) \equiv E_{j}(B)$ (see also the extended interpretation for the basis given in Eq.~\eqref{channelsdefinition}). We indicate the short-range collision eigenstates by $|(s_{A} s_{B})S (i_{A},i_{B})I f m_{f} \rangle \equiv | \alpha \rangle$.
We can now write out the final FT matrix $U_{i\alpha }$ between the long- and short-range channels, which is needed in Eq.~\eqref{eq:Frametransform}. Recall that in the present notation, the long-range scattering channels in the presence of a magnetic field $B\ $\ are written as $i=\{m_{A},k_{A},m_{B},k_{B}\},$ and the short-range collision eigenchannels are $\alpha =\{(s_{A},s_{B})S(i_{A},i_{B})I,fm_{f}\},$ and the unitary transformation between these is given explicitly in terms of standard angular momentum coefficients (Clebsch-Gordan and Wigner 9-j symbol) and the Breit-Rabi eigenvectors such as $\left\langle k_{A}|f_{A}\right\rangle ^{(m_{A})}$, etc. as:
\begin{equation}
\begin{split}
U_{i\alpha }= \sum\limits_{f_{A}f_{B}f}\left\langle
k_{A}|f_{A}\right\rangle ^{(m_{A})}\left\langle k_{B}|f_{B}\right\rangle^{(m_{B})} \times\\
\left\langle f_{A}m_{A},f_{B}m_{B}|fm_{f}\right\rangle \times\\
\left\langle
(s_{A}i_{A})f_{A}(s_{B}i_{B})f_{B}|(s_{A},s_{B})S(i_{A},i_{B})I\right\rangle
^{(f)}
\end{split}
\end{equation}
Note that in the FT approximation, this matrix is
independent of $l$, so this quantum number is not explicitly represented. The transformation mentioned for the ABM is constructed in the same way.
Note that the final step of computing scattering or bound state observables such as the FRs at zero incident energy in various scattering channels requires solving the MQDT equations as a function of energy and/or magnetic field. As usual in MQDT studies, this is the step where exponential decay of the large-$R$ closed-channel radial solutions is imposed.
The determinantal condition for a resonance to occur at an energy just above an open-channel threshold is $det(K^{sr}_{QQ}+\cot \gamma)=0,$ where the notation $K^{sr}_{QQ}$ indicates just the closed-channel partition of the full short-range $K$-matrix. In this equation, $\gamma$ is a diagonal matrix of long-range negative energy phase parameters as mentioned above for the construction of the $S$-matrix from the MQDT.
The energy- and field-analytic nature of the single-channel solutions allows them to be constructed on a very coarse mesh of energy and magnetic field. In its most simple form, the energy dependence of the quantum defects can be dropped, and the quantum defects, which are calculated at a specific energy only once, can be used throughout the entire energy and magnetic field range of interest. Ref. \cite{Burke1998,Gao2005} demonstrate how the quantum defects can be represented by only two parameters, namely $a_{S}$ and $a_{T}$, in the FT formalism. This yields the crudest, but also computationally lightest realization of the MQDT.
The details for a calculation of ${}^{6}\text{Li }$-${}^{133}\text{Cs }$~FRs using the MQDT are given in Sect.~\ref{sec:MQDT}. Additionally, we introduce a slightly modified MQDT-FT approach, which allows us to calculate FRs for systems that are lacking a detailed microscopic model.
\section{Application to the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ system}
\label{sec:application}
In this chapter we apply the models described in Sect.~\ref{sec:nutshell} as a case study to the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$~system, where FRs have been measured recently \cite{Repp2013}. Throughout the entire section, we use the $C_{6}$ coefficient from Derevianko et al. \cite{Derevianko2001} for the description of the van der Waals interaction, which has been calculated with sufficient accuracy. In order to compare the models among themselves and with experiment, we calculate the weighted rms deviation $\delta B^{rms}$ on the resonance positions, which is defined as
\begin{equation}
\label{eq:rms}
\delta B^{rms}=\frac{\sqrt{(\sum_{i}^{N}\delta_{i}^{2}/\delta B_{i}^{2})/N}}{\sqrt{\sum_{i}^{N}\delta B_{i}^{-2}}}.
\end{equation}
The summation is performed over $N$ resonances, $\delta=B_{\text{res}}^{\text{exp}} - B^{\text{theo}}_{\text{res}}$ is the deviation of experimental ($B_{\text{res}}^{\text{exp}}$) and theoretical ($B^{\text{theo}}_{\text{res}}$) resonance positions, and $\delta B$ contains the experimental uncertainty of the measured resonance positions, which are given in Table~\ref{tab: List2}, and a $200\,\mathrm{mG}$ drift of the magnetic field for all resonances.
\subsection{Coupled Channels Calculation}
\label{sec:channels}
We have provided details of the CC calculation for a mixture of ${}^{6}\text{Li }$ and ${}^{133}\text{Cs }$ atoms elsewhere (see Ref.\cite{Repp2013} and references therein). Here, we review the method of our CC calculation and summarize its results, as they are used as benchmark for the other approximate models presented in the subsections below.
For the CC matrix, the Hamiltonian of Eq.~\eqref{eq:Hamiltonian} is employed, where the effective form of $H_{dd}$ (Eq.~\eqref{eq:Vdipole}) is used. Only basis states with partial waves up to $l=2$ are included, which is sufficient for the descriptions of alkali atoms in the $\mu\mathrm{K}$ regime. Besides the atomic constants, which are readily available in the literature \cite{Arimondo1977}, accurate potentials are crucial in order to precisely determine the position of FRs. For this purpose, the relevant potential curves for the $a ^3\Sigma^+$ and $X^1\Sigma^+$ states of ${}^{6}\text{Li} {}^{133}\text{Cs}$~are expanded in a power series of the internuclear separation $R$ (similar to Ref. \cite{Gerdes2008}), where $R$ is mapped onto a Fourier grid following \cite{Tiesinga1998}. Then, the expansion coefficients, which were initially determined via Fourier-transform spectroscopy \cite{Staanum2007}, are modified iteratively in such a way that both the calculated maxima of binary collision rates and the rovibrational transition frequencies are in agreement with the measured FRs and with the 6498 previously observed molecular transitions \cite{Staanum2007}, respectively. The potential parameters are summarized in the online material, the parameters of the bound states, which are involved in the observed FRs, are given in Table~\ref{tab: List1} and the resulting resonance positions in Table~\ref{tab: List2}. The molecular energy levels for the $^6\text{Li}\left|F=1/2,m_{F}=-1/2\right\rangle\oplus ^{133}\text{Cs}\left|3,3\right\rangle\ $ channel with respect to the incoming channel are given in Fig.~ \ref{fig:EnergyLevels}. The rms deviation for this model is 39 mG.
\begin{table*}[t]
\newcommand{\mc}[3]{
\multicolumn{#1}{#2}{#3}}
\begin{ruledtabular}
\begin{center}
\begin{tabular}{l|cc|c|cc|c|cc}
Model & $\epsilon^{0}_{0}$ (MHz) & $\epsilon^{0}_{1}$ (MHz) & $\zeta_{0}$ & $\epsilon^{1}_{0}$ (MHz) & $\epsilon^{1}_{1}$ (MHz) & $\zeta_{1}$& $a_{S}$ ($a_{0}$) & $a_{T}$ ($a_{0}$) \\
\hline
CC & 1566 & 3942 & 0.866 & 1159 & 3372 & 0.866 & 30.3(1) & -34.3(2) \\
bABM& 1592 & 4189 & 0.866 & 1191 & 3641 & 0.860 & \it{29.6} & \it{-42.4} \\
dABM & 1543 & 4155 & 0.870 & 1191 & 3641 & 0.860 & \it{30.7} & \it{-40.8} \\
MQDT & 1565 & 3945 & 0.866 & 1158 & 3375 & 0.862 & 30.3 & -34.4 \\
MQDT-FT & - & - & - & - & - & - & 30.1 & -39.2 \\
\end{tabular}
\caption{List of bound state energies $\epsilon^{l}_{S}$, wave function overlaps $\zeta_{l}$ and background scattering lengths $a_{S}$ for singlet ($S=0$) and triplet ($S=1$) potentials. The fit results for the CC calculation, the bare ABM (bABM), the dressed ABM (dABM), the MQDT, and the MQDT-FT are tabulated. Note that the $l=1$ values for the bABM and the dABM are taken from the same fit. The scattering lengths indicated for the ABM are derived from the binding energies using the accumulated phase method.
}
\label{tab: List1}
\end{center}
\end{ruledtabular}
\end{table*}
\begin{figure}[t]
\includegraphics[width=1.0\columnwidth]{testfig3.pdf}
\caption{(Color online) Molecular energy levels for the $^6\text{Li}\left|F=1/2,m_{F}=-1/2\right\rangle\oplus ^{133}\text{Cs}\left|3,3\right\rangle\ $ channel for $l$=0, s-waves. The energies with respect to the open channel asymptote for the bare ABM (blue line), dressed ABM (black line), MQDT (red dash-dotted line) and CC (green dashed line) are depicted. The horizontal line at zero energy represents the continuum threshold. The crossings of the molecular channels with the threshold mark the positions of the FRs. Calculated quasi-bound levels and box states from the CC model are removed for clarity. The inset shows a zoom into the region of the resonance at $\sim 889$~G, where the differences between the models is clearly visible. E.g. the energy level of the bABM is not shifted, as it neglects the coupling to the continuum.}%
\label{fig:EnergyLevels}
\end{figure}
\begin{table*}[t]
\newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}}
\begin{ruledtabular}
\begin{center}
\begin{tabular}{c|l|cc|c|c|c|c|l}
Entrance channel & l& $ \text{B}_{\text{res}}^{\text{exp}} $(G) & $\Delta \text{B}^{\text{exp}}$ (G)& $\delta_{CC}$ (G)&$\delta_{bABM}$ (G)&$\delta_{dABM}$ (G) &$\delta_{MQDT}$ (G)& $\delta_{MQDT-FT}$ (G)\\
\hline
${}^{6}\text{Li }$$\left|1/2,+1/2\right\rangle \ $ & 1 & 662.79(1)& 0.10(2)&-0.04 & -0.04 & -- & -0.11& -0.26 \\
$ \oplus \ ^{133}\text{Cs} \left|3,+3\right\rangle$ & 1 & 663.04(1)& 0.17(2) & -0.02 & -0.37 & -- & -0.01& 0.01 \\
& 1 & 713.63(2) & 0.10(3) & -0.05 & -0.82 & -- & -0.09 & -0.22\\
& 1 & 714.07(1) & 0.14(3) & -0.05 & -1.35 & -- & 0.10 & 0.22\\
& 0 & 843.5(4) & 6.4(1)& 0.51& 8.32 & -0.64 & 0.38 & 0.00\\
& 0 & 892.87(7) & 0.4(2) &-0.11 &-7.03 & -1.20 & -0.04 & -0.39 \\
\hline
${}^{6}\text{Li }$$\left|1/2,-1/2\right\rangle $ & 1 & 658.21(5)& 0.2(1) & 0.07 & -3.02 & -- & 0.04 & -0.06 \\
$ \oplus \ $${}^{133}\text{Cs }$ $\left|3,+3\right\rangle$ & 1 &708.63(1) & 0.10(2)& -0.05 & 1.24 & -- & -0.11 & -0.19 \\
& 1 & 708.88(1) & 0.18(2) &-0.03 & 0.91 & -- & -0.01 & 0.06 \\
& 1 &764.23(1) & 0.07(3) & -0.06 & 0.83 & -- & -0.06 & -0.09 \\
& 1 &764.67(1) & 0.11(3) &-0.05 & 0.29 & -- & 0.12 & 0.35 \\
& 0 & 816.24(2) & 0.20(4) & -0.12 & -5.19 & 1.51 & -0.26 & -0.02\\
& 0 & 889.2(2) & 5.7(5) &0.46 & 9.31 & 0.31 & 0.34 & 0.00 \\
& 0 & 943.26(3) & 0.38(7) &-0.12 & -5.68 & 0.10 & -0.04 & -0.34 \\
\hline
${}^{6}\text{Li }$$\left|1/2,+1/2\right\rangle \ $ & 1 & 704.49(3) & 0.35(9) & 0.07 & 0.67 & -- & 0.01 & -0.01 \\
$\oplus \ $${}^{133}\text{Cs }$ $\left|3,+2\right\rangle$ & 0 & 896.6(7) & 10(2) & 0.68 & 19.51 & -0.35 & 0.07 & -0.21 \\
\hline
${}^{6}\text{Li }$$\left|1/2,-1/2\right\rangle \ $ & 1 & 750.06(6) & 0.4(2) &0.06 & 1.47 & -- & -0.01 & 0.01 \\
$ \oplus \ $${}^{133}\text{Cs }$ $\left|3,+2\right\rangle$ & 0 & 853.85(1) & 0.15(3) & -0.17 & -7.34 & -0.29 & -0.41 & 0.15\\
& 0 & 943.5(1.1) & 15(3)& 2.21 & 21.69 & 1.59 & 1.64 & 1.4 \\
\hline
$\delta B^{rms}$ (G)] & & & & 0.039 & 0.965 & 0.263 & 0.040 & 0.048\\
\end{tabular}
\caption{Comparison of the resulting ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ FR positions from the various models to the observed resonances.
The experimental positions $B_{\text{res}}^{\text{exp}}$ and widths $\Delta B^{\text{exp}}$ are taken from Ref. \cite{Repp2013}. The resonance positions $B^{\text{theo}}_{\text{res}}$ derived from CC calculation ($\delta_{CC}$), bare ABM ($\delta_{bABM}$), dressed ABM ($\delta_{dABM}$), MQDT ($\delta_{MQDT}$) and MQDT-FT ($\delta_{MQDT-FT}$) are given as deviations $\delta$=$B_{\text{res}}^{\text{exp}} - B^{\text{theo}}_{\text{res}}$
with respect to the observations. We also state the rms deviation $\delta B^{rms}$ (see Eq.~\ref{eq:rms}) for all models. For the dressed ABM the $p$-wave values are identical to the bare ABM, and therefore not repeated in the table. The splitting of the $p$-wave resonances has not been considered in $\delta_{MQDT-FT}$.}
\label{tab: List2}
\end{center}
\end{ruledtabular}
\end{table*}
\subsection{Asymptotic Bound State Model}
\label{sec:ABM}
For the ABM calculation of ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ FR positions, we begin using the ABM in its simplest form starting from the Hamiltonian of Eq.~\eqref{eq:Hamiltonian}, replacing $T+V$ by the bound-state energies and neglecting $H_{dd}$. The latter is incorporated at a later stage. Because the spacing of the vibrational states in the ${}^{6}\text{Li} {}^{133}\text{Cs}$~potential is large compared to the hyperfine energy, we only include the least bound vibrational state of the singlet and triplet potential and neglect the role of deeper bound states. This yields a fit of only three parameters per partial wave. However, the three fit parameters are not independent with the present set of data, as will be explained in the following discussion.
As a prelude to the new fits below, we start with the ABM as practiced in Ref. \cite{Repp2013}, where the ABM was applied leaving five parameters ($\epsilon_0^0$, $\epsilon_1^0$, $\zeta_0$, $\epsilon_0^1$, $\epsilon_1^1$) as free fit parameters, while $\zeta_1$ was taken to be equal to $\zeta_0$ \cite{[{Also, the atomic masses in the calculation where not accurate enough, which also had minor effects on the fitting results. In the present work we use atomic masses from~}][{ (version 3.0). [Online] Available: http://physics.nist.gov/Comp [2013, 12 09]. National Institute of Standards and Technology, Gaithersburg, MD.}]atomicmasses2012}.
In this work we redo the fit, utilizing $\zeta_{1}$ also as a free parameter, thus using six parameters as fit parameters to minimize $\delta B^{rms}$ (see Eq.~\eqref{eq:rms}). This quantity gives intuitive and quantitative insight into the deviations of calculated from measured resonance positions. In ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ the hyperfine interaction gives rise to a very strong singlet-triplet mixing, which is indicated by an expectation value for the total spin $S$ of $\left\langle S \right\rangle \simeq 0.6-0.7$ on the resonances. This results in a strong correlation of the fit parameters and the resonances can be fitted with similar rms deviations over a large range (within a few GHz) of binding energies.
The best fit has a rms deviation of 877~mG with the parameters $\epsilon_0^0=5824$~MHz, $\epsilon_1^0=2995$~MHz, $\zeta_0=0.559$, $\epsilon_0^1=1844$~MHz, $\epsilon_1^1=3575$~MHZ and $\zeta_1=0.821$. However, since the singlet and triplet binding energies and their overlap parameter are related to each other through the interaction potential, the obtained overlap parameter of $\zeta_{0}=0.559$ is unphysical for the fitted binding energies. Additionally, the $p$-wave shift is unreasonably large for the singlet channel binding energy, while it has opposite sign for the triplet binding energy, which are clear indications that the fit results are unphysical.
In order to obtain a fit restrained to physical parameters, we demonstrate how these discrepancies of binding energies and overlap parameters can be reduced in the bare ABM (bABM) in three steps. The first step is to reduce the number of independent fit parameters by deriving the wave function overlaps from the two binding energies via the accumulated phase method (as described in Ref.~\cite{Verhaar2009}) instead of leaving them as a free parameter, thereby restricting ourselves to only the physical range of the fit parameters and reducing the fit to two parameters.
The coupling of the bound state to the continuum, which is neglected at this stage, results in significant shifts for $s$-wave resonances. Therefore, we use only the narrow $p$-wave resonances for the initial fit, because their widths are not acquired by coupling to the continuum but rather by tunneling through the centrifugal barrier which is suppressed at low collision energies.
The fit results of $\epsilon^{1}_{0}=1193 \ \text{MHz}$, $\epsilon^{1}_{1}=3638 \ \text{MHz}$ and the calculated $\zeta_{1}=0.861$
agree much better with the CC values, which are shown in Table~\ref{tab: List1}.
The $p$-wave resonances are reproduced with a rms deviation of 560 mG, where the mean was used for resonances which are split due to the magnetic spin-spin and second-order spin-orbit coupling. This demonstrates how the bare ABM model, which extensively simplifies the spatial part of the scattering problem, satisfactorily reproduces resonances which are not shifted due to coupling to the open channel scattering wave function.
Since we obtain the asymptotic wave functions in the procedure described above, in the second step it is now also possible to include the magnetic dipole-dipole and second-order spin-orbit coupling term into the ABM Hamiltonian (similar to Ref. \cite{Goosen2010}) in its effective form (see Eq.~\eqref{eq:Vdipole}). With a rms deviation of $375$\,mG, a fit containing $H_{dd}$ improves the prediction of the $p$-wave resonances by about $185$\,mG, which is the expected order of magnitude, considering that the splitting is only on the order of a few hundred mG at most. For calculating the expectation value of $H_{dd}$ only the long range part of the wave function was used. Thus the spin-orbit contribution does not play a role and $a_\mathrm{SO}$ could be set to zero.
In order to include the $s$-wave resonances in the third and final step of the bABM,
the $s$-wave binding energies are deduced from the fitted $p$-wave binding energies using the accumulated phase method as follows. We numerically solve the Schr\"odinger equation containing only the van der Waals term in the interaction potential and using the phases of the obtained $p$-wave functions at $R_{i}$ and $\psi \rightarrow 0$ for $R\rightarrow \infty$ as boundary conditions. Here, $R_{i}$ is the radius where the van der Waals energy is larger than the hyperfine energy, and the exchange energy is large enough to split the singlet-triplet manifold~\cite{Verhaar2009}.
This approach neglects the $l$-dependence of the phase-shift at $R_{i}$ which is a small correction in our case \cite{Verhaar2009}.
The rms deviation for all resonances in the case where the $p$-wave resonances are fitted and the $s$-wave resonances are calculated is 1.26 G when $H_{dd}$ is neglected.
Including $H_{dd}$ into this fit results in the lowest attainable rms deviation value of 965 mG for a physically meaningful bare ABM fit. The resulting molecular energy levels for the $^6\text{Li}\left|F=1/2,m_{F}=-1/2\right\rangle\oplus ^{133}\text{Cs}\left|3,3\right\rangle\ $ are shown in Fig.~\ref{fig:EnergyLevels}. As one can see, the positions where the bound state energies cross the threshold are not shifted due to interactions, as is the case for the other models. Thus, compared to the measured values, the broad FRs are systematically shifted to lower magnetics fields, as illustrated in Table~\ref{tab: List2}, where the resonance positions are given in the "bABM" column. The positions of the narrow resonances are shifted to higher values most likely because of the application of the accumulated phase methods, which introduces errors in the determination of the $s$-wave binding energies. The latter are given together with the $p$-wave binding energies as "bABM" values in Table~\ref{tab: List1}. If one needs a more accurate prediction for the narrow resonances, one could also fit them using $\epsilon^{0}_{0}$ and $\epsilon^{0}_{1}$ as additional parameters.
The binding energies can be used to derive background scattering lengths via the accumulated phase method \cite{Verhaar2009}, by propagating the wavefunction with the known phase at $R_{i}$ to large internuclear separations and then comparing to a long-range wavefunction which is not shifted by an interaction potential. This procedure introduces additional errors on the order of $\sim 10 \%$ that depend somewhat on the precise choice of $R_{i}$. These errors are related to the accumulated phase method and not to the ABM. By including the energy dependence of the accumulated phase as described in Ref.~\cite{Verhaar2009} the scattering length might be calculated with better accuracy. However, calculating these derived quantities allows for a comparison with the MQDT-FT (see Sect.~\ref{sec:MQDT}) where no binding energies were derived directly, and yields an additional test of consistency among the three different models. The results are shown in Table~\ref{tab: List1}.
In order to achieve higher accuracy, we include the coupling of the closed channel responsible for the FR to the open channel, which is referred to as dressed ABM (dABM). In Sect. \ref{sec:NutshellABM} we discussed the limiting cases, where the coupling of the closed channel bound state to either the least bound state, or to a virtual state in the open channel can be used as an estimate for the shift of the resonance position. The ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ system, however, is in an intermediate regime, where both of the approaches do not deliver satisfactory results. The background scattering length of the triplet channel of $a_{T}=-34.3(2) \ a_{0}$ indicates this regime. On the one hand, it is significantly far from the van der Waals range of $r_{0}=45 \ a_{0}$, which would indicate a non-resonant open channel, but on the other hand, it is also not dominating the scattering process, which would lead to a much larger magnitude of the background scattering length.
Therefore, we introduce an extension to the ABM similar to the approach presented in Park, et al. \cite{Park2012}, which includes both effects in a phenomenological manner. We use the complex energy shift of Eq.~\eqref{eq:VScomplexShift}, just as in a system with a resonant open channel \cite{Marcelis2004,Kempen2004}. To calculate $A_{vs}=A\zeta_{vs}$, we multiply the square of the appropriate matrix element of $\mathcal{H}_{PQ}$ taken from the matrix $\underline{M'}_{ABM}$ in the form where $\mathcal{H}_{QQ}$ is diagonalized (denoted by $\mathcal{K}^2$) with an additional scaling factor $\zeta_{vs}$, which handles the spatial part of the matrix element and is equal for all FRs. However, in contrast to Ref. \cite{Marcelis2004}, the molecular energy $\epsilon_{Q}$ is not taken to be the bare energy from the submatrix $\mathcal{H}_{QQ}$, but we rather diagonalize the full matrix $\underline{M}_{ABM}$ and replace $\epsilon_{Q}$ in the $S$-matrix with the dressed resonant molecular state $\epsilon_{ABM}$. In doing so, $\epsilon_{ABM}$ contains the coupling to the open channel bound state. Therefore, both the influence of the coupling to the bound and the virtual states in the $P$-channel are accounted for. The FR positions are now simply obtained at magnetic fields where the following identity is satisfied:
\begin{equation}
\epsilon_\mathrm{ABM} = \frac{\mu}{\hbar^2}\frac{\zeta_{vs} \mathcal{K}^2}{\kappa_{vs}^2}.
\label{eq:New_ABM_energy}
\end{equation}
$\kappa_{vs}$ is obtained as described in Sect.~\ref{sec:NutshellABM} and $\zeta_{vs}$ is left as a free fit parameter. Within this approach the FRs, including coupling to the (near-resonant) scattering states, can be found by simple matrix operations and linear equations.
We note that the full scattering properties around the resonance (including the resonance width and for the case of overlapping resonances \cite{Park2012}), can be obtained from the $S$-matrix by using the complex energy shift $A_{vs}$ as given above.
We start the calculation by only fitting the narrow $s$-wave resonances, where the coupling to the open channel is small, in order to derive $\epsilon^{0}_{0}$ and $\epsilon^{0}_{1}$. Their overlap $\zeta_{0}$ is obtained with the same method as for the bare ABM. The results of this fit are given in Table~\ref{tab: List1} as "dABM", where the background scattering lengths for the specific incoming channels are deduced in the same manner as described above and are used to calculate $\kappa_{vs}$ for each channel needed for Eq.~\eqref{eq:New_ABM_energy}. The next step is the fitting of the scaling factor $\zeta_{vs}$ by performing a weighted least squares minimization on all observed $s$-wave resonances, additionally allowing the overlap parameter to vary by $<0.1\%$. The result of $\zeta_{vs}=0.0255$ yields a rms deviation of 310 mG on all $s$-wave resonances.
Combining with the results of the bare ABM for the $p$-wave resonances we obtain a total rms deviation of $\delta B^{rms} = 263\,\mathrm{mG}$ on all resonances. The resonance positions are listed in Table~\ref{tab: List2}, and selected molecular energy levels are given as a function of magnetic field in Fig.~\ref{fig:EnergyLevels} for comparison. The largest deviation between the CC and ABM approach is seen at the broad resonances at 890~G (see inset Fig.~\ref{fig:EnergyLevels}). The coupling to the continuum is obvious by the nonlinear function of the energy with respect to the magnetic field. This is in contrast to the bare ABM, where the influence of the continuum is neglected.
While the ABM assigns the FRs with a sub-1 G accuracy, there is a significant deviation of the triplet binding energies $\epsilon_1^0$ from the CC value. This could arise from the fact that the coupling to the (near resonant) triplet channel is only included phenomenologically by adding a single virtual state. Also, only one bound state is taken into account and we treat the resonances as being non-overlapping.
An additional approximation is introduced by using the accumulated phase method to derive the overlap parameters, and relating the $s$-wave and $p$-wave binding energies enlarges the uncertainty. We characterize the accuracy of the accumulated phase method by comparing the obtained binding energies with the bound states of the full ${}^{6}\text{Li }$-${}^{133}\text{Cs }$~potentials. Using the boundary conditions at $R_i$ from the $s$-wave binding energies we find that the accumulated phase method reproduces the $p$-wave binding energies to $\sim 1-2\%$ for both the singlet and triplet potentials. Changing the binding energies on the order of $\sim 1-2\%$ increases $\delta B^{rms}$ on the FRs with more than a factor 2, indicating that at this level of accuracy the inner part of the potential has a significant effect. Extending the ABM to use more information of the full potentials to link the $s$- and $p$-wave binding energies is straightforward, however, by this step we would lose the advantage of the simple calculations of the ABM. We also note that a more rigorous method to include both the bound and the virtual state is the Resonant State Model as presented in Ref. \cite{Goosen2011}. Also, the accumulated phase methods leads to an error in the derivation of the background scattering lengths. While this is on the order of $\sim 10 \%$, a significant part of the deviation of $a_{T}$ is a result of the systematic shift of $\epsilon^{0}_{1}$, in both the bare and the dressed ABM.
\subsection{Multichannel Quantum Defect Theory}
\label{sec:MQDT}
We apply the {\it ab initio} MQDT treatment, using the potentials of Ref.~\cite{Repp2013} as input for the calculation. We then slightly modify the inner wall of these potentials in order to minimize the deviation from the experiment. The present calculation only requires solving the coupled differential equations out to $r_0=$40 a.u., and very little difference is seen if this matching radius to the long-range single-channel QDT solutions is reduced to $30$ a.u. Table~\ref{tab: List2} shows the accuracy of the FR positions in comparison with the experimentally determined resonances of Repp et al.\cite{Repp2013}, and Fig.~\ref{fig:EnergyLevels} plots three of the obtained energy levels for comparison with the other models. The close agreement between CC and MQDT is very satisfying. The bound state energies (see Table~\ref{tab: List1}), which do not play the same central role in the MQDT calculation as in the ABM model, can be extracted from the underlying modified potentials for comparison. They show excellent agreement with the CC values and give a measure as to what degree the potentials from the CC calculation have been modified. This agreement and the small rms deviation of the FR positions from the experimental values, as given in Table~\ref{tab: List2} demonstrate, that MQDT and CC calculation are asymptotically consistent with regard to $\delta B^{rms}$ but for the description of individual resonances the two models deviate up to $\sim 100$~mG. This might also indicate the limit for predicting new FRs.
In the present study we test an alternative way to utilize the FT plus MQDT formulation; the idea is to empirically fit the single-channel singlet and triplet quantum defects so as to achieve optimum agreement with a few measured FRs. In this treatment, if the long-range van der Waals coefficient is already known to sufficient accuracy, as is believed to be the case for ${}^{6}\text{Li }$-${}^{133}\text{Cs }$, then with two fit parameters it is possible to achieve good agreement with all of the $s$-wave resonances that have been measured to date, and to predict additional resonances. The $l$-dependence of the fitted quantum defects is approximately known, but to achieve better accuracy on other partial waves, it appears to be necessary to fit one small additional correction for $p$-waves (see Eq.~\eqref{eq:defects}). While the MQDT has been shown in a number of studies to give a highly efficient way to calculate ultracold scattering observables when the interaction potentials are known, there is an increasing demand for a robust method for analyzing new, complex systems where FRs have been measured but not yet analyzed to the level of yielding a detailed microscopic model. The present test of the semi-empirical MQDT frame transformation (MQDT-FT) is encouraging in its potential for such problems, as is seen from the results presented below for the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ interaction.
In our implementation of the frame-transformed version of MQDT utilized for the present study, the long-range parameters ($A$, $\mathcal{G}$, $\gamma$ and $\eta$) are determined once and for all for a pure van der Waals potential at long-range, $-C_6/R^6$. The long-range MQDT parameters are standard and can be used for any alkali atom collision, because they are tabulated as functions of the single dimensionless variable which is the product of the van der Waals length and the wavenumber $k$ (see e.g. Ref.~\cite{Ruzic2013}). Two energy-independent and field-independent short-range quantum defects, namely $\mu^{sr}_{S,l}$ for $S=0,1$, were adjusted until optimum agreement was achieved with the experimental resonance positions. Note that a global search was not carried out over all values of the $0 \le \mu^{sr}_S < 1$ (mod 1). The starting values of the search came from quantum defects extracted from the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$~potentials provided in Ref. \cite{Repp2013} and only small adjustments of those values were needed in the MQDT-FT fit to achieve the quoted level of agreement with the experimental resonance locations. In contrast with the MQDT calculation and the full CC calculation, this MQDT-FT calculation did not include the magnetic dipole-dipole interaction nor the second-order spin-orbit interaction term. Nevertheless, the fit with three adjustable parameters (adding $\alpha_1$ of Eq.~\eqref{eq:defects}) gives a small rms deviation with experimental resonance positions in Table~\ref{tab: List2}, namely 47 mG. The fitted $p$-wave correction is $\alpha_1=0.00208$ and the $l=0$ quantum defects are $ \left \lbrace \mu^{sr}_{0},\mu^{sr}_{1} \right\rbrace=\left\lbrace 0.092115, 0.346848 \right\rbrace $. These can be used to derive the background scattering lengths of the singlet and triplet potential, which are given in Table~\ref{tab: List1}. Because the bound state energies are only expected to be accurate when the binding is quite small, no comparisons with bound levels obtained in the other methods are presented here. In Table~\ref{tab: List1} one sees that the largest deviations of the scattering lengths appear for the triplet state. Despite the very different qualities of the fit for the ABM and MQDT-FT approach, their results for the triplet scattering lengths are fairly close but deviate significantly from the result of CC and MQDT. Similarly, the fit quality of MQDT and MQDT-FT are comparable but the derived scattering lengths deviate strongly. No physical reason for this behavior is known at present.
\section{Conclusion}
\label{sec:Conclusion}
In this work we have applied three different models for the assignment of the measured FRs. All three models describe the observed resonances with a sub-1 G accuracy. However, depending on the desired degree of precision and the availability of accurate interaction potentials, the models serve different purposes.
In some cases a phenomenon under investigation requires highly accurate knowledge of scattering observables that are not measured but rather deduced from theory, as for example the scattering length in dependence of the magnetic field. In these cases it is inevitable to use the CC calculation. This demands either accurate \textit{ab initio} potentials or sufficient experimental data to construct such potentials. The high accuracy of the CC calculation stems from the fact that it incorporates the least amount of assumptions out of the three models. The rms deviation of 39 mG is the lowest of all three models. It is a rigorous and straightforward numerical approach, which comes at the cost of computational power and complex codes. The potentials for the CC calculations were mainly determined by spectroscopic data and here the data set for the triplet state is fairly sparse. The minimum is not well characterized because the vibrational levels from $v=0-4$ are not yet observed. Thus, predictions of FR for $^7$Li-${}^{133}\text{Cs }$ may be less accurate than for ${}^{6}\text{Li }$-${}^{133}\text{Cs }$. Measurements would be very desirable.
There are situations where the required precision of scattering observables is less stringent. One example is evaporative cooling of ultracold gases or sympathetic cooling of ultracold mixtures to quantum degeneracy by means of a FR, or for initial characterization of FRs. In these cases, the processes are often optimized experimentally, and it is not necessary to know the exact value of the scattering length for the start of the optimization. Under these circumstances the two other simplified models are much more appropriate.
With a rms deviation of 965 mG, the bare ABM explains the FR structure already on the level of $\sim$ 1 G. Only two parameters are sufficient for the description of the FR positions. The relatively large deviation is related to the fact that couplings to continuum states are neglected, which results in shifts on the order of the FR width of broad resonances. The fact that the rms deviation of $p$-wave resonances is only 560 mG, and can be further reduced to 375 mG when the spin-spin interaction is included, shows that narrow resonances are predicted sufficiently well. An advantage of the bare ABM is that no molecular information is required, as it builds solely on atomic constants and few fit parameters. Additionally, the code for the calculation is extremely simple since it only involves the numerical diagonalization of a small matrix, which is included in standard computational software programs. Therefore, it can be applied at low programming expense for all systems, in order to assign or predict FRs, or quickly map out all resonances of a system. In fact, it was used to estimate whether there are any FRs expected in experimentally achievable field regions for the ${}^{6}\text{Li }$-${}^{133}\text{Cs }$ ~system before the experiment was set up. Also, it can be used to optimize the starting conditions for a CC calculation \cite{Li2008}. We remark that the open channels for $M=5/2$ and $M=3/2$ have overlapping continua, $\left|1/2,-1/2\right\rangle \oplus \ $ $\left|3,+3\right\rangle$ with $\left|1/2,1/2\right\rangle \oplus \ $ $\left|3,+2\right\rangle$ or $\left|1/2,-1/2\right\rangle \oplus \ $ $\left|3,+2\right\rangle$ with $\left|3/2,-3/2\right\rangle \oplus \ $ $\left|3,+3\right\rangle$, respectively. Thus the two-state approach for the $S$-matrix might not be completely justified.
For the calculation of scattering properties and an accurate description of broad resonances, the dressed ABM has to be applied. With a rms deviation of 263 mG it is somewhat less accurate than the MQDT-FT. Yet, since it does not solve the Schr\"odinger equation numerically but rather utilizes an analytical expression of the $S$-matrix, it is still computationally straightforward. In cases where the background scattering lengths for the incoming channel are known, the implementation is simple, since a comparison with the van der Waals length directly shows whether the analytical expressions for bound or virtual state or --as is the case for ${}^{6}\text{Li }$-${}^{133}\text{Cs }$-- for an intermediate regime are appropriate. If this information is not available, we suggest first fitting the narrow resonances, in order to deduce background scattering lengths from the fitted bound state energies via the accumulated phase method, as explained in Sect.~\ref{sec:ABM}. Then the right choice of the analytical expression to be used will become evident.
In order to gain accurate information from the MQDT, the interaction potentials have to be known sufficiently well. However, compared to the CC calculation, it reduces the complexity of the problem enormously. As it is still a full scattering physics approach, employing a coupled-channel solution at short-range, the code is more complex and lengthy than the ABM code. Yet, once this code is available, it can be used for any alkali system without adaptation to the system specifics and solves the scattering problem efficiently. The accuracy of the final results with a rms deviation of 40 mG is statistically indistinguishable from that of the full CC calculation. Also, both yield smaller values for the FR positions of the broad resonances as compared to the observations. This deviation stems from the fact that for the experimental determination of the position, a Gaussian profile was fit to the loss spectrum, which neglects the asymmetric line shape of a broad resonance and therefore returns slightly larger values than the actual resonance positions. As a result, the current investigation does not indicate a model problem of CC and MQDT for the broad resonances.
Many of the above mentioned properties of the MQDT are also true for the MQDT-FT. The latter is especially useful for systems with little knowledge of interaction potentials and only a few experimentally measured FRs. A two parameter fit for only $s$-waves (three-parameters for $s+p$ etc.) allows to assign the resonances and to investigate the existence of possibly broader or for specific applications more appropriate FRs. While the rms deviation of 48 mG is comparable to MQDT and CC models, the predicted values of the bare singlet and triplet scattering lengths is less accurate. Because the variation of the quantum defects compensates for deviations introduced by the assumptions of the MQDT-FT (see Sect.~\ref{sec:MQDT}) in order to recreate the FR positions, the accuracy of other scattering properties should be tested in future studies. As it relies on only three parameters for the prediction or assignment of FR, it is appropriate in systems that are currently lacking accurate interaction potentials.
In conclusion, depending on the knowledge of molecular parameters, the required accuracy of the predicted scattering parameters, the complexity of code and the computational expense, each model has its own strength in applicability.
\section{Acknowledgments}
The work carried out at Colorado and at Purdue has been supported by the U.S. Department of Energy, Office of Science. The work carried out in Heidelberg was supported by the CQD. R.P. acknowledges support by the IMPRS-QD. J.U. acknowledges support by the DAAD.
\section{Appendix}
\begin{figure}[h]
\subfigure[$\Delta\approx-0.167$ G, and $a_{1,\text{bg}}^3(B)$ is constant.]{
\includegraphics[width=.99\columnwidth]{res_658.pdf}
\label{fig:polea}
}
\subfigure[$\Delta\approx-1.65$ $\mu$G, and $a_{1,\text{bg}}^3(B)$ is linear.]{
\includegraphics[width=.99\columnwidth]{res_773.pdf}
\label{fig:poleb}
}
\caption{(Color online) These graphs show the $l=1$ resonances near (a) 658.2 G and (b) 773.1 G. The (red) curves are the best fit form of Eq.\eqref{eq:pole}. The (blue) dots are the MQDT calculation of $a_1^3(B)$. The (black) dashed line is the function $a_{1,\text{bg}}^3(B)$.}
\end{figure}
In this Appendix we demonstrate the predictive power of the MQDT calculation by determining all the $s$- and $p$-wave FRs for the initial states measured experimentally, in the magnetic field range 0-1500 Gauss. For brevity we only describe $p$-wave FRs for a single value of incident $m_l$ for each incident spin state. We also describe how the theory extracts resonance widths.
Within MQDT finding and identifying FRs is straightforward. Approximate FR locations are quickly determined by searching for roots of det($K^\text{sr}_{QQ}+\cot\gamma$), and the eigenstate of $K^\text{sr}_{QQ}+\cot\gamma$ whose eigenvalue crosses zero near an FR identifies the quantum numbers of the resonant state. Table \ref{tab:key} reports FRs in the range of magnetic field $B=0-1500$ G, labeled by their incident spin state, $\ket{f_{\text{Li}}, m_{f_{\text{Li}}}, f_{\text{Cs}}, m_{f_{\text{Cs}}}}$, and resonant-state quantum numbers, $m_{f_{\text{Li}}}+m_{f_\text{Cs}}$, $l$, and $m_l$. The $m_l$ quantum number of each incident channel is easily inferred from Table \ref{tab:key} by conservation of total angular momentum, $m_\text{tot}^\text{inc}=m_\text{tot}^\text{res}$.
To characterize each FR in terms of a position $B_0$ and a width $\Delta$ in magnetic field, we calculate one of two quantities: the real part of the scattering length, $a_0^1$, or the real part of the scattering volume, $a_1^3$. We refer to these two quantities simultaneously as $a_l^{2l+1}$. MQDT quickly generates $a_l^{2l+1}$ on a fine grid in magnetic field, and we fit $a_l^{2l+1}(B)$ to one of three different functional forms described below.
For the majority of FRs in Table \ref{tab:key}, the resonant state is much more strongly coupled to the incident scattering channel than to any inelastic (exoergic) channel, and a clear pole emerges in $a_l^{2l+1}(B)$. In this case $a_l^{2l+1}(B)$ takes the conventional form,
\begin{equation}
\label{eq:pole}
a_l^{2l+1}(B)=a_{l,\text{bg}}^{2l+1}(B)\left(1-\frac{\Delta}{B-B_0}\right),
\end{equation}
where $\Delta$ and $B_0$ are constants. $\Delta$ is the field width, and $B_0$ is the resonance location.
When $|\Delta|$ is relatively large ($|\Delta|>0.1$ G), we let $a_{l,\text{bg}}^{2l+1}(B)$ be constant in $B$. This allows for an excellent fit of $a_l^{2l+1}(B)$. However, when $|\Delta|$ is relatively small ($|\Delta|<0.1$ G), we let $a_{l,\text{bg}}^{2l+1}(B)$ be linear in $B$ to achieve an equivalent fit. Figures \ref{fig:polea} and \ref{fig:poleb} demonstrate fits of resonances described by Eq.~\ref{eq:pole} when $|\Delta|>0.1$ and $|\Delta|<0.1$, respectively.
For several FRs in Table \ref{tab:key}, the resonant state is comparably coupled to both the incident channel and an inelastic channel, and the variation of $a_l^{2l+1}(B)$ becomes less drastic than for pure elastic scattering. In this case we fit $a_l^{2l+1}(B)$ to the form~\cite{Hutson2007},
\begin{equation}
\label{eq:inelastic}
a_l^{2l+1}(B)= a_{l,\text{bg}}^{2l+1}(B) + \frac{\alpha \big(2(B-B_0)/\Gamma\big) + \beta}{\big(2(B-B_0)/\Gamma\big)^2+1},
\end{equation}
where $\alpha$, $\beta$, $\Gamma$, and $B_0$ are constants. $\Gamma$ is the inelastic field width.
Since the variation in $a_l^{2l+1}(B)$ near these FRs can be very small, we fit these FRs by letting $a_{l,\text{bg}}^{2l+1}(B)$ be a high order (order=9) polynomial in $B$. This high order fit is appropriate as the coefficients decrease by orders of magnitude with successive powers of $B$, and the best fit $a_{l,\text{bg}}^{2l+1}(B)$ is not oscillatory in the vicinity of $B_0$. For example, Figure \ref{fig:inelastic} shows the fit of the resonance near 760.4 G.
\begin{figure}[h]
\includegraphics[width=.99\columnwidth]{res_760.pdf}
\caption{\label{fig:inelastic} (Color online) This graph shows the $l=1$ resonance near 760.4 G. The (red) curve is the best fit $a_1^3(B)$ from Eq.~\ref{eq:inelastic}. The (blue) dots are the MQDT calculation of $a_1^3(B)$. The (black) dashed line is the high order polynomial $a_{1,\text{bg}}^3(B)$. This fit excludes data around the narrow resonance near 773.1 G.}
\end{figure}
We calculate all FRs at 1 $\mu$K incident collision energy. $a_l^{2l+1}$ is approximately independent of energy on this ultralow energy scale,
\begin{equation}
a_l^{2l+1}\xrightarrow{k\rightarrow0}-\tan\delta_l/k^{2l+1},
\end{equation}
where the phase shift, $\delta_l$, obeys the Wigner threshold laws, $\delta_l\xrightarrow{k\rightarrow0}\propto k^{2l+1}$. We numerically determine that the simple threshold behavior of $\delta_l$ leads to energy independent field widths, $\Delta$ and $\Gamma$, for $l=0$ and $l=1$ resonances.
Table \ref{tab:key} summarizes the behavior of each FR in terms of the small set of parameters: $B_0$, $\Delta$, $a_{l,\text{bg}}^{2l+1}(B_0)$, and $\Gamma$. We use the parameter $\Delta$ in order to directly compare all FRs, regardless of their character. As fitting the FRs to the form of Eq.~\ref{eq:inelastic} does not determine $\Delta$, we suggest an approximate relation between $\Delta$ and $\Gamma$. By comparing the large $(B-B_0)/\Gamma$ limit of Eq.~\ref{eq:inelastic} to Eq.~\ref{eq:pole}, we obtain,
\begin{equation}
\Delta\approx -\frac{\alpha \Gamma/2}{a_{l,\text{bg}}^{2l+1}(B_0)}.
\end{equation}
Listing a value for $\Gamma$ in Table \ref{tab:key} indicates this approximation for $\Delta$.
Five of the resonances whose locations are identified by MQDT do not exhibit an appreciable variation in $a_{l}^{2l+1}$ with $B$. For these FRs Table \ref{tab:key} gives the predicted resonance location from the root of det($K^\text{sr}_{QQ}+\cot\gamma$) but does not report a width $\Delta$. Our method has found FRs with a $\Delta$ as small as $10^{-9}$ G; therefore, the uncharacterized resonances are most likely heavily suppressed by inelastic scattering or extremely narrow.
\begin{widetext}
\begin{table*}[t!]
\caption{\label{tab:key} This table characterizes FRs in $^6$Li + $^{133}$Cs obtained from the MQDT by reporting the incident spin state, $\ket{f_{\text{Li}}, m_{f_{\text{Li}}}, f_{\text{Cs}}, m_{f_{\text{Cs}}}}$; resonant state quantum numbers, $m_{f_{\text{Li}}}+m_{f_\text{Cs}}$, $l$, and $m_l$; resonance location, $B_0$; field width, $\Delta$; background value of $a_l^{2l+1}(B)$, $a_{l,\text{bg}}^{2l+1}(B_0)$ (in atomic units); and inelastic field width, $\Gamma$. The incident collision energy in each case is 1 $\mu$K. All magnetic field values are in units of gauss. When $l=0$ the background scattering length, $a_{0,\text{bg}}(B_0)$, has units of $a_0$. When $l=1$ the background scattering volume, $a_{1,\text{bg}}^3(B_0)$, has units of $a_0^3$.
}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
$\ket{f_{\text{Li}}, m_{f_{\text{Li}}}, f_{\text{Cs}}, m_{f_{\text{Cs}}}}$ &$m_{f_{\text{Li}}}+m_{f_\text{Cs}}$ & $l$ & $m_l$ & $B_0$ & $\Delta$ & $a_{l,\text{bg}}^{2l+1}(B_0)$ & $\Gamma$ \\
\hline
$\ket{1/2, 1/2, 3, 3}$ & 5/2 & 1 & 1 & 634.2 & $ -1.39 \times 10^{-4} $ & $ -6.89 \times 10^{4} $ & $ - $ \\
$ $ & 7/2 & 1 & 0 & 662.9 & $ -9.55 \times 10^{0} $ & $ -1.02 \times 10^{5} $ & $ - $ \\
$ $ & 5/2 & 1 & 1 & 682.3 & $ -3.98 \times 10^{-6} $ & $ -1.52 \times 10^{5} $ & $ - $ \\
$ $ & 9/2 & 1 & -1 & 690.6 & $ -2.50 \times 10^{-5} $ & $ -1.36 \times 10^{5} $ & $ - $ \\
$ $ & 7/2 & 1 & 0 & 713.7 & $ -5.92 \times 10^{-1} $ & $ -1.23 \times 10^{5} $ & $ - $ \\
$ $ & 5/2 & 1 & 1 & 737.6 & $ -2.04 \times 10^{-9} $ & $ -1.20 \times 10^{5} $ & $ - $ \\
$ $ & 7/2 & 0 & 0 & 843.1 & $ -6.56 \times 10^{1} $ & $ -2.64 \times 10^{1} $ & $ - $ \\
$ $ & 7/2 & 0 & 0 & 892.9 & $ -2.07 \times 10^{0} $ & $ -6.40 \times 10^{1} $ & $ - $ \\
$\ket{1/2, -1/2, 3, 3}$ & 3/2 & 1 & 1 & 632.5 & $ -2.01 \times 10^{-6} $ & $ -9.01 \times 10^{4} $ & $ - $ \\
$ $ & 5/2 & 1 & 0 & 658.2 & $ -1.67 \times 10^{-1} $ & $ -8.42 \times 10^{4} $ & $ - $ \\
$ $ & 3/2 & 1 & 1 & 676.0 & $ -9.57 \times 10^{-5} $ & $ -7.46 \times 10^{4} $ & $ - $ \\
$ $ & 7/2 & 1 & -1 & 687.4 & $ - $ & $ - $ & $ - $ \\
$ $ & 5/2 & 1 & 0 & 708.7 & $ -9.32 \times 10^{0} $ & $ -1.03 \times 10^{5} $ & $ - $ \\
$ $ & 3/2 & 1 & 1 & 728.8 & $ -3.21 \times 10^{-6} $ & $ -1.50 \times 10^{5} $ & $ - $ \\
$ $ & 7/2 & 1 & -1 & 740.9 & $ -1.54 \times 10^{-5} $ & $ -1.31 \times 10^{5} $ & $ 4.40 \times 10^{-1} $ \\
$ $ & 5/2 & 1 & 0 & 764.3 & $ -5.69 \times 10^{-1} $ & $ -1.21 \times 10^{5} $ & $ - $ \\
$ $ & 5/2 & 0 & 0 & 816.5 & $ -2.37 \times 10^{0} $ & $ -4.30 \times 10^{0} $ & $ - $ \\
$ $ & 5/2 & 0 & 0 & 888.9 & $ -6.37 \times 10^{1} $ & $ -2.70 \times 10^{1} $ & $ - $ \\
$ $ & 5/2 & 0 & 0 & 943.3 & $ -2.03 \times 10^{0} $ & $ -6.10 \times 10^{1} $ & $ - $ \\
$\ket{1/2, 1/2, 3, 2}$ & 5/2 & 1 & 1 & 704.5 & $ -1.79 \times 10^{1} $ & $ -9.94 \times 10^{4} $ & $ 1.70 \times 10^{-1} $ \\
$ $ & 7/2 & 1 & 0 & 734.6 & $ - $ & $ - $ & $ - $ \\
$ $ & 5/2 & 1 & 1 & 760.4 & $ -6.03 \times 10^{-1} $ & $ -1.33 \times 10^{5} $ & $ 1.20 \times 10^{1} $ \\
$ $ & 9/2 & 1 & -1 & 773.1 & $ -1.65 \times 10^{-6} $ & $ -1.32 \times 10^{5} $ & $ - $ \\
$ $ & 7/2 & 1 & 0 & 798.3 & $ -1.89 \times 10^{-6} $ & $ -1.22 \times 10^{5} $ & $ 8.67 \times 10^{-1} $ \\
$ $ & 5/2 & 1 & 1 & 824.7 & $ -1.39 \times 10^{-3} $ & $ -1.17 \times 10^{5} $ & $ 7.87 \times 10^{-1} $ \\
$ $ & 5/2 & 0 & 0 & 896.2 & $ -1.39 \times 10^{2} $ & $ -2.07 \times 10^{1} $ & $ 7.23 \times 10^{-1} $ \\
$ $ & 5/2 & 0 & 0 & 939.6 & $ -2.00 \times 10^{0} $ & $ -9.08 \times 10^{1} $ & $ 2.10 \times 10^{1} $ \\
$ $ & 5/2 & 0 & 0 & 1019.1 & $ -1.30 \times 10^{-3} $ & $ -5.03 \times 10^{1} $ & $ 7.55 \times 10^{-1} $ \\
$\ket{1/2, -1/2, 3, 2}$ & 3/2 & 1 & 1 & 694.8 & $ -3.90 \times 10^{-1} $ & $ -6.86 \times 10^{4} $ & $ - $ \\
$ $ & 5/2 & 1 & 0 & 728.5 & $ - $ & $ - $ & $ - $ \\
$ $ & 3/2 & 1 & 1 & 750.1 & $ -1.75 \times 10^{1} $ & $ -1.00 \times 10^{5} $ & $ 1.48 \times 10^{-1} $ \\
$ $ & 7/2 & 1 & -1 & 761.5 & $ - $ & $ - $ & $ - $ \\
$ $ & 5/2 & 1 & 0 & 784.8 & $ - $ & $ - $ & $ - $ \\
$ $ & 3/2 & 1 & 1 & 811.2 & $ -5.68 \times 10^{-1} $ & $ -1.30 \times 10^{5} $ & $ 1.28 \times 10^{1} $ \\
$ $ & 7/2 & 1 & -1 & 828.0 & $ -2.27 \times 10^{-6} $ & $ -1.28 \times 10^{5} $ & $ 8.42 \times 10^{-1} $ \\
$ $ & 5/2 & 1 & 0 & 853.8 & $ -1.33 \times 10^{-6} $ & $ -1.20 \times 10^{5} $ & $ 8.37 \times 10^{-1} $ \\
$ $ & 3/2 & 0 & 0 & 854.3 & $ 1.43 \times 10^{0} $ & $ 9.74 \times 10^{0} $ & $ - $ \\
$ $ & 3/2 & 0 & 0 & 941.6 & $ -1.33 \times 10^{2} $ & $ -2.17 \times 10^{1} $ & $ 6.01 \times 10^{-1} $ \\
$ $ & 3/2 & 0 & 0 & 989.9 & $ -1.89 \times 10^{0} $ & $ -8.42 \times 10^{1} $ & $ 2.12 \times 10^{1} $ \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{widetext}
|
1,314,259,996,699 | arxiv | \section{Introduction}
A great variety of algorithms and methods have been designed for various optimization problems. In classic \emph{Combinatorial Optimization}, the algorithm knows the complete input of the problem, and its goal is to produce an optimal or near optimal solution. However, in many modern applications the input of the problem is spread among a set of \emph{selfish agents}, where eachone owns a different part of the input as private knowledge. Hence, every agent is capable to manipulate the algorithm by miss-reporting its part of the input in order to maximize its personal payoff. In their seminal paper Nisan and Ronen \cite{NR99} were the first to study the impact of the ``strategic'' behavior of the agents on the difficulty of an optimization problem.
Since then \emph{Algorithmic Mechanism Design} studies optimization problems in presence of selfish agents with private knowledge of the input and potentially conflicting individual objective functions. The goal is to know whether it is possible to propose a \emph{truthful (or incentive compatible) mechanism}, i.e., an algorithm solving the optimization problem together with a set of incentives/payments for the agents motivating them to report honestly their part of the input.
As an illustrating example, consider the problem of finding the maximum of a set of values $v_1,v_2,\ldots,v_n$. In the classic setting, computing the maximum value is trivial. Let us consider now the case when the inputs are strategic. It means that
each of the $n$ selfish agents $i$, for $1\leq i\leq n$,
has a private value $v_i$ (not known to the algorithm) for
being selected (as the maximum), and may report any value $b_i$. If the agents know that the maximum value will be computed using the classic algorithm: $\max_i b_i$, then each agent will have an incentive
to cheat by declaring $+\infty$ instead of her true value.
In such a strategic setting, we need a mechanism that is capable to give incentive to the agents to report eachone their true value. For doing that we may use Vickrey's (also known as \emph{second-price}) mechanism \cite{Vickrey}. In this setting, the maximum $\max_i b_i$ is still
computed, but the agent $i^*$=arg max $b_i$ is charged the second
highest reported value $p^*:=\max_{j \neq i^*} b_j$. Her utility is
therefore $v_{i^*}-p^*$. It can be shown that in such a setting each
agent $i$ will have an incentive to report $b_i=v_i$, and henceforth
this mechanism is able to compute the maximum value in this strategic environment \cite{D16}.
In this paper, we consider the problem of \emph{maximum vectors}, i.e., the problem of finding the maxima of a set of vectors in a strategic environment. The classic problem of computing the maxima of a
set of vectors can be stated as follows: we are given a set $V$ of $n$ $k$-dimensional vectors $v_1,v_2,\ldots,v_n$ with $v_i=(v_i^1, v_i^2, \ldots, v_i^k)$ for $i=1,2,\ldots,n$. Given two vectors
$v_i=(v_i^1, v_i^2, \ldots, v_i^k)$ and
$v_j=(v_j^1, v_j^2, \ldots, v_j^k)$, we say that $v_i$ is {\em dominated}
by $v_j$ if $v_i^l \leq v_j^{l}$ for $1\leq l\leq k$,
with at least one strict inequality among the $k$ inequalities. The problem consists in computing $MAX(V)$, i.e. the set of all \emph{nondominated} vectors among the $n$ given vectors. This problem is related to other known problems as the \emph{Pareto curve computation} in multiobjective optimization \cite{Ehr00,PY,S86}, the \emph{skyline} problem in databases \cite{KRR,PTFS03}, or the \emph{contour} problem \cite{M74}.
In a ``strategic'' setting the problem is as follows:
there are $n$ selfish agents $1,2,\ldots,n$ and the value of
agent $i$ is described by a vector $v_i=(v_i^1, v_i^2, \ldots, v_i^k)$ for being \emph{selected}\footnote{An agent is \emph{selected} if its bid belongs to the set of nondominated vectors.}. The vector $v_i$ is a \emph{private information} known only by agent $i$. Computing the set of nondominated vectors by using one of the classic algorithms gives incentive to the agents to cheat by declaring $+\infty$ in all the coordinates of their vectors instead of their true values per coordinate. Our work explores under which conditions it is
possible to incentivize agents to report their true values. In order to
precisely answer this question, it is useful to distinguish two cases.
In the strongest case, the mechanism is able to enforce truthtelling
for each agent regardless of the reports of the others (\emph{truthfulness}).
In the second case, the mechanism is able to enforce truthtelling for each
agent assuming that the others report their true values
(\emph{equilibria truthfulness}).
\emph{Previous works}
The Artificial Intelligence (AI) community is faced with many real-world problems involving multiple, conflicting and noncommensurate objectives in path planning \cite{Donald,Khouadjia,Quemy}, game search \cite{Dasgupta}, preference-based configuration \cite{Benabbou}, ... Modeling such problems using a single scalar-valued criterion may be problematic (see for instance \cite {Zeleny}) and hence multiobjective approaches have been studied in the AI literature \cite{Hart,Mandow}.
Some multiobjective problems have been considered in the mechanism design framework. However, these works apply a budget approach where instead of computing the set of all Pareto solutions (or an approximation of this set), they consider that among the different criteria, one is the main criterion to be optimized while the others are modeled via budget constraints \cite{Bilo,Grandoni}.
Another family of related works concern \emph{auction theory}.
In the
classical setting, the item as well as the valuation of the bidders are characterized
by a scalar representing the price/value of the item. However, in many situations
an item is characterized, besides of its price, by quality measures, delivery times,
technical specifications etc. In such cases, the valuation of the bidders for the
item are vectors. Auctions where the item to sell or buy are characterized by a vector
are known as \emph{multi-attribute auctions} \cite{Bellosta,Bellosta2,Bichler,Bichler2,Bichler3,Branco,Che,Smet,desmet-new}.
In most of these works, a scoring rule is used for combining the values of the different attributes in order to determine the winner of the auction.
\emph{Our contribution} We first show that neither truthfulness nor equilibria truthfulness are achievable. However, if one assumes that the agents have distinct values in each of the dimensions, we show that it
is possible to design an equilibria truthful mechanism for the {\em Maximum Vector problem}. We also show that the payments that our algorithm computes are the only payments that give this guarantee. In order to go beyond the negative result concerning ties in the valuations of agents, we show that it is possible to get an equilibria truthful mechanism for the
{\em Weakly Maximum Vector problem} in which one looks for weakly nondominated vectors instead of nondominated ones \cite{Ehr00}.
\section{Problem definition}\label{sec-prbdef}
The following definition and notations will be useful in the sequel of the paper.
\begin{definition} Given two vectors $x,y \in \mathbb{R}_+^k$ we say that:
\begin{itemize}
\item $x$ {\em weakly dominates} $y$, denoted by $x \succeq y$, iff $x^j \ge y^j$ for all $j \in \{1, \ldots ,k\}$;
\item $x$ {\em dominates} $y$, denoted by $x \succ y$, iff $x \succeq y$ and $x^j > y^j$ holds for at least one coordinate $j \in \{1, \ldots ,k\}$;
\item $x$ {\em strongly dominates} $y$, denoted by $x \gg y$, iff
$x^j > y^j$ holds for all coordinates $j \in \{1, \ldots ,k\}$;
\item $x$ and $y$ are {\em incomparable}, denoted by $x \sim y$,
iff there exist two coordinates, say $j$ and $j'$, such that $x^j < y^j$ and $x^{j'} > y^{j'}$.
\end{itemize}
\end{definition}
We denote by $\mathbb{R}^k_{+}$ (resp. $\mathbb{R}^k_{*+}$)
the set of vectors $v\in \mathbb{R}^k$ such that $v\succ \vec{0}$ (resp.
$v\gg \vec{0}$), with $\vec{0}:=(0,\ldots ,0)$ the zero vector.
Given a set $F \subset \mathbb{R}^k_+$, as stated before, we denote by $MAX(F)$ the subset of all nondominated vectors, i.e. $MAX(F) := \{v\in F\: :\: \not\exists v_*\in F, v_*\succ v \}$.
Such a set is composed of pairwise incomparable vectors.
In a similar way, we will denote by $MIN(F)$ the set
$\{v\in F\: :\: \not\exists v_*\in F, v\succ v_* \}$. We will also consider the subset of all weakly nondominated vectors, i.e. $WMAX(F) := \{v\in F\: :\: \not\exists v_*\in F, v_*\gg v \}$.
The Maximum Vector problem has been studied in the classical framework, and the following proposition is known:
\begin{proposition}
(from \cite{KLP75})\label{lem-par} The set $MAX(F)$ can be computed in
$O(|F| \log |F|)$ time for $k=2,3$ and at most
$O(|F| (\log |F|)^{k-2})$ for $k\geq 4$.
\end{proposition}
Following the mechanism design framework~\cite{IntroMD}, we aim to design
a mechanism, that we call {\em Pareto mechanism}, such that no agent has an incentive to misreport its vector in order to increase her utility.
The set of agents is denoted by $N$.
Each agent $i$ has a private {\em vector} $v_i=(v_i^1, v_i^2, \ldots, v_i^k)$ representing the agent valuations on $k$ numerical criteria for being selected. In the following, we consider that $k$ is a fixed constant.
We denote by $V$ the set of private vectors.
Each agent $i$ reports a vector (a bid) $b_i=(b_i^1, b_i^2,\ldots, b_i^k)$. We denote by $B$ the set of all reported vectors. Based on the set of reported vectors, the mechanism computes for each agent $i$ a vector-payment $p_i=(p_i^1,p_i^2,\ldots ,p_i^k)$.
For each agent $i$, if $b_i$ belongs to $MAX(B)$ she has to pay $p_i$ and so her utility is $u_i := v_i-p_i$, while if $b_i$ does not belong to $MAX(B)$ her utility is $u_i=\vec{0}$ (zero vector).
Since no agent has an incentive to misreport her vector in order to increase her utility, we will be able to correctly compute $MAX(V)$ by computing $MAX(B)$ since we will have $MAX(B)=MAX(V)$.
If we consider $WMAX$ instead of $MAX$ we use the term of a weakly Pareto mechanism.
\section{Preliminaries}
The Pareto mechanism we want to design must satisfy several properties.
\begin{definition}[multiobjective individual rationality]
A Pareto mechanism satisfies the multiobjective individual rationality (MIR)
constraint iff $u_i \succeq \vec{0}$ for all agents $i$.
\end{definition}
By the MIR constraint, it is always better for an agent to participate in
the mechanism (i.e. reports a vector) than not participating. In the following we will always assume that the mechanism satisfies the MIR constraint.
We want that the Pareto mechanism incentivize agents to report their true values. This leads to the two following formal definitions.
\begin{definition}[multiobjective truthfulness]
For any fixed set of reported vectors $b_{i'}$, $i'\neq i$, let $u_i$ be agent $i$'s utility if she reports $b_i=v_i$ and let $u'_i$ denotes her
utility if she reports $b_i\neq v_i$ (the reported vectors of all the
other agents remaining unchanged). A Pareto mechanism is said to be
multiobjective {\em truthful} iff
$u_i \succeq u'_i$ or $u_i \sim u'_i$ for any agent $i \in
\{1, \ldots, n\}$.
\end{definition}
\begin{definition}[multiobjective equilibria truthfulness]
As in the previous definition, let $u_i$ be agent $i$'s utility if $b_i=v_i$ and let
$u'_i$ denotes her utility if $b_i\neq v_i$. A Pareto mechanism is said to be
multiobjective {\em equilibria truthful} iff
$u_i \succeq u'_i$ or $u_i \sim u'_i$ for any agent $i \in
\{1, \ldots, n\}$, assuming that $b_{i'}=v_{i'}$ for all $i'\neq i$.
\end{definition}
Honestly reporting her valuation is a dominant strategy for any agent if the mechanism is truthful.
We will also need some additional definitions in the context of multicriteria
optimization, along with some technical lemmas. The missing proofs can be found in
the Appendix Section. In the sequel, all sets $S$ have a finite size.
\begin{definition}\label{def-t1}
Let $S\subset \mathbb{R}^k_{*+}$ be a finite set of $k$-dimensional vectors.
We define the {\em reference points}\footnote{This set is known in multiobjective optimization as the set of \emph{local upper bounds} \cite{Vander}.} of $S$, denoted by ${\cal T}(S)$,
as the minimum subset of $\mathbb{R}^k_{+}$ such that
for any $v\in \mathbb{R}^k_{*+}$ with $v\not\in MAX(S)$,
one has $v\in MAX(S\cup \{v\})$ iff
$\exists t\in {\cal T}(S)$ such that $v\gg t$.
\end{definition}
Such a set can be easily computed in dimension 2. For $k=2$, an example is depicted in
Figure~\ref{fig-refpoints}.
Let $S\subset \mathbb{R}^2_{*+}$. By Proposition~\ref{lem-par}, we compute $MAX(S) = \{s_1,\ldots ,s_r\}$, where the solutions
$s_i$, $1\leq i\leq r$, are pairwise incomparable.
Without loss of generality we assume that
$s_1^1 < s_2^1 < \ldots < s_r^1$ and $s_1^2 > s_2^2 > \ldots > s_r^2$.
Then one has ${\cal T}(S)=\{t_1,\ldots t_{r+1}\}$
with $t_1 = (0,s_1^2)$, $t_{r+1} = (s_r^1,0)$
and $t_l=(s_{l-1}^1,s_l^2)$ for $2\leq l\leq r$.
The overall complexity to compute ${\cal T}(S)$ is therefore $O(|S| \log |S|)$ in dimension 2.
The existence and uniqueness of such a set for any dimension follows from
Proposition~\ref{prop-ref}.
Let $D_S^j:=\{0\} \cup \{s^j \, : \, s\in S\}$ for $j=1,\ldots ,k$,
and $\Omega_S := D_S^1 \times D_S^2 \times \cdots \times D_S^k$.
\begin{proposition} \label{prop-ref}
For any finite set $S\subset \mathbb{R}^k_{*+}$,
one has ${\cal T}(S) = MIN( \{t \in \Omega_S \, : \,
\forall s \in S, \; s \not \gg t \}).$
\end{proposition}
Notice that $|\Omega_S| \leq (|S|+1)^k$ and by using Proposition~\ref{lem-par}
we obtain that for any $S\subset \mathbb{R}^k_{*+}$ its set of reference
points ${\cal T}(S)$ can be computed in polynomial time with respect to
$|S|$ ($k$ is assumed to be a constant).
For example, with $k=3$ and
$S=\{(2,2,2),(1,3,3),(3,1,1)\}$, using Proposition~\ref{prop-ref} one
obtains:
${\cal T}(S)=\{(3,0,0),(2,1,0),(2,0,1),(1,2,0),(1,0,2),(0,3,0),(0,0,3)\}.$
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=.5]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw (7,0) node[below right] {};
\draw[->] (0,0) -- (0,7);
\draw (0,8) node[above] {};
\draw[step=1cm,gray,thin,dotted] (0,0) grid (7,6);
\draw (1,5) node[sommet]{};
\node[above right] at (1,5) {$s_1$};
\draw (3,4) node[sommet]{};
\node[above right] at (3,4) {$s_2$};
\draw (4,2) node[sommet]{};
\node[above right] at (4,2) {$s_3$};
\draw (6,1) node[sommet]{};
\node[above right] at (6,1) {$s_4$};
\draw (0,5) node[rect]{};
\node[above right] at (0,5) {$t_1$};
\draw (1,4) node[rect]{};
\node[above right] at (1,4) {$t_2$};
\draw (3,2) node[rect]{};
\node[above right] at (3,2) {$t_3$};
\draw (4,1) node[rect]{};
\node[above right] at (4,1) {$t_4$};
\draw (6,0) node[rect]{};
\node[above right] at (6,0) {$t_5$};
\end{tikzpicture}
\caption{\label{fig-refpoints}The reference points in dimension 2. One has
$S=\{s_1,s_2,s_3,s_4\}$ and ${\cal T}(S)=\{t_1,t_2,t_3,t_4,t_5\}$.}
\end{center}
\end{figure}
\section{Impossibility results}
Because of the following Proposition, achieving an equilibria truthful Pareto
mechanism is the best we can hope for.
\begin{proposition}\label{not-truth}A Pareto mechanism cannot be truthful.
\end{proposition}
\begin{proof}
Let us consider an instance in two dimensions, with three agents reporting $b_1=(3,1)$,
$b_2=(1,3)$ and $b_3=(2,2)$. Then $b_3\in MAX(B)$, and agent 3 is charged
some payment $p_3\preceq (2,2)$ by the MIR assumption.
Let us define the following region of payments ${\cal R} = ([1,2]\times [1,2])
\setminus \{(1,\alpha)\cup (\alpha,1)\}$ for $1\leq \alpha\leq 2$, depicted
in Figure~\ref{fig-just}.
We claim that we can not have $p_3\in {\cal R}$.
Indeed if it was the case, then if $v_3=(2,2)$ agent 3's interest would
be to lie and report $b_3=((1+p_3^1)/2,(1+p_3^2)/2)$. She would still be in $MAX(B)$ and get charged $p'_3\preceq b_3\prec p_3$.\\
Now, since $p_3\preceq (2,2)$ and $p_3\not\in {\cal R}$ it means that
either $p_3\preceq (1,2)$ (case 1) or $p_3\preceq (2,1)$ (case 2).
In the first case, if $v_3=(1,2.5)$ then agent 3's interest would
be to lie and report $b_3=(2,2)$. She would belong to $MAX(B)$ and
her utility would be $(1,2.5)-p_3\succ (0,0)$, whereas if she reports $b_3=v_3$
she would not belong to $MAX(B)$ and therefore gets a utility $(0,0)$.
In the second case, in a similar way if $v_3=(2.5,1)$ then agent 3's interest
would be to lie and report $b_3=(2,2)$.
\end{proof}
\begin{definition}
An instance satisfies the {\em DV} property (distinct values) if
for every couple of distinct agents $i$, $i' \in N$, and
every $j \in \{1,\ldots ,k\}$, $v_i^j \neq v_{i'}^j$ holds.
\end{definition}
Let us motivate the introduction of this property.
\begin{proposition}\label{dvprop} Without the DV property, a Pareto mechanism cannot be
equilibria-truthful.
\end{proposition}
\begin{proof}
The proof is very similar to the one for Proposition~\ref{not-truth}.
We only need to assume that agents 1 and 2 are reporting their true vectors, i.e.
$b_1=v_1=(3,1)$ and $b_2=v_2=(1,3)$, and notice that the DV assumption does
not hold in cases 1 and 2.
\end{proof}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=.95]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (4,0);
\draw (6,0) node[below right] {};
\draw[->] (0,0) -- (0,4);
\draw (0,6) node[above] {};
\draw[step=1cm,gray,thin,dotted] (0,0) grid (3,3);
\draw (3,1) node[sommet]{};
\node[above right] at (3,1) {$b_1$};
\draw (1,3) node[sommet]{};
\node[above right] at (1,3) {$b_2$};
\node[above right] at (2,2) {$b_3$};
\draw (2,2) node[sommet] (B) {};
\node[left] at (1,2.5) {$v_3$};
\node[below] at (2.5,1) {$v_3$};
\draw (1,2.5) node[sommet] (A1) {};
\draw (2.5,1) node[sommet] (A2) {};
\draw [->] (A1) to [bend left] node[midway,above,scale=.7]{case 1} (B);
\draw [->] (A2) to [bend right] node[midway,right,scale=.7]{case 2} (B);
\draw[dashed,red,pattern=north east lines,pattern color=red] (1,1) --
(2,1) -- (2,2) -- (1,2) -- (1,1);
\draw[color=red] (1,2) -- (2,2);
\draw[color=red] (2,1) -- (2,2);
\node[below] at (1,0) {$1$}; \node[below] at (2,0) {$2$};
\node[below] at (3,0) {$3$};
\node[left] at (0,1) {$1$}; \node[left] at (0,2) {$2$};
\node[left] at (0,3) {$3$};
\end{tikzpicture}
\caption{\small \label{fig-just}Illustration of the proof of Proposition~\ref{not-truth}.}
\end{center}
\end{figure}
\section{A Pareto mechanism for the Maximum Vector problem}
We are going to present a Pareto mechanism, denoted by $\cal M$, which
satisfies the MIR constraint and which is equilibria-truthful under the hypothesis DV.
The mechanism is described in Algorithm~\ref{figpmeca}. In the initial step,
we remove all identical vectors. This means that if there is a set of
agents with the same reported vector $b$, this vector is removed from the set $B$ and
all such agents will not be considered anymore in the mechanism.
Notice that this case will not occur, since we are in the context of
a equilibria truthfulm mechanism and we have the DV assumption. Not having
two identical vectors is a formal requirement used in the proof of
Lemma~\ref{lem-prop1}.
The mechanism computes for all agents $i$ such that $b_i\in MAX(B)$ a set of possible payments, denoted by $PAY(i)$, and can charge agent $i$ any payment from this set.
\begin{algorithm}
\begin{algorithmic}[1]
\STATE Remove all identical vectors and corresponding agents.
\STATE Compute $MAX(B)$.
\STATE For all $b_i\in MAX(B)$, set $PAY(i) := \{t \in
{\cal T}(B\setminus \{b_i\}) \: |\: b_i \succeq t\}$, and choose any
$p_i\in PAY(i)$.
\STATE For all $b_i\notin MAX(B)$, we set $p_i=\vec{0}$.
\caption{\label{figpmeca}The Pareto mechanism ${\cal M}$.}
\end{algorithmic}
\end{algorithm}
\begin{lemma} \label{lem-prop1}
For any agent $i$ such that $b_i\in MAX(B)$, one has $PAY(i)\neq \emptyset$.
\end{lemma}
\begin{proof}
We need the following notations.
Let $D_B^j:=\{0\} \cup \{b^j \, : \, b\in B\}$ for $j=1,\ldots ,k$.
Given $a \in \mathbb{R}_{*+}$, let us denote by
$dec_{B,j}(a)$ the quantity $\max\{x \in D_B^j \, : \, x<a\}$.\\
Let us consider a vector $r =(dec_{B,1}(b_i^1),\ldots,dec_{B,k}(b_i^k))$.
If there exists an agent $z\neq i$ such that $b_z \gg r$ then
$b_z \succ b_i$ (since no two reported vectors are identical), which is a
contradiction with $b_i \in MAX(B)$.
Therefore $r \in \{t \in \Omega_{B\setminus\{b_i\}} \, : \, \forall s \in B
\setminus \{b_i\}, \, s \not \gg t\}$.
By definition of the $MIN$ operator, there exists
$w \in MIN(\{t \in \Omega_{B\setminus\{b_i\}} \, : \, \forall s \in B
\setminus \{b_i\}, \, s \not \gg t\}) =
{\cal T}(B \setminus \{b_i\})$ such that $w \preceq r$.
Since $r \prec b_i$ (by construction), we get $w \prec b_i$.
Therefore $w \in PAY(i)$ and $PAY(i)\neq \emptyset$.
\end{proof}
\begin{theorem} The Pareto mechanism ${\cal M}$ satisfies the MIR constraint.
\label{theo-mir}
\end{theorem}
\begin{proof}
Given an agent $i$ such that $b_i\not\in MAX(B)$ her utility $u_i$ is the zero vector $\vec{0}$ by definition.
Now given an agent $i$ such that $b_i\in MAX(B)$, we have to prove that $b_i \succeq p_i$ for all
$p_i \in PAY(i)$. By Lemma~\ref{lem-prop1}, $PAY(i)$ is nonempty, and according
to the mechanism $\cal M$, $PAY(i)$ contains vectors dominated by $b_i$, thus
the property holds.
\end{proof}
In what follows, we use the following standard notation in game theory.
Given a set of reported vectors $B := \{b_j\:|\:j\in N\}$, we denote by
$(b_{-i},v_i)$ the set in which each agent $j\in N\setminus \{i\}$
reports $b_j$, excepted the agent $i$ which reports $v_i$ instead, and we denote by
$(b_{-i},b_i)$ the set in which each agent $j\in N$ reports $b_j$
including the agent $i$ which reports $b_i$.
Before proving Theorem~\ref{theo-eqt} we need the following two lemmas:
\begin{lemma}\label{lem-la}
Let $S\subset \mathbb{R}^k_{*+}$ be a finite set.
Then, $\forall t\in {\cal T}(S)$ and $\forall j\in \{1,\ldots k\}$, then
$t^j=0$ or $\exists s\in S$ such that $t^j=s^j$.
\end{lemma}
\begin{lemma}\label{lem-lt}
Let $S\subset \mathbb{R}^k_{*+}$ be a finite set.
Then, ${\cal T}(S)$ is composed of mutually noncomparable vectors, i.e.
$\forall t,t'\in {\cal T}(S)$, one has $t=t'$ or $t\sim t'$.
\end{lemma}
\begin{theorem}\label{theo-eqt}The Pareto mechanism ${\cal M}$ is equilibria-truthful.
\end{theorem}
\begin{proof}
Let $u_i$ be agent $i$'s utility if $b_i=v_i$ and let $u'_i$ denotes her
utility if $b_i\neq v_i$.
We need to show that $u_i \succeq u'_i$ or $u_i \sim u'_i$.
Recall that we always assume that $b_i \gg \vec{0}$ and $v_i \gg \vec{0}$.\\
We have two cases to consider. First, let assume that
$v_i\not\in MAX(b_{-i}, v_i)$.
The utility $u_i$ of agent $i$ is $\vec{0}$. In that case, agent $i$ may have an
incentive to report a vector $b_i\neq v_i$ such that $b_i\in MAX(b_{-i},b_i)$.
According to the mechanism $\cal M$, agent $i$ will be charged some
$t\in {\cal T}(B\setminus \{b_i\})$.
Since $v_i\not\in MAX(b_{-i}, v_i)$ we get from
Definition~\ref{def-t1} with $S=B\setminus \{b_i\}$ and $v=v_i$
that $v_i\not\gg t$. From the DV (distinct values) hypothesis
and Lemma~\ref{lem-la} and using that $v_i\gg \vec{0}$, we can conclude
that either $v_i\sim t$ t or $t\gg v_i$. Indeed, if it was not the case, then
$v_i\not\gg t$, $v_i\not\sim t$ t and $t\not\gg v_i$ implies that
$\exists j$ such that $v_i^j = t^j$ and by Lemma~\ref{lem-la} we know that
either $t^j=0$ or $\exists s\in B\setminus \{b_i\}$ such that $t^j=s^j$, meaning that
either $v_i^j=0$ or $v_i^j=s^j$ with $s$ the reported vector of a agent different from $i$.
But recall that we have assumed that $v_i \gg \vec{0}$ and moreover since the other agents
report their true values and by the DV hypothesis this is not possible.
This case is illustrated in Figure~\ref{fig-eqt-c1}.
Therefore, the utility
$u'_i = v_i - t$ of agent $i$ will satisfy $\vec{0}\sim u'_i$ or $\vec{0}\gg u'_i$.\\
Assume now that $v_i\in MAX(b_{-i}, v_i)$. According to the mechanism
${\cal M}$, if agent $i$ declares her true value, she will be charged some
$t$ such that $t\in {\cal T}(B\setminus \{b_i\})$. Her utility $u_i$ will
be $v_i-t$.
If agent $i$ reports $b_i$ such that $b_i\in MAX(b_{-i},b_i)$, then
she will be charged $t'$ for some
$t'\in {\cal T}(B\setminus \{b_i\})$ and her utility $u'_i$ will be
$v_i - t'$. Since by Lemma~\ref{lem-lt} one has $t=t'$ or $t\sim t'$, the
utilitie will satisfy $u'_i=u_i$ or $u'_i\sim u_i$.
This case is illustrated in Figure~\ref{fig-eqt-c2}.
Finally if agent $i$ reports $b_i$ such that $b_i\not\in MAX(b_{-i},b_i)$,
her utility will be zero, i.e. $u'_i=\vec{0}$ whereas $u_i\succeq \vec{0}$ according to
Theorem~\ref{theo-mir}.
\begin{figure}[t]
\begin{center}
\subfloat[\label{fig-eqt-c1}First Case - Here $t$ can be either $t_2$, $t_3$
or $t_4$. The situation corresponding to the dashed line for which one has
neither $v_i\sim t_2$ nor $t_2\gg v_i$ is not possible.]{
\begin{tikzpicture}[scale=.65, every node/.style={transform shape}]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw[->] (0,0) -- (0,8);
\draw[step=1cm,gray,thin,dotted] (0,0) grid (8,8);
\draw (1,7) node[sommet]{}; \node[above right] at (1,7) {$b_1$};
\draw (3,5) node[sommet]{}; \node[above right] at (3,5) {$b_2$};
\draw (5,3) node[sommet]{}; \node[above right] at (5,3) {$b_3$};
\draw (7,1) node[sommet]{}; \node[above right] at (7,1) {$b_4$};
\draw (6,6) node[sommet] (B) {}; \node[above right] at (6,6) {$b_i$};
\draw[dashed] (0,6) -- (6,6); \draw[dashed] (6,0) -- (6,6);
\draw (0,7) node[rect]{}; \node[below left] at (0,7) {$t_1$};
\draw (1,5) node[rect]{}; \node[below left] at (1,5) {$t_2$};
\draw (3,3) node[rect]{}; \node[below left] at (3,3) {$t_3$};
\draw (5,1) node[rect]{}; \node[below left] at (5,1) {$t_4$};
\draw (7,0) node[rect]{}; \node[below left] at (7,0) {$t_5$};
\draw (0,7) -- (1,7) -- (1,5) -- (3,5) -- (3,3) -- (5,3) -- (5,1) -- (7,1) -- (7,0);
\draw (4.4,2.4) node[sommet] (A) {}; \node[left] at (4.4,2.4) {$v_i$};
\draw (2,2) node[sommet] (A2) {}; \node[left] at (2,2) {$v_i$};
\draw (1,3.5) node[sommet] (A3) {}; \node[left] at (1,3.5) {$v_i$};
\draw[->] (A) to [bend left] (B);
\draw[->] (A2) to [bend left] (B);
\draw[dashed,->] (A3) to [bend left] (B);
\end{tikzpicture}
} \ \ \ \
\subfloat[\label{fig-eqt-c2}Second Case - The various possible payments
according to the vector the agent $i$ declares are indicated with a dashed
line.]{
\begin{tikzpicture}[scale=.65, every node/.style={transform shape}]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw[->] (0,0) -- (0,8);
\draw[step=1cm,gray,thin,dotted] (0,0) grid (8,8);
\draw (1,7) node[sommet]{}; \node[above right] at (1,7) {$b_1$};
\draw (3,5) node[sommet]{}; \node[above right] at (3,5) {$b_2$};
\draw (5,3) node[sommet]{}; \node[above right] at (5,3) {$b_3$};
\draw (7,1) node[sommet]{}; \node[above right] at (7,1) {$b_4$};
\draw (0,7) node[rect]{}; \node[below left] at (0,7) {$t_1$};
\draw (1,5) node[rect] (T2) {}; \node[below left] at (1,5) {$t_2$};
\draw (3,3) node[rect] (T3) {}; \node[below left] at (3,3) {$t_3$};
\draw (5,1) node[rect] (T4) {}; \node[below left] at (5,1) {$t_4$};
\draw (7,0) node[rect]{}; \node[below left] at (7,0) {$t_5$};
\draw (0,7) -- (1,7) -- (1,5) -- (3,5) -- (3,3) -- (5,3) -- (5,1) -- (7,1) -- (7,0);
\draw (4,4) node[sommet] (B0) {}; \node[above right] at (4,4) {$v_i$};
\draw (2,6) node[sommet] (B1) {}; \node[above right] at (2,6) {$b_i$};
\draw (6,6) node[sommet] (B2) {}; \node[above right] at (6,6) {$b_i$};
\draw (6,2) node[sommet] (B3) {}; \node[above right] at (6,2) {$b_i$};
\draw[->] (B0) to [bend left] (B1);
\draw[->] (B0) to [bend right] (B2);
\draw[->] (B0) to [bend right] (B3);
\draw[red,dashed,->] (B1) to [bend right] (T2);
\draw[red,dashed,->] (B3) to [bend right] (T4);
\draw[red,dashed,->] (B0) to [bend right] (T3);
\draw[red,dashed,->] (B2) to [bend right] (T2);
\draw[red,dashed,->] (B2) to [bend right] (T3);
\draw[red,dashed,->] (B2) to [bend right] (T4);
\end{tikzpicture}
}
\caption{\small \label{fig-eqt}Illustration of the proof of Theorem~\ref{theo-eqt}.}
\end{center}
\end{figure}
\end{proof}
We are now going to prove that the Pareto mechanism we introduced is the
unique way of achieving equilibria truthfulness.
Let $\pi$ be a truthful payment function, i.e. given a set of reported vectors $B$, for any agent $i$, $\pi_i(B)$ is the amount charged to the agent $i$, such that no agent has an incentive to declare a false
vector.
Recall that by the MIR constraint, we assume that
\begin{eqnarray}
b_i \succeq \pi_i(b_{-i},b_i) \mbox{ for any vector } b_i. \label{ineg1}
\end{eqnarray}
\begin{lemma}\label{lem1}
For any two different reported vectors
$b_i$ and $b'_i$, such that $b_i\in MAX(b_{-i}, b_i)$ and
$b'_i\in MAX(b_{-i}, b'_i)$, either
$\pi_i(b_{-i},b_i)\sim \pi_i(b_{-i},b'_i)$ or
$\pi_i(b_{-i},b_i) = \pi_i(b_{-i},b'_i)$.
\end{lemma}
\begin{proof}
Let us assume that $\pi_i(b_{-i},b'_i)\succ \pi_i(b_{-i},b_i)$. Then
if $v_i=b'_i$, agent $i$ would have an incentive to report $b_i$ instead
of her true value $v_i$.
The same line of reasoning shows that one cannot have
$\pi(b_{-i},b_i) \succ \pi(b_{-i},b'_i)$ neither. \\$\Box$
\end{proof}
We need additional lemmas.
\begin{lemma}\label{lem-t01}
Let $S\subset \mathbb{R}^k_{*+}$ be a finite set.
Then, $\forall s\in MAX(S)$, $\exists t\in {\cal T}(S)$ such that
$s \succ t$.
\end{lemma}
\begin{lemma}\label{lem-t5}Let $x,y,z \in \mathbb{R}^k_+$ be three
$k$-dimensional vectors, such that $y\succ x$ and $y\sim z$.
Then, $x\not\succeq z$
\end{lemma}
\begin{proof}
The proof is by contradiction. Assume that $x\succeq z$. Since
$y\sim z$, one can assume without loss of generality that
$y^1 > z^1$ and $y^2 < z^2$.
Since $y\succ x$, one has $y^2 \geq x^2$. Therefore, $z^2 > x^2$, which is
in contradiction with $x\succeq z$.
\end{proof}
\begin{lemma}\label{lem-t4}Let $S\subset \mathbb{R}^k_+$ be a set of
$k$-dimensional vectors, mutually non comparable, i.e.
$\forall s,s'\in S$, one has $s\sim s'$.
Let $p\in \mathbb{R}^k_+$ such that $s^*\succ p$, for some $s^*\in S$.
Then $\forall s\in S$, $s\succ p$ or $s\sim p$.
\end{lemma}
\begin{proof}
Let us consider any $s\in S$ with $s\neq s^*$. We know that $s\sim s^*$,
and by using Lemma~\ref{lem-t5} with $x=p$, $y=s^*$ and $z=s$, we obtain
that $p\not\succeq s$, i.e. $s\gg p$ or $s\sim p$.
\end{proof}
\begin{lemma}\label{lem-t2}Let $S\subset \mathbb{R}^k_+$ be a set of
$k$-dimensional vectors, and let $p\in \mathbb{R}^k_+$ such that
$\forall s\in S$, $p\sim s$. Then, there exist $\delta \in \mathbb{R}^k_{*+}$
such that $\forall s\in S$, $p+\delta \sim s$.
\end{lemma}
\begin{proof}
Let $S=\{s_1,\ldots ,s_{|S|}\}$. Since $p\sim s$, $\forall s\in S$, it means
that $\forall s_i\in S$, $\exists l_i,l'_i \in \{1,\ldots ,k\}$ such that
$s_i^{l_i} > p^{l_i}$ and $s_i^{l'_i} < p^{l'_i}$.
Let $I_j = \{ i \: | \: l_i = j \}$.
For all $j\in \{1,\ldots ,k\}$, let
$\varepsilon^j = \min_{i\in I_j} (s_i^{l_i} - p^{l_i})$ if
$I_j \neq \emptyset$, and let $\varepsilon^j$ be any positive real number
otherwise.
Then it is easy to see that the vector
$\delta=(\varepsilon^1/2,\ldots ,\varepsilon^k/2)$ satisfies the lemma.
Indeed, observe that $\varepsilon^j>0$ for $1\leq j\leq k$.
Now let us consider any $s_r\in S$ with $1\leq r\leq |S|$.
One has $s_r^{l_r} > p^{l_r}$ and $s_r^{l'_r} < p^{l'_r}$.
Therefore $s_r^{l'_r} < p^{l'_r} + \varepsilon^{l'_r}/2 = (p+\delta)^{l'_r}$.
We are going to show that $s_r^{l_r} > (p+\delta)^{l_r}$.
One has $r\in I_{l_r}$, hence $\varepsilon_{l_r} \leq s_r^{l_r}-p^{l_r}$, and
$\delta^{l_r}=\varepsilon_{l_r}/2 < s_r^{l_r}-p^{l_r}$, which means that
$s_r^{l_r} > (p+\delta)^{l_r}$.
\end{proof}
\begin{lemma}\label{lem-t3}Let $S\subset \mathbb{R}^k_+$ a set of
$k$-dimensional vectors, and let $p\in \mathbb{R}^k_+$ such that
one has $\forall s\in S_1$, $s\sim p$, and one has
$\forall s\in S_2$, $s\succ p$, with $S_1$ and $S_2$ a bipartition of $S$,
i.e. $S_1\cup S_2=S$ and $S_1\cap S_2=\emptyset$.
Then, there exist $\delta\in \mathbb{R}^k_{*+}$ such that
$\forall s\in S$, $s\sim p+\delta$ or $s\succ p+\delta$.
\end{lemma}
\begin{proof}
This result can be considered as a generalization of Lemma~\ref{lem-t2}
and the proof is quite similar.
For the set $S_1$ we define $\epsilon^j$ as previously. For the set
$S_2$, we define $\varepsilon'^j = \min_{\{s\in S_2\, : \, s^j-p^j>0\}} s^j-p^j$
if the set $\{s\in S_2\, : \, s^j-p^j>0\}$ is not empty, otherwise
$\varepsilon'^j$ is any positive value.
Then it is easy to see that the vector
$\delta$ with $\delta^j = \min \{\varepsilon^j, \varepsilon'^j\}/2$,
for $1\leq j\leq k$ satisfies the lemma.
For example, let us consider any $s\in S_2$.
Since $s\succ p$, $\exists l$, $1\leq l\leq k$, such that $s^l>p^l$.
Then $s^l \geq p^l+\varepsilon'^l > p^l + \delta^l$.
It implies that $s\sim p+\delta$ or $s\succ p+\delta$.
If $s\in S_1$ the proof is the same than Lemma~\ref{lem-t2}.
\end{proof}
\begin{lemma}\label{lem-infinity}
Let $z_j$ and $u_j$, for $j\geq 1$, be two infinite sequences of points in
$\mathbb{R}^k_{+}$ such that $z_j \gg z_{j+1}$, $z_j \succeq u_j\succ t$, and
$\lim_{j\to \infty} z_j = t$. Then $\exists l,l'$ such that $u_l \succ u_{l'}$.
\end{lemma}
\begin{proof}
First, we assume there exist a point $u_i$ such that $u_i\gg t$.
Let $\delta=\min_{1\leq j\leq k} u_i^j - t >0$.
Since $\lim_{j\to \infty} z_j = t$, $\exists N$ such that $\forall i\geq N$
one has $||z_i-t||_{\infty} = \max_{1\leq j\leq k} z_i^j-t^j \leq \delta/2$.
We have $u_i^j \geq t+\delta > t+\delta/2 \geq z_N^j \geq u_N^j$, meaning
that $u_i \succ u_N$.\\
Now assume that $\forall i$, $u_i\not\gg t$.
Since $u_i\succ t$, it means that $\exists \alpha_i$ such that
$u_i^{\alpha_i}=t^{\alpha_i}$. Without loss of generality by considering
an (infinite) subsequence of $u_i$ we can assume that $\forall i$, $\alpha_i=1$.
Again by considering an (infinite) subsequence we can assume that
$\exists K$ with $1\leq K\leq k-1$ such that
$\forall j$, $1\leq j\leq K$, one has $u_i^j =t^j$, and
$\forall j$, $K+1\leq j\leq k$, one has $u_i^j > t^j$. Now we can apply
the same line of reasoning than in the first case, by considering only
the coordinates between $K+1$ and $k$.
\end{proof}
\begin{theorem}\label{theo-all}The Pareto mechanism $\cal M$ gives all the equilibria
truthful payments.
\end{theorem}
\begin{proof}
The proof is by contradiction. We assume that the payment is computed in a different way than in our Pareto mechanism, and we show that there exists a configuration for which an agent has an incentive to lie. We consider any agent $i$, and any vector $b_i$ such that
$b_i\in MAX(b_{-i}, b_i)$.\\
Let first assume that the payment computed $\pi(b_i)$ is incomparable with
the set of points ${\cal T}(B\setminus \{b_i\})$, i.e.
$\forall t\in {\cal T}(B\setminus \{b_i\})$ one has $\pi(b_i)\sim t$.
Using Lemma~\ref{lem-t2}, there exists $\delta\gg \vec{0}$ such that
$\forall t\in {\cal T}(B\setminus \{b_i\})$, $\pi(b_i)+\delta\sim t$.
Let us assume that $v_i=\pi(b_i)+\delta$. Since
$\forall t\in {\cal T}(B\setminus \{b_i\})$, $v_i\sim t$, we know by
Definition~\ref{def-t1} that
$v_i\not\in MAX(b_{-i}, v_i)$ or
$v_i\in MAX(B\setminus \{b_i\})$.
One cannot have $v_i\in MAX(B\setminus \{b_i\})$, because
Lemma~\ref{lem-t01} would contradict the fact that
$\forall t\in {\cal T}(B\setminus \{b_i\})$, $v_i\sim t$.
Therefore one has $v_i\not\in MAX(b_{-i}, v_i)$,
and the utility of agent $i$ is $\vec{0}$ if $i$ reports her true value $v_i$.
If agent $i$ reports $b_i$ her utility will be $u'_i=v_i-\pi(b_i)=\delta\gg \vec{0}$,
meaning that she has an incentive to lie.
This case is illustrated in the Figure~\ref{fig-theo-all} (Case 1).\\
We assume now that the payment computed $\pi(b_i)$ is dominated by
at least one point from ${\cal T}(B\setminus \{b_i\})$.
Using Lemma~\ref{lem-t4} it means that there exists a bipartition
${\cal T}_1,{\cal T}_2$ of ${\cal T}(B\setminus \{b_i\})$ such that
$\forall t\in {\cal T}_1$, $t \succ \pi(b_i)$ and
$\forall t\in {\cal T}_2$, $t \sim \pi(b_i)$.
Using Lemma~\ref{lem-t3} there exists $\delta\gg \vec{0}$ such that
$\forall t\in {\cal T}(B\setminus \{b_i\})$, $t \succ \pi(b_i)+\delta$ or
$t \sim \pi(b_i)+\delta$.
Let us assume that $v_i=\pi(b_i)+\delta$. According to the
Definition~\ref{def-t1}, $v_i\not\in MAX(b_{-i}, v_i)$ or
$v_i\in MAX(B\setminus \{b_i\})$. One cannot have
$v_i\in MAX(B\setminus \{b_i\})$, because
Lemma~\ref{lem-t01} would contradict the fact that
$\forall t\in {\cal T}_1$, $t \succ v_i$ and
$\forall t\in {\cal T}_2$, $t \sim v_i$.
Therefore $v_i\not\in MAX(b_{-i},v_i)$ and the utility of agent $i$ is
$\vec{0}$ if $i$ reports her true value $v_i$.
If agent $i$ reports $b_i$ her utility will be $u'_i=v_i-\pi(b_i)=\delta\gg \vec{0}$,
meaning that she has an incentive to lie. This case is illustrated in the
Figure~\ref{fig-theo-all} (Case 2).\\
We assume now that the payment computed $\pi(b_i)$ strictly dominates at least
one point $t$ from ${\cal T}(B\setminus \{b_i\})$.
Now there are two cases to consider, either
$\pi(\pi(b_i))\neq \pi(b_i)$ or $\pi(\pi(b_i))=\pi(b_i)$.
If $\pi(\pi(b_i))\neq \pi(b_i)$, by using (\ref{ineg1}) one has
$\pi(b_i)\succ \pi(\pi(b_i))$.
If we assume that $v_i=b_i$ then agent $i$ would have an incentive to report
$\pi(b_i)$ instead of her true value $b_i$. Indeed, since $\pi(b_i)\gg t$,
one can conlude from Definition~\ref{def-t1} that
$\pi(b_i)\in MAX(b_{-i},\pi(b_i))$.
Moreover, agent $i$ would pay $\pi(\pi(b_i))$ instead of $\pi(b_i)$. This case
is illustrated in the Figure~\ref{fig-theo-all} (Case 3).
Assume now that $\pi(\pi(b_i))=\pi(b_i)$. Consider any vector $b'_i$ such that
$\pi(b_i)\succ b'_i$ and $b'_i\in MAX(b_{-i}, b'_i)$. For example,
one can take $b'_i = \pi(b_i) - (\pi(b_i)-t)/2$ (since $\pi(b_i)\gg t$, one
has $b'_i\gg t$ and hence by Definition~\ref{def-t1} one has
$b'_i\in MAX(b_{-i}, b'_i)$).
Using Lemma~\ref{lem1}, either $\pi(b'_i)$ is incomparable with
$\pi(\pi(b_i))=\pi(b_i)$, or $\pi(b'_i)=\pi(\pi(b_i))=\pi(b_i)$.
If $\pi(b'_i)\sim \pi(b_i)$ then by using Lemma~\ref{lem-t5} with
$x=b'_i$, $y=\pi(b_i)$ and $z=\pi(b'_i)$, one has
$b'_i\not\succeq \pi(b'_i)$. However this contradict the assumption that
$b'_i\succeq \pi(b'_i)$ (see (\ref{ineg1})).
If $\pi(b'_i) = \pi(b_i)$ then since $\pi(b_i)\succ b'_i$, we have
$\pi(b'_i)\succ b'_i$, and there is again a contradiction with the
assumption (\ref{ineg1}). This case is illustrated in the
Figure~\ref{fig-theo-all} (Case 4).\\
According to the previous discussion, the only remaining case we need to
consider is when $\pi(b_i)$ dominates, but not strictly dominates, at least
one point from ${\cal T}(B\setminus \{b_i\})$.
Since $b_i\in MAX(b_{-i}, b_i)$ (and
$b_i\not\in MAX(B\setminus \{b_i\})$) we know by
Definition~\ref{def-t1} that $\exists t'\in {\cal T}(B\setminus \{b_i\})$
such that $b_i\gg t'$.
We are going to consider a set of vectors
$b_{i,l}$, with $l\geq 1$, such that $\forall l\geq 1$,
$b_{i,l}\gg b_{i,l+1}$, $b_{i,l}\gg t'$ and $\lim_{l\to \infty} b_{i,l} = t'$.
Since ${\cal T}(B\setminus \{b_i\})$ has a finite number of elements,
we can assume without loss of generality that all the vectors $\pi(b_{i,l})$
dominates the same point $t\in {\cal T}(B\setminus \{b_i\})$.
Recall also that, by inequality~(\ref{ineg1}),
$b_{i,l} \succeq \pi(b_{-i},b_{i,l})= \pi(b_{i,l})$.
One can easily see that necessarily $t=t'$. Now using Lemma~\ref{lem-infinity}
we obtain that, $\exists l,l'$ such that $\pi(b_{i,l}) \succ \pi(b_{i,l'})$.
If we assume that $v_i=b_{i,l}$ then agent $i$ would have an incentive to report
$b_{i,l'}$ instead of her true value $b_{i,l}$.
\end{proof}
\begin{figure}[t]
\begin{center}
\subfloat[Case 1]{
\begin{tikzpicture}[scale=.5, every node/.style={transform shape}]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw[->] (0,0) -- (0,8);
\draw[step=1cm,gray,thin,dotted] (0,0) grid (8,8);
\draw (1,7) node[sommet]{}; \node[above right] at (1,7) {$b_1$};
\draw (3,5) node[sommet]{}; \node[above right] at (3,5) {$b_2$};
\draw (5,3) node[sommet]{}; \node[above right] at (5,3) {$b_3$};
\draw (7,1) node[sommet]{}; \node[above right] at (7,1) {$b_4$};
\draw (6,6) node[sommet] (B) {}; \node[above right] at (6,6) {$b_i$};
\draw[dashed] (0,6) -- (6,6); \draw[dashed] (6,0) -- (6,6);
\draw (0,7) node[rect]{}; \node[below left] at (0,7) {$t_1$};
\draw (1,5) node[rect]{}; \node[below left] at (1,5) {$t_2$};
\draw (3,3) node[rect]{}; \node[below left] at (3,3) {$t_3$};
\draw (5,1) node[rect]{}; \node[below left] at (5,1) {$t_4$};
\draw (7,0) node[rect]{}; \node[below left] at (7,0) {$t_5$};
\draw (0,7) -- (1,7) -- (1,5) -- (3,5) -- (3,3) -- (5,3) -- (5,1) -- (7,1) -- (7,0);
\draw (3.7,1.7) node[sommet] {}; \node[below left] at (3.7,1.7) {$\pi(b_i)$};
\draw (4.4,2.4) node[sommet] (A) {}; \node[left] at (4.4,2.4) {$v_i = \pi(b_i)+\delta\:$};
\draw[->] (A) to [bend left] (B);
\end{tikzpicture}
}
\subfloat[Case 2]{
\begin{tikzpicture}[scale=.5, every node/.style={transform shape}]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw[->] (0,0) -- (0,8);
\draw[step=1cm,gray,thin,dotted] (0,0) grid (8,8);
\draw (1,7) node[sommet] {}; \node[above right] at (1,7) {$b_1$};
\draw (3,5) node[sommet] {}; \node[above right] at (3,5) {$b_2$};
\draw (5,3) node[sommet]{}; \node[above right] at (5,3) {$b_3$};
\draw (7,1) node[sommet]{}; \node[above right] at (7,1) {$b_4$};
\draw (6,6) node[sommet] (B) {}; \node[above right] at (6,6) {$b_i$};
\draw[dashed] (0,6) -- (6,6); \draw[dashed] (6,0) -- (6,6);
\draw (0,7) node[rect] (T1) {}; \node[below left] at (0,7) {$t_1$};
\draw (1,5) node[rect] (T2) {}; \node[below left] at (1,5) {$t_2$};
\draw (3,3) node[rect] (T3) {}; \node[below left] at (3,3) {$t_3$};
\draw (5,1) node[rect] (T4) {}; \node[below left] at (5,1) {$t_4$};
\draw (7,0) node[rect] (T5) {}; \node[below left] at (7,0) {$t_5$};
\draw (0,7) -- (1,7) -- (1,5) -- (3,5) -- (3,3) -- (5,3) -- (5,1) -- (7,1) -- (7,0);
\draw (2,1) node[sommet] {}; \node[below left] at (2,1) {$\pi(b_i)$};
\draw (2.5,1.5) node[sommet] (A) {}; \node[left] at (2.5,1.5) {$v_i = \pi(b_i)+\delta\:$};
\draw[->] (A) to [bend left=10] (B);
\draw[line width=35pt, line cap=round, rounded corners, opacity=0.1] (T1) -- (T2);
\draw[line width=35pt, line cap=round, rounded corners, opacity=0.1] (T3) -- (T4);
\draw[line width=35pt, line cap=round, rounded corners, opacity=0.1] (T5) -- (T5);
\node at (1.5,6.5) {${\cal T}_2$}; \node at (7.5,0.5) {${\cal T}_2$};
\node at (5.5,2) {${\cal T}_1$};
\end{tikzpicture}
}\\
\subfloat[Case 3]{
\begin{tikzpicture}[scale=.5, every node/.style={transform shape}]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw[->] (0,0) -- (0,8);
\draw[step=1cm,gray,thin,dotted] (0,0) grid (8,8);
\draw (1,7) node[sommet]{}; \node[above right] at (1,7) {$b_1$};
\draw (5,3) node[sommet]{}; \node[above right] at (5,3) {$b_3$};
\draw (3,5) node[sommet]{}; \node[above right] at (3,5) {$b_2$};
\draw (7,1) node[sommet]{}; \node[above right] at (7,1) {$b_4$};
\draw (6,6) node[sommet] (B) {}; \node[above right] at (6,6) {$v_i=b_i$};
\draw[dashed] (0,6) -- (6,6); \draw[dashed] (6,0) -- (6,6);
\draw (0,7) node[rect]{}; \node[below left] at (0,7) {$t_1$};
\draw (1,5) node[rect]{}; \node[below left] at (1,5) {$t_2$};
\draw (3,3) node[rect]{}; \node[below left] at (3,3) {$t_3$};
\draw (5,1) node[rect]{}; \node[below left] at (5,1) {$t_4$};
\draw (7,0) node[rect]{}; \node[below left] at (7,0) {$t_5$};
\draw (0,7) -- (1,7) -- (1,5) -- (3,5) -- (3,3) -- (5,3) -- (5,1) -- (7,1) -- (7,0);
\draw (4.5,4.5) node[sommet] (pibi) {}; \node[above] at (4.5,4.5) {$\pi(b_i)$};
\draw (2,2) node[sommet] {}; \node[below left] at (2,2) {$\pi(\pi(b_i))$};
\draw[->] (B) to [bend left] (pibi);
\end{tikzpicture}
}
\subfloat[Case 4]{
\begin{tikzpicture}[scale=.5, every node/.style={transform shape}]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw[->] (0,0) -- (0,8);
\draw[step=1cm,gray,thin,dotted] (0,0) grid (8,8);
\draw (1,7) node[sommet]{}; \node[above right] at (1,7) {$b_1$};
\draw (5,3) node[sommet]{}; \node[above right] at (5,3) {$b_3$};
\draw (3,5) node[sommet]{}; \node[above right] at (3,5) {$b_2$};
\draw (7,1) node[sommet]{}; \node[above right] at (7,1) {$b_4$};
\draw (6,6) node[sommet] (B) {}; \node[above right] at (6,6) {$b_i$};
\draw[dashed] (0,6) -- (6,6); \draw[dashed] (6,0) -- (6,6);
\draw (0,7) node[rect]{}; \node[below left] at (0,7) {$t_1$};
\draw (1,5) node[rect]{}; \node[below left] at (1,5) {$t_2$};
\draw (3,3) node[rect]{}; \node[below left] at (3,3) {$t_3$};
\draw (5,1) node[rect]{}; \node[below left] at (5,1) {$t_4$};
\draw (7,0) node[rect]{}; \node[below left] at (7,0) {$t_5$};
\draw (0,7) -- (1,7) -- (1,5) -- (3,5) -- (3,3) -- (5,3) -- (5,1) -- (7,1) -- (7,0);
\draw (4.5,4.5) node[sommet] (pibi) {}; \node[right] at (4.5,4.5) {$\:\pi(\pi(b_i))=\pi(b_i)$};
\draw (3.75,3.75) node[sommet] (bip) {}; \node[below] at (3.75,3.75) {$b'_i$};
\draw (4,5) node[sommet] (pibip) {}; \node[above right] at (4,5) {$\pi(b'_i)$};
\end{tikzpicture}
}
\caption{\small \label{fig-theo-all}An illustration of the proof of
Theorem~\ref{theo-all}.}
\end{center}
\end{figure}
\section{A Pareto mechanism for the Weakly Maximum Vector problem}
We are going to present a Pareto mechanism, denoted by $\cal M'$, which
satisfies the MIR constraint and which is equilibria-truthful.
For doing so, we modify the mechanism ${\cal M}$ in order to
remove the DV condition.
The modified mechanism is given
in Table~\ref{figpmecarelaxed} and illustrated in
Figure~\ref{fig-mprime}. It can be shown that $\cup_{i\in W} b_i = WMAX(B)$.
\bigskip
\begin{algorithm}[h]
\begin{algorithmic}[1]
\STATE Remove all identical vectors and corresponding agents.
\STATE For all $i\in B$, set $PAY(i) := \{t \in
{\cal T}(MAX(B)\setminus \{b_i\}) \: |\: b_i \succeq t\}$.
\STATE $W$ is the set of agents $i$ such that $PAY(i)\neq \emptyset$.
\STATE For all $i\in W$ choose any $p_i\in PAY(i)$.
\STATE For all $i\notin W$, we set $p_i=\vec{0}$.
\caption{\label{figpmecarelaxed}The weakly Pareto mechanism ${\cal M'}$.}
\end{algorithmic}
\end{algorithm}
\bigskip
\begin{figure}[h]
\begin{center}
{
\begin{tikzpicture}[scale=.8, every node/.style={transform shape}]
\tikzstyle{sommet}=[circle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\tikzstyle{cercle}=[circle,draw,minimum size=9pt,inner sep=0pt]
\tikzstyle{rect}=[rectangle,draw,fill=black,minimum size=5pt,inner sep=0pt]
\draw[->] (0,0) -- (8,0);
\draw[->] (0,0) -- (0,4);
\draw[step=1cm,gray,thin,dotted] (0,0) grid (8,4);
\draw (1,1) node[sommet]{}; \node[below] at (1,1) {$b_1$};
\draw (2,1) node[sommet]{}; \node[below] at (2,1) {$b_2$};
\draw (4,1) node[sommet]{}; \node[below] at (4,1) {$b_3$};
\draw (5,1) node[sommet]{}; \node[below] at (5,1) {$b_4$};
\draw (6,1) node[sommet]{}; \node[below] at (6,1) {$b_5$};
\draw (6,1) node[cercle]{};
\draw (1,2) node[sommet]{}; \node[left] at (1,2) {$b_6$};
\draw (3,2) node[sommet]{}; \node[right] at (3.1,2) {$b_7$};
\draw (3,2) node[cercle]{};
\draw (2,3) node[sommet]{}; \node[right] at (2.1,3) {$b_8$};
\draw (2,3) node[cercle]{};
\draw (0,3) node[rect]{}; \node[above right] at (0,3) {$t_1$};
\draw (2,2) node[rect]{}; \node[above right] at (2,2) {$t_2$};
\draw (3,1) node[rect]{}; \node[above right] at (3,1) {$t_3$};
\draw (6,0) node[rect]{}; \node[above right] at (6,0) {$t_4$};
\end{tikzpicture}
}
\caption{\small \label{fig-mprime}An illustration of the mechanism
${\cal M'}$. One has $MAX(B)=\{b_8,b_7,b_5\}$,
${\cal T}(MAX(B))=\{t_1,t_2,t_3,t_4\}$, and
$WMAX(B)=\{b_3,b_4,b_5,b_7,b_8\}$. Indeed, the vectors
$b_1,b_2$ and $b_6$ do not belong to the weakly Pareto curve since
neither $t_1$, nor $t_2,t_3,t_4$ is dominated by them. In contrary,
$b_3$ and $b_4$ belong to the weakly Pareto curve since they
dominate $t_3$.
}
\end{center}
\end{figure}
\newpage
|
1,314,259,996,700 | arxiv | \section{Introduction}
Flow boiling in microchannel has been extensively researched in recent times due to its higher heat removal capacity, caused by its high heat transfer surface area to fluid flow volume ratio coupled with the utilization of latent heat of vaporization. \citet{Harirchian2010} observed single phase, bubbly region, slug - plug, churn and annular flow patterns based on the operating conditions. The heat transfer capabilities are highly affected by the flow patterns. In general, the heat transfer coefficient associated with the slug-plug or the elongated bubble is higher due to the presence of thin liquid film surrounding the bubble. \citet{Mirmanto2016} experimentally investigated flow boiling in a single mini/micro-channel and observed a decrease in heat transfer coefficient with the increase in the local thermodynamic quality. \citet{Yin2016} observed an increasing and then a decreasing trend in heat transfer coefficient with local thermodynamic quality. \citet{sobierska2007heat} observed single phase, bubble and slug flow during boiling in a narrow channel. The maximum heat transfer coefficient was found in the slug region.
\subsection{Transient variation}
\citet{Jagirdar2016} investigated the transient local heat transfer coefficient at different locations in a single rectangular channel with dimensions as \(0.42 \times 2.54 \times 25.4 \, mm\) during subcooled flow boiling. The synchronized flow visualization showed that the variation in the heat transfer coefficient is due to the passing of liquid, vapour bubble (surrounded by liquid) and dry-out, cyclically over the given location. The increase in the local heat transfer coefficient was observed due to the passing of bubble with evaporating thin film. Similar observations were also reported by \citet{Bigham2015} for flow boiling experiment conducted in a microchannel with nanostructures. A sudden drop was observed in wall temperature during the passing of the bubble. \citet{Jafari2015} performed Computational Fluid Dynamics (CFD) simulations using the phase-field method to study the hydrodynamics and heat transfer characteristics of an elongated bubble (present in superheated liquid) inside the microchannel. The results showed high heat transfer coefficient on the bottom and side walls in the confined bubble region. \citet{sun2018transient} demonstrated and formulated the effect of shear stress on the evaporating thin film inside a microtube. The results showed variation in the heat transfer coefficient due to the cyclic process of elongated bubble growth, dry out and liquid slug. The residence time of each region was also measured and it was found that with the increase in the heat flux, liquid slug time decreases and the elongated bubble or dry region time increases. Transient pressure fluctuations and heat transfer coefficient were observed by \citet{Barber2011} for a single microchannel using n-Pentane as the working fluid. The confined bubble was observed due to the small aspect ratio and the bubble expanded both in the upstream and downstream directions. Pressure fluctuations were observed due to confinement of the bubble inside the channel. \citet{Rao2015a} experimentally studied transient heat transfer coefficient in multiple flow regimes such as bubbly, slug and annular in a rectangular microchannel. The periodic variation in the heat transfer coefficient was observed with the peak due to explosive bubble growth and presence of evaporating thin film which is followed by a sharp decrease due to the emergence of local dry out.
Rapid confined bubble growth causes pressure fluctuations during flow boiling in mini/micro-channels. These fluctuations in the presence of upstream compressibility such as trapped non-condensables, pump characteristics, subcooled boiling in the upstream region/preheater \citet{Gedupudi2011} and interaction between the neighbouring channels (in the case of parallel microchannels), as summarized by \citet{Prajapati2017}, leads to flow reversal. \citet{Liu2013} extensively studied the effect of upstream compressibility on flow boiling. Heat transfer coefficient was found to increase with increase in the compressible volume, for certain range of operating conditions. On the other hand, the inlet pressure was found to be oscillating with lower frequency. On further increasing the compressible volume, there was a decrease in pressure fluctuation frequency. \citet{Karayiannis2017} pointed out the lack of data on the effect of flow reversal on heat transfer coefficient. \citet{kandlikar2001high} observed pressure fluctuation and flow reversal due to the interaction between the neighbouring channels in a parallel microchannel heat sink. \citet{Kenning2001} and \citet{Yan1999} observed pressure fluctuations under both negligible and high upstream compressibility conditions. \citet{Barber2010} studied bubble dynamics inside a rectangular microchannel with FC-72 as the working fluid and reported a sharp rise in pressure along with flow reversal. The experimental investigations carried by \citet{Gedupudi2011} indicates that in the presence of upstream compressibility, the upstream end moved towards the inlet of the channel. On the other hand, no flow reversal was observed when the compressibility was removed. However, pressure fluctuations were observed in both the cases. \citet{Yin2017} experimentally observed pressure fluctuations in the slug and bubbly flow patterns. \citet{Wang2014} and \citet{Singh2009a} experimentally observed transient two-phase pressure drop, with the amplitude and frequency dependent on heat flux and mass flux.
\subsection{Average heat transfer coefficient}
\citet{Wang2012} experimentally studied the effect of heat flux, vapor quality and hydraulic diameter on flow boiling heat transfer in a rectangular microchannel using FC-72 as the working fluid. The local heat transfer coefficient initially increases due to thin film evaporation and then decreases due to the emergence of dry patches. \citet{Diaz2007} studied the variation of local heat transfer coefficient for different heat fluxes and working fluids. The heat transfer coefficient was found to increase with heat flux, indicating the dominance of the nucleation boiling regime. Temperature fluctuation of high amplitude was also observed for water in the low-quality region. \citet{Mirmanto2016} experimentally investigated the local pressure drop and heat transfer coefficient for flow boiling of water in a micro-channel. Non-linear variation in pressure drop was observed along the channel length. Local heat transfer coefficient was found to increase and then decrease along the channel. Effect of local pressure on the estimation of local heat transfer coefficient was also studied.
\subsection{Prediction of heat transfer coefficient}
\citet{Thome2004} proposed a three-zone model consisting of the liquid zone, bubble with the thin liquid film and vapor zone for small tubes supplied with constant heat flux on the boundary and evaluated time-averaged heat transfer coefficient. The model utilized three adjustable parameters for initial film thickness, nucleation frequency and minimum film thickness, which were obtained from the experimental data set as explained by \citet{Dupont2004}. This model does not consider the effect of local transient pressure during the bubble growth and the effect of shear stress on the thin film depletion. \citet{Wang2010} extended the three-zone model (\citet{Thome2004}) to a rectangular channel. The four zones they considered are liquid zone, elongated bubble zone, partial dry-out with liquid present in the corners and full dry-out region. The model was verified with the experiments conducted in a single rectangular channel with constant heat flux supplied through the channel walls. This model eliminated minimum thin film thickness as a parameter. This model also neglected the effect of local pressure conditions and effect of shear stress on the depletion of the thin film. The three-zone and four-zone models mentioned above were used to predict heat transfer coefficient over the entire range of thermodynamic quality. \citet{Harirchian2012} estimated heat transfer coefficient based on the flow patterns observed during flow boiling. For the bubbly region, the correlation is used. For confined annular flow, the model has been developed based on the conservation of mass, momentum (along with the interfacial stresses) and energy equation. Similarly, three-zone model developed by \citet{Thome2004} was utilized for slug-plug region but with a different initial film thickness relationship and the associated parameters. A similar flow pattern based approach was also utilized for the prediction of pressure drop across the channel. \citet{antonsen2015pragmatic} developed numerical models for the evaporation and condensation of refrigerants in a microtube with the combination of correlations, numerical methods and analytical model. Bubble growth model based on the heat flux is calculated for the cases with and without axial conduction. The model utilized the three-zone model for the estimation of heat transfer coefficient. Effects of different parameters such as wall material, nucleation frequency and mass flux have been studied. \citet{Jafari2016} simulated flow boiling process in a microchannel considering an artificial cavity at the channel wall using Cahn - Hilliard method. The bubble is nucleated from the artificial cavity and it grows up till it reaches the channel height and then departs from the nucleation site. The velocity, temperature, and pressure distributions were observed inside the channel. The heat transfer performance increased by 8 times the single phase value due to the presence of the thin liquid film layer, which, later, upon the dry-out, led to a reduction in heat transfer capability. Periodic fluctuations in heat transfer coefficient were observed based on the bubble location. The heat removal capacity was observed to increase with the increase in the mass flux. \citet{cerza2007analytical} performed an analytical investigation of the bubble growth and estimation of heat transfer. The model assumed a moving liquid layer along with the bubble with non-zero velocity and presence of the developing thermal profile. The model utilized the film thickness for the estimation of the heat transfer coefficient under various bubble velocity conditions. The model estimated the temperature profile of the liquid film below the bubble which caused the bubble growth. \citet{Magnini2017} improvised the three-zone model initially developed by \citet{Thome2004} for the two-phase system with refrigerant as a fluid. The improved model considered the time dependent variation of the local film thickness along with the inclusion of thermal inertia of the liquid film. The recirculation of the liquid in between the trapped bubbles is also considered for the liquid slug heat transfer coefficient calculation. \citet{li2017effect} proposed a model for the bubble dynamics in a micro-tube under flow reversal condition and its effect on the heat transfer coefficient for R134a.
\subsection{Objective of the current work}
Most of the heat transfer and pressure drop models in the literature focus on microtubes (circular in cross section). But rectangular microchannel heat sinks, machined in copper or aluminum or etched in silicon, are important from the application point of view, considering the dissipation of high heat fluxes and the ease of manufacture. The four-zone model proposed by \citet{Wang2010} for a rectangular channel does not consider partially confined bubble region and the influence of local pressure fluctuations, shear stress and flow reversals on heat transfer characteristics. The current work is aimed at predicting the transient heat transfer coefficient under the influence of local pressure fluctuation caused due to the confined growth of the bubble inside a rectangular micro-channel. Transient and time-averaged pressure drop and heat transfer coefficient are compared with the experimental data available in the literature for a single rectangular microchannel. Flow reversal caused due to explosive bubble growth and presence of upstream compressible volume and its effect on the heat transfer coefficient is studied. Effect of shear stress on transient heat transfer characteristics is also modeled.
\section{Model description}
The current model consists of two modules i.e., the bubble growth module and the heat transfer module. The first module estimates bubble length, bubble velocity, and pressure drop and the second module calculates the heat transfer coefficient based on the four zones, namely, (1) the liquid zone, (2) elongated bubble zone, (3) partial dry-out zone and (4) full dry-out zone as proposed by \citet{Wang2010} and additionally considers partially confined bubble zone, as shown in Fig. \ref{fig:1a}. The liquid zone consists of the liquid trapped between the two bubbles. The second zone (partially confined bubble zone) consists of liquid bulk on either side the bubble as shown in Fig. \ref{fig:1a}. The third zone consists of thin liquid film surrounding bubble along with the bulk liquid present in the corner of the rectangular channel between the bubble and the wall. As the thin film depletes the third zone appears with the dry patch area along with the bulk liquid in the corner. Finally, complete evaporation of the corner liquid leads to full dry-out zone/vapor zone.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{zonesnew5.pdf}
\caption{Various zones considered in the present study.}
\label{fig:1a}
\end{figure}
\subsection{Assumptions}
In order to simplify the model few assumptions are considered. These assumptions are in line with the assumptions made by \citet{Thome2004}, \citet{Wang2010} and \citet{Gedupudi2011}.
\begin{enumerate}
\item Current model is 1-dimensional i.e. variations are only considered in the axial direction, i.e., along the flow direction.
\item The heat sink consist of a single channel with constant heat supplied through the three sides (two sides and the base) of the channel.
\item The bubble in the elongated region (liquid as thin film and liquid in the corner) is considered as a rectangle with rounded corners for simplification.
\item Fluid is considered to be saturated from the point of nucleation; this is possible for the cases with the flow boiling in the near saturated condition.
\item The bubbles are considered to be in the saturated condition, hence properties corresponding to local pressure are considered.
\item Initial bubble is assumed to be a cube with the length, breadth and height equal to the minimum dimension of the cross-section as assumed by \citet{Gedupudi2011}.
\item Single nucleation site is considered in the present study. In the current case, the bubble grows in axial direction exponentially leading to increased velocity which reduces the thermal boundary layer thickness suppressing the bubble growth. Moreover passing of a long bubble in a shorter channel also decreases the possibility of multiple nucleation sites. The models developed by \citet{Thome2004} and \citet{Wang2010} also consider single nucleation site.
\item The length of liquid slug is equivalent to the liquid passing over the nucleation site during the interval between the successive nucleation, i.e., the liquid slug length is equal to the liquid velocity multiplied by the nucleation time period.
\item Minimum film thickness before the occurrence of the dry-out is of the order of surface roughness. This value varies with surface, so, if available, the specified value would be used. Else, it would be considered as a constant similar to the value proposed by \citet{Thome2004}.
\item Change in property of fluid due to viscous dissipation as explained by \citet{Han2007} does not affect the properties of liquid significantly.
\item The simulation ends if there is no bubble in the channel, or the upstream end of the bubble reaches the channel entrance, possible in the case of inlet compressibility.
\item The contribution of surface tension to the pressure drop is negligible. The surface tension of water at \ang{100}C is 0.058 N/m and the minor dimension of the channel considered in the study is 0.00024 m. Therefore the approximate pressure drop due to surface tension is \(\sigma/R = 0.058/0.00012 = 0.48 \, kPa\), which is negligible compared to the pressure drop due to friction and acceleration for the range of the parameters considered in the present study.
\end{enumerate}
\subsection{Properties determination}
Variation of fluid properties with pressure is not considered in the models for the bubble growth presented by \citet{He2016} and \citet{Gedupudi2011}. \citet{Thome2004} and \citet{Wang2010} have also not considered the fluid property variation with local transient pressure in their heat transfer models. In the case of water, there is a significant difference in the amplitudes of pressure fluctuation if the variable properties are not considered (presented in the current study). This is due to the fact that as the pressure drop increases across the channel, the local pressure changes, which leads to a change in vapor properties that defines the bubble growth rate. These, in turn, influence the local velocity and pressure drop. The fluid properties are taken from \citet{Bell2014}. The saturated properties corresponding to the local pressure are estimated at each time step. The pressure at the channel outlet is assumed to be 101 kPa. In the current study, the working fluid is water unless otherwise specified.
\subsection{Location of nucleation site}
The current transient model requires a nucleation site as an input parameter. This is one of the major challenges in the area of flow boiling as nucleation site is not deterministic and it depends on surface characteristics. Therefore, it is difficult to estimate the nucleation site accurately. Though, \citet{Liu2005} summarizes relationships available in the literature and proposes the condition for nucleation based on wall superheat. In the present study, Eq. (\ref{eq:1}) is used to determine nucleation site.
\begin{equation} \label{eq:1}
\sqrt{T_{wall}} - \sqrt{T_{sat}} = \sqrt{\frac{2 \, (1+\cos(\theta))\, \sigma \, q}{\rho_{v} \, k_{l} \, i_{lv}}}
\end{equation}
The location of nucleation site is determined iteratively. Initially for the specified inlet temperature and for the assumed channel inlet pressure equal to the exit pressure, the location where the wall temperature obtained from single-phase heat transfer coefficient correlation matches the wall temperature obtained from Eq. (\ref{eq:1}) is chosen as the initial nucleation site. Time-averaged two-phase pressure drop is then calculated. Saturation temperature corresponding to the obtained pressure at the assumed nucleation site is determined, and then the new wall temperature required for nucleation from Eq. (\ref{eq:1}) is obtained. Then the new location of the nucleation site is determined using single phase correlation based on the required wall temperature for nucleation. The estimation of the location of nucleation site and two-phase pressure drop is performed iteratively till the required wall temperature from Eq. (\ref{eq:1}) matches the obtained wall temperature from single phase correlation. Contact angle varies with the substrate and should be chosen accordingly. For example, it is around \ang{90} for copper.
\subsection{Nucleation frequency}
The nucleation time period consists of two components i.e., waiting time and growth time. Bubble growth and its departure from the nucleation site disturbs the local wall temperature. It takes a while for the reformation of the thermal boundary layer, i.e., for the wall to attain the superheat required for nucleation. This duration is called as waiting period. This has been determined with the help of transient single phase simulation carried out using ANSYS FLUENT 16.1. Another component is bubble growth period and is determined using the correlation given by \citet{Mikic1970}. The initial theoretical value obtained using this correlation is quite small and is of the order of \(100 \,\mu s\). The experimental data obtained from \citet{Kadam2018}, \citet{Lee2004} and \citet{Wang2010} shows that the nucleation time period varies in the range 1-500 ms for the water based on both heat flux and mass flux. In the present case, with the increase in heat flux, the time period decreases as both the waiting and bubble growth time decreases. As the heat flux increases the wall reaches the condition of superheat at a faster rate. Similarly, with the increase in mass flux two forces are opposing one is the inertial force which detaches the bubble from the nucleation site this decreases the time period as also observed by \citet{Jafari2016} and another is that the increases the time periods as due to large mass flux the thermal boundary layer
thickness decreases.
\begin{equation} \label{eq:2a}
R = \sqrt{\frac{12 \, k_{l} \, t_{grow}}{C1 \,\rho_{l} \, C_{p,l}\pi}} \, Ja
\end{equation}
where,
\begin{equation*} \label{eq:2b}
Ja = \frac{C_{p,l} \, \Delta T_{sup} \, \rho_{l}}{\rho_{v} \, i_{lv}}
\end{equation*}
Here,
\(C1\) is the adjustable factor to determine the growth time period. In the present study \(C1 = 210\) is used.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{channel1.pdf}
\caption{Stages of bubble growth in the one dimensional model.}
\label{fig:1}
\end{figure}
\subsection{Bubble growth}
In the case of the flow boiling the bubble is initiated at the channel wall based on the wall superheat requirement. The initial stage consists of the free (unconfined) growth similar to the nucleate boiling. Once the bubble grows to the minor dimension of the channel the growth rate changes as observed by \citet{Yin2016}. \citet{Barber2010} and \citet{Gedupudi2011} also observed two different growth stages, namely partially confined growth where the bubble is confined by the minor dimension and grows widthwise and lengthwise and fully confined growth where the bubble is confined by the minor and major dimensions of the cross-section and the bubble grows only along the length. The stage where the vapour bubble passes through the channel outlet is called vapour venting.
Energy balance for the bubble growth is given in Eq. (\ref{eq:3}).
\begin{equation} \label{eq:3}
q \; A_{heat} = \frac{d}{dt} \left(\rho_{v} \, A_{base} \, h \, i_{lv}\right)
\end{equation}
In the partially confined region, the area \(A_{base}\) differs from the partially confined condition and fully confined condition. \(A_{base} = L_{bub} \times L_{bub}\) for partially confined stage. \(A_{base} = L_{bub} \times w\) for fully confined stage. \(A_{heat} = L_{bub} \times L_{bub}\) for partially confined stage and \(A_{heat} = p_{h} \times L_{bub}\) for fully confined stage. \(p_{h}\) is the heated perimeter equal to \(w\) for single-sided heating and equal to \(2h\, +\, w\) for three-sided heating.
\subsubsection*{Time constant}
Time constant is defined in Eq. (\ref{eq:4}).
\begin{equation}\label{eq:4}
\tau = \frac{w \, h\, i_{lv} \, \rho_{v}}{p_{h} \, q}
\end{equation}
where \(p_{h}\) is the heated perimeter. The value of $\tau$ is calculated for each time step and substituted in the equations to obtain the downstream end velocity, position and the bubble length.
\subsubsection{Partially confined bubble growth}
The growth rate is obtained by substituting the expression for \(A_{base}\) in Eq. (\ref{eq:3}) and is given as
\begin{equation} \label{eq:5a}
\frac{d L_{bub}}{dt} = \frac{L_{bub}}{2 \, \tau }
\end{equation}
Using Taylor series expansion, length of the bubble at t+\(\Delta\)t is estimated as in Eq. (\ref{eq:5}) where \textit{`t'} refers to the time from the start of the simulation.
\begin{equation} \label{eq:5}
(L_{bub})_{t + \Delta t} = (L_{bub})_{t} + \left( \frac{L_{bub}}{2 \, \tau} \right)_{t} \, \Delta t
\end{equation}
From continuity, the velocity of the downstream end of the bubble is obtained as shown in Eq. (\ref{eq:6a}) and (\ref{eq:6}).
\begin{equation} \label{eq:6a}
U_{d} = U_{u} + \left(\frac{d A_{base}}{w \,dt}\right)
\end{equation}
\begin{equation} \label{eq:6}
U_{d} = U_{u} + \left(\frac{L_{bub}^{2}}{w \,\tau}\right)
\end{equation}
Similarly, the location of the downstream end of the bubble is calculated by Eq. (\ref{eq:7}).
\begin{equation}{\label{eq:7}}
z_{d} = z_{u} + L_{bub}
\end{equation}
\subsubsection{Fully confined bubble growth}
Similar to the partially confined bubble growth, the length, the velocity and the position of the bubble in fully confined growth are determined using Eq. (\ref{eq:8}), (\ref{eq:9}) and (\ref{eq:10}) respectively.
\begin{equation} \label{eq:8}
(L_{bub})_{t + \Delta t} = (L_{bub})_{t} + \left( \frac{L_{bub}}{ \tau} \right)_{t} \, \Delta t
\end{equation}
\begin{equation} \label{eq:9}
U_{d} = U_{u} + \left(\frac{L_{bub}}{\tau}\right)
\end{equation}
\begin{equation}{\label{eq:10}}
z_{d} = z_{u} + L_{bub}
\end{equation}
\subsubsection{Vapour venting}
During vapour venting stage, as the bubble passes through the channel outlet, the bubble is driven by the velocity on the upstream end and is given by
\begin{equation} \label{eq:11}
\frac{d \,L_{bub}}{dt} = -U_{u}
\end{equation}
The exit vapour velocity is determined using either Eq. (\ref{eq:6}) or (\ref{eq:9}) as the case may be.
\subsection{Upstream compressibility}
Due to the presence of sources of upstream compressibility, there can be flow reversal in the channel. In the present study, trapped non-condensable in the channel inlet plenum is considered, with pressure \(P_{c}\) and volume \(V_{c}\). Plenum is supplied with constant flow rate \(\dot{V}_{in}\) . The channel inlet velocity can be calculated from the principle of continuity as.
\begin{equation}{\label{eq:fr1}}
\dot{V}_{in} + \frac{d V_{c}}{dt} = A_{cs} \, U_{i}
\end{equation}
For the current case, non condensable gas is assumed to undergo isothermal compression.
\begin{equation}{\label{eq:fr2}}
P_{c} \, V_{c} = P_{0, ini} \, V_{0, ini} = constant
\end{equation}
The rate of change of gas volume is determined using Eq. (\ref{eq:fr3}).
\begin{equation}{\label{eq:fr3}}
\frac{d V_{c}}{dt} = - \frac{V_{c}}{P_{c}} \, \frac{d P_{c}}{dt}
\end{equation}
Stagnation pressure \((P_{0})\) is given by
\begin{equation}{\label{eq:fr4}}
P_{0} = P_{1} + (1.5K) \, \frac{\rho_{l} \, U^{2}_{i}}{2}
\end{equation}
Here \(P_{c}\) is equal to \(P_{0}\), and \(U_{i}\) is channel inlet velocity and \(P_{1}\) is channel inlet pressure. \(K = 1\) for the forward flow and \(K = 0\) for the flow reversal.
Differentiating the pressure equation Eq. \ref{eq:fr4} gives
\begin{equation}{\label{eq:fr6}}
\frac{d P_{0}}{dt} = \frac{d P_{1}}{dt} + (1.5K ) \, U_{i} \, \frac{d U_{i}}{dt}
\end{equation}
Substituting the Eq. \ref{eq:fr3} in the Eq. \ref{eq:fr6}.
\begin{equation}{\label{eq:fr7}}
\frac{d P_{1}}{dt} = - \frac{P_{c}}{V_{c}} \, \frac{d V_{c}}{dt} - (1.5K) \, U_{i} \, \frac{d U_{i}}{dt}
\end{equation}
Substituting the value of \(dV_{c}/dt\) from the Eq. \ref{eq:fr1}.
\begin{equation}{\label{eq:fr8}}
\frac{d P_{1}}{dt} = -\frac{P_{0,ini}}{V_{0,ini}}(A_{cs}\, U_{i} \, - \, \dot{V}_{in}) - (1.5K) \, U_{i} \, \frac{d U_{i}}{dt}
\end{equation}
The initial compressible volume of the non-condensable gas is assumed to be a variable parameter and would be varied in order to study its effect.
\subsection{Pressure drop}
The channel pressure drop consists of the viscous pressure drop and the acceleration pressure drop across the liquid slugs. Pressure drops due to momentum change and friction across vapour bubble or vapour slug are assumed to be negligible \citet{Gedupudi2011} as \(\rho_{v} << \rho_{l}\). As explained in the \textit{Assumptions section}, pressure drop across the vapor bubble and pressure drop due to surface tension are neglected.
\subsubsection{Viscous pressure drop}
The frictional pressure drop arises due to the motion of liquid slugs present in the channel. The total frictional pressure drop is the sum of the frictional pressure drops calculated for the individual liquid slugs. Apart from this, there will also be viscous pressure drop due to the liquid slug between the channel inlet and upstream end of the newest bubble.
\begin{equation} \label{eq:13}
\Delta P_{vis} = \frac{2 \, \rho_{l}\, f \, z_{n,u} \, U^{2}_{i}}{D_{h}} \, + \, \sum_{i = m}^{n} \left(\frac{2 \, \rho_{l} \, f \, L_{slug, i} \,U_{slug, i}^{2}}{D_{h}}\right)_{i}
\end{equation}
where,
where \textit{`m'} represents the liquid slug in between the exit and the last bubble, \textit{`n'} represents the liquid slug between the the newest bubble \textit{`n'} and \textit{`n-1'} bubble, as shown in Fig. \ref{fig:1}.
The Fanning friction factor \textit{f} used in Eq. (\ref{eq:13}) is for a rectangular cross section with the aspect ratio \(\alpha \) and is obtained from Eq. (\ref{eq:14}). If the value \textit{f} goes below 0.01, then \textit{f} is taken as a constant equal to 0.01 to account for the turbulent flow, as considered in the previous study by \citet{Gedupudi2011}.
\begin{equation}\label{eq:14}
\left(f \, Re\right) = 24 \left(1 -1.3553 \alpha + 1.9476 \alpha^{2} - 1.7012 \alpha^{3} + 0.9564 \alpha^{4} - 0.2537 \alpha^{5}\right)
\end{equation}
\subsubsection{Acceleration pressure drop}
The acceleration pressure drop is calculated using the momentum conservation applied for all the liquid slugs.
\begin{equation}\label{eq:15}
\Delta P_{acc} = \rho_{l} \, z_{n,u} \frac{d U_{i}}{dt} \, + \,\sum_{i = m}^{n} \left(\rho_{l} \, L_{slug ,i} \, \frac{d U_{slug, i}}{dt}\right)_{i}
\end{equation}
The first term in the above expression will be zero if there is no upstream compressibility.
\subsection{Heat transfer coefficient}
In the present work, the time-averaged heat transfer coefficient is calculated at each location. The heat transfer coefficient is calculated for each zone i.e. the liquid zone, partially confined bubble zone, elongated bubble zone, partial dry-out zone and full dry-out zone individually as they pass over a particular position cyclically. These are then time-averaged based on their residence time to obtain the local time-averaged heat transfer coefficient. The estimation of the heat transfer coefficients for different zones is described below.
\subsubsection{Liquid slug}
This is the liquid present between the two successive bubbles. The liquid slug moves with the velocity equal to the downstream end velocity of the bubble present on the upstream end or with the inlet velocity. In the literature various correlations have been used to estimate the heat transfer coefficient. \citet{Thome2004} estimates the liquid slug heat transfer coefficient (HTC) with the help of the laminar and turbulent correlations for the circular tube case. In order to avoid the jump from the laminar to turbulent region, asymptotic approach is used. This method averages the laminar and turbulent region HTC. In the case of four zone model developed by \citet{Wang2010}, single phase laminar fully developed region correlation is used. \citet{He2010} studied the heat transfer characteristics of liquid slug trapped between the two bubbles. Due to the movement of the liquid and no slip condition at the wall, internal vortex is seen. This vortex increases the convection and therefore increases the HTC. In the present study the fluid flow is considered laminar for \(Re < \, 1600\) and in the transition zone if \(Re\) is in between 1600 and 3000 and turbulent for \(Re \, > \,3000\) as used by the \citet{Kandlikar2004}. For the transition range, based on \(Re\), linear average of the heat transfer coefficient is used. Eq. (\ref{eq:25}) is used for the calculation of HTC for the rectangular channel in laminar region with the aspect ratio \((\alpha = h/w)\).
For the laminar fully developed condition, the Nusselt number expressions for four-sided and three-sided heating configurations are as given in Eq. (\ref{eq:16}) and Eq. (\ref{eq:17}) respectively.
\begin{equation}{\label{eq:16}}
Nu_{4,\infty} = 8.235 (1-2.0421 \, \alpha + 3.0853 \alpha^{2} \, - 2.4765 \alpha^{3} +1.0578 \alpha^{4} -0.1861 \alpha^{5})
\end{equation}
\begin{equation}{\label{eq:17}}
Nu_{3,\infty} = 8.235 (1-1.883 \, \alpha + 3.767 \alpha^{2} \, - 5.814 \alpha^{3} +5.361 \alpha^{4} -2 \alpha^{5})
\end{equation}
\citet{Lee2006} suggested heat transfer correlation for laminar thermally developing region for the rectangular channel with the aspect ratio \((\alpha)\) for the constant heat flux condition for four sided heat condition which was further modified by \citet{phillips1988microchannel} for three side heating condition as given in Eq. (\ref{eq:25}).
\begin{equation}\label{eq:25}
Nu_{3} = \frac{Nu_{3,\infty}}{Nu_{4,\infty}} \, Nu_{4}
\end{equation}
In the case of turbulent flow the relation, Eq. (\ref{eq:26}), proposed by \citet{wu1984measurement} for a microchannel is used.
\begin{equation}\label{eq:26}
Nu = 0.00222 \, Re^{1.09} \, Pr^{0.4}
\end{equation}
\subsubsection{Thin film region}
As elucidated in the literature, evaporation of the thin film is the major contributor of the heat transfer coefficient. Hence it is very important to estimate the thin film thickness. Initial film thickness plays an important role in this process. Various researchers have proposed the relationship based on the experimental observations. \citet{Thome2004} used the basic relationship provided by \citet{moriyama1996thickness} and used an asymptotic method to estimate the film thickness. A factor was also used to correct the above relationship based on the experimental data available in the literature. \citet{Wang2010} also used the similar equation as by \citet{Thome2004} but the factor is based on the experiments conducted. \citet{Han2010} proposed the relationship for the initial film thickness for the bubble accelerating through the channel. The initial film thickness decreases with the increase in the acceleration due to the thinning of boundary layer. Film thickness at any location or at any instant of time is influenced by initial film thickness and rate of thinning. The latter depends on the rate of evaporation and shear stress. \citet{Thome2004} and \citet{Wang2010} considered only the evaporation of the thin liquid film, deducing the new film thickness with the help of the energy balance. On the contrary, \citet{consolini2008convective} considered the effect of shear stress on the film thickness for a bubble growing inside a circular tube. \citet{sun2018transient} considered the thinning of the film due to evaporation and proposed an expression for the additional thinning of the thin film due to the shear stress. The relationship was derived analytically and a constant was used in order to match the experimental film thickness. \citet{younes2012analytical} also employed the analytical approach by considering the shear stress at the liquid-vapour interface to calculate the heat transfer coefficient. The heat transfer coefficient is calculated by Eq. \ref{eq:27}.
\begin{equation}\label{eq:27}
HTC = \frac{k_{l}}{\delta_{film}}
\end{equation}
The initial film thickness Eq. (\ref{eq:28}) for a vapour bubble in the two phase region is:
\begin{equation}\label{eq:28}
\frac{\delta_{film}}{D_{h}} = C_{\delta} \, \left(3 \sqrt{\frac{\nu_{l}}{U \, D_{h}}}\right)^{0.84} \, \left[\left(0.07 \, \left(\frac{\rho_{l} \, D_{h} \, U^{2}}{\sigma}\right)^{0.41}\right)^{-8} \, + \, 0.1^{-8} \right]^{-1/8}
\end{equation}
\citet{Thome2004} obtained the value of the constant \(C_{\delta}\) equal to 0.29, by matching the model results with the experimental data for refrigerants boiling in a circular tube. The initial film thickness based on this constant is of the order of few microns. \citet{Wang2010} used a value equal to 0.8117 by matching their model results with their experimental heat transfer data for water boiling in a rectangular channel (\(D_{h}\) = 0.137 mm). The initial film thickness based on this constant is in the range \(1-2 \, \mu m\) for the bubble velocity in the range \(0.2 - 0.5 \, m/s\) considered in their study. But in the present study with water as a working fluid, for the range of the parameters considered (\(D_{h}\) = 0.324 mm and 0.658 mm), the value of the constant \(C_{\delta}\) equal to 3.48 results in a good match between the modeling results and the experimental heat transfer data from the literature. The initial film thickness based on this constant is in the range \(10-30 \, \mu m\) for the bubble velocity in the range \(0.5 - 2 \, m/s\). The initial film thickness predictions made using this constant is in line with the experimental thin film thickness measured by \citet{sun2018transient} for water. Their experimental observations for flow boiling indicate initial film thickness of 25 and 31 \(\mu m\) for the bubble velocities 3.09 m/s and 3.63 m/s and tube diameter 0.94 mm. The corresponding values obtained using the proposed constant (equal to 3.48) is 17 and 16 \(\mu m\) respectively. Further, their experimental observations show that from the point of initial film thickness, there is a gradual decrease in local film thickness until it reaches dryout thickness which is around 2-3 \(\mu m\). The film thickness estimated by \citet{Gedupudi2011} is in the range 20-30 \(\mu m\) and that observed by \citet{Balasubramanian2005} is near 50 \(\mu m\), all for water boiling in a mini/microchannel close to atmospheric pressure.
In the present study, both the heat transfer and shear stress effect are taken into account to evaluate the depletion of the thin film thickness. Thin film exists in partially confined bubble region, elongated bubble region and partial dryout region.
\subsubsection*{Partially confined bubble region}
In this region, the bubble is partially confined by the minor channel dimension. The heat transfer coefficient is calculated considering the heat transfer through the thin film between the bottom wall and the bubble and the heat transfer through the liquid on either side of the bubble. The heat transfer coefficient is calculated using Eq. (\ref{eq:pc_1}) to Eq. (\ref{eq:pc_3}). The liquid heat transfer coefficient is estimated using the equations presented in \textit{Liquid slug} section considering the effective liquid flow cross-section obtained after subtracting the bubble cross-sectional area from the channel cross-section. Hydraulic diameter is calculated using the effective cross-sectional area and the corresponding wetted perimeter.
\begin{figure}[H]
\centering
\includegraphics[width=0.3\linewidth]{partialconf.pdf}
\caption{Paritally confined bubble zone.}
\label{fig:zone_partial_conf}
\end{figure}
\begin{equation}\label{eq:pc_1}
(T_{w} - T_{sat}) \times (0.5 \times w \, + h) = \int_{0}^{0.5 \times L_{bub}} (T_{w} - T_{sat}) ds
+ \int_{0}^{h} (T_{w} - T_{sat}) ds
+ \int_{0}^{0.5 \times ( w - L_{bub})} (T_{w} - T_{sat}) ds
\end{equation}
\begin{equation}
\frac{q \times (0.5 \times w + h)}{HTC_{pc}} = \frac{q \times 0.5 \times L_{bub}}{HTC_{film}} + \frac{q \times (0.5 \times w + h - 0.5 \times L_{bub} )}{HTC_{l}}
\end{equation}
\begin{equation}\label{eq:pc_3}
\frac{1}{HTC_{pc}} = \frac{0.5 \times w + h - 0.5 \times L_{bub}}{0.5 \times w + h} \times \frac{1}{HTC_{l}} + \frac{0.5 \times L_{bub}}{0.5 \times w + h} \times \frac{1}{HTC_{film}}
\end{equation}
The \(HTC_{film}\) is given by the Eq. \ref{eq:pc_htc_1}.
\begin{equation}\label{eq:pc_htc_1}
HTC_{film} = \frac{k_{l}}{\delta_{eq}}
\end{equation}
The \(\delta_{film}\) is the film thickness over the flat region and its depletion is given by Eq. \ref{eq:pc_htc_2}.
\begin{equation} \label{eq:pc_htc_2}
\left(\delta_{film}\right)_{t + \Delta t} = \left(\delta_{film}\right)_{t} - \frac{q \, L_{bub} \, \Delta t}{\rho_{l} \, i_{lv} \, L_{bub} }
\end{equation}
The \(\delta_{eq}\) is the equivalent thickness over the horizontal region which is given by Eq. \ref{eq:pc_htc_3}.
\begin{equation} \label{eq:pc_htc_3}
\delta_{eq} = \frac{h \times L_{bub} - A_{cs,bub}}{2 \times L_{bub}}
\end{equation}
\(A_{cs,bub}\) is calculated for hyperellipse with \(n1 = 4\) , \(a = 0.5 \times L_{bub}\) and \(b = 0.5 \times h - \delta_{film}\) in Eq. \ref{eq:pc_htc_4} presented by \citet{Tamayol2010}.
\begin{equation}\label{eq:pc_htc_4}
A_{cs,bub} = 4 a \, b \frac{\sqrt{\pi} \: \Gamma \left(\frac{n1+1}{n1}\right)}{4^{1/n1} \, \Gamma \left(\frac{n1+2}{2n1}\right)}
\end{equation}
Here, \(\Gamma\) is the gamma function.
\subsubsection*{Elongated bubble region}
This is the region where the cross- section has liquid film over the entire perimeter of the channel as shown in Fig. \ref{fig:zone_elong}. Vapour bubble is assumed to be of rectangular shape with rounded corners, obtained with \(n1 = 4\) in hyperellipse formula \citet{Tamayol2010}. The region between the curved portion at the corner and the channel walls form the bulk liquid region.
\begin{figure}[H]
\centering
\includegraphics[width=0.3\linewidth]{Elongatedbubble1.pdf}
\caption{Elongated bubble zone.}
\label{fig:zone_elong}
\end{figure}
Here \(\delta_{film}\) is the film thickness at the flat region of the channel.
\begin{equation} \label{eq:29a}
\left(\delta_{film}\right)_{t + \Delta t} = \left(\delta_{film}\right)_{t} - \frac{q \, p_{h} \, \Delta t}{\rho_{l} \, i_{lv} \, p_{lv} }
\end{equation}
Here \(p_{lv}\) is the perimeter of liquid and vapor interface at any cross-section used by \citet{Tamayol2010}. \(p_{ch}\) is the channel perimeter and \(p_h\) is the heated perimeter.
\begin{equation}\label{eq:29b}
\delta_{eq} = \frac{A_{l}}{p_{ch}} = \frac{w \times h - A_{cs,bub}}{2 \, \times (w + h)}
\end{equation}
The cross-sectional area for a rectangle with rounded corner is given by following equation and used by \citet{Tamayol2010}. Here \(n1 = 4 \) for rectangle with rounded corners.
\(a = 0.5 \times w - \delta_{film}\) and \(b = 0.5 \times h - \delta_{film}\)
\begin{equation}\label{eq:29c}
A_{cs,bub} = 4 a \, b \frac{\sqrt{\pi} \: \Gamma \left(\frac{n1+1}{n1}\right)}{4^{1/n1} \, \Gamma \left(\frac{n1+2}{2n1}\right)}
\end{equation}
Here, \(\Gamma\) is the gamma function.
The Eq. \ref{eq:31} is used to determine the heat transfer coefficient in the elongated bubble region.
\begin{equation}\label{eq:31}
HTC_{elb} = \frac{k_{l}}{\delta_{eq}}
\end{equation}
\subsubsection*{Partial dryout region}
Once the thin film in the flat region of the elongated bubble is evaporated fully, i.e., \(\delta_{film} = 0 \), it leads to partial dryout as shown in Fig. \ref{fig:2part}. The corners of the channel still have the liquid which is not evaporated. Thus the equivalent heat transfer coefficient is the average of the heat transfer coefficient of the dryout region i.e. over the wall in contact with vapour and liquid heat transfer coefficient due to liquid in the corners of the channel. In the present case, the cross-sectional area occupied by vapour bubble is of rectangular in shape with rounded corners. In partial-dryout region, perimeter of bubble cross-section intersects with the channel wall. and the corner liquid area and lengths are calculated.
\begin{equation}\label{eq:32_1}
(T_{w} - T_{sat}) \times (0.5 \times w \, + h) = \int_{0}^{l_{a}} (T_{w} - T_{sat}) ds
+ \int_{0}^{l_{b}} (T_{w} - T_{sat}) ds
+ \int_{0}^{l_{c}} (T_{w} - T_{sat}) ds
+ \int_{0}^{0.5 \times w + h - l_{l}} (T_{w} - T_{sat}) ds
\end{equation}
\begin{equation}
\frac{q \times (0.5 \times w + h)}{HTC_{pdr}} = \frac{q \times l_{a}}{HTC_{l}} + \frac{2 \,q \times l_{b}}{HTC_{l}} + \frac{q \times (0.5 \times w + h - l_{l} )}{HTC_{v}}
\end{equation}
\begin{equation}\label{eq:32}
\frac{1}{HTC_{pdr}} = \frac{0.5 \times w + h - l_{l}}{0.5 \times w + h} \times \frac{1}{HTC_{v}} + \frac{l_{l}}{0.5 \times w + h} \times \frac{1}{HTC_{l}}
\end{equation}
Here \(l_{l}\) refers to the length of the wall in contact with the liquid over the heated perimeter. \(l_{l} \, = \, l_{a} \, + \,l_{b} \, + \, l_{c} = l_{a} \, + \, 2 l_{b}\).
\begin{figure}[H]
\centering
\includegraphics[width=0.3\linewidth]{partialdry.pdf}
\caption{Partial dryout zone.}
\label{fig:2part}
\end{figure}
\begin{equation}\label{eq:32a}
\left(\delta_{corner}\right)_{t + \Delta t} = \left(\delta_{corner}\right)_{t} - \frac{q \, p_{h} \, \Delta t}{ \rho_{l} \, i_{lv} \, p_{lv}}
\end{equation}
\(A_{l}\) is the area enclosed by the sides of the channel and the curve formed by the liquid-vapour interface at the corner.
This curve is obtained by assuming a hyperellipse (\citet{Tamayol2010}) with \((0.5 \times w + \delta_{corner})\) and \((0.5 \times h + \delta_{corner})\) as major axis (a) and minor axis (b), as described in \citet{Tamayol2010}. The enclosed area is calculated by numerical integration.
\begin{equation}\label{eq:32c}
\delta_{eq} = \frac{A_{l}}{p_{lv}}
\end{equation}
And the corner bulk liquid heat transfer coefficient is calculated by the Eq. (\ref{eq:32d}).
\begin{equation}\label{eq:32d}
HTC_{l} = \frac{k_{l}}{\delta_{eq}}
\end{equation}
\subsubsection{Full dry-out region}
This is the region developed due to the complete evaporation of the thin film. The HTC can be calculated by means of the Eq. (\ref{eq:16}) to (\ref{eq:26}) using vapor properties.
\subsubsection{Shear stress effect}
\citet{sun2018transient} experimentally investigated the evolution of thin film with time, for adiabatic and diabatic cases. The depletion of the film thickness was observed to be faster in diabatic case than that in adiabatic case. They also derived the depletion rate of thin film thickness for circular tube. In the present study, effect of shear stress is extended to a rectangular channel.
From continuity,
\begin{equation}
\rho_{l} \, \delta_{eq, z} \, p_{lv} \, U_{l, mean, z} \, = \,\rho_{l} \, \delta_{eq, z+\Delta z} \, p_{lv} \, U_{l, mean, z+\Delta z}
\end{equation}
\begin{equation}
\delta_{eq} \, U_{l,mean} = constant
\end{equation}
Differentiating the above equation with time and using \(dz = U_{l,mean}dt\) results in Eq. (\ref{eq:33}).
The rate of thin film depletion is defined by the Eq. (\ref{eq:33}).
\begin{equation}\label{eq:33}
\frac{d \delta_{eq}}{dt} = -\delta_{eq} \, \frac{d U_{l,mean}}{dz}
\end{equation}
This equation requires the variation of the mean liquid velocity with the axial position. In order to estimate the value, the shear stress on the vapour side is equated to the shear stress on the liquid side at liquid-vapour interface, which is given by Eq. \ref{eq:34}.
\begin{equation}\label{eq:34}
\tau_{l} = \tau_{v}
\end{equation}
The shear stress on the vapor side is given by the following Eq. (\ref{eq:35}).
\begin{equation}\label{eq:35}
\tau_{v} = \mu_{v} \frac{d U_{v}}{dr}
\end{equation}
Similarly the shear stress due on the liquid is given by the following Eq. (\ref{eq:36}).
\begin{equation}\label{eq:36}
\tau_{l} = \mu_{l} \, \frac{U_{l}}{\delta_{eq}}
\end{equation}
Equating the shear stress relationship mentioned in Eq. (\ref{eq:35}) and (\ref{eq:36}) to obtain Eq. (\ref{eq:37}).
\begin{equation}\label{eq:37}
U_{l, mean} = \frac{\mu_{v}}{\mu_{l}} \delta_{eq} \frac{d U_{v}}{dr}
\end{equation}
In the Eq. \ref{eq:37}, the spatial variation of the vapor velocity needs to be determined.
\begin{equation}\label{eq:38}
\frac{d U_{v}}{dr} = \frac{d(u^{*} U_{max})}{dr} = U_{max} \frac{du^{*}}{dr}
\end{equation}
In the Eq. (\ref{eq:38}), \(u^{*}\) is a non-dimensional velocity calculated as described by \citet{Tamayol2010}.
The equation below shows the relationship between the mean velocity and maximum velocity with the help of the velocity distribution along the cross-section.
\begin{equation}\label{eq:39}
U_{max} = \frac{U_{v,mean} \times A_{cs,bub}}{\iint u^{*} r \, dr \, d\theta}
\end{equation}
Differentiating Eq. (\ref{eq:37}) w.r.t dz results in the Eq. (\ref{eq:39a}).
\begin{equation}\label{eq:39a}
\frac{dU_{l, mean}}{dz} = \frac{\mu_{v}}{\mu_{l}} \delta_{eq} \, \frac{dU_{max}}{dz} \frac{du^{*}}{dr}
\end{equation}
Substituting the value of \(U_{max}\) from Eq. (\ref{eq:39}) into Eq. (\ref{eq:39a}) results in Eq. \ref{eq:39b}.
\begin{equation}\label{eq:39b}
\frac{dU_{l, mean}}{dz} = \frac{\mu_{v}}{\mu_{l}} \delta_{eq} \, \frac{A_{cs,bub}}{\iint u^{*} r \, dr \, d\theta} \times \frac{dU_{v, mean}}{dz} \frac{du^{*}}{dr}
\end{equation}
From the energy balance the mean velocity variation vapor velocity is given by the Eq. (\ref{eq:40}).
\begin{equation}\label{eq:40}
\frac{d U_{v, mean}}{dz} = \frac{p_{h} \, q}{A_{cs,bub} \, i_{lv} \, \rho_{v}}
\end{equation}
\(dU_{v,mean} / dz\) from Eq. (\ref{eq:40}) is substituted into Eq. (\ref{eq:39b}) and the resulting value of \(dU_{v,mean} / dz\) into Eq. (\ref{eq:33}) to obtain the rate of depletion of the thin film due to shear stress.
In the case of partial dryout region , liquid is present in the corners and the effect of the shear stress on the depletion of the thin film is modeled as described below.
\begin{equation}\label{eq:40a}
\frac{d \delta_{corner}}{dt} = -\delta_{corner} \, \frac{d U_{l,mean}}{dz}
\end{equation}
\(dU_{l}/dz\) is calculated using the Eq. (\ref{eq:37}) to (\ref{eq:39a}). \(dU^{*}/dr\) is calculated using hyperellipse with exponent as 4 and major and minor axes as mentioned in the section on partial dryout region. The limits of \(\theta\) and \(r\) in Eq. (\ref{eq:39}) and (\ref{eq:39b}) correspond to intersection made by the hyperellipse with the rectangle.
\vspace{5mm}
In the present study, for pressure drop module, a single vapour bubble is assumed that comprises of all stages-partially confined, elongated, partial dryout and full dryout-considered in the heat transfer module. This will not affect pressure drop calculation as pressure drop across vapour bubbles is negligible, as mentioned in \textit{Pressure drop} section. But these stages do influence the heat transfer coefficient and hence are considered in heat transfer module.
\section{Solution procedure}
In order to solve the equations presented in the bubble growth and heat transfer coefficient sections the channel is divided into several discretized elements with the length as \textit{dz}. Forward time step marching is used with time step \textit{dt}. Sequence of steps followed to obtain the solution is given below.
\begin{enumerate}
\item The first step involves the determination of the location of nucleation site as explained in section 2.3.
\item The nucleation time period is calculated by the procedure described in section 2.4.
\item Based on the time period and nucleation site, a partially confined bubble of dimension \(h \times h \times h\) is placed at the nucleation site, as mention in \textit{Assumptions} section.
\item Velocity at the downstream end of the bubble is calculated using Eq. (\ref{eq:6}) or (\ref{eq:9}) and bubble length based on Eq. (\ref{eq:5}) or Eq. (\ref{eq:8}) depending on the on the bubble growth stage.
\item Then frictional and acceleration pressure drops are calculated using Eq. (\ref{eq:13}) and Eq. (\ref{eq:15}) respectively.
\item Inlet stagnation pressure is calculated and then the inlet velocity using Eq. (\ref{eq:fr4}) if inlet compressibility is present.
\item Local heat transfer coefficients are calculated based on the zones as explained in section 2.8, considering the fluid property variation with the local transient pressure.
\item Time step is then incremented and the steps (4)-(7) are repeated until periodic behavior is achieved.
\item Time averaged and channel averaged pressure drop and heat transfer coefficient are calculated.
\end{enumerate}
\section{Results and discussion}
\subsection{Validation and verification}
\subsubsection{Time step and grid size independence study}
Grid independence and time-step independence studies are carried out to make sure that the numerical errors do not affect the solution. Transient heat transfer coefficient and pressure drop variations are observed and it can be seen from Fig. \ref{fig:2} and \ref{fig:3} that the change is negligible for the grid size and time step smaller than \(2 \times 10^{-5}\) m and \(2 \times 10^{-5}\) s respectively.
\begin{figure}[H]
\centering
\subfloat[Transient pressure drop.\label{fig:2a}]{\includegraphics[width=0.5\textwidth]{griddp.pdf}}\hfill
\subfloat[Transient heat transfer coefficient.\label{fig:2b}] {\includegraphics[width=0.5\textwidth]{gridhtc.pdf}}
\caption{Grid size independence study for channel \( 0.24 \times 0.50 \times 25\) mm, \(q = 400 \, kW/m^{2} \), \(G = 500 \, kg/m^{2}s\), \(T_{in} = 373 \, K\) and \(P_{e} = 101 \, kPa\).} \label{fig:2}
\end{figure}
\begin{figure}[H]
\centering
\subfloat[Transient pressure drop.\label{fig:3a}]{\includegraphics[width=0.5\textwidth]{timedp.pdf}}\hfill
\subfloat[Transient heat transfer coefficient.\label{fig:3b}] {\includegraphics[width=0.5\textwidth]{timehtc.pdf}}
\caption{Time step independence study for channel \( 0.24 \times 0.50 \times 25\) mm, \(q = 400 \, kW/m^{2} \), \(G = 500 \, kg/m^{2}s\), \(T_{in} = 373 \, K\) and \(P_{e} = 101 \, kPa\).} \label{fig:3}
\end{figure}
\subsubsection{Transient pressure and heat transfer coefficient fluctuations}
Comparison of the modeled pressure fluctuation with the experimental data reported by \citet{Brutin2004} for n-Pentane is shown in Fig. \ref{fig:transp_a}. The amplitude predicted by the model closely matches the experimental observation. The nucleation time period used in the model is the same as the observed fluctuation period. The nucleation time period may also depend on the surface characteristics which has not been taken into account in the present study. The location of the nucleation site is calculated as described in section 2.3. The modeled transient pressure has also been compared with the CFD prediction made by \citet{Zu2011} for water and is shown in Fig. \ref{fig:transp_b}. The trend matches with the 3-dimensional CFD result, but the amplitude obtained from model is lower than that predicted by CFD study due to the fact that the modeling considers fluid property variation with local pressure, but the CFD study assumes constant fluid property. The increase in pressure drop is due to the acceleration of the liquid slugs and the corresponding frictional pressure drop. As the liquid slug leaves the channel, the amount of accelerated liquid slug present in the channel decreases and hence the drop in pressure.
\begin{figure}[H]
\centering
\subfloat[Comparison with \citet{Brutin2004}. \label{fig:transp_a}]{\includegraphics[width=0.5\textwidth]{pflucbrutin.pdf}}\hfill
\subfloat[Comparison with \citet{Zu2011}.\label{fig:transp_b}] {\includegraphics[width=0.5\textwidth]{pflucCFD.pdf}}
\caption{Transient pressure fluctuation variation comparison. } \label{fig:transp}
\end{figure}
Fig. \ref{fig:transhtc} shows the comparison of the modeled local transient heat transfer coefficient with the experimental data of \citet{Jagirdar2016} for water, under different operating conditions. The modeled heat transfer coefficient is in reasonably good agreement with the experimental data, both in trend and magnitude.
The local heat transfer coefficient variation from A to B is due to the acceleration of the liquid slug caused by the bubble growth on the upstream side. The jump from B to C is due to the change in zone from liquid slug to vapour bubble that causes a sudden increase in heat transfer coefficient. The gradual increase in heat transfer coefficient from C to D is attributed to the reduction in the thin film thickness caused by evaporation and shear stress. The drop from D to E is due to the change in zone from vapour bubble to liquid slug at that location. The heat transfer coefficient remains constant from E to F, as the liquid slug with constant velocity passes over the location. This continues until a bubble nucleates, which then causes an increase in the liquid velocity and hence the heat transfer coefficient (from A to B).
\begin{figure}[H]
\centering
\subfloat[Comparison with \(q = 172 \, kW/m^{2}\) and \(G = 200 \, kg/m^{2}s\). \label{fig:transhtc_a}]{\includegraphics[width=0.5\textwidth]{htcfluccase1.pdf}}\hfill
\subfloat[Comparison with \(q = 320 \, kW/m^{2}\) and \(G = 400 \, kg/m^{2}s\).\label{fig:transhtc_b}] {\includegraphics[width=0.5\textwidth]{htcfluccase2.pdf}}
\caption{Comparison of transient heat transfer coefficient variation with \citet{Jagirdar2016} at 19.05 mm from inlet.} \label{fig:transhtc}
\end{figure}
\subsubsection{Time averaged pressure drop and heat transfer coefficient }
In this section, time-averaged pressure drop and heat transfer coefficients are compared with the experimental data reported by \citet{jayaramu2018experimental}. Fig. \ref{fig:4} shows the comparison for a channel of dimension \(0.24 \times 0.50 \times 40 \) mm, for the specified operating conditions. The modeled time-averaged pressure drop is within 25\% of the experimental average pressure drop, as shown in Fig. \ref{fig:4a}, and the time-averaged heat transfer coefficient within 24\%, as in Fig. \ref{fig:4b}. The maximum deviations for a channel of dimension \(0.49 \times 1.00 \times 40 \) mm are 20\% and 5\% for pressure drop and heat transfer coefficient respectively, as shown in Fig. \ref{fig:4}. The deviations may perhaps be attributed to the change in flow patterns.
\begin{figure}[H]
\centering
\subfloat[Time average pressure drop.\label{fig:4a}]{\includegraphics[width=0.5\textwidth]{prasdphtcaverage.pdf}}\hfill
\subfloat[Time and spactially heat transfer coefficient\label{fig:4b}.] {\includegraphics[width=0.5\textwidth]{prashtcvalidaverage.pdf}}
\caption{Comparison of pressure drop and heat transfer coefficient with the experimental data \citet{jayaramu2018experimental}, channel dimension 0.24 \(\times\) 0.50 \(\times\) 40 mm.} \label{fig:4}
\end{figure}
\begin{figure}[H]
\centering
\subfloat[{Time average pressure} drop.\label{fig:4a6}]{\includegraphics[width=0.5\textwidth]{papdpvalid658.pdf}}\hfill
\subfloat[Time and spatially heat transfer coefficient\label{fig:4b6}.] {\includegraphics[width=0.5\textwidth]{paphtcvalid658.pdf}}
\caption{Comparison of pressure drop and heat transfer coefficient with the experimental data \citet{jayaramu2018experimental}, channel dimension 0.49 \(\times\) 1.0 \(\times\) 40 mm.} \label{fig:46}
\end{figure}
In the present model, the calculated quality is different from the thermodynamic quality. Liquid slugs are assumed to be superheated due to the heat transfer from the wall and the heat conducted between the liquid slug and the vapour bubble is neglected as the area of contact between the liquid slug and vapour bubble is assumed to be small compared to the total heat transfer area of liquid slug. The quality is estimated by the Eq. \ref{eq:q_est}.
\begin{equation}\label{eq:q_est}
x_{est} = \frac{\sum \rho_{v} U_{v}}{\sum \rho_{v} U_{v} + \sum \rho_{l} U_{l}}
\end{equation}
The summation is over the time period at the channel exit. The variation in quality with heat flux is presented in Fig. \ref{fig:q}. There is a reduction in the percentage of deviation between the thermodynamic quality and the actual quality as the heat flux increases. This is due to the decrease in the amount of liquid slug with the increase in frequency caused by the increase in heat flux. There is also a reduction in the percentage of deviation with the decrease in mass flux, due to the decrease in the amount of liquid slugs. \citet{Magnini2015} performed CFD simulations for confined bubble growth in a microtube under constant heat flux condition and reported 82\% utilization of the heat flux for the bubble growth, which perhaps indicates the presence of superheated liquid and hence the deviation between the actual and thermodynamic quality. The observations from the present study are in line with their results.
\begin{figure}[H]
\centering
\subfloat[Channel dimension \(0.49 \times 1.0 \times 40 \) mm. \label{fig:qa}]{\includegraphics[width=0.5\textwidth]{quality628.pdf}}\hfill
\subfloat[Channel dimension \(0.24 \times 0.5 \times 40 \) mm.\label{fig:qb}] {\includegraphics[width=0.5\textwidth]{quality324.pdf}}\hfill
\caption{Variation of estimated quality and thermodynamic quality.} \label{fig:q}
\end{figure}
Fig. \ref{fig:corr} makes a comparison between the spatio-temporally averaged heat transfer from the model and that obtained from the existing steady flow correlations (Table \ref{table}) in the literature. For lower mass flux, the model prediction lies in between the ones obtained using \citet{Sun2009} model and \citet{li2010general} model, and for higher it is in between \citet{Sun2009} and \citet{mahmoud2013heat}. The possible reasons for deviation are the conditions used for the development of correlations and the influence of heating surface characteristics which has not been taken into account in the present model and also in the development of the correlations.
\begin{figure}[H]
\centering
\subfloat[Correlation comparison for \(G = 600 \,kg/m^{2}s.\) \label{fig:corra}]{\includegraphics[width=0.5\textwidth]{corr600.pdf}}\hfill
\subfloat[Correlation comparison for \(G = 1000 \,kg/m^{2}s.\)\label{fig:corrb}] {\includegraphics[width=0.5\textwidth]{corr1000.pdf}}\hfill
\caption{Comparison of estimated heat transfer coefficient with the existing correlations for channel \(0.49 \times 1.0 \times 40 \, mm \), \(P_{e} = 101 \, kPa\) and \(T_{in} = 373 \,K\).} \label{fig:corr}
\end{figure}
\subsection{Different zones}
The occurrence of five different zones during flow boiling depends on the channel dimension, inlet condition, mass flux and heat flux. Fig. \ref{fig:zonea} shows different zones passing cyclically over the location that is 1.25 mm away from the inlet. Initially the transients are affected by the initial condition during simulation and later the cycle repeats itself. Zone A-B is the liquid slug zone (un-accelerated), B-C is the liquid slug region (accelerated), the jump C-D represents transition from liquid to partial confinement, D-E the partially confined bubble region, E-F the jump from partial confinement to full confinement, F-G the fully confined or elongated bubble stage and G-H the transition from the bubble to the liquid slug zone. Fig. \ref{fig:zoneb} shows different zones over a location that is 3.75 mm from inlet, for with and without shear and for different minimum liquid film thickness values for dryout. Zones A-B, B-C, C-D and D-E indicate the liquid slug zone (un-accelerated), the liquid slug zone (accelerated by partially confined bubble), the liquid slug zone (accelerated by fully confined bubble) and the transition from the liquid slug to the elongated bubble region, respectively. The zones E-F, E-F' and E-F'' represent elongated bubble region for the case with shear (3 \(\mu m\)), with shear (0.3 \(\mu m\)) and without shear (3 \(\mu m\)) respectively. The minimum liquid film thickness values considered in the present study are based on the experimental observations by \citet{sun2018transient}. Zones F-G, G-H, H-I, I-J and J-K represent the transition to partial dryout, the partial dryout region, the transition to full dryout, the full dryout region and the transition to the liquid slug region, respectively. Zones F'-K and F''-K indicate the transition from elongated bubble to liquid slug region. The differences between the two figures are the presence of partially confined bubble region and the absence of partial and full dryout in Fig. \ref{fig:zonea} as against Fig. \ref{fig:zoneb}. The effect of shear stress is discussed in detail in the section 4.3.
\begin{figure}[H]
\subfloat[) Local heat transfer coefficient variation with time. \label{fig:zonea}] {\includegraphics[width=0.5\textwidth]{zones1.pdf}}\hfill
\subfloat[Local heat transfer coefficient variation with time.\label{fig:zoneb}] {\includegraphics[width=0.5\textwidth]{zones3.pdf}}
\caption{Various zones during flow boiling, \( 0.20 \times 0.60 \times 25\) mm with \(G = 500 \, kg/m^{2}s\) , \(q = 200 \, kW/m^{2}\) , \(P_{e} = 101 \, kPa\) and \(T_{in} = 373 \, K\).} \label{fig:zone}
\end{figure}
\subsection{Shear stress effect}
In the process of bubble growth, the thin liquid film gets depleted due to the evaporation of film and also due to the shear stress acting on the liquid-vapor interface. The models presented by \citet{Thome2004} and \citet{Magnini2017} for circular channel and \citet{Wang2010} for rectangular channel, do not take into account the influence of shear. Fig. \ref{fig:14} shows the effect of shear stress on the variations of heat transfer coefficient and thin film thickness. Dashed lines indicate the heat transfer coefficient with shear and solid lines without shear. The spatially averaged and the time-averaged heat transfer coefficients with shear stress are higher than that without shear, as shown in Fig. \ref{fig:14a} and Fig. \ref{fig:14b}, respectively. This can be explained from Fig. \ref{fig:14c} that shows the variation of local heat transfer coefficient with time for two different locations (\(0.2 \, \times \, L_{ch}\) and \(0.4 \, \times \, L_{ch}\) from the inlet). With shear, the heat transfer coefficient during the passage of elongated bubble is higher compared to that without shear due to the higher depletion rate of thin film with shear as shown in Fig. \ref{fig:14d}. The reduction in the thin film thickness is larger for the upstream location as the rate of depletion of thin film is directly proportional to the film thickness as indicated by the Eq. (\ref{eq:33}). This leads to a larger increase in the heat transfer coefficient for the upstream location.
\begin{figure}[H]
\centering
\subfloat[Variation of spatially averaged heat transfer coefficient with time. \label{fig:14a}]{\includegraphics[width=0.5\textwidth]{shearhtctime.pdf}}\hfill
\subfloat[Variation of time-averaged heat transfer coefficient with location. \label{fig:14b}] {\includegraphics[width=0.5\textwidth]{shearhtcloc.pdf}}\hfill
\end{figure}
\begin{figure}[H]
\subfloat[ Local heat transfer coefficient variation with time. \label{fig:14c}] {\includegraphics[width=0.5\textwidth]{shearhtc4L6L.pdf}}\hfill
\subfloat[Local thin film thickness variation with time.\label{fig:14d}] {\includegraphics[width=0.5\textwidth]{shearfilm4L6L.pdf}}
\caption{Effect of shear stress, \( 0.10 \times 0.20 \times 25\) mm with \(G = 500 \, kg/m^{2}s\), \(q = 100 \, kW/m^{2}\), \(P_{e} = 101 \, kPa\) and \(T_{in} = 373 \, K\).} \label{fig:14}
\end{figure}
\subsection{Effect of inlet compressibility}
This section presents the influence of inlet compressibility caused by trapped non-condensable gas on pressure fluctuations, flow reversals and heat transfer coefficients. Modeling the location of nucleation site and nucleation frequency with flow reversals is difficult due to the complex thermo-fluidics and hence the approaches presented in Sections 2.3 and 2.4 are not valid for the cases with flow reversals. Here, for simplicity, the nucleation site is assumed to be in the middle of the channel to account for the bubble growth in the upstream direction during flow reversal. Bubble is assumed to nucleate at regular intervals provided there is no bubble passing over the nucleation site at that instant of time. With inlet compressibility, there is a flow reversal, i.e., the bubble grows in both the upstream direction and the downstream direction as shown in Fig. \ref{fig:17a}. The flow reversal can be explained using Eq. (\ref{eq:fr1}) and (\ref{eq:fr8}). The flow, after reversing for certain duration, changes its direction and moves forward. The extent of flow reversal is coupled with the change in the inlet (plenum or stagnation) pressure. Fig. \ref{fig:17b} shows that there is a reduction in the amplitude of pressure fluctuation with the increase in compressible volume, which is in line with the experimental observations made by \citet{Liu2013}. This can be attributed to the decrease in the net acceleration (and hence velocities) due to flow reversal. Fig. (\ref{fig:17c}) indicates higher amplitudes of heat transfer coefficient fluctuation and enhanced average heat transfer coefficient in the presence of instabilities. The spatial and temporal averaged values of heat transfer coefficient are 30.8, 37.0 and 39.3 \(kW/m^{2}K\) for no compressible volume, 4 \(V_{ch}\) and 8 \(V_{ch}\) respectively. The maximum heat transfer coefficient is higher for the case with flow reversal due to larger residence period of the confined bubble leading to larger contribution from the thin film evaporation as shown in Fig. \ref{fig:9a} to \ref{fig:9d}. The minimum heat transfer coefficient with flow reversal is lower than that without flow reversal, due to lower velocities (caused by reduction in the net acceleration) of liquid slugs with flow reversals. Fig. \ref{fig:9a} and \ref{fig:9b} show the variation of local heat transfer coefficients with time on the upstream side of the nucleation site and Fig. \ref{fig:9c} and \ref{fig:9d} show the variation on downstream side of the nucleation site. It is clear that the thin film evaporation exists for a longer duration over a larger length for the cases with flow reversals, resulting in higher heat transfer coefficients. Higher heat transfer coefficients associated with the flow reversal cases are in line with the experimental observations made by \citet{Gedupudi2011}. \citet{Kenning2001} reported no appreciable change in the heat transfer coefficient due to compressible volume for the range of parameters and dimensions considered in their experimental study. \citet{Liu2013} reported higher heat transfer coefficients with compressible volume for saturated boiling of water for certain heat flux and mass flux and enhanced heat transfer coefficient only at thermodynamic quality greater than 0.05 for a different heat flux and mass flux. \citet{li2017effect} carried out the experimental investigation of the influence of periodic flow reversal on the performance of microchannel heat exchanger and reported higher heat transfer coefficients, mainly in the upstream part. The results obtained from the present model, shown in Fig. \ref{fig:9a} and \ref{fig:9b}, are very similar to their experimental observations.
\begin{figure}[H]
\centering
\subfloat[Comparison of position of upstream and downstream end of bubble.\label{fig:17a}] {\includegraphics[width=0.5\textwidth]{xupos.pdf}}\hfill
\subfloat[Variation of pressure drop with time. \label{fig:17b}]{\includegraphics[width=0.5\textwidth]{flowreversaldpglobal.pdf}}\hfill
\subfloat[Variation of spatial-averaged heat transfer coefficient with time. \label{fig:17c}] {\includegraphics[width=0.5\textwidth]{flowreversalhtcglobal.pdf}}\hfill
\caption{Effect of inlet compressibility on bubble position, pressure drop and heat transfer coefficient, 0.24 x 0.50 x 40 mm with \(G = 500 \, kg/m^{2}s\), \(q = 200 \, kW/m^{2}\), \(L1 = 0.5 \, L_{ch}\), \(P_{e} = 101 \, kPa\) and \(T_{in} = 373 \,K\).}\label{fig:17}
\end{figure}
\begin{figure}[H]
\centering
\subfloat[At \(L = 0.2 \times L_{ch}\). \label{fig:9a}]{\includegraphics[width=0.5\textwidth]{flowrev02L.pdf}}\hfill
\subfloat[At \(L = 0.4 \times L_{ch}\).\label{fig:9b}] {\includegraphics[width=0.5\textwidth]{flowrev04L.pdf}}\hfill
\end{figure}
\begin{figure}[H]
\subfloat[At \(L = 0.6 \times L_{ch}\).\label{fig:9c}] {\includegraphics[width=0.5\textwidth]{flowrev06L.pdf}}\hfill
\subfloat[At \(L = 0.8 \times L_{ch}\).\label{fig:9d}] {\includegraphics[width=0.5\textwidth]{flowrev08L.pdf}}
\caption{Variation of heat transfer coefficient at various locations under different inlet compressibility, \(0.24 \times 0.50 \times 40\) mm with \(G = 500 \, kg/m^{2}s\) , \(q = 200 \, kW/m^{2}\),\(L1 = 0.5 \times L_{ch}\), \(P_{e} = 101 \, kPa\) and \(T_{in} = 373 \,K\).} \label{fig:9}
\end{figure}
Fig. \ref{fig:zone_reva} and Fig. \ref{fig:zone_revb} show the local transient variation of heat transfer coefficient and thin film thickness
respectively. Larger residence time of the bubble for the case of flow reversal, as shown in Fig. \ref{fig:zone_revb}, causes complete dryout on the flat surfaces leaving the liquid film only at the corners, termed as partial dryout that results in lower local heat transfer coefficient as shown in Fig. \ref{fig:zone_reva}. In Fig. \ref{fig:zone_revb}, A-B indicates transition from liquid slug to elongated bubble, B-C depletion of thin film on flat face in elongated bubble region, C-D transition to partial dryout and D-E partial dryout. \citet{tuo2013periodic} observed decrease in heat transfer coefficient due to the emergence of dryout during flow reversal. C'-D' denotes transition from vapour bubble to liquid slug. The film thickness at C reaches minimum film thickness whereas at C' it is higher than minimum film thickness. A-E and A'-D' indicate the bubble residence period for flow reversal case and no flow reversal case respectively.
\begin{figure}[H]
\subfloat[Local heat transfer coefficient variation with time.\label{fig:zone_reva}]
{\includegraphics[width=0.5\linewidth]{zonesflowrev.pdf}}\hfill
\subfloat[Local thin film thickness variation with time.\label{fig:zone_revb}]
{\includegraphics[width=0.5\linewidth]{zonesflowrevfilm.pdf}}
\caption{Local heat transfer coefficient under flow reversal and no flow reversal case, \( 0.20 \times 0.40 \times 25\) mm with \(G = 900 \, kg/m^{2}s\), \(q = 300 \, kW/m^{2}\), \(L1 = 0.5 \times L_{ch}\) and \(P_{e} = 101 \, kPa\).}
\label{fig:zone_rev}
\end{figure}
\subsection{Effect of local pressure}
In this section, the effect of local pressure on the pressure drop and the heat transfer coefficient is studied. In the models previously developed by \citet{kew1996pressure}, \citet{Wang2010}, \citet{Thome2004} and \citet{He2016}, fluid property variation with local transient pressure was not considered. Fig. \ref{fig:6a} makes a comparison between the transient variation of channel pressure drop calculated considering the property variation with local pressure and assuming constant pressure equal to exit pressure. The peak pressure obtained considering the property (vapour density, latent heat, liquid density and liquid viscosity) variation with local pressure is much lower and the bubble residence time is higher than that with constant property case. This is due to the reduction in the bubble acceleration due to property variation with local pressure. Similarly, Fig. \ref{fig:6b} the transient variation of channel heat transfer coefficient. The heat transfer coefficient obtained with variable property is lower than that with constant property, due to the changes in initial film thickness and the velocities of accelerated liquid slugs. Fig. \ref{fig:7a} to \ref{fig:7d} shows the transient variation of local heat transfer coefficient at different axial locations. The difference between the constant property case and variable property case, especially the peak values, increases along the flow direction, due to the increase in the difference between the accelerations (which influence thin film depletion rate and liquid slug velocity) obtained with constant property case and variable property case.
\begin{figure}[H]
\centering
\subfloat[Variation of pressure drop with time. \label{fig:6a}]{\includegraphics[width=0.5\textwidth]{constvardptime.pdf}}\hfill
\subfloat[Variation of spatial averaged heat transfer coefficient with time.\label{fig:6b}] {\includegraphics[width=0.5\textwidth]{constvarhtctime.pdf}}\hfill
\caption{Effect of local pressure condition on pressure drop and heat transfer coefficient, \(0.10 \times 0.20 \times 25 \) mm with \(G = 500 \, kg/m^{2}s\), \(q = 100 \, kW/m^{2}\), \(P_{e} = 101 \, kPa\) and \(T_{in} = 373 \, K\).} \label{fig:6}
\end{figure}
\begin{figure}[H]
\centering
\subfloat[At \(L = 0.2 L_{ch}\). \label{fig:7a}]{\includegraphics[width=0.5\textwidth]{constvar02L.pdf}}\hfill
\subfloat[At \(L = 0.4 L_{ch}\).\label{fig:7b}] {\includegraphics[width=0.5\textwidth]{constvar04L.pdf}}\hfill
\end{figure}
\begin{figure}[H]
\subfloat[At \(L = 0.6 L_{ch}\).\label{fig:7c}] {\includegraphics[width=0.5\textwidth]{constvar06L.pdf}}\hfill
\subfloat[At \(L = 0.8 L_{ch}\).\label{fig:7d}] {\includegraphics[width=0.5\textwidth]{constvar08L.pdf}}
\caption{Effect of local pressure condition on heat transfer coefficient, \(0.10 \times 0.20 \times 25\) mm with \(G = 500 \, kg/m^{2}s\), \(q = 100 \, kW/m^{2}\), \(T_{in} = 373 \,K\) and \(P_{e} = 101 \, kPa\).} \label{fig:7}
\end{figure}
\subsection{Effect of fluid properties}
Fig. \ref{fig:13} shows the effect of fluid properties on the channel transient pressure drop and heat transfer coefficient for flow boiling of water (\(P_{e}\) = 101 and 19.8 kPa) and R134a (\(P_{e}\) = 76.6 kPa). For R134a, the expressions for nucleation frequency and initial film thickness are taken from \citet{Thome2004}. The pressure drop and its amplitude for R134a is almost negligible as compared to that for water, due to the higher value of the product of vapour density and latent heat of vaporization for R134a. Hence, the assumption of constant property for R134a can be justified, but the same assumption cannot be used for water. The amplitude of pressure fluctuation for water with \(P_{e}\) = 101 kPa is lower than that for water with \(P_{e}\) = 19.8 kPa , as the product of vapour density and latent heat of vaporization is higher for water with \(P_{e}\) = 101 kPa. However, the heat transfer coefficient for R134a is comparable, though lower than that for water. The heat transfer coefficient for water with \(P_{e}\) = 76.6 kPa is higher than that for water with \(P_{e}\) = 101 kPa due to the higher nucleation frequency and acceleration for the former.
\begin{figure}[H]
\centering
\subfloat[Variation of pressure drop with time. \label{fig:13a}]{\includegraphics[width=0.5\textwidth]{fluidpressdp.pdf}}\hfill
\subfloat[Variation of heat transfer coefficient with time.\label{fig:13b}] {\includegraphics[width=0.5\textwidth]{fluidpresshtc.pdf}}
\caption{Effect of fluid properties on pressure drop and heat transfer coefficient, 0.24 \(\times\) 0.50 \(\times\) 40 mm with \(G = 500 \,kg/m^{2}s\), \(q = 100 \,kW/m^{2}\).} \label{fig:13}
\end{figure}
\section{Conclusions}
In the present study, a 1-dimensional semi-mechanistic model combining pressure and heat transfer coefficient is proposed for flow boiling in a rectangular mini/micro-channel. The major contributions of the present study, based on the available literature, are the evaluation of transient local heat transfer coefficient in conjunction with the local transient pressure, inclusion of the effect of shear stress at liquid-vapour interface on heat transfer coefficient, addition of partial-confined bubble zone to heat transfer model and applying the developed model to evaluate the heat transfer characteristics under flow reversal condition, for flow boiling in a rectangular microchannel. The proposed model is validated with the transient and time-averaged data available in the literature. The major conclusions that can be drawn from the present work are as follows.
\begin{enumerate}
\item For high aspect ratio (\(w>>h\)) and shorter channels, the duration over which the partial confinement occurs can be significant and therefore the present model incorporates the same into the heat transfer model. The local heat transfer coefficient is evaluated considering the heat transfer through the thin film and the surrounding bulk liquid.
\item The proposed model takes in account the evaporation as well as the shear stress at liquid-vapour interface for the evaluation of rate of thin film depletion during flow boiling in a rectangular mini/micro-channel. Results indicate that the inclusion of shear stress increases the heat transfer coefficient and also leads to early local dryout.
\item The present model has been applied to determine the heat transfer characteristics under flow reversal condition caused by inlet compressibility. Results show that the heat transfer coefficient slightly increases due to the larger contribution from the thin film evaporation resulting from the longer residence period of the elongated bubble. The model also demonstrates that the flow reversal can lead to partial / full dryout based on the operating conditions.
\item The proposed model incorporates the variation of fluid properties with local pressure. The estimated transient and time-averaged pressure drop and heat transfer coefficients are lower than those evaluated with constant pressure equal to the channel exit pressure. The effect of variable vapour properties is significant for water and is found to be negligible for refrigerants.
\item The present model for a rectangular channel neglects the transient conduction in the thin film. This can be a future study. The present model does not impose thermodynamic equilibrium, hence the estimated exit qualities are lower than the thermodynamic qualities, which seem to match with the CFD prediction in the literature. The presence of superheated liquid slugs, responsible for lowering the quality during flow boiling in mini/microchannels, needs further investigation.
\end{enumerate}
\clearpage
\section*{Nomenclature}
\begin{tabbing}
xxxxxxxxxxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \kill
xxxxxxxxxxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \kill
\(a\) \> major dimension in hyperellipse \\
\(A\) \> area, \(m^{2}\) \\
\(b\) \> minor dimension in hyperellipse \\
\(Bo\) \> Boiling number \((Bo = q / G \, i_{lv})\) \\
\(Bd\) \> Bond number \((Bd = \frac{g \, \Delta \rho \, D_{h}^{2}}{\sigma})\) \\
\(C1\) \> growth factor \\
\(C_{p}\) \> specific heat, \(J/kg \, K\) \\
\(C_{\delta}\) \> factor for initial film thickness \\
\(Co\) \> confinement number \(\Big(Co = \frac{(\sigma / g \Delta \rho)^{0.5}}{D_{h}})\)\\
\(cp\) \> channel perimeter, \(m\) \\
\(D_{h}\) \> hydraulic diameter, \(m\) \\
\(\Delta T_{sup}\) \> wall superheat \((T_{wall} - T_{sat})\), K \\
\(dt\) \> time step, \(s\) \\
\(dz\) \> grid size, \(m\) \\
\(f\) \> Fanning friction factor \\
\(F_{new}\) \> factor used in correlation \\
\(g\) \> acceleration due to gravity, \(m/s^{2}\) \\
\(G\) \> mass flux, \(kg/m^{2}s\) \\
\(h\) \> height of the channel, \(m\) \\
\(i_{lv}\) \> latent heat of vaporization, \(J/kg\) \\
\(Ja\) \> Jacob number \((Ja = \frac{C_{p,l} \, \Delta T_{sup} \, \rho_{l}}{\rho_{v} \, i_{lv}})\) \\
\(k\) \> conductivity, \(W/m \, K\) \\
\(K\) \> flow reversal factor \\
\(L\) \> length, \(m\) \\
\(l_{a}\,\, l_{b}, \, l_{c}\) \> liquid length intercepts, \(m\) \\
\(L1\) \> nucleation site, \(m\) \\
\(L_{ch}\) \> length of channel, \(m\) \\
\(m\) \> last bubble near the exit \\
\(M_{w}\) \> molecular weight, \(kg/mol\) \\
\(n\) \> newest bubble in channel \\
\(n1\) \> factor used in shape determination \\
\(N_{co}\) \> convection number \((N_{co} = (\frac{1-x}{x})^{0.8} \, (\frac{\rho_{v}}{\rho_{l}})^{0.5})\) \\
\(Nu\) \> Nusselt number \\
\(p\) \> perimeter \\
\(P\) \> pressure, \(Pa\) \\
\(\Delta P\) \> pressure drop, \(Pa\) \\
\(Pr\) \> Prandtl number \\
\(P_{crit}\) \> critical pressure , \(Pa\) \\
\(P_{r}\) \> reduced pressure \((P_{r} = P_{sat} / P_{crit})\) \\
\(P_{e}\) \> exit pressure,\(Pa\) \\
\(q\) \> heat flux, \(W/m^{2}\) \\
\(r\) \> radial distance, \(m\) \\
\(R\) \> radius of bubble, \(m\) \\
\(R_{a}\) \> average roughness, \(\mu \, m\) \\
\(Re\) \> Reynolds number \((Re = \frac{\rho \, U \, D_{h}}{\mu})\) \\
\(R_{p}\) \> maximum valley depth, \(\mu \, m\) \\
\(S_{new}\) \> factor \\
\(T\) \> temperature, \(K\) \\
\(t\) \> time, \(s\) \\
\(\overline{T}\) \> mean temperature, \(K\) \\
\(u^{*}\) \> non dimensional velocity, Eq.\ref{eq:38} \\
\(U\) \> velocity, \(m/s\) \\
\(V\) \> volume, \(m^{3}\) \\
\(\dot{V}_{in}\) \> inlet volume flow rate, \(m^{3}/s\) \\
\(w\) \> width of the channel, \(m\) \\
\(We\) \> Weber number \((We = \frac{G^{2}\, D_{h}}{\sigma \rho})\) \\
\(x\) \> quality \\
\(X\) \> Matinelli parameter \\
\(z\) \> position, \(m\) \\
\end{tabbing}
\subsection*{Greek letters}
\begin{tabbing}
xxxxxxxxxxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \kill
xxxxxxxxxxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \kill
$\alpha$ \>aspect ratio \((h/w)\)\\
$\delta$ \> thin film thickness, \(m\)\\
$ \Gamma$ \> gamma function \\
$\rho$ \> density, $kg/m^{3}$\\
$\mu$ \>dynamic viscosity, $Pa-s$\\
$\theta$ \> angle, $rad$\\
$\sigma$ \> surface tension, $N/m$ \\
$\tau$ \> time constant or shear stress\\
$\nu$ \> kinematic viscosity, $m^{2}/s$
\end{tabbing}
\subsection*{Subscripts}
\begin{tabbing}
xxxxxxxxxxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \kill
\(0\) \>stagnation point\\
\(1\) \> inlet point\\
\(3\) \> three sided heating\\
\(4\) \> four sided heating\\
\(4,\infty\) \> developed four sided heating\\
\(3,\infty\) \>developed three sided heating\\
\(acc\) \>acceleration pressure drop\\
\(Base\) \> base\\
\(bub\) \> bubble\\
\(c\) \> compressible\\
\(corner\) \>corner\\
\(ch\) \>channel\\
\(cs\) \> cross-section\\
\(d\) \>downstream end\\
\(e\) \>exit\\
\(eq\) \> equivalent \\
\(elb\) \> elongated bubble \\
\(est\) \> estimated \\
\(film\) \> film\\
\(h\) \> heated length\\
\(grow\) \> growth \\
\(heat\) \> heated \\
\(i\) \> channel inlet\\
\(ini\) \> initial \\
\(l\) \>liquid\\
\(lv\) \> liquid-vapor \\
\(lo\) \>liquid only\\
\(max\) \>maximum\\
\(mean\) \>mean\\
\(pc\) \> partially confined \\
\(pdr\) \> partial dryout region \\
\(slug\) \>liquid slug between two bubbles\\
\(sp\) \>single phase\\
\(sat\) \>saturated\\
\(sub\) \>sub-cooled\\
\(sup\) \>superheat\\
\(tp\) \>two-phase\\
\(th\) \> thermodynamic \\
\(u\) \>upstream end\\
\(v\) \>vapor\\
\(vis\) \>viscous pressure drop\\
\(wall\) \>wall\\
\(wait\) \> waiting
\end{tabbing}
\subsection*{Abbreviations}
\begin{tabbing}
xxxxxxxxxxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \kill
\(CFD\) \> Computational fluid dynamics \\
\(HTC\) \> Heat transfer coefficient
\end{tabbing}
\newpage
\begin{table}[H]
\caption{Various correlations for heat transfer coefficient}
\begin{center}
\label{table}
\renewcommand\arraystretch{0.9}
\resizebox{1.1\textwidth}{!}{
\begin{tabular}{|C{2.5cm}|c|c|}
\hline
Reference & Correlation & Comments \\
\hline
\citet{lazarek1982evaporative} & \(HTC_{tp} = \frac{k_{l}}{D_{h}} \, (30 \, Re_{lo}^{0.857} \, Bo^{0.714})\) & Mini channel correlations based on 728 points,\\ & & \(D_{h} = 3.1 \, mm\) \\ & & \(G = 125 - 750 \, kg/m^{2}s, Bo = 2.3 - 76 \times 10^{-4}\) \\ & & \(P = 1.3 - 4.1 \, bar\) \\
\hline
\citet{Kew1997} & \(HTC_{tp} = \frac{k_{l}}{D_{h}} \, (30 \, Re_{lo}^{0.857} \, Bo^{0.714}) \, (1-x)^{-0.143} \) & Mini/micro channel correlation \\ & & Based on circular data \(D_{h} = 1.39 - 3.69 \, mm\) , \\ & & working fluid R141b \\
\hline
\citet{Sun2009} & \(HTC_{tp} = \frac{6 \, Re_{lo}^{1.05} \, Bo^{0.54}}{We_{l}^{0.191} (\rho_{l}/\rho_{v})^{0.142}} \, \frac{k_{l}}{D_{h}}\) & Mini/micro channel correlation \\ & & Based on 11 different fluid and 2505 data points \\ & & \(D_{h} = 0.21 - 6.05 \, mm\) \\ & & \(G = 44 - 1500\, kg/m^{2}s\, , q = 5 - 109 \, kW/m^{2}\) \\
\hline
\citet{li2010general} & \(HTC_{tp} = 334 \, Bo^{0.3} \, \left(Bd \, Re_{l}^{0.36} \right)^{0.4} \, \frac{k_l}{D_{h}}\)& Mini-micorchannel correlation \\ & & Based on 3700 data points, 13 different fluids \\ & & \(D_{h} = 0.2 - 3 \, mm\) \\
\hline
\citet{mohamed2012statistical} & For \(D_{h} = 0.52 \, mm\) , \(x \leq 0.3\) & Mini/micro channel correlation \\ & \(HTC_{tp} = 3320 \frac{Bo^{0.63} \, We_{l}^{0.2} \, Re_{l}^{0.11}}{Co^{0.6}} \, \frac{k_{l}}{D_{h}}\) & Based on 8561 data points and R134a - working fluid
\\ & For \(D_{h} = 0.52 \, mm\) , \(x > 0.3\) & \(D_{h} = 0.52 - 4.26 \, mm \), \(G = 100-500 \, kg/m^{2}s\)
\\ & \(HTC_{tp} = 5324 \, \left[\frac{Bo^{0.3} \, We_{l}^{0.25}}{N_{co}^{0.25}}\right]^{1.79} \, \frac{k_{l}}{D_{h}}\) & \(q = 2.4 - 175.4 \, kW/m^{2}\), \(P = 6- 14 \, bar\) \\
\hline
\citet{mahmoud2013heat} & \(HTC_{tp} = S_{new} \, HTC_{Cooper} + F_{new} \, HTC_{l}\) & Mini/micro channel correlation
\\ & & Based on 5152 data points and R134a-working fluid
\\ & \(HTC_{Cooper} = 55 \, P_{r}^{0.12- 0.434\, ln (R_{p})} (-log\, P_{r})^{-0.55} \, M_{w}^{-0.5} \, q^{0.67} \) & \(D_{h} = 0.52 - 4.26 \, mm \), \(G = 100-700 \, kg/m^{2}s\)
\\ & \(HTC_{l} = 4.36 \, k_{l}/D_{h} \) for \(Re_{l} < \, 2000\) & \(q = 1.7 - 158 \, kW/m^{2}\), \(P = 6- 14 \, bar\)
\\ & \(HTC_{l} = 0.023 \, Re_{l}^{0.8} \, Pr_{l}^{0.4} \, \frac{k_{l}}{D_{h}} \) for \(Re_{l} > \, 3000\) &
\\ & \(Re_{l} = \frac{(1-x) \, G \, D_{h}}{\mu_{l}}\) &
\\ & \(F_{new} = (1 + \frac{2.812 \, Co^{-0.408}}{X})^{0.64}\) &
\\ & \(S = \frac{1}{1 + 2.56 \times 10^{-6} (Re_{l} F_{new}^{1.25})^{1.17}}\) &
\\ & \(X = (\frac{f_{l}}{f_{v}})^{0.5} \, (\frac{\rho_{v}}{\rho_{l}})^{0.5} \, \frac{(1-x)}{x}\) & \\
\hline
\citet{Lee2005} & For \(0 < \,x \leq \,0.05\), \(HTC_{tp} = 3.856 X^{0.267} \, HTC_{l}\) & Microchannel correlation \\
& \(HTC_{l} = Nu_{3,\infty} \, k / D_{h}\) & Based on water - 207 data points, \\
& \(X_{vv} = (\frac{\mu_{l}}{\mu_{v}})^{0.5} \, (\frac{1-x}{x})^{0.5} \, (\frac{\rho_{v}}{\rho_{l}})^{0.5}\) & R134a - 111 data points \\
& \(X_{vt} = (\frac{f_{l} \, Re^{0.2}_{v} }{0.079})^{0.5} \, (\frac{1-x}{x}) \, (\frac{\rho_{v}}{\rho_{l}})^{0.5}\)& \(D_{h} = 349 \, \mu m\) \\
& \(Re_{v} = G \, x \, D_{h}/ \mu_v\) & \\
& For \(0.05 < x \leq 0.55\) , \(HTC_{l} = 436.48 \, Bo^{0.522} \, We^{0.351}_{l} \, X^{0.665} \, HTC_{l} \) & \\
& For \(0.55 < x \leq 1, \, HTC_{l} = MAX(108.6 \, X^{1.665} \, HTC_{v} , \, HTC_{v}) \) &\\
&\(HTC_{v} = Nu_{v} k_{v}/ D_{h}\) & \\
& \(Nu_{v} = 0.023 \, Re^{0.8}_{v} \, Pr^{0.4}_{v}\) & \\
\hline
\end{tabular}}
\end{center}
\end{table}
\clearpage
|
1,314,259,996,701 | arxiv | \section{Introduction}
\label{s:intro}
\textit{Clustering problems} are widely studied in Combinatorial Optimization literature due to their vast applications in Operational Research,
Machine Learning, Data Science and Engineering
\cite{WS11,LIN,CGTS99,VGRMMV01,CG99,JV01,K12,Y00,LS16,CL12,KSS10,SS18}. Typically a fixed number of centers must be placed in a metric space such that a set of clients is served the best possible way. The quality of a clustering solution is captured through the \textit{$p$-norm} of the vector consisting of the distance of each client to its closest center, for some $p\geq 1$ or $p = \infty$. For example \textit{$k$-median} and \textit{$k$-means} assume $p=1$ and $2$ respectively, while \textit{$k$-center} assumes $p=\infty$ \cite{LIN,KSS10,SS18}.
Today's access on vast data (that may be frequently updated over time) has motivated
the study of
clustering problems in case of \textit{time-evolving clients}, which dynamically change positions over time \cite{KW18,FKKLSZ19,EMS14,ANS17}. In time-evolving clustering problems, centers may also change position over time so as to better capture the clients' trajectories. For example, a city may want to reallocate the units performing rapid tests for Covid-19 so as to better serve neighborhoods with more cases,
the distribution of which may substantially change from day to day. Other interesting applications
of dynamic clustering include viral marketing, epidemiology, facility location (e.g. schools, hospitals), conference planning etc. \cite{JV11,EMS14,N3,PS01,CBK07}.
Our work is motivated by the fact that in most settings of interest, clients can move in fairly complicated and unpredictable ways, and thus, an \textit{a-priori knowledge} on such trajectories is heavily under question (most of the previous work assumes perfect knowledge on clients' positions over time \cite{EMS14,ANS17,KW18,FKKLSZ19}). To capture this lack of information we cast clustering problems under the perspective of \textit{online learning} \cite{H16}. We study an online learning problem called \textit{Dynamic $k$-Clustering} in which a \textit{learner} selects at each round $t$, the positions of $k$ centers trying to minimize the connection cost of some clients, the positions of which are unknown to the learner prior to the selection of the centers.
\begin{online_problem}[Dynamic $k$-Clustering] Given a metric space $d:V \times V \mapsto \mathbb{R}_{\geq 0}$. At each round $t$,
\begin{enumerate}
\item The learner selects a set $F_t \subseteq V$, with $|F_t| = k$, at which centers are placed.
\item The adversary selects the positions of the clients, denoted as $R_t$ (after the selection of the positions of the centers by the learner).
\item The learner suffers the connection cost of the clients,
\[C_{R_t}(F_t) = \left(\sum_{j \in R_t} d(j,F_t)^p\right)^{1/p}\]
where $d(j,F_t)$ is the distance of client $j$ to the closest center, $d(j,F_t) = \min_{i \in F_t}d_{ij}$.
\end{enumerate}
\end{online_problem}
Based on the past positions of the clients $R_1,R_2,\ldots, R_{t-1}$ an \textit{online learning algorithm} must select at each round $t$, a set of $k$ centers $F_t \subseteq V$ such that the
connection cost of the clients over time is close to the connection cost of the \textit{optimal (static) solution} $F^\ast$. If the cost of the online learning algorithm is at most $\alpha$ times the cost of $F^\ast$, the algorithm is called $\alpha$-regret, whereas in case $\alpha = 1$, the algorithm is called \textit{no-regret} \cite{H16}. Intuitively, a low-regret online learning algorithm
converges to the optimal positions of the centers (with respect to the overall trajectories of the clients) by just observing the clients' dynamics.
\begin{example}\label{ex:1}
The clients are randomly generated according to a time-varying uniform distribution with radius $0.3$ and center following the periodic trajectory $\left(\sin ( \frac{2\pi \cdot t}{T}),\cos ( \frac{2\pi \cdot t}{T})\right)$ for $t=1,\ldots,T$.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.7\linewidth]{images/uniform_circle/k=0.png}\label{fig:sub1}}\hfill
\label{f:circle}
\end{figure}
The centers placed by a (sufficiently) low-regret algorithm would converge to positions similar in structure to the ones illustrated in Figure~\ref{f:circle2} (for $k=1,2,4$ and $k=8$) which are clearly close to the optimal (static) solution for the different values of $k$.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.45\linewidth]{images/uniform_circle/k=1.png}\label{fig:sub1}}\hfill
{\includegraphics[width=0.45\linewidth]{images/uniform_circle/k=2.png}\label{fig:sub2}}\hfill
{\includegraphics[width=0.45\linewidth]{images/uniform_circle/k=4.png}\label{fig:sub2}}\hfill
{\includegraphics[width=0.45\linewidth]{images/uniform_circle/k=8.png}\label{fig:sub2}}\hfill
\caption{The figure depicts the actual centers at which a low-regret algorithm, that we subsequently propose, converges. For further details see Section~\ref{s:experiments}.}
\label{f:circle2}
\end{figure}
\end{example}
\textbf{Efficient Online Learning for Dynamic $k$-Clustering.}
The existence of no-regret online learning algorithms for Dynamic $k$-Clustering immediately follows by standard results in online learning literature \cite{H16}. Dynamic $k$-Clustering is a special case of \textit{Learning from Expert Advice} problem for which the famous \textit{Multiplicative Weights Update Algorithm} achieves no-regret \cite{H16}. Unfortunately using the $\mathrm{MWU}$ for Dynamic $k$-Clustering is not really an option due to the huge time and space complexity that $\mathrm{MWU}$ requires. In particular $\mathrm{MWU}$ keeps a different weight (probability) for each of the possible ${|V|}\choose{k}$ possible placements of the $k$ centers, rendering it inapplicable even for small values of $|V|$ and $k$.
Our work aims to shed light on the following question.
\begin{question}\label{q:main}
Is there an online learning algorithm
for Dynamic $k$-Clustering that runs in polynomial time and achieves $\alpha$-regret?
\end{question}
\smallskip
\textbf{Our Contribution and Techniques.}
We first show that constant regret cannot be achieved in polynomial time for Dynamic $k$-Clustering. In particular we prove that any $O(1)$-regret polynomial-time online learning algorithm for Dynamic $k$-Clustering
implies the existence of an $O(1)$-approximation algorithm for the \textit{Minimum-$p$-Union problem} \cite{CDKKR16}. Recent works on the theory of computational complexity establish that unless well-established cryptographic conjectures fail, there is no $O(1)$-approximation algorithm for $\mathrm{Min}$-$p$-$\mathrm{Union}$ \cite{CDKKR16,A12,CDM17}. This result narrows the plausible regret bounds achievable in polynomial time, and reveals an interesting gap between Dynamic $k$-Clustering and its offline counterparts, which admit polynomial-time $O(1)$-approximation algorithms.
Our main technical contribution consists of polynomial-time online learning algorithms for Dynamic $k$-Clustering with non trivial regret bounds. We present a $\Theta(k)$-regret polynomial-time deterministic online learning algorithm and a $\Theta(r)$-regret polynomial-time randomized online learning algorithm,
where $r$ is the maximum number of clients
appearing in a single round ($r = \max_{1\leq t \leq T}|R_t|$). Combining these algorithms, one can achieve $\Theta\left( \min(k,r) \right)$-regret for Dynamic $k$-Clustering, which (to the best of our knowledge) is the first guarantee on the regret achievable in polynomial time. The regret bounds above are independent of the selected $p$-norm, and hold for any $p \geq 1$ and for $p = \infty$.
At a technical level, our approach consists of two major steps. In the first step, we consider an online learning problem, that can be regarded as the \textit{fractional relaxation} of the Dynamic $k$-Clustering (see Section~\ref{s:fractional}), where the \textit{fractional connection cost} is given by the optimal value of an appropriate convex program and
the action space of the learner is the $|V|$-dimensional simplex. For this intermediate problem, we design a \textit{no-regret} polynomial-time online learning algorithm through the use of the subgradients of the fractional connection cost. We show that such subgradient vectors can be computed in polynomial time via the solution of
the dual program of the fractional connection cost. In the second step of our approach (see Section~\ref{s:det} and Section~\ref{s:rand}), we provide computationally efficient online (deterministic and randomized) rounding schemes converting a vector lying in the
$|V|$-dimensional simplex (the action space of Fractional Dynamic $k$-Clustering) into $k$ locations for the centers on the metric space $V$ (the action space of Dynamic $k$-Clustering).
In Section~\ref{s:det}, we present a deterministic rounding scheme that, combined with the no-regret algorithm for Fractional Dynamic $k$-Clustering, leads to a $\Theta(k)$-regret polynomial-time deterministic online learning algorithm for the original Dynamic $k$-Clustering.
Interestingly, this regret bound is approximately optimal for all deterministic
algorithms. In Section~\ref{s:rand}, we show that
combining the no-regret algorithm for Fractional Dynamic $k$-Clustering with a
randomized rounding scheme proposed in \cite{CL12}\footnote{This randomized rounding scheme was part of a $4$-approximation algorithm for $k$-median \cite{CL12}} leads to a $\Theta(r)$-regret randomized algorithm running in polynomial time. Combining these two online learning algorithms, we obtain a $\Theta(\min(k,r))$-regret polynomial-time online learning algorithm for Dynamic $k$-Clustering, which is the main technical contribution of this work. Finally, in Section~\ref{s:experiments}, we present the results of an experimental evaluation, indicating that for client locations generated in a variety of natural and practically relevant ways, the realized regret of the proposed algorithms is way smaller than $\Theta\left( \min(k,r) \right)$.
\begin{remark}
Our two-step approach provides a structured framework for designing polynomial-time low-regret algorithms in various combinatorial domains. The first step
extends far beyond the context of Dynamic $k$-Clustering and provides a systematic approach to the design of \textit{polynomial-time no-regret online learning algorithms} for the \textbf{fractional relaxation} of the combinatorial online learning problem of interest.
Combining such no-regret algorithms with
online rounding schemes, which convert fractional solutions into integral solutions of the original online learning problem, may lead to polynomial time low-regret algorithms for various combinatorial settings. Obviously, designing such rounding schemes is usually far from trivial, since the specific combinatorial structure of each specific problem must be taken into account.
\end{remark}
\textbf{Related Work.}
Our work relates with the research line of Combinatorial Online Learning. There exists a long line of research studying low-regret online learning algorithms for various combinatorial domains such that
online routing \cite{HS97,AK08}, selection of permutations \cite{TW00,YHKSTT11,FLPS20,A14,HW07}, selection of binary search trees \cite{TM03}, submodular optimization \cite{HK12a,JB11,SG08}, matrix completion \cite{HKS12},
contextual bandits \cite{ALLS14,DHKKLRZ11} and many more. Finally, in combinatorial games agents need to learn to play optimally against each other over complex domains \cite{ITLMPT11,dehghani2016price}.
As in the case of Dynamic $k$-Clustering in all the above online learning problems, MWU is not an option, due to the exponential number of possible actions.
Another research direction of Combinatorial Online Learning studies \textit{black-box reductions} converting polynomial time offline algorithm (full information on the data) into polynomial time online learning algorithms. \cite{kalai03} showed that any (offline) algorithm solving optimally and in polynomial time the objective function, that the \textit{Follow the Leader framework} suggests, can be converted into a no-regret online learning algorithm. \cite{kakade07} extended the previous result for specific class of online learning problems called \textit{linear optimization problems} for which they showed that any $\alpha$-approximation (offline) can be converted into an $\alpha$-regret online learning algorithm. They also provide a surprising counterexample showing that such black-box reductions do not hold for general combinatorial online learning problems. Both the
time efficiency and the regret bounds of the reductions of \cite{kalai03} and \cite{kakade07} were subsequently improved by \cite{rahmanian17,suehiro12,koolen10,balcan06,syrganis17,hazan16,fujita13,garber17,wei18}. We remark that
the above results do not apply in our setting since
Dynamic $k$-Clustering can neither be optimally solved in polynomial-time nor is a linear optimization problem.
Our works also relates with the more recent line of research studying clustering problems with \textit{time-evolving clients}. \cite{EMS14} and \cite{ANS17} respectively provide $\Theta\left( \log (nT)\right)$ and $O(1)$-approximation algorithm for a generalization of the facility location problem in which clients change their positions over time. The first difference of Dynamic $k$-Clustering with this setting is that
in the former case there is no constraint on the number of centers that can open
and furthermore, crucially
perfect knowledge of the positions of the clients is presumed.
More closely related to our work are \cite{KW18,FKKLSZ19}, where the special case of Dynamic $k$-Clustering on a line is studied (the clients move on a line over time). Despite the fact that both works study online algorithms, which do not require knowledge on the clients' future positions, they only provided positive results for $k=1$~and~$2$.
\section{Preliminaries and Our Results}\label{s:prelim}
In this section we introduce notation and several key notions as long as present the formal Statements of our results.
We denote by $D$ the diameter of the metric space, $D = \max_{i \in V, j \in V} d_{ij}$. We denote with $n$ the cardinality of the metric space $\left(|V| =n\right)$ and with $r$ the maximum number of clients appearing in a single round, $r = \max_{1\leq t \leq T}|R_t|$. Finally we denote with $\Delta_{n}^k$ the $n$-dimensional simplex, $\Delta_{n}^k= \{y \in \mathbb{R}^n:~ \sum_{i \in V} y_i = k ~\mathrm{and}~y_i \geq 0\}$.
Following the standard notion of regret in online learning \cite{H16}, we provide the formal definition of an \textit{$\alpha$-regret} online learning algorithm for \textit{Dynamic $k$-Clustering}.
\begin{definition}\label{d:regret}
An online learning algorithm for the \textit{Dynamic $k$-Clustering} is $\alpha$-regret if and only if for any sequence of clients' positions $R_1,\ldots,R_T \subseteq V$,
\[\sum_{t=1}^T C_{R_t}(F_t) \leq \alpha \cdot \min_{|F^\ast| \leq k} \sum_{t=1}^T C_{R_t}(F^\ast) + \Theta\left(\mathrm{poly}(n,D) \cdot T^\beta\right)\]
where $F_1,\ldots,F_T$ are the positions of the centers produced by the algorithm for the sequence $R_1,\ldots,R_T$ and $\beta < 1$.
\end{definition}
Next, we introduce the \textit{Minimum-$p$-Union} problem,
the inapproximability results of which allow us to establish that constant regret cannot be achieved in polynomial time for Dynamic $k$-Clustering.
\begin{problem}[$\mathrm{Min-}p\mathrm{-Union}$]
Given a universe of elements $\mathbb{E}$ and a collection of sets
$\mathbb{U} =\{S_1, \ldots, S_m\}$ where $S_i \subseteq \mathbb{E}$. Select $\mathbb{U}' \subseteq \mathbb{U}$ such that $|\mathbb{U'}| =p$ and $|\cup_{S_i \in \mathbb{U}'}S_i|$ is minimized.
\end{problem}
As already mentioned, the existence of an $O(1)$-approximation algorithm for $\mathrm{Min-}p\mathrm{-Union}$ violates several widely believed conjectures in computational complexity theory\cite{CDKKR16,A12,CDM17}. In Theorem~\ref{t:hardnes} we establish the fact that the exact same conjectures are violated in case there exists an online learning algorithm for \textit{Dynamic $k$-Clustering} that runs in polynomial-time and achieves $O(1)$-regret.
\begin{theorem}\label{t:hardnes}
Any $c$-regret polynomial-time online learning algorithm for the Dynamic $k$-Clustering implies a $(c+1)$-approximation polynomial-time algorithm for $\mathrm{Min-}p\mathrm{-Union}$.
\end{theorem}
In Section~\ref{s:det}, we present a polynomial-time deterministic online learning algorithm achieving \textit{$\Theta(k)$}-regret.
\begin{theorem}\label{t:det-regret}
There exists a $6k$-regret deterministic online learning algorithm for Dynamic $k$-Clustering that runs in polynomial time (Algorithm~\ref{alg:det}). More precisely,
\[\sum_{t=1}^T\mathrm{C}_{R_t}(F_t) \leq 6k \cdot \min_{|F^\ast|=k} \sum_{t=1}^T\mathrm{C}_{R_t}(F^\ast) + \Theta \left (k D n \sqrt{\log n T} \right)\]
where $F_1,\ldots,F_T$ are the positions in which Algorithm~\ref{alg:det} places the centers for the sequence of clients' positions $R_1,\ldots,R_T$.
\end{theorem}
In Theorem~\ref{t:lower_bound_det} we prove that the $\Theta(k)$ bound on the regret of Algorithm~\ref{alg:det} cannot be significantly ameliorated with deterministic online learning algorithm even if the algorithm uses exponential time and space.
\begin{theorem}\label{t:lower_bound_det}
For any deterministic online learning algorithm for Dynamic $k$-Clustering problem, there exists a sequence of clients $R_1,\ldots,R_T$ such as the regret is at least $k+1$.
\end{theorem}
In Section~\ref{s:rand} we present a randomized online learning algorithm the regret of which depends on the parameter $r$.
\begin{theorem}\label{t:rand-regret}
There exists a $\Theta(r)$-regret randomized algorithm that runs in polynomial time (Algorithm~\ref{alg:rand}).
For any sequence of clients' positions $R_1,\ldots,R_T$ with $|R_t| \leq r$,
\begin{equation*}
\begin{split}
\sum_{t=1}^T \mathbb{E}\left[C_{R_t}(F_t)\right] & = 4r \cdot \min_{|F^\ast|=k} \sum_{t=1}^T\mathrm{C}_{R_t}(F^\ast)\\ &+ \Theta \left (k D n \sqrt{\log n T} \right)
\end{split}
\end{equation*}
where $F_t$ is the random variable denoting the $k$ positions at which Algorithm~\ref{alg:rand} places the centers at round $t$.
\end{theorem}
By combining Algorithm~\ref{alg:det} and Algorithm~\ref{alg:rand} we can achieve $\Theta \left(\min(k,r)\right)$-regret in polynomial time.
\begin{theorem}\label{t:main}
There exists an online learning algorithm for Dynamic $k$-Clustering that runs in polynomial-time and achieves $\min\left(6k,4r \right)$-regret.
\end{theorem}
\begin{remark}
In case the value $r = \min_{1\leq t \leq T}|R_t|$ is initially known to the learner, then Theorem~\ref{t:main} follows directly by Theorem~\ref{t:det-regret}~and~\ref{t:rand-regret}. However even if $r$ is not initially known, the learner can run a Multiplicative Weight Update Algorithm that at each round follows either Algorithm~\ref{alg:det} or Algorithm~\ref{alg:rand} with some probability distribution depending on the cost of each algorithm so far. By standard results for MWU \cite{H16}, this meta-algorithm admits time-average cost less than the best of Algorithm~\ref{alg:det}~and~\ref{alg:rand}.
\end{remark}
\section{Fractional Dynamic $k$-Clustering
}\label{s:fractional}
In this section we present the \textit{Fractional Dynamic $k$-Clustering} problem for which we provide a polynomial-time no-regret online learning algorithm. This online learning algorithm serves as a primitive for both Algorithm~\ref{alg:det} and Algorithm~\ref{alg:rand} of the subsequent sections concerning the original Dynamic $k$-Clustering.
The basic difference between Dynamic $k$-Clustering and Fractional Dynamic $k$-Clustering is that in the second case the learner can \textit{fractionally} place a center at some point of the metric space $V$. Such a fractional opening is described by a vector $y \in \Delta_{n}^k$.
\begin{online_problem}\label{pr:frac}
[Fractional Dynamic $k$-Clustering]At each round $t \geq 1$,
\begin{enumerate}
\item The learner selects a vector $y_t \in \Delta_{n}^k$. The value $y_i^t$ stands for the fractional amount of center that the learner opens in position $i \in V$.
\item The adversary selects the positions of the clients denoted by $R_t \subseteq V$ (after the selection of the vector $y_t$).
\item The learner incurs fractional connection cost $\mathrm{FC}_{R_t}(y_t)$ described in Definition~\ref{d:frac_cost}.
\end{enumerate}
\end{online_problem}
\begin{definition}[Fractional Connection Cost]\label{d:frac_cost}
Given the positions of the clients $R \subseteq V$, we define the fractional connection cost $\mathrm{FC}_{R}(\cdot)$ of
a vector $y \in \Delta_n^k$ as
the optimal value of the following convex program.
\begin{equation}
\begin{array}{lr@{}ll}
\mbox{\emph{minimize}} \left(\sum_{j \in R}\beta_j^p \right)^{1/p}
\\
\\
\mathrm{}{s.t.}~~~~ \beta_j = \sum\limits_{i \in V} d_{ij} \cdot x_{ij} \,\,~~~~\forall j \in R\\
~~~~~~~~~ \sum\limits_{i \in V} x_{ij} = 1 \,\,~~~~~~~~~~~~~~~\forall j \in R\\
~~~~~~~~ x_{ij} \leq y_i \,\,~~~~~~~\forall j \in R,~\forall i \in V\\
~~~~~~~~ x_{ij} \geq 0 \,\,~~~~~~~~\forall j \in R,~\forall i \in V
\end{array}
\end{equation}
\end{definition}
It is not hard to see that once the convex program of Definition~\ref{d:frac_cost} is formulated with respect to an \textit{integral vector} $y\in \Delta_n^k$ ($y_i$ is either $0$ or $1$) the
fractional connection cost $\mathrm{FC}_{R}(y)$ equals the original connection cost $\mathrm{C}_{R}(y)$. As a result, the cost of the optimal solution $y^\ast \in \Delta_{k}^n$ of Fractional Dynamic $k$-Clustering is upper bounded by the cost of the optimal positioning of the centers $F^\ast$ in the original Dynamic $k$-Clustering.
\begin{lemma}\label{l:frac_int}
For any sequence of clients' positions $R_1,\ldots,R_T$, the cost of the optimal fractional solution $y^\ast$ for Fractional Dynamic $k$-Clustering
is smaller than the cost of the optimal positioning $F^\ast$ for Dynamic $k$-Clustering,
\[ \min_{y^\ast \in \Delta_n^k}\sum_{t=1}^T \mathrm{FC}_{R_t}(y^\ast) \leq \min_{ |F^\ast| =k}\sum_{t=1}^T \mathrm{C}_{R_t}(F^\ast)\]
\end{lemma}
Lemma~\ref{l:frac_int} will be used in the next sections where the online learning algorithms for the original Dynamic $k$-Clustering are presented. To this end, we dedicate the rest of this section to design a polynomial time no-regret algorithm for Fractional Dynamic $k$-Clustering. A key step towards this direction is the use of the subgradient vectors of $\mathrm{FC}_{R_t}(\cdot)$.
\begin{definition}[Subgradient]\label{d:subgradients}
Given a function $f:\mathbb{R}^n \mapsto \mathbb{R}$, a vector $g \in \mathbb{R}^n$ belongs in the subgradient of $f$ at point $x\in \mathbb{R}^n$,$g \in \partial f(x)$, if and only if $f(y) \geq f(x) + g^\top (y -x)~$, for all $y \in \mathbb{R}^n$.
\end{definition}
Computing the subgradient vectors of functions, as complicated as $\mathrm{FC}_{R_t}(\cdot)$, is in general a computationally hard task. One of our main technical contributions consists in showing that the latter can be done through the solution of an adequate convex program corresponding to the dual of the convex program of Definition~\ref{d:frac_cost}.
\begin{lemma}\label{l:dual}
Consider the convex program of Definition~\ref{d:frac_cost} formulated with respect to a vector $y \in \Delta_n^k$ and the clients' positions $R$. Then the following convex program is its dual.
\begin{equation}\label{eq:ALP}
\begin{array}{lr@{}ll}
\mbox{\emph{maximize}}~~~ \sum_{j \in R}A_j - \sum_{i \in V}\sum_{j \in R}k_{ij}\cdot y_i\\
\\
\mathrm{s.t.}~~~~ ||\lambda||_{p}^\ast \leq 1 \\
~~~~~~~~~ d_{ij} \cdot \lambda_j + k_{ij} \geq A_j \,\,~~~~\forall i \in V, j \in R\\
~~~~~~~~~~k_{ij} \geq 0 \,\,~~~~~~~~~~~~~~~~~~~~~~~\forall i \in V, j \in R\\
\end{array}
\end{equation}
where $|| \cdot||_{p}^\ast$ is the dual norm of $||\cdot ||_p$
\end{lemma}
In the following lemma we establish the fact that a subgradient vector of $\partial \mathrm{FC}_{R_t}(\cdot)$ can be computed through the optimal solution of the convex program in Lemma~\ref{l:dual}.
\begin{lemma}\label{l:subgradients}
Let $k_{ij}^\ast$ denote the value of the variables $k_{ij}$ in the optimal solution of the convex program in Lemma~\ref{l:dual} formulated with respect to vector $y \in \Delta_n^k$ and the clients' positions $R$. Then for any vector $y' \in \Delta_n^k$,
\[\mathrm{FC}_{R_t}(y') \geq \mathrm{FC}_{R_t}(y) + \sum_{i \in V} \left(-\sum_{j \in R}k_{ij}^{\ast} \right)\cdot \left( y_i' - y_i \right) \]
Moreover there exits an $\Theta(r \cdot |V|)$ algorithm for solving the dual program (Algorithm~\ref{alg:dual}) and additionally $|k_{ij}^\ast| \leq D$.
\end{lemma}
\begin{algorithm}[H]
\caption{A time-efficient algorithm for solving the dual program of Lemma~\ref{l:dual}}
\begin{algorithmic}[1]
\State\textbf{Input:} A vector $y \in \Delta_{n}^k$ and a set of clients $R \subseteq V$.
\State \textbf{Output:} An optimal solution for the convex program of Lemma~\ref{l:dual}.
\For{ each client $j \in R$,}
\State Sort the nodes $i \in V$ in increasing order according to $d_{ij}$.
\State $\mathrm{Rem} \leftarrow 1$
\For{each each $i \in V$}
\State $x_{ij} \leftarrow \min(y_i , \mathrm{Rem})$.
\State $\mathrm{Rem} \leftarrow \mathrm{Rem} - x_{ij}$.
\EndFor
\EndFor
\For{ each client $j \in R$}
\State $V_j^+ \leftarrow \{i \in V:~ x_{ij} > 0\}$ and $D_j \leftarrow \max_{i \in V_{j}^+} d_{ij}$.
\State $\beta_j \leftarrow \sum_{i \in V}d_{ij} \cdot x_{ij}$
\State $\lambda_j \leftarrow \left[ \frac{\beta_j}{||\beta||_p} \right]^{p-1}$
\State $A_j \leftarrow \lambda_j \cdot D_j$
\State $k_{ij} \leftarrow \min \left(\lambda_j\cdot \frac{x_{ij}}{y_i} \cdot \left(D_j - d_{ij}\right) , 0 \right)$
\EndFor
\end{algorithmic}
\label{alg:dual}
\end{algorithm}
\begin{remark}
Algorithm~\ref{alg:dual} is not only a computationally efficient way to solve the convex program of Lemma~\ref{l:dual}, but most importantly guarantees that the value $k_{ij}^\ast$ are bounded by $D$ (this is formally Stated and proven in Lemma~\ref{l:dual}). The latter property is crucial for developing the no-regret algorithm for Fractional Dynamic $k$-Clustering.
\end{remark}
Up next we present the no-regret algorithm for Fractional Dynamic $k$-Clustering.
\begin{algorithm}[H]
\caption{A no-regret algorithm for Fractional Dynamic $k$-Clustering}
\begin{algorithmic}[1]
\State Initially, the learner selects $y^1_i = k/n$ for all $i \in V$.
\For{ rounds $t = 1 \cdots T$}
\State The learner selects $y_t \in \Delta_{n}^k$.
\State The adversary selects the positions of the clients $R_t \subseteq V$.
\State The learner receives cost, $\mathrm{FC}_{R_t}(y_t)$.
\State The learner runs Algorithm~\ref{alg:dual} with input $y_t$ and $R_t$ and sets $g_i^t = -\sum_{ j \in R_t}k_{ij}^t$
\For{ each $i \in V$}
\State
\[y_i^{t+1} = \frac{ y_i^t \cdot e^{-\epsilon g_i^t}}{\sum_{i\in V}y_i^t \cdot e^{- \epsilon g_i^t}}\]
where $\epsilon =\frac{\sqrt{ \log n}}{D r \sqrt{T}}$
\EndFor
\EndFor
\end{algorithmic}
\label{alg:frac_no_regret}
\end{algorithm}
We conclude the section with Theorem~\ref{t:no-regret-frac} that establishes the no-regret property of Algorithm~\ref{alg:frac_no_regret} and the proof of which is deferred to the Appendix~\ref{app:fractional}.
\begin{theorem}\label{t:no-regret-frac}
Let $y_1,\ldots,y_T$ be the sequence of vectors in $\Delta_n^k$ produced by Algorithm~\ref{alg:frac_no_regret} for the clients' positions $R_1,\ldots,R_T$. Then,
\[\sum_{t=1}^T\mathrm{FC}_{R_t}(y_t) \leq \min_{y^\ast \in \Delta_k} \sum_{t=1}^T\mathrm{FC}_{R_t}(y^\ast) + \Theta \left (k D n \sqrt{\log n T} \right)\]
\end{theorem}
\section{A $\Theta(k)$-Regret Deterministic Online Learning Algorithm}\label{s:det}
In this section we show how one can use Algorithm~\ref{alg:frac_no_regret} described in Section~\ref{s:fractional} to derive $\Theta(k)$-regret for the Dynamic $k$-Clustering in polynomial-time.
The basic idea is to use a rounding scheme that given a vector $y\in \Delta_n^k$ produces a placement of the $k$ centers $F_y \subseteq V$ (with $|F_y| \leq k$) such that \textit{for any set of clients' positions $R$}, the
connection cost $C_{R}(F_y)$ is approximately bounded by the factional connection cost $\mathrm{FC}_{R}(y)$. This rounding scheme is described in Algorithm~\ref{alg:rounding}.
\begin{algorithm}
\caption{Deterministic Rounding Scheme}
\begin{algorithmic}[1]
\State \textbf{Input}: A vector $y \in \Delta_{n}^k$.
\State \textbf{Output}: A set $F_y \subseteq V$ at which centers are opened.
\State Run Algorithm~\ref{alg:dual} with input $y$ and $R = V$.
\State Sort the positions $i \in V$ according to the values $\beta_i$ produced by Algorithm~\ref{alg:dual}.
\State $F_y \leftarrow \emptyset$
\For{ $i = 1$ \bfseries{ to } $V$}
\If{ $\min _{j \in F_y}d_{ij} > 6k \cdot \beta_i$}
\State $F_y \leftarrow F_y \cup \{i\}$
\EndIf
\EndFor
\end{algorithmic}
\label{alg:rounding}
\end{algorithm}
\begin{lemma}\label{l:rounding_lemma}[Rounding Lemma] Let $F_y$ denote the positions of the centers produced by Algorithm~\ref{alg:rounding} for input $y \in \Delta_n^k$. Then the following properties hold,
\begin{itemize}
\item For any set of clients $R$,
\[\mathrm{C}_R(F_y)~\leq 6k \cdot \mathrm{FC}_{R}(y)\]
\item The cardinality of $\mathrm{F}_y$ is at most $k$, $|\mathrm{F}_y| \leq k$.
\end{itemize}
\end{lemma}
Up next we show how the deterministic rounding scheme described in Algorithm~\ref{alg:rounding} can be combined with Algorithm~\ref{alg:frac_no_regret} to produce an $\Theta(k)$-regret deterministic online learning algorithm that runs in polynomial-time. The overall online learning algorithm is described in Algorithm~\ref{alg:det} and its regret bound is formally Stated and proven in Theorem~\ref{t:det-regret}.
\begin{algorithm}[H]
\caption{A $\Theta(k)$-regret deterministic online learning algorithm
for Dynamic $k$-Clustering}
\label{alg:det}
\begin{algorithmic}[1]
\For{ rounds $t = 1 \cdots T$}
\State The learner computes the vector $y_t \in \Delta_{n}^k$ by running Algorithm~\ref{alg:frac_no_regret} for the sequence of clients' positions $(R_1,\ldots,R_{t-1})$.
\State The learner places centers to the positions $F_{y_t}$ produced by Algorithm~\ref{alg:rounding} given input $y_t$.
\State The adversary selects the clients' positions $R_t \subseteq V$.
\State The learner suffers connection cost $C_{R_t}(F_{y_t})$
\EndFor
\end{algorithmic}
\label{alg:det}
\end{algorithm}
We conclude the section with the proof of Theorem~\ref{t:det-regret} in which the regret bounds of Algorithm~\ref{alg:det} are established.
\begin{proof}[Proof of Theorem~\ref{t:det-regret}]
The second case of Lemma~\ref{l:rounding_lemma} ensures that $|F_{t}|\leq k$ and thus Algorithm~\ref{alg:det} opens at most $k$ facilities at each round. Applying the first case of Lemma~\ref{l:rounding_lemma} for $R=R_t$ we get that $C_{R_t}(F_t)\leq 6k \cdot \mathrm{FC}_{R_t}(y_t)$. As a result,
\begin{eqnarray*}
&&\sum_{t=1}^T C_{R_t}(F_t) \leq \sum_{t=1}^T 6k \cdot \mathrm{FC}_{R_t}(y_t)\\
&&\leq 6k \min_{y^\ast \in \Delta_k}
\sum_{t=1}^T \mathrm{FC}_{R_t}(y^\ast) + \Theta \left (k D n \sqrt{\log n T} \right)
\end{eqnarray*}
where the last inequality follows by Theorem~\ref{t:no-regret-frac}. However Lemma~\ref{l:frac_int} ensures that \[\min_{y^\ast \in \Delta_k} \sum_{t=1}^T \mathrm{FC}_{R_t}(y^\ast) \leq \min_{F^\ast: |F^\ast|=k} \sum_{t=1}^T \mathrm{C}_{R_t}(F^\ast)\]
\end{proof}
\section{A \textbf{$\Theta(r)$}-Regret Randomized Online Learning Algorithm}\label{s:rand}
In this section we present a $\Theta(r)$-regret
randomized online learning algorithm.
This algorithm is described in Algorithm~\ref{alg:rand} and is based on the randomized rounding developed by Charikar and Li for the $k$-median problem \cite{CL12}.
\begin{lemma}[\cite{CL12}]\label{l:Charikar-Lin}
There exists a polynomial-time randomized rounding scheme that given a vector $y \in \Delta_n^k$
produces a probability distribution, denoted as $\mathrm{CL}(y)$, over the subsets of $V$ such that,
\begin{enumerate}
\item with probability $1$ exactly $k$ facilities are opened, $\mathbb{P}_{F \sim \mathrm{CL}(y)}\left[|F| = k\right] = 1$.
\item for any position $j \in V$,
\[\mathbb{E}_{F \sim \mathrm{CL}(y)}\left[C_{\{j\}}(F_y) \right] \leq 4 \cdot \mathrm{FC}_{\{j\}}(y).\]
\end{enumerate}
\end{lemma}
Similarly with the previous section, combining the randomized rounding of Charikar-Li with Algorithm~$1$ produces a $\Theta(r)$-regret randomized online learning algorithm that runs in polynomial-time.
\begin{algorithm}[H]
\caption{A $\Theta(r)$-regret randomized online learning algorithm}
\label{alg:rand}
\begin{algorithmic}[1]
\For{ rounds $t = 1 \cdots T$}
\State The learner computes the vector $y_t \in \Delta_{n}^k$ by running Algorithm~\ref{alg:frac_no_regret} for the sequence of clients' positions $(R_1,\ldots,R_{t-1})$.
\State The learner places centers to the positions $F_t \subseteq V$ produced by the Charikar-Li randomized rounding with input $y_t$, $F_t \sim \mathrm{CL}(y_t)$.
\State The adversary selects a request $R_t \subseteq V$.
\State The learner suffers connection cost $C_{R_t}(F_t)$
\EndFor
\end{algorithmic}
\end{algorithm}
The proof of Theorem~\ref{t:rand-regret} that establishes the regret bound of Algorithm~\ref{alg:rand} follows by Lemma~\ref{l:Charikar-Lin} and Theorem~\ref{t:no-regret-frac} and is deferred to the Appendix~\ref{app:rand}.
\section{Experimental Evaluations}\label{s:experiments}
In this section we evaluate the performance of our online learning algorithm against adversaries that select the positions of the clients according to time-evolving probability distributions. We remark that the
regret bounds established in Theorem~\ref{t:det-regret} and Theorem~\ref{t:rand-regret} hold even if the adversary \textit{maliciously} selects the positions of the clients at each round so as to maximize the connection cost. As a result, in case clients arrive according to some (unknown and possibly time-varying) probability distribution that does not depend on the algorithm's actions, we expect the regret of to be way smaller.
In this section we empirically evaluate the regret of Algorithm~\ref{alg:det} for Dynamic $k$-Clustering in case $p =\infty$. We assume that at each round $t$, $20$ clients arrive according to several static or time-varying two-dimensional probability distributions with support on the $[-1,1] \times [-1,1]$ square and the possible positions for the centers being the discretized grid with $\epsilon = 0.1$. In order to monitor the quality of the solutions produced by Algorithm~\ref{alg:det}, we compare the time-average connection cost of Algorithm~\ref{alg:det} with the time-average \textit{fractional connection cost} of Algorithm~\ref{alg:frac_no_regret}. Theorem~\ref{t:no-regret-frac} ensures that for $T=\Theta(k^2 D^2/\epsilon^2)$ the time-average fractional connection cost of Algorithm~\ref{alg:frac_no_regret} is at most $\epsilon$
greater than the time-average connection cost of the optimal static solution for Dynamic $k$-Clustering. In the following simulations we select $\epsilon = 0.1$ and track the ratio between the time-average cost of Algorithm~\ref{alg:det} and of Algorithm~\ref{alg:frac_no_regret} which acts as an upper bound on the regret.
\textbf{Uniform Square} In this case the $20$ clients arrive \textit{uniformly at random} in the $[-1,1] \times [-1,1]$ square. Figure~\ref{f:uniform_square} illustrates the solutions at which Algorithm~\ref{alg:det} converges for $k=2,3$ and $8$ as long as the achieved regret.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.49\linewidth]{images/uniform_square/k=2,T=400,Clients=20.png}\label{fig:sub2}}\hfill
{\includegraphics[width=0.49\linewidth]{images/uniform_square/k=2.png}\label{fig:sub3}}\hfill
{\includegraphics[width=0.49\linewidth]{images/uniform_square/k=3,T=9000,Cleints=20.png}\label{fig:sub2}}\hfill
{\includegraphics[width=0.49\linewidth]{images/uniform_square/k=3.png}\label{fig:sub3}}\hfill
{\includegraphics[width=0.49\linewidth]{images/uniform_square/k=8,T=20000,Clients=20.png}\label{fig:sub2}}\hfill
{\includegraphics[width=0.49\linewidth]{images/uniform_square/k=8.png}\label{fig:sub3}}\hfill
\caption{The \textcolor{green} {green curve} depicts the time-average connection cost Algorithm~\ref{alg:det}, the \textcolor{red}{red curve} depicts the time-average fractional connection cost of Algorithm~\ref{alg:frac_no_regret} and the \textcolor{blue}{blue curve} depicts their ratio that acts as an upper bound on the regret.
}\label{f:uniform_square}
\end{figure}
\textbf{Uniform Distribution with Time-Evolving Centers} In this case
the $20$ clients arrive according to the uniform distribution with radius $0.3$ and a time-varying center that periodically follows the trajectory described in Example~\ref{ex:1}. Figure~\ref{f:circle2} depicts the centers at which Algorithm~\ref{alg:det} converges after $100k^2$ rounds which are clearly close to the optimal ones.
\textbf{Moving-Clients on the Ellipse}
In this case the $20$ clients move in the ellipse $\left(\frac{x}{1.2}\right)^2 + \left(\frac{y}{0.6}\right)^2=1$ with different speeds and initial positions. The position of client $i$ is given by $\left(x_i(t),y_i(t)\right) = \left(1.2 \cos ( 2\pi f_i t + \theta_i ) , 0.6 \sin \left( 2\pi f_i t + \theta_i \right)\right)$ where
each $f_i,\theta_i$ was selected uniformly at random in $[0,1]$. Figure~\ref{fig:circle} illustrates how Algorithm~\ref{alg:det} converges to the underlying ellipse as the number of rounds increases.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.45\linewidth]{images/ellipse/ellipse_T=1000.png}\label{fig:sub1}}\hfill
{\includegraphics[width=0.45\linewidth]{images/ellipse/ellipse_T=10000.png}\label{fig:sub1}}\hfill
{\includegraphics[width=0.5\linewidth]{images/ellipse/ellipse_T=100000.png}\label{fig:sub1}}\hfill
\caption{The solution produced by Algorithm~\ref{alg:det} for $k=8$ after $100$, $1000$ and $10000$ rounds.}
\label{fig:circle}
\end{figure}
\textbf{Mixture of Multivariate Guassians} In this case 15 clients arrive according to the Gaussian with $\mu_1 = (-0.7,0.7)$ and $\Sigma_1
=[[0.3,0],[0,0.3]]$ and $5$ according to the Gaussian with $\mu_2 = (0.7,-0.7)$ and $\Sigma_2
=[[0.3,0],[0,0.3]]$. All the clients outside the $[-1,1]\times [-1,1]$ are projected back to the square. Figure~\ref{f:gaussian} illustrates the
solutions at which Algorithm~\ref{alg:det} converges for $k=2,8$ and $16$.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.49\linewidth]{images/Gaussian/k=2,Gaussian,T=2000}\label{fig:sub2}}\hfill
{\includegraphics[width=0.49\linewidth]{images/Gaussian/k=2,cost.png}\label{fig:sub3}}\hfill
{\includegraphics[width=0.49\linewidth]{images/Gaussian/Gaussian,k=8,T=20000}\label{fig:sub2}}\hfill
{\includegraphics[width=0.49\linewidth]{images/Gaussian/k=8,cost.png}\label{fig:sub3}}\hfill
{\includegraphics[width=0.49\linewidth]{images/Gaussian/k=16}\label{fig:sub2}}\hfill
{\includegraphics[width=0.49\linewidth]{images/Gaussian/k=16,cost.png}\label{fig:sub3}}\hfill
\caption{On the left, the solutions which Algorithm~\ref{alg:det} converges for $k=2,8$ and $k=16$. On the right, the time-average cost of Algorithm~\ref{alg:det}, Algorithm~\ref{alg:frac_no_regret} and the regret bounds.}
\label{f:gaussian}
\end{figure}
\section{Conclusion}
This work studies polynomial-time low-regret online learning algorithms for Dynamic $k$-Clustering, an online learning problem capturing clustering settings with time-evolving clients for which no information on their locations over time is available. We show that, under some well-established conjectures, $O(1)$-regret cannot be achieved in polynomial time and we provide a $\Theta(\min(k,r))$-regret polynomial time algorithm with $r$ being the maximum number of clients appearing in a single round. At a technical level, we present a two-step approach where in the first step we provide a no-regret algorithm for the Fractional Dynamic $k$-Clustering while in the second step we provide online rounding scheme converting the sequence of fractional solutions, produced by the no-regret algorithm, into solutions of Dynamic $k$-Clustering. Applying the same approach to other combinatorial online learning problems is an interesting research direction.
\bibliographystyle{plain}
|
1,314,259,996,702 | arxiv | \section{Introduction}
One of the most important issues in the high $T_c$ Fe-based
superconductors (FeSC) is their pairing symmetry
~\cite{ding:47001,sato:047002,liu:177005}. Theories based on
antiferromagnetic (AF) spin fluctuations have predicted $s_{\pm}$
pairing, where the superconducting (SC) order parameters on the hole
and electron Fermi pockets have opposite
signs~\cite{mazin:057003,wang:047005}. The proposed symmetry is
consistent with a number of experiments, such as the spin resonance
peak in neutron scattering~\cite{christianson:930}, sensitive SC
junction data~\cite{chen:260,teague:087004}, and quasiparticle
interference in tunneling experiments~\cite{hanaguri:23042010}.
However, the pairing symmetry in FeSC may not be universal, and there
are evidences for different pairing structures as discussed in a
recent review \cite{hirschfeld:124508}.
The effect of disorder to the superconductivity is an important test
to the pairing symmetry. According to Anderson's theorem, the
conventional s-wave superconductivity is insensitive to non-magnetic
impurities. The sign reversed $s_{\pm}$ superconductivity is,
however, sensitive to non-magnetic impurities which scatter inter-band
electrons. Replacement of part of Fe-atoms by Co or Ni in a parent
compound of FeSC leads to superconductivity. However, the role of the
Co or Ni doping is more subtle and remains controversial. One
scenario is that the doping introduces additional electron carriers. This scenario is supported by the
angle resolved photoemission spectroscopy, which indicates the shrinking of
the hole pockets~\cite{kaminski:419}. On the other hand, recent resonant
photoemission spectroscopy and density functional calculations
indicate that Co doping is covalent and introduces
disorder~\cite{sawatzky:077001}. It is plausible that the Co doping introduces
both carriers and disorder~\cite{ku:207003}. Zn-ion has a $3d^{10}$
configuration, hence a very high electric potential to charge
carriers. Replacing a Fe-atom by Zn in FeSC introduces inter-band
scattering and is expected to severely suppress the $s_{\pm}$
superconductivity. Therefore, the Zn-doping is an effective test to
the $s_{\pm}$ pairing in FeSC. There have been several experiments on
the Zn-doping effect on FeSC, including so-called 1111 compounds
LaFe$_{1-x}$Zn$_x$AsO$_{1-y}$F$_y$~\cite{li:053008,li:083008}, and more
recently 122 compounds
BaFe$_{2(1-x-y)}$Zn$_{2x}$Co$_{2y}$As$_2$ and SrFe$_{1.8-2x}$Zn$_{2x}$Co$_{0.2}$As$_2$~\cite{li:671}.
The results are
mixed at present, which appears to be strongly dependent of material and charge carrier content. The
experimental data on the 1111 compounds may be divided into two
categories. The optimally doped LaFeAsO$_{0.9}$F$_{0.1}$
\cite{li:053008} is insensitive, but the over-doped
LaFeAsO$_{0.85}$F$_{0.15}$ is very sensitive to the
Zn-impurities~\cite{li:083008}. The effect of Zn-doping on Co-doped
122 compounds clearly shows the suppression of superconducting
transition temperature $T_c$, but the reduction is much slower than
the theory predicted~\cite{li:671}. A careful examination indicates
that the suppression of $T_c$ may be saturating at large Zn-doping to
some of the compounds. Note that it is not easy to dope Zn into the Fe lattices
uniformly even under high pressure, and reliable data is only available up to
$6\%$ Zn-doping at present. Therefore the experimental data are not complete.
Nevertheless, the available experiments on Zn-doping indicate
complexity of the effect, and suggest possible competition of sign
changed $s_{\pm}$ and sign unchanged $s_{++}$ pairings in FeSC.
In this paper, we use a two-orbital model for FeSC including both
on-site (or $s_{++}$) pairing coupling $g_0$ and next nearest neighbor
(NNN) intersite (or $s_{\pm}$) pairing coupling $g_2$ to study
Zn-impurity effect, which may help understand the complex result of
the Zn doping effect on 1111 and 122 compounds. We apply Bogliubov
de-Gennes (BdG) equation to study the model on a finite-size
system. The two SC pairings in the multi-band system show interesting
interplay. They may mix but also compete with each other. The disorder
strongly suppresses the intersite pairing, and its effect to the
superconductivity depends on the strength of $g_0$. For large $g_0$,
$g_2$ plays little role and the pairing is $s_{++}$ and is robust
against the disorder. For small $g_0$, the pairing is $s_{\pm}$ and
the disorder strongly suppresses superconductivity. For moderate value
of $g_0$, the disorder may enhance the on-site pairing and induce a
transition from $s_{\pm}$ to $s_{++}$ superconductivity. We further
study the interplay between $g_0$ and $g_2$ in a clean system and show
that the disorder effect on the gap functions is similar to the
reduction of $g_2$. Our theory is consistent with the Zn-doped
impurity experiments on 1111 and 122 compounds, and suggests multi-pairing
couplings in some of the FeSC. We present our new experimental data
of the Zn-impurity effect on the very high $T_c=50$K Sm-1111
compound. The lattice constant measurement show that the Zn-atoms are
doped into the Fe-lattice uniformly up to $6\%$. The results appear to
indicate possible saturation of $T_c$ under the Zn doping, consistent
with the present theory.
\section{Model Hamiltonian and Mean Field Theory}
We consider a model Hamiltonian
\begin{equation}
H=H_0 + H_{\mathrm{pair}} + H_{\mathrm{imp}},
\end{equation}
which includes a tight-binding kinetic term $H_0$, a pairing
interaction $H_\mathrm{pair}$, and a disordered term $H_{\mathrm{imp}}$. For $H_0$,
we consider a two-orbital model with $d_{xz}$ (orbital
1) and $d_{yz}$ (orbital 2) as proposed by Raghu et
al.~\cite{raghu:220503}.
\begin{equation}
H_0=\sum_{\left < i{\alpha},j{\beta}\right >\sigma}
C_{i\sigma}^{\dagger}\hat{h}_{ij}C_{j\sigma},
\end{equation}
where $C_{i\sigma}^{\dagger}= (c_{i1,\sigma}^{\dagger},
c_{i2,\sigma}^{\dagger})$, and $\hat{h}_{ij}^{\alpha, \beta}=
t_{i,j}^{\alpha\beta}$ is the electron hopping term between orbital
$\alpha$ at site $i$ and orbital $\beta$ at site $j$ on a
2-dim. square lattice of Fe-atoms (lattice constant $a=1$). While this
model may be an over-simplified one to describe many detailed material
properties of FeSC, it should capture the basic feature of the
disorder effect to the pairing in systems with multi-Fermi
surfaces. The non-vanishing hopping matrix elements are
$t_{i,i}^{\alpha \alpha} = -\mu$, $t_{i,i+\hat{y}}^{11} = t_2$,
$t_{i,i+\hat{y}}^{22} = t_1$, $t_{i,i+\hat{x}+\hat{y}}^{\alpha \alpha}
= t_3$, $t_{i,i+\hat{x}+\hat{y}}^{12} = t_{i,i+\hat{x}+\hat{y}}^{21}
=-t_4$. We choose $t_1=1$ as the energy unit, and $\mu=1.6$,
$t_2=-1.3$, $t_3=t_4=0.85$, which gives Fermi surfaces with hole
pockets near the $\Gamma$- and $M$ points, and electron pockets near
the $X$- and $Y$ points in an extended Brillouin zone as plotted in
Fig. 1.
\begin{figure}[htbp]
\centering \includegraphics[width=0.25\textwidth]{fig1.eps}
\caption{(Color online) Hole (red) and electron (blue) Fermi pockets obtained in
the two-orbital model Eq. (2). Points A and B are the
representative $\vec k$ points for the hole and electron pockets,
respectively.}
\end{figure}
We consider randomly distributed impurities on the lattice and
introduce an on-site repulsive potential on the Zn-impurity site,
\begin{equation}
H_{\mathrm{imp}}= I\sum_{i \in{\mathrm{imp}}} \sum_ {\sigma}C_{i
\sigma}^{\dagger}C_{i\sigma},
\end{equation}
where $i$ sums over all the impurity sites, and we
consider the large $I$ case ($I=24t_1$ in the actual calculation) to model the large repulsion to an electron at the Zn site. The
pairing Hamiltonian is modeled by
\begin{equation}
\label{eq:4} H_\mathrm{pair} =
-\sum_{<ij>}(V_{ij}c^{\dagger}_{i\alpha\uparrow}c^{\dagger}_{j\beta\downarrow}c_{j
\beta\downarrow} c_{i \alpha \uparrow}+h.c.),
\end{equation}
where the pairing coupling $V_{ij}$ includes an on-site term
$g_0>0$ and an NNN intersite term $g_2$,
\begin{equation}
V_{ij}=g_0\delta_{i,j}+
g_2\sum_{\vec{\tau}}\delta_{j,i+\vec{\tau}}. \label{realv}
\end{equation}
with $\vec{\tau}$ the vector of the two NNN site displacement.
Note that $g_0$ term favors $s_{++}$ and $g_2$ term
favors $s_{\pm}$ symmetry.
We introduce a mean field gap function $\Delta_{ij}^{\alpha
\alpha}=V_{ij} \left \langle c_{j \alpha \downarrow} c_{i \alpha
\uparrow} \right \rangle$. Our calculations show that the
inter-orbital pairing $\Delta_{ij}^{12}$ is very tiny and will be
neglected below. The BdG equation for the mean field Hamiltonian then
reads
\begin{eqnarray}
\label{bdg}
\sum_j \left(
\begin{array}{cc}
\hat{h}_{ij} & \hat{\Delta}_{ij} \\
\hat{\Delta}^{\ast}_{ij} & -\hat{h}^{\ast}_{ij,\bar{\sigma}}
\end{array}
\right)
\left(
\begin{array}{c}
\mathbf{u}^{n}_{j,\sigma} \\
\mathbf{v}^{n}_{j,\bar{\sigma}} \\
\end{array}
\right)= E_{n}\left(
\begin{array}{c}
\mathbf{u}^{n}_{i,\sigma} \\
\mathbf{v}^{n}_{i,\bar{\sigma}} \\
\end{array}
\right),
\end{eqnarray}
with $\hat{\Delta}_{ij}=\Delta_{ij} \hat{I}$, and $\hat{I}$ an
identity matrix.
\begin{small}$\mathbf{u}_{i,\sigma}=\left(\begin{array}{c}
u_{i1,\sigma} \\ u_{i2,\sigma}\end{array}\right)$\end{small}. The
self-consistent equation for the gap function is
\begin{eqnarray}
\label{sc}
\Delta_{ij}^{\alpha \alpha}&=&\frac{V_{ij}}{4}\sum_{n}(u^{n}_{i\alpha,\sigma}
v^{n\ast}_{j\alpha,\bar{\sigma}}
+v^{n\ast}_{i\alpha,\bar{\sigma}}u^{n}_{j\alpha,\sigma})\times \nonumber\\
&&\tanh(\frac{E_{n}}{2k_{B}T})
\end{eqnarray}
For the form of $V_{ij}$ in Eq.~\eqref{realv}, we define
$\Delta_{0}^{\alpha \alpha}(i)= \Delta_{ii}^{\alpha \alpha}$, and $\Delta_{2}^{\alpha \alpha}(i)= \sum_{\vec \tau}
\Delta_{i,i+\vec \tau}^{\alpha \alpha}/4$.
\section{Numerical Results}
We now discuss the numerical solutions of $H$. In our calculations,
for each impurity content, the impurity positions are randomly
distributed and the statistical averages are taken over 400 times. We
consider three typical cases: (i) $g_0$ is large and dominant; (ii)
$g_2$ is large and $g_0$ is weak; and (iii) $g_2$ is dominant but $g_0$ is moderately
large. In case (i), the SC pairing is always $s_{++}$ and the
superconductivity is robust against the impurity as we expect from the
Anderson theorem.
In Fig. 2 (a) and (b), we show the spatially averaged on-site and NNN
inter-site pairing amplitudes $\Delta_0=\Delta_0^{\alpha\alpha}$ and
$\Delta_2=\Delta_2^{\alpha\alpha}$ as functions of the impurity
concentration $n_{\mathrm{imp}}$ for cases (iii) and (ii). Also shown
are the gaps at the hole pocket (point A [$(0,0.22\pi)$]) and at the
electron pocket (point B [$(0.62\pi,0)$]), which are the Fourier
transform of the impurity averaged gaps in real space. In the case
(ii) of weak on-site pairing, the impurities strongly suppress
$\Delta_2$ as shown in the Fig. 2(b). $\Delta_0$ is tiny and the SC
gap functions $\Delta_A$ and $\Delta_B$ monotonically decrease as
$n_{\mathrm{imp}}$ increases. Because of the finite lattice size, our
study is limited to the short coherence length or the strong pairing
coupling cases, which require $n_{\mathrm{imp}} \approx 0.15$ to
destroy the superconductivity. We expect this value to be much
smaller in weaker pairing coupling cases.
Case (iii) is most interesting, and our theory shows an impurity
driven phase transition from $s_{\pm}$ to $s_{++}$ pairings. In the
absence of impurity, $g_2$ dominates and the pairing is $s_{\pm}$. As
shown in Fig. 2(a), the pairing symmetry remains to be $s_{\pm}$ at
$n_{\mathrm{imp}} < 0.02$, and the gap amplitudes on $k$ points A and
B are monotonically suppressed as $n_{\mathrm{imp}}$ increases. At $
0.02 < n_{\mathrm{imp}} < 0.05$, $\left| \Delta_2 \right|$ decreases, and
$\left| \Delta_0 \right|$ increases. At $n_{\mathrm{imp}}> 0.05$, both $\Delta_A$
and $\Delta_B$ are positive and we have $s_{++}$ pairing. It is
interesting to note that the on-site pairing may be enhanced by the
impurities due to the suppression of the NNN pairing.
\begin{figure}[htbp]
\center
\epsfig{figure=fig2.eps,width=0.45\textwidth}
\caption{(Color online) Upper panel: The gap functions at hole and
electron Fermi pockets $\Delta_h$ and $\Delta_e$ as functions of
impurity density $n_{\mathrm{imp}}$, obtained in the mean field
solution for $H$. (a): for modestly strong on-site pairing
coupling $g_0=1.8$; (b): for weak on-site coupling
$g_0=0.8$. Insets: spatially averaged gap functions $\Delta_0$
(on-site) and $\Delta_2$ (NNN inter site). In both cases, the NNN
coupling $g_2=1.6$. Lower panel: SC gaps calculated by the
simplified BCS formalism, with $N_e(0) = 0.12$, $N_h(0) = 0.1$,
and $\omega_D = 0.8$. (c): $g_0=1.8$; (d): $g_0=0.8$.}
\end{figure}
In the case of weak on-site pairing, as shown in Fig. 2(b), the impurities
strongly suppresses $\Delta_2$. We have examined the
SC order parameters in real space and found that the disorder does not
result in severe pair-breaking effect to the on-site pairing measured
by $\Delta^{\alpha \alpha}_{i,i}$, whose peak amplitude is almost unaltered by the
impurities. On the other hand, the non-magnetic impurities not only
destroy NNN SC pairing order parameter $\Delta^{\alpha \alpha}_{i,i+\hat{x}+\hat{y}}$
in larger spatial areas, but also weaken the peak amplitude of the SC
pairing immensely.
As we have demonstrated, the impurities suppress
the NNN pairing order parameter $\Delta_2$. This effect is similar to
the reduction of $g_2$ in the clean sample. Therefore, tuning
$n_{\mathrm{imp}}$ in the disorder system is similar to tuning $g_2$
in a clean system~\cite{tuning_note}. Below we shall study SC order
parameters and $T_c$ in the model Hamiltonian $H$ as functions of
$g_2$ in the absence of disorder to mimic the impurity effect. This
enables us to further reveal the interplay between the SC pairings of
$s_{++}$ and $s_{\pm}$.
\section{$T_c$ Reduction: Theory and Experiments}
For a clean system, we have lattice translational symmetry, and the
gap function Eq. (7) becomes
\begin{equation}
\label{gapeq}
\Delta_m(\mathbf{k}) = - \sum_{\mathbf{k}^{\prime}}
V_{mn}(\mathbf{k}, \mathbf{k}^{\prime})\frac{\tanh(\beta
E_{n\mathbf{k}^{\prime}}/2)}{2E_{n\mathbf{k}^{\prime}}}\Delta_n(\mathbf{k}^{\prime})
\end{equation}
where $m, n$ are the band indices, $E_{n\mathbf{k}} =
\sqrt{\Delta_n({\mathbf{k}})^2 + \epsilon_n(\mathbf{k})^2}$,
$\epsilon_n(\mathbf{k})$ is the single particle energy. The summation
is taken only in the vicinity of Fermi pockets with an energy cut-off
$\omega_D$. The pairing potential $V_{m, n}(\mathbf{k}, \mathbf{k}')$
describes the coupling between gap function on various Fermi pockets,
and with Eq.~\eqref{eq:4}, we have
\begin{align}
\label{eq:1}
V_{mn}(\mathbf{k}, \mathbf{k}') &= \sum_{\alpha} U_{m \alpha}(-\mathbf{k}) U_{m \alpha}(\mathbf{k}) U_{n
\alpha}(\mathbf{k}') U_{n \alpha} (-\mathbf{k}')
\nonumber\\
& \phantom{=} \times \left( g_0 + 4 g_2 \cos q_x \cos q_y \right),
\end{align}
where $m,n$ are band indices, $\alpha$ is orbital index,
$U(\mathbf{k})$ is the transformation matrix between bands and
orbitals.
In our two-orbital model, there are four Fermi pockets, two for hole
bands at $\Gamma$ and M points respectively and two for electron bands
at X and Y points respectively, which makes it very difficult to solve
Eq.~\eqref{eq:1} analytically. So in the following, we will ignore
the size of the pockets and assume there are four point-like Fermi
surfaces at $\Gamma$, X, Y and M with finite density of states. And
we also assume the summation in Eq.~\eqref{gapeq} are only over the
four momentum $\Gamma = (0,0)$, $Y = (\pi, 0)$, $X = (0, \pi)$, and $M
= (\pi, \pi)$.
Then we consider the transformation matrix under this approximation.
In the two-orbital model, the two orbitals, $d_{xz}$ and $d_{yz}$,
mixes strongly in the hole Fermi pockets. On the other hand, the two
orbitals can be connected by a C$_4$ rotation. So in the case of
point-like hole Fermi surface, it is obviously that the two orbitals
contribute equally to the hole pockets,
i.e. $U_{h,xz(yz)}[{\Gamma}(M)] = \frac{1}{\sqrt{2}}$, where $h$
denotes the hole band and $xz$ and $yz$ denote the two orbitals. On
the other hand, the two electron pockets are dominated by d$_{xz}$ and
d$_{yz}$ orbital respectively. So under the small pocket
approximation, we have $U_{e, xz}(Y) = U_{e, yz}(X) = 1$, and $U_{e,
yz}(Y) = U_{e, xz}(X) = 0$. And the nonzero pairing potentials are
$V_{hh}(\Gamma, \Gamma) = V_{hh}(\Gamma, M) = V_{hh}(M, M) =
\frac{v_0}{2}$, $V_{ee}(X, X) = V_{ee}(Y, Y) = v_0$, and
$V_{he}[\Gamma(M), X(Y)] = \frac{v_2}{2}$, where $v_0 = g_0 + 4 g_2$
and $v_2 = g_0 - 4 g_2$.
In the small pocket approximation, the gaps on the two electron
pockets should be same because of the C$_4$ rotational invariance of
the iron pnictide. Though the gaps on the hole pockets may be
different, we still assume they are equal for simplicity. So with the
above pairing potentials, we can solve the gap equation \eqref{gapeq}
and get the critical temperature
\begin{align}
\label{eq:2}
k_B T_c = 1.14 \omega_D e^{-1/N_h(0)\tilde{v}}
\end{align}
with
$\tilde{v}=\frac{1}{2}[(1+\lambda)(g_0+4g_2)+\sqrt{(1+\lambda)^2(g_0+4g_2)^2-16\lambda
g_0g_2}]$, where $\lambda = N_e(0)/ N_h(0)$, and $N_e(0)$ and
$N_h(0)$ denote the density of state at the Fermi level of electron
and hole pockets respectively.
\begin{figure}[htbp]
\center
\epsfig{figure=fig3.eps,width=0.45\textwidth}
\caption{\label{fig:tc}(Color online) Left: The critical temperature $T_c$ as a function
of $g_2$ with strong, weak, and moderate on-site pairing coupling $g_0$.
Right: Zn-impurity effect on $T_c$ in various Fe-based superconductors observed in experiments. The references of the data are listed. }
\end{figure}
Our calculation on the critical temperature for various impurity
concentrations, depicted at the left panel of Fig.~\ref{fig:tc}, reveals
that the different impurity-doping behaviors observed in FeSC
~\cite{li:083008,li:671} may be characterized by the strength of the
effective on-site pairing potential $g_0$. There are three types of
cases for the disorder effect. In the case of large $g_0$, where the
on-site pairing dominates, $T_c$ is hardly suppressed by the Zn
doping. In the case of weak $g_0$, superconductivity is
destroyed by the impurity. When $g_0$ is comparable with $g_2$, as Zn impurity
concentration increases, $T_c$ is initially suppressed rapidly and
then saturate. The experimental facts seem to support the
above scenarios and the effect of Zn doping depends on the material
and the charge carrier concentration . In
LaFe$_{1-x}$Zn$_x$AsO$_{0.9}$F$_{0.1}$ (Ref.~\cite{li:053008}), $T_c$
are insensitive to the Zn-impurity, and may be explained due to large
$g_0$. In the over-doped LaFe$_{1-x}$Zn$_x$AsO$_{0.85}$F$_{0.15}$
(Ref.~\cite{li:083008}) and LaFeAsO$_{0.85}$ (Ref.~\cite{guo:054506}),
in BaFe$_{2(1-x-y)}$Zn$_{2x}$Co$_{2y}$As$_2$ (Ref.~\cite{li:671}),
and in LaFe$_{1-x-y}$Co$_y$Zn$_x$O (Ref.~\cite{yukeliunpublished}),
$T_c$ decreases rapidly with the Zn doping, and may belong to the
category of weak $g_0$. In SrFe$_{1.8-2x}$Zn$_{2x}$Co$_{0.2}$As$_2$
(Ref.~\cite{li:671}), $T_c$ was found to decrease slowly and has the
tendency to saturate although higher Zn-doping will be needed to
confirm the speculation. These scenarios are summarized in
Fig.~\ref{fig:tc}, which shows the critical temperature vs $g_2$ at
different $g_0$ compared with the experimental data of the three types
of materials that behaves differently upon Zn doping.
The moderate value of $g_2$ case is most interesting, for it reflects
the competition between the two SC pairings. To further explore
this possibility, we have prepared
SmFe$_{1-x}$Zn$_x$AsO$_{0.9}$F$_{0.1}$ system with $T_C=50K$ and studied
systematically the Zn-impurity effect to $T_c$ experimentally. The
results are summarized in Fig. 4. We have measured the change of the
lattice constant due to Zn-doping and confirmed that Zn-atoms are
indeed doped into the iron sites up to $6\%$ of Zn doping, see
Fig. 4(b)~\cite{zn-doping}. Beyond this doping, our data indicate
that some Zn-impurity may not enter into Fe-lattice so the measurement
of $T_c$ may not correspond to the uniformly doped Zn-impurities. The
main experimental result of $T_c$ versus Zn concentration on this very
high $T_c$ material is plotted in Fig. 4(a). As we can see, as Zn is
introduced, $T_c$ reduces from 50K continuously down to 40K at $6\%$
of Zn. The slow reduction in $T_c$ may suggest that the
superconductivity saturates at large Zn doping. It will be
interesting to confirm this by doping high Zn concentration under high
pressure, which remains a challenge in material preparation.
\begin{figure}
\center
\epsfig{figure=fig4.eps,width=0.5\textwidth}
\caption{(Color online) (a) $T_c$ versus Zn-doping concentration for
SmFe$_{1-x}$Zn$_x$AsO$_{0.9}$F$_{0.1}$. (b) Lattice constants
$a$ and $c$ as
functions of Zn-doping content in
SmFe$_{1-x}$Zn$_x$AsO$_{0.9}$F$_{0.1}$.}
\end{figure}
In addition to the critical temperature, another important feature during
the transition from $s_{\pm}$ to $s_{++}$ is the change of low energy
density of states (DOS). As shown in Fig.~\ref{fig:dos}(a), the gap
amplitude reduces accompanying with the increase of low energy DOS
when the system approaches the transition point $n_{imp} \approx 0.04$
from clean limit. And if one further increases the impurity
concentrations, the low energy DOS will be suppressed again due to the
reopening of the gap. The non monotonic behavior of DOS with
impurities concentration should be able to be observed by integrated
photoemission spectroscopy.
\begin{figure}
\center
\epsfig{figure=fig5.eps,width=0.6\textwidth}
\caption{\label{fig:dos}(Color online) (a) DOS for various impurity
concentrations. $n_{imp} = 0, 0.015, 0.025, 0.035, 0.045, 0.065$ and
$0.15$ from bottom to top. (b) The low temperature specific heat
at various impurity concentrations.}
\end{figure}
The change of DOS with impurity concentration may also be observed by
the experiments which can measure the low energy DOS, for example the
specific heat. In the superconducting state, the electron specific
heat can be calculated with
\begin{eqnarray}
\label{eq:6}
C(T) &=& \frac{\partial}{\partial T} \int_{-\infty}^{\infty} E N(E)f(E)dE,
\end{eqnarray}
where $N(E)$ is the DOS, and $f(E)$ is the Fermi distribution
function. In the low temperature regime, the temperature dependence
of the superconducting order parameter is very weak and can be
neglected. So we use the zero-temperature DOS to calculate the
specific heat with Eq.~\eqref{eq:6}. In the calculation, we use
$\Delta_{\mathrm{coh}} = 0.18 t_1 = 6\mathrm{meV}$ as the energy
scale, and the result is depicted in Fig.~\ref{fig:dos}(b). We find
that the electron specific heat below $10K$ is small in the clean
limit, and shows a significant increase with approaching the
transition point by increasing impurity concentrations. When the
system is stabilized in the $s_{++}$ state, $C(T)$ drops to a low
value again. And the absolute value shown in Fig.~\ref{fig:dos}(b) is
in the same order or even larger than the experimental measurement of
1111 material \cite{wen:174501}. So it should be able to be observed in
experiments.
We note the recent work of Efremov et al. \cite{efremov:3840}, who
applied T-matrix method to study the non-magnetic effect on FeSC. Our
microscopic theory shares some similarities with their. In their
phenomenological theory the impurity-doping behavior is found to be
associated with the averaged pairing coupling strength. In our theory,
the decisive role of on-site pairing on the impurity effect is
identified.
\section{summary}
In summary, we have studied the disorder-induced pair-breaking effect
on the Fe-based superconductors using a model that incorporates both
the on-site and NNN pairings. We show that the Zn-impurity largely
suppresses NNN pairing. Its effect to the superconductivity depends
strongly on the on-site pairing coupling strength $g_0$. The
superconductivity can be robust, or evolves a transition from
$s_{\pm}$ to $s_{++}$, or is strongly suppressed in the presence of
the disorder. Our theory qualitatively explains different reductions
of T$_c$ in various iron pnictide superconductors observed in the
experiments on the Zn-impurity effect. We also predict the possible
Zn-impurity doping induced transition from $s_{\pm}$ to $s_{++}$
pairing states in certain samples. Furthermore, we have
systematically prepared Sm-1111 samples with $T_c=50$K under the
Zn-doping and show that the reduction of $T_c$ could be consistent with
the scenario of a moderate on-site s-wave pairing. It will be highly
interesting and important to prepare systematic controlled samples
with higher Zn-doping to experimentally confirm or falsify the theory.
The reduction of the gap in DOS during the transition from $s_{\pm}$
to $s_{++}$ can be observed by integrated photoemission spectroscopy
or specific heat experiments. Finally we remark that the explicit
paring forms are unlikely to be universal in Fe-based superconductors,
in contrast to the universal $d$-wave pairing in cuprates.
\begin{acknowledgements}
This work is partly supported by Hong Kong RGC grant and NSFC/RGC
Joint Research Scheme (No. 10931160425 and No. N$\_$HKU 726/09).
\end{acknowledgements}
|
1,314,259,996,703 | arxiv | \section{Introduction}
\label{section_intro}
Measurements of luminosity distances to Type Ia supernovae ({\small {SN}}e) have
played a central role in cosmology, leading two independent groups to the
remarkable discovery of an unknown, presently-dominant component of the
universe, dark energy, and strong evidence for an accelerating universe
\citep{riess98,perlmutter99}. Current surveys that target high-redshift
{\small {SN}}e\ from the ground -- the Canada-France-Hawaii Telescope Supernova Legacy
Survey ({\small {SNLS}}\footnote{http://www.cfht.hawaii.edu/SNLS}; \citealt{astier06})
and the Equation of State: Supernovae Trace Cosmic Expansion
({\small {ESSENCE}}\footnote{http://www.ctio.noao.edu/essence}; \citealt{wood-vasey07,
miknaitis07}) -- and from space using the {\it {\small {HST}}}\
\citep{riess04a,riess07,barbary06} have substantially increased the sample of
high-$z$ {\small {SN}}e, and have provided much-improved statistical constraints on
the expansion history of the universe.
The discovery of cosmic acceleration was made possible in part through
extensive observations of nearby Type Ia {\small {SN}}e\ by the Cal\'an/Tololo
Supernova Search \citep{hamuy93, hamuy96a, hamuy96b, hamuy96c} and by the
{\small {CfA}}\ follow-up program \citep{riess95, riess96a, riess99, jha06, jha07},
and by studies pioneered by \citet{pskovskii77} and by \citet{phillips93} of
the relationship between peak brightness and light curve decline rate
\citep{hamuy96d, riess96b, phillips99}. Current low-redshift {\small {SN}}\ surveys
and follow-up programs (Lick Observatory Supernova Search --
{\small {LOSS}}\footnote{http://astro.berkeley.edu/\~bait/kait.html};
\citealt{filippenko01}, Carnegie Supernova Project --
{\small {CSP}}\footnote{http://csp1.lco.cl/~cspuser1/PUB/CSP.html}; \citealt{hamuy06},
Nearby Supernova Factory -- {\small {SNF}actory}\footnote{http://snfactory.lbl.gov/};
\citealt{aldering02}, and the {\small {CfA}}\ {\small {SN}}\
Group\footnote{http://cfa-www.harvard.edu/oir/Research/supernova/SNgroup.html}),
are continuing to discover {\small {SN}}e\ and compile a large number of high-quality
multicolor light curves as well as multi-epoch optical spectra of {\small {SN}}e{\small{~I}}a\ to
expand the library of local training data used as ``templates''. These
high-quality data sets will be indispensable for calibrating the
brightness-decline relation to high precision. Obtaining and studying
multi-epoch spectra are also important for computing improved
$K$-corrections and minimizing systematic uncertainties \citep{kim96,
nugent02, hsiao07}. Recent spectroscopic modeling efforts have also led to
a better understanding of the physical mechanism responsible for the
observed brightness-decline relation (see, e.g., \citealt{kasen07} and
references therein).
As one of the three primary scientific components of the Sloan Digital Sky
Survey-II ({\small {SDSS-II}}), the Supernova Survey takes repeated imaging scans of
the same 300~square degrees of the sky during the Fall seasons of 2005 --
2007 to search for and measure light curves of {\small {SN}}e. The imaging survey is
complemented by an extensive spectroscopic follow-up program to confirm the
{\small {SN}}\ type and measure redshifts, and to study the detailed spectral
properties of a sample of selected events.
This program exploits the unique capabilities of the {\small {SDSS}}\ 2.5m telescope
\citep{gunn06} and its {\small {CCD}}\ imaging camera \citep{gunn98} to survey a large
volume of space at moderately high cadence. The survey complements and
improves upon other low-z and high-z surveys in several important ways. The
wide field-of-view camera operating in drift scan mode allows for efficient
discoveries of type Ia {\small {SN}}e\ at $0.05 \la z \la 0.4$, a redshift interval
that is not easily probed by other existing surveys that target either known
nearby galaxies (low-$z$) or narrow pencil-beam volumes (high-$z$). Its
well-calibrated multi-band photometric system ($ugriz$;
\citealt{fukugita96}) enables precise measurements of supernova light curves
with controlled systematics. The absolute magnitude scale is accurate to
better than $\sim 2$\% in $r$ and $\sim 2 - 3$\% in the colors \citep{dr5},
and a factor of $\sim 2$ improvement has been obtained from repeat imaging
of the equatorial region \citep{ivezic07}. Finally, the survey is sensitive
to the redshift interval that, given a large enough sample, enables
cosmological distance measurements with data from a {\it single telescope},
eliminating the need for cross-calibration across two or more photometric
systems.
This paper is part of a series describing the {\small {SDSS-II}}\ Supernova Survey.
Here we present a technical description of the search algorithm, data
processing, photometric typing of {\small {SN}}\ candidates, and spectroscopic target
selection. \citet{frieman07} presents an overview of the program.
Photometry and light curves of the full sample of spectroscopically
confirmed {\small {SN}}e\ from the 2005 season are presented in \citet{holtzman07}.
Spectroscopic data and their analysis results are described in
\citet{zheng07}. \citet{kessler07} presents the {\small {SN~I}}a\ Hubble diagram and
cosmological analysis from the 2005 season. The measurement of the
low-redshift Type Ia {\small {SN}}\ rate is presented in \citet{dilday07}. Detailed
studies of two peculiar {\small {SN}}e\ discovered by the {\small {SDSS-II}}\ {\small {SN}}\ Survey,
\sn2005hk and \sn2005gj, are presented in \citet{phillips07} and
\citet{prieto07}, respectively.
The main body of the paper is separated into two broad sections. The first
part (\S\ref{sec:obs}) presents the details of the real-time on-mountain
data processing and the mechanics of the search pipeline, which identifies
new transient events. The second part (\S\ref{sec:spectro}) describes the
procedures for supernova candidate identification, photometric {\small {SN}}\ typing,
and the algorithm adopted for selecting targets for spectroscopy. A brief
discussion of follow-up imaging observations of a sample of
spectroscopically confirmed {\small {SN}}e\ is presented in \S\ref{sec:imaging}. The
general results from the 2005 and 2006 seasons are presented in
\S\ref{sec:results}. We briefly summarize in \S\ref{sec:summary}.
\section{SDSS 2.5m Observations and Data Processing}
\label{sec:obs}
The {\small {SDSS-II}}\ Supernova Survey has been allocated the bulk of the time during
the Fall seasons (September 1 - November 30) of 2005 -- 2007 on the
{\small {SDSS}}\ telescope at Apache Point Observatory ({\small {APO}}). Imaging observations
are scheduled on most nights excluding a 5-day period around full moon.
Some nights are shared with the {\small SEGUE} program especially during
stretches of consecutive nights with good observing conditions. An overview
of the {\small {SDSS}}\ is given by \citet{york00}; see \citet{frieman07} for an
overview of the {\small {SN}}\ Survey.
\subsection{Survey Area: Stripe 82}
\label{subsec:stripe82}
The survey area is Stripe 82, the 300~deg$^2$ southern equatorial stripe of
the {\small {SDSS}}\ footprint, which covers the approximate coordinate ranges
$-60^\circ \la \alpha \la +60^\circ$ (20~hrs to 4~hrs in right ascension,
{\small {RA}}\ or $\alpha$) and $-1.25^\circ \la \delta \la +1.25^\circ$ in
declination ($\delta$). The detailed Stripe 82 footprint of each camera
column is shown in Table~\ref{tbl:stripe82}. This survey area was selected
for three primary reasons: (1) extensive repeat observations were acquired
as part of the {\small {SDSS-I}}\ survey, before the start of the {\small {SN}}\ Survey, (2) the
area is easily accessible to most telescopes, and (3) it is accessible at
low airmass during the Fall months, when most of the northern {\small {SDSS}}\ area is
not. The advantages of having repeat observations from {\small {SDSS-I}}\ are
three-fold. First, the data provide a catalog of known variable sources,
which is crucial for distinguishing a small number of new {\small {SN}}\ candidates
from a large population of foreground variable stars and background active
galactic nuclei. Second, the deep coadded images constructed from the
individual scans serve as references for image subtraction, enabling a more
sensitive search of new transient events (the template is essentially
noiseless in the detection process). Finally, the deep coadds also allow
identification of faint host galaxies that are otherwise undetected in the
single-scan images, which are frequently useful for prioritizing follow-up
observations. The stripe contains over 3 million cataloged galaxies that
are brighter than $r \sim 22.5$~mag.
\begin{figure}[!tb]
\begin{center}
\includegraphics[angle=0, width=.65\textwidth]{f1.eps}
\end{center}
\caption{Number of imaging scans of the northern (black) and southern (red)
strips for the 2005 (top) and 2006 (bottom) seasons as a function of right
ascension. Note that, in contrast to the 2005 season, the 2006 scans are
more evenly distributed in right ascension with very little difference
between the northern and southern strips.} \label{fig:runarea}
\end{figure}
Due to gaps between the six {\small {CCD}}\ columns, the {\small {SDSS}}\ imaging camera is
capable of scanning approximately half of the stripe (or one strip) in a
single night. During the 2005 season, the observations alternated between
the northern strip (82N) and the southern strip (82S) from night to night.
In 2006, however, a significant effort was made to avoid long temporal gaps
in any given part of the strip, so pieces of {\small {RA}}\ ranges from both strips
were sometimes observed in a single night. We also note that approximately
10\% of Stripe 82 is covered by both the northern and southern strips due to
overlapping {\small {CCD}}\ columns, so $\sim 30$~deg$^2$ of the sky is observed on a
cadence of 1 day (modulo weather losses). The lists of all
{\small {SDSS-II}}\ {\small {SN}}\ runs, i.e., continuous imaging scans, and their corresponding
{\small {RA}}\ ranges taken during the 2005 and 2006 seasons are given in
Table~\ref{tbl:runs2005} and Table~\ref{tbl:runs2006}, respectively. We
also show in Figure~\ref{fig:runarea} the number of visits made to each of
the strips in 2005 and 2006 as a function of the {\small {RA}}. The complete set of
corrected frames and the uncalibrated object catalogs from 2005 and 2006 are
available online as part of the first {\small {SN}}\ data release ({\small
DRSN}1\footnote{http://www.sdss.org/drsn1/DRSN1\_data\_release.html}).
These data are also accessible from the Data Archive Server of the Sixth
Data Release\footnote{http://www.sdss.org/dr6/access/index.html}
\citep{dr6}.
\subsection{On-Mountain Processing}
\label{subsec:processing}
The {\small {SDSS}}\ 2.5m telescope nominally performs observations every night during
the search seasons, so the data must be processed within 24~hours to avoid a
backlog of unprocessed data and to rapidly identify {\small {SN}}\ candidates for
spectroscopic follow-up observations. A single night of observing produces
up to $\sim 200$~Gb of imaging data, so it is highly desirable to process
the data on the mountain and significantly reduce the number of images that
are transferred over the internet for visual inspection.
The data are processed on a dedicated 10 dual-processor (20 processors
total) computer cluster that runs at {\small {APO}}. There are nine computers that
process data; each of the 18 processors is responsible for processing all
images from a single combination of filter ({\it {gri}}) and camera column (1 --
6). Each computer has 400~Gb of {\small {RAID}} 5 disk and 2~Gb memory. The 10th
monitor/backup computer has 3~Tb of disk and 4~Gb memory and is used for
process-management, data transfer, and software development. This setup was
specifically chosen to meet our requirement of processing a full night's
worth of data within 24~hours. An 11th computer is available as a spare but
so far has not been needed -- during the first two years of operation, there
were no serious failures. Several disks failed, but the {\small {RAID}}\ system
worked properly to prevent any loss of processing time while waiting for a
disk to be replaced. There was one curious glitch that may be related to
operation at high altitude (2800~m). On occasion one of the computers would
hang up, and cycling the power was the only way to revive the system. There
were about 20 such incidents per season, although the probability seemed
higher in the month of November, possibly due to better weather and longer
nights resulting in more computing time. Since people are present at
{\small {APO}}\ day and night, minimal processing time was lost from these hang-ups.
There are three main tasks for the process manager: (i) start jobs, (ii)
monitor processing progress, and (iii) monitor disk space. Due to the
structure of the data acquisition system, the image processing can start
only after an imaging run has finished, and the master script is usually
executed manually by a person in the morning. While this script could have
been automatically started, it is better that a human reviews the observer
logs and checks that processing starts smoothly. The master script
schedules and allocates resources for copying data from the data acquisition
computer to a local disk, photometric reduction in $ugriz$
(\S~\ref{subsub:photo}), and frame subtraction (\S~\ref{subsub:framesub})
and object detection (\S~\ref{subsub:doobjects}) in $gri$. Several monitor
scripts are used to check the status throughout the $\sim 20$~hours needed
to process the data. Since data processing continues virtually
round-the-clock, a few people often shared the monitoring burden.
Continuous monitoring was necessary because there were two common problems
that could interupt the processing. First, poor observing conditions caused
the photometric reduction software to abort. In this case, the photometric
reduction must be reprocessed with a more restrictive {\small {RA}}\ range. The
second source of interruption was the computer hang-ups discussed above.
The last process-manager issue concerns disk space. Rather than clearing
disk space after each night's data have been processed, we kept all of the
subtracted images at {\small {APO}}\ for 1--2 weeks. The reason for keeping
subtracted images is that for interesting candidates we could go back to
earlier epochs and process $u$ and $z$: $u$ band was particularly useful to
distinguish Type Ia and Type II supernovae. In addition, since the
candidate position is known, we re-ran $gri$ photometry at the known
location in previous epochs to get a better estimate of the early-epoch
fluxes and upper limits. The disk space was managed with a priority system
so that clearing disk was mostly based on following a pre-defined algorithm
and rarely involved a hasty decision. To avoid risks with automated
disk-cleaning, a human-entered command was required.
Below we describe the search pipeline and the reduction procedures adopted
to take the raw images to the point where they are transferred to the central
{\small {SN}}\ database server at Fermilab.
\subsubsection{PHOTO}
\label{subsub:photo}
The raw data are first processed through the {\tt {Photo}}\ pipeline
\citep{lupton01,lupton02,stoughton02}. The software produces the corrected
frames and generates bad-pixel maps, position-dependent {\small {PSF}}, and
astrometric solution \citep{pier03} that are used in subsequent processing
stages. Our version of {\tt {Photo}}\ does not identify sources in the images or
perform photometric measurements \citep{hogg01,smith02,ivezic04,tucker06},
as they are not appropriate for efficiently identifying {\small {SN}}\ candidates,
which are usually blended with their host galaxies. As with all standard
{\small {SDSS}}\ data, images from each camera column are written in frames of $2048
\times 1361$ unique pixels or 13.51\arcmin\ $\times~8.98$\arcmin\ on the
sky. The pipeline software was slightly tweaked to run on data taken under
poor observing conditions (bright moon, low atmospheric transparency, and
poor seeing). In 2005 and 2006, approximately 34\% and 30\%, respectively,
of the frames were acquired when the moon was above the horizon.
Figure~\ref{fig:seeing} shows the distributions of the $r$-band {\small {PSF}}\ of
frames successfully processed through {\tt {Photo}}\ during the first two seasons.
\begin{figure}[tb]
\begin{center}
\includegraphics[angle=270, width=.65\textwidth]{f2.eps}
\end{center}
\caption{Cumulative distributions of the $r$-band {\small {PSF}}\ during the 2005
(black) and 2006 (red) search seasons. More than half of the frames were
acquired at $< 1.2$\arcsec. Only frames successfully processed through
{\tt {Photo}}\ are included.} \label{fig:seeing}
\end{figure}
\subsubsection{Image Subtraction}
\label{subsub:framesub}
To identify new transients in the search data, the images are run through a
difference imaging pipeline, which matches the {\small {PSF}}\ of the search and
template frames, adjusts for the difference in photometric zeropoint,
accurately registers their pixels, and performs the subtraction. The image
templates were divided into four separate {\small {RA}}\ ranges so that human scanning
and target selection could begin well before the entire night of data was
finished processing. The subtracted frames in each filter are searched for
positive fluctuations that consist of at least 2 contiguous pixels each
above $3.0~\sigma$ of the noise. The software is derived from version 7.1
of the {\tt Photpipe} software used by the Super{\small MACHO} and {\small {ESSENCE}}\
collaborations \citep{smith02}. Extensive changes have been implemented to
operate successfully on processed {\small {SDSS}}\ data. The modified application is
called {\tt Framesub}.
The pipeline is operated in three modes: 1) {\tt sdssred} mode, where the
input of images is prepared for difference imaging. 2) {\tt sdssdiff},
which performs a difference imaging analysis of the search data compared to
the template data. To conserve computing time, only the {\it {gri}}\ frames are
differenced and used for object detection. This is the default mode that
produces the objects for manual inspection, and subsequently the {\small {SN}}\
candidates described below. 3) {\tt sdssforce} mode, in which, after
reliable detection of a supernova candidate, we difference {\it all} data on
the object, including the $u$ and $z$ bands, and perform forced-positional
photometry at the location of the candidate. The measurements of magnitudes
by {\tt sdssdiff} and {\tt sdssforce} are referred to as {\it search} and
{\it forced} photometry. The overall structure of the software and detailed
descriptions of each mode are provided in Appendix~\ref{app:software}.
\subsubsection{Object Detection and Filtering Algorithm}
\label{subsub:doobjects}
The individual peaks found by {\tt Framesub} are filtered through a software
called {\tt doObjects} to remove statistical fluctuations and identify true
astronomical sources. First, the single-filter peaks are matched by
position to identify sources that are detected in at least two of the three
{\it {gri}}\ filters within $0.8$\arcsec. These sources are flagged as {\it
objects}. Significant negative fluctuations are not flagged. The list of
objects is compared against the list of known variable sources (the veto
catalog) constructed from previous scans of Stripe 82. Any object that
matches the position of a known variable is filtered out. Most of the
sources in this catalog are variable stars, active galactic nuclei ({\small {AGN}}),
and other persistently varying sources. Approximately 10\% (20\%) of the
objects identified by {\tt doObjects} in 2005 (2006) were associated with
sources in the veto catalog.
Given the large area covered by the survey and the overlap of the central
{\small {RA}}\ range of Stripe~82 with the ecliptic plane, our detections are
overwhelmingly dominated by solar system objects. The challenge is to
remove as many of these objects as possible prior to handscanning, without
filtering out real {\small {SN}}\ candidates. Objects with proper motions in excess
of $\sim 1$\arcsec\ per minute are easily rejected by requiring that the
object is detected in the $r$ and $i$ bands within $0.8$\arcsec\ (the
filters are imaged in the order $riuzg$ and the effective time difference
between adjacent filters is 71.7 seconds). Objects that move as slowly as
$\sim 0.2$\arcsec\ per minute can be identified through motion between the
$g$ and $r$ bands. Objects that move at a slower rate are slightly more
difficult to identify; when they pass in front of a background galaxy, they
can look like perfectly good {\small {SN}}\ candidates. Approximately 35\% (40\%) of
the objects found in 2005 (2006) were tagged as moving objects and removed
prior to human evaluation.
New transient sources are entered into a dedicated {\tt MySQL} database and
the cutout images are transferred to Fermilab for visual inspection.
\subsection{Handscanning, Autoscanning, and SN Candidate Selection}
\label{subsec:handscan}
In addition to epochs of {\small {SN}}\ light-curves, the difference imaging and
object detection algorithms described above result in the detection of many
background sources of variation, both physical and non-physical. In order
to robustly reject background and allow us to focus further analysis on
promising {\small {SN}}\ candidates, we require that a human visually inspect images
of the objects, a process which we refer to as {\it handscanning}.
In addition to moving solar system objects, other major sources of
contamination include artifacts caused by (1) subtraction of slightly
misaligned images, which creates shapes with clusters of positive and
negative counts (objects that we call dipoles) in the differenced images,
(2) diffraction spikes, and (3) bright saturated stars. Satellite trails
that are not properly masked by the software also contribute to the
background. A small fraction of the dipoles are high proper-motion stars
that have drifted between the epochs of the template and search images.
Many of these background objects, however, can be quickly rejected by visual
inspection. In Figure~\ref{fig:object_mosaic}, we show a gallery of cutout
search, template, and differenced images of various types of objects
evaluated by scanners -- none, artifact, dipole, variable, transient,
{\small {SN}}\ gold, {\small {SN}}\ bronze, and {\small {SN}}\ other. All images shown are from the $r$
band. In Figure~\ref{fig:object_moving}, we show {\it {gri}}\ images of a moving
object correctly flagged by the software. These images show the source
moved between the $g$ and $r$ exposures, which are separated by about 5
minutes.
As a convenient way for humans to handscan the detected objects, a web
interface was constructed that queries the database and displays all of the
relevant information about the object, including {\it {gri}}\ cutout images of the
search, template, and differenced images, measured magnitudes, the mean sky
coordinates and the relative positions measured in each filter, and a list
of all previous detections (if any) within 0.8$\arcsec$ of the object under
inspection. The scanner evaluates the information and decides to either tag
the object as a possible {\small {SN}}\ or to reject it as background. If an object
is tagged as a {\small {SN}}\ and there are no previous detections of objects within
0.8$\arcsec$, it becomes a new {\small {SN}}\ {\it candidate} with a unique
{\small {SN}}\ {\small ID} number. A candidate will always remain a candidate unless
it is manually vetoed by one of the scanners (which rarely occurs). All of
the candidates can be accessed through a public web
site\footnote{http://sdssdp47.fnal.gov/sdsssn/candidates/candTable.php}.
\begin{figure}[!tb]
\begin{center}
\includegraphics[angle=0, width=1.00\textwidth]{f3.eps}
\end{center}
\caption{A gallery of various types of objects that a scanner evaluates.
The panels show $r$-band search (left columns in each panel), template
(central columns), and differenced (right columns) images of objects
classified by scanners as ``none'', ``artifact'', ``dipole'',
``variable'', ``transient'', ``{\small {SN}}\ gold'', ``{\small {SN}}\ bronze'', and
``{\small {SN}}\ other''. Objects classified as ``none'' are typically detections
near the threshold, and are often not visible to the human eye. Most
``artifacts'' are caused by different orientations of diffraction spikes
in the search and template images. A ``dipole'' results from imperfect
image registration as well as high proper-motion stars. A ``variable'' is
an object associated with a persistent source and usually exhibits random
temporal variability. These sources are newly variable sources that are
not part of the veto catalog. A ``transient'' object is usually seen for
the first time and {\it not} associated with a galaxy. The source can be
a true {\small {SN}}\ with an undetected host galaxy or a slow-moving asteroid. An
``{\small {SN}}\ gold'' is a good {\small {SN}}\ candidate that is well-separated from the
host. An ``{\small {SN}}\ bronze'' is a possible {\small {SN}}\ that lies close to the core
of the galaxy and can, therefore, also be a variable active galactic
nucleus. An ``{\small {SN}}\ other'' is also a possible {\small {SN}}, but is most likely a
different type of variable source -- in the case shown above, the source
is a variable star in NGC1068 (M77). There is another category called
``{\small {SN}}\ silver'' (not shown), which is used for objects that are similar to
a ``transient'', but detected in more than a single epoch (e.g., an
{\small {SN}}\ with an undetected host galaxy). The distinction between
gold/silver/bronze/other {\small {SN}}\ candidates is not important; all objects
tagged as an ``{\small {SN}}'' become a {\small {SN}}\ candidate. Each cutout image is
20\arcsec\ on the side.}
\label{fig:object_mosaic}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[angle=0, width=0.4\textwidth]{f4.eps}
\end{center}
\caption{An example of a moving object that was correctly flagged by the
autoscanner. As in Figure~\ref{fig:object_mosaic}, the search, template,
and differenced images are shown in the left, middle, and right columns,
respectively. In addition to the $r$-band images (top row), we also
display the $g$ (middle row) and $i$-band (bottom row) images to show the
relative position of the source in each filter. Note that the faint
source in the $g$-band differenced image appears slightly below and to the
left of the center of the image. The images are acquired in the order
$ri(uz)g$ with a 71.7-second delay between each filter.}
\label{fig:object_moving}
\end{figure}
During the 2005 observing season, we required that every object that is not
rejected by {\tt doObjects} be handscanned by a human. An average of about
3000 -- 5000 objects were inspected per full night of imaging during the
Fall 2005 season. Six scanners were on duty on a given night, each scanner
responsible for inspecting all objects from one of the six camera columns,
or 500 -- 800 objects. Based on our experience in 2005, we made a number of
changes to the handscanning procedure for the 2006 season described below.
\subsubsection{Autoscanner}
\label{subsec:autoscan}
To reduce the number of objects to be scanned by humans, prior to the 2006
season we implemented a new software filter, called the {\it autoscanner}.
The software performs two primary tasks: 1) identifies all objects detected
in more than one epoch as well as bright ($g$ or $r<21$~mag) objects
detected for the first time, and 2) uses statistical classification
techniques to identify and filter out first-epoch background non-{\small {SN}}\
objects.
The reasons for performing 1) are as follows. First, the selection of
objects detected in more than one epoch provides a very robust way of
eliminating moving objects, one of the major contaminants of the 2005 {\small {SN}}\
candidates. Second, the selection of bright first-epoch objects enables us
to discover nearby {\small {SN}}e; we can thereby obtain spectroscopic observations
well before maximum light and provide rapid alerts to the {\small {SN}}\ community.
In addition, bright, single-epoch candidates provide a back-up list of
spectroscopic targets for nights when the spectroscopy queue is not filled
by promising multi-epoch candidates. Although most of our targets were
spectroscopically observed after two or more epochs of photometry, in a few
instances, we did obtain spectroscopic observations of {\small {SN}}\ candidates based
on a single epoch of detection. In the selection of these objects, the
autoscanner also identifies objects associated with either a known variable
in the veto catalog or a known {\small {SN}}\ candidate. In the former case, the
object is filtered out. In the latter case, since there is no need for a
human to scan the object, it is instead flagged by the autoscanner as a
known {\small {SN}}\ candidate, stored in the database, and used in the light curve
analysis (see Section~\ref{subsec:snphoto}).
For 2), the autoscanner attempts to identify and filter out bright
first-epoch objects that belong to one of three classes of background;
unmasked diffraction spikes, artifacts of imperfect image registration
(dipoles), and moving objects. The software treats the set of objects,
along with their evaluations by a human, from the 2005 search as a training
set, and compares the observed quantities for newly detected objects against
this set. The method used to classify an object is the {\it histogram
method of probability density function ({\small {PDF}}) estimation}. In this method
the observable quantities, or attributes, of an object form a
multi-dimensional space, and a classification decision is made by
considering the number count of objects, in the training set, from each
class in a bin centered at the point describing the object we wish to
classify. This method has the advantages of being non-parametric, allowing
the decision boundaries in the observable space to be arbitrarily
complicated, and of explicitly retaining the correlations between
observables. A caveat of using the histogram method of {\small {PDF}}\ estimation is
that one must have a sufficient number of objects in the training set to
sample the {\small {PDF}}\ well in all regions of interest. With $\approx 90,000$
objects in the training set, this is not a significant limitation for the
{\small {SDSS-II}}\ {\small {SN}}\ Survey. A technical discussion of the algorithm is presented
in Appendix~\ref{app:autoscanner}.
During the 2006 observing season, the autoscanner was used to reject moving
objects from among the set of bright first-epoch objects that were to be
handscanned. For such bright objects, only $\approx 0.7$\% of them are
incorrectly tagged as moving objects by the autoscanner. In contrast to the
2005 observing season, we chose not to handscan epochs of known candidates
or objects that showed variation in more than one observing season, and the
autoscanner was used to reject such objects. Although some objects were
classified as artifacts or dipoles, we chose not to reject these from the
handscanning as they are a small fraction of the background in comparison to
moving objects, and the relatively small reduction in handscanning that
would be accomplished was determined to be outweighed by the risk of missing
an early epoch of a nearby {\small {SN}}.
This new handscanning strategy and the autoscanner enabled a reduction in
the number of objects scanned by more than an order of magnitude between
2005 and 2006, with no reduction in the quality or quantity of confirmed
{\small {SN}}e. While scanning a single camera column in 2005 typically took $2 -
3$~hours per person for a full night of data, in 2006 a scanner could cover
two columns in only $10 - 20$~minutes. The reduction of the 2006
handscanning by the autoscanner is summarized in Table~\ref{tbl:scanning}.
A comparison of the objects classified by the autoscanner as artifacts to
their human classification shows that the autoscanner is extremely efficient
at recognizing artifacts, and that our concern about potentially rejecting
nearby {\small {SN}}e\ was overly cautious. During the 2006 observing campaign, 3753
objects had been classified by the autoscanner as artifacts. Humans
classified 2668 of these objects as artifacts and 3710 of them as some type
of non-{\small {SN}}\ background. Of the remaining 43 objects classified by a humans
as {\small {SN}}\ candidates, 41 were background (non-{\small {SN}}) erroneously classified by
the scanner, one was a bright nova outburst, and one was the first epoch of
a {\small {SN}}\ that was later (after the 2nd epoch detection) classified as a {\small {SN}}.
Therefore, in cases where they disagree on classification of artifacts, the
autoscanner appears to be more reliable than the human scanners.
\section{Spectroscopic Target Selection and Follow-up Observations}
\label{sec:spectro}
Spectroscopic follow-up is an essential component of any current supernova
survey, for both confirming the type of the {\small {SN}}\ candidate and measuring the
cosmological redshift (preferably from host galaxy emission/absorption
lines). For the {\small {SDSS-II}}\ {\small {SN}}\ survey, the amount of scheduled spectroscopic
telescope time is larger than the amount of imaging time on the {\small {SDSS}}\ 2.5m
telescope by a large factor (see also \citealt{frieman07}). The telescopes
used for spectroscopy in 2005 and 2006 included the 9.2m {\small {HET}}, 3.6m {\small {NTT}},
3.5m {\small {ARC}}, 8.2m Subaru, 2.4m {\small {MDM}}\ Hiltner, 4.2m {\small {WHT}}, 4m {\small {KPNO}}\ Mayall, 10m
Keck, 2.5m {\small {NOT}}, and the 11m {\small {SALT}}.
On any given night during the fall season, there could be several telescopes
that are simultaneously scheduled and observing a set of our {\small {SN}}\
candidates. This requires a significant amount of coordination between the
various observers at each of the telescopes to avoid observing the same
candidate. A single person (the first author of this paper) was responsible
for coordinating all follow-up imaging and spectroscopic observations in
2005 and 2006. Telescopes separated by several time zones are relatively
easy to coordinate, but those that have similar longitudes (e.g., {\small {MDM}},
{\small {ARC}}, {\small {KPNO}}) require real-time communication. In general, the brighter
targets are parsed to telescopes with smaller aperture, but the magnitude
limits must be adjusted depending on the observing conditions at each site.
We have made every effort to avoid taking duplicate spectra, but on occasion
the same object was observed nearly simultaneously on two telescopes.
A good night of imaging of half of Stripe 82 typically yields $100 - 300$
objects that are tagged as {\it new} {\small {SN}}\ candidates, which are publicly
accessible through the
web\footnote{http://sdssdp47.fnal.gov/sdsssn/candidates/candTable.php}
immediately after they are entered into the database. This number
significantly exceeds the number of targets we can observe
spectroscopically, and choices must be made. The following subsections
describe the further prioritization of {\small {SN}}\ targets for spectroscopic
observations. The algorithm and strategy depends on the amount of resources
available, which is something that evolved quite substantially between the
2005 and 2006 search seasons.
\subsection{SN Photometric Typing}
\label{subsec:snphoto}
After each night of imaging on the {\small {SDSS}}\ 2.5m telescope, the {\it {gri}}\ light
curves of all active candidates are compared against a library of light
curve templates of different {\small {SN}}\ types. The purpose of this procedure is
twofold -- 1) to quickly identify the best-matching template and provide
estimates of the redshift, extinction, the approximate date of maximum
light, and current apparent magnitudes, and 2) to compute other quantities
that are useful for prioritizing follow-up spectroscopy, such as the amount
of host galaxy contamination. The process is essential since the number of
active {\small {SN}}\ candidates (a good fraction of which are not {\small {SN}}e) at any given
time exceeds the number that can be observed spectroscopically by a large
factor. A reliable system is necessary to make efficient use of the
valuable spectroscopic resources. Several techniques and algorithms for
{\small {SN}}\ photometric identification have been introduced previously by
\citet{poznanski02, poznanski06}, \citet{johnson06}, \citet{riess04b},
\citet{sullivan06}, \citet{kunz07}, and \citet{kuznetsova07}. The method
adopted here is most similar to that used by the {\small {SNLS}}\ spectroscopic follow
up described in \citet{sullivan06}. Below, we describe the details of the
algorithm.
The measured light curves are compared against a library of light curve
templates, which are grouped into three types -- {\small {SN~I}}a, {\small {SN~I}}b/c, and {\small {SN~II}} --
and are generated from multi-epoch spectra constructed and compiled by
P. Nugent\footnote{http://supernova.lbl.gov/~nugent/nugent\_templates.html}
and also from the {\small {SUSPECT}}\
database\footnote{http://bruford.nhn.ou.edu/~suspect/index1.html}.
Specifically, for the Type Ia, we use Nugent's Branch-normal, 1991T-like,
and 1991bg-like templates. The Branch-normal spectra are used for computing
synthetic light curves of {\small {SN}}e{\small{~I}}a\ with a range of luminosities parameterized
by the decline rate. The model is described in Appendix~\ref{app:snia_lc}.
The set of templates used for the Type Ib/c are Nugent's normal and
hypernova Ib/c spectra, as well as spectra and light curves of {\small {SN}} 1999ex
and {\small {SN}} 2002ap from the {\small {SUSPECT}}\ database. Similarly, we use Nugent's
II-P, II-L, and IIn spectra, and {\small {SUSPECT}} 's 1993J (IIb), 1998S (IIn), and
{\small {SN}} 1999em (II-P) to generate a set of Type II light curves.
The light curves in the observed {\it {ugriz}}\ filters are calculated on a grid of
four parameters ($z$, $A_V$, $T_{\rm{max}}$, [$\Delta m_{15}(B)$, template {\small {SN}}]), where
$z$ is the redshift, $A_V$ is the host galaxy extinction in the $V$ band and
assumes $R_V = 3.1$, and $T_{\rm{max}}$ is the time of rest-frame $B$-band
maximum light. The last parameter refers to either the decline rate
parameter for the Branch-normal Ia models ($\Delta m_{15}(B)$), or the particular
{\small {SN}}\ template for the peculiar {\small {SN}}e{\small{~I}}a\ (1991T-like and 1991bg-like) and the
core-collapse models. We do not attempt to fit or correct for the Milky Way
extinction. In this procedure, we {\it assume} a cosmology to convert the
redshift to a luminosity distance, similar to the method adopted by the
{\small {SNLS}}\ described in \citet{sullivan06}. The adopted cosmological parameters
are $\Omega_m = 0.3$ and $\Omega_\Lambda = 0.7$. For the {\small {SN~I}}a\ templates,
we also assume a fiducial peak $B$-band absolute magnitude of $M_B = -19.0 +
5 \log (H_0/70)$~mag, where $H_0$ is the Hubble constant in units of
km/s/Mpc, for a standard $\Delta m_{15}(B)$ = 1.1 {\small {SN~I}}a. As shown by \citet{sullivan06},
a particular set of assumed cosmological parameters in the computation of
the model light curves does not significantly bias the population of targets
for spectroscopic observations. At low redshift ($z \la 0.2$), the
luminosity distances are not sensitive to our choice of $\Omega_m$ and
$\Omega_\Lambda$. Above $z \sim 0.2$, the statistical uncertainties in the
fluxes and the combination of varying $z$ and $A_V$ can compensate for
differences in luminosity distance that might result from a different set of
cosmological parameters. The assumption, however, could systematically bias
the estimated photometric redshifts, but that is not a concern for the
purposes of target selection.
The templates are grouped into three {\small {SN}}\ types -- Ia, Ibc, and II -- and
the fitter records the best-fit parameters and the minimum value of the
$\chi^2$ statistic on the four-dimensional parameter grid within each of the
three types. In addition to identifying the {\small {SN}}\ type with the lowest
$\chi^2$ value, which we refer to as ``type-best'', we examine the relative
values of the $\chi^2$ and determine whether the {\small {SN}}\ candidate should be
considered as ``typed'' according to one or both of the following two
criteria. If we denote the value of the $\chi^2$ of the best-matching type
by $\chi^2_{\rm{min}}$, the next best type as $\chi^2_1$, and the worst one
as $\chi^2_2$ (i.e., $\chi^2_{\rm{min}} < \chi^2_1 < \chi^2_2$), the ``A''
criterion is satisfied if $\chi^2_{\rm{min}} < \chi^2_1 -
\chi^2_{\rm{min}}$. If ``type-best'' is a Ia {\it and} satisfies the above
criterion, the candidate is said to have a ``type-A of Ia''. The ``B''
criterion is satisfied if $\chi^2_{\rm{min}} < N (\chi^2_{\rm{min}} +
\chi^2_1 + \chi^2_2)/3$ with $N = 0.5$. Similarly, if the best-fit type is
a {\small {SN~I}}a\ and satisfies the above $\chi^2$ criterion, the candidate is said
to have a ``type-B of Ia''. The value of 0.5 adopted for $N$ works well for
{\small {SN}}\ candidates with low {\small {S/N}}, and was empirically determined from a
sample of spectroscopically confirmed {\small {SN}}e\ from our 2004 engineering run
\citep{sako04}. Also computed are the estimated current $g$ and $r$
magnitudes from the model light curves. We also search for the nearest
galaxy within $10\arcsec$ from the {\small {SN}}\ position in the {\small {SDSS}}\ galaxy
catalog and refit and retype the light curves using the best estimate of its
redshift as a prior. We adopt galaxy photometric redshifts from
\citet{oyaizu07} and spectroscopic redshifts from the {\small {SDSS}}\ {\small DR}5
\citep{dr5}.
We note that the absolute $\chi^2$ values for the model fits do not appear
to be very meaningful. The models have not been calibrated against real
data and there are no errors associated with the light curves (see, e.g.,
Appendix~\ref{app:snia_lc}). The current implementation also does not
attempt to reject outlier points due to poor zeropointing and imperfect
image registration, which are common features of the search photometry. The
best-fit $\chi^2$ value for the {\small {SN~I}}a\ model of a true {\small {SN~I}}a\ may be large
compared to the number of degrees of freedom, but the confidence can be high
if that value is well below the values of the other {\small {SN}}\ types. The {\it
relative} values of the $\chi^2$ are, therefore, important and useful
discriminators. In Figure~\ref{fig:sn13135_chisq}, we show the values of
$\chi_{\rm{min}}^2$, $\chi^2_1 < \chi^2_2$, and $0.5~(\chi^2_{\rm{min}} +
\chi^2_1 + \chi^2_2)/3$ as functions of the number of epochs for {\small {SN}} 2006fz,
a Type Ia at z=0.105, with a flat galaxy photometric redshift prior. For
this candidate, the model with the minimum $\chi^2$ corresponds to the
{\small {SN~I}}a\ model, and both the ``A'' and ``B'' criteria are satisfied at all
epochs. Also shown in Figure~\ref{fig:sn13135_lc} are the light curve fits
to {\small {SN}} 2006fz using the first 2, 4, 6, and 8 epochs of search photometry.
\begin{figure}[!tb]
\begin{center}
\includegraphics[angle=270, width=.7\textwidth]{f5.eps}
\end{center}
\caption{Fit results for \sn2006fz ({\small {SN~I}}a\ at z=0.105; internal candidate
{\small {ID}}\ 13135) as functions of the number of epochs in the light curve. The
bottom curve (labeled with filled circles) shows the minimum $\chi^2$
values, which at every epoch correspond to the Ia model. The top two
curves labeled with filled stars and triangles show values relevant for
the ``A'' and ``B'' typing criteria, respectively. For this particular
{\small {SN}}, $\chi^2_{\rm{min}}$ is always below these two curves, which implies
that this {\small {SN}}\ is a Type Ia at high-confidence.}
\label{fig:sn13135_chisq}
\end{figure}
\begin{figure}[!tb]
\begin{center}
\includegraphics[angle=0, width=1\textwidth]{f6.eps}
\end{center}
\caption{Light curves of \sn2006fz (same {\small {SN}}\ as shown in
Figure~\ref{fig:sn13135_chisq}) and fits to the best-fit {\small {SN~I}}a\ models
after 2, 4, 6, and 8 epochs of detections without a galaxy photometric
redshift prior. Search photometry from {\it {gri}}\ are shown. Model light
curves in $u$ and $z$ are shown as well.}
\label{fig:sn13135_lc}
\end{figure}
The overall performance of the photometric typing software is demonstrated
in Figure~\ref{fig:type_snia}, where we plot the fraction of the
spectroscopically confirmed {\small {SN}}e{\small{~I}}a, whose best-fit model is that of a {\small {SN~I}}a\
as a function of the number of the light curve epochs. The figure also
shows the fraction of {\small {SN~I}}a\ that satisfy one of the ``A'' or ``B'' criteria
with or without a nearest-neighbor host galaxy redshift prior. Note that
the fractions increase between 2 to $\sim 4$~epochs, but the improvement
beyond $\sim 5$~epochs is marginal. The fraction with the best type equal
to a Ia does not reach unity because the spectroscopic sample includes 1)
peculiar {\small {SN}}e{\small{~I}}a, 2) candidates with poorly subtracted images and photometry
epochs, 3) {\small {SN}}e{\small{~I}}a\ discovered well after maximum light, and 4) {\small {SN}}e\ with
low-{\small {S/N}}\ light curves whose best-fit type drifts with the number of
epochs. Candidates that fall into the first three categories were targeted
for spectroscopy because they were nearby and bright enough to be observed
on 3-m class telescopes, and those candidates were observed even if their
best-fit photometric type was not a {\small {SN~I}}a\ (see
\S~\ref{subsec:spectro-strategy}). The candidates in the fourth category
were observed because they were at one time typed as a {\small {SN~I}}a. We also show
in Figure~\ref{fig:peakmjd} the time of $B$-band maximum light estimated
from 2 -- 4 light curve epochs in comparison with the ``final'' estimate of
the peak using the full light curves. Again, the reliability improves
substantially between 2 and $\sim 4$~epochs.
\begin{figure}[tb]
\begin{center}
\includegraphics[angle=270, width=0.7\textwidth]{f7.eps}
\end{center}
\caption{The fraction of spectroscopically confirmed {\small {SN}}e{\small{~I}}a\ whose best-fit
model is that of a {\small {SN~I}}a\ as a function of the number epochs (top curve
with stars). Also shown is the fraction of {\small {SN}}e{\small{~I}}a, which satisfy one of
the ``A'' or ``B'' criteria with or without a galaxy redshift prior
(bottom curve with triangles). A large fraction of the sources ($>
80$\%) are typed as {\small {SN~I}}a\ after only 2 epoch of imaging.}
\label{fig:type_snia}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[angle=270, width=0.55\textwidth]{f8.eps}
\end{center}
\caption{Distribution of the difference between the time of $B$-band
maximum estimated using the full light curves and that using the first 2
(top), 3 (middle), and 4 (bottom) epochs of the spectroscopically
confirmed {\small {SN}}e{\small{~I}}a. Values are taken from fits with a flat redshift prior.
The distributions are fit with a single gaussian, and the $\sigma$ values
are listed in each panel. The width of the distribution stays
approximately constant above 4 epochs. The vertical dashed lines
represent the x-axis origin; not the centroid of the distributions.}
\label{fig:peakmjd}
\end{figure}
Studies of the software and handscanning efficiencies as functions of the
object brightness with the fakes (see Section~\ref{sec:fakes}) show that the
{\small {SN}}\ candidate identification and spectroscopic follow-up observations are
essentially complete out to $z \sim 0.12$ \citep{dilday07}. The one
exception to this is our bias against highly-extincted {\small {SN}}e, since the grid
of light curve models did not extend beyond $A_V = 1.0$ in 2005 and $A_V =
3.0$ in 2006. A post-season analysis of the 2005 candidates with an
extended $A_V$ range recovered two additional photometric {\small {SN~I}}a\ candidates
with large extinction in this redshift range. There is no evidence for an
additional population of low-$z$ {\small {SN}}e{\small{~I}}a\ that were missed due to the models
adopted.
The light curve fits generally run on the {\it {gri}}\ search photometry, which is
usually accurate to within $\sim 0.1$~mag. Whenever the computer cluster at
{\small {APO}}\ is idle, we run the pipeline in {\tt sdssforce} mode (see,
\S\ref{subsub:framesub}) to produce $u$ and $z$-band differenced images from
all runs available on the mountain for each of the {\small {SN}}\ candidates. This
typically includes frames from several pre-discovery epochs. The $u$-band
photometry is sometimes useful for distinguishing {\small {SN}}e{\small{~I}}a\ from core-collapse
{\small {SN}}e\ that are generally bluer ($u - g \la 0.5$) during the early phases
after explosion.
\subsection{Observability Index}
\label{subsec:observability}
After the {\small {SN}}\ photometric typing process, the software computes three
quantities that help prioritize the target list for spectroscopic
observations. These quantities are referred to as ``weights'' and they are
constructed specifically to identify candidates that are 1) near or before
peak brightness, 2) less contaminated by galaxy light, and 3) not heavily
extincted by intervening dust. The product of the three weights defines the
``observability index''; candidates with higher observability index are
generally assigned higher priority.
For 1), we define the ``time weight'' as,
\begin{equation}
\label{equ:W_T}
W_T = \left\{ \begin{array}{ll}
e^{|t|\{\Delta m_{15}(B)/[10(1+z)]\}} & {\rm{if}}~t < 0~\rm{days}; \\
e^{-t\{\Delta m_{15}(B)/[20(1+z)]\}} & {\rm{if}}~t \geq 0~\rm{days},
\end{array} \right .
\end{equation}
where $t$ is the current estimate of the {\small {SN}}\ epoch in days relative to
$B$-band maximum brightness. Note that $W_T$ is large when the {\small {SN}}\ is
young, as {\small {SN}}\ are most easily classified near maximum light
\citep{filippenko97}, and decays exponentially with a characteristic time
scale of 20 days. The decline-rate parameter $\Delta m_{15}(B)$\ and redshift $z$ are
also adopted from the best-fit values obtained from the light curve fits.
If the best-fit type is a core-collapse {\small {SN}}, $\Delta m_{15}(B)$\ is set to unity.
The second weight computed by the software is the ``contamination weight''
defined as,
\begin{equation}
\label{equ:W_C}
W_C = e^{-2/\theta} \times
\left(\frac{F_{{\rm{SN}},r}}{F_{{\rm{Gal}},r}}\right)^{1/2},
\end{equation}
where $\theta$ is the distance in arcseconds between the {\small {SN}}\ and the
centroid of the nearest-neighbor host galaxy. The quantity
$F_{{\rm{SN}},r}$ is proportional to the {\it current} $r$-band flux of the
{\small {SN}}, again, as estimated from the best-fit light curve model,
\begin{equation}
\label{equ:F_SN}
F_{{\rm{SN}},r} = 10^{-0.4~r_{\rm{SN}}}
\end{equation}
and $F_{{\rm{Gal}},r}$ is the estimated local galaxy $r$-band flux at the
position of the {\small {SN}}\ defined to be,
\begin{equation}
\label{equ:F_gal}
F_{{\rm{Gal}},r} = 10^{-0.4(r_{\rm{Gal}} + d/d_{{\rm{eff}}})},
\end{equation}
where $d$ is the angular separation between the {\small {SN}}\ and the centroid of the
nearest-neighbor source and $d_{\rm{eff}}$ is the effective radius of that
source along position angle of the {\small {SN}}. This quantity is computed according
to,
\begin{equation}
\label{equ:d_eff}
d_{\rm{eff}} = \frac{{\tt isoA}^2 (1-e^2)}{1-e^2\cos^2\phi}.
\end{equation}
Here, $\phi$ is the position angle of the {\small {SN}}\ measured from the position
angle of host galaxy's semimajor axis ({\tt isoPhi}) with ellipticity given
by,
\begin{equation}
\label{equ:ellip}
e^2 = 1 - \left(\frac{{\tt isoB}}{{\tt isoA}}\right)^2.
\end{equation}
The isophotal galaxy parameters ({\tt isoA}, {\tt isoB}, and {\tt isoPhi}),
which are used to measure the galaxy's ellipticity of the 25 magnitudes per
arcsecond isophote, are adopted from the {\small {SDSS}}\ {\small DR}4
catalog\footnote{http://www.sdss.org/dr4/algorithms/classify.html}.
Clearly, candidates with larger values of $W_C$ suffer less contamination
from the host galaxy light.
The third and final weight is the ``dust weight'', which estimates the
amount of dust extinction from both host galaxy morphology and color, and is
defined to be,
\begin{equation}
\label{equ:W_D}
W_D = e^{-d_{\rm{eff}}/d} \times (1 + 3~f_{\rm{deV}} + 5~c_{\rm{ellip}}),
\end{equation}
The quantity $f_{\rm{deV}}$ is the fractional probability of the source
being well represented by a de Vaucouleurs profile and is again computed
from values in the {\small {SDSS}}\ catalog, i.e.,
\begin{equation}
\label{equ:f_deV}
f_{\rm{deV}} = \frac{{\tt deV\_L}}{{\tt deV\_L} + {\tt exp\_L} + {\tt
star\_L}},
\end{equation}
where {\tt deV\_L}, {\tt exp\_L}, {\tt star\_L} are the null hypothesis
probabilities of the object being well-represented by a de Vaucouleurs
profile, an exponential function, and a stellar profile ({\small {PSF}}),
respectively. The last quantity in parentheses of Equation~\ref{equ:W_D}
characterizes the color of the galaxy, and is defined to be,
\begin{equation}
c_{\rm{ellip}} = \left\{ \begin{array}{ll}
(r_{\rm{Gal}} - i_{\rm{Gal}} - 0.4)/0.2 & {\rm{if}}~(r_{\rm{Gal}} -
i_{\rm{Gal}}) > 0.4~{\rm{and}}~(g_{\rm{Gal}} - r_{\rm{Gal}}) > 0.9; \\
0 & {\rm{otherwise}},
\end{array} \right .
\end{equation}
where $g_{\rm{Gal}},r_{\rm{Gal}},i_{\rm{Gal}}$ are the host galaxy $g$, $r$,
and $i$-band model magnitudes. This quantity is non-zero only for red
elliptical galaxy candidates, which typically populate the color-color space
bounded by $(r - i) > 0.4$ and $(g - r) > 0.9$. Note that the light curves
also provide an independent (and usually more reliable) estimate of the
amount of dust extinction.
The final observability index is defined as the product of the three weights,
\begin{equation}
{\rm{observability~index}} = W_T \times W_C \times W_D.
\end{equation}
In practice, the observability index is rarely used for the bright ($r \la
20.5$~mag) low-$z$ {\small {SN}}\ candidates, since there are enough spectroscopic
resources to observe nearly all of those targets. At high $z$, however, the
number of {\small {SN~I}}a\ candidates exceeds the number we can observe given our
limited resources on large-aperture telescopes. In order to select the best
targets from a large number of candidates that satisfy the light curve
selection criteria, the sources are ranked based on the value of the
observability index. Manual screening and selection of the sources by the
spectroscopic coordinator is still required. We describe our strategy in
detail below.
\subsection{Spectroscopic Follow-up Strategy}
\label{subsec:spectro-strategy}
Spectra are obtained by the various spectroscopic teams, who independently
apply for observing time during each of the search seasons. The teams
agreed to confirm types and obtain redshifts of {\small {SN~I}}a\ candidates at the
highest priority, but each team generally has one or more of their own
{\small {SN}}\ projects that they pursue in parallel. Some of these include 1)
multi-epoch spectroscopy of peculiar {\small {SN}}e, 2) {\small {SN}}e\ in underluminous host
galaxies, 3) {\small {SN~I}}b/c\ and hypernova candidates, 4) detailed spectroscopic
properties of {\small {SN~I}}a\ hosts, and 5) multi-epoch studies of line features and
their diversity in {\small {SN}}e{\small{~I}}a. Whenever possible, we also try to obtain spectra
of other variable objects if a scanner has classified them as
``interesting''. An example of this is a dwarf nova discovered in Aquarius
\citep{prieto06}.
In 2005, the primary telescopes used for observing the low-redshift
candidates were the {\small {ARC}}\ 3.5m, {\small {WHT}}\ 4.2m and the {\small {MDM}}\ 2.4m telescopes.
The high-z candidates were observed with either the {\small {HET}}\ or Subaru. The
resources were sufficient to obtain spectra of most $z \la 0.15$ {\small {SN~I}}a\
candidates as well as core-collapse {\small {SN}}e\ at $z \la 0.06$, but only a
fraction of the $z \ga 0.15$ candidates were observed.
During the Fall 2006 campaign, the various spectroscopic teams were awarded
sufficient 3m-class telescope time to obtain a spectrum of essentially {\it
all} {\small {SN}}\ candidates with $r \la 20.5$~mag (an average {\small {SN~I}}a\ at $z \la
0.15$ and core-collapse {\small {SN}}\ at $z \la 0.06$ near maximum light) {\it and}
obtain multi-epoch spectra of a large sample of nearby ($z \la 0.1$) {\small {SN}}e{\small{~I}}a.
We generally require at least two epochs of detection before we target the
candidate for spectroscopic observation, to guarantee that the source is not
an asteroid. This also enables a better estimate of the epoch of peak
brightness of the {\small {SN}}. Since the primary goal of our survey is to obtain
well-sampled light curves of Type Ia {\small {SN}}e, {\small {SN}}\ Ia candidates that are
discovered early are generally given higher priority for spectroscopy. This
is especially true for the very low-$z$ candidates ($z \la 0.1$) to enable
rapid confirmation and dissemination to the public.
In practice, we initially select {\small {SN~I}}a\ candidates that satisfy at least one
of the ``A'' or ``B'' criteria either with or without a galaxy photometric
redshift prior. There are times, however, during stretches of poor weather
at {\small {APO}}\ and/or when several 3m-class spectroscopic telescopes are online,
when there are not enough new and bright {\small {SN~I}}a\ candidates to observe. This
was not an issue during most of the 2005 season, but it occurred frequently
in 2006. In such cases, we loosen the selection criteria and inspect all
currently-bright sources irrespective of their best-fit {\small {SN}}\ type; there is
only small number ($\sim 30 - 50$) of sources with $r \la 20.5$~mag at any
given time. When necessary, we also target candidates that have only a
single epoch of detection. Extremely red single-epoch targets that are not
associated with a host galaxy are generally avoided (to minimize the chance
of observing an asteroid).
On the other hand, there is never a shortage of higher-$z$ {\small {SN~I}}a\ targets at
$z \ga 0.2$, so the ``A'' or ``B'' criterion with or without a galaxy
photometric redshift prior is always used in the pre-selection process. At
these redshifts, we generally avoid candidates that are located within $\sim
0.5$\arcsec\ of the centroid of a host galaxy, since they have a relatively
higher chance of being an active galactic nucleus. Candidates at $z \sim
0.3$ that are well-separated from the host and suffer a relatively low level
of extinction (as estimated from the multi-band light curve fits) are
high-priority (priority 0 and 1) targets for {\small {HET}}, which is a
queue-scheduled telescope \citep{shetrone07}. Lower-$z$ candidates ($z \sim
0.2$) are assigned lower {\small {HET}}\ priority (mostly priority 2, but some at
priority 3). Targets are generally kept active in the {\small {HET}}\ queue until
they are $\sim 20$ days past maximum light as estimated from the light curve
fits. The Subaru telescope has focused on {\small {SN~I}}a\ candidates in a similar
redshift range, but, to take advantage of its superb image quality, the
Subaru telescope has generally been assigned targets that suffer a
relatively large amount of host contamination (i.e., a low value of $W_C$;
see Section~\ref{subsec:observability}).
The spectra are often analyzed in real time by the observers on site, and
feedback is given to the spectroscopic coordinator. Depending on the
observing conditions, the target list is adjusted in real time. At the end
of the night, the observers provide a preliminary spectroscopic type of each
of the candidates observed. The type and the preliminary redshift is
entered immediately into the database, and the results are disseminated to
the community through the Central Bureau Electronic Telegrams. All nearby
{\small {SN}}e\ ($z \la 0.08$), however, are also announced to a list of subscribers
via email usually within hours from confirmation, allowing rapid
complementary follow up on other telescopes. Information on all
spectroscopically confirmed {\small {SN}}e\ is placed on a public web
page\footnote{http://sdssdp47.fnal.gov/sdsssn/snlist\_confirmed.php} as soon
as the {\small {SN}}\ type has been entered into the database. A list of all
spectroscopically confirmed {\small {SN}}e\ from 2005 and 2006 is presented in
Tables~\ref{tbl:sn2005} and \ref{tbl:sn2006}, respectively.
\section{Imaging Follow-up}
\label{sec:imaging}
Throughout the 2005 search, we used some of our {\small {ARC}}\ 3.5m and {\small {MDM}}\ 2.4m
time, as well as the imager on the {\small NMSU} 1m telescope, University of
Hawaii 88-inch telescope, the 1.8m Vatican Advanced Technology Telescope
({\small {VATT}}), and the 3.5m {\small {WIYN}}\ telescope to (1) augment the light curve points
of spectroscopically confirmed {\small {SN}}e{\small{~I}}a\ during periods of poor weather
conditions at {\small {APO}}\ and (2) obtain late-time photometry of {\small {SN}}e\ that had
faded below the {\small {SDSS}}\ detection limit. Some imaging observations were
performed by the {\small {VATT}}, the {\small {WIYN}}, and the 1.5m telescope at Maidanak
Observatory during the 2006 season. Additional imaging data were obtained
on the {\small {ARC}}\ 3.5m during November of 2006 when the Dual Imaging Spectrograph
was not functional. This effort of follow-up imaging was not carried over
for the {\small {MDM}}\ and for {\small {ARC}}\ during September and October, since we concluded
that spectroscopy is the more valuable use of our resources. In December of
both 2005 and 2006, however, imaging observations were performed to get
light curves of {\small {SN}}e\ discovered in November past maximum light. Deep
multi-band imaging of several host galaxies of {\small {SN}}e{\small{~I}}a\ were also obtained by
the 2.5m Isaac Newton Telescope.
A few high-profile {\small {SDSS-II}}\ {\small {SN}}\ targets were also observed extensively by
several follow-up programs -- optical imaging and spectroscopy by the CfA
{\small {SN}}\ Group, optical and infrared imaging and optical spectroscopy by the
{\small {CSP}}, and optical spectroscopy by the {\small {SNF}actory}.
\section{Results from Fall 2005 and 2006}
\label{sec:results}
The 2005 season resulted in 73 unique {\small {SDSS}}\ imaging runs acquired on 59
different nights. Approximately half of the frames were taken under
non-photometric conditions, bright moon, and/or poor seeing. A total of
155,616 objects were visually inspected during this season, of which 24,402
were tagged as potential {\small {SN}}e, resulting in 11,385 unique {\small {SN}}\ candidates
(see Table~\ref{tbl:summary}). Interestingly, 6,618 of those candidates
were detected in only a single epoch, and are mostly likely moving solar
system objects that were not filtered properly. A few of these single-epoch
candidates may be fast transients that rise and fade on timescales shorter
than $\sim 1$~day \citep{becker04}. Most of the remaining 4767 candidates
are true variables or transient sources\footnote{A very small fraction of
the 4767 multi-epoch candidates are artefacts of image subtraction -- e.g.,
faint dipoles that appear in the same place at two or more different
epochs.}. A short summary of the 2005 and 2006 seasons is shown in
Table~\ref{tbl:summary}.
For the 2005 search season, we acquired 248 spectra from 187 distinct {\small {SN}}\
candidates under various observing conditions. A total of 130 unique
candidates were spectroscopically confirmed to be of Type Ia. One of these
{\small {SN}}e{\small{~I}}a\ ({\small {SN}} 2005hj) was co-discovered and spectroscopically observed by the
Texas Supernova Search \citep{quimby07}. The number of confirmed
core-collapse {\small {SN}}e\ were 7 and 13 for Type Ib/c and II, respectively. One
of these Type II {\small {SN}}e\ ({\small {SN}} 2005mj) was spectroscopically confirmed by the
{\small {ESSENCE}}\ Group. An additional 40 candidates were observed in the
post-season during December 2005 -- January 2006. A sample of 41 {\small {SN}}\
targets including {\small {SN}} 2005hk \citep{phillips07} and {\small {SN}} 2005gj
\citep{prieto07} for which we acquired 16 and 9 spectra, respectively, were
observed spectroscopically more than once. The host galaxy of one 2005
candidate ({\small {SN}} 7017) was observed during the 2006 search season.
Interestingly, that supernova was still active and visible a year after
discovery; the spectrum was identified as a peculiar {\small {SN}}\ similar to
\sn2002ic and \sn2005gj.
In 2006, 90 {\small {SDSS}}\ imaging runs were taken on 60 nights. The number of
visually-inspected objects was reduced by more than an order of magnitude to
14,430. This improvement is due to the modified moving object finder and
the choice of scanning only second-epoch and bright first-epoch objects. We
identified a total of 3694 unique {\small {SN}}\ candidates. Surprisingly, there is
still a somewhat large sample (599 out of 3694) of slow-moving single-epoch
objects that were tagged by a scanner as a possible {\small {SN}}\ and made the
candidates list.
During the second search season, a total of 449 spectra were acquired, $\sim
80$\% more than the 2005 season, from 285 unique candidates. Though the
majority of these were observations of new 2006 candidates, a handful of
host galaxies of the 2005 candidates were observed as well. A sample of 197
candidates, including five candidates that were observed by the CfA {\small {SN}}\
Group and the {\small {ESSENCE}}\ Survey, were spectroscopically confirmed to be
{\small {SN}}e{\small{~I}}a\ and 14 candidates were identified as spectroscopically probable
{\small {SN}}e{\small{~I}}a. The number of confirmed core-collapse {\small {SN}}e\ were 7 and 19 for Type
Ib/c and II, respectively.
Table~\ref{tbl:spectro} summarizes the spectroscopic observations grouped
into candidates discovered in 2005 and 2006. We performed 172 (241)
observations of new {\small {SN~I}}a\ candidates discovered in 2005 (2006), of which
130 (197) were spectroscopically confirmed to be {\small {SN}}e{\small{~I}}a, and 16 (14)
candidates were identified as spectroscopically probable {\small {SN}}e{\small{~I}}a; the latter
are denoted {\small {SN}}\ Ia? in Tables~\ref{tbl:sn2005} and \ref{tbl:sn2006}. In
addition, observations of 7 (7) additional candidates in 2005 (2006)
resulted in galaxy spectra (either the {\small {SN}}\ had faded or the galaxy was too
bright relative to the {\small {SN}}) with spectroscopic redshifts consistent with
photometric redshifts estimated from the multi-band light curves of the
{\small {SN}}e. Therefore, in the first two years, 90\% of {\small {SN~I}}a\ targets resulted in
spectra of {\small {SN}}e{\small{~I}}a, probable {\small {SN}}e{\small{~I}}a, or host galaxies of photometric {\small {SN~I}}a\
candidates. A total of 12 {\small {SN~I}}a\ targets (3\%) were classified instead as
core-collapse {\small {SN}}e, 4 targets (1\%) were classified as either an {\small {AGN}}\ or a
flaring M-dwarf, and 9 targets (2\%) resulted in galaxy spectra with
spectroscopic redshifts inconsistent with photometric redshifts estimated
from the {\small {SN~I}}a\ model fits. Finally, there is a sample of 18 spectra (4\%),
most of which were taken under poor weather conditions, that are
unidentified.
Evaluation of the full light curves after the end of each search season
enables identification of additional photometric {\small {SN~I}}a\ candidates that, for
various reasons, were not spectroscopically observed during the search. We
have identified at least $\sim 200$ high-quality {\small {SN~I}}a\ candidates from the
2005 candidates list, and have already acquired host galaxy spectra for a
significantly-sized subsample. These targets are also good fillers during
periods of poor observing conditions at {\small {APO}}\ that result in a lack of good
new {\small {SN}}\ candidates. As of the writing of this paper, we have measured
redshifts for 81 host galaxies of candidates with Ia-like light curves. An
additional 13 candidates were identified to be good Ia candidates with host
galaxy redshifts from the {\small {SDSS}}\ redshift survey.
\section{Summary}
\label{sec:summary}
The search pipeline of the {\small {SDSS-II}}\ {\small {SN}}\ Survey has enabled efficient
discoveries of variable and transient astronomical sources, filtering over
375,000 objects detected each season into several thousands of {\small {SN}}\
candidates. A reliable photometric {\small {SN}}\ typing system and spectroscopic
follow-up algorithm have allowed spectroscopic observations of $\sim 150$
{\small {SN}}e{\small{~I}}a\ per season with 90\% targeting efficiency. After two seasons, the
search and spectroscopic follow-up algorithms have reached a relatively
mature stage, and it is unlikely that major changes will be made for the
third and final season of 2007. If the observing conditions resemble those
of the previous two years, we expect to increase our sample of
spectroscopically confirmed {\small {SN~I}}a\ by an additional $\sim 150 - 200$ events,
reaching a sample of $\sim 500$ confirmed {\small {SN}}e{\small{~I}}a\ for the completed survey.
A majority of the low-redshift sources will have well-sampled multi-band
light curves that can serve as templates for future studies.
Spectroscopic target selection by a human is appropriate (and preferred) for
a relatively small survey like the {\small {SDSS-II}}\ {\small {SN}}\ survey, but this is
unlikely to be feasible for future large-scale surveys that will discover
thousands or tens of thousands of {\small {SN}}e\ over the period of a few years.
Based on our experience, however, we believe that much of the candidate
identification process can be automated, and with just 2 -- 4 epochs of
multi-band imaging, {\small {SN}}\ candidates can be assigned probabilities that are
reliable enough for performing follow-up spectroscopy. More quantitative
studies of {\small {SN}}\ identification using photometric data alone will be
presented in a future article.
\clearpage
\acknowledgements Funding for the {\small {SDSS}}\ and {\small {SDSS-II}}\ has been provided by
the Alfred P. Sloan Foundation, the Participating Institutions, the National
Science Foundation, the U.S. Department of Energy, the National Aeronautics
and Space Administration, the Japanese Monbukagakusho, the Max Planck
Society, and the Higher Education Funding Council for England. The
{\small {SDSS}}\ Web Site is \verb9http://www.sdss.org/9.
The {\small {SDSS}}\ is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the American
Museum of Natural History, Astrophysical Institute Potsdam, University of
Basel, Cambridge University, Case Western Reserve University, University of
Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, Johns Hopkins University, the Joint Institute for
Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and
Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences
({\small LAMOST}), Los Alamos National Laboratory, the Max-Planck-Institute
for Astronomy ({\small MPIA}), the Max-Planck-Institute for Astrophysics
({\small MPA}), New Mexico State University, Ohio State University,
University of Pittsburgh, University of Portsmouth, Princeton University,
the United States Naval Observatory, and the University of Washington.
The Hobby-Eberly Telescope ({\small {HET}}) is a joint project of the University of
Texas at Austin, the Pennsylvania State University, Stanford University,
Ludwig-Maximillians-Universit\"at M\"unchen, and Georg-August-Universit\"at
G\"ottingen. The {\small {HET}}\ is named in honor of its principal benefactors,
William P. Hobby and Robert E. Eberly. The Marcario Low-Resolution
Spectrograph is named for Mike Marcario of High Lonesome Optics, who
fabricated several optics for the instrument but died before its completion;
it is a joint project of the Hobby-Eberly Telescope partnership and the
Instituto de Astronom\'{\i}a de la Universidad Nacional Aut\'onoma de
M\'exico. The Apache Point Observatory 3.5-meter telescope is owned and
operated by the Astrophysical Research Consortium. We thank the observatory
director, Suzanne Hawley, and site manager, Bruce Gillespie, for their
support of this project. The Subaru Telescope is operated by the National
Astronomical Observatory of Japan. The William Herschel Telescope is
operated by the Isaac Newton Group, and the Nordic Optical Telescope is
operated jointly by Denmark, Finland, Iceland, Norway, and Sweden, both on
the island of La Palma in the Spanish Observatorio del Roque de los
Muchachos of the Instituto de Astrofisica de Canarias. Observations at the
{\small ESO} New Technology Telescope at La Silla Observatory were made
under programme {\small ID}s 77.A-0437, 78.A-0325, and 79.A-0715. Kitt Peak
National Observatory, National Optical Astronomy Observatory, is operated by
the Association of Universities for Research in Astronomy, Inc. ({\small
AURA}) under cooperative agreement with the National Science Foundation.
The {\small WIYN} Observatory is a joint facility of the University of
Wisconsin-Madison, Indiana University, Yale University, and the National
Optical Astronomy Observatories. The W.M.\ Keck Observatory is operated as
a scientific partnership among the California Institute of Technology, the
University of California, and the National Aeronautics and Space
Administration. The Observatory was made possible by the generous financial
support of the W.M.\ Keck Foundation. The South African Large Telescope of
the South African Astronomical Observatory is operated by a partnership
between the National Research Foundation of South Africa, Nicolaus
Copernicus Astronomical Center of the Polish Academy of Sciences, the
Hobby-Eberly Telescope Board, Rutgers University, Georg-August-Universit\"at
G\"ottingen, University of Wisconsin-Madison, University of Canterbury,
University of North Carolina-Chapel Hill, Dartmough College, Carnegie Mellon
University, and the United Kingdom {\small {SALT}}\ consortium.
\clearpage
|
1,314,259,996,704 | arxiv | \section{Introduction}
\label{secintro}
Random sequential adsorption (RSA) is a classical model for various
physical, chemical or biological problems \cite{Evans:1993:RCS}. In the
simplest form, RSA processes can be formulated as sequential addition to
a structure of
objects that cannot overlap, and once inserted, cannot move or leave the
structure \cite{Brilliantov:1996:FFO}.
In this paper, we present a pharmacological example in which application of
the RSA model can provide meaningful qualitative insights. Motivated by
pharmacological applications, we present
a slight generalization of the classical RSA model to enable us to study the
effects of polydispersity and partial overlap of adsorbing macromolecules
on the surface of a virus. We also study the dependence of the adsorption
process on interactions (reactions) of the adsorbing macromolecules
with the solvent.
The paper is organized as follows. In Section \ref{secbiology}, we will
introduce the motivating pharmacological example and our questions of
interest. In Section \ref{secmodel}, we will introduce the generalized
random sequential adsorption (gRSA) model suitable for capturing essential
features of the pharmacological problem from Section \ref{secbiology},
which we formulate in one dimension. The analytical results for
this model are presented in Section \ref{secanalysis}. We derive the
governing integro-differential equation for the evolution of gaps between
polymers, and compute the asymptotical properties
of the quantities of interest, namely the number of macromolecules adsorbed
and the total area (in one dimension,
length) they cover. In Section \ref{seceqfree}, we apply equation-free
methods to the computational study of the system.
The main idea that underlies this
equation-free computer-assisted analysis is the design and execution of
appropriately-initialized short bursts of stochastic simulations; the
results of these are processed to estimate coarse-grained quantities of
interest - in this case the self-similarly evolving shape of the gap
statistics in the problem.
Finally, in Section \ref{secdiscussion}, we discuss the higher dimensional
case and summarize the connections between the theory and the experimental
data.
\section{Pharmacological background}
\label{secbiology}
Many medical conditions such as cancer, heart disease and heritable
disorders (hemo\-phi\-lia, cystic fibrosis, muscular dystrophy etc) have
faulty, mutant genes as an underlying cause. Healthy, normal genes can be
readily synthesized in the laboratory but introducing them into diseased
cells remains a challenge. Many research groups are studying viruses,
such as adenovirus, as a means to introduce normal genes into diseased
cells. For therapeutic use, the virus' own DNA is usually partially or
completely replaced by the gene of interest. The most common adenovirus
strain used for this purpose is adenovirus type 5 (Ad5), because it is easy
to manipulate and is non-pathogenic in humans \cite{Vorburger:2002:AGT}.
Ad5 has been used with great success to treat diseases in laboratory animals
but the results have not been replicated in humans. One of the greatest
problems with using Ad5 in humans is the presence of neutralizing
antibodies. In addition, the viruses often infect non-target cells,
particularly the liver, causing unwanted toxicity.
Our laboratory is exploring the use of hydrophilic polymers such as
poly[N-(2-hydroxypropyl)methacrylamide] (pHPMA) to coat virus particles and
protect them from neutralising antibodies by steric shielding. This
technique is very effective at protecting the virus and permitting it
to be retargeted to specific cell types
\cite{Fisher:2001:PCA,Fisher:2000:VSR,Pouton:1998:KIN}.
The polymer has multiple esters along its length that are used to bind to
the amino side chain of lysine residues on the virus surface. In a coating
reaction the polymers bind randomly to the virus surface until (a) all of
the lysine residues are occupied or (b) lysine residues are rendered
inaccessible
(obscured) by polymer chains. We know little about the orientation of
polymers on the virus surface or how to optimise the coating reaction
because there are no techniques to visualise the orientation of polymers on
the virus surface.
Representative questions one would like to answer are:
How many polymer molecules will become attached to the viral surface
by a given time? How large is the
surface area covered by the polymer coat at that time? In this
paper, the theoretical approach is chosen to address these
questions for simplified models of the adsorption process.
Since the adsorption process is driven by the diffusion of molecules to the
surface of the virus, and since the adsorption is effectively irreversible,
a suitable modification of the classical RSA might be applied to model the
process.
It is important to take into account polydispersity in the polymer size.
Even if we prepare the polymer molecules with a specified target molecular
weight, some relatively small molecules of the polymer will always be
present, and they will diffuse faster than the larger molecules.
A smaller molecule can reach the surface of the virus at a higher rate. We
will consider the adsorption of polymers whose diameters are distributed
according to a probability distribution function $P(z).$ The solution is
assumed to be well-mixed. We also define $p(z)$ as the probability
distribution function of the particles which can reach the surface in a
single time step. An important modelling issue lies in a good choice of
$P(z)$ (e.g. it might be the Gamma distribution) and in a realistic relation
between $P(z)$ and $p(z)$. In Section \ref{secmodel}, we simply specify
$p(z)$ (avoiding the above questions). The long term dynamics of the
polydisperse adsorption depends on the behaviour of $p(z)$ close to zero.
We will use different distributions $p(z)$ given by (\ref{distrlength}).
Finally, the reactive groups of the polymer molecules can also react with
the solvent before reaching the surface.
This is the case for commonly used biocompatible pHPMA reactive polymers
\cite{Subr:2005:CDC}. If all reactive groups of a polymer are already
hydrolyzed then the polymer cannot covalently bind to the surface of the
virus. Hence, we have to consider that only a fraction $r(t)$ of the
polymers is still reactive at time $t$.
Depending on the form of $r(t)$, different polymer coats may be created.
This issue will be discussed in more detail in Section
\ref{sectimedependent}.
Polymers are long flexible molecules \cite{Doi:1996:IPP}.
The pHPMA polymer molecule has (one or more) reactive
group(s) which can react with the primary amino groups on the viral surface.
As a result, a polymer molecule becomes covalently
(irreversibly) attached to the surface at a point.
The rest of the polymer is not attached (unless another covalent bond is
created) and it freely ``wiggles" in the space above the viral surface.
Having in mind that the ``wiggling tail"
does not perfectly shield the underlying surface, we generalize the
classical RSA model to allow partial overlap of the adsorbing
macromolecules, i.e. we allow some squeezing of the polymers.
Let $N(t)$ be the number of polymers attached to the surface at time $t$.
Let $A(t)$ be the total area of the surface covered by adsorbed polymers at
time $t$.
Since we cover the surface by polymers
of different sizes, there is no obvious relation between $A(t)$ and $N(t)$.
However, both variables $A(t)$ and $N(t)$ are of practical interest as
discussed below.
We cover the surface of the virus by polymers to protect the surface from
unwanted interactions. Hence, the number $A(t)$ gives us the simple
characterization of the area of the surface which is protected by the
polymer coat.
The unwanted interactions are not the only problem which one has to overcome
in order to use viruses as a drug delivery system. Another important task is
to retarget the virus to infect the cells of interest (e.g. cancer cells)
via new receptors. Assuming that we put one ``targeting" group per polymer
molecule, the number of targeting molecules will be equal to $N(t).$
If we considered the adsorption of the same-size nonoverlapping objects
of the area $a$,
then we would have $A(t)=a N(t)$. In our case, the adsorbing molecules have
different sizes. There is no obvious relation between $A(t)$ and $N(t)$ and
both quantities are of interest. In the following sections, we will present
theoretical approaches to compute the time evolution of $A(t)$ and $N(t)$.
\bigskip
\section{A simple generalization of random sequential adsorption}
\label{secmodel}
Random sequential adsorption has been extensively
studied during the last several decades \cite{Evans:1993:RCS}.
The theoretical work is more mature in one
dimension with information in higher dimensions mostly
coming from numerical simulations \cite{Krapivsky:1992:KRS}.
If we consider fixed size objects, then RSA usually starts
from an empty surface and continues until the time
when no further object can be placed, the so-called
``jamming limit".
If the objects to be covered have spherical symmetry
\cite{Swendsen:1981:DRS} then the coverage approaches the
jamming limit as $t^{-1/d}$ where $t$ is time and $d$ is the
dimension.
The asymptotic behaviour can be more complicated
for objects of different shape \cite{Evans:1993:RCS}.
As argued in Section \ref{secbiology}, polydispersity
is often present in real systems.
If we allow adsorbing
particles (in one spatial dimension)
of arbitrarily small length, then the coverage
approaches the full coverage as $t \to \infty.$
Relatively less is known for polydisperse
adsorption. One-dimensional analytical results
can be found in \cite{Krapivsky:1992:KRS},
where it is assumed that the
attached polymers prevent binding of other
polymer molecules that would overlap with them.
In reality, the polymer molecules are stretching
during the adsorption process, creating a polymer
brush (for semitelechelic polymers) after sufficiently
long time \cite{Milner:1988:TGP,Gennes:1980:CPA}. Thus
each molecule ``covers" a smaller surface area
at later times.
Consequently, it is
possible to adsorb more molecules onto the surface.
Here, we take this fact into account and
we modify the random sequential adsorption algorithm
accordingly.
We state our generalized random sequential
algorithm (gRSA) in one dimension as follows.
\medskip
\leftskip 8mm
\rightskip 8mm
\noindent
{\bf gRSA algorithm:}
{\it
We consider adsorption of small intervals of different
sizes onto the interval $[0,1]$, the adsorbing domain.
At each
time step, the size of a small interval is chosen randomly
according to the probability distribution function $p(z).$
We select randomly the position of the center $w$ inside the
adsorbing domain $[0,1]$ and we make an attempt to place the
small interval of length $z$ there.
If the center $w$
of the segment to be adsorbed lies inside a segment already placed, the
adsorption is rejected.
If the position of the center $w$ is chosen
in the gap $(x_1,x_2)$ between attached polymers, then the new
polymer segment is adsorbed with probability $\xi(z,w-x_1,x_2-w)$
where $w-x_1$ and $x_2-w$ are distances of the center $w$ of
the polymer from the endpoints of the gap $(x_1,x_2)$.
}
\leftskip 0mm
\rightskip 0mm
\medskip
\noindent
The parameters of gRSA which have to be specified include the
probability distribution function $p(z)$ and the
probability $\xi(z,w-x_1,x_2-w)$.
In what follows,
we assume that the lengths of polymers are distributed according
to the formula
\begin{equation}
p(z) =
\left\{
\begin{array}{ll}
(\alpha+1) \varepsilon^{-\alpha-1} z^{\alpha} \quad
& \mbox{for} \; z < \varepsilon, \\
0 & \mbox{for} \; z \ge \varepsilon,
\end{array}
\right.
\label{distrlength}
\end{equation}
for $\alpha>-1$ and small $\varepsilon \ll 1.$
Let us assume that the position of the center $w$ of the
new polymer is chosen in the gap $[x_1,x_2],$
i.e. $w \in [x_1,x_2]$.
Then we take the probability (per unit time) of adsorbing the
polymer segment of length $z \le x = x_1 - x_2$ as
\begin{equation}
\xi(z,w-x_1,x_2-w)
=
\left\{
\begin{array}{ll}
\displaystyle \frac{2 (w - x_1)}{z} \quad &
\mbox{for} \; w \in \left[ x_1, x_1 + \displaystyle \frac{z}{2} \right]; \\
1 \raisebox{-3.6mm}{\rule{0pt}{10mm}} \quad &
\mbox{for} \; w \in
\left[ x_1 + \displaystyle \frac{z}{2},
x_2 - \displaystyle \frac{z}{2} \right]; \\
\displaystyle \frac{2 (x_2 - w)}{z} \quad &
\mbox{for} \; w \in \left[ x_2 - \displaystyle \frac{z}{2}, x_2 \right];
\end{array}
\right.
\label{defxi1}
\end{equation}
and the probability of adsorbing the polymer segment of length
$z > x$ as
\begin{equation}
\xi(z,w-x_1,x_2-w)
=
\left\{
\begin{array}{ll}
\displaystyle \frac{2 (w - x_1)}{z} \raisebox{-3.6mm}{\rule{0pt}{10mm}} \quad &
\mbox{for} \; w \in \left[ x_1, \displaystyle \frac{x_1 + x_2}{2} \right]; \\
\displaystyle \frac{2 (x_2 - w)}{z} \raisebox{-2.6mm}{\rule{0pt}{10mm}} \quad &
\mbox{for} \; w \in \left[ \displaystyle \frac{x_1 + x_2}{2}, x_2 \right].
\end{array}
\right.
\label{defxi2}
\end{equation}
In the latter case, the maximum probability of adsorption is achieved
for $w=\frac{x_1 + x_2}{2}$, for which
$\xi(w) = \frac{x}{z}$.
The formulas (\ref{defxi1}) and (\ref{defxi2}) give
the same probability density function $\xi(\cdot)$ for $z=x$ as is
desirable.
The plot of $\xi$ as a function of $w$ is given in Figure
\ref{figprobxi}.
\begin{figure}
\picturesAB{./figs/modRSAprobA.eps}
{./figs/modRSAprobB.eps}
{1.3in}
\caption{{\it The probability $\xi$ as a function of $w$ for
gRSA model $(\ref{defxi1})$ -- $(\ref{defxi2})$:}
(a) {\it for the case $z \le x=x_2-x_1$;}
(b) {\it for the case $z > x=x_2-x_1$}.}
\label{figprobxi}
\end{figure}
Formula (\ref{defxi1}) is shown
in Figure \ref{figprobxi}(a) where the gap size $x=x_2-x_1$
is greater than the length of the new polymer segment $z$.
Formula (\ref{defxi2}) is shown
in Figure \ref{figprobxi}(b) where the gap size $x=x_2-x_1$
is less than the length of the new polymer $z$.
To explain the motivation behind formula (\ref{defxi1}),
three possible cases of the relative position of the new
(red) interval of the length $z \le x$ and the gap $(x_1,x_2)$
are shown in Figure \ref{figmodRSA}.
\begin{figure}
\picturesABCm{./figs/modRSApolA.eps}
{./figs/modRSApolB.eps}
{./figs/modRSApolC.eps}{0.83in}
\caption{{\it Schematic of gRSA.} (a) {\it Polymer is refused;}
(b) {\it polymer is adsorbed with the
probability $\xi(z,w-x_1,x_2-w)$};
(c) {\it polymer is adsorbed.}}
\label{figmodRSA}
\end{figure}
In Figure \ref{figmodRSA}(a), the red interval is rejected
because its middle point $w$ lies inside a polymer segment which
is already adsorbed to the surface.
Hence, the probability
of adsorption is $0$, the same probability as in the
classical RSA model.
In Figure \ref{figmodRSA}(c), the red
segment of the length $z$ does not overlap with neighbouring
polymers, and we allow it to be adsorbed with probability $\xi=1$.
Cases in Figure \ref{figmodRSA}(a) and
Figure \ref{figmodRSA}(c) are treated as in the classical RSA
model.
In Figure \ref{figmodRSA}(b), the center of the red polymer is
inside the gap but the red polymer overlaps with neighboring
polymers segments.
This polymer would be rejected by the classical RSA model.
We believe it is more realistic to consider that such a polymer will
be adsorbed with some nonzero probability which continuously
interpolates between the cases shown in Figures \ref{figmodRSA}(a) and
\ref{figmodRSA}(c), i.e. between zero for $w-x_1=0$ and 1 for
$w-x_1=z$.
Formula (\ref{defxi1}) takes this fact into
account, using simple linear interpolation.
Formula
(\ref{defxi2}) naturally extends the formula (\ref{defxi1})
for polymer segment lengths greater than the gaps (see also
Figure \ref{figprobxi}).
Having explained the new rules for adsorbing the polymer,
we must also specify what part of the surface is actually
covered.
We will assume that the new polymer covers only
the intersection of the intervals
\begin{equation}
\left[ w-\frac{z}{2},w+\frac{z}{2} \right]
\bigcap
\left[ x_1, x_2 \right].
\label{covered}
\end{equation}
This guarantees that a possibly long, newly
adsorbed polymer will not ``spill over" and cover any
part of the neighboring gaps.
\bigskip
\section{Analysis of gRSA}
\label{secanalysis}
Let $G(x,t)$ be the concentration of gaps (holes) of length
$x$ at time $t$ and let $C(x,t)$ be the corresponding
cumulative probability
distribution function; that is,
\begin{equation}
C(x,t) = \frac{1}{\int_0^\infty G(y,t) \mbox{d}y} \int_0^x G(y,t) \mbox{d}y.
\label{forcdf}
\end{equation}
The total length of the surface that is covered by polymers
at time $t$, $A(t)$, is directly related to $G(x,t)$
by
\begin{equation}
A(t)
=
1 - \int_0^1 x G(x,t) \mbox{d}x.
\label{evolA}
\end{equation}
The number of polymers attached to the surface at time $t$, $N(t)$,
can be also related to $G(x,t)$, as we will see in Section \ref{secevolN}.
Thus, the starting point of the analysis of the system is
the derivation of the evolution equation for
the distribution function of gaps $G(x,t)$.
A gap of length $x$ can be created from a larger gap
(of length $y>x$) by adsorbing a suitable interval to the
system.
Thus the evolution of the concentration of gaps
$G(x,t)$ is given by the equation
\begin{eqnarray}
\frac{\partial G}{\partial t} (x,t)
& = &
-
\;
G(x,t)
\int_0^\infty
\left[
\int_0^x \xi(z,u,x-u)
\mbox{d}u
\right]
p(z)
\mbox{d}z
\;
+
\label{evolGgen}
\\
&&
+
\int_x^{\infty}
\left[
\int_0^{2(y-x)}
2 \,
\xi \left(z, x+\frac{z}{2}, y-x-\frac{z}{2} \right)
p(z) \mbox{d}z
\right]
G(y,t) \mbox{d}y.
\nonumber
\end{eqnarray}
Using (\ref{defxi1}) and (\ref{defxi2}), equation
(\ref{evolGgen}) can be rewritten in the following form
$$
\frac{\partial G}{\partial t} (x,t)
=
- G(x,t)
\int_0^{x}
\left[
\int_0^{z/2}
\frac{2 u}{z}
\mbox{d}u
+
\int_{z/2}^{x-z/2}
1 \,
\mbox{d}u
+
\int_{x-z/2}^{x}
\frac{2 (x-u)}{z}
\mbox{d}u
\right]
p(z)
\mbox{d}z
-
$$
\begin{equation}
-
G(x,t)
\int_x^\infty
\left[
\int_0^{x/2}
\frac{2 u}{z}
\mbox{d}u
+
\int_{x/2}^x
\frac{2 (x-u)}{z}
\mbox{d}u
\right]
p(z)
\mbox{d}z
+
\label{evolGgen2}
\end{equation}
$$
+
\int_x^{\infty}
\int_0^{y-x}
2
G(y,t) p(z)
\mbox{d}z \mbox{d}y
+
\int_x^{\infty}
\int_{y-x}^{2(y-x)}
2
\frac{2 (y-x) - z}{z}
G(y,t) p(z)
\mbox{d}z \mbox{d}y.
$$
Hence,
\begin{equation}
\frac{\partial G}{\partial t} (x,t)
=
- G(x,t)
\int_0^{x}
\left[
x
-
\frac{z}{2}
\right]
p(z)
\mbox{d}z
-
G(x,t)
\int_x^\infty
\left[
\frac{x^2}{2 z}
\right]
p(z)
\mbox{d}z
+
\label{evolGgen3}
\end{equation}
$$
+
\int_x^{\infty}
\int_0^{y-x}
2
G(y,t) p(z)
\mbox{d}z \mbox{d}y
+
\int_x^{\infty}
\int_{y-x}^{2(y-x)}
2
\left[
\frac{2(y-x)}{z} - 1
\right]
G(y,t) p(z)
\mbox{d}z \mbox{d}y.
$$
We assume that the lengths of polymers are
distributed according to formula (\ref{distrlength})
for $\alpha>-1$ and small $\varepsilon \ll 1.$
Moreover, we assume
that there are already
no holes of the length greater than
$\varepsilon/2$ in the system, i.e. $G(x,t)=0$ for
$x>\varepsilon/2.$
Then (using (\ref{distrlength})),
equation (\ref{evolGgen3}) can be rewritten
(for $x < \varepsilon/2$ and $\alpha \ne 0$) as
$$
\frac{\partial G}{\partial t} (x,t)
=
-
\frac{G(x,t)(\alpha+1)}{\varepsilon^{\alpha+1}}
\int_0^{x}
\left[
x
-
\frac{z}{2}
\right]
z^{\alpha}
\mbox{d}z
-
\frac{G(x,t)(\alpha+1)}{\varepsilon^{\alpha+1}}
\int_x^{\varepsilon}
\left[
\frac{x^2}{2 z}
\right]
z^{\alpha}
\mbox{d}z
+
$$
$$
+
\frac{2 (\alpha+1)}{\varepsilon^{\alpha+1}}
\int_x^{\infty}
G(y,t)
\int_0^{y-x}
z^{\alpha}
\mbox{d}z \mbox{d}y
+
$$
$$
+
\frac{2 (\alpha+1)}{\varepsilon^{\alpha+1}}
\int_x^{\infty}
G(y,t)
\int_{y-x}^{2(y-x)}
\left[
\frac{2(y-x)}{z} - 1
\right]
z^{\alpha}
\mbox{d}z \mbox{d}y
$$
which implies
\begin{equation}
\frac{\partial G}{\partial t} (x,t)
=
\frac{x^{\alpha+2} G(x,t)}{\alpha (\alpha+2)\varepsilon^{\alpha+1}}
-
\frac{x^2 G(x,t)(\alpha+1)}{2 \alpha \varepsilon}
+
\frac{2^{\alpha+2}-4}{\alpha \varepsilon^{\alpha+1}}
\int_x^{\infty}
G(y,t)
(y-x)^{\alpha+1}
\mbox{d}y.
\label{evolGgen4}
\end{equation}
If $\alpha=0$, equation (\ref{evolGgen3}) implies
(for $x < \varepsilon/2$)
\begin{equation}
\frac{\partial G}{\partial t} (x,t)
=
-
\frac{x^2 G(x,t)}{2 \varepsilon}
\left(
\frac{3}{2}
+
\ln \left[ \frac{\varepsilon}{x} \right]
\right)
+
\frac{4 \ln 2}{\varepsilon}
\int_x^{\infty}
G(y,t)
(y-x)
\mbox{d}y.
\label{evolGgen5}
\end{equation}
Equation (\ref{evolGgen4}) (or (\ref{evolGgen5}))
is the desired integro-differential equation for $G(x,t)$.
If we solve (\ref{evolGgen4}), we can compute the evolution
of $A(t)$ by (\ref{evolA}).
The equation for the evolution of $N(t)$ is
given in the next section.
\subsection{Evolution of $N(t)$}
\label{secevolN}
At each time step, an interval of length
between $(z,z+\mbox{d}z)$ is chosen with probability $p(z)\mbox{d}z.$
This interval can be placed in any gap of size
$x$ with probability $\int_0^x \xi(z,u,x-u) \mbox{d}u.$
There exist $G(x,t)\mbox{d}x$ gaps whose size lies
in the interval $(x,x+\mbox{d}x).$
Hence, the integral
$\int_0^\infty [\int_0^x \xi(z,u,x-u) \mbox{d}u] G(x,t) \mbox{d}x$ gives the
probability that the randomly chosen position of the
polymer of length $z$ will be accepted.
Thus the probability of attaching a polymer of any length
at one time step is equal to
\begin{equation}
\int_0^\infty \int_0^\infty
\left[\int_0^x \xi(z,u,x-u) \mbox{d}u \right] G(x,t) p(z) \mbox{d}x \mbox{d}z.
\label{probabilityofattachment}
\end{equation}
Using a continuous approximation for $N(t)$, we find
that $N(t)$ satisfies the following ordinary
differential equation
\begin{equation}
\frac{\mbox{d}N}{\mbox{d}t}
=
\int_0^\infty \int_0^\infty
\left[\int_0^x \xi(z,u,x-u) \mbox{d}u \right]
G(x,t) p(z) \mbox{d}z \mbox{d}x.
\label{equationforN}
\end{equation}
Taking $p(z)$ to be given by (\ref{distrlength})
and $\xi(z,u,x-u)$ to be given by (\ref{defxi1}) -- (\ref{defxi2}),
and considering the regime where all gaps are
already less than $\varepsilon$ (i.e. $G(x,t) = 0$
for $x > \varepsilon$), we obtain
$$
\int_0^\infty \int_0^\infty
\left[\int_0^x \xi(z,u,x-u) \mbox{d}u \right]
G(x,t) p(z) \mbox{d}z \mbox{d}x
=
$$
$$
\int_0^\infty \int_0^x
\left[
\int_0^{z/2}
\frac{2 u}{z}
\mbox{d}u
+
\int_{z/2}^{x-z/2}
1
\mbox{d}u
+
\int_{x-z/2}^x
\frac{2 (x -u)}{z}
\mbox{d}u
\right]
G(x,t) p(z) \mbox{d}z \mbox{d}x
+
$$
$$
+
\int_0^\infty \int_x^\infty
\left[
\int_0^{x/2}
\frac{2 u}{z}
\mbox{d}u
+
\int_{x/2}^x
\frac{2 (x -u)}{z}
\mbox{d}u
\right]
G(x,t) p(z)
\mbox{d}z \mbox{d}x
=
$$
$$
=
\int_0^\varepsilon
G(x,t) \int_0^x
\left[
x - \frac{z}{2}
\right]
p(z) \mbox{d}z \mbox{d}x
+
\frac{1}{2}
\int_0^\varepsilon
G(x,t) x^2
\int_x^\infty
\frac{p(z)}{z}
\mbox{d}z
\mbox{d}x
=
$$
$$
=
\frac{\alpha+1}{\varepsilon^{\alpha+1}}
\int_0^\varepsilon
G(x,t) \int_0^x
\left[
x - \frac{z}{2}
\right]
z^{\alpha} \mbox{d}z \mbox{d}x
+
\frac{\alpha+1}{2 \varepsilon^{\alpha+1}}
\int_0^\varepsilon
G(x,t) x^2
\int_x^\varepsilon
z^{\alpha-1}
\mbox{d}z
\mbox{d}x
=
$$
$$
=
-
\frac{1}{\alpha(\alpha+2)\varepsilon^{\alpha+1}}
\int_0^\varepsilon
G(x,t) x^{\alpha+2}
\mbox{d}x
+
\frac{\alpha+1}{2 \alpha \varepsilon}
\int_0^\varepsilon
G(x,t) x^2
\mbox{d}x.
$$
Hence
\begin{equation}
\frac{\mbox{d}N}{\mbox{d}t}
=
-
\frac{1}{\alpha(\alpha+2)\varepsilon^{\alpha+1}}
\int_0^\varepsilon
G(x,t) x^{\alpha+2}
\mbox{d}x
+
\frac{\alpha+1}{2 \alpha \varepsilon}
\int_0^\varepsilon
G(x,t) x^2
\mbox{d}x.
\label{equationforN2}
\end{equation}
Before analyzing (\ref{evolGgen4}) and (\ref{equationforN2})
further, we summarize some results from the literature
on classical RSA.
\bigskip
\subsection{Some results for the classical RSA}
\label{classicalRSA}
If we consider particles of the same length
$\varepsilon$ so that $p(z) = \delta (z - \varepsilon)$,
and if we choose $\xi(z,w-x_1,x_2-w)$ of the form
\begin{equation}
\xi(z,w-x_1,x_2-w)
=
\left\{
\begin{array}{ll}
1 \raisebox{-3.6mm}{\rule{0pt}{8mm}} \quad &
\mbox{for} \; \;
x_1 + \displaystyle \frac{z}{2} \le w \le
x_2 - \displaystyle \frac{z}{2}; \\
0 \raisebox{-2.6mm}{\rule{0pt}{8mm}} \quad &
\mbox{otherwise};
\end{array}
\right.
\label{defxi3}
\end{equation}
then our gRSA algorithm reduces to the classical RSA algorithm.
The evolution equation
(\ref{evolGgen}) can be used to verify known one-dimensional results
about fixed segment size, non-overlapping random sequential
adsorption \cite{Krapivsky:1992:KRS},
namely one can show that the jamming limit
is asymptotically approached as $t^{-1}$ \cite{Swendsen:1981:DRS}.
Random sequential adsorption with a probability
distribution $p(z)$ given by (\ref{distrlength}) and
probability $\xi(z,w-x_1,x_2-w)$ given by (\ref{defxi3})
has been studied in \cite{Krapivsky:1992:KRS}.
Then equation (\ref{evolGgen}) for
$x<\varepsilon$ reads as follows (assuming that initially there exist no
holes of length greater than $\varepsilon$
in the system, i.e. $G(x,t)=0$ for $x>\varepsilon$)
\begin{equation}
\frac{\partial G}{\partial t} (x,t)
=
-
\frac{x^{\alpha+2} G(x,t)}{(\alpha+2)\varepsilon^{\alpha+1}}
+
\frac{2}{\varepsilon^{\alpha+1}}
\int_x^{x+\varepsilon} G(y,t) (y-x)^{\alpha+1} \mbox{d}y.
\label{evolGn2}
\end{equation}
The scaling ansatz \cite{Krapivsky:1992:KRS}
for the concentration $G(x,t)$ can be written as
\begin{equation}
G(x,t) \sim t^{a} \Phi \left( x \, t^b \, \right)
\quad
\mbox{for}
\quad
x \ll 1,
\quad
t \gg 1,
\quad
\mbox{and}
\quad
x t^b \; \mbox{finite}.
\label{scalingansatz}
\end{equation}
Defining the moments
\begin{equation}
M_\gamma (t) = \int_0^\infty x^{\gamma} G(x,t) \mbox{d}x,
\qquad
m_\gamma = \int_0^\infty \xi^{\gamma} \Phi(\xi) \mbox{d}\xi
\label{defmoments}
\end{equation}
and using (\ref{scalingansatz}), we obtain
\begin{equation}
M_\gamma (t) \sim t^{a - b - b \gamma} m_\gamma.
\label{scalmoments}
\end{equation}
Moreover, multiplying equation (\ref{evolGn2}) by $x^\gamma$
and integrating over $x$, one can derive
the equation for moments,
\begin{equation}
\frac{\partial M_\gamma}{\partial t} (x,t)
=
\frac{F(\gamma,\alpha)}{\varepsilon^{\alpha+1}}
M_{\gamma + \alpha + 2}
\quad
\mbox{where}
\quad
F(\gamma,\alpha)
=
2
B(\gamma+1,\alpha+2)
-
\frac{1}{\alpha+2},
\label{evolGn3}
\end{equation}
where $B(\cdot,\cdot)$ is Beta function.
We define the function $\widehat{\gamma}(\alpha)$ implicitly
by the equation $F(\widehat{\gamma},\alpha) = 0$.
If $\gamma$ is equal
to $\widehat{\gamma}(\alpha)$, then the moment $M_\gamma$
is independent of time.
Hence, using (\ref{scalmoments}),
we obtain the relation
$a = b + b \, \widehat{\gamma}(\alpha)$
between the coefficients of the scaling ansatz (\ref{scalingansatz})
and the parameter $\alpha$ of the model.
Finally, substituting the scaling ansatz (\ref{scalingansatz})
in (\ref{evolGn2}), we find that
$b = (\alpha+2)^{-1}.$
Thus, the scaling of moments
(\ref{scalmoments}) can be rewritten in the form
\begin{equation}
M_\beta (t) \sim
t^{\mu}
\qquad
\mbox{where}
\quad
\mu
=
\frac{\widehat{\gamma}(\alpha) - \beta}{\alpha+2}.
\label{scalmomentsrevised}
\end{equation}
Using (\ref{evolA}) and (\ref{scalmomentsrevised}), we obtain
\begin{equation}
1 - A(t)
=
\int_0^1 x G(x,t) \mbox{d}x
\sim
t^{-\omega(\alpha)}
\qquad
\mbox{where}
\quad
\omega(\alpha)
=
\frac{1-\widehat{\gamma}(\alpha)}{\alpha+2}.
\label{scalingAt}
\end{equation}
The graph of the function $\omega(\alpha)$
is given in Figure \ref{figgraphomegasigma}(a).
\begin{figure}
\picturesAB{./figs/graphomega.eps}{./figs/graphsigma.eps}{2.2in}
\caption{(a) {\it
The graph of the exponent $\omega(\alpha)$ given
by $(\ref{scalingAt})$.}
(b) {\it
The graph of the exponent $\sigma(\alpha)$ given
by $(\ref{scalingNt})$.}
}
\label{figgraphomegasigma}
\end{figure}
The equation (\ref{equationforN}) for $p(z)$ given by
(\ref{distrlength}) and probability $\xi(z,w-x_1,x_2-w)$
given by (\ref{defxi3}) reads as follows:
\begin{equation}
\frac{\mbox{d}N}{\mbox{d}t}
=
\frac{\varepsilon^{-\alpha-1}}{\alpha+2}
\int_0^\varepsilon
x^{\alpha+2}
G(x,t) \mbox{d}x.
\label{equationforN3}
\end{equation}
Using (\ref{scalmomentsrevised}), we obtain (for $\sigma(\alpha)>0$)
\begin{equation}
N(t)
\sim
t^{\sigma(\alpha)}
\qquad
\mbox{where}
\; \;
\sigma(\alpha)
=
\frac{\widehat{\gamma}(\alpha)}{\alpha+2}.
\label{scalingNt}
\end{equation}
The graph of the function $\sigma(\alpha)$
is given in Figure \ref{figgraphomegasigma}(b).
\subsection{Evolution of gRSA}
\label{secalphaminus}
The temporal evolution of the gRSA model is more complex than the cases
discussed in Section \ref{classicalRSA}.
To see this, we use the moments $M_\gamma(t)$ defined by (\ref{defmoments}).
Multiplying equation (\ref{evolGgen4}) by $x^\gamma$
and integrating over $x$, we can derive
the equation for moments (for $\alpha \ne 0$),
$$
\frac{\partial M_\gamma}{\partial t} (x,t)
=
\frac{1}{\alpha (\alpha+2)\varepsilon^{\alpha+1}}
M_{\gamma + \alpha + 2}
-
\frac{\alpha+1}{2 \alpha \varepsilon}
M_{\gamma + 2}
\;
+
$$
$$
+
\;
\int_0^\infty
\frac{2^{\alpha+2}-4}{\alpha \varepsilon^{\alpha+1}}
\int_x^{\infty}
G(y,t)
x^{\gamma} (y-x)^{\alpha+1}
\mbox{d}y \mbox{d}x
=
$$
$$
=
\frac{1}{\alpha (\alpha+2)\varepsilon^{\alpha+1}}
M_{\gamma + \alpha + 2}
-
\frac{\alpha+1}{2 \alpha \varepsilon}
M_{\gamma + 2}
+
\int_0^{\infty}
\frac{2^{\alpha+2}-4}{\alpha \varepsilon^{\alpha+1}}
G(y,t)
\int_0^{y}
x^{\gamma} (y-x)^{\alpha+1}
\mbox{d}x \mbox{d}y
=
$$
$$
=
\frac{M_{\gamma + \alpha + 2}}{\alpha (\alpha+2)\varepsilon^{\alpha+1}}
+
\frac{2^{\alpha+2}-4}{\alpha \varepsilon^{\alpha+1}}
\int_0^\infty
G(y,t) y^{\gamma+\alpha+2} \mbox{d}y
\int_0^{1} \xi^{\gamma} (1-\xi)^{\alpha+1} \mbox{d}\xi
-
\frac{\alpha+1}{2 \alpha \varepsilon}
M_{\gamma + 2}
=
$$
$$
=
\frac{1}{\varepsilon^{\alpha+1}}
\left(
\frac{2^{\alpha+2}-4}{\alpha}
B(\gamma+1,\alpha+2)
+
\frac{1}{\alpha (\alpha+2)}
\right)
M_{\gamma + \alpha + 2}
-
\frac{\alpha+1}{2 \alpha \varepsilon}
M_{\gamma + 2}
=
$$
\begin{equation}
=
\frac{1}{\varepsilon^{\alpha+1}}
H(\gamma,\alpha)
M_{\gamma + \alpha + 2}
-
\frac{\alpha+1}{2 \alpha \varepsilon}
M_{\gamma + 2},
\label{evolmom1}
\end{equation}
where $B(\cdot,\cdot)$ is Beta function and $H(\gamma,\alpha)$ is
defined as
\begin{equation}
H(\gamma,\alpha)
=
\frac{2^{\alpha+2}-4}{\alpha}
B(\gamma+1,\alpha+2)
+
\frac{1}{\alpha (\alpha+2)}.
\label{defHgammaalpha}
\end{equation}
First, consider the case $\alpha<0$; the dominant term on the right-hand
side of (\ref{evolmom1}) is the term $\varepsilon^{-\alpha-1}
H(\gamma,\alpha) M_{\gamma + \alpha + 2}$.
The zeroth order moment,
$$
M_0(t) = \int_0^\infty G(x,t) \mbox{d}x,
$$
gives the total number of gaps at time $t$.
At leading order, we have
$$
\frac{\partial M_0}{\partial t} (x,t)
=
\varepsilon^{-\alpha-1}
H(0,\alpha) M_{\alpha + 2}.
$$
There is a constant $\overline{\alpha} \doteq -0.415$ such that
$H(0,\alpha)$ is positive for $\alpha<\overline{\alpha}$
and negative for $\alpha>\overline{\alpha}$.
We immediately see that
\begin{equation}
\int_0^\infty G(x,t) \mbox{d}x \quad \to \quad 0
\qquad \qquad
\mbox{for} \quad \alpha>\overline{\alpha}.
\label{decaygaps}
\end{equation}
To illustrate the result (\ref{decaygaps}), we will execute two
stochastic simulations with the gRSA algorithm.
We will use (\ref{distrlength}),
(\ref{defxi1}) and (\ref{defxi2})
where $\alpha=-0.1$ or $\alpha=-0.3$.
We choose $\varepsilon=10^{-3}$.
The results are given
in Figure \ref{figalphasmall}, where
the time evolution of the number of gaps and
the number of adsorbed polymers are shown.
Note that we use a
logarithmic scale on the time axis because the long-term dynamics
are very slow.
\begin{figure}
\centerline{{\Large $\alpha = - 0.1$}
\qquad \qquad \qquad \qquad \qquad \qquad {\Large $\alpha = - 0.3$}}
\centerline{
\psfig{file=./figs/numberpolymersalM01.eps,height=2.2in}
\quad
\psfig{file=./figs/numberpolymersalM03.eps,height=2.2in}
}
\smallskip
\centerline{
\psfig{file=./figs/numbergapsalM01.eps,height=2.2in}
\quad
\psfig{file=./figs/numbergapsalM03.eps,height=2.2in}
}
\smallskip
\centerline{
\psfig{file=./figs/numberinvgapsalpM01.eps,height=2.2in}
\quad
\psfig{file=./figs/numberinvgapsalpM03.eps,height=2.2in}
}
\caption{{\it gRSA model for $\alpha=-0.1$ (panels on
the left) and $\alpha=-0.3$ (panels on the right).
We plot the time evolution of the number of adsorbed
polymers (top panels) and the time evolution of the number
of gaps (middle panels). The time axis of the top and middle panels
is logarithmic. We also plot the time
evolution of the quantity $[1-A(t)]^{-1}$ (bottom panels)
where time is scaled according to $(\ref{scalingAtgenRSA})$.
}}
\label{figalphasmall}
\end{figure}
For $\alpha=-0.1$, the stochastic
simulation was stopped when $99.9999\%$ of the
surface was covered.
For $\alpha=-0.3$, the stochastic
simulation was stopped when $99.9994\%$ of the
surface was covered.
Next, we will study the behaviour of the system for
$\alpha < \overline{\alpha}$.
Here, we will assume the scaling ansatz (\ref{scalingansatz}).
Differentiating (\ref{defHgammaalpha})
with respect of $\gamma$, we obtain
\begin{equation}
\frac{\partial H}{\partial \gamma}
(\gamma,\alpha)
=
\frac{2^{\alpha+2}-4}{\alpha}
B(\gamma+1,\alpha+2)
\Big[ \psi_0 (\gamma+1) - \psi_0 (\gamma+\alpha+3) \Big]
\label{derHgammaalpha}
\end{equation}
where $\psi_0$ is the polygamma function. For each
$\alpha < \overline{\alpha}$, the equation
\begin{equation}
H(\overline{\gamma},\alpha) = 0
\label{equationHzero}
\end{equation}
defines implicitly
the function $\overline{\gamma}(\alpha)$.
If $\gamma$ is equal
to $\overline{\gamma}(\alpha)$, then the moment $M_\gamma$
is independent of time. Hence, using (\ref{scalmoments}),
we obtain the relation
$$
a = b + b \, \overline{\gamma}(\alpha)
$$
between the coefficients of the scaling ansatz (\ref{scalingansatz}) and
the parameter $\alpha$ of the model.
Finally, substituting the scaling ansatz (\ref{scalingansatz})
into (\ref{evolmom1}), one can find that
$b = (\alpha+2)^{-1}.$
Hence, the scaling of moments
(\ref{scalmoments}) can be rewritten in the form
\begin{equation}
M_\beta (t) \sim
t^{\overline{\mu}}
\qquad
\mbox{where}
\quad
\overline{\mu}
=
\frac{\overline{\gamma}(\alpha) - \beta}{\alpha+2}.
\label{scalmomentsrevisedgenRSA}
\end{equation}
Using (\ref{evolA}) and (\ref{scalmomentsrevisedgenRSA}), we obtain
\begin{equation}
1 - A(t)
=
\int_0^1 x G(x,t) \mbox{d}x
\sim
t^{-\overline{\omega}(\alpha)}
\qquad
\mbox{where}
\quad
\overline{\omega}(\alpha)
=
\frac{1-\overline{\gamma}(\alpha)}{\alpha+2}.
\label{scalingAtgenRSA}
\end{equation}
The graph of the function $\overline{\omega}(\alpha)$
is given in Figure \ref{figgraphomegasigmabar}(a).
We also plot $\omega(\alpha)$ given by (\ref{scalingAt})
for comparison.
\begin{figure}
\picturesAB{./figs/graphomegagenRSA.eps}{./figs/graphsigmagenRSA.eps}{2.2in}
\caption{(a) {\it
The graph of the exponent $\overline{\omega}(\alpha)$ given
by $(\ref{scalingAtgenRSA})$. The dashed line shows the
exponent $\omega(\alpha)$ given
by $(\ref{scalingAt})$.}
(b) {\it
The graph of the exponent $\overline{\sigma}(\alpha)$ given
by $(\ref{scalingNtgenRSA})$. The dashed line shows the
exponent $\sigma(\alpha)$ given
by $(\ref{scalingNt})$.}
}
\label{figgraphomegasigmabar}
\end{figure}
Using (\ref{equationforN2}) and (\ref{scalmomentsrevisedgenRSA}),
we also obtain
\begin{equation}
N(t)
\sim
t^{\overline{\sigma}(\alpha)}
\qquad
\mbox{where}
\;
\overline{\sigma}(\alpha)
=
\frac{\overline{\gamma}(\alpha)}{\alpha+2}.
\label{scalingNtgenRSA}
\end{equation}
The graph of the function $\overline{\sigma}(\alpha)$ is given in Figure
\ref{figgraphomegasigmabar}(b); we also plot
$\sigma(\alpha)$ given by (\ref{scalingNt})
for comparison.
To illustrate the results (\ref{scalingAtgenRSA}) and
(\ref{scalingNtgenRSA}), we will execute two gRSA
stochastic simulations.
We will use
(\ref{distrlength}), (\ref{defxi1}) and (\ref{defxi2})
where $\alpha=-0.5$ or $\alpha=-2/3$. We choose
$\varepsilon=10^{-3}$.
The results are given
in Figure \ref{figalphalarge1} where
the time evolution of the number of gaps and
the number of adsorbed polymers are shown.
The time is scaled according to (\ref{scalingAtgenRSA}) and
(\ref{scalingNtgenRSA}); we solve
(\ref{equationHzero}) to obtain the desired exponents
\begin{equation}
\overline{\sigma}(-0.5) =0.0872,
\quad
\overline{\omega}(-0.5) =0.5795,
\quad
\overline{\sigma}\left(- \frac{2}{3}\right) =0.2875,
\quad
\overline{\omega}\left(- \frac{2}{3}\right) =0.4625,
\label{scalingexample}
\end{equation}
and then we plot the results of stochastic simulations using the
corresponding scaling (\ref{scalingexample}).
\begin{figure}
\centerline{{\large $\alpha = - \displaystyle\frac{1}{2}$}
\qquad \qquad \qquad \qquad \qquad \qquad \qquad
{\large $\alpha = - \displaystyle\frac{2}{3}$}}
\centerline{
\psfig{file=./figs/polymersgenM05.eps,height=2.2in}
\quad
\psfig{file=./figs/polymersgenM067.eps,height=2.2in}
}
\centerline{
\psfig{file=./figs/gapsgenM05.eps,height=2.2in}
\quad
\psfig{file=./figs/gapsgenM067.eps,height=2.2in}
}
\centerline{
\psfig{file=./figs/CDFgenM05.eps,height=2.2in}
\quad
\psfig{file=./figs/CDFgenM067.eps,height=2.2in}
}
\caption{{\it gRSA model for $\alpha=-0.5$ (panels on
the left) and $\alpha=-2/3$ (panels on the right).
We plot the time evolution of the number of adsorbed
polymers (top panels) and the time evolution of the
inverse gap size
$[1-A(t)]^{-1}$ (middle panels). Time is scaled according
to $(\ref{scalingexample})$ (top and middle panels).
We also plot the cumulative distribution function
$C(x,t)$ at different times for both simulations
(bottom panels).
}}
\label{figalphalarge1}
\end{figure}
\begin{figure}
\centerline{{\large $\alpha = - \displaystyle\frac{1}{2}$}
\qquad \qquad \qquad \qquad \qquad \qquad \qquad
{\large $\alpha = - \displaystyle \frac{2}{3}$}}
\centerline{
\psfig{file=./figs/scaledCDFgenM05.eps,height=2.2in}
\quad
\psfig{file=./figs/scaledCDFgenM067.eps,height=2.2in}
}
\caption{{\it Scaled cumulative
distribution function of gaps for gRSA model for $\alpha=-0.5$
(panel on the left) and $\alpha=-2/3$ (panel on the right).
}}
\label{figalphalarge2}
\end{figure}
In Figure \ref{figalphalarge1}, we also
plot the cumulative distribution function $C(x,t)$
for different times (i.e. for different numbers of polymers
attached).
Using the suitable rescaling $\widehat{C}(x,t) = C(kx,t)$,
the curves collapse to a single curve as shown in
Figure \ref{figalphalarge2}.
Finally, one can easily show that formula (\ref{scalingAtgenRSA})
works also for the case $\overline{\alpha} \le \alpha < 0.$
To illustrate it, we plot the time evolution of the quantity
$[1-A(t)]^{-1}$ for $\alpha=-0.1$ and $\alpha=-0.3$
in Figure \ref{figalphasmall} (bottom panel) using the scaling
(\ref{scalingAtgenRSA}). On the other hand, formula
(\ref{scalingNtgenRSA}) is no longer correct because
integrating of (\ref{equationforN2}) implies that
$N(t)$ is approaching a constant value. More precisely,
$N(t) \sim C + t^{\overline{\sigma}(\alpha)}$ for
$\overline{\sigma}(\alpha)<0.$ If $\alpha$ is positive than
(\ref{decaygaps}) is valid, i.e. we have (\ref{decaygaps})
for any $\alpha>\overline{\alpha}$.
\subsection{Time dependent concentration of reactive polymers}
\label{sectimedependent}
As discussed before, the reactive groups on our polymers
are capable of reacting with the solvent before reaching the
surface \cite{Subr:2005:CDC}.
It may therefore be more realistic
to consider that only a fraction of polymers
$r(t)$ is still reactive at time $t$.
Here, $r(t) \in [0,1]$,
$r(0)=1$ and $r(t)$ is a decreasing function of time.
The random sequential adsorption algorithm has to be modified as
follows: at each time step, we generate the random number
uniformly distributed in the interval $(0,1)$.
If this number is greater than $r(t)$, then the selected
polymer has lost its binding site through reaction
with the solvent (it cannot
be adsorbed), and we continue with the next step.
Otherwise,
we choose randomly a position on the interval and we attempt
to place the polymer there.
Depending on the form of function $r(t)$, different dynamics
can be observed.
First, let us suppose that
\begin{equation}
r(t) = \frac{1}{t^{\lambda}}
\qquad
\mbox{for} \; \lambda \in [0,1).
\label{formr1}
\end{equation}
In this case, we can find a relation between the modified random
sequential algorithm and the previous results.
At each time $t$, we can compute the average waiting
time $\Delta t$ before a reactive polymer
hits the surface as the solution of the
equation
\begin{equation}
\int_{t}^{t+\Delta t} \frac{1}{\tau^{\lambda}} \; \mbox{d}\tau = 1.
\label{equationforDeltat}
\end{equation}
Solving (\ref{equationforDeltat}), we find
$$
(t+\Delta t)^{1-\lambda} = t^{1-\lambda} + 1 - \lambda.
$$
Hence,
$$
\Delta t = \Big[t^{1-\lambda} + 1 - \lambda \Big]^{1/(1-\lambda)} - \, t
\quad
\sim
\quad
t^{\lambda}.
$$
Consequently, we can make use of the formulas (\ref{scalingAt})
and (\ref{scalingNt}), or formulas (\ref{scalingAtgenRSA})
and (\ref{scalingNtgenRSA}), in the case (\ref{formr1}).
For example, using
(\ref{scalingAt}) and (\ref{scalingNt}), we obtain
that the quantities $A(t)$ and $N(t)$
satisfy the following asymptotic behaviour
\begin{equation}
A(t)
\sim
t^{(1-\lambda)\omega(\alpha)}
\qquad
\mbox{and}
\qquad
N(t)
\sim
t^{(1-\lambda)\sigma(\alpha)}.
\label{scalingANtlambda}
\end{equation}
To illustrate the formula (\ref{scalingANtlambda}),
we stochastically simulate the
gRSA model with the probability distribution $p(z)$ given
by (\ref{distrlength}) and the probability $\xi(z,w-x_1,x_2-w)$ given
by (\ref{defxi3}), where the fraction of the reactive polymers
in the system decreases with time according
to (\ref{formr1}).
We select $\varepsilon=10^{-3}$ and verify the asymptotic behaviour
(\ref{scalingANtlambda}) for
$\alpha=-0.5$ and $\lambda=0.5$.
Then (\ref{scalingANtlambda}) implies
$$
1 - A(t) \sim t^{-0.1014}
\qquad
\mbox{and}
\qquad
N(t) \sim t^{0.2319}.
$$
The time evolution
of $A(t)$ and $N(t)$ is given in Figure
\ref{figpolattgap2} (top panels).
We also present results for
$\alpha = 0$ and $\lambda=0.33$
in Figure \ref{figpolattgap2} (bottom panels).
Again, we scale the time according to
(\ref{scalingANtlambda}).
\begin{figure}[t]
\centerline{
\psfig{file=./figs/polattexM05lam05.eps,height=2.1in}
\qquad
\psfig{file=./figs/gapexM05lam05.eps,height=2.1in}
}
\centerline{
\psfig{file=./figs/polattex0lam033.eps,height=2.1in}
\qquad
\psfig{file=./figs/gapex0lam033.eps,height=2.1in}
}
\caption{{\it Modified RSA model from Section $\ref{sectimedependent}.$
The time evolution of the number of polymer molecules attached
to the surface $N(t)$} (left). {\it Time evolution of the inverse of the
total gap size $[1-A(t)]^{-1}$} (right). {\it Time is scaled according
to $(\ref{scalingANtlambda})$.
}}
\label{figpolattgap2}
\end{figure}
The reactive group is lost by chemical reaction with the solvent.
It might be more natural to consider (instead of
(\ref{formr1})) that the fraction of
reactive polymers is exponentially decreasing, i.e.
\begin{equation}
r(t) = e^{- \lambda t}
\qquad
\mbox{for} \; \lambda > 0.
\label{formr2}
\end{equation}
The formula (\ref{formr2}) gives rise to qualitatively different
dynamics for the system, as opposed to the dynamics associated with
(\ref{formr1}).
For simplicity, let us assume that every
polymer with a functional reactive group can be adsorbed
(which will give a bound on $N(t)$ from above).
Then the average number of adsorbed polymers $N(t)$ is
$$
N(t)=\frac{1}{\lambda} \Big(1 - e^{- \lambda t} \Big),
$$
which implies that the number of adsorbed polymers does not approach
infinity as in the previous case.
\bigskip
\section{Equation-free analysis of gRSA}
\label{seceqfree}
In the previous theory, we assumed the scaling ansatz (\ref{scalingansatz})
(see also \cite{Krapivsky:1992:KRS})
for $G(x,t)$
and we computed the time dependence of the quantities of interest
$A(t)$ and $N(t)$.
A related interesting question is whether we can also compute the profile
$\Phi$ from (\ref{scalingansatz}). One possibility is to
substitute (\ref{scalingansatz}) in equation (\ref{evolGgen4})
and solve it numerically for $\Phi$ but we will not proceed
this way.
Instead, we demonstrate the computation
of self-similar profile $\Phi$ using only short-time
appropriately initialized simulations of the stochastic gRSA model.
In this equation-free context \cite{Kevrekidis:2003:EFM,Chen:2005:EFD},
it is easier to work with the
cumulative distribution function $C(x,t)$, which can be obtained
from $G(x,t)$ through (\ref{forcdf}); $C(x,t)$ is less noisy than $G(x,t)$
(e.g. \cite{Gear:2001:PIM}).
Using (\ref{forcdf}) and
(\ref{scalingansatz}), we obtain
$$
C(x,t)
=
\frac{1}{\int_0^\infty G(y,t) \mbox{d}y} \int_0^x G(y,t) \mbox{d}y
=
$$
$$
=
\frac{1}{\int_0^\infty \Phi \left( y \, t^b \, \right) \mbox{d}y}
\int_0^x \Phi \left( y \, t^b \, \right) \mbox{d}y
=
\frac{1}{\int_0^\infty \Phi \left( \xi \right) \mbox{d}\xi}
\int_0^{x t^{b}} \Phi \left( \xi \right) \mbox{d}\xi.
$$
Hence, we see that the cumulative density function
$C(x,t)$ scales as
\begin{equation}
C(x,t) \equiv \overline{C}(x t^b).
\label{forcdfscaling}
\end{equation}
To compute the profile $\overline{C}$, we can use
an equation-free iterative fixed point algorithm which is shown
schematically in Figure \ref{figeqfreescheme}.
\begin{figure}
\centerline{\psfig{file=./figs/eqfreescheme.eps,height=1.7in}}
\caption{{\it
Schematic of the equation-free mapping $\Psi$.}
}
\label{figeqfreescheme}
\end{figure}
Starting with the initial guess $\overline{C}_0$,
we compute the sequence of profiles
$\overline{C}_K$, $K=1, 2, 3, \dots$, where
\begin{equation}
\overline{C}_{K+1} = \Psi (\overline{C}_{K}),
\qquad
\mbox{for} \; K=0, 1, 2, 3, \dots,
\label{formulapsi}
\end{equation}
and where the mapping $\Psi$ is obtained as the composition
of the following four steps:
\bigskip
\leftskip 1cm
\noindent
(a) Given the cumulative density profile $\overline{C}_K$,
create one or more microscopic realizations of gaps in the unit interval
such that the initial cumulative density function
is $C(\cdot,0)=\overline{C}_K$.
\smallskip
\noindent
(b) Use the microscopic simulator (i.e. use the gRSA algorithm)
for a short time $\Delta t.$
\smallskip
\noindent
(c) Compute the new cumulative distribution function
$C(\cdot,\Delta t)$ at time $\Delta t.$
\smallskip
\noindent
(d) Rescale $C(\cdot,\Delta t)$ to compute $\overline{C}_{K+1}$.
\leftskip 0cm
\bigskip
\noindent
One possible way to rescale $C(\cdot,\Delta t)$ is to compute the
average gap size $a_0$ from $C(\cdot,0)$ and the
average gap size $a_{\Delta t}$ from $C(\cdot,\Delta t)$.
Then the $\overline{C}_{K+1}$ can be computed by
\begin{equation}
\overline{C}_{K+1}(x)
=
C \left( \frac{a_{\Delta t}}{a_0} x ,\Delta t \right).
\label{rescaling}
\end{equation}
We now present illustrative results obtained by
this fixed point computation (\ref{formulapsi}) using the gRSA algorithm.
We will use
(\ref{distrlength}), (\ref{defxi1}) and (\ref{defxi2})
where $\alpha=-0.5$ or $\alpha=-2/3$.
We choose $\epsilon = 10^{-3}$.
The results of long
term simulations for these parameter values were already
shown in Figure \ref{figalphalarge1}.
Our goal is to use
the iterative formula (\ref{formulapsi}) to compute
the scaled cumulative distribution function profile which
was shown in Figure \ref{figalphalarge2}.
This algorithm allows us to find the self-similar shape
by performing simulations while simulating at a scale
(at relatively larger average gap sizes) where the evolution
is relatively fast, compared to the long-term dynamics
close to jamming.
The initial guess is given as
$$
\overline{C}_0(x)
=
\left\{
\begin{array}{ll}
0 \quad & \mbox{for} \; x \le 1.5 \times 10^{-4}; \\
1 \quad & \mbox{for} \; x > 1.5 \times 10^{-4}; \\
\end{array}
\right.
$$
which means that initially all our gaps have the same size
$1.5 \times 10^{-4}$.
At each iteration step (see
Figure \ref{figeqfreescheme}), we place 1000 gaps
according to the cumulative distribution function
$\overline{C}_K$ to the interval $[0,1].$
We evolve the
simulation until 100 new polymers are placed.
We then {\it rescale} the new cumulative distribution according
to (\ref{rescaling}) and we compute $\overline{C}_{K+1}$.
Several first iterations are shown in Figure \ref{figeqfreeresults}
(top panels).
We see that after 20 iterations, we have effectively
reached the steady state (the stationary shape of the self-similarly
evolving gap distribution).
More precisely, the error between
iterations is small and it is not further systematically
decreasing.
The comparison of the equation-free 20th iteration
with the results obtained by the long-time simulations
are also shown in Figure \ref{figeqfreeresults} (bottom panels).
\begin{figure}
\centerline{{\large $\alpha = - \displaystyle\frac{1}{2}$}
\qquad \qquad \qquad \qquad \qquad \qquad \qquad
{\large $\alpha = - \displaystyle \frac{2}{3}$}}
\centerline{
\psfig{file=./figs/eqfreeitergenM05.eps,height=2.2in}
\quad
\psfig{file=./figs/eqfreeitergenM067.eps,height=2.2in}
}
\smallskip
\centerline{
\psfig{file=./figs/eqfreegenM05.eps,height=2.2in}
\quad
\psfig{file=./figs/eqfreegenM067.eps,height=2.2in}
}
\caption{{\it Equation free gRSA computational results
for $\alpha=-0.5$ (left panels) and $\alpha=-2/3$
(right panels).
Iterations of the equation-free dynamic renormalization algorithm
$(\ref{formulapsi})$ (top panels).
Comparison with the steady steate profile obtained through long time
simulations (bottom panels).}
}
\label{figeqfreeresults}
\end{figure}
Finally, we note that many other algorithmic possibilities
for the computation of the profile $\overline{C}$ exist.
The equation
(\ref{formulapsi}) seeks a fixed point
of the mapping $\Psi$.
Instead of successive substitution, other fixed point algorithms
implemented in a matrix-free fashion through short simulation bursts
can be used to find stationary solutions - for example,
Newton-GMRES iterations \cite{Chen:2005:EFD}; these would be
able to converge on even dynamically unstable self-similarly
evolving distributions.
\section{Discussion}
\label{secdiscussion}
In this paper, motivated by a pharmacological example involving
polymer coating of a virus surface,
we studied certain aspects of polydisperse adsorption
of macromolecules in one spatial dimension.
We presented an extension of the
classical random sequential adsorption algorithm
to capture better certain essential properties of the
pharmacological model system currently used
in drug development research.
We introduced partial
overlapping of adsorbing macromolecules, i.e.
we considered that the polymers are not
rigid objects but they can be deformed while
attaching to the surface.
We found two distinct asymptotic regimes.
Depending on the
parameters of the processes involved, we can observe that either
(a) the number of gaps between polymers asymptotically approaches zero, or
that
(b) the number of gaps asymptotes to infinity and the gap
distribution acquires an asymptotically self-similar profile.
We also briefly discussed the impact of a possible reaction
of the polymers with the
solvent on gRSA dynamics.
Again, two possibilities exist.
If the decay of the reactive groups is relatively weak, then
the dynamics of the system remains qualitatively unchanged and the
system only evolves on a slower time scale.
On the other hand,
if the reactive groups decay exponentially, this decay
ultimately wins over the polynomial time asymptotics
of gRSA.
From the applications point of view, it therefore becomes crucial
to know the corresponding rate constants in order to reliably predict
what type
of behaviour one might expect over the time scales of interest.
Typically, the coating process is performed overnight in the laboratory
and different reactive groups have different half lives; measuring
these rates becomes an important task.
In this paper we worked in one spatial dimension and provided analytical
results about the long time behaviour of gRSA models.
The analytical approach was based on
two important facts: we knew what the good macroscopic observables
for describing the system behaviour were and we were able to write down
analytically tractable equations for these observables.
The good observable for our system was a distribution
of gaps $G(x,t)$ between adsorbed polymers.
If we know the
initial distribution of gaps $G(0,t)$ one could easily
predict $G(x,t)$ at future times.
On the other hand, if we know (hope) that the gap distribution
$G(x,t)$ is a good observable for the system of interest
but we do not know the evolution equation for $G(x,t)$, then it is
still possible to use the equation-free
methods \cite{Kevrekidis:2003:EFM}.
The main idea of the equation-free methods is to use the short bursts
of appropriately initialized microscopic/stochastic
computations to {\it estimate} macroscopic
quantities of interest on demand.
Hence, if one does not have an explicit coarse-grained evolution
equation for the system statistics, one can in principle
avoid long, brute-force simulations.
This might be the case for one-dimensional adsorption
problems with more complicated microscopic evolution rules.
The situation becomes significantly more difficult in the higher dimensional
case.
Here, the analytical theory is far behind in development,
and the literature contains mostly computational results.
The first question for higher dimensional adsorption is the
nature of the ``good" coarse-grained observables for the system.
Good
observables (the variables
in terms of which the unavailable effective model would be written)
are necessary for developing a useful analytical
theory.
Knowing appropriate coarse-grained observables
is also an important feature of equation-free algorithms.
Having one-dimensional analogues in mind, we see that one needs
an effective way to describe the statistics of ``gaps" (free space) in higher
dimensions.
If we cannot estimate (by intuition or by
suitable algorithms for the detection of
low-dimensionality in high-dimensional data) effectively
good observables for the system, then the direct, brute-force computationally
intensive simulations might be the only modelling option.
In this paper
we showed cases where we could do more than brute-force simulation
and provided
analytical results giving insights into the dynamics of gRSA.
The problems studied in this paper were motivated
by the pharmacological example mentioned above,
and realistic predictive modelling of the problem
clearly requires extensive model parameter information that must
come from experimental data.
As we showed, we can
expect different dynamics of the problem depending
on the values of the parameters of the polymer and the virus
which are used.
Obtaining reliably such parameters and bounds on their uncertainty
for our particular model problem is non-trivial,
and we are not yet ready to report about it.
\bibliographystyle{amsplain}
|
1,314,259,996,705 | arxiv | \section{Introduction}
The detailed suppression pattern, $R_{AA}(p_T)$, and elliptic flow,
$v_2(p_T)$, of high-transverse-momentum hadrons is an important
experimental signature of the quark-gluon plasma creation in heavy ion
collisions~\cite{Gyulassy:2003mc}. Jet quenching for light mesons, such
as $\pi$, $K$ and $\eta$, at RHIC is well explained by radiative energy
loss calculations~\cite{Vitev:2005he}. It also gives the dominant
contribution to the azimuthal asymmetry of hard
probes~\cite{Gyulassy:2003mc}.
In contrast, models~\cite{Wicks:2005gt} with a physically reasonable set
of QGP temperatures and densities, predict a QCD heavy-quark energy loss
which is too small compared to the measured suppression of single
non-photonic electrons~\cite{Adare:2006nq,Abelev:2006db}. Therefore, it
is critical to investigate new interaction mechanisms in the QGP that
may be specific to heavy
flavor~\cite{vanHees:2004gq,vanHees:2005wb,prep,Adil:2006ra}.
\section{Heavy-flavor suppression in a combined transport +
quenching approach }
Thermalization of heavy quarks in the QGP-heat bath has been recently
studied in the framework of the parton-transport
approach~\cite{vanHees:2004gq,vanHees:2005wb,Molnar:2006ci}. Large
nuclear suppression and elliptic flow $v_2$ result when employing a
Fokker-Plank equation,
\begin{equation}
\frac{\partial f(\vec{p},t)}{\partial t} =
\frac{\partial}{\partial p_i} \left[ p_i A(\vec{p},t) +
\frac{\partial}{\partial p_j} B_{ij}(\vec{p},t) \right] \; ,
\label{fp}
\end{equation}
solved via an equivalent Langevin simulation. In Eq.~(\ref{fp})
$f(\vec{p},t)$ is the distribution of $c$- and $b$-quarks and
$A(\vec{p},t), B(\vec{p},t)$ are the drag / diffusion coefficients,
respectively. It has been argued that strong coupling between the $c$-
and $b$-quarks and the QGP medium may be generated via quark-resonance
interactions near the QCD-phase transition, $T \sim
T_c$~\cite{vanHees:2004gq}. It is, therefore, important to study the
interplay between such non-perturbative effects and the
radiative~\cite{Gyulassy:2000er} and collisional~\cite{Wicks:2005gt}
heavy-quark energy loss. We evaluate~\cite{prep} the contribution of
these processes to the drag and diffusion coefficients,
\begin{equation}
A(\vec{p},t) = \frac{1}{p_i}
\frac{ \langle \delta p_i \rangle}{\delta t} \;,
\qquad
B_{ij}(\vec{p},t) = \frac{1}{2}
\frac{ \langle \delta p_i \delta p_j \rangle} { \delta t} \; ,
\label{coefs}
\end{equation}
which are then applied in the relativistic Fokker-Planck equation.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=6.2cm]{quark-minbias-RAA-with-rad.eps}
\hspace*{.2cm} \includegraphics[width=6.2cm]{quark-minbias-v2-with-rad.eps}
\end{center}
\vspace*{-0.3cm}
\caption{ Left panel: preliminary results on the nuclear modification
$R_{AA}(p_T)$ for heavy $c$- and $b$-quarks~\cite{prep} from
collisional~\cite{Wicks:2005gt} and
radiative energy loss~\cite{Gyulassy:2000er} and quark-resonance
interactions~\cite{vanHees:2004gq,vanHees:2005wb}.
For charm quarks, the PQCD $\Delta E$ contribution is shown separately.
Right panel: elliptic flow $v_2(p_T)$ for heavy c- and b-quarks
for the same physics mechanisms~\cite{prep}. }
\label{figure1}
\vspace*{-2mm}
\end{figure}
Results on the $p_T$-dependent suppression pattern
of heavy quarks, $R_{AA}(p_T)$, are shown in the left panel of
Fig.~\ref{figure1}. Drag coefficients are easily evaluated from
the fractional momentum loss of heavy quarks, see Eq.~(\ref{coefs}).
The diffusion coefficients in this preliminary study were constrained
from the fluctuation-dissipation relation. We observe that the
suppression
of charm quarks can be very large even in minimum-bias
reactions of large nuclei and $R_{AA}({\rm charm}) \ll
R_{AA}({\rm bottom})$. The high-$p_T$ azimuthal asymmetry
for minimum-bias Au+Au collisions is shown in the right
panel of Fig.~\ref{figure1}. We note that the generated
$v_2$ for $b$-quarks is much smaller than that for $c$-quarks.
One of the reasons for the large suppression in our current energy-loss
implementation is that the Einstein fluctuation-dissipation relation
induces minimal Gaussian fluctuations. These are significantly
different from the ones in the probabilistic treatment of PQCD-energy
loss~\cite{Gyulassy:2003mc,Vitev:2005he,Wicks:2005gt}. Future Langevin
simulations of $c$- and $b$-quark diffusion should include momentum
fluctuations beyond the Einstein's relation and the decay of the heavy
quark / hadron spectra into $(e^+ + e^-)$ for direct comparison to the
non-photonic electron observables at RHIC~\cite{prep}.
\section{QGP-induced dissociation of heavy mesons}
In the perturbative QCD-factorization approach, the cause of
the limited single non-photonic electron quenching is
identified as the small suppression of $B$-mesons,
which dominate the high-$p_T$ $e^+ e^-$ yields.
Such models assume that the hard jet hadronizes in vacuum,
having fully traversed the region of hot and dense nuclear matter,
$L_T^{\rm QGP} \leq 6$~fm, and lost energy via radiative and
collisional
processes~\cite{Gyulassy:2003mc,Vitev:2005he,Wicks:2005gt}.
In Ref.~\cite{Adil:2006ra} we examined the validity of this
assumption for different species of final-state partons and
decay hadrons. For a $p_T = 10$~GeV pion at mid-rapidity
$\tau_{\rm form} \approx 25$~fm $\gg L_T^{\rm QGP}$, consistent with
the jet-quenching assumptions~\cite{Gyulassy:2003mc,Vitev:2005he}.
In contrast, $B$- and $D$-mesons of the same $p_T$ have
formation times $\tau_{\rm form} \approx 0.4, 1.6$~fm,
respectively, $\ll L_T^{\rm QGP}$.
Therefore, at the finite $p_T$ range
accessible at RHIC and LHC a conceptually different approach
to the description of $D$- and $B$-meson quenching in A+A
collisions is required, when compared to light hadrons.
\begin{figure}[!t]
\vspace*{+0.cm}
\begin{center}
\includegraphics[width=6.3cm]{AuCuLHC.eps}
\hspace*{.2cm}\includegraphics[width=6.3cm]{CentSupp-New.eps}
\end{center}
\vspace*{-0.3cm}
\caption{Left panel: suppressions of $D$- and $B$-meson production via
collisional dissociation in the QGP in central Au+Au and Cu+Cu
reactions at RHIC~\cite{Adil:2006ra}
for gluon rapidity densities $dN^g/dy = 1175$ and
$350$, respectively~\cite{Vitev:2005he}. Right panel:
quenching of the non-photonic electrons from the softened
$D$- and $B$-meson spectra in central Au+Au
collisions~\cite{Adil:2006ra}. Data is from
PHENIX~\cite{Adare:2006nq,ralf} and
STAR~\cite{Abelev:2006db,alex,zhong}. }
\label{figure2}
\vspace{-2mm}
\end{figure}
Motivated by this finding, in the framework of the GLV theory,
we derive the collisional dissociation probability of heavy
mesons in the QGP~\cite{Adil:2006ra}:
\begin{eqnarray}
&& P_d (\chi\mu^2 \xi) =
\left[ 1- P_s (\chi\mu^2 \xi) \right] \geq 0 \;,
\quad P_d (\chi\mu^2 \xi = 0) = 0 \;.
\label{sprob}
\end{eqnarray}
In Eq.~(\ref{sprob}) $2 \chi \mu^2 \xi = 2 (\mu^2 L / \lambda) \xi$
is the cumulative 2D transverse momentum squared per parton. The
dissociation probability also depends on the detailed heavy meson
light cone wave function. The dynamics of open heavy flavor
production and modification in this model
is represented by a set of coupled rate equations that
describe the competition between $b$- and $c$-quark
fragmentation and $D$- and $B$-meson dissociation~\cite{Adil:2006ra}:
\begin{eqnarray}
\label{rateq1}
&& \hspace*{-2.5cm} \partial_t f^{Q}({p}_{T},t) =
- \frac{f^{Q}({p}_{T},t)}{\langle \tau_{\rm form}(p_T, t) \rangle}
+ \, \frac{1}{\langle
\tau_{\rm diss}(p_T/\bar{x}, t) \rangle}
\int_0^1 dx \, \frac{1}{x^2} \phi_{Q/H}(x)
f^{H}({p}_{T}/x,t) \;, \qquad \\
&& \hspace*{-2.5cm} \partial_t f^{H}({p}_{T},t) =
- \frac{f^{H}({p}_{T},t) }{\langle \tau_{\rm diss}(p_T, t) \rangle}
+\, \frac{1}{\langle
\tau_{\rm form}(p_T/\bar{z}, t) \rangle}
\int_0^1 dz \, \frac{1}{z^2} D_{H/Q}(z)
f^{Q}({p}_{T}/z,t) \;. \qquad
\label{rateq2}
\end{eqnarray}
In Eqs. (\ref{rateq1}) and (\ref{rateq2}) $f^i(p_T,t) =
d\sigma^i/dy d^2 p_T$. For further details, see~\cite{Adil:2006ra}.
We solve this system of coupled rate equations numerically,
using the same initial soft-gluon rapidity density $dN^g/dy$ as
in the calculation of the $\pi^0$ quenching~\cite{Vitev:2005he}
in central Au+Au and Cu+Cu collisions at RHIC. Our results,
including a study of a range of anticipated QGP densities at
the LHC, are shown in the left panel of Fig.~\ref{figure2}.
Contrary to calculations that emphasize radiative and collisional
heavy quark energy loss~\cite{Wicks:2005gt,vanHees:2005wb,prep},
QGP-induced dissociation predicts $B$-meson suppression comparable
to or larger than that of $D$-mesons at transverse momenta as low as
$p_T \sim 10$~GeV~\cite{Adil:2006ra}. The heavy meson spectra
are decayed into electrons $(e^++e^-)$ using the PYTHIA event
generator. Our results are shown in the right panel of
Fig.~\ref{figure2}. The predicted $R_{AA}(p_T)$, which does not
neglect the large $B$-meson contribution, describes well the most
recent heavy flavor quenching measurements at
RHIC~\cite{alex,zhong,ralf}.
We emphasize that such agreement between theory and experiment
is not achieved at the cost of neglecting the contribution of
the $B$-mesons to the non-photonic $e^+ e^-$ spectra.
\section{Conclusions}
In these proceedings, we compared two new theoretical
approaches~\cite{prep,Adil:2006ra} to open heavy flavor
modification in the QGP. Preliminary results on Langevin
simulations of heavy quark diffusion, which include radiative
energy loss, collisional energy loss and quark-resonance
interactions, were shown. While an improved implementation of
momentum fluctuations is required for quantitative
comparison between data~\cite{Adare:2006nq,Abelev:2006db}
and theory~\cite{prep},
we find normal suppression, $R_{AA}(c)\ll R_{AA}(b)$,
and elliptic flow, $v_{2}(c)\gg v_{2}(b)$, hierarchies as
function of the heavy quark mass.
Results on QGP-induced collisional dissociation of
heavy mesons~\cite{Adil:2006ra} were also shown.
A good description~\cite{alex,zhong,ralf} of the large
quenching of the inclusive non-photonic
electrons~\cite{Adare:2006nq,Abelev:2006db}
is achieved by this model. A natural consequence of the approach
developed in Ref.~~\cite{Adil:2006ra} is
that $B$-mesons are attenuated as much as $D$-mesons at
transverse momenta as low as $p_T \sim 10$~GeV.
We conclude that robust experimental determination of the dominant
mechanism for in-medium modification of open heavy flavor would require
direct and separate measurements of the $B$- and $D$-meson
$R_{AA}$ and $v_2$ distributions versus $p_T$ and centrality in
collisions of heavy nuclei.
\vspace*{0.2cm}\noindent\textbf{Acknowledgment:} This work is
supported by the U.S. Department of Energy under contract no.
DE-AC52-06NA25396 (IV) and grant no. DE-FG02-93ER40764 (AA) and
by the U.S. National Science Foundation under grant no. PHY-0449489
(HvH).
\vspace*{-.2cm}
\section*{References}
|
1,314,259,996,706 | arxiv | \section{Introduction}
We consider the problem of 3D hand pose estimation from single depth images. Hand pose estimation has important applications in human-computer interaction (HCI) and augmented reality (AR). Estimating the freely moving hand has several challenges including large viewpoint variance, finger similarity and self occlusion and versatile and rapid finger articulation.
Methods for hand pose estimation from depth generally fall into two camps. The first is frame-to-frame model based tracking~\cite{forth-pso,forth-quasi,msra_modelbased,Liang-tip}. Model-based tracking approaches can be highly accurate if given enough computational resources for the optimization. The second camp, where our work also falls, is single frame discriminative pose estimation~\cite{cascaded-hand,Tang-LRF,Li2015-sip,xuHand_13}. These methods are less accurate than model-based trackers but much faster and are targeted towards real-time performance without GPUs. Model-based tracking and discriminative pose estimation are complementary to each other and there have been notable hybrid methods ~\cite{Tang2015-blackbox,msr-chi,detection_guided,collatorative,Ballan-hand} which try to maintain the advantages of both camps.
Earlier methods for discriminative hand pose estimation tried to estimate all joints directly~\cite{Keskin2012-ns,forth-oneshot} though such approaches tend to fail with dramatic view-point changes and extreme articulations. Following the lead of several notable methods~\cite{cascaded-hand,Tang-LRF,Li2015-sip,Tang2015-blackbox},
we cast pose estimation as a hierarchical regression problem.
The idea is to start with easier parent parts such as the wrist or palm, and then tackle subsequent and more difficult children parts such as the fingers. The assumption is that the children parts, once conditioned on the parents, will exhibit less variance and simplify the learning task.
Furthermore, by constraining the underlying graphical model to follow the tree-structured topology of the hand, hierarchical regression implicitly captures the skeleton constraints and therefore shares some advantages of model-based tracking that are otherwise not present when directly estimating all joints independently.
Our framework starts with estimating the surface normals of given point clouds. The normal direction establishes the local reference frames used in later conditional regression and serves as features
We then apply our
\emph{Frame Conditioned Regression Forest} (FCRF) to hierarchically regress hand joints down from the wrist to the finger tips. At each stage, the frame of reference is established based on previously estimated local surface normal or joint positions. The regression forest considers offsets between input points and joints of interest with respect to the local reference frame and also conditions the feature with respect to these local frames. Our use of conditioned features is inspired by~\cite{cascaded-hand}, though we consider angular differences between local surface normals, which is far more robust to rigid transformations than the original depth difference feature.
\begin{figure}[t]
\centering
\includegraphics[width=.9\textwidth]{framework_v2.pdf}
\caption{Framework. (a) shows the hand skeleton model used in our work. (b) sketches our hierarchical regression framework, with each successive stage denoted by a shaded box
We first estimate a reference frame for every input point encoding all information from previous stages and use that reference frame as input to estimate the location of children joints. The sub-figure around the depth map amplifies a local region from the initial depth map and shows the corresponding frame for a specific point. To save space, only thumb and index finger cases are shown and finger tip points(TIP) estimation is omitted as it is identical to that of DIP \textbf{(best viewed in colour)} }
\label{fig:framework}
\vspace{-0.5cm}
\end{figure}
Our proposed method has the following contributions:
\begin{enumerate}
\item We are the first to incorporate local surface normals for pose estimation. To this end, we propose an extremely efficient normal estimation method based on regression trees adapted to handle unit vector distributions, different from vector space properties.
\item We extend the commonly used depth difference feature\cite{cascaded-hand,Tang-LRF,Li2015-sip,Tang2015-blackbox,Shotton2013_pami,Vitruvian} to an angular difference feature between two normal directions. Our normal difference feature is highly robust to 3D rigid transformation. In particular, the feature is invariant to in-plane rotations, which means we can dispense with data augmentation and have more efficient training and testing routines.
\item We propose a flexible conditional regression framework, encoding all previously estimated information as a part of the local reference frame. This includes local point properties such as the normal direction and global properties such as the estimated joint position.
\end{enumerate}
We validate our method on two real-world challenging hand pose estimation datasets, ICLV\cite{Tang-LRF} and MSRA\cite{cascaded-hand}. On ICLV, we achieve the state-of-art performance against all previous discriminative based methods~\cite{cascaded-hand,Tang-LRF,Li2015-sip} with a large margin. On MSRA, our method is on-par with the state-of-art methods~\cite{collatorative,cascaded-hand} at the threshold of 40mm, and with some minor modifications outperforms ~\cite{collatorative,cascaded-hand}.
\section{Related Works}
\label{sec:relate}
We limit our discussion to the most relevant issues and works, and refer readers to~\cite{review15,review07} for more comprehensive reviews on hand pose estimation in general. \vspace{-0.3cm}
\paragraph{Hierarchical Regression}
Several methods have adopted some form of hierarchical treatment of the pose estimation problem. For example, in~\cite{msr-chi,Keskin2012-ns,Tang-sf}, the hand is first classified into several classes according to posture or viewpoint; further pose estimation is then conditioned on such initial class. Obviously, such an approach cannot generalize to unseen postures and viewpoints.
Other works~\cite{cascaded-hand,Tang-LRF,Li2015-sip,xuHand_13,Tang2015-blackbox} hierarchically follow the tree-structured hand topology. In~\cite{Tang-LRF,Li2015-sip}, data points are recursively partitioned into subsets and only corresponding subsets of points are considered for subsequent joint estimation. In~\cite{Tang2015-blackbox}, estimated parent joints are used as inputs for regressing children joints; a final energy minimization is applied to refine the estimation. In~\cite{cascaded-hand,xuHand_13}, predictions are made based on previously estimated reference frames. Our work is similar in spirit to~\cite{cascaded-hand,xuHand_13}, as we also make estimations based on reference frames. However, unlike \cite{cascaded-hand,xuHand_13}, we utilize the normal direction to establish the reference frame and take local point properties into consideration. Further explanations on the differences between our work and \cite{cascaded-hand,xuHand_13} are given in Section~\ref{sec:ndf} and~\ref{sec:rf}.
\paragraph{Viewpoint Handling}
The free moving hand can exhibit large viewpoint changes and a variety of techniques have been proposed to handle these. For example,~\cite{Tang-sf,dantone2012real} discretize viewpoints into multiple classes and estimate pose in the view-specific classes. Unfortunately, these methods may introduce quantization errors and cannot generalize to unseen viewpoints. In~\cite{xuHand_13}, the regression for hand pose is conditioned on an estimated in-plane rotation angle. This is extended in~\cite{cascaded-hand},
which regresses the pose residual iteratively, conditioned on the estimated 3D pose at each iteration. Such a method is highly sensitive to the pose initialization and may get trapped in local minima.
\paragraph{Point Cloud Features
Depth difference features are widely used together with random forests in body pose~\cite{Shotton2013_pami,Vitruvian} and hand pose~\cite{cascaded-hand,Tang-LRF,Li2015-sip,xuHand_13,Tang2015-blackbox,Keskin2012-ns,Tang-sf} estimation.
Depth differences, however, ignore many local geometric properties of the point cloud, \emph{e.g.~} local surface normals and curvatures, and are not robust to rigid transformations and sensor noise.
In ~\cite{msra_modelbased,Liang-tip} geodesic extreme points such as finger tip candidates are used to guide later estimation.
Rusu \emph{et al.~}~\cite{Rusu-pfh} proposed a histogram feature describing different local properties. Inspired by~\cite{Rusu-pfh}, we establish local Darboux frames and using angular differences as feature values, but unlike~\cite{Rusu-pfh}, our features are based on random offsets and retain the efficiency of~\cite{Shotton2013_pami}. Most recently, convolutional neural networks (CNNs) have been used to automatically learn point cloud features~\cite{graz-cnn,nyu_hand}. Due to the heavy computational burden, CNNs can still not be used in real-time without a GPU.
\section{Random Normal Difference Feature}
\label{sec:normal}
\subsection{Random difference features}
One of the most commonly used features in depth-based pose estimation frameworks, for both body pose estimation~\cite{Shotton2013_pami,Vitruvian} and hand pose estimation~\cite{cascaded-hand,xuHand_13}, is the random depth difference feature~\cite{Shotton2013_pami}. Formally, the random difference feature $f_{\mathcal{I}}$ for point $\mathbf{p}_i \in \mathcal{R}^3$ from depth map $\mathcal{I}$ is defined as follows,
\begin{equation}
\label{equ:rdf}
f_{\mathcal{I}}(\mathbf{p}_i, \mathbf{\delta}_1, \mathbf{\delta}_2) = \Delta(\phi_{\mathcal{I}}(r(\mathbf{p}_i, \mathbf{\delta}_1)), \phi_{\mathcal{I}}(r(\mathbf{p}_i, \mathbf{\delta}_2))),
\end{equation}
\noindent
where $\mathbf{\delta}_j \in \mathcal{R}^3, j=\{1,2\}$ is a random offset, $r(\mathbf{p}_i, \mathbf{\delta}_j) \in \mathcal{R}^3$ calculates a random position given point $\mathbf{p}_i$ and offset $\mathbf{\delta}_j$. $\phi_{\mathcal{I}}(\mathbf{q})$ is the local feature map for position $\mathbf{q}\in \mathcal{R}^3$ on the point cloud and $\Delta(\cdot, \cdot)$ returns the local feature difference. In the case of random depth difference features~\cite{cascaded-hand,xuHand_13,Shotton2013_pami}, $\phi_{I}$ is the recorded depth, though the same formalism applies for other features.
Random difference features are well suited for random forest frameworks; the many possible combinations of offsets perfectly utilize their feature selection and generalization power. In addition, every dimension of the feature is calculated independently, which gives rise to parallelization schemes and allows for both temporal and spatial efficiency in training and testing. One of the main drawbacks of the depth-difference feature, however, is its inability to cope with transformations. Since random offsets in $r(\mathbf{p}_i, \mathbf{\delta}_1)$ are determined either \textit{w.r.t.} the camera frame~\cite{Shotton2013_pami} or to a globally estimated frame~\cite{cascaded-hand,xuHand_13}, the depth difference for the same offset can vary widely under out of plane rotations.
\subsection{Pose conditioned random normal difference feature}
\label{sec:ndf}
Surface normals are an important local feature for many point-cloud based applications such as registration~\cite{Rusu-pfh} and object detection~\cite{Rusu-kitchen,linemod,dl-rgbd}. Surface normals would seem a good cue for hand pose estimation too, since the direction of the surface helps to establish the local reference frame, as will be described in~\ref{sec:rf}. For two given points, the angular difference between their normal directions remains unchanged after rigid transformations. Hence, we propose a pose-conditioned
normal difference feature which is highly robust towards 3D rigid transformations.
To make random features invariant to 3D rigid transformations \textit{i.e.},
\begin{equation}
f_{\mathcal{I}}(\mathbf{p}_i, \mathbf{\delta}_1, \mathbf{\delta}_2) = f_{\mathcal{I}'}(\mathbf{p}'_i, \mathbf{\delta}_1, \mathbf{\delta}_2),
\label{equ:invariant}
\end{equation}
\noindent where $\mathcal{I'}$ and $\mathbf{p'}_i \in \mathcal{R}^3$ are the depth map and point position after transformation, it is necessary to satisfy the following two conditions:
\begin{description}
\item[i] The random offset generator $r(\cdot, \cdot)$ should be invariant to rigid transformations, \emph{i.e.~}
\begin{equation}
T(r(\mathbf{p}_i, \mathbf{\delta}_j)) = r(T(\mathbf{p}_i), \mathbf{\delta}_j),
\end{equation}
where $T(\mathbf{q}) = \mathbf{R}\cdot \mathbf{q} + \mathbf{t}$ is the rigid transformation with $\mathbf{R} \in \mbox{SO(3)}$\footnote{Readers unfamiliar with Lie group matrix notations may refer to http://ethaneade.com/lie.pdf for more details. In short, SO(3) represents a 3D rotation while SE(3) represents a 3D rigid transformation.} and $\mathbf{t}$ as its rotation and translation respectively. This condition is equivalent to guaranteeing that the relative position between $\mathbf{p}_i$ and $r(\mathbf{p}_i, \mathbf{\delta}_j)$ remains unchanged after transformation, \textit{i.e.}, $ T(\mathbf{p}_i - r(\mathbf{p}_i, \mathbf{\delta}_j)) = T(\mathbf{p}_i) - r(T(\mathbf{p}_i), \mathbf{\delta}_j)$.
\item[ii] The feature difference $\Delta(\cdot, \cdot)$ should be invariant to rigid transformation, \emph{i.e.~}
\begin{equation}
\Delta(\phi_{\mathcal{I}}(\mathbf{q}_1), \phi_{\mathcal{I}}(\mathbf{q}_2)) = \Delta(\phi_{\mathcal{I'}}(\mathbf{q'}_1), \phi_{\mathcal{I}}(\mathbf{q'}_2)),
\end{equation}
\noindent
where $\mathbf{q}_j'= T(\mathbf{q}_j), j\in\{1,2\}$ is the transformed offset position.
\end{description}
\noindent
To meet condition \textbf{i}, we extend the random position calculation $r(\mathbf{p}_i, \mathbf{\delta}_j)$ as
\begin{equation}
\label{equ:offset}
r(\mathbf{p}_i, \mathbf{\delta}_j, \mathbf{R}_i) = \mathbf{p}_i + \mathbf{R}_i\cdot \mathbf{\delta}_j,
\end{equation}
\noindent
where $\mathbf{R}_i \in \mbox{SO(3)}$ is a latent variable
representing the pose of local reference frame~\ref{sec:rf}. For any rigid transformation $\mathbf{T} =
\begin{bmatrix}
\overline{\mathbf{R}} & \overline{\mathbf{p}} \\
0 & 1 \end{bmatrix}$,
Equ. \ref{equ:offset} satisfies condition \textbf{i} \textit{iff}
\begin{equation}
\label{equ:cond1}
\mathbf{R}'_i = \overline{\mathbf{R}}\mathbf{R}_i,
\end{equation}
where $\mathbf{R}_i$ and $\mathbf{R}'_i$ are the estimated latent variable before and after rigid transformation respectively. In comparison to~\cite{cascaded-hand}, which also uses a latent variable $\mathbf{R}$, the $\mathbf{R}$ is estimated globally and therefore can be sensitive to the initialization. For us, the local Darboux frame is established through the local surface normal direction (see Section~\ref{sec:regression}) and has no such sensitivity.
To meet condition \textbf{ii}, given the random positions $\mathbf{q}_1$ and $\mathbf{q}_2$, we use the direction of the normal vector as our local feature map. The feature difference is cast as the angle between two normals, \emph{i.e.~}
\begin{equation}
\Delta(\phi_{\mathcal{I}}(\widetilde{\mathbf{q}_1}), \phi_{\mathcal{I}}(\widetilde{\mathbf{q}_2})) = n(\widetilde{\mathbf{q}_1})\cdot n(\widetilde{\mathbf{q}_2}),
\end{equation}
\noindent
where $\widetilde{q}$ denotes the 2D projection of the random position onto the image plane, since the input 2.5D point cloud is indexed by the 2D projection coordinates. $n(\cdot)\in \mathcal{R}^3$ denotes the corresponding normal vector. Since the angle between two normal vectors remains unchanged under a rigid transformation for any two given surface points, our feature also fulfills condition \textbf{ii}. In comparison, the depth difference feature, as used in~\cite{cascaded-hand,xuHand_13,Shotton2013_pami}, does not fulfill this condition.
Our proposed normal difference feature can be computed based on any surface normal estimate. We describe a conventional method based on eigenvalue decomposition in~\ref{sec:normal:accu} and then propose an efficient approximation alternative in~\ref{sec:normal:appro}.
\begin{figure}[b]
\centering
\includegraphics[width=1.\textwidth]{normal.pdf}
\caption{Estimated surface normal. From (a) to (c) the x, y, z-axis coordinate of the normal vector, resp. The first row is the regressed surface normal by the random forest and the second row is estimated by PCA. \textbf{(Best viewed in colour)}}
\label{fig:normal}
\end{figure}
\subsection{Surface normal estimation based on eigenvalue decomposition}
\label{sec:normal:accu}
For an input 2.5D point cloud, we distinguish between inner points that lie inside the point cloud and edge points on the silhouette of the point cloud. For edge points, normal estimation degenerates to 2D curve normal estimation since the normal direction is constrained to lie in the image plane.
For inner points, the local surface can be approximated by the $k$-neighbourhood surface direction\cite{Rusu-kitchen}. The eigenvector corresponding to the smallest eigenvalue of the neighbourhood covariance matrix can be considered the normal direction. The sign of the normal direction is further constrained to be the same as the projection ray. In our implementation, we set $k$ as 10 mm. We show estimated normals in the second row of Fig. \ref{fig:normal}. Our preliminary experiments show that $k$ values from 5mm to 15mm all return comparable normal estimates; above 15mm, performance starts to deteriorate, presumably due to points from multiple fingers being grouped together into the same neighbourhood.
\subsection{Surface normal regression with random forests}
\label{sec:normal:appro}
Estimating the normal at every inner point in the point cloud can become very computationally expensive, with an eigenvalue decomposition per point. Alternatively, we can take advantage of the efficiency of random forests and regress an approximate normal direction.
Directly regressing the normal vectors in vector space does not maintain unit length
so we parameterize the normal vector with spherical coordinates $(\theta, \varphi)$ where $\theta$ and $\varphi$ are the polar and azimuth angles, resp. $\theta$ and $\varphi$ are independent and can be regressed separately.
We model the distribution of a set of angular values $\mathcal{S} = \{\theta_1, \cdots \theta_n \}$ as a Von Mises Distribution, which is the circular analogue of the normal distribution. The distribution is expressed as
\begin{equation}
p_{VM}(\theta_i | \mu, \kappa) = \frac{e^{\kappa cos(\theta_i-\mu)}}{2\pi I_0({\kappa})},
\label{equ:vm}
\end{equation}
\noindent
where $\mu$ is the mean of the angles, $\kappa$ is inversely related to the variance of the approximated Gaussian and $I_0({\kappa})$ is the modified Bessel function of order 0.
To estimate the mean and variance of the distribution, we first define
\begin{equation}
\overline{C} = \sum_{i}\cos(\theta_i),~
\overline{S} = \sum_{i}\sin(\theta_i),~
\overline{R} = (\overline{C}^2 + \overline{S}^2)^{\frac{1}{2}}.
\end{equation}
\noindent
Then the maximum likelihood estimates of $\mu$ and $\kappa$ are
\begin{equation}
\mu = \mbox{atan2}(\overline{S}, \overline{C}) \qquad \text{and} \qquad
\overline{R} = \frac{I_1({\kappa})}{I_0({\kappa})}.
\end{equation}
During training, each split node is set by maximizing the information gain as
\begin{equation}
I = H(\mathcal{S}) - \sum_{i\in\{L,R\}}\frac{|\mathcal{S}^i|}{|\mathcal{S}|}H(\mathcal{S}^{i}),
\end{equation}
\noindent
where the entropy of the Von Mises Distribution is defined as
\begin{equation}
H(\mathcal{S}) = \mbox{ln}(2\pi I_0(\kappa)) - \kappa\frac{I_1({\kappa})}{I_0({\kappa})}.
\end{equation}
The training procedure for the random forest that estimates the normal is almost identical to~\cite{Shotton2013_pami} with the exception that the random offsets are restricted to lie within the region of the same $k$-nearest neighbourhood that was used for the eigenvalue decomposition based normal estimation in~\ref{sec:normal:accu}. The mean of the angular values propagated to each leaf node is selected as the leaf node's prediction value. In practice, to make the normal regression even more efficient, we combine the estimation of $\theta$ and $\varphi$ into one forest by regressing the $\theta$ in the first 10 layers and $\varphi$ in the later 10 layers, rather than estimating them independently.
Since the random offset is limited to a small area, which restricts the randomness of the trees, we find that the average error between approximated and true normal directions only goes up from~\mytilde12\degree{} to~\mytilde14\degree{} when decreasing the number of trees from 10 to 1. As the normal difference feature is not sensitive to such minor errors, we use only 1 tree for all experiments in this paper. The proposed method is extremely efficient; normals for input point clouds can be estimated in \mytilde4 ms on average, compared to \mytilde14 ms based on eigenvalue decompositions on the same machine.
\section{Frame conditioned regression forest}
\label{sec:rf}
We formulate hand joint estimation as a regression problem by regressing the 3D offsets between an input 3D point and a subset of hand joints. Directly regressing all joints of the hand at once, as has been done in previous works~\cite{forth-oneshot,Keskin2012-ns} is difficult, given the highly articulated nature of the hand and the many ambiguities due to occlusions and local self-similarities of the fingers. Instead, we prefer to solve for the joints in a hierarchical manner, as state-of-the-art results~\cite{cascaded-hand,Tang2015-blackbox} have demonstrated the benefits of solving the pose progressively down the kinematic chain.
In this section, we propose a conditional regression forest, namely the \textit{Frame Conditioned Regression Forest} (FCRF) which performs regression conditioned on information estimated in the previous stages. The hand joints are regressed hierarchically by following the kinematic chain from wrist down to the finger joints.
At each stage, we first estimate the reference frame based on results of previous stages and then regress the hand joints relevant to that stage with the FCRF.
There are three main benefits to using the FCRF. First of all, offsets between input points and finger joints are transformed into the local reference frame. This reduces the variance of the offsets and simplifies the training. It also implicitly incorporates skeleton constraints provided by the training data. Secondly, the related normal difference feature, as described in section \ref{sec:normal}, is conditioned on the estimated reference frame and makes the joint regression highly robust to 3D rigid transformations. Finally,
FCRF is in-plane rotation-invariant, and does not need manually generated in-plane rotated training samples for training as in \cite{cascaded-hand,Tang-LRF,Li2015-sip}, so the training time and resulting tree size can be reduced significantly.
Specifically, given input point $\mathbf{p}_i \in \mathcal{R}^3$ from the point cloud, the FCRF for the $j^\text{th}$ stage solves the following regression
\begin{equation}
\label{equ:fcrf}
\mathbf{O}_{j}^{(i)} = r_{j}(\mathcal{I}, \mathbf{C}_{j}^{(i)}),
\end{equation}
\noindent
where $\mathbf{O}_{j}^{(i)} \in \mathcal{R}^{3\times n}$ is the offsets between input point $\mathbf{p}_i$ and the $n$ joints to be estimated in $j^\text{th}$ stage, $\mathcal{I}$ denotes the input depth map and $\mathbf{C}_{j}^{(i)} \in \mbox{SE(3)}$ is the corresponding local frame. We define the position of the input point $\mathbf{p}_i$ as the origin of the local reference frame, \emph{i.e.~}
\begin{equation}
\label{equ:frame}
\mathbf{C}_{j}^{(i)} =
\left[
\begin{array}{c|c} \mathbf{R}_{j}^{(i)} & \mathbf{p}_i \\
\hline 0 & 1 \end{array}
\right],
\end{equation}
\noindent
where $\mathbf{R}_{j}^{(i)} = \begin{bmatrix}\mathbf{x}, \mathbf{y}, \mathbf{z}\end{bmatrix}\in \mbox{SO(3)}$ is a rotation matrix representing the frame pose, and $\mathbf{x},\mathbf{y},\mathbf{z}\in \mathcal{R}^3$ are the corresponding axis directions. Both $\mathbf{R}_i$ and $\mathbf{p}_i$ are defined with respect to the camera frame.
The regression $r_j(\mathcal{I}, \mathbf{C}_{j}^{(i)})$ is done by a random forest.
During training, $\mathbf{o}_{ik} \in \mathcal{R}^3$, the offset between point $\mathbf{p}_i$ and joint $\mathbf{l}_k$ to be estimated, is first rotated to the local reference frame $\mathbf{C}_j^{(i)}$ as $\widetilde{\mathbf{o}_{ik}}$, \emph{i.e.~}
\begin{equation}
\widetilde{\mathbf{o}_{ik}} = (\mathbf{R}_j^{(i)})^{T}\cdot \mathbf{o}_{ik}.
\end{equation}
The distribution of offset samples are modeled as a uni-modal Gaussian as in \cite{Shotton2013_pami}. For each split node of the tree, the normal difference feature which results in the maximum information gain from a random subset of features is selected. For each leaf node, mean-shift searching~\cite{meanshift} is performed and the maximal density point is used as the leaf prediction value.
During testing, given the estimated local frame $\mathbf{C}_j^{(i)}$, the resulting offset $\mathbf{o}_{ik}$ can be re-projected to the camera frame as
\begin{equation}
\mathbf{o}_{ik} = (\mathbf{R}_j^{(i)})\cdot \widetilde{\mathbf{o}_{ik}}.
\end{equation}
\section{Hierarchical hand joint regression}
\label{sec:regression}
In this section, we detail the design of reference frames used by FCRFs in every stage, given the estimated local surface normal and the parent joint positions from previous stages. Free moving hand pose estimation faces two major challenges, \textit{i.e.}, large variations of viewpoints, and self-similarities of different fingers. We decompose hand pose estimation into two sub-problems that explicitly tackle these two challenges: first, we estimate the reference frame of the palm and second, we estimate the finger joints.
In Sections \ref{sec:rf_wrist} and \ref{sec:rf_mcp} the palm estimation is introduced by first estimating the wrist joint (palm position) followed by MCP joints(Fig.~\ref{fig:framework}(a))
for all 5 fingers (palm pose), in which the Darboux frame for every input point is established by taking the estimated wrist joint as reference point. In Sections \ref{sec:rf_pip} and \ref{sec:rf_tip} the joints for each finger are estimated, progressively conditioned on the previously estimated joint position.
\subsection{Wrist estimation}
\label{sec:rf_wrist}
We consider only edge points on the hand silhouette as inputs for estimating the wrist joint. Our rationale is that we cannot find unique reference frames for non-edge points, since knowing only the direction of the normal, \emph{i.e.~} the z-axis, is insufficient to uniquely determine the x- and y-axis on the tangent plane. We assume orthographic projection for the point cloud, \emph{i.e.~} the tangent plane of edge point is orthogonal to the image plane, then the local reference frame of edge point $\mathbf{p}_i$ can be defined uniquely as follows,
\begin{equation}
\label{equ:wrist}
\begin{split}
&\mathbf{x}_{wrist}^{(i)} = \mathbf{n},\\
&\mathbf{y}_{wrist}^{(i)} = \mathbf{z}_{wrist}^{(i)} \times \mathbf{x}_{wrist}^{(i)},\\
&\mathbf{z}_{wrist}^{(i)} = \mathbf{n}_i,
\end{split}
\end{equation}
\noindent
where {\bf n} is the image plane normal direction, ${\bf n}_i$ is the normal to the silhouette at point $i$.
The resulting local reference frame is not only invariant to 2D rotations in the image plane but to some degree also robust to out-of-plane rotations, provided that the hand silhouette does not change too much.
\subsection{Metacarpophalangeal (MCP) Joint Estimation}
\label{sec:rf_mcp}
Given the estimated wrist point position as a reference point, we assume its relevant position under the local frame $C_{MCP}^{(i)}$ is unchanged then the local reference frame for point $\mathbf{p}_i$ is established as follows
\begin{equation}
\label{equ:mcp}
\begin{split}
&\mathbf{x}_{MCP}^{(i)} = \mathbf{y}_{MCP}^{(i)}\times \mathbf{z}_{MCP}^{(i)}, \\
&\mathbf{y}_{MCP}^{(i)} = \frac{\mathbf{n}_i\times(\mathbf{p}_{wrist}-\mathbf{p}_i)}{\|\mathbf{n}_i\times(\mathbf{p}_{wrist}-\mathbf{p}_i)\|_{2}}, \\
&\mathbf{z}_{MCP}^{(i)} = \mathbf{n}_i,
\end{split}
\end{equation}
\noindent
where the z-axis of the local reference frame is defined as the normal direction $\mathbf{n}_i$, and the y-axis is defined by taking the wrist location $\mathbf{p}_{wrist}$ as a reference point.
The MCP joints from all five fingers are then regressed simultaneously, \textit{i.e.}, $\mathbf{O}_{MCP}^{(i)} \in \mathcal{R}^{3\times 5}$ using our previously defined FCRF.
The estimated MCP joints are then replaced by the transformed MCP position from a template palm to reduce the accumulated regression error.
We first find a closed form solution of the palm pose using a variation of ICP~\cite{aicp}. The palm pose matrix $\mathbf{R}_{palm}$'s y-axis is defined as the direction from the wrist to the MCP joint of the middle finger, the z-axis is defined as the palm normal.
\subsection{Proximal Interphalangeal (PIP) Joint Estimation}
\label{sec:rf_pip}
In the estimation of the PIP joint for finger $k$, all input reference frames share the same pose as the rotated palm reference frame as follows,
\begin{equation}
\mathbf{C}_{PIP_{k}}^{(i)} =
\left[
\begin{array}{c|c} \mbox{Rot}_k(\mathbf{R}_{palm}) & \mathbf{p_i} \\
\hline 0 & 1 \end{array}
\right],
\label{equ:pip}
\end{equation}
\noindent
where $\mbox{Rot}_k(\cdot)$ is an in-plane rotation to align the reference frame's y-axis to the $k-^\text{th}$ finger's empirical direction Fig.~\ref{fig:framework} (a).
Given the local self-similarity between fingers, it can be easy to double-count evidence. To avoid this, we adopt two simple measures. First, we use points only from the neighbourhood of the parent MCP joint as input for regressing each PIP joint, since these points best describe the local surface distortion raised by the parent joint articulation~\cite{Kovalsky-learnArticulation}.
Secondly we limit the offset of the FCRF to lie along the direction of the finger to maintain robustness to noisy observations from nearby fingers.
\subsection{Distal Interphalangeal Joint (DIP) and Finger Tip (TIP) Estimation}
\label{sec:rf_tip}
The ways to estimate DIP and TIP joints are identical, since their parents are both 1-DoF joints.
The local reference frame for each joint is defined as follows
\begin{equation}
\begin{split}
&\mathbf{x}_l= \mathbf{z}_{palm}\times \mathbf{y}_l,\\
&\mathbf{y}_l = \mathbf{p}(l)-\mathbf{g}(l), \\
&\mathbf{z}_l = \mathbf{x}_l\times \mathbf{y}_l,
\end{split}
\end{equation}
\noindent
where $\mathbf{z}_{palm}$ is the normal direction of palm, $\mathbf{p}(l)$ and $\mathbf{g}(l) \in \mathcal{R}^3$ denote the parent and grandparent joint of $l$ respectively. To avoid double counting of local evidence, we adopt the same techniques as in section~\ref{sec:rf_pip} .
\section{Experiments}
\label{sec:exp}
We apply our proposed hand estimation method to two publicly available real-world hand pose estimation datasets: ICLV~\cite{Tang-LRF} and MSRA~\cite{cascaded-hand}. The performance of our method is evaluated both quantitatively and qualitatively. For quantitative evaluation, two {evaluation metrics}, per-joint error (in mm) averaged over all frames and percentage of frames in which all joints are below a threshold~\cite{Vitruvian}, are used. We show qualitative results in Fig.~\ref{fig:qual} and encourage the reader to watch the accompanying supplementary videos.
All experiments are conducted on an Intel 3.40 GHz I7 machine and the \textit{average run time} is 29.4fps or 33.9ms per image. The \textit{maximum depth} of all the trees is set to 20. The \textit{number of trees} for all joint regression forests are set to 5 and 1 for normal estimation (see Section \ref{sec:normal:appro}).
To highlight the effectiveness of our proposed normal difference feature, we first apply our frame conditioned regression forests with the same hierarchical structure but based on the standard depth difference feature~\cite{Shotton2013_pami}. We denote this variation using the depth difference feature as our \textit{baseline method}. It should be noted that the baseline does depend on normal estimation for the establishment of the local wrist frame. We also compare to methods directly regressing the wrist and MCP joint positions without establishing the frame~\cite{Tang-LRF,Li2015-sip} or based on an initial guess and the subsequent, iterative regression of the error~\cite{cascaded-hand}.
\subsection{ICLV hand dataset}
The ICLV hand dataset~\cite{Tang-LRF} has 20K images from 10 subjects and an additional 160K in-plane rotated images for training. Since our method is invariant to in-plane rotation, we train with only the initial 20K.
The test set is composed of 2 sequences with continuous finger movement but little viewpoint change.
\begin{figure}[!htbp]
\centering
\includegraphics[width=1.0\textwidth]{icl.pdf}
\caption{Quantitative evaluation on ICLV dataset. From (a) to (c), success rates over different thresholds on sequence A, B and both respectively. (d) pre-joint average error on both sequences (R:root, T:tip)}
\label{fig:icl_res}
\vspace{-0.5cm}
\end{figure}
We compare our method (both the baseline and the version with the normal difference feature) against the state-of-art methods Latent Regression Forest (LRF)~\cite{Tang-LRF}, Segmentation Index Points(SIP)~\cite{Li2015-sip}, and Cascaded Regression (Cascaded)~\cite{cascaded-hand}. Fig.~\ref{fig:icl_res}(a)-(c) shows that both variations of our proposed method outperform LRF~\cite{Tang-LRF} and SIP~\cite{Li2015-sip} by a large margin on both test sequences. In comparison to the Cascaded method of~\cite{cascaded-hand}, shown in Fig. \ref{fig:icl_res}(c), our baseline is comparable or better at almost all allowed distances, while the variation with the normal difference feature boosts performance by another $5-10\%$. As shown in Fig.~\ref{fig:icl_res}(d), our method significantly out-performs~\cite{Tang-LRF}, and it outperforms \cite{cascaded-hand} by \mytilde2mm in terms of the mean error.
These results confirm that conditioning finger localization on the wrist pose, as we have done and as is done in~\cite{cascaded-hand}, can significantly boost accuracy. Furthermore, our proposed normal difference feature is able to better handle 3D rigid transformations.
\subsection{MSRA hand dataset}
The MSRA hand dataset~\cite{cascaded-hand} contains 76.5K images from 9 subjects with 17 hand gestures. We use a leave-one-subject-out training/testing split and average the results over the 9 subjects. This dataset is complementary to the ICLV dataset since it has much larger viewpoint changes but limited finger movements. The sparse gesture set does not come close to reflecting the range of hand gestures in real-world HCI applications and as such, is not suitable for evaluating how well a method can generalize towards unseen hand gestures. Yet, this dataset is very good for evaluating the robustness of pose estimation methods to 3D rigid transformations; for HCI applications, this offers flexibility for mounting the camera in different locations.
\begin{figure}[!htbp]
\centering
\includegraphics[width=1.\textwidth]{msra.pdf}
\caption{Quantitative evaluation on MSRA dataset. (a) to (b): average joint error as a function of pitch and yaw angle of the palm pose with respect to camera frame; (c) success rates over different thresholds.}
\label{fig:msra_res}\vspace{-0.5cm}
\end{figure}
As is shown in Fig.~\ref{fig:msra_res}(a)-(b), using the normal difference exhibits less variance to viewpoint changes than using the depth difference. This is more prominent in the pitch angle due to the elongated hand shape. For a given pair of points, their depth difference exhibits larger variation \emph{w.r.t.~} pitch angle viewpoint changes. Nevertheless, the performance of the normal difference does decrease under large viewpoint changes. We attribute this to the errors in surface normal estimation due to point cloud noise and to the fact that a 2.5D point cloud only partially represents the full 3D surface.
We compare our proposed method against the state-of-the-art Cascaded Regression (Cascaded)~\cite{cascaded-hand} and the Collaborative Filtering (Filtering)~\cite{collatorative} approaches. Above an allowed distance of 40mm to the ground truth, our approach is comparable to the others. Below the 40mm threshold, our baseline and the normal difference feature version has around~\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}$14\%$ less frames than competing methods. We attribute the difference to the fact that both the Cascaded and the Filtering approach consider the finger as a whole, in the former case for regression, and in the latter as a nearest neighbour search from the training data.
While our method generalizes well to unseen finger poses by regressing each finger joint progressively, it is unable to utilize the sparse (albeit similar to testing) set of finger poses in the training. Nevertheless, in an HCI scenario, a user is often asked to first make calibration poses which are important to improve accuracy. As such, we propose two minor modifications to make more comparable evaluations.
For the first modification, we first regress the palm pose, normalize the hand, and then classify the hand pose as a whole. Based on the classification, we assign a corresponding pose sampled from the training set, transformed accordingly to the palm pose. This modification, which we denoted as \textit{pose classification} is similar to Filtering~\cite{collatorative} as both methods consider the hand as a whole. By classifying the 17 gesture classes as provided by the MSRA dataset we now outperform \cite{collatorative} over a large interval of thresholds larger than 22mm. We attribute the increased performance to our accurate estimate of the palm pose
For the second modification, we regress each finger (\emph{i.e.~} the 3 finger joints PIP, DIP, TIP) as a whole given the estimated palm pose. This is similar in spirit to the regression strategy in \cite{cascaded-hand} which takes each finger as a whole. Our method outperforms~\cite{cascaded-hand} by \raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}$5\%$ in the 25-30mm threshold interval. We attribute this improvement to our palm pose estimation scheme which avoids sensitivity to initialization~\cite{cascaded-hand}.
Despite our modifications, it should be noted that regressing the finger as a whole cannot generalize to unseen joint angle combinations for one finger, which is usually the case in real-world HCI scenarios, \emph{e.g.~} grasping a virtual object, where one finger may exhibit various joint angle combinations according to the shapes of different objects. However, the two strategies are complementary, \emph{i.e.~} regressing finger joints progressively can generalize to unseen finger poses while regressing the finger as a whole can capture finger joint correlations in training samples. Given enough computational resources, the two strategies can be performed in parallel, with the best estimation being selected according to an energy function as in model-based tracking. We leave this as our future work.
\begin{figure}[!htbp]
\centering
\includegraphics[width=.95\linewidth]{qualitative.pdf}
\caption{Examples of successful and failed pose estimates on the ICLV~\cite{Tang-LRF} and the MSRA~\cite{cascaded-hand} dataset. Failures are due to extreme view point, wrongly estimated normal direction, etc. (best viewed in colour)}
\label{fig:qual}
\vspace{-1.0cm}
\end{figure}
\section{Conclusion and Future Work}
\label{sec:con}
We have presented a hierarchical regression scheme conditioned on local reference frames. We utilize the local surface normal both as a feature map for regression and to establish the local reference frame. We also proposed an efficient surface normal estimation method based on random forests. Our system shows excellent results on two real-world, challenging datasets and is either comparable or outperforms state-of-the-art methods in hand pose estimation.
The surface normal serves as an important local property of the point cloud. While random forests are an efficient way of estimating the normal, they are only one way and other methods could be developed to be more accurate. Given the success of using surface normals in our work, we expect that there will be benefits for model-based tracking as well.
In our current work, we follow a tree-structured model of the hand. Given the flexibility of our proposed conditioned regression forest, one can also perform hierarchical regressions with other underlying graphical models. With different models, one could take into account the correlations and dependencies between fingers, especially with respect to grasping objects. We leave this as future work in improving the current system.
\bibliographystyle{splncs}
|
1,314,259,996,707 | arxiv | \section{Introduction}
\subsection{Literature Review}
Kelly (1956) obtained the eponymous Kelly rule (``Fortune's Formula,'' Poundstone 2010) by maximizing the asymptotic growth rate of one's capital when gambling on repeated horse races where the posted odds diverge from the true win probabilities. Famously (cf. with Thorp 2017), the Kelly rule was employed by card counter Edward O. Thorp to size his bets at the Nevada blackjack tables. Thorp went on to use the same principle (of the log-optimal constant-rebalanced portfolio) in money management on Wall Street. For the general discrete time portfolio problem, the Kelly investor willingly foregoes the tangency portfolio (of maximum Sharpe ratio) in exchange for the highest possible asymptotic capital growth rate. Breiman (1961) showed that a Kelly gambler will almost surely outperform any essentially different strategy (by an exponential factor), and he has the shortest mean waiting time to reach a distant wealth goal.
\par
In a pair articles, Bell and Cover (1980, 1988) proved a short-term optimality property of the discrete time Kelly rule. They show that the Kelly criterion emerges as the solution of a wide class of ``investment $\phi$-games'' where the goal is for one investor to outperform the other (in the sense of an increasing function $\phi(\bullet)$ of the ratio of the two players' final wealths). Both papers use an artifice whereby, before the game itself, each player is allowed make a ``fair randomization'' of his initial dollar, by exchanging it for any random variable distributed over $[0,\infty)$ whose mean is at most 1.
\subsection{Contribution}
This paper studies a similar game in continuous time, where each player commits to a rebalancing rule that must be used continuously over the interval $[0,t]$. The unique Nash equilibrium (that constitutes a saddle point of the expected ratio of wealths at $t$) is for both players to use the continuous time Kelly rule. This result, which matches that of Bell and Cover (1988), holds for the general market with $n$ correlated stocks $(i=1,...,n)$ in geometric Brownian motion. This being done, we show that the continuous time Kelly rule is the basis for the solution of a ``continuous time investment $\phi$-game'' that is analogous to the discrete time version solved by Bell and Cover.
\section{Model}
We consider a continuous time trading game between two players. There is a risk-free bond whose price $B_t:=e^{rt}$ evolves according to $dB_t=rB_t\,dt$ and a single stock whose price $S_t$ follows the geometric Brownian motion
\begin{equation}
dS_t:=S_t(\mu\,dt+\sigma dW_t),
\end{equation}where $\mu$ is the drift, $\sigma$ is the volatility, and $W_t$ is a standard Brownian motion. At $t=0$ each player chooses a constant rebalancing rule $b\in\mathbb{R}$ that he must adhere to for $0\leq t\leq T$. A rebalancing rule $b$ is a fixed-fraction betting scheme that maintains the fraction $b$ of wealth in the stock and $1-b$ in the bond at all times. Let $V_t(b)$ denote the wealth at $t$ of a $\$1$ deposit into the rebalancing rule $b$. At instant $t$, the trader holds $bV_t(b)/S_t$ shares of the stock and $(1-b)V_t(b)e^{-rt}$ units of the bond. This portfolio will be held over the differential time step $[t,t+dt]$, after which point it must be rebalanced again. The players are free to use any amount of leverage ($b>1$ or $b<0$), if desired.
\par
Player 1 (the ``numerator player'') chooses the rebalancing rule $b\in\mathbb{R}$ and Player 2 (the ``denominator player'') chooses a rebalancing rule $c\in\mathbb{R}$. We consider the two-person, zero-sum game with payoff kernel $\pi(b,c):=\mathbb{E}[V_T(b)/V_T(c)]$. The numerator player seeks to maximize the expected ratio of his final wealth to that of the opponent's. The denominator player seeks to minimize this quantity.
\subsection{Payoff Computation}
Each player's wealth follows a geometric Brownian motion
\begin{equation}
\frac{dV_t(b)}{V_t(b)}=b\frac{dS_t}{S_t}+(1-b)\frac{dB_t}{B_t}=[r+b(\mu-r)]dt+b\sigma dW_t.
\end{equation}Solving, we obtain
\begin{equation}
V_t(b)=\exp\{[r+b(\mu-r)-\sigma^2b^2/2]t+b\sigma W_t\}.
\end{equation} The ratio of final wealths is
\begin{equation}
\frac{V_t(b)}{V_t(c)}=\exp\{[(\mu-r)(b-c)+(c^2-b^2)\sigma^2/2]t+(b-c)\sigma W_t\}.
\end{equation}
Thus, since the ratio of final wealths is log-normally distributed (cf. Shonkwiler 2013), we have, after simplification,
\begin{equation}\boxed{
\mathbb{E}\bigg[\frac{V_t(b)}{V_t(c)}\bigg]=\exp\{(\mu-r-\sigma^2c)(b-c)t\}.}
\end{equation}
After a monotonic transformation, we may re-write the payoff kernel as
\begin{equation}
\boxed{\pi(b,c):=(\mu-r-\sigma^2c)(b-c),}
\end{equation}which is the exponential growth rate of $\mathbb{E}[V_t(b)/V_t(c)]$.
\subsection{Equilibrium}
Player 1's best response correspondence is
\[ b^*(c)=\begin{cases}
+\infty & \text{if }c<(\mu-r)/\sigma^2 \\
\mathbb{R} & \text{if }c=(\mu-r)/\sigma^2 \\
-\infty & \text{if }c>(\mu-r)/\sigma^2
\end{cases}.
\]Player 2's best response function is
\begin{equation}
c^*(b)=\frac{1}{2}\bigg(b+\frac{\mu-r}{\sigma^2}\bigg).
\end{equation}Thus, the unique Nash equilibrium is $b^*=c^*=(\mu-r)/\sigma^2$, which happens to be the continuous time Kelly rule (cf. Luenberger 1998). Ordinarily, the Kelly (1956) rule is derived by maximizing the asymptotic continuously-compounded capital growth rate
\begin{equation}\boxed{
\text{Growth Rate}(b):=\underset{t\to\infty}{\lim}\,\,\frac{1}{t}\log\,V_t(b)=r+(\mu-r)b-\frac{\sigma^2}{2}b^2.}
\end{equation}Hence, even over very short intervals of time $[0,t]$, the desire to outperform other traders in the market dictates the use of the Kelly rule $b^*:=(\mu-r)/\sigma^2$. We have thus derived a short-term optimality property of the continuous time Kelly rule that matches the results obtained by Bell and Cover (1988) in discrete time.
\subsection{Several Correlated Stocks}
We extend the above result to the general stock market with $n$ correlated stocks ($i=1,...,n$) whose prices $S_{it}$ follow the geometric Brownian motions (cf. Bj\H{o}rk 1998)
\begin{equation}
dS_{it}:=S_{it}(\mu_i\,dt+\sigma_i \,dW_{it}),
\end{equation}where $\mu:=(\mu_1,...,\mu_n)'$ is the drift vector, $\sigma:=(\sigma_1,...,\sigma_n)'$ is the vector of volatilities, and $\Sigma$ is the covariance matrix of instantaneous returns per unit time, e.g. $\Sigma_{ij}=\text{Cov}(dS_{it}/S_{it},dS_{jt}/S_{jt})/dt.$ The $W_{it}$ are correlated standard Brownian motions, with $\rho_{ij}:=\text{Corr}(dW_{it},dW_{jt})$ and $\Sigma_{ij}=\rho_{ij}\sigma_i\sigma_j$. We assume that $\Sigma$ is invertible. In this context, a rebalancing rule is a vector $b:=(b_1,...,b_n)'\in\mathbb{R}^n$, where the gambler continuously maintains the fixed fraction $b_i$ of wealth in stock $i$ at all times. He keeps the fraction $1-\sum\limits_{i=1}^nb_i$ of wealth in bonds. As in the univariate case, this permits the freest possible use of leverage, if desired.
\par
Each player's final wealth $V_t(b)$ follows the geometric Brownian motion
\begin{equation}
\frac{dV_t(b)}{V_t(b)}=\sum\limits_{i=1}^nb_i\frac{dS_{it}}{S_{it}}+\bigg(1-\sum\limits_{i=1}^nb_i\bigg)\frac{dB_t}{B_t}=[r+(\mu-r\textbf{1})'b]dt+\sum\limits_{i=1}^nb_i\sigma_idW_{it}.
\end{equation}
The solution of this stochastic differential equation is
\begin{equation}
V_t(b)=\exp\bigg\{[r+(\mu-r\textbf{1})'b-b'\Sigma b/2]t+\sum\limits_{i=1}^nb_i\sigma_i W_{it}\bigg\}.
\end{equation}
This can be verified directly by applying It\^{o}'s Lemma for several diffusion processes (cf. Wilmott 2001) to the function $F(W_1,...,W_n,t):=\exp\{[r+(\mu-r\textbf{1})'b-b'\Sigma b/2]t+\sum\limits_{i=1}^nb_i\sigma_i W_{i}\}.$
The ratio of final wealths is
\begin{equation}
\frac{V_t(b)}{V_t(c)}=\exp\bigg\{(\mu-r\textbf{1})'(b-c)+(c'\Sigma c-b'\Sigma b)/2]t+\sum\limits_{i=1}^n(b_i-c_i)\sigma_i W_{it}\bigg\}.
\end{equation} Thus, the ratio of final wealths is log-normally distributed, with
\begin{equation}\boxed{
\mathbb{E}\bigg[\frac{V_t(b)}{V_t(c)}\bigg]=\exp\{(\mu-r\textbf{1}-\Sigma c)'(b-c)t\}.}
\end{equation} After monotonic transformation, we obtain the simplified payoff kernel
\begin{equation}\boxed{
\pi(b,c):=(\mu-r\textbf{1}-\Sigma c)'(b-c).}
\end{equation}
Player 1's best response correspondence is
\[ b_i^*(c)=\begin{cases}
+\infty & \text{if }(\Sigma c)_i<\mu_i-r \\
\mathbb{R} & \text{if }(\Sigma c)_i=\mu_i-r \\
-\infty & \text{if }(\Sigma c)_i>\mu_i-r,
\end{cases}
\] where $(\Sigma c)_i:=\sum\limits_{j=1}^n\rho_{ij}\sigma_i\sigma_jc_j$ is the $i^{th}$ coordinate of the vector $\Sigma c$. Assuming that $\Sigma$ is invertible, Player 2's best response function is
\begin{equation}
c^*(b)=\frac{1}{2}[b+\Sigma^{-1}(\mu-r\textbf{1})].
\end{equation}Intersecting the best responses, we find that the unique Nash equilibrium is $b^*=c^*=\Sigma^{-1}(\mu-r\textbf{1})$, which is the multivariate Kelly rule in continuous time. We thus have the identity
\begin{equation}
\underset{b\in\mathbb{R}}{\max}\,\,\underset{c\in\mathbb{R}}{\min}\,\,\mathbb{E}\bigg[\frac{V_t(b)}{V_t(c)}\bigg]=\underset{c\in\mathbb{R}}{\min}\,\,\underset{b\in\mathbb{R}}{\max}\,\,\mathbb{E}\bigg[\frac{V_t(b)}{V_t(c)}\bigg]=1.
\end{equation}Thus, since the Kelly rule $b^*$ is Player 1's maximin strategy, we have $\mathbb{E}[V_t(b^*)/V_t(c)]\geq1$ for all $c$, and since the Kelly rule $c^*$ is Player 2's minimax strategy, we have $\mathbb{E}[V_t(b)/V_t(c^*)]\leq1$ for all $b$.
\section{Investment $\phi$-Game}
Based on the fact that the Kelly rule $b^*=c^*$ guarantees $\mathbb{E}[V_t(b^*)/V_t(c)]\geq1$ for all $c$ and $\mathbb{E}[V_t(b)/V_t(c^*)]\leq1$ for all $b$, we can obtain a general result analogous to that of Bell and Cover (1988). First, we need some definitions.
\begin{definition}
By a ``fair randomization'' of the initial dollar is meant a random variable $\textbf{W}$ with support $[0,\infty)$ and $\mathbb{E}[\textbf{W}]\leq 1$.
\end{definition}
\begin{definition}
For any increasing function $\phi(\bullet)$, the ``primitive $\phi$-game,'' with value $v_\phi$, is the two-person, zero-sum game with payoff kernel $\mathbb{E}[\phi(\textbf{W}_1/\textbf{W}_2)]$, where player 1 chooses a fair randomization $\textbf{W}_1$ and player 2 chooses a fair randomization $\textbf{W}_2$. The value of the primitive $\phi$-game is $v_\phi:=\underset{\textbf{W}_1}{\sup}\,\,\underset{W_2}{\inf}\,\,\mathbb{E}[\phi(\textbf{W}_1/\textbf{W}_2)]=\underset{\textbf{W}_2}{\inf}\,\,\underset{\textbf{W}_1}{\sup}\,\,\mathbb{E}[\phi(\textbf{W}_1/\textbf{W}_2)]$. The random wealths $\textbf{W}_1$ and $\textbf{W}_2$ are independent of each other.
\end{definition}
\begin{definition}
For any increasing function $\phi(\bullet)$, the ``investment $\phi$-game'' is the two-person, zero-sum game with payoff kernel $\mathbb{E}[\phi\{\textbf{W}_1V_t(b)/(\textbf{W}_2V_t(c))\}]$, where player 1 chooses a rebalancing rule $b$ and a fair randomization $\textbf{W}_1$ of the initial dollar, and player 2 chooses a rebalancing rule $c$ and a fair randomization $\textbf{W}_2$ of his initial dollar. The random wealths $\textbf{W}_1$ and $\textbf{W}_2$ are independent of all stock prices and independent of each other.
\end{definition}
\begin{theorem}
The investment $\phi$-game has the same value $v_\phi$ as the primitive $\phi$-game. In equilibrium, both players use the continuous-time Kelly rule $b^*:=\Sigma^{-1}(\mu-r\textbf{1})$, and the players use the same minimax randomizations $(\textbf{W}_1^*,\textbf{W}_2^*)$ that solve the primitive $\phi$-game.
\end{theorem}
\begin{proof}
First, we show that $\mathbb{E}[\phi\{\textbf{W}_1^*V_t(b^*)/(\textbf{W}_2V_t(c))\}]\geq v_\phi$ for any fair randomization $\textbf{W}_2$ and any rebalancing rule $c$, where $b^*$ is the Kelly rule. Note that the quantity $\textbf{W}_2V_t(c)/V_t(b^*)\geq0$ is a fair randomization, since $\mathbb{E}[V_t(c)/V_t(b^*)]\leq1$. The inequality $\mathbb{E}[V_t(c)/V_t(b^*)]\leq1$ follows from direct substitution of $b^*:=\Sigma^{-1}(\mu-r\textbf{1})$ into the expected wealth ratio. Thus, since $\textbf{W}_1^*$, is Player 1's minimax solution in the primitive $\phi$-game, we must have $\mathbb{E}[\phi\{\textbf{W}_1^*V_t(b^*)/(\textbf{W}_2V_t(c))\}]\geq v_\phi$.
\par
Similarly, we show that $\mathbb{E}[\phi\{\textbf{W}_1V_t(b)/(\textbf{W}_2^*V_t(c^*))\}]\leq v_\phi$ for any fair randomization $\textbf{W}_1$ and any rebalancing rule $b$, where $c^*$ is the Kelly rule. Note that the quantity $\textbf{W}_1V_t(b)/V_t(c^*)\geq0$ is a fair randomization, since $\mathbb{E}[V_t(b)/V_t(c^*)]\leq1$. Thus, since $\textbf{W}_2^*$, is Player 2's minimax solution of the primitive $\phi$-game, we must have $\mathbb{E}[\phi\{\textbf{W}_1V_t(b)/(\textbf{W}_2^*V_t(c^*))\}]\leq v_\phi$.
\par
Thus, we have shown that $(\textbf{W}_1^*,b^*)$ forces the payoff to be $\geq v_\phi$ and $(\textbf{W}_2^*,c^*)$ forces the payoff to be $\leq v_\phi$ when $b^*$ and $c^*$ are equal to the Kelly rule and $(\textbf{W}_1^*,\textbf{W}_2^*)$ are the minimax strategies from the primitive $\phi$-game. This proves the theorem.
\end{proof}
\begin{example}
As in Bell and Cover (1980), we let $\phi(x):=\textbf{1}_{[1,\infty)}(x)$ be the indicator function of $[1,\infty)$. This turns the payoff kernel into $\text{Prob}\{\textbf{W}_1V_t(b)\geq \textbf{W}_2V_t(c)\}$. The equilibrium amounts to the Kelly rule $b^*=c^*$ and the fair exchange of the initial dollar for a $\text{uniform}(0,2)$ variable. The value of the game is $1/2.$
\end{example}
\section{Simulation of a Sample Play of the Game}
To illustrate, we use the example of ``Shannon's Demon'' in continuous time. In Shannon's classic discrete time example, there is cash (that pays no interest) and a ``hot stock'' that each period either doubles or gets cut in half in price, each with $50\%$ probability. The continuous time analog is to set $r:=0$ and
\begin{equation}
dS_t:=\sigma S_t\bigg(\frac{\sigma}{2}\,dt+dW_t\bigg),
\end{equation}with $\sigma:=\log2\approx0.693.$ The unique pure strategy Nash equilibrium of this game is for both players to use the rebalancing rule $b^*:=0.5$; the players' best response correspondences are plotted in Figure \ref{bestresponse}. For the sake of argument, assume that Player 1 behaves correctly, but Player 2 (perhaps confused by the $24\%$ annual drift rate) chooses to put all his money into the stock, and hold.
\par
Player 1's wealth at $t$ is $\exp(0.06t+0.3465W_t)$, and Player 2's wealth at $t$ is $\exp(0.693W_t)$. The expected wealth ratio is $\exp(0.12t)$. In Figure \ref{simulation} we have simulated a single play of the game, with a horizon of $T:=300$. At time $t$, the probability that Player 1 has more wealth than Player 2 is $N(0.173\sqrt{t}),$ where $N(\bullet)$ is the cumulative normal distribution function. At $t:=50$, there is an $89\%$ chance that Player 1 has more wealth. At $t:=100$ this number rises to $96\%$.
\begin{figure}[t]
\begin{center}
\includegraphics[height=185px]{bestresponse.png}
\caption{\sc{The best response correspondences $b^*(c)$ and $c^*(b)$ that obtain for the parameter values $r:=0, \sigma:=\log2, \text{and }\mu:=\sigma^2/2$.}}
\label{bestresponse}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=425px]{ctsDemon.png}
\caption{\sc{Simulation of one play of the game ($b:=0.5$ and $c:=1$), for the parameter values $r:=0, \sigma:=\log2, \mu:=\sigma^2/2, T:=300$.}}
\label{simulation}
\end{center}
\end{figure}
\section{The General Stochastic Differential Game}
Finally, we show that the restriction to constant rebalancing rules entails no loss of generality. We do this below for the one-stock case; the proof for several stocks is similar. Let $M_{1t}$ and $M_{2t}$ be the wealths of the numerator and denominator player, respectively. We now allow the players' portfolios to depend on the most general state vector, which is $(S_t,t,M_{1t},M_{2t})$. Player 1's trading strategy is now denoted $b(S,t,M_1,M_2)$, and Player 2's strategy is $c(S,t,M_1,M_2)$. We show that in equilibrium, both players still adhere to the constant rebalancing rule $b(S,t,M_1,M_2)=c(S,t,M_1,M_2):=(\mu-r)/\sigma^2$.
\par
First, assume that the denominator player uses the Kelly rule $c:=(\mu-r)/\sigma^2.$ We show that the numerator player's best response is to use the same control policy. Let $J(S,t,M_1,M_2)$ be the numerator player's maximum value function. His HJB equation is
\begin{multline}\label{HJB}
-\frac{\partial J}{\partial t}=\underset{b\in\mathbb{R}}{\max}\,\,\bigg\{\mu S\frac{\partial J}{\partial S}+[r+b(\mu-r)]M_1\frac{\partial J}{\partial M_1}+[r+c(\mu-r)]M_2\frac{\partial J}{\partial M_2}\\
+\frac{\sigma^2}{2}S^2\frac{\partial^2 J}{\partial S^2}+\frac{b^2\sigma^2}{2}M_1^2\frac{\partial^2 J}{\partial M_1^2}+\frac{c^2\sigma^2}{2}M_2^2\frac{\partial^2 J}{\partial M_2^2}\\
+b\sigma^2SM_1\frac{\partial^2J}{\partial S\partial M_1}+c\sigma^2SM_2\frac{\partial^2J}{\partial S\partial M_2}+bc\sigma^2M_1M_2\frac{\partial^2J}{\partial M_1\partial M_2}\bigg\}.
\end{multline} The boundary condition is $J(S,T,M_1,M_2):=M_1/M_2$. We guess that $J(S,t,M_1,M_2)\equiv M_1/M_2$, which obviously satisfies the boundary condition. Under this guess, Player 1's HJB equation simplifies to
\begin{equation}
\underset{b\in\mathbb{R}}{\max}\,\,(\mu-r-\sigma^2c)(b-c)=0,
\end{equation}where $c:=(\mu-r)/\sigma^2.$ This value of $c$ makes the maximand identically 0, so of course $b^*:=c$ is a maximizer. Thus, substitution of $J\equiv M_1/M_2$ has turned the HJB equation into an identity. This proves that the numerator player's best response to the Kelly rule is to play the Kelly rule himself.
\par
We can repeat the above calculation, this time assuming that the numerator player's policy is $b(S,t,M_1,M_2)\equiv(\mu-r)/\sigma^2.$ Using $J$ again to denote the denominator player's (minimum) value function, we get the same HJB equation and boundary condition, except that $\underset{b\in\mathbb{R}}{\max}\,\,\{\bullet\}$ is replaced by $\underset{c\in\mathbb{R}}{\min}\,\,\{\bullet\}$. We again make the guess $J\equiv M_1/M_2$, which turns Player 2's HJB equation into the identity
\begin{equation}
\underset{c\in\mathbb{R}}{\min}\,\,(\mu-r-\sigma^2c)(b-c)=0.
\end{equation}The unique minimizer is $c=b=(\mu-r)/\sigma^2$. This completes the proof that the constant control policies $b(S,t,M_1,M_2)=c(S,t,M_1,M_2)=(\mu-r)/\sigma^2$ are best responses to each other. The proof for several stocks is similar, except that $(\mu-r-\sigma^2c)(b-c)$ is replaced by $(\mu-r\textbf{1}-\Sigma c)'(b-c)$.
\section{Conclusion}
For the continuous time two-person trading game whereby Player 1 seeks to maximize the expected ratio of his wealth to that of Player 2 (and Player 2 seeks to minimize this ratio), the unique Nash equilibrium is for both players to use the (possibly leveraged) Kelly rebalancing rule $b^*:=\Sigma^{-1}(\mu-r\textbf{1})$. More generally, we showed that the Kelly rule is the basis for the solution of a ``continuous-time investment $\phi$-game'' that is the analog of the discrete time version solved by Bell and Cover (1980, 1988). For practically \textit{any} criterion $\phi\{\textbf{W}_1V_t(b)/(\textbf{W}_2V_t(c))\}$ of short-term relative performance, the correct behavior is for both players to use the Kelly rule $b^*=c^*$ in conjunction with appropriate fair randomizations $(\textbf{W}_1^*,\textbf{W}_2^*)$ of the initial dollar. Thus, the continuous time Kelly rule (which is renowned for its optimal asymptotic growth rate) is desirable even for a trader whose goal is to perform well relative to other traders over very short periods of time.
|
1,314,259,996,708 | arxiv | \section{Introduction}
\label{intro}
Superconductivity in metals is a consequence of pairing between electrons \cite{Cooper56} and formation of a new macroscopic coherent state made of electron pairs \cite{Bardeen57}. The microscopic Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity explains all key experimental features of the superconducting state. Recent developments in trapping, cooling, controlling, and detecting of atoms allowed to investigate superfluidity in neutral fermionic systems, cf. reviews:~\cite{Jak05,Blo08}. In particular, a crossover between Bose-Einstein condensation (BCS) limit, where fermionic pairs overlap significantly, and the BEC limit, where tightly bound pairs form a coherent state of Bose-Einstein condensed bosons \cite{Leggett80,Noz85}, was demonstrated experimentally \cite{Reg03}. Since both fermionic as well as bosonic atoms are available in experiments it is natural now to investigate pairing between bosons and a coherent superfluid state of paired bosons.
Pairing and phase transitions of bosons with zero spin and with an attractive interaction was discussed in Ref.~[\onlinecite{Eva65}] in the context of superfluid helium four. It was found that collective excitations of a coherent condensed state of paired bosons can undergo another BEC type condensation into a one-particle condensed state, now known as the Evans-Rashid transition \cite{Eva65,Eva73,Sto94}. However, already in Ref.~[\onlinecite{Sto94}] it was found that both the homogeneous coherent paired phase and the homogeneous phase due to the Evans-Rashid transition are unstable against a mechanical collapse. I.e., the bosons with an attractive interaction tend to form clusters of particles. Moreover, extending the mean-field result of [\onlinecite{Eva65,Eva73,Sto94}] by the leading-order fluctuation contributions \cite{Mue99,Jeo02} or higher order corrections \cite{Man10} to the thermodynamic potential due to the interaction between particles does not stabilize homogeneous phases of bosons with pairing potential. It is observed in \cite{Zin08} that by confining bosons with paring interaction in a trap, which produces a gap between the ground and the excited states, one can protect the system from the mechanical instability. On the other hand, in Ref.~\cite{Sto08} a narrow region at finite temperatures on a phase diagram is found, where many-body effects can stabilize the homogeneous normal phase of bosons with pairing interaction.
In the present paper we employ a mean-field theory to solve a problem of pairing between bosons with spin $S=1$ moving on an optical lattice. We show that the coherent BCS type phase of paired bosons induced by attractive interaction in the singlet channel is stable povided that the interaction in a quintet channel is repulsive and sufficienlty strong. Since bosons with the nonzero hyperspin are available in experiments with cold atoms \cite{Jak05,Blo08} and both sign and strength of the interaction between them can be tuned by the optical Feshbach resonance \cite{Tha05} it is interesting to look for such a paired bosonic state in a laboratory.
The paper is organized as follows: In Section II we introduce the model. In Section III spinor pair condensed phases are classified by the matrix BCS type order parameter and the mean-field theory solution of the Hamiltonian is presented. Section IV is devoted to the symmetry classification for the phases characterized by the complex matrix order parameter following the standard approaches \cite{Vol86,Vol00,Yip07}. It should be noted that the finite spin of bosons with repulsive interaction in all channels \cite{Ohm98,Ho98} leads to many nontrivial ground and excited states of the spinor condensates \cite{Blo08,Ued10}. They include topologically nontrivial phases \cite{Koa00,Ued02,Yip07}, skyrmion excitations \cite{Kha01}, and even nonabelian vortices \cite{Sem07}. The symmetry classification presented in this Section may help in the future for detailed analysis of non-trivial excitations in paired phases of bosons. In Section V we present numerical solutions to the mean-field equations and in Section VI stability of particular phases is discussed. Conclusions are in Section VII and details on our derivations are presented in Appendices.
\section{The model}
\label{model}
We consider the Hubbard model for spin-one bosons,
which are trapped on an optical lattice. The grand canonical
Hamiltonian \cite{Fis89,Oos01}
\begin{equation}
H=H^0+H^{\rm int}-\mu N\label{hubbard}
\end{equation}
contains the kinetic part $H^0$ and the interaction part $H^{\rm int}$.
We introduce the chemical potential $\mu$ to fix the average total number
of bosons in the lattice.
The kinetic part is
\begin{equation}
H^0=-t\sum_{\langle i,j\rangle \sigma} b_{i\sigma}^{\dagger} b_{j\sigma}, \label{kinetic}
\end{equation}
where $b_{i\sigma}$ ($b_{i\sigma}^{\dagger}$) is an annihilation (creation) operator of
a boson at the lattice site $i$ with spin $\sigma=-1,0,\;{\rm or} \;1$,
and $\langle i,j\rangle$ denotes the summation over nearest
neighbor sites. We also introduce here the hopping integral $t$. We
absorb a constant single site occupation energy into the definition of $\mu$.
The effective parameter of our model
$t$ can be derived from the microscopic details of the optical lattice
\cite{Jak05,Maz06}, assuming that the lattice
site orbitals correspond to localized Wannier functions with one level
per site. We neglect the harmonic trap confinement in the following.
The interaction part $H^{\rm int}$ is constructed under an assumption
that the total spin of the system is conserved and that the interaction amplitude
is local \cite{Pet00,Ued10}.
While the first requirement
is natural due to general conservation laws,
the second assumption is justified for cold
atoms due to their neutrality and short-range character of interacting forces.
The total spin $S$ of two interacting spin-one bosons
attains three possible values $S=0,1,2$.
Because of the bosonic symmetry of the wave functions, in the
presence of the local interaction only $S=0$ and $S=2$ terms contribute
in $H^{\rm int}$.
The resulting
interaction amplitudes $U_S$ are proportional to the scattering lengths $a_S$ for each $S$ channel \cite{Jak05}.
Following Ref.\ \cite{Ima03} we write $H^{\rm int}$ in terms of the number operator
$n_i$ and spin operator ${\bf S}_i$ at the lattice site $i$:
\begin{equation}
H^{\rm int} = \frac{g_n}{2}\sum_in_i(n_i - 1) + \frac{g_s}{2} \sum_i(\mathbf{S}^2_i - 2 n_i), \label{hint1}
\end{equation}
were $g_n= (2U_2 + U_0)/3$ and $g_s= (U_2 - U_0)/3$.
Both the hopping integral $t$ and the
interaction strength $U_S$ can be tuned to
become of comparable magnitude by manipulating the laser light producing the
optical lattice.
In this paper we are interested in the effects of
attractive interaction giving rise to pairing between spin-one bosons.
We introduce an auxiliary annihilation (and creation)
operator for a Cooper pair of bosons
$B^{S,M}_{ij}= \sum_{\sigma\sigma'} C^{S,M}_{\sigma\sigma'} b_{i \sigma} b_{j \sigma'}$,
where $C^{S,M}_{\sigma\sigma'}$ is the Clebsch-Gordan coefficient for the total spin $S$
and with spin projection $M$.
The explicit
form of the pair operators can be found in Ref. \cite{Ued10}.
The Hamiltonian \eqref{hint1} takes a new, compact form
\begin{eqnarray}
H^{\rm int}=\sum_i \left(U_0 B^{0,0\dagger}_{ii} B^{0,0}_{ii}+ U_2 \sum_{M=-2}^2 B^{2,M\dagger}_{ii} B^{2,M}_{ii} \right),\nonumber \\ \label{hint2}
\end{eqnarray}
which is more appropriate here since it shows directly all structures of bosonic pair
correlations and hints to possible order parameters.
In the next Section we solve the model \eqref{hubbard} with \eqref{kinetic} and \eqref{hint1}
within a Hartree-Fock mean-field approximation (MFA) and discuss
possible condensed phases of bosonic Cooper pairs.
\section{Mean-field approximation}
\label{solution}
The kinetic part \eqref{kinetic} of our model Hamiltonian
is diagonal in the momentum representation
\begin{equation}
H^0=\sum_{k \sigma} \xi_k b_{k\sigma}^{\dagger} b_{k\sigma},
\label{diag}
\end{equation}
where $b_{k\sigma}^+$ ($b_{k\sigma}$) is the creation (annihilation) operator for a
particle with the lattice momentum ${\mathbf k}$ and the single particle kinetic energy
is denoted by $\xi_k$. We keep a general form of the dispersion relation $\xi_k$
in our derivation of the self-consistent equations and use a
specific model later in Section \ref{singlet}.
The construction of the appropriate mean-field Hamiltonian can
be done within a textbook rule \cite{Bru04}
by splitting two-body operators into
paring operators and their
non-vanishing expectation values. This approximation consequently
neglects fluctuations.
The form of our model Hamiltonian \eqref{hint2}
suggests the following choice for the pair expectation value
\begin{equation}
\Lambda^{S,M}=\langle B^{S,M}_{ii}\rangle,
\label{lambdaSM}
\end{equation}
which can also be expressed as
$\Lambda^{S,M}=\sum_{\sigma\sigma'} C^{S,M}_{\sigma\sigma'} \Lambda_{\sigma\sigma'}$,
with $\Lambda_{\sigma\sigma'}= \tfrac{1}{N_s} \sum_k \langle b_{k \sigma} b_{-k \sigma'}\rangle$.
The expectation values are taken at thermal equilibrium with the inverse temperature $\beta$ and
$N_s$ denotes the number of lattice sites.
We also allow for nonzero normal density expectation values by defining
an average site occupation matrix
$n_{\sigma\sigma'}= \tfrac{1}{N_s} \sum_k \langle b_{k \sigma}^\dagger b_{k \sigma'}\rangle$. Throughout the paper we deal with quantities
described by $3\times 3$ matrices in the spin index,
such as $n_{\sigma\sigma'}$ or $\Lambda_{\sigma\sigma'}$.
Therefore, we introduce here a more compact matrix notation $\hat n$ and $\hat \Lambda$ for those quantities.
Within MFA \cite{Eva65,Bru04} the Hamiltonian \eqref{hint1} takes the following form
\begin{equation}
H^{\mathrm{int}}_{\mathrm{MF}} =
\sum_{k\sigma\sigma'}\left( b^\dagger_{k \sigma} w_{\sigma\sigma'} b_{k \sigma'} +
\tfrac{1}{2}(b^\dagger_{k\sigma} \Delta_{\sigma\sigma'} b^\dagger_{-k\sigma'} +
\mathrm{h.c.})\right)-E_0N_s.\label{hintMF}
\end{equation}
In the above equation a matrix valued order parameter appears
\begin{equation}
\hat \Delta = U_0 \hat C^{0,0} \Lambda^{0,0} + U_2\sum^2_{M=-2} \hat C^{2,M} \Lambda^{2,M},
\label{deldef}
\end{equation}
which describes spontaneous symmetry breaking due to BCS-type paring.
The quantity
\begin{equation}
\hat w = 2 \left(U_0 \hat C^{0,0} \hat n \hat C^{0,0} + U_2 \sum^2_{M=-2} \hat C^{2,M} \hat n \hat C^{2,M} \right)\label{wdef}
\end{equation}
describes the effective Hartee-Fock potential. Note that the Clebsch coefficients
for fixed $S,M$ are also represented by a $3\times 3$ matrix $\hat C^{S,M}$.
We keep the additive constant
\begin{eqnarray}
E_0&=&
\tfrac{U_0 - U_2}{2}\left[2\mathrm{Tr}(\hat n^T\hat C^{0,0}\hat n\hat C^{0,0}) +
|\mathrm{Tr}(\hat\Lambda \hat C^{0,0})|^2\right]\nonumber\\
&&+\tfrac{U_2}{2}\left[\mathrm{Tr} (\hat n^2 + \hat\Lambda^\dagger \hat\Lambda ) + n^2\right],
\label{ezero}
\end{eqnarray}
which is necessary in the discussion of phase stabilities
presented in the Section \ref{stability}.
We finally arrive at the mean field self-consistent equations by calculating
the normal $n_{\sigma\sigma'}$ and anomalous $\Lambda_{\sigma\sigma'}$ averages
in the grand canonical ensemble with the quadratic interaction Hamiltonian \eqref{hintMF}.
This procedure is equivalent \cite{Bru04} to the requirement of attaining a minimum
of the free energy with the Hamiltonian \eqref{hintMF}, when $\hat \Delta$ and $\hat w$
are variational parameters. The technical details
of the derivation are given in the Appendix \ref{diagapp}.
Here we present the final result obtained from \eqref{aver} and \eqref{averM}
\begin{equation}
\begin{pmatrix}
\openone + \hat n^* & -\hat \Lambda \\
\hat \Lambda^* & -\hat n
\end{pmatrix}
= \tfrac{1}{N_s}\sum_{k}\left(1 - e^{-\beta M_k}\right)^{-1},
\label{selfcons}
\end{equation}
where $M_k$ is a Bogoliubov--de Gennes matrix
\begin{equation}
M_k = \begin{pmatrix}
(\xi_k-\mu) \openone + \hat w & \hat \Delta \\ -\hat \Delta^* & - (\xi_k-\mu) \openone - \hat w^*
\end{pmatrix}.
\label{defM}
\end{equation}
For the purpose of solving the self-consistency equations
\eqref{selfcons} and \eqref{defM} in practice it is convenient to simplify the expression for
$\hat \Delta$ given in \eqref{deldef} and for $\hat w$ in \eqref{wdef}. With the help of general algebraic identities \cite{ident} applied to the matrices $\hat \Lambda$ and $\hat n$ we arrive at
\begin{equation}
\hat \Delta = (U_0-U_2)\hat C^{0,0}\mathrm{Tr}(\hat C^{0,0}\hat \Lambda)+U_2\hat \Lambda,
\label{delsimpl}
\end{equation}
\begin{equation}
\hat w = 2(U_0-U_2)\hat C^{0,0}\hat n\hat C^{0,0}+U_2(\hat n^T+n \hat \openone),
\label{wsimpl}
\end{equation}
where all $\hat C^{S,M}$ have been eliminated except of $\hat C^{0,0}$.
\section{Symmetry classification of ordered states}
\label{symmetry}
The accepted strategy, which allows to classify the solutions for the matrix order
parameter from the self-consistent equations, relays on symmetry considerations \cite{Vol00}.
The symmetry arguments alone allow to identify stationary states of the free energy,
as it was done recently
for the spinor condensates \cite{Ho99,Yip07}. Here we need not only to identify
the symmetry classified states, but we also want to investigate the
phase diagram as a function of the interaction parameters.
Therefore, we have to compare
free energy of symmetry classified phases to find the minimal one.
In the investigation
of superfluid $^3$He it was observed \cite{Vol86}, but not strictly proven,
that the phase possessing the
highest remaining symmetry
corresponds indeed to a local, and very often to the global free energy minimum.
We follow the standard symmetry classification approach.
We start by determining the highest allowed symmetry phase, and then we consider
the solutions with a lower symmetry. For the sake of completeness of
the presentation we give below a more detailed account of this derivation.
We will use the classification introduced in this Section
to determine numerically the phase diagram,
by solving the non-linear mean field equations within
a given symmetry class.
The full symmetry of our system (in a generic case $U_2\neq U_0$)
involves the gauge and the spin rotation symmetry, so it is $U(1)\times SO(3)$.
This symmetry is smaller then in the superfluid $^3$He case, which has
$U(1)\times SO(3)\times SO(3)$ symmetry group.
The possibility of breaking the gauge invariance is crucial
in our search of the phases with pairing. Symmetry of our
system allows
the gauge symmetry to be broken not only independently,
but also in a combination with the spin symmetry operation.
Thus we have to consider also the possibility of gauge-spin
symmetry breaking, similar to superfluid $^3$He.
\subsection{Symmetry transformations}
We start the discussion of symmetry with the global $U(1)$ gauge symmetry transformation
$b_{k \sigma} \rightarrow e^{i\psi} b_{k \sigma}$,
where $\psi$ is a constant phase.
The single site occupation
matrix $\hat n$ is gauge invariant, so from \eqref{wdef}
it follows that $\hat w$ is gauge invariant as well. The pair expectation
value transforms as $\hat \Lambda \rightarrow e^{2i\psi} \hat \Lambda$, which
substituted to \eqref{deldef}
leads to the order parameter transformation $\hat \Delta \rightarrow e^{2i\psi} \hat \Delta$.
It is easy to check that this gauge transformation is a symmetry of our mean field
equation \eqref{selfcons} with \eqref{defM}.
The spin rotation SO(3) is described by a unitary matrix $\hat r$, which acts as follows:
$b_{k\sigma} \rightarrow \sum_{\sigma'} r_{\sigma\sigma'} b_{k\sigma'}$. The general
rotation matrix $\hat r$ can be parameterized by three Euler angles of elementary rotations
generated by three components of the spin-one operator. From the definitions
of the averages $\hat n$ and $\hat \Lambda$ we obtain the
transformation rules
\begin{equation}
\hat n \rightarrow \hat r^* \hat n \hat r^T,\qquad
\hat \Lambda \rightarrow \hat r \hat \Lambda \hat r^T.
\label{rotn}
\end{equation}
The above transformations substituted to \eqref{wsimpl} and \eqref{delsimpl}
give the following spin rotation of the effective potential
and the pairing order parameter:
\begin{equation}
\hat w \rightarrow \hat r \hat w \hat r^\dagger,\qquad
\hat \Delta \hat \eta \rightarrow \hat r \hat \Delta \hat \eta \hat r^\dagger,
\label{rotDel}
\end{equation}
where $\hat \eta=\sqrt{3}\hat C^{0,0}$. We have used
the identity $\hat \eta \hat r^* \hat \eta =\hat r$, which
follows from the explicit form $\eta_{\sigma\sigma'}=-(-1)^{\sigma}\delta_{\sigma,-\sigma'}$.
One can check that the right-hand side of the mean field
equation Eq.\ \eqref{selfcons} consequently transforms as
$M_k\rightarrow R M_k R^\dagger$, with
a unitary $R=\mathrm{diag}(\hat r, \hat r^*)$, where $\mathrm{diag}$
stands for a block diagonal matrix.
The left-hand side of this equation transforms upon \eqref{rotn} in
the same manner, thus verifying the $SO(3)$ spin rotation
symmetry of our mean-field formulation.
\subsection{Continuous symmetry phases}
\label{chapcont}
{\em No broken symmetry.} The requirement of invariance upon the full symmetry
transformation $U(1)\times SO(3)$ applied
to $\hat w$ and $\hat \Delta$ gives as the only solution
$\hat w=w \hat \openone$ and $\hat \Delta=0$,
where $w=\frac{2}{9}(U_0+5U_2) n $. The single site density
is $n$ and the occupation matrix
reads $\hat n=\frac{1}{3}n\hat \openone $.
This describes a free boson gas with a renormalized chemical
potential due to the Hartree-Fock treatment of the contact interaction.
{\em Singlet phase.} The highest possible symmetry phase with non-zero pairing amplitude
arises when we break the $U(1)$ gauge symmetry, but leave the
spin rotation symmetry. We derive from the invariance condition
\begin{equation}
\hat \Delta \hat \eta = \hat r \hat \Delta \hat \eta \hat r^\dagger
\end{equation}
that for a general $\hat r$ the order parameter has to be
$\hat \Delta \hat \eta =\Delta \hat \openone $ with some complex $\Delta$ and
$\hat w = w \hat \openone$, with $w$ the same as in the free case discussed above.
Going back to Eq.\ \eqref{deldef}
we find that the expectation values of the bosonic pair operators $\Lambda^{S,M}$
are non-zero only for $S=0$ in this $SO(3)$ symmetric phase. We will
call this
phase the \emph{singlet phase} as pairing happens only in the singlet channel,
with the finite order parameter $\hat \Delta = U_0\hat C^{0,0}\Lambda^{0,0}$.
{\em Quintet phase.} We search now for paired phases, which allow for
non-vanishing $S=2$ (i.e. quintet) components of the order parameter. The simplest way to achieve this is by
lowering the spin rotation $SO(3)$ symmetry to an axial $U(1)$ symmetry.
We choose an arbitrary quantization axis and express
the spin rotations around this axis as $\hat r(\varphi)=e^{i\varphi \hat S_z} $,
with some
angle $\varphi$ and $\hat S_z=\hat C^{2,2}- \hat C^{2,-2}$.
We require now a more general spin-gauge invariance
condition for the order parameter
\begin{equation}
\hat \Delta \hat \eta = e^{2i\psi} \hat r(\varphi) \hat \Delta \hat \eta \hat r(\varphi)^\dagger,
\label{spingauge}
\end{equation}
where the gauge symmetry breaking phase $\psi$ can now
depend on the spin rotation angle $\varphi$. We
obtain three different solutions, which are presented below:
\begin{subequations}
\label{all}
\begin{align}
U(1)_{S_z-\varphi}:\quad \psi&=-\varphi & \hat \Delta &= \Delta \hat C^{2,2}, \label{phi}\\
U(1)_{S_z-\frac{\varphi}{2}}:\quad \psi&=-\varphi/2& \hat \Delta &= \Delta \hat C^{2,1}, \label{phi2}\\
U(1)_{S_z}:\quad \psi&=0& \hat \Delta &= \Delta \hat C^{0,0} + \Delta' \hat C^{2,0}.
\label{zero}
\end{align}
\end{subequations}
We follow the
notation of Ref.~\cite{Vol86} to label the above spin-rotation
breaking axial phases.
Remaining solutions with $+\varphi$, and $+\varphi/2$
can be obtained by changing the direction of the quantization
axis, so they do not describe a different symmetry phase.
The only non-zero pair expectation amplitude
is $\Lambda^{2,2}$ for the $U(1)_{S_z-\varphi}$ phase
and $\Lambda^{2,1}$ for $U(1)_{S_z-\frac{\varphi}{2}}$, which follows from
the comparison of $\hat \Delta$ definition in \eqref{deldef} with the result \eqref{all}.
In these two axial phases
the symmetry allows for pairing only in the \emph{quintet channel}. The remaining $U(1)_{S_z}$ phase
has a mixed singlet--quintet pairing order parameter, which has to be parameterized
by two (complex) numbers $\Delta$ and $\Delta'$.
The Hartree--Fock potential $\hat w$ in all the axial phases is
restricted by the symmetry to
be diagonal. This brings a possibility of magnetic order,
coexisting with the pairing,
marked by spin rotation symmetry breaking in the spin dependent site occupation.
\subsection{Discrete symmetry phase}
Within only $3\times 3$
matrix representations one cannot construct the icosahedral or octahedral
symmetry, without allowing for generation of all possible rotations.
The biggest non-trivial discrete symmetry is thus $T$ -- the symmetry group
of tetrahedron without reflections. The
set of group generators can be
explicitly expressed as $\{\openone, e^{i\frac{2\pi}{3} \hat S_z},
e^{i\pi(\hat S_z+\sqrt{2}\hat S_x)/\sqrt{3})}\}$,
where we use $3\times 3$ matrix representation of spin one with
$\hat S_x= \hat C^{2,1} + \hat C^{2,-1}$.
Substituting these generators for $\hat r$ in the
invariance condition \eqref{spingauge} we obtain as the only
solution $\psi=\tfrac{2\pi}{3}$ and
$\hat \Delta=\Delta (\hat C^{2,2}+\sqrt{2} \hat C^{2,-1})$.
\section{Mean-field solution for singlet phase}
\label{singlet}
The singlet phase, introduced from the symmetry arguments
in Section \ref{chapcont} is our natural candidate for a physically
attainable phase.
The system in the singlet phase has a maximal remaining
symmetry of all the phases with non-zero pairing. The singlet phase is
unitary, meaning that the matrix order parameter $\hat \Delta$
is proportional to a unitary matrix.
Stable phases of liquid $^3$He were previously found to be
unitary \cite{Vol86} as well.
Simple form of the order parameter $\hat \Delta=\Delta \hat \eta$
in the singlet phase leads to an identity $M_k^2=e_k^2\openone_{6\times 6}$,
where the Bogoliubov--de Gennes matrix $M_k$ was defined in \eqref{defM}.
The quasi-particle excitation energy
\begin{equation}
e_k=\sqrt{(\xi_k-\mu+w)^2-|\Delta|^2}
\label{spec}
\end{equation}
is a triple degenerate eigenvalue of $M_k$ as defined
in Eq.\ \eqref{diagM}.
We can now directly calculate the generalized occupation factor
in the mean field equation \eqref{selfcons}
\begin{equation}
\left(1 - e^{-\beta M_k}\right)^{-1}=f(e_k)M_k+\tfrac{1}{2},
\label{cosh}
\end{equation}
with $f(e_k)=\frac{\coth (\beta e_k/2)}{2 e_k}$. We recall that $M_k$
depends on $\hat w$ and $\hat \Delta$, which are related to $\hat n$ and $\hat \Lambda$:
\begin{subequations}
\label{w_del}
\begin{align}
\hat w&=\tfrac{2}{3}(U_0+5U_2)\hat n,\\
\hat \Delta&=U_0\hat \Lambda,
\end{align}
\end{subequations}
in the singlet phase, as obtained in section \ref{chapcont}. We substitute \eqref{w_del} into $M_k$ in
\eqref{cosh} and then equate to
the left-hand side of Eq.\ \eqref{selfcons}. The resulting self-consistent
equations in the singlet phase take a simple form
\begin{subequations}
\label{selfbcs}
\begin{align}
\frac{n}{3}&= \tfrac{1}{N_s} \sum_{k} \left( \sqrt{e_k^2+|\Delta|^2}f(e_k) - \tfrac{1}{2} \right), \\
-\frac{1}{U_0}&= \tfrac{1}{N_s} \sum_{k}f(e_k).
\end{align}
\end{subequations}
The first equation provides a relation between the average occupation $n$
and the chemical potential $\mu$, while the second
guarantees a non-zero paring amplitude. The form of this second
equation is similar to the gap equation in the BCS theory, but with
a different function $f(e_k)$ due to boson statistics of condensating
quasiparticles.
Interestingly, these equations are formally equivalent to the one obtained
in the case of scalar attracting bosons in Ref. \cite{Sto94}. The only difference
is that the optical lattice provides a natural ultraviolet cutoff in our model.
The existence of the BCS type singlet solutions depends only on the strength $U_0$ of attraction
in the singlet channel and is insensitive to scattering in the quintet
channel. We will show in the next Section that the singlet paired phase of attracting
bosons can be stabilized by a repulsive quintet interaction. This is in a
marked difference to the scalar case, where the system always undergoes
a mechanical collapse before reaching Evans-Rashid transition \cite{Sto94}.
We note that the quasiparticles in BCS type bosonic condensate may undergo
a statistical (Bose-Einstein) condensation \cite{Eva65,Sto94}. The transition
occurs when the excitation spectrum in Eq. \eqref{spec} becomes gapless \cite{Cao07}.
The singular condition $e_{k=0}=0$ can be satisfied in a thermodynamic limit
for
\begin{equation}
|\Delta|=\xi_{k=0}-\mu+w,
\label{const}
\end{equation}
which fixes the chemical potential similarly to a standard BEC.
We separate the $k=0$ terms to obtain
\begin{equation}
\tfrac{1}{N_s}\sum_k f(e_k) = \frac{n_{\text{BEC}}}{3|\Delta|}+
\tfrac{1}{N_s}\sum_{k\neq 0} f(e_k),
\label{decomp}
\end{equation}
where we introduce $n_{\text{BEC}}$ --
a finite average density for quasi-particles with $k=0$ only.
The self-consistent equations for the BEC quasiparticle phase follow
\begin{subequations}
\label{selfbec}
\begin{align}
\frac{n}{3}&= \frac{n_{\text{BEC}}}{3}+\tfrac{1}{N_s} \sum_{k\neq 0} \left( \sqrt{e_k^2+|\Delta|^2}f(e_k) - \tfrac{1}{2} \right), \\
-\frac{1}{U_0}&= \frac{n_{\text{BEC}}}{3|\Delta|} + \tfrac{1}{N_s} \sum_{k\neq 0}f(e_k),
\end{align}
\end{subequations}
when we substitute the
decomposition \eqref{decomp} into \eqref{selfbcs}.
The chemical potential $\mu$ is fixed by \eqref{const}, so $n_\text{BEC}$
becomes a new thermodynamic parameter, which we have to determine.
Therefore, we distinguish two different phases: i) BCS phase where $n_{\rm BEC}=0$ and $\Delta \neq 0$,
and ii) BEC phase where $n_{\rm BEC}\neq 0$ and $\Delta \neq 0$.
The condition for the BCS/BEC borderline is obviously $n_\text{BEC}\rightarrow 0$, it is when
\eqref{selfbec} reduces to \eqref{selfbcs}.
We note that the transition between BCS and BEC in the boson case cannot be interpreted as being a counterpart of the BCS-BEC crossover known in the fermionic condensed systems \cite{Leggett80,Noz85}.
\begin{figure}[tb]
\centerline{%
\includegraphics[width= 0.85\linewidth]{fig_stab.eps}}%
\caption{The diagram of the singlet phase
presented
in the density $n$ -- temperature $T$ coordinates at
$U_0 = -0.33W$.
The solid line denotes the boarder of BCS type
phase with pairing (`NP' stands for no--pairing, normal phase),
the dashed line marks the borderline of BEC quasiparticle
condensate.
For $U_2/|U_0|=0.59$ both phases in (c) (blank) region fulfill
the standard thermodynamic stability conditions,
but are unstable in (a) (grey) and (b) (light-grey) regions. For $U_2/|U_0|=0.64$
the regions (b) and (c) are thermodynamically stable, (a) region is unstable.
}
\label{phaseweak}
\end{figure}
\begin{figure}[tb]
\centerline{%
\includegraphics[width= 0.85\linewidth]{fig2_stab.eps}}%
\caption{The diagram of the singlet phase for strong interaction
$U_0 = -W$ in the density $n$ -- temperature $T$ coordinates.
The zero temperature
critical density is $n^*\approx 0.15$.
The solid line denotes the border of BCS type
paired phase,
the dashed line marks the borderline of BEC quasiparticle
condensate. Stability regions (a), (b) and (c) are
defined in the same way as in Fig.\ \ref{phaseweak}.
}
\label{phasestrong}
\end{figure}
In order to solve numerically Eqs.\ \eqref{selfbcs} and
\eqref{selfbec}
we only need to provide the density of states
$\rho(E)=\sum_k\delta(E-\xi_k)$
in the optical lattice.
We assume a simple elliptic model for the density
$\rho(E)=\frac{8N_s}{\pi W^2}\sqrt{(W/2)^2-E^2}$,
where $W=(32\pi)^{2/3}t$ is chosen to fit a low-energy density profile obtained
from the dispersion relation $\xi_k$.
We are able to gain some analytical insight into the solution for this BEC singlet
phase. The integrals appearing in \eqref{selfbec} can be performed
in the limit
$T\rightarrow 0$ leading to
\begin{subequations}
\label{tzero}
\begin{align}
n &= n_{\text{BEC}}+ \tfrac{3}{\pi}\left[(1 - \omega^2)\arctan{\tfrac{1}{\sqrt{\omega}}} + \sqrt{\omega}(1+ \omega)\right] - \tfrac{3}{2},\\
-\frac{W}{U_0} &= \frac{2n_{\text{BEC}}}{3\omega} + \tfrac{4}{\pi}\left[(1 + \omega)\arctan{ \tfrac{1}{\sqrt{\omega}} } - \sqrt{\omega}\right],
\label{omtzero}
\end{align}
\end{subequations}
where $\omega=\frac{2|\Delta|}{W}$. These two non-linear algebraic equations
determine $n_{\text{BEC}}$ and $|\Delta|/W$ for a given $n$. We define a critical
density $n^*$ by setting $n_{\text{BEC}}=0$ in \eqref{tzero}. The
BEC condensate solution with
finite $n_{\text{BEC}}$ exists for densities larger than $n^*$ at $T=0$.
For weak interactions
$|U_0|/W<\frac{1}{2}$ we find only $n^*=0$ solution,
which means that there is only the BEC phase at $T=0$.
For stronger interactions $|U_0|/W>\frac{1}{2}$ we find a region
of BCS phase extending down to zero temperature.
We present the resulting finite temperature phase diagram in the weak
interaction case
in Fig.\ \ref{phaseweak} for a fixed attractive interaction $U_0=-0.33W$.
The situation with strong interaction is illustrated in Fig.\ \ref{phasestrong} for $U_0=-W$.
Additionally, one can obtain an analytic solution $n^*=\frac{32}{3\pi^2}|U_0|/W$
for $|U_0|/W\gg 1$.
\section{Stability}
\label{stability}
We discuss below the standard thermodynamic stability conditions expressed by:
i) positivity of pressure
$p$, ii) positivity of constant volume specific heat $c_V$, and iii)
positivity of isothermic compressibility
$\kappa_T$ (for the calculation see Appendix B).
{\it Singlet phase.}
We find that $c_V$ is
positive in the singlet phase and is
independent of $U_2$. The pressure $p$ and the
inverse compressibility $\kappa_T^{-1}$ have a following linear dependence
on $U_2$:
\begin{subequations}
\label{pres}
\begin{align}
p(U_0,U_2)&=p(U_0)+U_2\tfrac{5n^2}{9a^3},\\
\kappa_T^{-1}(U_0,U_2)&=\kappa_T^{-1}(U_0)+ U_2\tfrac{10n^2}{9a^3}.
\end{align}
\end{subequations}
This means that for any point on the phase diagram in Fig.\ \ref{phaseweak}
or Fig.\ \ref{phasestrong}
we can find $U_2$ large enough to stabilize the BCS or BEC singlet phase.
The shadowed regions in these figures exemplify stability
for $|U_2|/U_0=0.59$ and $0.64$, respectively.
\begin{figure}[tb]
\centerline{%
\includegraphics[width= 0.85\linewidth]{fig_cv_v2.eps}}%
\caption{The specific heat versus temperature for a fixed
density $n=1.5$. The solid line illustrates the sequence of
transitions for interaction strength
$U_0 = -W$, the dashed line is for $U_0=-0.33W$.
}
\label{cvfig}
\end{figure}
We illustrate the nature of subsequent transition by plotting
the specific heat $c_V$ in Fig. \ref{cvfig}. We show the specific
heat dependence on the temperature $T$ at fixed density $n=1.5$,
which is representative both for strong and weak
attraction and contains all three phases: non-pairing, BCS-like
and the BEC quasiparticle condensate.
The plot does not depend on the strength $U_2$, provided
it is strong enough to stabilize the phases.
The specific heat exhibits a jump at the onset of pairing
(marked by a triangle in Fig. \ref{cvfig}),
indicating that the corresponding transition is of second order.
The transition to BEC quasiparticle condensate is contiunuos
with a cusp in the specific heat dependence (marked by a square),
the behaviour being known in the usual Bogoliubov theory of BEC condensation \cite{Sto99}.
We inspect further the details of stability lines shown in Fig. \ref{phasestrong}.
We find generically two stable phases separated by an unstable
one for the system at fixed temperature. The system will then
have a tendency to spontaneously separate into the
dense and dilute phases. We make this statement qualitative
by considering the thermodynamic spinodal decomposition \cite{Spino95}
into the dense BEC and dilute BCS or normal phase.
The results are presented in Fig. \ref{spinodalfig}. The spinodal
stability lines are redrawn from Fig. \ref{phasestrong}, region (b). The line
marked by triangles is given by the compressibility condition
$\kappa_T^{-1}=0$, while the one marked by the squares
is given by the pressure $p=0$.
We find that the region denoted by light gray shadowing corresponds to
a metastable state, which undergoes the spinodal dcomposition. The
regions denoted BCS or BEC above the solid binodal line are stable
against such a thermal fluctuation.
This result indicates that the thermal fluctuations around the mean field solution
do not change qualitatively our phase diagram, they are of importance at the vicinity of the
stability borderlines. The role of quantum fluctuations is left
for future research.
\begin{figure}[tb]
\centerline{%
\includegraphics[width= 0.85\linewidth]{fig4_v1.eps}}%
\caption{
The phase diagram of the singlet phase with the same parameters as in Fig.~\ref{phasestrong}.
The binodal line is indicated by a thin solid curve. The system in both shadowed regions
(d) within BCS phase and (e) within BEC phase undergoes a spinodal decomposition.
The BCS phase above (d) is stable, same as BEC to the left of (e) region.
}
\label{spinodalfig}
\end{figure}
We find that the most unstable point of our diagram is located at the
BCS/BEC borderline (marked by the thick dashed line in Fig. \ref{spinodalfig}).
We can solve the following conditions $p>0$ and $\partial p/\partial n>0$
at zero temperature on this corssover line, corresponding to
the density $n^*$ (compare Fig. \ref{phasestrong}). We thus find $U_2^c$ -- the
critical strength of repulsion in the quintet channel at which the whole
BCS and BEC phases become stable. In the weak interaction regime
\begin{equation}
U_2^c=\frac{|U_0|}{10}\left(2+\frac{3}{1-|U_0|/2W}\right),
\end{equation}
which is valid for $|U_0|/W<\frac{1}{2}$, while in the strong interaction
regime
\begin{equation}
U_2^c=\frac{|U_0|}{20}\left(1+\frac{3\,\mathrm{arctan}(1/\sqrt{\omega})}{\sqrt{\omega}-\omega\mathrm{arctan}(1/\sqrt{\omega})}\right),
\end{equation}
valid for $|U_0|/W>\frac{1}{2}$ and with $\omega$ calculated from Eq.~\eqref{omtzero}
at $n_{\mathrm{BEC}}=0$. For the interaction strength $U_2>U_2^c$ there is
always a non-collapsing phase, which can be either normal,
BCS or BEC homogeneous, or an inhomogeneous mixture of the dilute and dense phases.
{\it Other symetry phases.} We have carried out a detailed, both analytical and numerical study of
stability \cite{Pel10} for the other phases. We find that both magnetic phases $U(1)_{S_z-\varphi}$
and $U(1)_{S_z-\varphi/2}$ are mechanically unstable. The tetrahedral T-phase with
the quintet pairing occurs for $U_2<0$ and $U_0>0$.
It can be made thermodynamically stable by increasing the
singlet repulsion $U_0$. We find however, that the ferromagnetic
phase $U(1)_{S_z-\varphi}$ has lower free energy in the parameter regions,
where T-phase becomes stable. Morover, we find that an infinitezimal
$\epsilon>0$ distortion of T-phase order paremeter by a magnetic contribution
$\Delta((1+\epsilon)\hat C^{2,2}+\sqrt{2}\hat C^{2,-1})$ leads to a
lower free energy. We conclude that the T-phase is not even metastable, as it
always corresponds to a saddle point of free energy.
\section{Conclusions and outlook}
\label{conclusions}
In summary, we applied the Hartree-Fock mean-field approximation to solve a problem of pairing between bosons with spin $S=1$ moving on optical lattices. The order parameter describing such paired bosons has a matrix form. Detailed classiffiation of possible solutions according to their symmetries was presented. In particular, we found that the self-consistent equations for the $SO(3)$ symmetric phase have the same form as those for the scalar bosons. We showed that the coherent BCS type phase of paired bosons induced by attractive interaction in the singlet channel is stable provided that the interaction in a quintet channel is repulsive. This finding might be usefull in experiments to stabilize bosons with attractive interaction against mechanical collapse.
The analyzed problem might be extended in the future in different ways. For example, it would be interesting to include local quantum correlation beyond the static Hartree-Fock approximation by using the bosonic dynamical mean-field theory developed recently \cite{Byczuk08}. Another line of research is to investigate inhomogeneous excited states of bosons with $S=1$ in the BCS or BEC phases, i.e. there should be generalized vortex states in the condensed phases because of the high remaining symmetry in the system. In the boson gas with hyper spin $S=2$, where a very reach variety of spinor BEC for repulsive interactions have been proposed \cite{Die06}, we expect stabilization of at least some of many symmetry allowed phases for attractive interactions.
\acknowledgments
The work of KB is supported by the grant N N202 103138 of Polish Ministry of Science and Education and, in part, by the grant the TRR80 of the Deutsche Forschungsgemeinschaft.
|
1,314,259,996,709 | arxiv | \section{Introduction}
Training machine learning models have always required an increasing amount of data for the increasing size of the architecture. This phenomenon has been exemplified in the NLP community with the recent explosion in use of the transformer \cite{vaswani2017attention}. Public datasets reflect this need for larger and larger data requirements in the recently released the 800GB Pile dataset \cite{gao2020pile}. In the absence of readily available data, an unsupervised way of text augmentation is needed.
Back translation is the unsupervised process of translating text from English to another language and then back to English. This text augmentation technique allows a variety of outputs for any input. A strategy might be to translate from English to many other languages and then back to English to create the most broad understanding of the input text. In this scenario, we envision that a system of back translations can provide transformers with the generalized data that they need to train larger and larger models.
To employ a complete text augmentation training strategy we need to understand the full effect of back translation. In this paper, we will investigate how each language's back translation effects various NLP metrics. We hope to decipher how and why some languages might be a better candidate for back translating to over any other language. We employ Google Translate's current Neural Machine Translator (NMT) \cite{wu2016google} to receive over 108 languages translations.
\subsection{Background}
Using text augmentation is a unsupervised way to expand your text training data. There are various text augmentations \cite{chaudhary2020nlpaugment}, for example the nlpaug package \cite{ma2019nlpaug} summarizes them in 5 categories: insert, substitute, swap, delete or crop. Some examples from the nlpaug pacakge include: RandomWordAug (Apply augmentation randomly), AntonymAug (Substitute opposite meaning word according to WordNet antonym), SplitAug (Split one word to two words randomly), WordEmbsAug (Leverage word2vec, GloVe or fasttext embeddings to apply augmentation).
Back translation is a version of the substitute augmentation. We take the imperfect system of translation in an attempt to increases generalizability of text models. This text augmentation has shown great performance for various tasks including text classification \cite{wei2019eda}, machine translation \cite{sennrich2015improving} \cite{edunov2018understanding}, and even in low data environments \cite{fadaee2017data}. While back translation has been show to improve various NLP tasks, in some situations is has been shown to provide marginal, if any, results in modern, large transformer \cite{longpre2020effective}.
In addition, to showing various NLP metrics for all 108 Google Translate supported languages, we also update the BLEU metrics for the Google's current NMT model. Previous scores generated by Aiken in 2019 can be found here \cite{aiken2019updated}. Of note is that all languages besides Latin use Google's NMT while Latin uses Google's Phrase-Based Machine Translation (PBMT).
\subsection{Reproducible}
To allow reproducible for the following experiment a Google Collaboratory \cite{google}. You can replicate the experiment in the following notebook. This colab includes all training data and models used. \smallskip
\begin{quote}
\textbf{Reproducible Google Colaboratory} \\
\url{https://colab.research.google.com/drive/1XpdkDrNruJ5TDZlwtdSRgZlkDRjcUM7d?usp=sharing}
\end{quote}
\section{Experiment}
To investigate the effect of different language translations on various text metrics we run a series of test as outlined in [Figure \ref{fig:exp}]. We took 1000 random English tweets from the Sentiment-140 \cite{go2009twitter} dataset, and run them through the Google Translate API \cite{googletranslate}. We translate to all 108 languages supported by Google Translate and then translated them back to English. We then analysed the differences using various text metrics: Bilingual Evaluation Understudy Score (BLEU) \cite{papineni2002bleu}, BERT embedding distance \cite{devlin2018bert}, BART embedding distance \cite{lewis2019bart}, GPT embedding distance \cite{radford2018improving}, XLNet embedding distance \cite{yang2019xlnet}, GloVe embedding distance \cite{pennington2014glove}, Doc2Vec embedding distance \cite{le2014distributed}, NLTK Vader \cite{hutto2014vader}, Textblob Polarity and Subjectivity \cite{loria2018textblob}, Flair \cite{akbik2018coling}, and common Text Statistics (Flesch-Kincaid Grade, Flesch-Reading Ease) \cite{flesch1943marks}. The following sections go into detail about each of these NLP metrics.
\begin{figure}[ht]
\centering
\includegraphics[width=.485\textwidth]{Figures/translation_augmentation.png}
\caption{Experimental flow diagram showing the languages used for translations followed by the metrics used to analysis the differences}
\label{fig:exp}
\end{figure}
\subsection{BLEU Scores}
BLEU is a metric for evaluating a generated sentence to a reference sentence. Scores a translation on a scale of 0 to 1, in an attempt to measure the adequacy and fluency of the machine translation output. Scored by n-grams, BLEU-1 to BLEU-4. We use NLTK's BLEU \cite{nltkbleu} score method which can weight each of the n-grams scores independently.
\subsection{BERT Embeddings}
Bidirectional Encoder Representations from Transformer (BERT) is an encoder encoder transformer architecture that trains by predicting masked words. By masking 15\% of the words, BERT can also predict the position of words in a sentence. This allows BERT to be a general language model which can predict word embeddings alongside their positional embeddings. BERT was originally pre-trained on the English Wikipedia and Brown Corpus. Our implementation used bert-base-uncased \cite{hugging_face}, 12-layer, 768-hidden, 12-heads, 110M parameters.
\subsection{BART Embeddings}
BART is a transformer that learns to train by generalizing the masking technique used by BERT to a random shuffle of the ordering of a sentence and a mask that spans many words. This unsupervised technique learns to map corrupted parts of a document and therefore performed SOTA on discriminative and text generation tasks in 2020. We use BART-Base \cite{facebook/bart-base}, 12-layer, 768-hidden, 16-heads, 139M parameters, for our analysis.
\subsection{GPT Embeddings}
GPT uses a decoder only transformer structure with masked self-attention to train the language mode. Originally published in 2018, GPT, once fine tuned for a task, was SOTA in many language tasks. This unsupervised pre-training transform architecture set the standard for the large transformer models to follow. We use open-gpt, GPT 1, \cite{gpt_huggingface}, 12-layer, 768-hidden, 12-heads, 110M parameters.
\subsection{XLNet Embeddings}
XLNet is a generalized autoregressive pretraining method that enables learning bidirectional contexts by maximizing the expected likelihood over all permutations of the factorization order. When released in 2020, XLNet outperformed BERT in many language tasks. We use xlnet-base-cased \cite{xlnet_huggingface}, 12-layer, 768-hidden, 12-heads, 110M parameters.
\subsection{GloVe Embeddings}
Global Vectors for Word Representation, GloVe, is an unsupervised learning algorithm for obtaining vector representations for words. Training is performed on aggregated global word-word co-occurrence statistics from a corpus, and the resulting representations showcase interesting linear substructures of the word vector space \cite{pennington}. GloVe was originally trained on 800 MB of text in Wikipedia 2014 and Gigaword 5. The model outputs a 100d vector.
\subsection{Doc2Vec Embeddings}
Doc2Vec in an extension of the Word2Vec \cite{mikolov2013efficient} architecture which uses either a skip-gram or continuous bag of words method. In addition to learning word vectors, Doc2Vec, also learns a paragraph vector which allows it to learn from documents of any length and format. We use the Associated Press News Skip-gram \cite{jhlau20168doc2vec} \cite{lau2016empirical} (0.6GB) which was trained on Wikipedia and AP News.
\subsection{Vader Compound Score}
Vader is a simple ruled based sentiment analysis tool original made for real-time social media. Vader is constructed from a generalizable, valence-based, human-curated gold standard sentiment lexicon.
They rule based approach was developed to be sensitive to both the polarity and the intensity of sentiments. To handle English idioms, a special rule set was constructed (the shit:+3, the bomb:+3, bad ass:+1.5, yeah right:-2, cut the mustard:+2, kiss of death:+1.5, hand to mouth:-2).
\subsection{Textblob Polarity/Subjectivity Score}
Textblob employs a lexicon of words \cite{Sloria} (with cornetto-synset-id and wordnet-id) and their part of speech, definition, polarity, subjectivity, and intensity. For example,
\begin{displayquote}
$<$ word form="great", pos="JJ", sense="very good", polarity="1.0", subjectivity="1.0", intensity="1.0", confidence="0.9" $>$
\end{displayquote}
Textblob uses the lexicon from the deprecated Pattern Library \cite{Pattern.web} (found via the WayBack Machine) which contains 2917 entries. We use both the Polarity score and the Subjectivity score in our analysis.
\subsection{Text Statistics}
Text statistics include Flesch-Reading Ease, Flesch-Kincaid Grade, lexicon count (number of words) and sentence count (number of sentences). Flesch metrics. Both Flesch-Reading Ease (Range 0-100) and Flesch-Kincaid Grade (Range 0-18) \cite{readable_2020} are metrics which uses total words, total sentences, and total syllables to calculate readability. The Flesch Reading Ease score is between 1 and 100, and the Flesch Kincaid Grade Level reflects the US education system.
\section{Evaluation}
We find the following 14 comparison metrics: BLEU, BERT Embeddings, BERT Embeddings, GPT Embeddings, XLNet Embeddings, Glove Embeddings, Doc2Vec Embeddings, Flair, Polarity, Subjectivity, Vader, Flesch-Kincaid Grade, Flesch Reading Ease, lexicon counts, and word counts for 108 languages. A summarization of all charts is located in the appendix alongside example back translations [Table \ref{tab:tweet}, Table \ref{tab:metrics1}, Table \ref{tab:metrics2}].
\begin{table*}[ht]
\centering
\begin{tabular}{||l|c|c|c|c|l|c|c|c|c||}
\hline
\multicolumn{1}{||c|}{\textbf{Language}} &
\textbf{BLEU-1} &
\textbf{BLEU-2} &
\textbf{BLEU-3} &
\textbf{BLEU-4} &
\multicolumn{1}{c|}{\textbf{Language}} &
\textbf{BLEU-1} &
\textbf{BLEU-2} &
\textbf{BLEU-3} &
\textbf{BLEU-4} \\ \hline
Afrikaans & 0.6619 & 0.4684 & 0.2483 & 0.2483 & Lithuanian & 0.5966 & 0.3759 & 0.1793 & 0.1793 \\ \hline
Albanian & 0.6637 & 0.4643 & 0.2565 & 0.2565 & Luxembourgish & 0.6229 & 0.4044 & 0.1845 & 0.1845 \\ \hline
Amharic & 0.4862 & 0.2171 & 0.0631 & 0.0631 & Macedonian & 0.6411 & 0.4356 & 0.2239 & 0.2239 \\ \hline
Arabic & 0.5174 & 0.3013 & 0.1215 & 0.1215 & Malagasy & 0.5058 & 0.2799 & 0.1051 & 0.1051 \\ \hline
Armenian & 0.6042 & 0.3794 & 0.1635 & 0.1635 & Malay & 0.6413 & 0.4255 & 0.2115 & 0.2115 \\ \hline
Azerbaijani & 0.5265 & 0.2764 & 0.0918 & 0.0918 & Malayalam & 0.3872 & 0.1668 & 0.0446 & 0.0446 \\ \hline
Basque & 0.6104 & 0.3465 & 0.1415 & 0.1415 & Maltese & 0.7265 & 0.5309 & 0.3185 & 0.3185 \\ \hline
Belarusian & 0.5683 & 0.3440 & 0.1443 & 0.1443 & Maori & 0.5603 & 0.3192 & 0.1295 & 0.1295 \\ \hline
Bengali & 0.5269 & 0.2744 & 0.0980 & 0.0980 & Marathi & 0.4848 & 0.2289 & 0.0704 & 0.0704 \\ \hline
Bosnian & 0.6025 & 0.3910 & 0.1842 & 0.1842 & Mongolian & 0.4601 & 0.2050 & 0.0502 & 0.0502 \\ \hline
Bulgarian & 0.5981 & 0.3862 & 0.1775 & 0.1775 & Myanmar (Burmese) & 0.4885 & 0.2321 & 0.0661 & 0.0661 \\ \hline
Catalan & 0.5504 & 0.3129 & 0.1292 & 0.1292 & Nepali & 0.4926 & 0.2297 & 0.0716 & 0.0716 \\ \hline
Cebuano & 0.6335 & 0.4292 & 0.2137 & 0.2137 & Norwegian & 0.7083 & 0.5470 & 0.3193 & 0.3193 \\ \hline
Chinese (Simplified) & 0.4817 & 0.2509 & 0.0853 & 0.0853 & Nyanja (Chichewa) & 0.5212 & 0.2871 & 0.1064 & 0.1064 \\ \hline
Chinese (Traditional) & 0.4830 & 0.2525 & 0.0860 & 0.0860 & Odia (Oriya) & 0.5146 & 0.2322 & 0.0625 & 0.0625 \\ \hline
Corsican & 0.6271 & 0.4044 & 0.1961 & 0.1961 & Pashto & 0.4685 & 0.2354 & 0.0659 & 0.0659 \\ \hline
Croatian & 0.6042 & 0.3959 & 0.1882 & 0.1882 & Persian & 0.4977 & 0.2967 & 0.1238 & 0.1238 \\ \hline
Czech & 0.6071 & 0.3877 & 0.1819 & 0.1819 & Polish & 0.5980 & 0.3653 & 0.1555 & 0.1555 \\ \hline
Danish & 0.7205 & 0.5678 & 0.3566 & 0.3566 & Portuguese & 0.6570 & 0.4500 & 0.2345 & 0.2345 \\ \hline
Dutch & 0.6674 & 0.4725 & 0.2516 & 0.2516 & Punjabi & 0.5354 & 0.2910 & 0.1042 & 0.1042 \\ \hline
English & 1.0000 & 0.9920 & 0.9500 & 0.9500 & Romanian & 0.6199 & 0.3973 & 0.1877 & 0.1877 \\ \hline
Esperanto & 0.6968 & 0.5320 & 0.3157 & 0.3157 & Russian & 0.5828 & 0.3611 & 0.1551 & 0.1551 \\ \hline
Estonian & 0.6044 & 0.3811 & 0.1696 & 0.1696 & Samoan & 0.5400 & 0.3133 & 0.1248 & 0.1248 \\ \hline
Finnish & 0.6156 & 0.4041 & 0.1882 & 0.1882 & Scots Gaelic & 0.6349 & 0.4170 & 0.1965 & 0.1965 \\ \hline
French & 0.6346 & 0.4226 & 0.2005 & 0.2005 & Serbian & 0.4944 & 0.2988 & 0.1206 & 0.1206 \\ \hline
Frisian & 0.7255 & 0.5640 & 0.3481 & 0.3481 & Sesotho & 0.5414 & 0.2961 & 0.1114 & 0.1114 \\ \hline
Galician & 0.5937 & 0.3644 & 0.1627 & 0.1627 & Shona & 0.5270 & 0.3064 & 0.1154 & 0.1154 \\ \hline
Georgian & 0.6055 & 0.3761 & 0.1570 & 0.1570 & Sindhi & 0.4755 & 0.2260 & 0.0597 & 0.0597 \\ \hline
German & 0.6345 & 0.4075 & 0.1875 & 0.1875 & Sinhala (Sinhalese) & 0.4646 & 0.1980 & 0.0506 & 0.0506 \\ \hline
Greek & 0.6374 & 0.4255 & 0.2104 & 0.2104 & Slovak & 0.5910 & 0.3779 & 0.1707 & 0.1707 \\ \hline
Gujarati & 0.5472 & 0.3017 & 0.1144 & 0.1144 & Slovenian & 0.5979 & 0.3754 & 0.1636 & 0.1636 \\ \hline
Haitian Creole & 0.6886 & 0.4966 & 0.2814 & 0.2814 & Somali & 0.5905 & 0.3468 & 0.1525 & 0.1525 \\ \hline
Hausa & 0.6460 & 0.4540 & 0.2395 & 0.2395 & Spanish & 0.6001 & 0.3697 & 0.1671 & 0.1671 \\ \hline
Hawaiian & 0.4993 & 0.2785 & 0.0978 & 0.0978 & Sundanese & 0.6523 & 0.4382 & 0.2261 & 0.2261 \\ \hline
Hebrew & 0.5618 & 0.3377 & 0.1446 & 0.1446 & Swahili & 0.6642 & 0.4616 & 0.2466 & 0.2466 \\ \hline
Hindi & 0.5345 & 0.2919 & 0.1050 & 0.1050 & Swedish & 0.6728 & 0.4985 & 0.2816 & 0.2816 \\ \hline
Hmong & 0.6311 & 0.4414 & 0.2253 & 0.2253 & Tagalog (Filipino) & 0.7298 & 0.5561 & 0.3349 & 0.3349 \\ \hline
Hungarian & 0.5593 & 0.3183 & 0.1202 & 0.1202 & Tajik & 0.5657 & 0.3265 & 0.1360 & 0.1360 \\ \hline
Icelandic & 0.6535 & 0.4635 & 0.2459 & 0.2459 & Tamil & 0.5012 & 0.2494 & 0.0792 & 0.0792 \\ \hline
Igbo & 0.5998 & 0.4059 & 0.1913 & 0.1913 & Tatar & 0.4109 & 0.1146 & 0.0122 & 0.0122 \\ \hline
Indonesian & 0.6353 & 0.4193 & 0.2092 & 0.2092 & Telugu & 0.5430 & 0.2867 & 0.1006 & 0.1006 \\ \hline
Irish & 0.6893 & 0.4715 & 0.2560 & 0.2560 & Thai & 0.4908 & 0.2598 & 0.0955 & 0.0955 \\ \hline
Italian & 0.6464 & 0.4451 & 0.2332 & 0.2332 & Turkish & 0.5029 & 0.2519 & 0.0786 & 0.0786 \\ \hline
Japanese & 0.4453 & 0.1957 & 0.0541 & 0.0541 & Turkmen & 0.4544 & 0.1843 & 0.0441 & 0.0441 \\ \hline
Javanese & 0.6430 & 0.4247 & 0.2033 & 0.2033 & Ukrainian & 0.5636 & 0.3300 & 0.1297 & 0.1297 \\ \hline
Kannada & 0.5162 & 0.2651 & 0.0914 & 0.0914 & Urdu & 0.4734 & 0.2453 & 0.0790 & 0.0790 \\ \hline
Kazakh & 0.4761 & 0.2144 & 0.0572 & 0.0572 & Uyghur & 0.4734 & 0.2012 & 0.0531 & 0.0531 \\ \hline
Khmer & 0.5303 & 0.2946 & 0.1142 & 0.1142 & Uzbek & 0.4630 & 0.2110 & 0.0559 & 0.0559 \\ \hline
Kinyarwanda & 0.5368 & 0.2983 & 0.1142 & 0.1142 & Vietnamese & 0.6403 & 0.4205 & 0.2098 & 0.2098 \\ \hline
Korean & 0.4448 & 0.1980 & 0.0578 & 0.0578 & Welsh & 0.6812 & 0.4651 & 0.2434 & 0.2434 \\ \hline
Kurdish & 0.5965 & 0.3650 & 0.1605 & 0.1605 & Xhosa & 0.5184 & 0.2804 & 0.1002 & 0.1002 \\ \hline
Kyrgyz & 0.4033 & 0.1617 & 0.0371 & 0.0371 & Yiddish & 0.6307 & 0.5009 & 0.3131 & 0.3131 \\ \hline
Lao & 0.5935 & 0.3590 & 0.1626 & 0.1626 & Yoruba & 0.6918 & 0.5222 & 0.3166 & 0.3166 \\ \hline
Latin & 0.4375 & 0.1882 & 0.0528 & 0.0528 & Zulu & 0.6375 & 0.4318 & 0.2238 & 0.2238 \\ \hline
Latvian & 0.6509 & 0.4510 & 0.2276 & 0.2276 & \multicolumn{5}{l||}{ } \\ \hline
\end{tabular}
\end{table*}
\null\newpage
\null\newpage
\subsection{BLEU Scores}
These scores remearkbly will follow the trend for Google's Reported BLEU score with some variance for the lack of "high quality" reference sentence that BLEU usaslly requires. Instead, we have publicly scraped Tweets which sometimes only include a pronoun (i.e. @user1234) which Google Translate would just return as is.
The top BLEU-1 scores are for Tagalog (BLEU-1=0.7298), Maltese (BLEU-1=0.7265), Frisian (BLEU-1=0.7255) while the top average BLEU scores across all n-grams (25\% weighted for each) are Danish (Weighted BLEU=0.4776), Frisian (Weighted BLEU=0.4719), Tagalog (Weighted BLEU=0.4619).
On the other side the bottom BLEU-1 Scores are Malayalam (BLEU-1=0.3872), Kyrgyz (BLEU-1=0.4033), Tatar (BLEU-1=0.4109) while the bottom weighted BLEU score are Tatar (Weighted BLEU=0.0515), Kyrgyz (Weighted BLEU=0.0973), Malayalam (Weighted BLEU=0.1065).
The lower the BLEU score the less likely the reference sentence matches the back translation.
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/bert_dict_compare.png}
\caption{BERT Euclidean Differences}
\label{fig:1}
\end{figure*}
\null\newpage
\subsection{BERT Embeddings}
Boxplot of absolute distance between BART embeddings. The closer the language is to English the closer the embeddings are to zero. BERT embeddings are of 768 dimensions and the distance is euclidean.
We find closest embeddings are Danish (2.0758 $\pm$ 1.6713), Frisian (2.1105 $\pm$ 1.7137), Yiddish (2.1669 $\pm$ 1.6737). The furthest embeddings from the English reference are Tatar (5.2999 $\pm$ 1.8633), Latin (4.9107 $\pm$ 2.0689), Sindhi (4.8234 $\pm$ 1.9959)
\null\newpage
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/bart_dist_compare.png}
\caption{BART Euclidean Differences}
\label{fig:2}
\end{figure*}
\null\newpage
\subsection{BART Embeddings}
Boxplot of absolute distance between BART embeddings. The closer the language is to English the closer the embeddings are to zero. BART embeddings are of 768 dimensions and the distance is euclidean.
We find closest embeddings are Danish (0.5903 $\pm$ 0.4142), Frisian (0.5954 $\pm$ 0.423), Tagalog (0.5976 $\pm$ 0.4236). The furthest embeddings from the English reference are Tatar (1.309 $\pm$ 0.5083), Myanmar (1.2359 $\pm$ 0.554), Malayalam (1.2327 $\pm$ 0.4586).
\null\newpage
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/gpt_dist_compare.png}
\caption{GPT Euclidean Differences}
\label{fig:2}
\end{figure*}
\null\newpage
\subsection{GPT Embeddings}
Boxplot of absolute distance between GPT embeddings. The closer the language is to English the closer the embeddings are to zero. GPT embeddings are of 768 dimensions and the distance is euclidean.
We find closest embeddings are Danish (4.2798 $\pm$ 4.5897), Norwegian (4.5235 $\pm$ 4.5455), Frisian (4.5661 $\pm$ 5.0261). The furthest embeddings from the English reference are Tatar (13.9721 $\pm$ 6.4386), Malayalam (12.2058 $\pm$ 6.3209), Latin (12.0613 $\pm$ 6.6757).
\null\newpage
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/xlnet_dist_compare.png}
\caption{XLnet Euclidean Differences}
\label{fig:2}
\end{figure*}
\null\newpage
\subsection{XLnet Embeddings}
Boxplot of absolute distance between XLnet embeddings. The closer the language is to English the closer the embeddings are to zero. XLnet embeddings are of 768 dimensions and the distance is euclidean.
We find closest embeddings are Danish (30.1903 $\pm$ 21.0538), Frisian (31.6068 $\pm$ 21.6522), Norwegian (31.6301 $\pm$ 20.7609). The furthest embeddings from the English reference are Tatar (55.9961 $\pm$ 21.5646), Malayalam (54.7046 $\pm$ 21.2162), Kyrgyz (52.7561 $\pm$ 21.0698).
\null\newpage
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/glove_dist_compare.png}
\caption{GloVe Pooled Euclidean Differences}
\label{fig:2}
\end{figure*}
\null\newpage
\subsection{Glove Embeddings}
Boxplot of absolute distance between pooled Glove embeddings. The closer the language is to English the closer the embeddings are to zero. Glove embeddings are of 100 dimensions and the distance is euclidean.
We find closest embeddings are Frisian (0.3772 $\pm$ 0.38), Maltese (0.3895 $\pm$ 0.3353), Yiddish (0.3952 $\pm$ 0.4116). The furthest embeddings from the English reference are Myanmar (0.9865 $\pm$ 0.7875), Kyrgyz (0.9349 $\pm$ 0.5172), Latin (0.8898 $\pm$ 0.5298).
\null\newpage
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/doc2vec_dict_zoom_compare.png}
\caption{Doc2Vec Euclidean Differences, Full size boxplot with zoomed in plot of the 25-75 percentile on the second x-axis}
\label{fig:2}
\end{figure*}
\null\newpage
\subsection{Doc2Vec Embeddings}
Boxplot of absolute distance between Doc2Vec embeddings and the English embedding. The closer the language is to English the closer the embeddings are to zero. Doc2Vec embeddings are of 300 dimension and the distance is euclidean. We show the overall graph in addition to the zoomed in portion on the 2nd x-axis.
The closest embeddings to English are much closer than the transformers embeddings regardless of the 300 dimension output. We find closest embeddings are Danish (0.0205 $\pm$ 0.0079), Frisian (0.0206 $\pm$ 0.0078), Tagalog (0.0207 $\pm$ 0.0077). The furthest embeddings from the English reference are Hmong (0.025 $\pm$ 0.0547), Tajik (0.0244 $\pm$ 0.04), Kazakh (0.024 $\pm$ 0.0328).
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/vader_dict_compare.png}
\caption{Vader Differences}
\label{fig:3}
\end{figure*}
\null\newpage
\subsection{Vader Compound Score}
Boxplot of 1D distance between Vader sentiment compound score
Compound score is a 'normalized, weighted composite score' is accurate by summing the valence scores of each word in the lexicon.
The Vader scores with means closest to English are Sesotho (0 $\pm$ 0.2624), Vietnamese (-0.0001 $\pm$ 0.196), Frisian (0.0002 $\pm$ 0.1850) while the scores further from the reference are Tatar (-0.0388 $\pm$ 0.3227), Japanese (-0.0326 $\pm$ 0.2575), Uyghur (-0.0251 $\pm$ 0.2666).
\null\newpage
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/polarity_dict_compare.png}
\caption{Polarity Differences}
\label{fig:4}
\end{figure*}
\null\newpage
\subsection{Textblob Polairty Score}
Boxplot of 1D distance between textblob polarity scores
The polarity score is a float within the range [-1.0, 1.0]. The subjectivity is a float within the range [0.0, 1.0] where 0.0 is very objective and 1.0 is very subjective.
The languages with the closest polarity scores are Icelandic (-0.0001 $\pm$ 0.1387), Galician (0.0001 $\pm$ .1738), Odia (Oriya) (0.0001 $\pm$ 0.2172) while the languages furthest from reference polarity are Samoan (-0.0256 $\pm$ 0.2834), Serbian (0.023 $\pm$ 0.1883), Latin (0.0191 $\pm$ 0.2477).
\null\newpage
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/subjectivity_dict_compare.png}
\caption{Subjectivity Differences}
\label{fig:5}
\end{figure*}
\null\newpage
\subsection{Textblob Subjectivity Score}
Boxplot of 1D distance between textblob subjectivity scores
The polarity score is a float within the range [-1.0, 1.0]. The subjectivity is a float within the range [0.0, 1.0] where 0.0 is very objective and 1.0 is very subjective.
The closest embeddings to English are much closer than the BERT embeddings. We find the closest languages are Swahili (0.0000 $\pm$ 0.1767), Swedish (-0.0001 $\pm$ 0.1455), Xhosa (-0.0001 $\pm$ 0.2472) while the languages furthest from the English reference are Tatar (-0.027 $\pm$ 0.2726), Hawaiian (0.0261 $\pm$ 0.2902), Odia (Oriya) (-0.0246 $\pm$ 0.2345).
\null\newpage
\begin{table*}[h]
\centering
\begin{tabular}{||l|c|c|c|c|l|c|c|c|c||}
\hline
\multicolumn{1}{||c|}{\textbf{Language}} &
\textbf{\begin{tabular}[c]{@{}c@{}}flesch\\ reading\\ ease\end{tabular}} &
\textbf{\begin{tabular}[c]{@{}c@{}}flesch\\ kincaid\\ grade\end{tabular}} &
\textbf{\begin{tabular}[c]{@{}c@{}}lexicon\\ count\end{tabular}} &
\textbf{\begin{tabular}[c]{@{}c@{}}sentence\\ count\end{tabular}} &
\multicolumn{1}{c|}{\textbf{Language}} &
\textbf{\begin{tabular}[c]{@{}c@{}}flesch\\ reading\\ ease\end{tabular}} &
\textbf{\begin{tabular}[c]{@{}c@{}}flesch\\ kincaid\\ grade\end{tabular}} &
\textbf{\begin{tabular}[c]{@{}c@{}}lexicon\\ count\end{tabular}} &
\textbf{\begin{tabular}[c]{@{}c@{}}sentence\\ count\end{tabular}} \\ \hline
Afrikaans & 61.5 & 11.3 & 12786 & 482 & Lithuanian & 63.63 & 10.4 & 12958 & 530 \\ \hline
Albanian & 62.21 & 11 & 13109 & 508 & Luxembourgish & 62.92 & 10.7 & 12969 & 516 \\ \hline
Amharic & 65.56 & 9.7 & 13264 & 590 & Macedonian & 62.92 & 10.7 & 12668 & 504 \\ \hline
Arabic & 67.08 & 9.1 & 13584 & 646 & Malagasy & 65.35 & 9.8 & 12309 & 542 \\ \hline
Armenian & 61.8 & 11.1 & 12822 & 489 & Malay & 67.01 & 11.2 & 12940 & 440 \\ \hline
Azerbaijani & 62.61 & 10.8 & 12796 & 503 & Malayalam & 65.66 & 9.7 & 13631 & 609 \\ \hline
Basque & 62.31 & 11 & 12844 & 500 & Maltese & 59.57 & 12 & 12710 & 448 \\ \hline
Belarusian & 62.11 & 11 & 13229 & 511 & Maori & 69.75 & 10.2 & 12512 & 469 \\ \hline
Bengali & 67.59 & 8.9 & 13118 & 641 & Marathi & 66.07 & 9.5 & 12899 & 587 \\ \hline
Bosnian & 62.31 & 11 & 12819 & 498 & Mongolian & 65.25 & 9.8 & 12648 & 554 \\ \hline
Bulgarian & 63.32 & 10.6 & 12717 & 514 & Myanmar (Burmese) & 66.37 & 9.4 & 12283 & 565 \\ \hline
Catalan & 63.12 & 10.6 & 12563 & 504 & Nepali & 75.54 & 7.9 & 13068 & 622 \\ \hline
Cebuano & 61.8 & 11.1 & 12332 & 470 & Norwegian & 69.96 & 10.1 & 13120 & 496 \\ \hline
Chinese (Simplified) & 66.78 & 9.2 & 12923 & 607 & Nyanja (Chichewa) & 62.72 & 10.8 & 12528 & 495 \\ \hline
Chinese (Traditional) & 66.67 & 9.3 & 12863 & 602 & Odia (Oriya) & 65.46 & 9.7 & 13572 & 601 \\ \hline
Corsican & 63.22 & 10.6 & 12457 & 503 & Pashto & 67.18 & 9.1 & 12529 & 600 \\ \hline
Croatian & 62.31 & 11 & 12891 & 502 & Persian & 68.1 & 8.7 & 12810 & 642 \\ \hline
Czech & 64.54 & 10.1 & 12745 & 543 & Polish & 63.12 & 10.6 & 12944 & 520 \\ \hline
Danish & 69.45 & 10.3 & 13103 & 486 & Portuguese (Portugal, Brazil) & 62.01 & 11.1 & 13050 & 502 \\ \hline
Dutch & 62.51 & 10.9 & 12721 & 498 & Punjabi & 66.67 & 9.3 & 12442 & 582 \\ \hline
English.1 & 65.29 & 11.9 & 12895 & 414 & Romanian & 64.14 & 10.3 & 12391 & 519 \\ \hline
Esperanto & 60.18 & 11.8 & 13204 & 475 & Russian & 70.57 & 9.9 & 13226 & 511 \\ \hline
Estonian & 63.63 & 10.4 & 12856 & 526 & Samoan & 60.69 & 11.6 & 12092 & 443 \\ \hline
Finnish & 61.6 & 11.2 & 13031 & 493 & Scots Gaelic & 63.63 & 10.4 & 12486 & 511 \\ \hline
French & 62.21 & 11 & 12708 & 493 & Serbian & 64.14 & 10.3 & 12851 & 537 \\ \hline
Frisian & 70.06 & 10 & 13097 & 496 & Sesotho & 62.51 & 10.9 & 12218 & 479 \\ \hline
Galician & 63.22 & 10.6 & 12610 & 509 & Shona & 64.34 & 10.2 & 12518 & 528 \\ \hline
Georgian & 62.92 & 10.7 & 12676 & 505 & Sindhi & 67.28 & 9 & 12956 & 622 \\ \hline
German & 69.96 & 10.1 & 12949 & 488 & Sinhala (Sinhalese) & 67.08 & 9.1 & 12689 & 603 \\ \hline
Greek & 63.83 & 10.4 & 12789 & 528 & Slovak & 63.93 & 10.3 & 12766 & 530 \\ \hline
Gujarati & 67.59 & 8.9 & 12871 & 629 & Slovenian & 63.43 & 10.5 & 12917 & 525 \\ \hline
Haitian Creole & 60.89 & 11.5 & 12913 & 477 & Somali & 61.9 & 11.1 & 12569 & 482 \\ \hline
Hausa & 61.4 & 11.3 & 12256 & 461 & Spanish & 62.92 & 10.7 & 12906 & 514 \\ \hline
Hawaiian & 64.95 & 9.9 & 12462 & 539 & Sundanese & 60.48 & 11.7 & 13048 & 475 \\ \hline
Hebrew & 67.38 & 9 & 12842 & 621 & Swahili & 59.47 & 12 & 12730 & 447 \\ \hline
Hindi & 66.78 & 9.2 & 12894 & 604 & Swedish & 70.67 & 9.8 & 12992 & 504 \\ \hline
Hmong & 70.87 & 9.7 & 12438 & 485 & Tagalog (Filipino) & 60.79 & 11.5 & 12915 & 475 \\ \hline
Hungarian & 64.54 & 10.1 & 12546 & 534 & Tajik & 64.04 & 10.3 & 12447 & 519 \\ \hline
Icelandic & 62.51 & 10.9 & 12935 & 507 & Tamil & 67.28 & 9 & 12941 & 621 \\ \hline
Igbo & 61.6 & 11.2 & 12267 & 465 & Tatar & 60.48 & 11.7 & 14279 & 520 \\ \hline
Indonesian & 60.38 & 11.7 & 12953 & 469 & Telugu & 74.42 & 8.4 & 13345 & 603 \\ \hline
Irish & 61.4 & 11.3 & 12833 & 483 & Thai & 68.13 & 10.8 & 12755 & 450 \\ \hline
Italian & 62.82 & 10.8 & 13129 & 521 & Turkish & 63.32 & 10.6 & 12923 & 524 \\ \hline
Japanese & 73.41 & 8.8 & 14919 & 645 & Turkmen & 68.13 & 10.8 & 13517 & 478 \\ \hline
Javanese & 59.37 & 12.1 & 12739 & 445 & Ukrainian & 63.43 & 10.5 & 12664 & 515 \\ \hline
Kannada & 66.67 & 9.3 & 13519 & 633 & Urdu & 76.76 & 7.5 & 12697 & 642 \\ \hline
Kazakh & 63.22 & 10.6 & 13077 & 527 & Uyghur & 65.76 & 9.6 & 13024 & 585 \\ \hline
Khmer & 66.27 & 9.4 & 12714 & 584 & Uzbek & 73.21 & 8.8 & 12528 & 537 \\ \hline
Kinyarwanda & 67.32 & 11.1 & 13929 & 478 & Vietnamese & 61.19 & 11.4 & 12926 & 482 \\ \hline
Korean & 67.49 & 9 & 13229 & 641 & Welsh & 59.67 & 12 & 12847 & 454 \\ \hline
Kurdish & 68.94 & 10.5 & 12892 & 469 & Xhosa & 60.99 & 11.5 & 12532 & 465 \\ \hline
Kyrgyz & 66.17 & 9.5 & 12232 & 559 & Yiddish & 66.67 & 9.3 & 13333 & 622 \\ \hline
Lao & 66.78 & 9.2 & 12315 & 577 & Yoruba & 60.79 & 11.5 & 12766 & 470 \\ \hline
Latin & 68.94 & 10.5 & 13874 & 504 & Zulu & \multicolumn{1}{l|}{60.38} & 11.7 & 12550 & 454 \\ \hline
Latvian & 61.9 & 11.1 & 13000 & 498 & \multicolumn{5}{l||}{ } \\ \hline
\end{tabular}
\end{table*}
\null\newpage
\subsection{Text Statistics}
Here we analysis Flesh-reading ease, Flesh-kincaid grade, lexicon count, and sentence count.
The English Flesch Reading Ease for reference is 65.29. On the top of the list is Urdu (76.76), Nepali (75.54), Telugu (74.42) while on the bottom of the list is Javanese (59.37), Swahili (59.47), Maltese (59.57).
The English Flesch Kincaid Grade for reference is 11.9. On the top of the list is Javanese (12.1), Swahili (12), Maltese (12) while on the bottom fo the list is Urdu (7.5), Nepali (7.9), Telugu (8.4).
The English Lexicon count for reference is 12895. On the top of the list is Japanese (14919), Tatar (14279), Kinyarwanda (13929) while on the bottom of the list is Samoan (12092), Sesotho (12218), Kyrgyz (12232).
The English Sentence count for reference is 414. On the top of the list is Arabic (646), Japanese (645), Urdu (642), while at the bottom of the list is Malay (440), Samoan (443), Javanese (445).
\null\newpage
\section{Discussion}
\label{sec:Discussion}
As shown the embedding space is quite consistently varied across all 108 languages. Languages with the furthest embeddings (and therefore the best for the most generalized models) are among the languages with the worst translations in the BLEU scores from the Google API. Embedding spaces for large scale transformers have more consistent movement in their feature space while most single metrics (unified sentiment scores) are mostly the same with only outliers not having the same sentiment score. The lexicon nature of those methods do not end up capturing the meaning of context of the sentence and therefore do not end up having their embedding space moved.
\section{Conclusion}
\label{sec:Conclusion}
Back translation show a significant ability to move the various NLP metrics in many transformer architectures. In this experiment, this text augmentation technique empirically shows back translation acts as a generalizable strategy. Specifically, the lack of good translation allows this technique to move the embedding space in various statistical ways. We show back translation to Tatar moves the embeddings the furthest while translating to Danish would not generalize as well.
\subsection{Next Steps}
Showing the embedding space is only the first step in deciding which languages are the best for back translation during training. Training transformer architectures on these embeddings is the next step to guarantee a more generalizable model.
\section*{Acknowledgment}
The authors PeopleTec Technical Fellows program for encouragement and project assistance. The views and conclusions contained in this
paper are those of the authors and should not be interpreted
as representing any funding agencies.
\begin{table*}[]
\centering
\caption{108 Back Translations of a Tweet from the Sentiment 140 Dataset}
\label{tab:tweet}
\begin{tabular}{||l|l|l|l||}
\hline
\multicolumn{1}{||c|}{\textbf{Language}} & \multicolumn{1}{c|}{\textbf{Translated Sentence}} & \multicolumn{1}{c|}{\textbf{Language}} & \multicolumn{1}{c||}{\textbf{Translated Sentence}} \\ \hline
Afrikaans & I love Fridays! Held by the pool & Lithuanian & I love Fridays! Laying by the pool \\ \hline
Albanian & I love Fridays! Laying by the pool & Luxembourgish & I love Fridays! Laying of pool \\ \hline
Amharic & I love Friday! Parking at the ready: & Macedonian & Love Friday! Set by the pool \\ \hline
Arabic & I love Fridays! Put on a swimming pool & Malagasy & I love Fridays! Laying by the pool \\ \hline
Armenian & I love Fridays! Laying by the pool & Malay & I love Fridays! Laying by the pool \\ \hline
Azerbaijani & I love Fridays! shoot pool & Malayalam & I love Fridays! laying pool \\ \hline
Basque & I love Fridays! Laying the pool & Maltese & I love Fridays! Placing the pool \\ \hline
Belarusian & I love Fridays! Moore poolside & Maori & I love Friday! Laying in the pit \\ \hline
Bengali & I love Friday! Laying by the pool & Marathi & I love Friday! Laying by the pool \\ \hline
Bosnian & I love Fridays! Laying by the pool & Mongolian & I love Fridays! Laying by the pool \\ \hline
Bulgarian & I love Friday! Laying the pool & Myanmar (Burmese) & I love Friday! Laying the lakes \\ \hline
Catalan & I love Friday! Lie by the pool & Nepali & I love Fridays! Laying by the pool \\ \hline
Cebuano & I love Fridays! Laying the lake & Norwegian & I love Fridays! Laying by the pool \\ \hline
Chinese (Simplified) & I love Fridays! Laying by the pool & Nyanja (Chichewa) & I love the snow! putting pool \\ \hline
Chinese (Traditional) & I love Fridays! Laying by the pool & Odia (Oriya) & I love Fridays! Laying by the pool \\ \hline
Corsican & I love Fridays! Set from pool & Pashto & I love Fridays! Placed by the pool \\ \hline
Croatian & I love Fridays! Laying by the pool & Persian & I'm in love Fridays! Laying by the pool \\ \hline
Czech & I love Fridays! Laying by the pool & Polish & I love Fridays! Lying by the pool \\ \hline
Danish & I love Fridays! Laying by the pool & Portuguese & I love Fridays! Laying by the pool \\ \hline
Dutch & I love Fridays! Laying by the pool & Punjabi & I love Friday! Pool \\ \hline
English & I love Fridays! Laying by the pool & Romanian & I love Fridays! Laying pool \\ \hline
Esperanto & I love Friday! Putting the pool & Russian & I love Fridays! Laying by the pool \\ \hline
Estonian & I love Fridays! laying the pool & Samoan & I love Friday! Laying by the pool \\ \hline
Finnish & I love Fridays! Lay in a pond & Scots Gaelic & I love Friday! Laying with that the swimming \\ \hline
French & I love Fridays! Laying the pool & Serbian & I love Fridays! Laying by the pool \\ \hline
Frisian & I love Fridays! Laying by the pool & Sesotho & I like Friday! Laying by the pool \\ \hline
Galician & I love Friday! Lying in pool & Shona & I love Fridays! putting pool \\ \hline
Georgian & I love Fridays! Laying pool & Sindhi & I love Fridays! Laying Pool \\ \hline
German & I love Fridays! Laying by the pool & Sinhala (Sinhalese) & I love Fridays! Shooting pool \\ \hline
Greek & I love Fridays! Mounting poolside & Slovak & I love Fridays! Laying by the pool \\ \hline
Gujarati & I love Fridays! Laying by the pool & Slovenian & I love heels! Laying by the pool \\ \hline
Haitian Creole & I love Fridays! Laying the pool & Somali & I love Fridays! Laying pool \\ \hline
Hausa & I love Fridays! Laying the pool & Spanish & I love fridays! Lie by the pool \\ \hline
Hawaiian & I love Fridays! Lying by the pool & Sundanese & I love Fridays! Laying by the pool \\ \hline
Hebrew & I love Fridays! And lie by the pool & Swahili & Let me Friday! Keep the pool \\ \hline
Hindi & Mujee like Friday! Laying by the pool & Swedish & I love Fridays! Laying by the pool \\ \hline
Hmong & I love Fridays! Laid by the pool & Tagalog (Filipino) & I love every Friday! Laying by the pool \\ \hline
Hungarian & I love Fridays! Laying the pool & Tajik & I love Fridays! Laying out by the pool \\ \hline
Icelandic & I love Fridays! Laying by the pool & Tamil & I love Fridays! Laying by the pool \\ \hline
Igbo & I love Fridays! To set the pool & Tatar & I am in love on Friday! Laying pool \\ \hline
Indonesian & I love Fridays! Laying by the pool & Telugu & I love Fridays! Laying by the pool \\ \hline
Irish & I love Fridays! To lay by the pool & Thai & I love Fridays! Place the pool \\ \hline
Italian & I love Fridays! Laying in the pool & Turkish & I love Fridays! The pool floor \\ \hline
Japanese & I love Friday! Laying by the pool & Turkmen & I love anna! the foundation stone of the pool \\ \hline
Javanese & I Friday! Laying by the pool & Ukrainian & I love Fridays! Laying pool \\ \hline
Kannada & I love Fridays! Laying by the pool & Urdu & I love Fridays! Laying by the pool \\ \hline
Kazakh & I love Fridays! the construction of a swimming pool & Uyghur & I love Fridays! Pavement with pool \\ \hline
Khmer & I love Fridays! basin & Uzbek & On Friday, I love you! swimming board \\ \hline
Kinyarwanda & I love the fifth! Laying the pool & Vietnamese & I love Friday! Put the pool \\ \hline
Korean & I love Fridays! Lying by the pool & Welsh & I love Fridays! Laying by the pool \\ \hline
Kurdish & I love Fridays! Lying by the pool & Xhosa & I love Fridays! To put it in the pool \\ \hline
Kyrgyz & I love Fridays! by the pool & Yiddish & I love Fridays! Installation by the pool \\ \hline
Lao & I love Friday! By the pool & Yoruba & I love Fridays! Laying by the pool \\ \hline
Latin & I love Fridays? Laying in the pool & Zulu & I love Fridays! Setting up the pool \\ \hline
Latvian & I love Fridays! Putting the pool & \multicolumn{2}{l||}{} \\ \hline
\end{tabular}
\end{table*}
\begin{table*}[t]
\centering
\caption{Language Metrics Mean and Standard Deviation}
\label{tab:metrics1}
\resizebox{\textwidth}{!}{%
\begin{tabular}{||l|c|c|c|c|c|c|c|c|c||}
\hline
\multicolumn{1}{||c|}{\textbf{Language}} & \textbf{BERT} & \textbf{xlnet} & \textbf{bart} & \textbf{gpt} & \textbf{glove} & \textbf{Doc2Vec} & \textbf{VADER} & \textbf{Polarity} & \textbf{Subjectivity} \\ \hline
Afrikaans & 2.6753 $\pm$ 1.7894 & 36.5315 $\pm$ 20.8224 & 0.7063 $\pm$ 0.4114 & 5.7252 $\pm$ 5.2914 & 0.5019 $\pm$ 0.4145 & 0.0219 $\pm$ 0.0060 & -0.0074 $\pm$ 0.2076 & -0.0008 $\pm$ 0.1559 & -0.0087 $\pm$ 0.1757 \\ \hline
Albanian & 2.7292 $\pm$ 1.8679 & 37.2732 $\pm$ 21.9942 & 0.7189 $\pm$ 0.4193 & 5.9409 $\pm$ 5.3134 & 0.4683 $\pm$ 0.4172 & 0.0216 $\pm$ 0.0064 & -0.0056 $\pm$ 0.2004 & 0.0041 $\pm$ 0.1674 & -0.0163 $\pm$ 0.1759 \\ \hline
Amharic & 4.2694 $\pm$ 1.8715 & 50.2214 $\pm$ 20.3938 & 1.0572 $\pm$ 0.4652 & 10.6803 $\pm$ 6.6559 & 0.6867 $\pm$ 0.4313 & 0.0229 $\pm$ 0.0038 & -0.0137 $\pm$ 0.2816 & 0.0002 $\pm$ 0.2007 & -0.0145 $\pm$ 0.2155 \\ \hline
Arabic & 4.3220 $\pm$ 1.8829 & 47.2316 $\pm$ 19.7599 & 1.0103 $\pm$ 0.4135 & 8.6844 $\pm$ 5.4613 & 0.7693 $\pm$ 0.4834 & 0.0232 $\pm$ 0.0030 & -0.0242 $\pm$ 0.2374 & 0.0035 $\pm$ 0.1754 & -0.0145 $\pm$ 0.1993 \\ \hline
Armenian & 3.3922 $\pm$ 2.0027 & 42.5497 $\pm$ 21.0910 & 0.8890 $\pm$ 0.4417 & 7.5857 $\pm$ 5.9067 & 0.5808 $\pm$ 0.4238 & 0.0224 $\pm$ 0.0051 & -0.0078 $\pm$ 0.2294 & -0.006 $\pm$ 0.1841 & -0.0072 $\pm$ 0.177 \\ \hline
Azerbaijani & 4.5472 $\pm$ 1.8635 & 50.2426 $\pm$ 22.0118 & 1.0690 $\pm$ 0.4685 & 10.3279 $\pm$ 6.4566 & 0.7133 $\pm$ 0.4366 & 0.0229 $\pm$ 0.0039 & -0.0066 $\pm$ 0.2559 & -0.0018 $\pm$ 0.1743 & -0.0081 $\pm$ 0.1693 \\ \hline
Basque & 3.4175 $\pm$ 1.8110 & 43.3499 $\pm$ 22.1301 & 0.8516 $\pm$ 0.4420 & 8.1463 $\pm$ 6.1391 & 0.5432 $\pm$ 0.3888 & 0.0225 $\pm$ 0.0050 & -0.005 $\pm$ 0.2389 & 0.0081 $\pm$ 0.1489 & 0.0008 $\pm$ 0.1524 \\ \hline
Belarusian & 3.4442 $\pm$ 1.8929 & 42.1689 $\pm$ 20.3298 & 0.8668 $\pm$ 0.4231 & 7.4848 $\pm$ 5.4101 & 0.6803 $\pm$ 0.4671 & 0.0225 $\pm$ 0.0049 & -0.005 $\pm$ 0.2368 & 0.0077 $\pm$ 0.2081 & 0.0026 $\pm$ 0.2123 \\ \hline
Bengali & 3.7937 $\pm$ 1.8846 & 48.1835 $\pm$ 21.9692 & 0.9935 $\pm$ 0.4678 & 9.3810 $\pm$ 6.5568 & 0.6196 $\pm$ 0.4292 & 0.0229 $\pm$ 0.0041 & -0.0022 $\pm$ 0.2327 & 0.0116 $\pm$ 0.1953 & 0.0011 $\pm$ 0.1850 \\ \hline
Bosnian & 3.2100 $\pm$ 1.9337 & 40.2262 $\pm$ 21.0845 & 0.8063 $\pm$ 0.4368 & 6.6938 $\pm$ 5.2725 & 0.6114 $\pm$ 0.4419 & 0.0223 $\pm$ 0.0054 & 0.0006 $\pm$ 0.2131 & 0.0112 $\pm$ 0.1806 & -0.0149 $\pm$ 0.1885 \\ \hline
Bulgarian & 3.2032 $\pm$ 1.8526 & 39.5151 $\pm$ 19.3191 & 0.8310 $\pm$ 0.4307 & 6.9228 $\pm$ 5.4705 & 0.6180 $\pm$ 0.4370 & 0.0224 $\pm$ 0.0050 & 0.0031 $\pm$ 0.2178 & 0.0036 $\pm$ 0.1820 & -0.0079 $\pm$ 0.1963 \\ \hline
Catalan & 3.7035 $\pm$ 1.8844 & 44.4201 $\pm$ 21.2358 & 0.9172 $\pm$ 0.4408 & 8.4720 $\pm$ 5.7828 & 0.6943 $\pm$ 0.4918 & 0.0226 $\pm$ 0.0047 & 0.0164 $\pm$ 0.2448 & -0.006 $\pm$ 0.2019 & -0.0137 $\pm$ 0.2155 \\ \hline
Cebuano & 3.1232 $\pm$ 1.9017 & 39.5497 $\pm$ 21.7279 & 0.7828 $\pm$ 0.4377 & 6.9678 $\pm$ 5.8215 & 0.5668 $\pm$ 0.3904 & 0.0221 $\pm$ 0.0057 & 0.0042 $\pm$ 0.2449 & 0.0047 $\pm$ 0.2130 & -0.0005 $\pm$ 0.1686 \\ \hline
Chinese (Simp.) & 3.9368 $\pm$ 1.9524 & 47.6320 $\pm$ 20.6808 & 1.0247 $\pm$ 0.4771 & 8.9017 $\pm$ 5.7305 & 0.7676 $\pm$ 0.4848 & 0.0229 $\pm$ 0.0039 & -0.0166 $\pm$ 0.2556 & 0.0032 $\pm$ 0.1982 & -0.0014 $\pm$ 0.2275 \\ \hline
Chinese (Trad.) & 3.9115 $\pm$ 1.9621 & 47.5426 $\pm$ 20.8729 & 1.0218 $\pm$ 0.4800 & 8.9060 $\pm$ 5.7579 & 0.7681 $\pm$ 0.4932 & 0.0229 $\pm$ 0.0040 & -0.0167 $\pm$ 0.2553 & 0.0031 $\pm$ 0.1986 & -0.0013 $\pm$ 0.2281 \\ \hline
Corsican & 3.3429 $\pm$ 1.9639 & 41.7332 $\pm$ 22.3249 & 0.8559 $\pm$ 0.4762 & 8.1193 $\pm$ 6.3887 & 0.5986 $\pm$ 0.4575 & 0.0219 $\pm$ 0.0060 & -0.002 $\pm$ 0.2604 & 0.0086 $\pm$ 0.1926 & 0.0088 $\pm$ 0.2209 \\ \hline
Croatian & 3.1456 $\pm$ 1.9034 & 40.8590 $\pm$ 21.5791 & 0.7901 $\pm$ 0.4028 & 6.5720 $\pm$ 5.2830 & 0.6174 $\pm$ 0.4898 & 0.0223 $\pm$ 0.0053 & 0.0054 $\pm$ 0.2067 & 0.0105 $\pm$ 0.1721 & -0.0098 $\pm$ 0.1646 \\ \hline
Czech & 3.1310 $\pm$ 1.8350 & 40.1054 $\pm$ 20.7646 & 0.8152 $\pm$ 0.4348 & 6.7120 $\pm$ 5.3675 & 0.5822 $\pm$ 0.4293 & 0.0224 $\pm$ 0.0052 & -0.0055 $\pm$ 0.2321 & -0.0013 $\pm$ 0.1812 & -0.009 $\pm$ 0.1804 \\ \hline
Danish & 2.0758 $\pm$ 1.6713 & 30.1903 $\pm$ 21.0538 & 0.5903 $\pm$ 0.4142 & 4.2798 $\pm$ 4.5897 & 0.4041 $\pm$ 0.4014 & 0.0205 $\pm$ 0.0079 & -0.0011 $\pm$ 0.1806 & 0.0013 $\pm$ 0.1331 & -0.0023 $\pm$ 0.1469 \\ \hline
Dutch & 2.6448 $\pm$ 1.7705 & 35.1354 $\pm$ 21.1426 & 0.6975 $\pm$ 0.4358 & 5.6755 $\pm$ 5.2079 & 0.4924 $\pm$ 0.4191 & 0.0215 $\pm$ 0.0066 & -0.0063 $\pm$ 0.1935 & 0.0009 $\pm$ 0.1618 & -0.0026 $\pm$ 0.1776 \\ \hline
English.1 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 & 0.0000 $\pm$ 0.0000 \\ \hline
Esperanto & 2.3467 $\pm$ 1.7330 & 33.1779 $\pm$ 21.0201 & 0.6306 $\pm$ 0.4066 & 4.8847 $\pm$ 4.8762 & 0.4573 $\pm$ 0.4409 & 0.0212 $\pm$ 0.0071 & -0.0137 $\pm$ 0.2001 & 0.0044 $\pm$ 0.1696 & -0.0198 $\pm$ 0.188 \\ \hline
Estonian & 3.1483 $\pm$ 1.8573 & 41.0236 $\pm$ 21.0431 & 0.8218 $\pm$ 0.4516 & 7.0421 $\pm$ 5.6867 & 0.5903 $\pm$ 0.4497 & 0.0222 $\pm$ 0.0055 & -0.002 $\pm$ 0.208 & 0.0019 $\pm$ 0.1815 & -0.0047 $\pm$ 0.183 \\ \hline
Finnish & 3.0003 $\pm$ 1.8270 & 39.4413 $\pm$ 20.4864 & 0.7904 $\pm$ 0.4394 & 6.4888 $\pm$ 5.2860 & 0.5781 $\pm$ 0.4414 & 0.0222 $\pm$ 0.0055 & 0.0032 $\pm$ 0.2167 & 0.0020 $\pm$ 0.1663 & -0.0027 $\pm$ 0.1833 \\ \hline
French & 3.0026 $\pm$ 1.8019 & 38.7591 $\pm$ 21.2701 & 0.7695 $\pm$ 0.4505 & 6.8370 $\pm$ 5.7311 & 0.5468 $\pm$ 0.4153 & 0.0220 $\pm$ 0.0059 & 0.0023 $\pm$ 0.1985 & -0.0048 $\pm$ 0.1815 & -0.0086 $\pm$ 0.1887 \\ \hline
Frisian & 2.1105 $\pm$ 1.7137 & 31.6068 $\pm$ 21.6522 & 0.5954 $\pm$ 0.4230 & 4.5661 $\pm$ 5.0261 & 0.3772 $\pm$ 0.3800 & 0.0206 $\pm$ 0.0078 & 0.0002 $\pm$ 0.1850 & 0.0003 $\pm$ 0.1522 & -0.0146 $\pm$ 0.1423 \\ \hline
Galician & 3.3084 $\pm$ 1.8283 & 41.5002 $\pm$ 20.6445 & 0.8464 $\pm$ 0.4474 & 7.3423 $\pm$ 5.6460 & 0.6065 $\pm$ 0.4407 & 0.0224 $\pm$ 0.0050 & -0.0054 $\pm$ 0.2207 & 0.0001 $\pm$ 0.1738 & -0.0119 $\pm$ 0.1991 \\ \hline
Georgian & 3.1359 $\pm$ 1.9296 & 41.5282 $\pm$ 21.2911 & 0.8393 $\pm$ 0.4507 & 6.9871 $\pm$ 5.8524 & 0.5619 $\pm$ 0.4255 & 0.0224 $\pm$ 0.0052 & -0.0041 $\pm$ 0.2067 & 0.0068 $\pm$ 0.1653 & -0.0051 $\pm$ 0.1654 \\ \hline
German & 3.0523 $\pm$ 1.8728 & 38.8812 $\pm$ 21.1129 & 0.7681 $\pm$ 0.4358 & 6.8274 $\pm$ 5.7338 & 0.5270 $\pm$ 0.4218 & 0.0222 $\pm$ 0.0056 & -0.0095 $\pm$ 0.2294 & 0.0009 $\pm$ 0.1652 & -0.0149 $\pm$ 0.1882 \\ \hline
Greek & 3.0388 $\pm$ 1.8310 & 38.5410 $\pm$ 21.5254 & 0.7858 $\pm$ 0.4563 & 6.6580 $\pm$ 5.5103 & 0.5591 $\pm$ 0.4279 & 0.0220 $\pm$ 0.0059 & -0.0043 $\pm$ 0.2288 & 0.0015 $\pm$ 0.1600 & -0.0057 $\pm$ 0.1789 \\ \hline
Gujarati & 3.5816 $\pm$ 2.0118 & 45.9618 $\pm$ 21.6391 & 0.9498 $\pm$ 0.4864 & 8.7021 $\pm$ 6.3993 & 0.5882 $\pm$ 0.4797 & 0.0228 $\pm$ 0.0043 & -0.0182 $\pm$ 0.2498 & 0.0122 $\pm$ 0.1941 & 0.0070 $\pm$ 0.1845 \\ \hline
Haitian Creole & 2.4935 $\pm$ 1.7943 & 35.1527 $\pm$ 22.1554 & 0.6506 $\pm$ 0.4146 & 5.4482 $\pm$ 5.2776 & 0.4396 $\pm$ 0.3751 & 0.0213 $\pm$ 0.0070 & -0.0002 $\pm$ 0.2061 & 0.0054 $\pm$ 0.1587 & -0.0066 $\pm$ 0.1645 \\ \hline
Hausa & 3.0370 $\pm$ 1.9173 & 38.3269 $\pm$ 21.4406 & 0.7694 $\pm$ 0.4485 & 6.6423 $\pm$ 5.5684 & 0.5737 $\pm$ 0.4244 & 0.0230 $\pm$ 0.0335 & -0.0156 $\pm$ 0.246 & -0.0056 $\pm$ 0.1898 & 0.0089 $\pm$ 0.2029 \\ \hline
Hawaiian & 4.3917 $\pm$ 2.0020 & 49.2824 $\pm$ 21.7270 & 1.0079 $\pm$ 0.4585 & 10.3520 $\pm$ 6.5060 & 0.8243 $\pm$ 0.4839 & 0.0228 $\pm$ 0.0042 & -0.0236 $\pm$ 0.327 & -0.0188 $\pm$ 0.2746 & 0.0261 $\pm$ 0.2902 \\ \hline
Hebrew & 3.3267 $\pm$ 1.8875 & 43.2773 $\pm$ 19.9098 & 0.8887 $\pm$ 0.4219 & 7.5287 $\pm$ 5.7295 & 0.6314 $\pm$ 0.4726 & 0.0229 $\pm$ 0.0038 & -0.0112 $\pm$ 0.2348 & 0.0037 $\pm$ 0.1807 & -0.0178 $\pm$ 0.1888 \\ \hline
Hindi & 3.7218 $\pm$ 1.8764 & 47.2855 $\pm$ 20.9455 & 0.9400 $\pm$ 0.4265 & 8.8569 $\pm$ 5.9814 & 0.6278 $\pm$ 0.4590 & 0.0230 $\pm$ 0.0038 & -0.0065 $\pm$ 0.2332 & 0.0050 $\pm$ 0.1950 & -0.0081 $\pm$ 0.2007 \\ \hline
Hmong & 3.0967 $\pm$ 1.8916 & 39.2700 $\pm$ 21.5981 & 0.7575 $\pm$ 0.4370 & 6.9259 $\pm$ 5.7573 & 0.5994 $\pm$ 0.5008 & 0.0250 $\pm$ 0.0547 & 0.0145 $\pm$ 0.2360 & 0.0095 $\pm$ 0.2231 & 0.0166 $\pm$ 0.2259 \\ \hline
Hungarian & 3.5887 $\pm$ 1.9075 & 44.1854 $\pm$ 21.7258 & 0.9007 $\pm$ 0.4560 & 8.0363 $\pm$ 5.7908 & 0.6660 $\pm$ 0.4395 & 0.0225 $\pm$ 0.0048 & -0.0005 $\pm$ 0.2233 & -0.0004 $\pm$ 0.1775 & -0.0155 $\pm$ 0.1822 \\ \hline
Icelandic & 2.6405 $\pm$ 1.8053 & 36.4142 $\pm$ 20.9481 & 0.7191 $\pm$ 0.4278 & 5.5187 $\pm$ 5.0761 & 0.4915 $\pm$ 0.4133 & 0.0217 $\pm$ 0.0064 & -0.0015 $\pm$ 0.2054 & -0.0001 $\pm$ 0.1387 & -0.009 $\pm$ 0.1561 \\ \hline
Igbo & 3.4281 $\pm$ 1.9496 & 41.2788 $\pm$ 21.5776 & 0.8291 $\pm$ 0.4534 & 7.7875 $\pm$ 6.0819 & 0.6407 $\pm$ 0.4314 & 0.0221 $\pm$ 0.0057 & -0.0035 $\pm$ 0.2895 & 0.0121 $\pm$ 0.2010 & 0.0225 $\pm$ 0.2272 \\ \hline
Indonesian & 2.9717 $\pm$ 1.8386 & 37.5045 $\pm$ 21.2866 & 0.7086 $\pm$ 0.4175 & 6.3412 $\pm$ 5.3137 & 0.5467 $\pm$ 0.4315 & 0.0230 $\pm$ 0.0337 & -0.0028 $\pm$ 0.2055 & -0.008 $\pm$ 0.1777 & -0.016 $\pm$ 0.1931 \\ \hline
Irish & 2.6636 $\pm$ 1.7391 & 36.4281 $\pm$ 21.8509 & 0.6890 $\pm$ 0.4175 & 6.2791 $\pm$ 5.7358 & 0.4246 $\pm$ 0.3723 & 0.0216 $\pm$ 0.0065 & -0.0106 $\pm$ 0.2102 & -0.0032 $\pm$ 0.1465 & -0.0043 $\pm$ 0.1464 \\ \hline
Italian & 2.8866 $\pm$ 1.8009 & 36.8250 $\pm$ 20.3264 & 0.7365 $\pm$ 0.4360 & 6.3374 $\pm$ 5.4678 & 0.5332 $\pm$ 0.4516 & 0.0217 $\pm$ 0.0063 & 0.0077 $\pm$ 0.1950 & -0.0042 $\pm$ 0.1596 & -0.0183 $\pm$ 0.1867 \\ \hline
Japanese & 4.4044 $\pm$ 1.8878 & 49.7830 $\pm$ 20.5813 & 1.1171 $\pm$ 0.4531 & 10.8755 $\pm$ 6.2951 & 0.7712 $\pm$ 0.4370 & 0.0231 $\pm$ 0.0032 & -0.0326 $\pm$ 0.2575 & -0.0104 $\pm$ 0.2126 & -0.0229 $\pm$ 0.2095 \\ \hline
Javanese & 3.1578 $\pm$ 1.9123 & 39.1957 $\pm$ 21.9393 & 0.7776 $\pm$ 0.4673 & 7.2494 $\pm$ 5.9900 & 0.5517 $\pm$ 0.3895 & 0.0218 $\pm$ 0.0062 & -0.0103 $\pm$ 0.2398 & -0.0003 $\pm$ 0.1801 & -0.0018 $\pm$ 0.1809 \\ \hline
Kannada & 3.9152 $\pm$ 1.8317 & 48.7288 $\pm$ 20.9314 & 1.0231 $\pm$ 0.4551 & 9.9590 $\pm$ 6.5459 & 0.6442 $\pm$ 0.4502 & 0.0229 $\pm$ 0.0038 & -0.0108 $\pm$ 0.2474 & 0.0119 $\pm$ 0.1797 & -0.0068 $\pm$ 0.2071 \\ \hline
Kazakh & 4.1914 $\pm$ 1.8486 & 50.2691 $\pm$ 21.5751 & 1.0625 $\pm$ 0.4802 & 10.6689 $\pm$ 6.6364 & 0.7273 $\pm$ 0.4659 & 0.0240 $\pm$ 0.0328 & -0.0043 $\pm$ 0.2616 & -0.0049 $\pm$ 0.225 & -0.0099 $\pm$ 0.2508 \\ \hline
Khmer & 3.6325 $\pm$ 1.8034 & 45.7669 $\pm$ 19.8676 & 0.9482 $\pm$ 0.4296 & 8.4474 $\pm$ 5.9054 & 0.6743 $\pm$ 0.4798 & 0.0238 $\pm$ 0.0246 & 0.0079 $\pm$ 0.2355 & 0.0110 $\pm$ 0.2183 & 0.0149 $\pm$ 0.2218 \\ \hline
Kinyarwanda & 3.9645 $\pm$ 2.0135 & 45.1762 $\pm$ 21.7999 & 0.9595 $\pm$ 0.4812 & 9.2847 $\pm$ 6.4706 & 0.7150 $\pm$ 0.4545 & 0.0228 $\pm$ 0.0044 & 0.0015 $\pm$ 0.2558 & -0.0035 $\pm$ 0.2228 & -0.0076 $\pm$ 0.2604 \\ \hline
Korean & 4.3485 $\pm$ 1.8834 & 52.2174 $\pm$ 21.6723 & 1.1133 $\pm$ 0.4722 & 11.1366 $\pm$ 6.4485 & 0.7793 $\pm$ 0.4660 & 0.0231 $\pm$ 0.0032 & -0.0074 $\pm$ 0.2906 & 0.0080 $\pm$ 0.2292 & 0.0022 $\pm$ 0.2383 \\ \hline
Kurdish & 3.5824 $\pm$ 2.0262 & 42.2055 $\pm$ 21.8016 & 0.8620 $\pm$ 0.4643 & 8.3174 $\pm$ 6.1876 & 0.6097 $\pm$ 0.4069 & 0.0222 $\pm$ 0.0055 & -0.0124 $\pm$ 0.2565 & -0.0043 $\pm$ 0.1914 & -0.002 $\pm$ 0.2121 \\ \hline
Kyrgyz & 4.7914 $\pm$ 1.9145 & 52.7561 $\pm$ 21.0698 & 1.1718 $\pm$ 0.4801 & 11.6158 $\pm$ 6.4069 & 0.9349 $\pm$ 0.5172 & 0.0236 $\pm$ 0.0158 & -0.0031 $\pm$ 0.3103 & 0.0116 $\pm$ 0.2699 & 0.0123 $\pm$ 0.2555 \\ \hline
Lao & 3.3175 $\pm$ 1.9198 & 43.6135 $\pm$ 21.6109 & 0.8391 $\pm$ 0.4251 & 7.4923 $\pm$ 5.9334 & 0.5750 $\pm$ 0.4177 & 0.0225 $\pm$ 0.0049 & 0.0022 $\pm$ 0.2218 & -0.003 $\pm$ 0.1703 & -0.0022 $\pm$ 0.2022 \\ \hline
Latin & 4.9107 $\pm$ 2.0689 & 52.5410 $\pm$ 21.2892 & 1.2010 $\pm$ 0.5471 & 12.0613 $\pm$ 6.6757 & 0.8898 $\pm$ 0.5298 & 0.0233 $\pm$ 0.0126 & -0.0033 $\pm$ 0.3289 & 0.0191 $\pm$ 0.2477 & -0.0006 $\pm$ 0.2383 \\ \hline
Latvian & 2.6936 $\pm$ 1.7852 & 37.3776 $\pm$ 21.3996 & 0.7204 $\pm$ 0.4138 & 5.6136 $\pm$ 5.0254 & 0.5063 $\pm$ 0.4181 & 0.0228 $\pm$ 0.0334 & -0.0049 $\pm$ 0.1962 & 0.0063 $\pm$ 0.1574 & -0.0083 $\pm$ 0.1664 \\ \hline
\end{tabular}%
}
\end{table*}
\begin{table*}[t]
\centering
\caption{Language Metrics Mean and Standard Deviation}
\label{tab:metrics2}
\resizebox{\textwidth}{!}{%
\begin{tabular}{||l|c|c|c|c|c|c|c|c|c||}
\hline
\multicolumn{1}{||c|}{\textbf{Language}} & \textbf{BERT} & \textbf{xlnet} & \textbf{bart} & \textbf{gpt} & \textbf{glove} & \textbf{Doc2Vec} & \textbf{VADER} & \textbf{Polarity} & \textbf{Subjectivity} \\ \hline
Lithuanian & 3.1533 $\pm$ 1.8887 & 40.9943 $\pm$ 20.4055 & 0.8371 $\pm$ 0.4415 & 6.9824 $\pm$ 5.6158 & 0.5906 $\pm$ 0.4520 & 0.0225 $\pm$ 0.0050 & -0.0073 $\pm$ 0.2184 & 0.0005 $\pm$ 0.1962 & 0.0016 $\pm$ 0.2072 \\ \hline
Luxembourgish & 3.3228 $\pm$ 2.0873 & 41.8774 $\pm$ 22.4604 & 0.8264 $\pm$ 0.4693 & 7.8428 $\pm$ 6.5077 & 0.5650 $\pm$ 0.4214 & 0.0220 $\pm$ 0.0058 & -0.0239 $\pm$ 0.2423 & -0.0054 $\pm$ 0.1864 & -0.016 $\pm$ 0.203 \\ \hline
Macedonian & 2.9089 $\pm$ 1.8076 & 38.2254 $\pm$ 20.8129 & 0.7846 $\pm$ 0.4521 & 6.2868 $\pm$ 5.4331 & 0.5389 $\pm$ 0.4325 & 0.0220 $\pm$ 0.0058 & 0.0033 $\pm$ 0.2172 & 0.0142 $\pm$ 0.1611 & -0.0083 $\pm$ 0.1569 \\ \hline
Malagasy & 3.8505 $\pm$ 1.9621 & 46.8842 $\pm$ 21.4363 & 0.9550 $\pm$ 0.4609 & 8.5786 $\pm$ 5.8147 & 0.7328 $\pm$ 0.4480 & 0.0227 $\pm$ 0.0045 & -0.0096 $\pm$ 0.2748 & -0.0048 $\pm$ 0.2342 & 0.0020 $\pm$ 0.2364 \\ \hline
Malay & 2.9645 $\pm$ 1.7530 & 37.6486 $\pm$ 21.2125 & 0.7135 $\pm$ 0.4255 & 6.6351 $\pm$ 5.4766 & 0.5458 $\pm$ 0.4164 & 0.0221 $\pm$ 0.0057 & 0.0003 $\pm$ 0.2124 & -0.0042 $\pm$ 0.1736 & -0.0039 $\pm$ 0.1938 \\ \hline
Malayalam & 4.8056 $\pm$ 1.7677 & 54.7046 $\pm$ 21.2162 & 1.2327 $\pm$ 0.4586 & 12.2058 $\pm$ 6.3209 & 0.7960 $\pm$ 0.4608 & 0.0233 $\pm$ 0.0022 & -0.0161 $\pm$ 0.3098 & -0.004 $\pm$ 0.2204 & 0.0007 $\pm$ 0.2301 \\ \hline
Maltese & 2.4760 $\pm$ 1.8293 & 33.9896 $\pm$ 22.4921 & 0.6439 $\pm$ 0.4210 & 5.6176 $\pm$ 5.4907 & 0.3895 $\pm$ 0.3353 & 0.0210 $\pm$ 0.0074 & 0.0067 $\pm$ 0.1889 & 0.0027 $\pm$ 0.1226 & -0.0123 $\pm$ 0.1361 \\ \hline
Maori & 3.8948 $\pm$ 1.9485 & 45.9441 $\pm$ 21.5976 & 0.9146 $\pm$ 0.4648 & 9.1761 $\pm$ 6.4592 & 0.6779 $\pm$ 0.4333 & 0.0234 $\pm$ 0.0228 & -0.0074 $\pm$ 0.2998 & -0.0098 $\pm$ 0.2442 & 0.0027 $\pm$ 0.2453 \\ \hline
Marathi & 4.1627 $\pm$ 1.8884 & 50.2687 $\pm$ 21.2404 & 1.0691 $\pm$ 0.4794 & 10.6952 $\pm$ 6.7298 & 0.7018 $\pm$ 0.4285 & 0.0230 $\pm$ 0.0035 & 0.0032 $\pm$ 0.2582 & 0.0026 $\pm$ 0.2057 & -0.0073 $\pm$ 0.227 \\ \hline
Mongolian & 4.2479 $\pm$ 1.9028 & 50.4806 $\pm$ 21.6242 & 1.0513 $\pm$ 0.4389 & 10.5640 $\pm$ 6.3119 & 0.7400 $\pm$ 0.4417 & 0.0229 $\pm$ 0.0040 & -0.0091 $\pm$ 0.2827 & -0.0008 $\pm$ 0.2194 & -0.0101 $\pm$ 0.2155 \\ \hline
Myanmar & 4.7016 $\pm$ 2.1506 & 51.2371 $\pm$ 21.4917 & 1.2359 $\pm$ 0.5540 & 11.9276 $\pm$ 6.6944 & 0.9865 $\pm$ 0.7875 & 0.0229 $\pm$ 0.0040 & -0.0165 $\pm$ 0.2696 & 0.0067 $\pm$ 0.2006 & -0.0082 $\pm$ 0.2115 \\ \hline
Nepali & 4.0612 $\pm$ 1.8466 & 49.6141 $\pm$ 21.5517 & 1.0406 $\pm$ 0.4593 & 10.4111 $\pm$ 6.5645 & 0.6683 $\pm$ 0.4215 & 0.0230 $\pm$ 0.0036 & -0.0231 $\pm$ 0.2732 & -0.0133 $\pm$ 0.1988 & -0.0076 $\pm$ 0.206 \\ \hline
Norwegian & 2.2990 $\pm$ 1.7932 & 31.6301 $\pm$ 20.7609 & 0.6194 $\pm$ 0.4124 & 4.5235 $\pm$ 4.5455 & 0.4440 $\pm$ 0.4034 & 0.0209 $\pm$ 0.0074 & -0.0033 $\pm$ 0.1962 & 0.0058 $\pm$ 0.1506 & -0.0113 $\pm$ 0.1498 \\ \hline
Nyanja & 3.8558 $\pm$ 1.9052 & 45.4260 $\pm$ 20.7191 & 0.9354 $\pm$ 0.4475 & 9.0426 $\pm$ 6.2212 & 0.7501 $\pm$ 0.4549 & 0.0227 $\pm$ 0.0044 & -0.0119 $\pm$ 0.2524 & -0.0116 $\pm$ 0.2206 & 0.0104 $\pm$ 0.2311 \\ \hline
Odia & 4.0704 $\pm$ 1.9244 & 49.8328 $\pm$ 22.4855 & 1.0118 $\pm$ 0.4827 & 10.1309 $\pm$ 6.6468 & 0.6151 $\pm$ 0.3989 & 0.0229 $\pm$ 0.0040 & -0.0105 $\pm$ 0.2847 & 0.0001 $\pm$ 0.2172 & -0.0246 $\pm$ 0.2345 \\ \hline
Pashto & 4.6559 $\pm$ 1.9994 & 51.1143 $\pm$ 21.3486 & 1.0847 $\pm$ 0.4492 & 10.2268 $\pm$ 6.4966 & 0.7500 $\pm$ 0.4439 & 0.0231 $\pm$ 0.0033 & 0.0089 $\pm$ 0.2680 & -0.0038 $\pm$ 0.2037 & -0.0081 $\pm$ 0.2333 \\ \hline
Persian & 4.4312 $\pm$ 1.9276 & 49.8650 $\pm$ 20.9929 & 1.0657 $\pm$ 0.4699 & 9.3400 $\pm$ 5.9010 & 0.7933 $\pm$ 0.5017 & 0.0232 $\pm$ 0.0030 & -0.0071 $\pm$ 0.2667 & 0.0109 $\pm$ 0.2062 & -0.0065 $\pm$ 0.2187 \\ \hline
Polish & 3.3162 $\pm$ 1.8666 & 42.0087 $\pm$ 20.5787 & 0.8431 $\pm$ 0.4436 & 7.3200 $\pm$ 5.5842 & 0.6051 $\pm$ 0.4539 & 0.0225 $\pm$ 0.0049 & 0.0076 $\pm$ 0.2281 & 0.0006 $\pm$ 0.1747 & -0.0077 $\pm$ 0.1833 \\ \hline
Portuguese & 2.8800 $\pm$ 1.8530 & 37.1307 $\pm$ 20.6883 & 0.7394 $\pm$ 0.4346 & 6.2778 $\pm$ 5.4059 & 0.5168 $\pm$ 0.4336 & 0.0219 $\pm$ 0.0060 & -0.003 $\pm$ 0.1948 & 0.0036 $\pm$ 0.1565 & -0.0101 $\pm$ 0.1773 \\ \hline
Punjabi & 3.6739 $\pm$ 1.9488 & 46.4990 $\pm$ 21.4981 & 0.9605 $\pm$ 0.4559 & 8.9806 $\pm$ 6.4451 & 0.6107 $\pm$ 0.4314 & 0.0230 $\pm$ 0.0038 & -0.0054 $\pm$ 0.2478 & 0.0076 $\pm$ 0.1807 & -0.0086 $\pm$ 0.1828 \\ \hline
Romanian & 3.1778 $\pm$ 1.8677 & 40.9882 $\pm$ 21.9662 & 0.8431 $\pm$ 0.4585 & 6.9520 $\pm$ 5.5552 & 0.5621 $\pm$ 0.4292 & 0.0223 $\pm$ 0.0054 & -0.0044 $\pm$ 0.2284 & 0.0015 $\pm$ 0.1592 & -0.0068 $\pm$ 0.1805 \\ \hline
Russian & 3.3236 $\pm$ 1.8926 & 41.1866 $\pm$ 19.9059 & 0.8444 $\pm$ 0.4155 & 7.1935 $\pm$ 5.3579 & 0.6554 $\pm$ 0.4803 & 0.0225 $\pm$ 0.0051 & -0.0062 $\pm$ 0.214 & 0.0073 $\pm$ 0.1843 & -0.0057 $\pm$ 0.2009 \\ \hline
Samoan & 3.8749 $\pm$ 1.9240 & 44.4024 $\pm$ 20.8871 & 0.9006 $\pm$ 0.4179 & 8.9794 $\pm$ 6.2978 & 0.7198 $\pm$ 0.4375 & 0.0225 $\pm$ 0.0049 & -0.0118 $\pm$ 0.2908 & -0.0256 $\pm$ 0.2834 & -0.002 $\pm$ 0.2353 \\ \hline
Scots Gaelic & 3.0683 $\pm$ 1.8886 & 40.3800 $\pm$ 21.6171 & 0.8097 $\pm$ 0.4434 & 6.9198 $\pm$ 5.7215 & 0.5381 $\pm$ 0.3810 & 0.0220 $\pm$ 0.0059 & -0.0002 $\pm$ 0.229 & -0.0067 $\pm$ 0.1535 & -0.0015 $\pm$ 0.178 \\ \hline
Serbian & 3.6279 $\pm$ 1.7905 & 45.0025 $\pm$ 20.4254 & 0.9349 $\pm$ 0.4339 & 8.3136 $\pm$ 5.4780 & 0.7145 $\pm$ 0.5159 & 0.0232 $\pm$ 0.0031 & 0.0112 $\pm$ 0.2407 & 0.0235 $\pm$ 0.1883 & -0.0098 $\pm$ 0.1977 \\ \hline
Sesotho & 3.9322 $\pm$ 1.9724 & 45.9501 $\pm$ 21.0958 & 0.9544 $\pm$ 0.4555 & 8.9511 $\pm$ 5.9904 & 0.7115 $\pm$ 0.4515 & 0.0227 $\pm$ 0.0045 & 0 $\pm$ 0.2624 & -0.0071 $\pm$ 0.2077 & -0.0055 $\pm$ 0.2282 \\ \hline
Shona & 3.8570 $\pm$ 1.8870 & 46.3480 $\pm$ 21.6009 & 0.9486 $\pm$ 0.4390 & 8.8672 $\pm$ 5.9634 & 0.7478 $\pm$ 0.4487 & 0.0227 $\pm$ 0.0045 & -0.0012 $\pm$ 0.2807 & -0.0069 $\pm$ 0.2265 & 0.0103 $\pm$ 0.2553 \\ \hline
Sindhi & 4.8234 $\pm$ 1.9959 & 51.6873 $\pm$ 21.0672 & 1.1107 $\pm$ 0.4948 & 10.9663 $\pm$ 6.5925 & 0.7565 $\pm$ 0.4359 & 0.0232 $\pm$ 0.0029 & -0.0118 $\pm$ 0.2885 & -0.0034 $\pm$ 0.1985 & 0.0059 $\pm$ 0.2285 \\ \hline
Sinhala & 4.4288 $\pm$ 1.8908 & 51.9869 $\pm$ 21.3391 & 1.0986 $\pm$ 0.4804 & 11.4927 $\pm$ 6.7297 & 0.7300 $\pm$ 0.4338 & 0.0230 $\pm$ 0.0037 & -0.0128 $\pm$ 0.2897 & -0.0004 $\pm$ 0.2239 & 0.0072 $\pm$ 0.2366 \\ \hline
Slovak & 3.2220 $\pm$ 1.8268 & 41.3408 $\pm$ 20.6178 & 0.8377 $\pm$ 0.4268 & 7.0250 $\pm$ 5.5250 & 0.6229 $\pm$ 0.4468 & 0.0224 $\pm$ 0.0051 & 0.0038 $\pm$ 0.2269 & -0.0009 $\pm$ 0.1864 & -0.0081 $\pm$ 0.1917 \\ \hline
Slovenian & 3.2271 $\pm$ 1.8310 & 40.4570 $\pm$ 20.0673 & 0.8125 $\pm$ 0.4246 & 6.8675 $\pm$ 5.2968 & 0.6132 $\pm$ 0.4481 & 0.0226 $\pm$ 0.0046 & -0.0011 $\pm$ 0.2357 & -0.0045 $\pm$ 0.1705 & -0.0178 $\pm$ 0.1876 \\ \hline
Somali & 3.6107 $\pm$ 1.9481 & 42.9309 $\pm$ 21.9753 & 0.8895 $\pm$ 0.4700 & 8.5719 $\pm$ 6.2760 & 0.6382 $\pm$ 0.4526 & 0.0222 $\pm$ 0.0056 & -0.0205 $\pm$ 0.2593 & -0.0014 $\pm$ 0.1875 & -0.0087 $\pm$ 0.1975 \\ \hline
Spanish & 3.3843 $\pm$ 1.8808 & 40.6751 $\pm$ 19.7936 & 0.8406 $\pm$ 0.4395 & 7.4043 $\pm$ 5.5196 & 0.6218 $\pm$ 0.4724 & 0.0225 $\pm$ 0.0050 & 0.0054 $\pm$ 0.2251 & -0.0023 $\pm$ 0.1972 & -0.0208 $\pm$ 0.2025 \\ \hline
Sundanese & 2.9151 $\pm$ 1.8701 & 38.0154 $\pm$ 21.1152 & 0.7447 $\pm$ 0.4526 & 6.7278 $\pm$ 5.8818 & 0.4759 $\pm$ 0.3581 & 0.0217 $\pm$ 0.0063 & 0.0014 $\pm$ 0.2095 & 0.0040 $\pm$ 0.1690 & -0.0127 $\pm$ 0.199 \\ \hline
Swahili & 2.8426 $\pm$ 1.8273 & 36.7274 $\pm$ 21.6291 & 0.6900 $\pm$ 0.4117 & 6.2376 $\pm$ 5.5779 & 0.5004 $\pm$ 0.3981 & 0.0217 $\pm$ 0.0064 & 0.0005 $\pm$ 0.2129 & 0.0071 $\pm$ 0.1719 & 0.0000 $\pm$ 0.1767 \\ \hline
Swedish & 2.4565 $\pm$ 1.7945 & 34.4340 $\pm$ 21.1383 & 0.6789 $\pm$ 0.4336 & 5.2523 $\pm$ 5.0355 & 0.4806 $\pm$ 0.4066 & 0.0225 $\pm$ 0.0311 & -0.0136 $\pm$ 0.2135 & 0.0005 $\pm$ 0.1513 & -0.0001 $\pm$ 0.1455 \\ \hline
Tagalog & 2.3192 $\pm$ 1.7408 & 31.9010 $\pm$ 21.4409 & 0.5976 $\pm$ 0.4236 & 5.0015 $\pm$ 4.9439 & 0.4048 $\pm$ 0.3955 & 0.0207 $\pm$ 0.0077 & -0.0007 $\pm$ 0.172 & -0.0025 $\pm$ 0.1454 & -0.0115 $\pm$ 0.1486 \\ \hline
Tajik & 3.6327 $\pm$ 1.8610 & 44.0494 $\pm$ 20.7996 & 0.9278 $\pm$ 0.4599 & 8.6401 $\pm$ 6.3315 & 0.6218 $\pm$ 0.4344 & 0.0244 $\pm$ 0.0400 & 0.0136 $\pm$ 0.2585 & 0.0178 $\pm$ 0.2109 & 0.0062 $\pm$ 0.2101 \\ \hline
Tamil & 4.0502 $\pm$ 2.0863 & 49.8349 $\pm$ 21.9587 & 1.0498 $\pm$ 0.5492 & 10.1832 $\pm$ 6.6707 & 0.6747 $\pm$ 0.5646 & 0.0229 $\pm$ 0.0040 & -0.0168 $\pm$ 0.2599 & -0.0021 $\pm$ 0.1816 & -0.0029 $\pm$ 0.1866 \\ \hline
Tatar & 5.2999 $\pm$ 1.8633 & 55.9961 $\pm$ 21.5646 & 1.3090 $\pm$ 0.5083 & 13.9721 $\pm$ 6.4386 & 0.8622 $\pm$ 0.4289 & 0.0232 $\pm$ 0.0030 & -0.0388 $\pm$ 0.3227 & -0.0181 $\pm$ 0.2454 & -0.027 $\pm$ 0.2726 \\ \hline
Telugu & 3.7709 $\pm$ 1.8097 & 47.7764 $\pm$ 21.5175 & 0.9947 $\pm$ 0.4750 & 9.6866 $\pm$ 6.7028 & 0.5838 $\pm$ 0.3905 & 0.0229 $\pm$ 0.0040 & -0.0151 $\pm$ 0.2313 & 0.0024 $\pm$ 0.1797 & -0.0041 $\pm$ 0.1871 \\ \hline
Thai & 4.2052 $\pm$ 1.9011 & 52.3345 $\pm$ 22.5921 & 1.1586 $\pm$ 0.4787 & 11.6755 $\pm$ 6.6255 & 0.7069 $\pm$ 0.4669 & 0.0231 $\pm$ 0.0032 & -0.0014 $\pm$ 0.2581 & 0.0023 $\pm$ 0.2101 & 0.0036 $\pm$ 0.2037 \\ \hline
Turkish & 4.2634 $\pm$ 1.8235 & 50.1228 $\pm$ 21.2174 & 1.0676 $\pm$ 0.4914 & 10.8648 $\pm$ 6.4401 & 0.7234 $\pm$ 0.4525 & 0.0230 $\pm$ 0.0038 & -0.0038 $\pm$ 0.2788 & 0.0021 $\pm$ 0.2076 & -0.0241 $\pm$ 0.2286 \\ \hline
Turkmen & 4.6553 $\pm$ 1.8657 & 52.6054 $\pm$ 22.3110 & 1.0889 $\pm$ 0.4439 & 11.8237 $\pm$ 6.7022 & 0.8106 $\pm$ 0.4712 & 0.0231 $\pm$ 0.0034 & -0.0115 $\pm$ 0.2965 & 0.0030 $\pm$ 0.2239 & -0.0224 $\pm$ 0.2394 \\ \hline
Ukrainian & 3.5216 $\pm$ 1.9053 & 42.8676 $\pm$ 20.1638 & 0.8892 $\pm$ 0.4196 & 7.7542 $\pm$ 5.6084 & 0.6736 $\pm$ 0.4630 & 0.0226 $\pm$ 0.0047 & 0.0040 $\pm$ 0.2281 & 0.0050 $\pm$ 0.1930 & 0.0020 $\pm$ 0.2158 \\ \hline
Urdu & 4.6288 $\pm$ 1.8598 & 51.8213 $\pm$ 21.1784 & 1.1069 $\pm$ 0.4547 & 10.3984 $\pm$ 6.1689 & 0.7847 $\pm$ 0.4910 & 0.0233 $\pm$ 0.0027 & -0.0079 $\pm$ 0.2553 & 0.0049 $\pm$ 0.2023 & -0.0079 $\pm$ 0.2119 \\ \hline
Uyghur & 4.4782 $\pm$ 2.1602 & 50.8991 $\pm$ 21.7484 & 1.1623 $\pm$ 0.5640 & 11.4469 $\pm$ 6.9056 & 0.7948 $\pm$ 0.6075 & 0.0229 $\pm$ 0.0039 & -0.0251 $\pm$ 0.2666 & -0.0173 $\pm$ 0.2182 & 0.0017 $\pm$ 0.2251 \\ \hline
Uzbek & 4.4620 $\pm$ 1.8179 & 51.5030 $\pm$ 21.9078 & 1.1060 $\pm$ 0.4634 & 11.0946 $\pm$ 6.6179 & 0.8107 $\pm$ 0.4898 & 0.0231 $\pm$ 0.0034 & -0.001 $\pm$ 0.284 & -0.0021 $\pm$ 0.222 & -0.0121 $\pm$ 0.253 \\ \hline
Vietnamese & 3.0241 $\pm$ 1.8411 & 38.3290 $\pm$ 20.3990 & 0.7485 $\pm$ 0.4310 & 6.8053 $\pm$ 5.7475 & 0.5340 $\pm$ 0.4371 & 0.0221 $\pm$ 0.0056 & -0.0001 $\pm$ 0.196 & -0.0021 $\pm$ 0.18 & -0.0063 $\pm$ 0.2022 \\ \hline
Welsh & 2.8045 $\pm$ 1.8026 & 36.3720 $\pm$ 20.9603 & 0.7044 $\pm$ 0.4320 & 6.6622 $\pm$ 5.8721 & 0.4618 $\pm$ 0.3712 & 0.0218 $\pm$ 0.0063 & 0.0021 $\pm$ 0.2133 & 0.0083 $\pm$ 0.1573 & -0.0056 $\pm$ 0.171 \\ \hline
Xhosa & 4.1014 $\pm$ 1.9087 & 48.1886 $\pm$ 21.5661 & 0.9588 $\pm$ 0.4316 & 9.3353 $\pm$ 5.9511 & 0.7828 $\pm$ 0.5315 & 0.0235 $\pm$ 0.0238 & -0.0245 $\pm$ 0.2816 & -0.0022 $\pm$ 0.2329 & -0.0001 $\pm$ 0.2472 \\ \hline
Yiddish & 2.1669 $\pm$ 1.6737 & 37.7814 $\pm$ 19.9013 & 0.7111 $\pm$ 0.3881 & 4.8267 $\pm$ 4.6606 & 0.3952 $\pm$ 0.4116 & 0.0222 $\pm$ 0.0055 & -0.0121 $\pm$ 0.1721 & 0.0025 $\pm$ 0.1317 & -0.0069 $\pm$ 0.1388 \\ \hline
Yoruba & 2.7452 $\pm$ 1.9641 & 34.5736 $\pm$ 21.7557 & 0.6621 $\pm$ 0.4389 & 6.0572 $\pm$ 5.6836 & 0.5101 $\pm$ 0.4320 & 0.0211 $\pm$ 0.0072 & -0.0102 $\pm$ 0.2006 & 0.0027 $\pm$ 0.1647 & 0.0033 $\pm$ 0.1639 \\ \hline
Zulu & 3.0442 $\pm$ 1.9465 & 38.9466 $\pm$ 21.0533 & 0.7747 $\pm$ 0.4438 & 6.7784 $\pm$ 5.7828 & 0.5749 $\pm$ 0.4456 & 0.0220 $\pm$ 0.0059 & -0.0023 $\pm$ 0.2255 & 0.0024 $\pm$ 0.1712 & 0.0030 $\pm$ 0.1826 \\ \hline
\end{tabular}%
}
\end{table*}
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\fi
\bibliographystyle{./bibtex/IEEEtran}
|
1,314,259,996,710 | arxiv | \section{Introduction}
As the impacts of climate change continue to be felt worldwide, policies to reduce greenhouse gas emissions and promote renewable energy generation are of increasing importance. One approach that encapsulates many policies is market-based solutions. The most well-known of the policies which fall under this umbrella are carbon cap-and-trade (C\&T) markets.
In carbon C\&T markets, regulators impose a limit on the amount of carbon dioxide ($\text{CO}_2$) that regulated firms can emit during a certain time period (referred to as a compliance period). They also distribute allowances (credits) to individual firms in the amount of this limit, each allowing for a unit of $\text{CO}_2$ emission, usually one tonne. Firms must offset each of their units of emissions with an allowance, or face a monetary penalty for each allowance they are lacking. These allowances are tradable assets, allowing firms who require more credits than what they were allocated to buy them, and firms who require less to sell them. In this way, C\&T markets aim to find an efficient way of allocating the costs of $\text{CO}_2$ abatement across the regulated firms.
In practice, these systems regulate multiple consecutive and disjoint compliance periods, which are linked together through mechanisms such as \textit{banking}, where unused allowances in period-$n$ can be carried over to period-$(n+1)$. Other linking mechanisms include \textit{borrowing} from future periods (where a firm may reduce its allotment of allowances in period-$(n+1)$ in order to use them in period-$n$) and \textit{withdrawal}, where non-compliance in period-$n$ reduces period-$(n+1)$ allowances by the amount of non-compliance (in addition to the monetary penalty previously mentioned).
A closely related alternative to these cap-and-trade markets are \textit{renewable energy certificate} markets (REC markets). A regulator sets a floor on the amount of energy generated from renewable sources for each firm (based on a percentage of their total energy generation), and provides certificates for each MWh of energy produced via these means\footnote{Not all generators of renewable energy who participate in REC markets are regulated Load Serving Entities (LSEs), though in this work, we largely focus on the decisions faced by those who are regulated.}. \textcolor{black}{This is also known as a Renewable Portfolio Standard (RPS).} To ensure compliance, each firm must surrender certificates totaling the floor at the end of each compliance period, with a monetary penalty paid for each lacking certificate. The certificates are traded assets, allowing regulated Load Serving Entities (LSEs) to make a choice about whether to produce electricity from renewable means themselves, or purchase the certificates on the market (or a mix of both).
REC markets can be used to encourage growth of a particular type of renewable energy. The most notable of these systems are Solar REC markets (SREC markets), which have been implemented in many areas of the northeastern United States\footnote{The largest and most mature SREC market in North America is the New Jersey SREC Market}, and are the focus of this work.
The similarities between carbon cap-and-trade markets and SREC markets are clear. However, there are also some notable differences. One key difference between the SREC market and traditional carbon cap-and-trade markets is the uncertainty in the former market is the supply of certificates (driven by some generation process), while in the latter, the uncertainty is in the demand for allowances (driven by an emissions process). In SREC markets, banking is typically implemented, but borrowing and withdrawal are not. Broadly speaking, SREC markets can be considered the inverse of a cap-and-trade system.
The existing literature on SREC markets largely focus on certificate price formation. \cite{coulon_khazaei_powell_2015} presents a stochastic model for SREC generation. They also calibrate it to the New Jersey SREC market, and ultimately solve for the certificate price as a function of economy-wide generation capacity and banked SRECs, and investigates the role and impact of regulatory parameters on these markets. The volatility of REC prices has been noted in other works, such as \cite{amundsen2006price} and \cite{hustveit2017tradable}. The latter focuses on the Swedish-Norwegian electricity certificate market and develops a stochastic model to analyze price dynamics and policy. \cite{khazaei2017adapt} studies an alternate design scheme for SREC markets and shows how it can stabilize SREC prices.
Additionally, there are extensive studies of the carbon cap-and-trade markets, particularly in developing stochastic equilibrium models for emissions markets. \cite{hitzemann2018equilibrium} presents a general stochastic framework for firm behaviour leading to the expression of allowance price as a strip of European binary options written on economy-wide emissions. Agents' optimal strategies and properties of allowance prices are also studied by \cite{carmona2010market} and \cite{seifert_uhrig-homburg_wagner_2008} within a single compliance period setup, with the former also making significant contributions through detailed analyses of potential shortcomings of these markets and their alternatives. \cite{carmona_fehr_hinz_2009} also proposes a stochastic equilibrium model to explain allowance price formation and develop a model where abatement (switching from less green to more green fuel sources) costs are stochastic. There is also significant work on structural models for financial instruments in emissions markets, such as \cite{howison_schwarz_2012} and \cite{carmona_coulon_schwarz_2012}.
Our contribution addresses a natural question in these systems; how should regulated LSEs behave? Here, we use stochastic control techniques to characterize firm specific optimal behaviour through generation and trading and discuss potential takeaways from a market design perspective. We believe these results are of interest to both regulators, the designers of SREC markets (\textcolor{black}{and REC markets in general}), and the firms regulated by them.
Specifically, we explore a cost minimization problem of a single regulated firm in a single-period SREC market with the goal of understanding their optimal behaviour as a function of their current level of compliance and the market price of SRECs. To this end, we pose the problem as a continuous time stochastic control problem. We provide the optimality conditions, and analyze the form of the optimal controls in feedback form to illuminate features of the solution. In addition, we numerically solve for the optimal controls of the regulated firm as generation and trading costs vary, including a detailed analysis of various scenarios and sample paths. We also explore the sensitivity of the optimal controls to the various parameters in the model. We extend these results to a single regulated firm in a multi-period SREC market.
There are several differences between our work and the extant literature. Firstly, we focus on the SREC market, which is a new and burgeoning market and there are few studies (in comparison to carbon C\&T markets). Secondly, we focus on the optimal behaviour of firms, something that has not been studied in SREC markets. In the carbon literature, prior works formulate a stochastic control problem in order to better understand the behaviour of the allowance prices, while we begin with an SREC price process (which regulated agents affect by trading and generation) and are interested in how the agent should optimally behave. We assume that agents affect the SREC price process in a manner similar to the permanent price impact models in the optimal execution literature (see \cite{almgren2001optimal}, \cite{cartea2015algorithmic}).
The remainder of this work is organized as follows. \textcolor{black}{\Cref{srec_markets} provides a background on REC markets in practice and SREC markets in particular.} \Cref{model} discusses our model and poses the general optimal behaviour problem in continuous time. \Cref{optimality} presents optimality results in a continuous time setting. \Cref{discrete} provides a discrete time formulation and numerically solves the dynamic programming equation to characterize the optimal behaviour of a regulated firm. Finally, in \Cref{results}, we present the results of our work including sensitivity analysis.
\section{SREC Market Overview} \label{srec_markets}
\textcolor{black}{RPS regulations have been instituted in numerous regions around the globe. In this section, we provide a brief overview of their use, with a particular focus on RPS regulations in the United States. These regulations aim to promote the production of electricity via solar energy (among potentially other energies) through the use of SREC markets. While we focus on the United States RPS regulations and their associated REC / SREC markets in this section, we note that RPS regulations have also been instituted around the globe, including China, Sweden, and Norway, among others. }
\textcolor{black}{Roughly 30 US states have enacted RPS regulations, see \cite{kolesnikoff_cleveland_shields_2019}. These regulations typically apply to investor owned utilities (IOUs) which are private LSEs (as opposed to municipal or state LSEs). Such IOUs supply electricity to the grid as part of their business-as-usual operations. They receive a (tradable) REC for each MWh of electricity they generate from renewable means. The RPS requires that regulated LSEs submit RECs annually in an amount proportional to their total electricity supply. They face a monetary penalty for any RECs they lack under the amount required by the RPS.}
\textcolor{black}{REC markets also include players who are not regulated by the RPS, but may have the ability to produce RECs. Often these are individuals who have attached solar panels to their place of residence, registered with the tracking authority, and sell the resulting RECs on the market.}
\textcolor{black}{RPS regulations can be stratified further. Many of the 30 US states that have RPS' have also instituted a `carve-out' for solar energy -- that is, a specification that a certain proportion of the renewable energy generated must come from solar means. This results in an SREC market, as opposed to a general REC market -- here, the certificates represent the solar nature of the generated energy. As such, the IOUs in such states must specifically generate solar energy (or purchase SRECs from an IOU / individual that has) in order to comply with the RPS regulation that applies to them. Unused certificates can be banked for a given amount of years before expiring \footnote{In New Jersey's SREC market, unused SRECs can be banked for four additional years, giving them a five-year life in total.}. This results in different `vintages' of SRECs in the market, depending on the year the SREC was produced (as that will impact how many years it can continue being banked into the future). In this work, we make a simplifying assumption that firms can bank SRECs indefinitely to avoid dealing with multiple vintages of SRECs and as including multiple vintages does not add more insight into the problem.
}
\textcolor{black}{The solar carve out is typically not large, relative to the overall distribution of electricity generation. New Jersey's SREC market, the largest and most mature in North America, has a solar carve-out of just 5.1\% of overall electricity sales in 2021, after which the state is transitioning to a new, currently undetermined solar energy generation incentive (see \cite{lane_2020}). In other states such as Washington D.C. (who are continuing their SREC programs for the foreseeable future), the solar carve out is planned to reach 10\% of overall electricity sales by 2041. }
\textcolor{black}{As New Jersey's market is the most mature of the North American SREC markets, we discuss it in more detail. While there is a dearth of academic literature regarding these markets, useful background information regarding it can be found on \cite{nj_power}. Despite the winding down of the New Jersey SREC market, discussion of it is nonetheless useful in order to better understand what a relatively mature SREC market looks like. In general, SREC markets still figure to be an important component of energy policy, with states like Maryland and Washington D.C. (as alluded to above) continuing to develop their own solar carve outs and associated SREC markets. Additionally, REC markets in general will continue to grow in importance in the future, and our work applies to REC markets in general.}
\textcolor{black}{Now focusing specifically on the NJ SREC market, we plot the number of SRECs issued in \cref{fig:issued_SRECs}. We retrieve this data from PJM-GATS, the administrator which tracks the New Jersey SREC market (see \cite{pjm}). The figure shows consistent increases in generated SRECs, as well as the seasonality effect. This latter property is natural due to reduced sunlight in winter months. As each SREC corresponds to a MWh of electricity generated from solar means, the figure suggests that market's monthly solar generation nears 40,000 MWh at its peak (around June 2019). From this, we can see the notable growth of the NJ SREC market. Over this time, the NJ SREC market has undergone numerous (significant) regulatory changes. These included changes to the requirement schedule, the penalty schedule, as well as the rules around banking of unused SRECs. The most notable of these changes occurred in 2012, where the regulatory body drastically decreased the non-compliance penalty, increased the SREC requirement, and allowed for extended banking of unused SRECs. We do not discuss these changes further in this section, except to remark that they did occur, and have contributed to the observed patterns.}
\begin{figure}[!t]
\begin{minipage}[t]{0.31\textwidth}
\centering
\includegraphics[align = c, width=\textwidth]{issued_srecs.pdf}
\caption{Issued SRECs in New Jersey SREC Market from 2008 - 2019}
\label{fig:issued_SRECs}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\textwidth}
\centering
\includegraphics[align = c, width=\textwidth]{traded_srecs.pdf}
\caption{Traded SRECs (all vintages) in New Jersey SREC Market from 2008 - 2019}
\label{fig:traded_SRECs}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\textwidth}
\centering
\includegraphics[align = c, width=\textwidth]{srec_price.pdf}
\caption{Weighted average SREC price (across all vintages) in New Jersey SREC Market from 2008 - 2019}
\label{fig:srec_price}
\end{minipage}
\end{figure}
\textcolor{black}{This growth is also apparent when we plot the magnitude of trading activity (of all SREC vintages) over time, as in \cref{fig:traded_SRECs}. Once again, we see notable seasonality, with the peaks occurring around June and October of each year; this has to do with the compliance dates of the energy year in the NJ SREC market (the energy year runs from June 1 - May 31), along with a `true-up' period. This is a period of six months, from June 1 to Nov 30, which is the time span firms have between the end of the energy year and the compliance date when they must submit their SRECs.}
\textcolor{black}{Next, we plot the weighted average monthly SREC price from 2008 onward, in \cref{fig:srec_price}. This is the average price of each SREC vintage sold in each month, weighted by the relative proportion of each vintage in the market. In practice, older vintages will typically trade at a slight discount to the current vintage, as they can be banked for less time in the future, on account of being older. We plot the weighted average for visual simplicity, because the difference between prices of older vintages and the current vintage tends to be rather small, and because the most recent vintage tends to have higher trading volumes.}
\textcolor{black}{From \cref{fig:srec_price}, we see a large amount of variance in the price in the early stages of the SREC market. The NJ SREC market initially sustained high SREC prices (close to the non-compliance penalty). However, as time passed, increased investment into solar generation led to oversupply, leading to lower prices. In conjunction with the aforementioned regulatory changes that lowered non-compliance penalties and increased the SREC requirement, this resulted in the notable drop in SREC price which occurred throughout 2011 and 2012. Since then, price changes have been less dramatic. For further information on the New Jersey SREC market and its price history, please see \cite{coulon_khazaei_powell_2015}, \cite{khazaei2017adapt}.}
\textcolor{black}{Finally, we remark on the number of agents regulated by SREC markets. As mentioned earlier, all electricity suppliers (equivalently, LSEs) in the licensed within the region over which the market is enforced must comply with the RPS obligations. In New Jersey, this includes hundreds of such firms (see \cite{supplier_list_2020} and \cite{nj_power}), in addition to the various other players who do not have RPS obligations, but have solar facilities that allow them to produce and sell SRECs.}
\section{The SREC Generation and Price Impact Model} \label{model}
\subsection{SREC Market Rules}
We assume the following rules for the SREC market, which are exogenously specified and fixed. In an $n$-period framework, a firm is obliged to submit $(R_1, ..., R_n)$ SRECs at the end of the compliance periods $[0, T_1], ..., [T_{n-1}, T_n]$, respectively. \textcolor{black}{As discussed in the previous section, the requirement that regulated firms are subject to is based on a proportion of the electricity they supply to the grid. We make a simplifying assumption, similar to \cite{coulon_khazaei_powell_2015}, that this requirement is instead exogenous.}
For the period $[T_{i-1}, T_i]$, firms pay $P_i$ for each SREC below $R_i$ at $T_i$. Firms receive an SREC for each MWh of electricity they produce through solar energy. We assume firms may bank leftover SRECs not needed for compliance into the next period, with no expiry on SRECs. This is a simplifying assumption we make -- many SREC markets have limitations on how long an SREC can be banked for (e.g., in New Jersey's SREC market, an SREC can be banked for a maximum of four years). This assumption reduces the dimensionality of the state space. After $T_n$, all SRECs are forfeited.
A single period framework follows the rules above with $n = 1$. For convenience, we remove the subscripts in the notation for the terms defined above when discussing a single-period framework. That is, the regulated firm is required to submit $R$ SRECs at time $T$, representing their required production for the compliance period $[0, T]$. A penalty $P$ is imposed for each missing SREC at time $T$. The firm considers any costs/profits arising from the SREC system after $T$ to be immaterial.
\subsection{Firm Behaviours}
We first consider a single firm who is optimizing their behaviour in a single compliance period SREC system. A regulated firm can control their planned generation rate (SRECs/year) at any given time $(g_t)_{{t\in\mfT}}$ (where ${\mathfrak{T}}:=[0,T]$) and their trading rate (SRECs/year) at any given time $(\Gamma_t)_{{t\in\mfT}}$. The processes $g$ and $\Gamma$ constitute the firm's controls.
The trading rate may be positive or negative, reflecting that firms can either buy or sell SRECs at the prevailing market rate for SRECs. Firms also incur a trading penalty of $\frac{1}{2}\gamma \Gamma_t^2$, $\gamma > 0$, per unit time. This induces a constraint on their trading speed. In general, the quadratic penalty could be replaced by any convex function of $\Gamma_t$.
In an arbitrary time period $[t_1, t_2]$, The firm aims to generate $\int_{t_1}^{t_2} g_t \,dt$, but actually generates $\int_{t_1}^{t_2} g_t^{(r)} dt = \int_{t_1}^{t_2} g_t dt + \int_{t_1}^{t_2} \nu_t dB_t^{(1)}$, where $\nu_t$ is a deterministic function of time, and $\nu_tdB_t^{(1)}$ may be interpreted as the generation rate uncertainty at $t$. We assume that a firm has a baseline deterministic generation level $h_t$ (SRECs/year), below which there is no cost of generation. Methods similar to \cite{coulon_khazaei_powell_2015} may be used to estimate $h_t$. We assume that $h_t < \infty$ for all $t$. Increases in planned generation from their baseline production incurs the cost $C(g, h) := \frac{1}{2}\zeta (g - h)_+^2$ per unit time.
\textcolor{black}{This choice of cost $C$ may be viewed as a `rental' cost for temporarily increasing SREC generation capacity, as opposed to an `investment' cost. It is as though the firm is renting additional capacity (i.e. solar panels) with which they are able to increase their SREC generation. However, this additional cost does not result in any long-run increases to their baseline production $h_t$. It is certainly possible to model instead expansion efforts rather than `rental' costs, but we leave such extensions for future work.
}
In \cite{aid2017coordination}, the authors utilize a similar cost structure in the context of expanding solar capacity. There, costs are quadratic, but not one-sided. Lastly, we note our choice of $C$ is both differentiable and convex; any choice of $C$ with these properties could be used instead in order for the analysis contained in this paper to be valid.
All processes are defined on the filtered probability space $(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t\geq0}, \mathbb{P})$, where $\mathbb{F}$ is the natural filtration generated by the SREC price. The set of admissible controls ${\mathcal{A}}$ equals the set of all progressively measurable (with respect to $\mathbb{F}$) processes $(g_t,\Gamma_t)_{t\in\mfT}$ such that $\mathbb{E}[\int_0^T g_t^2\, dt ] < \infty$, $\mathbb{E}[\int_0^T\Gamma_t^2\,dt] < \infty$, and $g_t \geq 0$ for all ${t\in\mfT}$.
At time $t$ the firm holds $b_t^{g, \Gamma}$ SRECs and the (controlled) SREC price is denoted $S_t^{g, \Gamma}$.
The various processes satisfy the stochastic differential equations (SDEs)
\begin{subequations}
\begin{align}
S_t^{g, \Gamma} &= S_0 + \int_0^t\left( \mu_u + \eta\, \Gamma_u - \psi \,g_u \right) du - \int_0^t \psi\, \nu_u\, dB_u^{(1)} + \int_0^t \sigma_u\, dB_u^{(2)} , \quad \text{and} \label{eq:S} \\
b_t^{g, \Gamma} &= b_0 + \int_0^t (g_u + \Gamma_u)\, du + \int_0^t \nu_u dB_u^{(1)},
\end{align}%
\label{eqn:S-and-b-SDE}
\end{subequations}%
where $B=(B^{(1)}_t,B^{(2)}_t)_{t\ge0}$ is a standard two-dimensional Brownian Motion
and $\mu, \sigma, \nu$ are deterministic functions. We further assume $\int_0^T \sigma_u^2 du < \infty$ and $\int_0^T \nu_u^2 du < \infty$. As the SDE above indicates, trading ($\int \Gamma_u du$) and realized generation ($\int g_u^{(r)} du$) impact the SREC price linearly. As such, our model is similar to the price impact models commonly studied in optimal execution problems. Buying (selling) of SRECs pushes the price up (down) and generation
pushes the price downwards. In this way, a firm's behaviour impacts the rest of the market. SREC inventory ($b_t^{g, \Gamma}$) accumulates by both trading and generation activity.
\textcolor{black}{We observe that this exogenous specification of $S_t^{g, \Gamma}$ does not necessarily converge to any particular value as $t \rightarrow T$. A hallmark of many equilibrium pricing models in both the C\&T and SREC literature is that the price of the certificate (allowance) converges to either the non-compliance penalty or $0$, depending on the compliance of the economy as a whole (see \cite{coulon_khazaei_powell_2015}, \cite{hitzemann2018equilibrium}, \cite{seifert_uhrig-homburg_wagner_2008}, among others). However, this work takes a different approach. While the value of an SREC to an \textit{individual} firm at time $T$ is $0$ (if the firm has complied) or $P$ (if the firm has not complied), this does not imply that the price that the market bears will necessarily be one of these two. This is because the SREC (or indeed, C\&T) market comprises of \textit{many} regulated firms, some of whom will find the certificate valueless, and some of whom will find it worth $P$. The relative proportions of agents who have complied or failed to comply will inform the certificate price and render it in the interval $[0,P]$ rather than $0$ or $P$ only.}
For any admissible strategy $g,\Gamma\in{\mathcal{A}}$, a regulated firm's performance criterion (at time $t$) for the single-period problem is
\begin{multline}
J^{g, \Gamma}(t, b, S) = -\mathbb{E}_{t,b,S}\left[\int_t^T C(g_u, h_u) \,du \right.
\\
+ \int_t^T \Gamma_u\, S_u^{g, \Gamma} \,du
+ \left. \tfrac{\gamma}{2} \int_t^T \Gamma_u^2\, du
+ P (R - b_T^{g, \Gamma})_+\right], \label{eq:pc}
\end{multline}
where $\mathbb{E}_{t,b,S}\left[\cdot\right]$ denotes taking expectation conditioned on $b_t = b$ and $S_t=S$ and in the sequel, $\mathbb{E}_t[\cdot]:=\mathbb{E}[\cdot|\mathcal{F}_t]$ and $\mathbb{P}_t[\cdot]:=\mathbb{P}[\cdot|\mathcal{F}_t]$.
The firm's cost minimization is the strategy which attains the sup (if it exists) below and the value of the optimal strategy is
\begin{equation}
V(t, b, S) = \sup_{(g_s, \Gamma_s)_{s\in[t,T]} \in {\mathcal{A}}} J^{g,\Gamma}(t, b, S). \label{eq:vf}
\end{equation}
In the next section, we characterize the optimal trading strategy and the relationship to SREC price using the stochastic maximum principle as well as the dynamic programming equation approach.
\section{Continuous time approach} \label{optimality}
\subsection{Stochastic Maximum Approach}
One approach to solving \eqref{eq:vf} is through
the Stochastic (Pontryagin) Maximum Principle (see the seminal works of \cite{kushner1972necessary} and \cite{peng1990general}). Here, we apply the stochastic maximum principle to our problem along the lines of \cite{hitzemann2018equilibrium}. In doing so, we characterize the optimal controls as a system of coupled equations. The key result is contained in the following proposition.
\begin{proposition}[Optimality Conditions]\label{opt_cond}
The processes $(g,\Gamma)=(g_t, \Gamma_t)_{{t\in\mfT}}$ satisfying
the forward-backward stochastic differential equations (FBSDEs)
\begin{align}
\Gamma_t &= \tfrac{1}{\gamma}\left(M_t - S_0 - \int_0^t (\mu_u + \psi \,g_u)\, du\right),\label{eq:eq_gam}
\\
\Gamma_T &= \tfrac{1}{\gamma} \left(P\, {\mathds 1}_{\{b_T^{g,\Gamma} < R\}} - S_T^{g,\Gamma}\right),
\\
g_t &= \left(h_t + \tfrac{1}{\zeta}\left(Z_t - \psi \int_0^t \Gamma_u du\right)\right){\mathds 1}_{\left\{P\, \mathbb{P}_t(b_T^{g,\Gamma} < R) \geq - \psi \,\mathbb{E}_t[\int_t^T \Gamma_u du]\right\}}\,,\label{eq:eq_g}
\\
g_T &= \tfrac{1}{\zeta}\left(P\, {\mathds 1}_{\left\{b_T^{g,\Gamma} < R\right\}} + \zeta\, h_t\right),
\end{align}
for all ${t\in\mfT}$, where the processes $(M,Z)=(M_t,Z_t)_{{t\in\mfT}}$ are martingales, are the optimal controls for problem \eqref{eq:vf}.
\end{proposition}
\begin{proof}
The Hamiltonian for the performance criterion \eqref{eq:pc} and state dynamics \eqref{eqn:S-and-b-SDE} is
\begin{multline}
\mathcal{H}(t, b, S, g, \Gamma, \boldsymbol{y}, \boldsymbol{z}) =
- \tfrac{\zeta}{2} ((g - h_t)_+)^2 - S \Gamma - \tfrac{\gamma}{2} \Gamma^2
\\
+ y_{b}\, (g + \Gamma) + y_{S} (\mu_t + \eta \Gamma - \psi g) + \sigma_t z_S - \psi \nu_t z_{S,b} + \nu_t z_b,
\label{eq:hamiltonian}
\end{multline}
where $\boldsymbol{y}=(y_b,y_S), \boldsymbol{z} =
\begin{bmatrix}
z_b & z_{bS} \\
z_{Sb} & z_S
\end{bmatrix}$.
This is concave in the controls $g,\Gamma$ and state variables $b,S$. Moreover, the adjoint processes $(y_b, y_S)=(y_{b,t},y_{S,t})_{{t\in\mfT}}$ satisfy the BSDEs
\begin{subequations}
\label{eqn:yb-and-ys-SDE}
\begin{align}
dy_{b, t} &= z_{b, t}\, dB_t^{(1)} + z_{bS, t}\, dB_t^{(2)},
&
y_{b, T} = P\, {\mathds 1}_{\{b_T^{g,\Gamma} < R\}}.
\\
dy_{S, t} &= \Gamma_t \,dt + z_{S,t}\, dB_t^{(2)} + z_{Sb, t}dB_t^{(1)},
&
y_{S, T} = 0.
\end{align}
\end{subequations}
The stochastic maximum principle implies that if there exists a solution $(\boldsymbol{\hat{y}}, \boldsymbol{\hat{z}})$ to \eqref{eqn:yb-and-ys-SDE}, then a strategy $(g, \Gamma)$ that maximizes $\mathcal{H}(t, b, S, g, \Gamma, \boldsymbol{\hat{y}}, \boldsymbol{\hat{z}})$ is the optimal control.
As both BSDEs have linear drivers, their solution is straightforward (see \cite{pham2009continuous}, Chapter 6) and given by
\begin{equation}
y_{b, t} = P\, \mathbb{P}_t(b_T^{g,\Gamma} < R)\,,
\qquad \text{and}
\qquad
y_{S, t} = -\mathbb{E}_t\left[\int_{t}^T \Gamma_u du\right]\,.
\label{eqn:y-adjoint-sol}
\end{equation}
Differentiating the Hamiltonian with respect to the controls, we obtain the first order conditions
\begin{subequations}
\begin{align}
\frac{\partial \mathcal{H}}{\partial \Gamma} &: \quad y_{b} + \eta\, y_{S} - S - \gamma\, \Gamma = 0,\qquad \text{and} \\
\frac{\partial \mathcal{H}}{\partial g} &: \quad y_{b} - \psi\, y_{S} - \zeta\, (g - h_t)_+ = 0,
\end{align}%
\end{subequations}%
and substituting the solutions to the adjoint processes \eqref{eqn:y-adjoint-sol}, we obtain the optimality conditions
\begin{subequations}
\begin{align}
P\,\mathbb{P}_t(b_T^{g,\Gamma} < R) - \eta\,\mathbb{E}_t\left[\int_{t}^T \Gamma_u du\right] - S_t^{g,\Gamma} - \Gamma_t \,\gamma &= 0, \qquad \text{and}
\label{eq:optGam}
\\
P \,\mathbb{P}_t(b_T^{g,\Gamma} < R) + \psi\, \mathbb{E}_t\left[\int_{t}^T \Gamma_u du\right] - \zeta\, (g_t - h_t)_+ &= 0.
\label{eq:optg}
\end{align}
\end{subequations}
We next, aim to solve these equations by isolating $g$ and $\Gamma$.
First, from \eqref{eq:optGam} we have
\begin{equation}
Y_t + \eta \int_0^t \Gamma_u du - S_t^{g,\Gamma} = \Gamma_t \gamma,
\label{eqn:GammaBSDE-step1}
\end{equation}
where $Y=(Y_t)_{{t\in\mfT}}$ is the Doob-martingale defined by
\begin{equation}
Y_t = P\, \mathbb{P}_t(b_T^{g,\Gamma} < R) - \eta\, \mathbb{E}_t\left[\int_0^T \Gamma_u du\right]\,.
\end{equation}
Rearranging \eqref{eqn:GammaBSDE-step1} and substituting in \eqref{eq:S}, we arrive at \eqref{eq:eq_gam} where the terminal condition follows immediately from \eqref{eq:optGam}, and $M= (M_t)_{t \in [0, T]}$ is the martingale defined by
\begin{equation}
M_t = Y_t - \int_0^t \sigma_u dB_u^{(2)} + \psi \int_0^t \nu_u dBu^{(1)}\,.
\end{equation}
For \eqref{eq:optg}, consider a modification $\mathcal{A}^U$ of the set of admissible controls $\mathcal{A}$ to controls that admit a finite upper bound $U > \sup_{{t\in\mfT}} h_t$\footnote{Any bound that is lower is practically meaningless as firms must be able to generate at or more than their `baseline' generation rate.}.
When $g_t \geq h_t$, the solution to \eqref{eq:optg} is
\begin{equation}
g_t = h_t + \tfrac{1}{\gamma}K_t, \qquad
\text{where}
\qquad
K_t = P\,\mathbb{P}_t(b_T < R) + \psi\,\mathbb{E}_t\left[\int_t^T \Gamma_u du\right]
\label{eq:g_soln}
\end{equation}
Define $g_t^\star:=h_t + \tfrac{1}{\gamma}K_t$.
When, $g_t<h_t$, the Hamiltonian is maximized at
\begin{equation}
g_t = \begin{cases}
U, & \text{if } K_t \geq 0, \\
0, & \text{otherwise}.
\end{cases} \label{eq:g_star2}
\end{equation}
Denote the sets
\begin{equation*}
A := \left\{g_t \ge h_t\right\}, \quad
B:=\left\{\{g_t < h_t\} \cap \left\{ K_t \geq 0\right\}\right\}, \quad \text{and}
\quad
C:= \left\{\{g_t < h_t\} \cap \left\{ K_t < 0\right\}\right\}.
\end{equation*}%
These sets satisfy the property that $A = (B \cup C)^c$. From \eqref{eq:g_soln} and \eqref{eq:g_star2}, the optimal generation rate is therefore
\begin{align}
g_t = g_t^\star\, {\mathds 1}_{A}+ U\, {\mathds 1}_{B}.
\label{g_t_star}
\end{align}
Consider the set $A^\star = \{g_t^\star \geq h_t\}$.
\begin{lemma}
If
$U>\sup_{{t\in\mfT}}~h_t$, then $A^\star = A$.
\end{lemma}
\begin{proof}
Take an event $\omega\in A$, by \eqref{g_t_star} $g_t(\omega) = g_t^\star(\omega)$ and so $g_t^\star(\omega) \ge h_t$, and hence $\omega\in A^\star$. Therefore $A \subset A^\star$.
Take an event $\omega\in A^\star$, so that $g_t^\star(\omega)\ge h_t$. As
$g_t(\omega) = g_t^\star(\omega) {\mathds 1}_{\{\omega\in A\}}+ U\, {\mathds 1}_{\{\omega\in B\}}$ and $U \geq\sup_{{t\in\mfT}} h_t$, we must have that $g_t(\omega) \geq h_t$, and thus $\omega\in A$. Therefore, $A^\star \subset A$.
\end{proof}
Therefore, we can rewrite \eqref{g_t_star} as follows:
\begin{equation}
g_t = g_t^\star\, {\mathds 1}_{A^\star} + U\, {\mathds 1}_{B}
\label{g_t_star_upd}
\end{equation}
Furthermore, $B=\emptyset$ because, from \eqref{g_t_star_upd}, $\omega\in B \implies g_t(\omega) = U > h_t$, so $\omega\notin B$.
Therefore, we obtain
\begin{equation}
g_t = \begin{cases}
h_t + \frac{1}{\gamma}K_t, & \text{if } K_t \ge 0, \\
0 & \text{otherwise}.
\end{cases} \label{eq:g_eqn}
\end{equation}
On the set ${K_t\ge 0}$, by adding and subtracting $\psi \int_0^t \Gamma_u du$ to $g_t$ and letting $Z=(Z_t)_{{t\in\mfT}}$ be the Doob-martingale defined by
\begin{equation}
Z_t = P \,\mathbb{P}_t(b_T < R) + \psi\, \mathbb{E}_t\left[\int_{0}^T \Gamma_u du\right]
\end{equation}
we obtain \eqref{eq:eq_g} with the terminal condition obtained from \eqref{eq:optg}. As (i) this solution is independent of $U$ for all $U>\sup_{{t\in\mfT}}h_t$, (ii) $\sup_{{t\in\mfT}}h_t<\infty$,
and (iii) $\mathcal{A}=\lim_{U\to\infty}\mathcal{A}^U$, this completes the proof.
\end{proof}
We end this subsection with a few comments regarding the results of this proposition and interpretations of the optimality conditions.
In comparison to \cite{hitzemann2018equilibrium}, where the authors develop optimality conditions in a carbon C\&T system, our results shows that the trading penalty and the impact of trading and generation on SREC prices modifies the optimality conditions. When $\eta = \psi = 0$, \eqref{eq:optg} reduces to $P\, \mathbb{P}_t(b_T < R) = \zeta (g_t - h_t)_+$. This is similar to the result that the marginal cost of generation is equal to the product of the penalty and probability of non-compliance found in \cite{hitzemann2018equilibrium}. Moreover, when $\eta = \psi = 0$, \eqref{eq:optGam} reduces to $P \,\mathbb{P}_t(b_T < R) - \gamma \Gamma_t = S_t$. Thus, in our setup, the SREC price equals the penalty scaled by the probability of non-compliance but modified by the optimal trading of the firm.
Similar behavior persists in the general case when $\eta > 0, \psi > 0$. From \eqref{eq:optGam}, the SREC price equals the penalty scaled by the probability of non-compliance, but modified by the time-$t$ marginal cost of the firm's trading and our expectations of their future trading. That is, low prices are associated with high rate of trading and high expected future rate of trading.
From \eqref{eq:optg}, the penalty scaled by the probability of non-compliance equals the difference between the marginal cost of generation and re-scaled (by $\psi$) expected future trading.
As well, from the form of $g_t$ in \eqref{eq:g_eqn}, the firm plans to either generate above their baseline or not at all. The decision is contingent on the inequality $P \mathbb{P}_t(b_T < R) \geq - \psi \mathbb{E}_t[\int_t^T \Gamma_u du]$. If the inequality is satisfied, then the firm will generate above their baseline, and if not, they cease to generate. Intuitively, this condition represents whether the firm benefits enough from generation to offset the potentially negative influence of their price impacts. The term $P \mathbb{P}_t(b_T < R)$ represents the expected non-compliance cost avoided by acquiring an additional SREC, while $- \psi \mathbb{E}_t[\int_t^T \Gamma_u du]$ represents the value lost through the impact of generating an additional SREC. Generation puts downwards pressure on $S$ (through \eqref{eq:S}), which the firm realizes through their expected trading level over the remainder of the compliance period. Note that $- \psi \mathbb{E}_t[\int_t^T \Gamma_u du]$ is only positive if expected future trading is negative - that is, the firm expects to be a seller of SRECs.
We outline two simple examples to demonstrate the effect of this property. Consider a firm that is far from compliance, and thus, $\mathbb{E}_t[\int_t^T \Gamma_u du] > 0$ (that is, the firm expects to purchase SRECs during $[t, T]$). The indicator in \eqref{eq:g_eqn} (and consequently, \eqref{eq:eq_g}) is always satisfied, and the firm will generate above the baseline $h$. This is consistent with the behaviour of a firm that has not reached compliance and is striving to acquire enough SRECs to hit the requirement $R$. Conversely, consider a firm that has SRECs well in excess of $R$ and plans to sell them over the remainder of the compliance period. Here, the indicator is not satisfied. That is, an additional generated SREC will not help the firm's compliance probability significantly (if at all), and additional generation will decrease the SREC price $S$, reducing the revenue generated by the firm through sales. As such, the firm chooses not to generate at all in such a scenario, and in doing so, mitigates the negative effect generation has on SREC prices.
We can solve Equations \eqref{eq:eq_gam}-\eqref{eq:eq_g} numerically using Least Square Monte Carlo techniques. Instead, we consider a dynamic programming approach to solving the original problem \eqref{eq:vf}.
\section{Discrete time version of problem} \label{discrete}
Thus far, we formulated the cost minimization problem of a single regulated firm using continuous time optimal control techniques to characterize the solution and tease out some essential features of the optimal strategy. To obtain numerical solutions, however, we solve a discrete time version of the problem which we find has better numerical stability. Indeed, a discrete time formulation more closely approximates practice, as regulated firms typically take actions only at discrete time points within a compliance period.
To this end, let $n$ be the number of decision points within a single compliance period,
which occur at $0 = t_1 < t_2 < ... < t_n < T = t_{n+1}$. For simplicity, we assume these are equally spaced so that $t_k=k\Delta t$.
The control processes $(g, \Gamma)$ are now piecewise constant within $[t_{i},t_{i+1})$, and the firm controls $\{g_{t_i}, \Gamma_{t_i}\}_{ i \in \mathfrak{N}}$ where $ \mathfrak{N}:=\{0,\dots,n\}$, so that at each time point, the regulated firm chooses their trading and generating behaviour over the next interval of length $\Delta t$. In this section, $(g, \Gamma)$ represent vectors whose elements are these controls.
Under the same assumptions as earlier, the performance criterion (corresponding to the total cost) for an arbitrary admissible control is
\begin{align}
J^{g, \Gamma}(m, b, S)
= \mathbb{E}_{t_m,b,S}\left[ \sum_{i=m}^n \left\{ \tfrac{\zeta}{2} ((g_{t_i} - h_{t_i})_+)^2 + \Gamma_{t_i} S_{t_i}^{g, \Gamma} + \tfrac{\gamma}{2} \Gamma_{t_i}^2\right\} \Delta t + P (R - b_T^{g, \Gamma})_+\right],
\label{eq:discretepc}
\end{align}
In the above, the dynamics of the state variables ($b, S$) are modified for discrete time to
\begin{subequations}
\begin{align}
S_{t_i}^{g, \Gamma} &= \min\left(\left(S_{t_{i-1}}^{g, \Gamma} + \left(\mu + \eta \,\Gamma_{t_{i-1}} - \psi\, g_{t_{i-1}}\right) \Delta t - \psi \nu \sqrt{\Delta t}\, \varepsilon_{t_i} + \sigma \sqrt{\Delta t}\, Z_{t_i}\right)_+,\; P\right) \label{eq:6}\\
b_{t_i}^{g, \Gamma} &= b_{t_{i-1}}^{g, \Gamma} + (g_{t_{i-1}} + \Gamma_{t_{i-1}})\Delta t + \nu \sqrt{\Delta t} \varepsilon_{t_i} \label{eq:7}
\end{align}%
\label{eqn:DiscreteStateEvolution}
\end{subequations}%
where $Z_{t_i}, \varepsilon_{t_i} \sim N(0, 1)$, iid, for all $i \in \mathfrak{N}$.
Note that \ref{eq:6} is the discrete time analogue of \eqref{eq:S} capped at $P$ and floored at $0$. The cap and floor ensures that SREC prices remain in the closed interval $[0,P]$ as prices outside this interval cannot occur in real markets.
We aim to optimize \eqref{eq:discretepc} with respect to $(g, \Gamma)$ and determine the value of the position of the regulated firm, as well as their optimal behaviour. Hence, we seek
\begin{align}
V(t, b, S) = \inf_{g, \Gamma\in{\mathcal{A}}} J^{g, \Gamma}(t, b, S), \label{eq:DiscreteValueFunc}
\end{align}
and the strategy that attains the inf, if it exists. Applying the Bellman Principle to \eqref{eq:DiscreteValueFunc} implies
\begin{subequations}
\begin{align}
\begin{split}
V(t_i, b, S) &= \inf_{g_{t_i}, \Gamma_{t_i}} \biggl\{
\left(\tfrac{\zeta}{2} ((g_{t_i} - h_{t_i})_+)^2
+ \Gamma_{t_i} S_{t_i}^{g, \Gamma}
+ \tfrac{\gamma}{2} \Gamma_{t_i}^2\right) \Delta t
\\ & \qquad\qquad\quad
+ \mathbb{E}_{t_i}\left[V\left(t_{i+1}, b_{t_{i+1}}^{g, \Gamma}, S_{t_{i+1}}^{g, \Gamma}\right)\right] \biggr\}, \qquad\qquad \text{and}
\label{eq:Discrete-Bellman}%
\end{split}%
\\
V(T, b, S) &= P (R - b)_+\,. \label{eq:Discrete-Bellman-terminal}
\end{align}%
\label{eqn:DiscreteBellman}%
\end{subequations}%
In the next section, we provide a numerical scheme for solving this optimization problem.
\section{Solution Algorithm and Results} \label{results}
\textcolor{black}{The remainder of the paper is contained in this section and we briefly pause here to provide an overview of its contents.}
\textcolor{black}{We begin by discussing the parameter choices for the numerical experiments we run. In the best scenario, we would calibrate our model to real-world data, where possible. This is difficult for several reasons. First, our framework requires the costs that regulated firms experience in the SREC market, through their planned generation and trading activities. Such information is, however, only known to industry insiders who trade and generate their SRECs. Instead, for the parameters that do not have clear real-world values to tether to, we choose what we feel are sensible values that allow us to understand the general behaviours and principles that govern this dynamical system.}
\textcolor{black}{Following the parameter choice, we present our algorithm to solve \eqref{eq:DiscreteValueFunc} in detail. From here, we transition into the presentation of the numerical simulations themselves. Naturally, there are many possible interesting scenarios to simulate. We present only a fraction of what is possible and theoretically interesting here, and leave to the reader to explore others. Instead, we select a handful of interesting scenarios that reveal important relationships between the firm's optimal behaviour and the state processes, and discuss them in depth. These include an overall summary of the regulated firm's optimal controls as a function of both state variables, simulated compliance periods, summary statistics, and sensitivity analysis. The code for this project can be found \href{https://github.com/AShrivats/SREC_Single_Player.git}{here}.}
\subsection{Parameter Choice}
For the first set of numerical experiments, we use the parameters reported in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.
\begin{table}[h]
\begin{minipage}[c]{0.54\textwidth}
\begin{center}
\begin{tabular}{ccccc}
\toprule\toprule
$n$& $T$ &$P$ (\textdollar/SREC) &$R$& ${h_t}$ (SREC/y) \\
\midrule
50 & 1& 300 & 500 & 500 \\
\bottomrule\bottomrule
\end{tabular}
\caption{Compliance parameters.
\label{tbl:ComplianceParams}}
\end{center}
\end{minipage}
\begin{minipage}[c]{0.45\textwidth}
\begin{center}
\begin{tabular}{ccccccc}
\toprule\toprule
$\mu$ & $\sigma$ & $\nu$ & $\psi$ &$\eta$ &$\zeta$ & $\gamma$ \\
\midrule
0 & 10 & 10 & 0.01 & 0.01 & 0.6 & 0.6 \\
\bottomrule\bottomrule
\end{tabular}
\caption{Model Parameters. \label{tbl:ModelParams}}
\end{center}
\end{minipage}
\end{table}
These parameters are chosen for illustrative purposes. As discussed in the preamble of this section, calibration to a particular firm is itself a non-trivial problem and requires proprietary knowledge of a firm's cost function and baseline production (which also varies significantly from firm to firm). Instead, we provide broad-level intuition regarding the optimal behaviour of a firm in a single-period SREC market with reasonable parameters. The penalty of $P = \$300$ is informed by the New Jersey SREC market, where the non-compliance penalty in compliance period ending May 2018 is $\$308$ \textcolor{black}{(see \cite{SRECTrade})}. \textcolor{black}{In practice, as the requirement $R$ is based on a proportion of sales for each regulated firm, the specific level of $R$ should be tied to sales. We choose $R$, however, to be exogenous as opposed to a stochastic process depending on electricity sales in order to simplify the analysis (this assumption is present in other works, such as \cite{coulon_khazaei_powell_2015}).} The choice ${h_t} = \frac{R}{T}$ implies the regulated firm has a probability of $0.5$ to comply if they simply plan to generate at their baseline rate and do not partake in the SREC market.
The values of $\zeta$ and $\gamma$ are motivated by the upper bounds they imply for $g_t, \Gamma_t$. Specifically, consider the case of a firm that cannot generate enough solar energy to meet the requirements, and hence will fail to comply. The benefit of generating SRECs is to reduce their non-compliance obligation, and with each generated SREC their obligation is reduced by $P$. Therefore, the costs and benefits of generation over a time-step are (independent of trading activity), respectively,
\begin{equation}
K_1(g_t) = \tfrac{1}{2}\zeta((g_t - {h_t})_+)^2\Delta t, \qquad\text{and} \qquad
B_1(g_t) = P g_t \Delta t\,.
\end{equation}
The firm generates energy in order to minimize $N_1(g_t) := K_1(g_t) - B_1(g_t)$ which occurs at $g_t^* = \tfrac{P}{\zeta} + {h_t}$. For the chosen parameters, $g^* = 1,000$ which is exactly twice the baseline rate ${h_t}$. In other words, this choice of $\zeta$ ensures the firm's maximum generation rate is bounded by twice their baseline.
We conduct a similar exercise for $\Gamma_t$. Consider a firm that will fail to comply.
In this scenario, a rational firm will purchase SRECs. As before, the benefit of a firm purchasing SRECs is to reduce their non-compliance obligation, with each generated SREC reducing the obligation by $P$. As such, the costs and benefits to purchase over the next time-step are (independent of generation activity):
\begin{equation}
K_2(\Gamma_t) = \left(\tfrac{1}{2}\gamma\,\Gamma_t^2 + S_t\, \Gamma_t\right)\Delta t,
\qquad\text{and}\qquad
B_2(\Gamma_t) = P\, \Gamma_t\, \Delta t,
\end{equation}
respectively. The firm purchases in order to minimize $N_2(\Gamma_t) := K_2(\Gamma_t) - B_2(\Gamma_t)$ which occurs at $\Gamma_t^* = \tfrac{P - S}{\gamma}$. For the chosen parameters, this is maximized when $S=0$ and results in $\Gamma^* = 500$. The significance of this computation is to show we have chosen parameters that result in a reasonable upper bound on the amount of trading a firm will partake in.
Repeating the same exercise for a firm that is guaranteed to comply (and thus is motivated to sell), we obtain $g_t^* = 0$ (due to price impacts of generation) and $\Gamma_t = -\tfrac{S}{\gamma}$ which is maximized (in absolute value) at $-500$ for the chosen parameters.
For the parameters in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}, this simple analysis shows that generation and trading rates are restricted to the range $g_t \in [0, 1000]$ and $\Gamma_t \in [-500, 500]$, which is a reasonable range of possible values given our choices of ${h_t}$ and $R$.
We set $\eta = 0.01, \psi = 0.01$ to demonstrate the effect of price impact. We justify these choices in a similar manner to the above. While $\eta$ and $\psi$ do not need to be equal in our model, generation and purchasing are substitutes for one another, and thus, it is logical to consider them as having equal market impacts. Above, we discussed natural bounds for the control processes, given the compliance and model parameters in $\cref{tbl:ComplianceParams}$ and $\cref{tbl:ModelParams}$. For these parameter choices, price impact parameters $\eta = 0.01, \psi = 0.01$ results in upper bounds for net price impacts of $500 \times 0.01 \times \tfrac{1}{50} = 0.1$ per time-step for trading and $1,000 \times \ 0.01 \times \tfrac{1}{50} = 0.2$ per time-step for generation. These are sizeable impacts for a single firm to have, but not so large that the model seems implausible.
In \Cref{sensitivity}, we consider other parameters. In particular, we explore how various levels of $\zeta, \gamma$ impact firm behaviour and the effect of other price impact parameters ($\psi \neq 0, \eta \neq 0$). In the following subsection, we detail our algorithm to solve the dynamic programming problem outlined in \Cref{discrete}.
\subsection{Numerical Scheme} \label{num_scheme}
We use the following numerical algorithm for solving \eqref{eqn:DiscreteBellman} with state variable dynamics in \eqref{eqn:DiscreteStateEvolution}:
\begin{enumerate}
\item Choose a grid of $b$ and $S$ values denoted by ${\mathfrak{G}}$. We use a uniform grid of $401$ points in $b$ from $0$ to $2R$, so that $R$ is on the grid, and a uniform grid of $S$ with $\Delta S = \sqrt{3 \Delta t} \sigma$ and lower and upper bounds of $0$ and $P$ respectively. In this manner, the number of grid points in $S$ is tuned to the volatility over a time-step\footnote{As with any numerical solution, there is a trade-off between grid size (accuracy of the dynamic program solution) and run-time. The grid we use provides an acceptable trade-off between these two, and we observed no further increase in accuracy by increasing the grid size.
\item Minimize \eqref{eq:Discrete-Bellman} at $i=n$ (corresponding to $t=T-\Delta t$) with respect to $(g_{t_n}, \Gamma_{t_n})$ for every point in ${\mathfrak{G}}$.
To do this, we require an estimate of $\mathbb{E}_{t_n}\left[V\left(t_{n+1}, b_{t_{n+1}}^{g_{t_n}, \Gamma_{t_n}}, S_{t_{n+1}}^{g_{t_n}, \Gamma_{t_n}}\right)\right]$
for each
$(S_{t_n},b_{t_n})\in{\mathfrak{G}}$. This is achieved by simulation as follows:
\begin{enumerate}[label=\emph{\Alph*.}]
\item Select a value $b_{t_n}\in{\mathfrak{G}}$. As the terminal condition is independent of $S_{t_{n+1}}^{g_{t_n}, \Gamma_{t_n}}$, the optimal controls and value function for $b_{t_n}$ will be the same for all values of $S_{t_n}$. That is, the evolution of the SREC price is unimportant at the last time-step.
\begin{enumerate}
\item Select a candidate pair $(g_{t_n},\Gamma_{t_n})
\begin{enumerate}[label=\emph{(\alph*)}]
\item Simulate $100$ scenarios of $b_{t_{i+1}}^{g_{t_i}, \Gamma_{t_n}}$ using \eqref{eq:7}, -- use the same set of random numbers for all points in ${\mathfrak{G}}$.
\item For each simulated $b_{t_{n+1}}^{g_{t_n},\Gamma_{t_n}}\,$, calculate the one-step-ahead
value function $V\left(t_{n+1}, b_{t_{n+1}}^{g_{t_i},\Gamma_{t_n}},S_{t_{n+1}}^{g_{t_n},\Gamma_{t_n}}\right)$ through the terminal condition \eqref{eq:Discrete-Bellman-terminal}
\item Use the empirical mean of the result of (c) as an estimate of the true mean at $b_{t_n}$, which will be the same regardless of $S_{t_n}$.
\end{enumerate}
\item Use Matlab's \texttt{fmincon} function to determine next candidate pair $(g_{t_i},\Gamma_{t_i})$ and repeat from (i) until converged, store optimal pair and value function.
\end{enumerate}
\item Go to next grid point in ${\mathfrak{G}}$ repeat from A.
\end{enumerate}
\item Step backwards from $i+1$ to $i$, by minimizing \eqref{eq:Discrete-Bellman} with respect to $(g_{t_i}, \Gamma_{t_i})$ at time $t_i$ for all points in ${\mathfrak{G}}$.
To do this, we require an estimate of $\mathbb{E}_{t_i}\left[V\left(t_{i+1}, b_{t_{i+1}}^{g_{t_i}, \Gamma_{t_i}}, S_{t_{i+1}}^{g_{t_i}, \Gamma_{t_i}}\right)\right]$
for each
$(S_{t_i},b_{t_i})\in{\mathfrak{G}}$. This is achieved by simulation as follows:
\begin{enumerate}[label=\emph{\Alph*.}]
\item Select a pair $(S_{t_i},b_{t_i})\in{\mathfrak{G}}$
\begin{enumerate}
\item Select a candidate pair $(g_{t_i},\Gamma_{t_i})
\begin{enumerate}[label=\emph{(\alph*)}]
\item Simulate $b_{t_{i+1}}^{g_{t_i}, \Gamma_{t_i}}$ using \eqref{eq:7}, -- use the same set of random numbers for all points in ${\mathfrak{G}}$.
\item Simulate $100$ scenarios of $S_{t_{i+1}}^{g_{t_i},\Gamma_{t_i}}$ by applying \eqref{eq:6} -- use the same set of random numbers for all points in ${\mathfrak{G}}$.
\item For each simulated pair of $(b_{t_{i+1}}^{g_{t_i},\Gamma_{t_i}}\,,\,S_{t_{i+1}}^{g_{t_i},\Gamma_{t_i}})$, estimate the one-step-ahead value function $V\left(t_{i+1}, b_{t_{i+1}}^{g_{t_i},\Gamma_{t_i}},S_{t_{i+1}}^{g_{t_i},\Gamma_{t_i}}\right)$ by interpolation.
\item Use the empirical mean of the result of (c) as an estimate of the true mean at $(b_{t_i},S_{t_i})$.
\end{enumerate}
\item Use Matlab's \texttt{fmincon} function to determine next candidate pair $(g_{t_i},\Gamma_{t_i})$ and repeat from (i) until converged, store optimal pair and value function.
\end{enumerate}
\item Go to next grid point in ${\mathfrak{G}}$ repeat from A.
\end{enumerate}
\end{enumerate}
This procedure provides an estimate of the value function at all grid points ${\mathfrak{G}}$ and at all times $\mathfrak{T}:=\{t_i\}_{i\in\mathfrak{N}}$, as well as the optimal generation and trading rates on ${\mathfrak{G}}\times\mathfrak{T}$.
\textcolor{black}{In the following subsections, we apply this scheme in a variety of simulation studies to learn more about the optimal behaviours of the regulated firms, and their associated implications.}
\subsection{Optimal Behaviours of a Regulated Firm}
A regulated firm's optimal behaviour is one of the key outputs from solving the Bellman equation. \Cref{fig:opt_beh} shows the dependence of the optimal trading and generation rate on banked SRECs for three SREC prices at six points in time.
\begin{figure}[!htp]
\centering
\includegraphics[align = c, width=0.6\textwidth]{sg_optimal_behaviour.pdf}
\includegraphics[align = c, width=0.1\textwidth]{legend.pdf}
\caption{Optimal firm behaviour (top panel: generation rate, bottom panel: trading rate) as a function of banked SRECs for various time-steps and SREC market prices. Parameters in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.}
\label{fig:opt_beh}
\end{figure}
The most notable feature is the distinct regimes of generation/trading. For low levels of banked SRECs and near the terminal date, the firm generates/purchases until the marginal cost of producing/purchasing another SREC exceeds $P$, as the firm is almost assured to fail to comply. This follows the classic microeconomic adage of conducting an activity until the marginal benefit from the activity equals the marginal cost. In this regime, the marginal benefit of an additional SREC to the firm is $P$, as each additional SREC lowers their non-compliance obligation by $P$.
As the banked amount increases, the firm reaches a point where the marginal benefit from an additional SREC decreases from $P$. This occurs as the probability of compliance becomes non-negligible, as additional SRECs in excess of $R$ provide smaller marginal benefit than $P$. This is a result of the sale price of an SREC being bounded above by $P$ and leads to a decrease in optimal generation and optimal trading. The firm adjusts its behaviour so that its marginal costs are in line with this marginal benefit. This eventually leads to the firm selling as opposed to purchasing SRECs, as the net proceeds from the sale exceed the marginal value of retaining those certificates.
This decrease continues until the firm no longer benefits from additional SRECs. That is, at a certain level of banked SRECs $b$, the marginal benefit of an additional SREC is zero. Specifically, having an additional SREC does not increase the firm's likelihood of compliance, nor can they sell the additional SREC to make a profit. Accordingly, at all time-steps except $t = 50$, we observe that optimal generation jumps downwards once a certain level of $b$ is achieved. This would not occur if price impacts were inactive ($\eta = \psi = 0$). This is consistent with the theoretical results in \Cref{optimality}, where we showed that the optimal generation is either (i) greater than or equal to $h_t$, or (ii) identically $0$ (see \eqref{eq:g_eqn}). Recall, the condition for the firm to choose to generate is $P \mathbb{P}_t(b_T^{g,\Gamma} < R) \geq - \psi \mathbb{E}_t[\int_t^T \Gamma_u du]$. Thus, the firm chooses to shut down after reaching a threshold level of $b$ (which depends on both $t$ and $S$). Intuitively, this threshold is point at which an additional SREC is worth less to the firm than the impact that additional generation would have on the firm through its effect on $S$. As generation lowers SREC price, a firm that has already complied and plans to sell off remaining SRECs is pushing the market against themselves by continuing to generate. As such, there is a point where it is instead optimal to shut down production entirely and sell. This does not occur at $t = 50$, as it is the last decision point, and thus the price impact of generation does not impact the firm in any way.
Trading is influenced by a change in SREC price, which is in accordance with our intuition and aligns with the theoretical results from \Cref{optimality}. As SREC prices increase, the regulated firm chooses to purchase less, regardless of banked SRECs. We also see that higher SREC prices generally imply higher generation, as the firm chooses to generate their own SRECs, either to avoid paying high prices for them in the market, or to sell in the market and capitalize on the high prices (which of these two factors is the larger contributor depends on how much is banked).
If we hold $b, S$ constant, generation and purchasing are increasing in $t$. This is natural when the firm's compliance is not guaranteed, as with less time until the end of a compliance period, the firm needs to accumulate more SRECs in order to comply. For values of $b$ and $S$ for which compliance is guaranteed, we note that this property will not always hold, and is dependent on the value of $\gamma$. This is covered in more detail in \Cref{sens_gen_trade}.
We also note that the optimal trading rate each of the `plateaus' varies slightly with time-step. This can be seen more clearly in \Cref{fig:sg_pi_demonstration},
where we enlarge the bottom-right plot in \Cref{fig:opt_beh} for two different areas of $b$ as an illustration of this property. That is, we plot optimal trading behaviour for each of the six time-steps detailed in \Cref{fig:opt_beh} when $S = 251$, $b \in [0, 40]$ (left) and $b \in [500, 1000]$ (right).
\begin{figure}[!t]
\centering
\includegraphics[align = c, width=0.4\textwidth]{sg_pi_demonstration1.pdf}
\includegraphics[align = c, width=0.4\textwidth]{sg_pi_demonstration2.pdf}
\includegraphics[align = c, width=0.1\textwidth]{legend.pdf}
\caption{Optimal trading behaviour with price impact parameters $\eta=0.01$, $\psi=0.01$ for $b \in [0, 40]$ (left), $b \in [500, 1000]$ (right) and $S_0 = 250$ as a function of banked SRECs for various time-steps. All remaining parameters in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.}
\label{fig:sg_pi_demonstration}
\end{figure}
As \Cref{fig:sg_pi_demonstration} illustrates, at high levels of banking ($b > R$), firms sell less at earlier time-steps than they do at later time-steps. Firms do this to mitigate the impact that their selling has on the SREC price and limit the extent to which the market moves against the firm as a result of their trading behaviour. The inverse behaviour occurs for low banking levels. Firms purchase less at earlier time-steps in order to keep prices down (relative to what would occur if they did not) and make compliance more attainable. At low banking levels, they also generate more, for the same reason (not shown to avoid repetition). These effects are proportional to the magnitude of $\eta$ and $\psi$, and increase with $S$. The generation rate for large values of $b$ does not vary with time-step due to the lower bound of generation being $0$.
\subsection{Sample Paths} \label{sample_paths}
In \Cref{fig:path}, we show the dynamics of optimal firm behaviour through the compliance period. Here, $S_0 = 150$, $b_0 = 0$, and we simulate a path for $S$ and $b$ and at each time-step along this path, we adopt the optimal firm strategy in accordance with their banked SRECs and the SREC price
\begin{figure}[t!]
\centering
\includegraphics[width=.75\textwidth]{sg_path.pdf}
\caption{A sample path of optimal firm behaviour with initial condition $S_0 = 150$ and $b_0 = 0$, and all remaining parameters in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.}
\label{fig:path}
\end{figure}
From \Cref{fig:path}, the regulated firm banks SRECs at what appears to be a steady rate, and in this sample path, the firm reaches compliance. We will see shortly that the latter does not always occur. While banked SRECs appear to be linear, there is some variation in the amount of SRECs the firm banks at each time-step, which is the result of the SREC production noise the firm experiences. If we were to plot $b_t - \tfrac{Rt}{T}$ as a measure of the firm's SREC inventory versus the pro-rated amount they would need to be on-track to comply, we would see a roughly similar shape to the firm's cumulative production noise over the course of the period.
Turning our attention to the other subplots, we see the generation and trading processes exhibit notable variation over time. In particular, the inverse relationship between SREC price and trading rate is evident at earlier points in the period. Similarly, we can observe a positive relationship between SREC price and planned generation rate during the same time frame. However, as the period progresses, generation and trading begin to move in the same direction, regardless of $S$ and its movements. This occurs as the randomness associated with SREC generation buffets the firm and changes their banked SRECs from one time-step to another in a way that cannot be foreseen. As $t$ approaches $T$, the firm has less time to adjust for this unforeseen noise resulting in the observed firm behaviour later in the period. The firm may have significantly more or less SRECs than what their planned generation and trading activity would suggest, and thus, they must determine whether they need to increase their SREC acquisition rate (increase planned generation and purchase more) or decrease their SREC acquisition rate (decrease planned generation and sell more) in order to behave optimally. We re-state that excess SRECs above $R$ expire valueless, so there is incentive for the firm to liquidate excess SRECs if in a strong position for compliance.
\textcolor{black}{The SREC price itself is also pushed downwards throughout the period by the actions of the agent. As the agent is generating SRECs and selling them, the SREC price is lower than what it would be if we had set $\eta = \psi = 0$.}
In Figure \ref{fig:path}, we see that cumulative production noise (the lowest subplot) decreases for the vast majority of the period. This means the firm generates less than planned in this time. As a reaction to this, they increase their planned generation and trading over the period in order to reach compliance, constantly reacting to their under-generation to put themselves back on track to achieve compliance. In general, towards the end of the period, increases (decreases) in cumulative production noise incite the firm to decrease (increase) planned generation and decrease (increase) purchasing of SRECs.
The fifth panel in \Cref{fig:path} shows instantaneous incurred costs (IIC), which is the running cost incurred to the firm at each time-step:
\begin{equation}
IIC_i = \left(\tfrac{\zeta}{2} ((g_{t_i} - h_{t_i})_+)^2
+ \Gamma_{t_i} S_{t_i}^{g, \Gamma}
+ \tfrac{\gamma}{2} \Gamma_{t_i}^2\right) \Delta t.
\end{equation}
For the parameters chosen in Figure \ref{fig:path} and the resulting optimal behaviours, IIC is negative at all time-steps, which signifies that the firm is making a profit in the system, due to their sale of SRECs.
Next, by performing multiple simulations, we investigate the distribution of various quantities of interest, including total SRECs $b_T$, total planned generation $\int_0^T g_u du$, total traded amount $\int_0^T \Gamma_u du$, and total profit (negative of costs). For the base-line parameter choice, and with initial condition $b_0=0$, $S_0 = 150$, we present summary statistics using $1,000$ simulated paths of $S$ and $b$ in \Cref{tbl:sum_stat_simple}.
\begin{table} [h]
\begin{center}
\begin{tabular}{crrrrrr}
\toprule\toprule
Statistic & Mean & Std.Dev &1st Quartile & 3rd Quartile & Skewness & Kurtosis \\
\midrule
$b_T$ & 501.61 & 1.62 & 500.50 & 502.74 & -0.02 & 2.69 \\
$\int_0^T g_u du$ &621.95 & 6.59 & 617.46 & 626.25 & -0.001 & 3.02 \\
$\int_0^T \Gamma_u du$ &-120.10 &6.31 & -124.34 & -115.79 & 0.02 & 2.86 \\
Profit & 8,730.00 & 940.00 & 8,080.00 & 9,360.00 & 0.02 & 2.91 \\
\bottomrule\bottomrule
\end{tabular}
\end{center}
\caption{Summary statistics using 1,000 sample paths of $S$ of banked amount, generation, trading, and profit following the optimal strategy with initial condition $S_0=150$, $b_0=0$ and all remaining parameters in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.} \label{tbl:sum_stat_simple}
\end{table}
In this one-period setup, the firm's optimal behaviour results in a symmetric distribution centred just above the requirement of $500$. There are cases (approximately 25\% of simulations) where the firm fails to comply ($b_T < 500$). Since $b$ (conditional on the firm's controls) is stochastic, and there is no advantage to additional SRECs above the requirement in a single-period framework, firms must strike a balance between being certain of compliance and wasting funds planning to generate or purchase SRECs over the requirement that may potentially end up unused. As such, for these parameters, the optimal firm plans to acquire (represented by $\int_0^T (g_u + \Gamma_u) du$) slightly more than the requirement of $500$, providing themselves with some buffer throughout the period in the event that they produce less than planned. However, this buffer is not so large that the firm is guaranteed to always comply.
\subsection{Parameter Sensitivity} \label{sensitivity}
In this section we investigate how varying parameters affect the optimal behaviour and resulting summary statistics, and explore the intuition behind the resulting effects.
\subsubsection{Sensitivity to Price Impact} \label{pimpacts}
In this section, we explore the impact of changing $\eta$ and $\psi$ on the optimal controls of the regulated firm. To do this, we compare an optimally behaving firm in a single-period model that is subject to various price impact scenarios to the baseline scenario of $\eta = \psi = 0.01$. We consider the $(\eta, \psi)$ pairs of $\{(0, 0), (0.01, 0.01), (0.02, 0.02)\}$. To do this, we simulate $1,000$ paths of $S$ in each price impact scenario, using the same random numbers in each scenario for $S$\footnote{While the random numbers used to generate paths are identical, the presence of price impact leads to different paths as impact varies. } and $\varepsilon$. In each path of $S$, we calculate total generation, total trading, and profit for the firm, and the difference between each quantity and their analogous amount under the baseline scenario. We calculate the mean and standard deviation of these differences across all paths, for each scenario. For example, for a pair $(\eta,\psi)$ we compute Profit($\eta,\psi$)-Profit($0.01,0.01$) across all scenarios and report the mean and standard deviation. In the first row of Table \ref{tab:p_impact_tbl} we report the raw results for the case $\eta=\psi=0.01$, while rows 2--3 report the results for the difference relative to the benchmark for the $2$ remaining pairs of $(\eta,\psi)$.
\begin{table}[htbp]
\color{black}
\centering
\begin{tabular}{rrrrrrrrrrr}
\toprule
\toprule
\multicolumn{1}{c}{$\eta$} & \multicolumn{1}{c}{$\psi$} & & \multicolumn{2}{c}{$\int_0^T g_u \,du$} & & \multicolumn{2}{c}{$\int_0^T \Gamma_u \,du$} & & \multicolumn{2}{c}{Profit} \\
\cmidrule{4-5}\cmidrule{7-8}\cmidrule{10-11} & & & \multicolumn{1}{l}{mean} & \multicolumn{1}{l}{std.dev.} & & \multicolumn{1}{l}{mean} & \multicolumn{1}{l}{std.dev.} & & \multicolumn{1}{l}{mean} & \multicolumn{1}{l}{std.dev.} \\
\cmidrule{1-2}\cmidrule{4-5}\cmidrule{7-8}\cmidrule{10-11}
0.01 & 0.01 & & 621.95 & 6.59 & & -120.10 & 6.31 & & 8,730 & 954.51 \\\midrule
0 & 0 & & 3.69 & 0.25 & & - 4.07 & 0.24 & & 440 & 58 \\
0.02 & 0.02 & & -4.04 & 0.25 & & - 3.92 & 0.27 & & -430 & 57 \\
\bottomrule
\bottomrule
\end{tabular}%
\caption{Mean and standard deviation of differences in quantities of interest between an optimally behaving firm under various price impact scenarios and an optimally behaving firm subject to the baseline scenario of $\eta = \psi = 0.01$. We use 1,000 sample paths of $S$, with initial condition $S_0 = 150$ and remaining parameters as in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}. }
\label{tab:p_impact_tbl}%
\end{table}%
\textcolor{black}{Decreasing $\eta$ and $\psi$ to 0, which removes the impact of the regulated firm in the market altogether, allows the firm to generate more and sell more without any fear of pushing the price downwards and the market against them. This results in a higher profit during the compliance period.}
Increasing $\eta$ and $\psi$ to 0.02 each results in lower generation, less selling, and consequently, lower profit. This is the result of the firm attempting to mitigate their price impact through sales, resulting in lower generation as a consequence (so as to not end up with a large amount of surplus SRECs). In general, price impacts lead to a feedback loop, as the firm's behaviour of generating and selling lowers prices, which further incentivize decreased selling and planned generation (as seen in \Cref{fig:opt_beh}).
\subsubsection{Sensitivity to Trading and Generation Costs}\label{sens_gen_trade}
To conclude our analysis of the single period model, we explore sensitivity to generation and trading speed costs ($\zeta$ and $\gamma$). \Cref{fig:TradingGenerationCost} shows how the optimal behaviour changes for various values of
$\zeta$ and $\gamma$, across six time-steps, for fixed SREC price level $S_t = 150$.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.26\textwidth}
\includegraphics[align = c,width=\textwidth]{zeta02gamma02.pdf}
\caption{\footnotesize$(\zeta, \gamma) = (0.2, 0.2)$}
\end{subfigure}
\begin{subfigure}{0.26\textwidth}
\includegraphics[align = c,width=\textwidth]{zeta06gamma06.pdf}
\caption{\footnotesize$(\zeta, \gamma) = (0.6, 0.6)$}
\end{subfigure}
\begin{subfigure}{0.26\textwidth}
\includegraphics[align = c,width=\textwidth]{zeta1gamma1.pdf}
\caption{\footnotesize$(\zeta, \gamma) = (1, 1)$}
\end{subfigure}
\begin{subfigure}{0.1\textwidth}
\includegraphics[align = c,width=\textwidth]{legend.pdf}
\end{subfigure}
\caption{Optimal generation and trading rates for differing levels of $\zeta$ (generation cost parameter) and $\gamma$ (trading speed penalty parameter) when $S_t=150$. Remaining parameters as in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.}
\label{fig:TradingGenerationCost}
\end{figure}
The middle subplots in \Cref{fig:TradingGenerationCost} show the firm's optimal behaviour in the default setting of $\zeta = 0.6, \gamma = 0.6$ as in \Cref{tbl:ModelParams}. Increasing/decreasing $\zeta, \gamma$ compresses/expands the range of optimal trading and planned generation. This is the result of higher/lower parameters corresponding to higher/lower costs and decreased/increased capacity of the firm to invest in generation and to trade.
Finally, \Cref{fig:TradingGenerationCost}(a) shows that when $b$ is above $R$, optimal planned generation and purchasing are larger at earlier time-steps than later time-steps. This is the result of small $\gamma$ leading to low trading costs, and the firm can aggressively sell excess SRECs before $T$. Hence, at earlier time-steps, the firm continues to generate above their baseline in order to acquire more SRECs to sell later in the period. Later in the period, the firm prefers to liquidate their excess SRECs in order to ensure they do not have excess inventories at time $T$, resulting in the observed behaviour. This does not happen in the cases where $\gamma = 0.6$ or $1$ as the firm is limited in how quickly it can viably liquidate excess SRECs by its trading speed penalty.
We do not include the plots of $(\zeta, \gamma)$ combinations where $\zeta \neq \gamma$ to avoid repetition. The results and interpretation are identical to those discussed above, with changes in $\zeta$ impacting optimal generation and changes in $\gamma$ impacting optimal trading.
\subsection{Multi-period model} \label{multi_per}
Thus far, we have considered a single period compliance framework. In practice, SREC markets consist of multiple periods. In this section, we present the results for an $N$-period SREC market, which is described in \Cref{model}. Much of the behaviour and intuition discussed in the earlier parts of this section carry over to the multi-period case. For the multi-period formulation, we assume there are $n$ (equally spaced) decision points within each compliance period denoted
\begin{equation}
0 = t_1 < \dots < t_n < T_1 = t_{n+1} < \dots < t_{2n} < T_2 = t_{2n +1} < \dots < t_{nN} < T_N = t_{nN +1},
\end{equation}
where $t_k = k\Delta t$. The last time-step $t_{Nn+1}$ is not a decision point. Therefore, there are $n\times N$ decision points, from $t_1, ..., t_{Nn}$. We will use the notation $\mathcal{T}:=\{T_1,\dots,T_N\}$ to denote the set of compliance times.
As before, we continue assuming $P$ and $R$ are constant across each of the $N$ periods, and the processes $g_t, \Gamma_t$ are piecewise constant within $[t_i, t_{i+1})$, with the firm controlling $\{g_{t_i}, \Gamma_{t_i}\}_{i \in \mathfrak{R}}$, where $\mathfrak{R} = \{0, ..., n\times N\}$. As in \Cref{discrete}, regulated firm choose their trading and generating behaviour at the start of the time interval.
The end points of the $i$-th period is $T_i$, $i=1,\dots,N$, and firms may bank unused certificates with no expiry. In real SREC markets, certificates generally have a finite life-time, but allowing indefinite banking reduces the dimensionality of the problem significantly and renders it computationally tractable.
The performance criterion (corresponding to the total cost) for an arbitrary admissible control is
\begin{align} \label{eq:discretepc_multi}
\begin{split}
J^{g, \Gamma}(k, b, S)=&
\mathbb{E}_{t_k,b,S}\biggl[\sum_{i=k}^{Nn} \left\{ \tfrac{\zeta}{2} ((g_{t_i} - h_{t_i})_+)^2 + \Gamma_{t_i} S_{t_i}^{g, \Gamma} + \tfrac{\gamma}{2} \Gamma_{t_i}^2\right\} \Delta t \\
&\quad\quad+ \sum_{j=1}^N P (R - b_{t_{nj}}^{g, \Gamma} - \Delta t (g_{t_{nj}} + \Gamma_{t_{nj}})-\nu \sqrt{\Delta t}\, \varepsilon_{t_{nj+1}})_+ \, {\mathds 1}_{\{t_k < t_{nj +1}\}}\biggr].
\end{split}
\end{align}
The dynamics of the state variables ($b, S$) are modified as follows
\begin{subequations}
\begin{align}
S_{t_i}^{g, \Gamma} &= \min\left(\left(S_{t_{i-1}}^{g, \Gamma} + \left(\mu + \eta \,\Gamma_{t_{i-1}} - \psi\, g_{t_{i-1}}\right) \Delta t - \psi \nu \sqrt{\Delta t}\, \varepsilon_{t_i} + \sigma \sqrt{\Delta t}\, Z_{t_i}\right)_+\;,\; P\right) \label{eq:S_discrete_multi}
\\
b_{t_i}^{g, \Gamma} &=
\left\{
\begin{array}{ll}
b_{t_{i-1}}^{g, \Gamma} + (g_{t_{i-1}} + \Gamma_{t_{i-1}})\Delta t + \nu \sqrt{\Delta t}\, \varepsilon_{t_i}, & t_i\notin \mathcal{T}
\\[0.5em]
\left( b_{t_{i-1}}^{g, \Gamma} + (g_{t_{i-1}} + \Gamma_{t_{i-1}})\Delta t + \nu \sqrt{\Delta t}\, \varepsilon_{t_i}-R\right)_+, & t_i\in \mathcal{T},
\end{array}
\right.
\label{eq:b_discrete_multi}
\end{align}%
\label{eqn:DiscreteStateEvolutionMulti}
\end{subequations}%
where $Z_{t_i}, \varepsilon_{t_i} \sim N(0, 1)$, iid, for all $i \in \mathfrak{N}$.
As in the single-period case, we seek
\begin{align}
V(t, b, S) = \inf_{g, \Gamma\in{\mathcal{A}}} J^{g, \Gamma}(t, b, S), \label{eq:DiscreteValueFuncMulti}
\end{align}
and the strategy that attains the inf, if it exists. Applying the Bellman Principle to \eqref{eq:DiscreteValueFuncMulti} implies
\begin{subequations}
\begin{align}
\begin{split}
V(t_i, b, S) &= \inf_{g_{t_i}, \Gamma_{t_i}} \biggl\{
\left(\tfrac{\zeta}{2} ((g_{t_i} - h_{t_i})_+)^2
+ \Gamma_{t_i} S_{t_i}^{g, \Gamma}
+ \tfrac{\gamma}{2} \Gamma_{t_i}^2\right) \Delta t
\\ & \qquad\qquad
+ \mathbb{E}_{t_i}\left[ P (R - b_{t_{i}}^{g, \Gamma} - \Delta t (g_{t_i} + \Gamma_{t_i}) - \nu \sqrt{\Delta t} \epsilon_{t_{i+1}})_+ \right] {\mathds 1}_{\{t_{i+1}\in\mathcal{T}\}}
\\ & \qquad\qquad
+ \mathbb{E}_{t_i}\left[V\left(t_{i+1}, b_{t_{i+1}}^{g, \Gamma}, S_{t_{i+1}}^{g, \Gamma}\right)\right] \biggr\}, \qquad\qquad \text{and}
\label{eq:Discrete-Bellman_multi}%
\end{split}%
\\
V(T_N, b, S) &= 0. \label{eq:Discrete-Bellman-terminal_multi}
\end{align}%
\label{eqn:DiscreteBellman_multi}%
\end{subequations}%
The dynamics of $b$ in the multi-period framework are such that $b_{T_j}$ represents the firm's SRECs \textbf{after} submitting the compliance requirement for the compliance period ending at $T_j$. We adjust our solution algorithm described in \Cref{num_scheme} to account for the assumptions stated above, using the same model parameters, and choosing $N = 5$. We denote the current period by $m$. As the algorithm for obtaining the optimal controls in the multi-period problem is very similar to that detailed in \Cref{num_scheme}, we omit it here.
\subsubsection{Sample results in the Multi-period model} \label{mpm_pi}
Analogous to \Cref{fig:opt_beh}, \Cref{fig:multi-period-impact}
shows the optimal behaviour of a regulated firm as a function of banked SRECs, across three different prices of $S$ and at six points in time during the first compliance period when there is price impact.
\begin{figure}[h]
\centering
\includegraphics[align = c, width=0.6\textwidth]{p_imp_p1o5.pdf}
\includegraphics[align = c, width=.1\textwidth]{legend.pdf}
\caption{Optimal firm behaviour as a function of banked SRECs across various time-steps (during the first of five compliance periods) and SREC market prices with parameters as in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.}
\label{fig:multi-period-impact}
\end{figure}
In \Cref{fig:multi-period-impact}, we plot the dependence of the optimal generation and trading rate of the firm in the first period ($m = 1$) of the $5$-period model against banked SRECs, for three SREC prices, at six points in time, with all remaining parameters as in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.
Much of the intuition surrounding \Cref{fig:opt_beh} applies here. There are, however, obvious differences between Figures \ref{fig:opt_beh} and \ref{fig:multi-period-impact}. As before, for low levels of banked SRECs, across all values of $S$, and near the end of the compliance period, the firm generates until the marginal cost of producing another SREC exceeds $P$, and purchases until the marginal cost of purchasing another SREC exceeds $P$, as the firm is almost assured to fail to comply. In this regime, the marginal benefit of an additional SREC is $P$, as each additional SREC lowers their non-compliance obligation by $P$.
As the banked amount increases, the firm reaches a point where the marginal benefit from an additional SREC decreases from $P$. This occurs as the probability of compliance becomes non-negligible, as additional SRECs in excess of $R$ provide smaller marginal benefit than $P$. This leads to a decrease in optimal generation and optimal trading, as the firm adjusts its behaviour so that its marginal costs are in line with this marginal benefit. Thus far, this is the same interpretation as the single-period setting.
As $b$ continues to increase, the firm holds sufficient banked SRECs such that they will be able to acquire surplus certificates above $R$. These surplus SRECs have little value in the current period to the firm, even including their use as insurance for extreme under-generation. They may, however, bank SRECs putting the firm in a better position for future compliance periods. In the single-period case, at the end of the compliance period, holding additional SRECs lack utility. The concept of banking means that this is not true in the multi-period case, and thus we see an abrupt change in the slope of the optimal controls, and a slower decay in generation and purchasing rate when compared to \Cref{fig:opt_beh}.
This decrease continues until the firm no longer benefits from additional SRECs. That is, at a certain level of $b$, the marginal benefit of an additional SREC is zero. Specifically, having an additional SREC does not increase the firm's likelihood of compliance in current or future periods, nor can the firm sell the additional SREC for a profit (taking into account their trading costs and $S$). As in \Cref{fig:opt_beh}, this results in optimal generation dropping to $0$ and optimal trading plateauing at the level where the marginal revenue from trading equals the marginal cost. This plateau is not visible in every subplot in \Cref{fig:opt_beh_mult} due to axis limits and the fact that $m = 1$. The impact of SREC price on generation and trading is similar to the single period case.
As $m$ increases, the firm has fewer future periods to position themselves for. Consequently, the firm's optimal planned generation and purchasing behaviour decays more quickly for larger $m$. See Appendix \ref{additional_figures}, \Cref{fig:multi-period-23_pi,fig:multi-period-45_pi} for the analogous figures for $m = 2$, $3$, $4$, and $5$. The optimal controls when $m=5$ are identical to the single-period case as they must be since the performance criterion is time-consistent.
\Cref{fig:5_per_path} shows a sample path of the optimal strategy for three firms (with the same cost functions, and experiencing the same randomness in $b$ and $S$) throughout the course of the 5-period SREC market, with each period lasting $1$ year. The firms differ in their initial banked amount: Firm 1 has $b_0 = 0$, Firm 2 has $b_0 = 250$ and Firm 3 has $b_0 = 500$. We set $S_0= 150$. At each time-step, each firm behaves optimally given their values of banked SRECs and the SREC price. Each firm exists in a separate `universe' and they do not have an impact on one another. Moreover, they are subject to the same realized randomness from the Brownian motions impacting the SREC price and their own SREC generation.
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\textwidth]{5_per_path.pdf}
\caption{Paths of three optimally behaving firms in a 5-period compliance system with $S_0 = 150, b_0 = 0$ (blue), $b_0 = 250$ (red), $b_0 = 500$ (yellow). Parameters as in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.}
\label{fig:5_per_path}
\end{figure}
We see the banked SRECs for all three firms converge roughly to $R=500$ as $t\rightarrow 5$. Consequently, Firm 3 accumulates SRECs at a slower rate than Firm 2, who accumulates SRECs at a slower rate than Firm 1. Even with the firm impacted by production noise, the path of $b$ appears steady within each compliance period for the firms, as before. The large drops are the effect of the firm submitting SRECs for compliance at the end of each period. This results in the \textit{converging saw-tooth} pattern in the first subplot of \Cref{fig:5_per_path}.
\textcolor{black}{The optimal behaviours of each firm follow roughly the same pattern, suggesting that they react similarly to changes in $S$. The difference in their behaviours is primarily due to their initial banked SRECs $b_0$. Firm 1 has no spare SRECs at $t = 0$, and generates the most and sells the least. Firm 3 has $500$ spare SRECs at $t = 0$ -- enough for an entire period of compliance. As such, they produce the least and sell the most. Firm 2 operates between Firm 1 and Firm 3. Naturally, Firm 3 profits the most from this system, due to their initial position. All three firms slow down generation and purchasing behaviour near the final time-steps, reacting to unexpected generation noise that has resulted in them generating more than planned in the time-steps immediately prior. This occurs at the ends of non-terminal compliance periods for Firm 1, as they are typically right on the border of compliance at each period, due to their small initial inventory. The other firms have SREC balances above $R$ and, as banking is allowed, there is no need for a firm to liquidate excess banked SRECs early. }
\textcolor{black}{The optimal behaviours of each firm also imply different SREC prices in each of the `universes' that each firm exists in. However, the magnitude of price impacts for an individual firm are small enough that visually, the price processes look almost identical. In fact, there is a difference of about \$0.07\footnote{At the terminal time-step} between the price path for Firms 1 and 2, and \$0.10 between the price path for Firms 2 and 3. Firm 3 has the highest price, as they are taking the least extreme generation and trading behaviour. Firm 1 has the lowest price, for the opposite reason. }
Finally, we simulate many paths of $S$ with $S_0 = 150, b_0 = 250$ in order to obtain summary statistics and learn about the distribution of various quantities for each firm. In Figure \ref{fig:histograms_mult_price_impact} we plot the histograms of total generated SRECs and total traded SRECs for a regulated firm in each period, based on $1,000$ such sample paths.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\textwidth]{histograms_mult_pi.pdf}
\caption{Histogram of firm generation and trading across each compliance period with $S_0 = 150, b_0 = 250$. Parameters as in Tables \ref{tbl:ComplianceParams} and \ref{tbl:ModelParams}.}
\label{fig:histograms_mult_price_impact}
\end{figure}
From the figure, we note that aggregate selling decreases as $m$ increases, while total planned generation is relatively more static. In particular, the static nature of $\int_0^T g_u du$ arises because lower values of $m$ are associated with higher levels of excess SRECs, as the firm begins with $b_0 = 250$ and thus has the freedom to plan to generate slowly. The change in trading is the result of the firm reacting to the (generally) lower SREC prices that occur when price impacts are active. We also see that the variance of the firm's aggregate behaviour increases as the periods progress. This is the result of simulating forward paths of $S_t$ conditioning on $\mathcal{F}_0$, as $\text{Var}(S_t | S_0)$ is increasing in $t$. As before, these patterns persist across various choices of $S_0$ and $b_0$. To avoid repetition, plots for other initial conditions are not included in this work.
\section{Conclusion}
In this work, we characterize the optimal behaviour of a single regulated LSE in a single-period SREC market. In particular, we characterize their optimal generation and trading behaviour as the solution to a continuous time stochastic control problem. In doing so, we characterize the solution and tease out essential features of the optimal strategy. We also numerically solve for the system in a discrete time setting for both single and multi-period SREC frameworks. Through this, we provide intuition and reasoning for the resulting optimal behaviour, including detailed analysis of various sample paths, summary statistics, strategies, and parameter choices.
Many further extensions are possible. Interactions between agents are a critical component of real SREC markets that are largely ignored in this single-firm setup. In particular, incorporating partial information of firms would be a very challenging but mathematically interesting problem that would more closely mimic the realities of SREC markets. This could potentially necessitate the use of a mean field games approach. Improved calibration to real world parameters would also increase the applicability of this work for use by regulated firms and regulators. \textcolor{black}{The privacy of the relevant data needed to accurately calibrate the cost parameters presents a significant challenge to this endeavour.}
However, even our simple model reveals salient facts about the nature of these systems and how firms should behave when regulated by them. Our single-period model reveals that the optimal generation and trading of regulated firms broadly exists in three regimes, depending on the marginal benefit received from holding an additional SREC. We observe that a firm's trading behaviour is more sensitive to changes in $S$ than its generation behaviour, and that higher SREC prices imply greater generation and lower purchasing (more selling). We show consistency between the numerical and theoretical solutions for our model. In particular, the interesting property that firms should generate above their baseline or shut down entirely is clearly demonstrated theoretically and empirically. Furthermore, we discuss sensitivity to selected other parameters in our model.
When extending to the multiple-period framework, we observe many similarities, but also the key difference that a fourth regime exists in the optimal generation and trading of regulated firms; that is, the regime where a marginal SREC does not provide value in the current period, but may be banked to provide value in the future. Additionally, we compare and contrast the optimal behaviours of firms throughout the multiple-period framework based on different initialization points, and study the changes in their aggregate behaviour across compliance periods.
In providing these results, we have produced a framework and numerical solution that would be of use for both regulated firms and regulatory bodies who both have immense interest in understanding the optimal behaviour of regulated LSEs in these systems.
\nocite{*}
\bibliographystyle{siamplain}
|
1,314,259,996,711 | arxiv | \section{Introduction}
Models for routing games stem from applications in road traffic \cite{wardrop1952road} and packet routing via Internet Protocol \cite{koutsoupias1999worst}. Many agents share a same routing network (a graph of vertices and edges) without any central authority, and each agent decides a path, with the individualistic goal to minimize her delay. However, by using the shared resources of the network, agents incur congestion to one another: they play a game.
\emph{Static} routing games, where every individual's decided path congests its edges all at once, made history as the first application for the Price of Anarchy (PoA), a measure for the loss of efficiency due to selfish behaviors \cite{roughgarden2002bad,roughgarden2005selfish,christodoulou2005price,awerbuch2005price,nisan2007algorithmic,roughgarden2009intrinsic}.
Static routing games, being potential games, also have the very satisfying property that a best-response dynamics always converges to a pure Nash equilibrium (PNE).
However, static models do not reflect the temporality of travels. Indeed, an agent, when on one edge of her path, should not be congesting the other edges. In routing games \emph{over time}, every agent travels through time as well as along the edges of her path. This more recent model was also deeply studied \cite{koch2009nash,anshelevich2009equilibria,hoefer2009competitive,werth2014atomic,harks2016competitive,ismaili2017,Harks:2018:CPR:3182630.3184137}.
Although natural, adding such temporality in delays calculations increases the complexity of games: pure-strategy Nash equilibria may not exist; computing a best-response or an equilibrium are computationally intractable; the PoA can be large.
Still, a last temporal aspect is missing from routing games over time. Although the locality of congestion was related with the time at which it occurs, agents might also make decisions along time. \emph{Sequential} routing games were introduced even more recently \cite{cao2017ec,cao2017arxiv}. Every agent, when she reaches any node, is allowed to observe previous actions, in order to decide the next course of actions, like the next edge on her path.
In this sequential setting, winning strategies and subgame perfect equilibrium (SPE) are standard concepts for individual rationality and a stable game outcome.
\textbf{Related Work.}
There is a rich literature on \emph{routing games over time}.
In \cite{harks2006competitive,harks2009competitive},
there are multiple asymmetric commodities that are routed in a sequence.
The cost of edges is an affine function of the load.
For splittable commodities, the PoA is almost four, and for unsplittable ones it is $3+2\sqrt{2}$.
The best-response problem is NP-hard.
\cite{koch2009nash,koch2011nash}
generalizes deterministic queuing into competitive flows over time.
In this non-atomic symmetric model,
an edge's outflow is determined by its capacity,
and also has a fixed transit delay.
An iterative algorithm is proposed, where $\varepsilon$-Nash flows converge to a Nash flow.
\cite{anshelevich2009equilibria} studies routing games with non-atomic asymmetric agents.
While for symmetric agents, a PNE is guaranteed to exist and efficiently computable,
in the asymmetric case and under FIFO, an equilibrium may not exist.
The PoA is bounded from below by the number of vertices.
\cite{hoefer2009competitive,hoefer2011competitive}
proposes temporal network congestion games.
Every edge has a processing speed $a_e\in\mathbb{R}_{>0}$, and various local policies are studied.
Under FIFO, edge $e$ processes one agent in time $a_e$, while the other agents wait.
In the unweighted symmetric case, despite the NP-hardness of computing a best-response, a PNE can always be efficiently computed. Otherwise, an equilibrium may not exist.
While costs in \cite{koch2009nash} where rationals, \cite{werth2014atomic} focuses on atomic agents and integer time-steps. Every edge is associated with a capacity (maximal flow-per-time-step) and a fixed transit delay.
A PNE may not exist;
the best-response problem is NP-complete;
PNE verification is coNP-complete;
PNE existence is at least NP-hard;
a bound is provided on the PoA.
\cite{harks2016competitive,Harks:2018:CPR:3182630.3184137} studies a model related to \cite{werth2014atomic}.
The focus is on local and overall priorities on the edges. (Crossovers may occur.)
Several bounds on the price of stability and PoA are shown.
It is APX-hard to compute optimal priority lists.
Under local priorities, the best-response and PNE problem are NP-hard.
\cite{ismaili2017} studies a model related to \cite{harks2016competitive} and similar to \cite{werth2014atomic} under FIFO.
Best-responses are inapproximable, and deciding whether a PNE exists is complete for class $\Sigma_2^P$.
Concerning \emph{sequential routing games},
\cite{cao2017arxiv,cao2017ec} are the first to propose a model where agents take new decisions on each vertex. In this very interesting setting, when several agents arrive on the same edge during the same round, ties are broken by an order on incoming edges. It allows for plays that do not depend on agents' IDs.
It also guarantees the existence of a subgame perfect equilibrium (when all agents share the same sink), in a constructive way: the iterative dominating path profile algorithm \cite[Alg. 1]{cao2017arxiv} outputs a path profile induced by an SPE. It is also reminded that an SPE might not exist when one uses an overall order on agents for tiebreaking \cite[Ex. 2]{cao2017arxiv}.
Besides, \cite{papadimitriou2010new} studies a new model where decisions are made by nodes instead of flows.
There is also a diverse literature on repeated routing games,
where sequentiality comes from a repetition of static routing games (see e.g. \cite{blum2006routing,roughgarden2009intrinsic,chien2011convergence} or \cite{monnot2017routing} in real life).
While it is well known that problem QSAT (also known as TQBF) is PSPACE-complete \cite{garey1979computers}, the Canadian Traveler Problem \cite{PAPADIMITRIOU1991} was an inspiration for settling our complexities.
\textbf{Our contributions.}
Here, we study sequential routing games that are played on a digraph, round after round.
W.l.o.g. on negative results, we assume that every edge has capacity and length one.
Every agent has to travel from her source-vertex to her sink-vertex in as few rounds as she can.
Congestion is modeled by a very natural FIFO queuing policy on every edge:
agents who enter the edge are queued, and an edge lets out one agent per round.
At every round, the top agents of all waiting lists pop out from their current edges, decide their next edges and queue there.
More precisely, we study an imperfect information setting where the top agents decide their next edges simultaneously,
and a perfect information setting where they decide in an ordered sequence.
We study the computational complexity of standard rationality and stability concepts,
and show the following results in both sequential and simultaneous action settings.
\begin{enumerate}[leftmargin=16mm]
\item[\bf Th. 1] Deciding whether an agent has a strategy that will guarantee her total delay below a given threshold is PSPACE-complete.
\item[\bf Th. 2] Let $n$ be the number of agents and $|V|$ the number of vertices. Approximating an agent's optimal strategy within $n^{1-\varepsilon}$ for any $\varepsilon>0$, and within any polynomial of $|V|$ are both PSPACE-hard.
\item[\bf Th. 3] Under perfect information, computing a subgame perfect equilibrium is PSPACE-hard and in FPSPACE. Under imperfect information, deciding SPE existence is PSPACE-complete.
\end{enumerate}
\vspace*{-6mm}
\section{Preliminaries}
\vspace*{-2mm}
We introduce relevant notation on routing games and sequential games.
Without loss of generality on negative results, edges have unitary length and capacity.
\footnote{A similar assumption (unitary length and capacity) is made in \cite{cao2017arxiv}. Since the present hardness results still hold in this particular case, it means that computational complexity does not come from numbers defining length and capacity.}
\begin{definition}[Sequential First-in-first-out Routing Game (FRoG)]~\\
A \textsc{FRoG}\ is characterized by a tuple $\left(G=(V,E),N,(s_i,s_i^\ast)_{i\in N}\right)$ where:
\begin{itemize}
\item $G=(V,E)$ is a finite digraph with vertex set $V$ and edges\footnote{An edge $e=(u_e,v_e)$ is a couple of vertices. Its tail (resp. head) is $u_e$ (resp. $v_e$).} $E\subseteq V\times V$,
\item finite set $N=\{1,\ldots,n\}$ is the set of agents, and
\item for every agent $i$, vertex $s_i\in V$ is her source and $s_i^\ast\in V\setminus\{s_i\}$ her sink. \hfill$\diamond$
\end{itemize}
\end{definition}
\noindent
For every edge $e$, set $F(e)\subseteq E$ denotes the \emph{successors} of $e$: all the edges whose tail is the head of $e$.\footnote{Vertices are mere connections. Edges are the only decided and time-costly resources.}
For every agent $i$, starting from her source $s_i$, the goal is to travel a path\footnote{A path is a finite list of edges such that for two consecutive edges the head of the former is the tail of the later. We allow edge repetitions and directed cycles. We assume that there always exists a path from $s_i$ to $s_i^\ast$.} $\pi_i$ that reaches her sink $s_i^\ast$, by deciding a successor edge each time she reaches a head.
Every edge $e$ acts as a FIFO \emph{queuing list}\footnote{A queuing list admits two operations: queuing and popping. Under FIFO ordering, queuing adds an agent at the end of the list, and popping takes the first agent.} $Q_e$ on agents:
every agent who enters $e$ is queued in $Q_e$.
A game is played sequentially over \emph{rounds} $r\in\mathbb{N}$.
Here is the main loop: once per round, every non-empty list $Q_e$ simultaneously pops its first agent $i$, who simultaneously decides her next edge $e'$ in successors $F(e)$, and enters waiting queue $Q_{e'}$. If the head of edge $e$ is already $i$'s sink $s_i^\ast$, she simply exits the system.
(If two agents arrive on the same edge at the same time, rules are precised later.)
Congestion comes from the fact that edges let out at most one agent per round.
If the game ends in a finite number of rounds\footnote{An agent can cycle or decide to get stuck somewhere with no way to reach her sink.},
every agent reached her sink,
and the outcome is
a path-profile $(\pi_1,\ldots,\pi_n)\in\mathcal{P}_1\times\ldots\times\mathcal{P}_n$, which we denote in bold by $\bm{\pi}\in\bm{\mathcal{P}}$, and where $\mathcal{P}_i$ denotes the set of finite paths from agent $i$'s source $s_i$ to her sink $s_i^\ast$.
Informally, the \emph{total-delay} (defined later) that an agent $i$ seeks to minimize is $C_i(\bm{\pi})=\sum_{\ell=1}^{|\pi_i|}w_i(\bm{\pi},\ell)$
where $w_{i}(\bm{\pi},\ell)\in\mathbb{N}_{\geq 1}$ is agent $i$'s position in waiting list $Q_{\pi_i(\ell)}$ when she enters edge $\pi_i(\ell)$.
\begin{example}
In this \textsc{FRoG}, vertices are circles, edges (with unitary capacities and delays) are arrows.
Agent $1$ (resp. $2$) start from $a$ (resp. $b$). Both go to $k$, under simultaneous actions and agent priority $2\succ 1$ (see tiebreaking rule (RO)).
\begin{center}
\begin{tikzpicture}
\draw[]
node[circle,draw=black,scale=0.67] (a) at (0,0) {$a$}
node[circle,draw=black,scale=0.67] (b) at (1.2,0) {$b$}
node[circle,draw=black,scale=0.67] (c) at (2,+1) {$c$}
node[circle,draw=black,scale=0.67] (d) at (2.3,0) {$d$}
node[circle,draw=black,scale=0.67] (e) at (2,-1) {$e$}
node[circle,draw=black,scale=0.67] (f) at (4,+1) {$f$}
node[circle,draw=black,scale=0.67] (g) at (4,-1) {$g$}
node[circle,draw=black,scale=0.67] (h) at (5,-0.5) {$h$}
node[circle,draw=black,scale=0.67] (i) at (6,1) {$i$}
node[circle,draw=black,scale=0.67] (j) at (6,0) {$j$}
node[circle,draw=black,scale=0.67] (k) at (7.5,0.5) {$k$};
\draw[]
(a) edge[-{Stealth[scale=0.6]}] (c)
(a) edge[-{Stealth[scale=0.6]}] (e)
(b) edge[-{Stealth[scale=0.6]}] (d)
(c) edge[-{Stealth[scale=0.6]}] (f)
(d) edge[-{Stealth[scale=0.6]}] (f)
(d) edge[-{Stealth[scale=0.6]}] (g)
(e) edge[-{Stealth[scale=0.6]}] (g)
(f) edge[-{Stealth[scale=0.6]}] (i)
(f) edge[-{Stealth[scale=0.6]}] (j)
(g) edge[-{Stealth[scale=0.6]}] (h)
(h) edge[-{Stealth[scale=0.6]}] (j)
(i) edge[-{Stealth[scale=0.6]}] (k)
(j) edge[-{Stealth[scale=0.6]}] (k);
\node[rectangle,draw=black,fill=black!20, thick,left = 0.25mm of a,scale=0.75]{$1$};
\node[rectangle,draw=black,fill=black!20, thick,left = 0.25mm of b,scale=0.75]{$2$};
\node[diamond,draw=black,fill=black!20, thick,right = 0.25mm of k,scale=0.75, inner sep=0.5mm]{$1$};
\node[diamond,draw=black,fill=black!20, thick,right = 5.25mm of k,scale=0.75, inner sep=0.5mm]{$2$};
\node[circle,draw=black,fill=white,inner sep=1mm] at (5,1) {~};
\end{tikzpicture}
\end{center}
\end{example}
\begin{definition}[Consecutive Configurations and History]
For every round $r\in\mathbb{N}$,
{configuration} $Q(r)=(Q_e(r))_{e\in E}$ is defined as the ordered contents of every waiting list, before tops are popped to end round $r$ and agents decide their next edges.\footnote{While $Q(0)$ is empty, for $r\geq 1$, configuration $(Q_e(r))_{e\in E}$ is a partition of the agents still in the system. Agents on top of queues in $Q(r)$ decide a new edge at round $r$.}
A history $H(r)=(Q(0),\ldots,Q(r))$ is a sequence of consecutive configurations:
from history $H(r)$, if configuration $Q(r)$ is non-empty, we obtain a {consecutive configuration} $Q(r+1)$ by the following decisions.
\begin{enumerate}
\item[a.] For round $r=0$, all the agents {simultaneously} decide their first edge (with tail $s_i$), and get queued there, defining $Q(1)$.
\item[b.] Now, we focus on the agents who need to make a decision:\\
For round $r\geq 1$, let set $M_{H(r)}$ be every agent $i$ on top of a queue $Q_e(r)$ s.t. $F(e)\neq\emptyset$ and the head of edge $e$ is not her sink $s_i^\ast$. All the agents in $M_{H(r)}$ simultaneously pop and decide their next edges $e'\in F(e)$, queuing in the corresponding waiting lists $Q_{e'}(r+1)$, and thus defining $Q(r+1)$.
\item[c.] For round $r\geq 1$, every agent $i$ on top of a queue $Q_e(r)$ s.t. the head of edge $e$ is $i$'s sink $s_i^\ast$ pops and exits the system with {total-delay} $C_i=r$. If $F(e)=\emptyset$ and the head is not her sink, she exists with total-delay $C_i=\infty$.
\end{enumerate}
For round $r\geq 0$,
the set of all possible histories is denoted by $\mathcal{H}(r)$.\hfill$\diamond$
\end{definition}
\begin{definition}[Game-tree]
The game-tree $\Gamma=(\mathcal{H}, \mathcal{Q})$ is defined as follows.
\begin{itemize}
\item Nodes are all possible histories: $\mathcal{H}=\bigcup_{r\geq 0} \mathcal{H}(r)$. History $H(0)$ is the root.
\item For any round $r\geq 0$, a transition from history $H(r)$ to $(H(r),Q(r+1))$ exists in $\mathcal{Q}$ if and only if $Q(r+1)$ is a consecutive configuration for $H(r)$.
\end{itemize}
Given game-tree $\Gamma=(\mathcal{H}, \mathcal{Q})$ and history $H(r)\in\mathcal{H}$, we define $\Gamma[H(r)]$ the subtree rooted on $H(r)$. It also defines the subgame starting from $H(r)$.\hfill$\diamond$
\end{definition}
The game-tree is a (possibly infinite) tree graph which entirely describes how the game can be played. On each node/history, agents make their decisions, branching into a consecutive history, until all agents reached their sink.
\begin{definition}[Strategy]
For agent $i$, a strategy $\sigma_i$ is a function which maps every history $H(r)$ in which agent $i$ has to decide a new edge (i.e. agent $i$ belongs to $M_{H(r)}$) to her next edge.
Let $\Sigma_i$ denote the set of possible strategies for $\sigma_i$.
A strategy-profile $(\sigma_1,\ldots,\sigma_n)\in\Sigma_1\times\ldots\times\Sigma_n$, which we denote in bold by $\bm{\sigma}\in\bm{\Sigma}$, defines a strategy $\sigma_i$ for every agent $i$, along with an adversary profile $\bm{\sigma}_{-i}$.
A \emph{strategical} (resp. \emph{trivial}) agent is an agent for who there is more than one (resp. exactly one) $s_i-s_i^\ast-$path.
\hfill$\diamond$
\end{definition}
A strategy could be a huge or infinite object, representing it entirely is intractable. Instead, one may assume that an agent has her own \emph{polynomial-time} algorithm for deciding her next edge, function of current configuration or history.
A \textsc{FRoG}\ ends in a finite number of rounds if an agent never decides to go where there is no path to her sink and never visits the same vertex/edge twice (e.g. if $G$ is acyclic or as a strategy).
\begin{definition}[Induced path-profile]
Given strategy-profile $\bm{\sigma}$, agents play their strategies on the game.
If the game ends in a finite number of rounds, then it generates one finite sequence of edges per agent: a unique path-profile $\bm{\pi}(\bm{\sigma})$.\hfill$\diamond$
\end{definition}
\begin{definition}[Total-delay]
For every agent $i$ and path-profile $\bm{\pi}$,
let $C_i(\bm{\pi})$ be the total-delay of agent $i$:
the round when she reaches her sink $s_i^\ast$ and exits the system.
It can be defined as $C_i(\bm{\pi})=\sum_{\ell=1}^{|\pi_i|}w_i(\bm{\pi},\ell)$
where $w_{i}(\bm{\pi},\ell)\in\mathbb{N}_{\geq 1}$ is agent $i$'s position in waiting list $Q_{\pi_i(\ell)}$ when she enters edge $\pi_i(\ell)$.
Function $C_i$ extends to strategy-profiles $\bm{\sigma}$ by mapping from path-profile $\bm{\pi}(\bm{\sigma})$ if the game ends in a finite number of rounds. Otherwise, an agent who reaches her goal in a finite number of rounds has this number as a total delay,
and an agent who doesn't get delay $C_i(\bm{\sigma})=\infty$.
Furthermore, function $C_i(\bm{\sigma} \mid H(r))$ is agent $i$'s total-delay when strategies $\bm{\sigma}$ are played starting from history $H(r)$, that is in subgame $\Gamma[H(r)]$.\hfill$\diamond$
\end{definition}
It is worth noting that a path-profile $(\pi_1,\ldots,\pi_n)$ can be mapped in polynomial time to total delays $(C_1(\bm{\pi}),\ldots,C_n(\bm{\pi}))$ by mean of a Dijkstra-style pop-the-next-event algorithm similar to \cite[Prop 2.2]{harks2016competitive} or \cite[Th. 2]{ismaili2017}.
\begin{definition}[Subgame perfect equilibrium (SPE)]
Given a \textsc{FRoG}, an SPE is a strategy-profile
$(\sigma_1,\ldots,\sigma_n)$
such that in game-tree $\Gamma=(\mathcal{H}, \mathcal{Q})$,
for any history $H(r)\in\mathcal{H}$, any agent $i$ in $M_{H(r)}$ and any deviation $\sigma_i'\in\Sigma_i$, one has:
$$C_i(\bm{\sigma}\mid H(r)) \quad \leq\quad C_i(\sigma_i',\bm{\sigma}_{-i}\mid H(r)).\vspace*{-8mm}$$
\hfill$\diamond$
\end{definition}
In other words, in any subgame, strategy-profile $\bm{\sigma}$ is a pure Nash equilibrium.
When an SPE is played, the game always ends in a finite number of rounds, since it is in players' interests.
\begin{example}
We depict below the game-tree of Example 1.
Every rectangle represents the set of deciding agents $M_{H(r)}$.
On every transition, we represent the edges decided.
Total delays are 2-vectors on leaves. SPEs are in bold.
\begin{center}
\begin{tikzpicture}
\draw[]
node[rectangle,draw=black] (A) {1}
node[rectangle,draw=black, below right= 3mm and 25mm of A] (B) {2}
node[rectangle,draw=black, below left= 3mm and 15mm of A] (C) {2}
node[rectangle,draw=black, below right= 4mm and 25mm of B] (D) {1,2}
node[rectangle,draw=black, below left= 4mm and 8mm of B] (E) {1}
node[rectangle,draw=black, below right= 4mm and 5mm of C] (F) {2}
node[below left= 4mm and 5mm of C,scale=0.67] (G) {$\bm{(6,5)}$}
node[below right= 5mm and 15mm of D,scale=0.67] (H) {$\bm{(6,5)}$}
node[below right= 8mm and -2mm of D,scale=0.67] (I) {$\bm{(5,4)}$}
node[below left= 8mm and -2mm of D,scale=0.67] (J) {$\bm{(4,5)}$}
node[below left= 5mm and 15mm of D,scale=0.67] (K) {$\bm{(5,4)}$}
node[below right= 5mm and -2mm of E,scale=0.67] (L) {$\bm{(5,5)}$}
node[below left= 5mm and -2mm of E,scale=0.67] (M) {$\bm{(4,5)}$}
node[below right= 5mm and -2mm of F,scale=0.67] (N) {$\bm{(5,5)}$}
node[below left= 5mm and -2mm of F,scale=0.67] (O) {$\bm{(5,4)}$};
\draw[]
(A) edge[-{Stealth[scale=0.6]},ultra thick] node[above,scale=0.75]{$(a,c)$} (B)
(A) edge[-{Stealth[scale=0.6]},ultra thick] node[above,scale=0.75]{$(a,e)$} (C)
(B) edge[-{Stealth[scale=0.6]}, ultra thick] node[above right,scale=0.75]{$(d,f)$} (D)
(B) edge[-{Stealth[scale=0.6]}] node[above left,scale=0.75]{$(d,g)$}(E)
(C) edge[-{Stealth[scale=0.6]}, ultra thick] node[above right,scale=0.75]{$(d,f)$}(F)
(C) edge[-{Stealth[scale=0.6]}] node[above left,scale=0.75]{$(d,g)$}(G)
(D) edge[-{Stealth[scale=0.6]}] node[above right,scale=0.75]{$(f,i)(f,i)$} (H)
(D) edge[-{Stealth[scale=0.6]}, ultra thick] node[right,scale=0.5]{$(f,i)(f,j)$}(I)
(D) edge[-{Stealth[scale=0.6]}] node[left,scale=0.5]{$(f,j)(f,i)$}(J)
(D) edge[-{Stealth[scale=0.6]}, ultra thick] node[above left,scale=0.75]{$(f,j)(f,j)$} (K)
(E) edge[-{Stealth[scale=0.6]}] node[above right,scale=0.75]{$(f,i)$}(L)
(E) edge[-{Stealth[scale=0.6]}, ultra thick] node[above left,scale=0.75]{$(f,j)$}(M)
(F) edge[-{Stealth[scale=0.6]}] node[above right,scale=0.75]{$(f,i)$} (N)
(F) edge[-{Stealth[scale=0.6]},ultra thick] node[above left,scale=0.75]{$(f,j)$} (O);
\end{tikzpicture}
\end{center}
\end{example}
\subsection{Discussion on Tie-breaking, Actions, Information and Existence}
In the simultaneous-action setting defined above,
several agents may simultaneously queue on a same waiting list $Q_e$,
during a \emph{same round}, hence a tie breaking rule for who queues first is required. Two exist in literature:
\begin{enumerate}[leftmargin=11mm]
\item[(RO)] There is an overall order $\succ$ on agents, and those with the higher priority queue first. However, in this case, an SPE is not guaranteed to exist.\footnote{Here, we do not consider mixed strategies.} (See e.g. \cite[Fig. 2]{cao2017arxiv}.)
\item[(RE)] For every vertex $u$, a priority $\succ_u$ is defined on incoming edges. If all agents share same sink, an SPE is guaranteed to exist \cite[Sec. 3]{cao2017arxiv}.
However, if sinks are different, an SPE is not guaranteed.\footnote{See e.g. the pursuer-follower counter-example in \cite[Fig. 1]{ismaili2017}. The same construction, but with priorities on incoming edges, is strategically equivalent. Moreover, since the only two strategical agents decide everything simultaneously on the first round, pure Nash equilibrium and SPE coincide (as an empty concept there).}
\end{enumerate}
The simultaneousness of decisions in each round, induces imperfect information:
agents in set $M_{H(r)}$ do not know what the others decide at round $r$.
In order to obtain a setting with sequential actions and perfect information, which guarantees existence of an SPE by backward induction, we use the same setting as above, with the following change that slices rounds into agent turns.
\begin{enumerate}[leftmargin=11mm]
\item[(RR)] Agents in $M_{H(r)}$ (instead of deciding simultaneously) take turn w.r.t. an agent order $1\succ\ldots\succ n$ for deciding their next edge and queue there. Hence, only one agent per (slice of) round makes a decision and acts.
\end{enumerate}
\subsection{Computational problems studied}
The main issue lies in settling the complexity\footnote{We assume as common knowledge: decision and function problem, length function, complexity classes P, NP, PSPACE and FPSPACE, many-to-one reduction, hardness, completeness and decision problems 3SAT and QSAT (also known as TQBF).} of
computing a subgame perfect equilibrium (when it always exists),
or of deciding whether one exists (if it may not).
While the length function of \textsc{FRoG} s is polynomial in $|V|$ and $n$,
a strategy $\sigma$ is by definition an output of intractable length.
Therefore, when SPE is guaranteed, computation of its induced path profile is a more formalizable problem. We explore this sequence of problems, where $\mathcal{R}$ refers to rules (RO) and (RR):
\begin{itemize}[leftmargin=3.5cm]
\setlength{\itemsep}{0.4em}
\item[\textsc{FRoG/$\mathcal{R}$/Br}:~~~~~]
Given a \textsc{FRoG} , an agent $i$, an adversary path-profile $\bm{\pi}_{-i}$ and a threshold $\theta\in\mathbb{N}$, decide whether there is a path $\pi_i$ with total delay $C_i(\pi_i,\bm{\pi}_{-i})\leq\theta$.
\item[\textsc{FRoG/$\mathcal{R}$/Win}:~~~]
Given a \textsc{FRoG} , an agent $i$ and a threshold $\theta\in\mathbb{N}$, decide whether there is a strategy $\sigma_i$ that guarantees her total delay $C_i(\sigma_i,\bm{\sigma}_{-i})\leq\theta$ for any adversary strategy $\bm{\sigma}_{-i}$.
\item[\textsc{FRoG/RO/Exist}:~]
Given a \textsc{FRoG} , does it admit an SPE?
\item[\textsc{FRoG/RR/Find}:]
Given a \textsc{FRoG} , find a path-profile induced by an SPE.
\end{itemize}
\section{The Complexity of Winning Strategies}
In this section, we show that deciding whether an agent has a winning strategy (one that grants her total delay below some threshold) is PSPACE-complete under simultaneous actions (RO) and sequential actions (RR).
\begin{theorem}\label{th:win}
\textsc{FRoG/$\mathcal{R}$/Win} is PSPACE-complete under (RO) and (RR).
\end{theorem}
The remainder of this section consists in the proof of Th. \ref{th:win}. In Def. \ref{def:loosener}, we first define gadget games that allow preference inversions which break Bellman's principle and cause computational intractability.
Then, for didactic reasons, in Lem. \ref{lem:br}, we consider a best-response decision problem related to \cite[Th. 3]{ismaili2017}, which lets us introduce smoothly an original many-one reduction from 3SAT that is then generalized to QSAT by reusing the same structure and gadget-games.
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.95]
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\node[circle,draw=black,inner sep=0.0em] (fnprime) at (11.9,0.1) {};
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\node[circle,draw=black,inner sep=0.0em] (f3a) at (8,1) {$~$};
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\node[circle,draw=black,inner sep=0.0em,fill=black] (f2d) at (6,4) {$~$};
\node[circle,draw=black,inner sep=0.0em,fill=black] (f3d) at (8,4) {$~$};
\node[circle,draw=black,inner sep=0.0em,fill=black] (fmd) at (10,4) {$~$};
\node[circle,draw=black,inner sep=0.0em,fill=black] (fnd) at (12,4) {$~$};
\draw[]
(e) edge[-{Stealth[scale=0.6]},ultra thick] node[below=2mm]{$e$} (f0)
(f0) edge[-{Stealth[scale=0.6]}] node[below=2mm]{$f_0$} (f1)
(f1) edge[-{Stealth[scale=0.6]}] node[below=2mm]{$f_1$} (f2)
(f2) edge[-{Stealth[scale=0.6]}] node[below=2mm]{$f_2$} (f3)
(f3) edge[-{Stealth[scale=0.6]},dashed] node[below=2mm]{$f_{j}$} (fm)
(fm) edge[-{Stealth[scale=0.6]}] node[below=2mm]{$f_m$} (fn);
\draw[]
(f2) edge[-{Stealth[scale=0.6]}] node[right=1mm]{$f_{1,\alpha}$}(f2a)
(f2a) edge[-{Stealth[scale=0.6]}] node[right=1mm]{$f_{1,\beta}$}(f2b)
(f2b) edge[-{Stealth[scale=0.6]},dashed] node[right=1mm]{$f_{1,\gamma}$} (f2c)
(f2c) edge[-{Stealth[scale=0.6]},ultra thick] node[right=1mm]{$g_1$} (f2d);
\draw[]
(f3) edge[-{Stealth[scale=0.6]}] node[right=1mm]{$f_{2,\alpha}$}(f3a)
(f3a) edge[-{Stealth[scale=0.6]}] node[right=1mm]{$f_{2,\beta}$}(f3b)
(f3b) edge[-{Stealth[scale=0.6]},dashed] node[right=1mm]{$f_{2,\gamma}$} (f3c)
(f3c) edge[-{Stealth[scale=0.6]},ultra thick] node[right=1mm]{$g_2$} (f3d);
\draw[]
(fm) edge[-{Stealth[scale=0.6]},dashed] node[right=1mm]{$f_{j,\alpha}$} (fma)
(fma) edge[-{Stealth[scale=0.6]},dashed] node[right=1mm]{$f_{j,\beta}$} (fmb)
(fmb) edge[-{Stealth[scale=0.6]},dashed] node[right=1mm]{$f_{j,\gamma}$} (fmc)
(fmc) edge[-{Stealth[scale=0.6]},dashed, thick] node[right=1mm]{$g_j$} (fmd);
\draw[]
(fn) edge[-{Stealth[scale=0.6]}] node[right=1mm]{$f_{m,\alpha}$}(fna)
(fna) edge[-{Stealth[scale=0.6]}] node[right=1mm]{$f_{m,\beta}$}(fnb)
(fnb) edge[-{Stealth[scale=0.6]},dashed] node[right=1mm]{$f_{m,\gamma}$} (fnc)
(fnc) edge[-{Stealth[scale=0.6]},ultra thick] node[right=1mm]{$g_m$} (fnd);
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(e) edge[-{Stealth[scale=0.6]}, bend left = 30] (f0);
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node[below = 1mm of e, rectangle,draw=black!50!blue,fill=black!50!blue,inner sep=0.2em] {~}
node[below = 2.3mm of e,scale=0.67, draw=black!50!blue, text=black!50!blue] {$r+1$}
node[below = 0.5mm of fn, diamond,draw=black!50!blue,fill=black!50!blue,inner sep=0.15em] {~}
(e) edge[-{Stealth[scale=0.6]}, bend right = 30] (f0)
(f0) edge[-{Stealth[scale=0.6]}, bend right = 20] (f1)
(f1) edge[-{Stealth[scale=0.6]}, bend right = 20] (f2)
(f2) edge[-{Stealth[scale=0.6]}, bend right = 20] (f3)
(f3) edge[-{Stealth[scale=0.6]}, bend right = 20,dashed] (fm)
(fm) edge[-{Stealth[scale=0.6]}, bend right = 20] (fn);
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node[rectangle,below right = 1mm and 0mm of f1,draw=black!50!green,fill=black!50!green,inner sep=0.2em] {~}
node[diamond,right = 1mm of f2b,draw=black!50!green,fill=black!50!green,inner sep=0.15em] {~}
node[rectangle,below = 2.2mm of f1,draw=black!50!green, text=black!50!green,scale=0.67] {$r+2+1$}
(f1) edge[-{Stealth[scale=0.6]}, bend left = 10] (f2prime)
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node[rectangle,below right = 1mm and 0mm of f2,draw=black!50!green,fill=black!50!green,inner sep=0.2em] {~}
node[diamond,right = 1mm of f3b,draw=black!50!green,fill=black!50!green,inner sep=0.15em] {~}
node[rectangle,below = 2.2mm of f2,draw=black!50!green,text=black!50!green,scale=0.67] {$r+2+2$}
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(f3a) edge[-{Stealth[scale=0.6]}, bend right = 25] (f3b);
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node[rectangle,below right = 1mm and 0mm of f3,draw=black!50!green,ultra thick,inner sep=0.2em] {~}
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node[rectangle,below = 2.5mm of f3,draw=black!50!green,text=black!50!green,scale=0.67] {$r+2+j$}
(f3) edge[-{Stealth[scale=0.6]}, bend left = 10,dashed] (fmprime)
(fmprime) edge[-{Stealth[scale=0.6]}, bend left =10,dashed] (fma)
(fma) edge[-{Stealth[scale=0.6]}, bend right = 25,dashed] (fmb);
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node[rectangle,below right = 1mm and 0mm of fm,draw=black!50!green,fill=black!50!green,inner sep=0.2em] {~}
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(fm) edge[-{Stealth[scale=0.6]}, bend left = 10] (fnprime)
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node[rectangle,left = 1mm of f2a,draw=black!30!orange,fill=black!30!orange,inner sep=0.2em] {~}
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node[rectangle,left = 2.3mm of f2a,draw=black!30!orange,text=black!30!orange,scale=0.67] {$r+4+1$}
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node[rectangle,left = 1mm of f3a,draw=black!30!orange,fill=black!30!orange,inner sep=0.2em] {~}
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node[rectangle,left = 2.3mm of f3a,draw=black!30!orange,text=black!30!orange,scale=0.67] {$r+4+2$}
(f3a) edge[-{Stealth[scale=0.6]}, bend left = 40] (f3b)
(f3b) edge[-{Stealth[scale=0.6]}, bend left = 40, dashed] (f3c)
(f3c) edge[-{Stealth[scale=0.6]}, bend left = 40] (f3d);
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node[rectangle,left = 1mm of fma,draw=black!30!orange,ultra thick,inner sep=0.2em] {~}
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node[rectangle,left = 2.5mm of fma,draw=black!30!orange,text=black!30!orange,scale=0.67] {$r+4+j$}
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node[above = 2mm of f0, rectangle,draw=black!30!red,fill=black!30!red,inner sep=0.2em] {~}
node[above right = 1mm and 0mm of f1, diamond,draw=black!30!red,fill=black!30!red,inner sep=0.1em] {~}
node[above = 3.3mm of f0, text=black!30!red,draw=black!30!red,inner sep=0.2em,scale=0.67] {$r+2$}
(f0) edge[-{Stealth[scale=0.6]}, bend left = 10] (f1)
(f0) edge[-{Stealth[scale=0.6]}, bend left = 20] (f1)
(f0) edge[-{Stealth[scale=0.6]}, bend left = 30] (f1)
(f0) edge[-{Stealth[scale=0.6]}, bend left = 40] (f1)
(f0) edge[-{Stealth[scale=0.6]}, bend left = 50] (f1)
(f0) edge[-{Stealth[scale=0.6]}, bend left = 60] (f1)
(f0) edge[-{Stealth[scale=0.6]}, bend left = 70] (f1);
\node[rectangle, draw=white, text width=6.4cm,scale=0.9] at (2.8,3.6) {\textbf{If} agent $i$ does not enter $e$ on round $r$:\\ $\bullet$ agent blue is on time,\\ $\bullet$ green agents are delayed by blue,\\ $\bullet$ gold agents are on time.};
\node[rectangle, draw=white, text width=6.4cm,scale=0.9] at (2.8,2.0) {\textbf{If} agent $i$ enters edge $e$ on round $r$:\\ $\bullet$ agent blue is delayed by her,\\ $\bullet$ green agents are on time,\\ $\bullet$ gold agents are delayed by greens.};
\end{tikzpicture}
\caption{Loosener for (RO) from trigger $(e,r)$ to consequences $(g_1,r_{g_1}),\ldots,(g_m,r_{g_m})$: Starting from edge $e$, there is a path of $m+1$ edges $e, f_0,f_1,\ldots,f_m$. Besides, from any edge $f_j$ ($1\leq j\leq m$), there is a path $f_j,f_{j,\alpha},f_{j,\beta},f_{j,\gamma_1},f_{j,\gamma_2},\ldots,g_j$ to consequence $g_j$. (How many $f_{j,\gamma_\ell}$ edges is specified later.) Agents are as follows. Every agent only has one possible path from his source to his sink. We represent their sources, starting times and sinks by respectively squares, a number near the square and a diamond. A \emph{dark} agent only travels edge $e$ on round $r$. A \emph{blue} agent travels from edge $e$ on round $r+1$ to edge $f_m$, in the best case crossing edge $f_j$ on round $r+2+j$. An arbitrary number of \emph{red} agents all try to only cross edge $f_0$ on same round $r+2$. On every path $f_j,f_{j,\alpha},f_{j,\beta}$, there is a different \emph{green} agent traveling it, starting from round $r+2+j$ on edge $f_j$. Finally, on every path $f_{j,\beta},f_{j,\gamma_1},f_{j,\gamma_2},\ldots,g_j$, there is a \emph{gold} agent traveling it, starting on edge $f_{j,\beta}$ from round $r+4+j$, and in the best case arriving on edge $g_j$ on round $r_{g_j}$. (Hence, there are $r_{g_j}-(r+5+j)$ edges $f_{j,\gamma_\ell}$.) The overall tiebreaking order on agents is defined so that: $\text{blue}\succ\text{green}\succ\text{gold}\succ\text{red}\succ\text{dark}\succ\text{agent } i$.}
\label{fig:loosener}
\end{figure}
\begin{definition}[Loosener]\label{def:loosener}
Given a \textsc{FRoG}, an agent $i$,
a \emph{trigger} couple $(e,r)$ of an edge and a round,
and $m$ \emph{consequence} couples $(g_1,r_{g_1}),\ldots,(g_m,r_{g_m})$ of an edge and a round,
a \emph{loosener} is a piece of \textsc{FRoG}\ defined under (RO) as in Figure \ref{fig:loosener},
with the minor assumption that $r_{g_j}\geq r+5+j$, for any $1\leq j\leq m$.\hfill$\diamond$
\end{definition}
\begin{lemma}\label{lem:loosener}
In a loosener, if agent $i$ reaches trigger-edge $e$ on trigger-round $r$,
then her delay on any consequence-edge $g_j$ in consequence-round $r_{g_j}$ is decreased by one round. (An agent with higher priority than $i$ and that was arriving on $g_j$ at $r_{g_j}$ is delayed by one.) Furthermore, the loosener is not a shortcut for $i$: her cost for going through it from round $r$ or later is arbitrarily large.
\end{lemma}
\begin{proof}[Lemma \ref{lem:loosener}, under (RO)]
If agent $i$ enters edge $e$ on round $r$, she gets queued before agent blue (but after dark) since she arrives one round earlier than blue. Hence, blue is delayed one round, and then much more by red agents. Therefore, there are no collisions between blue and the $m$ green agents, who are then on time. Consequently, when a green starts from edge $f_j$ on round $r+2+j$, then she enters edge $f_{j,\beta}$ on round $r+4+j$ and induces a delay on the corresponding gold agent who enters edge $g_j$ on round $r_{g_j}+1$.
If agent $i$ does not enter edge $e$ on round $r$, then blue enters edge $f_0$ on round $r+2$ before the red agents (by priority). Because of every collision blue and the greens on edges $f_j$ at times $r+2+j$, and because of priority $\text{blue}\succ\text{green}$, every green agent is delayed by one round. Consequently, no green agent collides with any gold agent. Therefore, every gold agent enters his edge $g_j$ on round $r_{g_j}$.
If agent $i$ tries to go through the loosener from round $r$ or later (in order to take a shortcut to some consequence edge), because of dark's priority on her, $i$ enters edge $f_0$ at round $r+2$ or later. Unfortunately, the arbitrarily large number of red agents have priority over her and she gets queued after them.
\qed
\end{proof}
\begin{lemma}\label{lem:br}
\textsc{FRoG/$\mathcal{R}$/Br} is NP-complete under (RO) and (RR).
\end{lemma}
\begin{figure}[t]
\centering
\begin{tikzpicture}
\node[circle,draw=black,inner sep=0.1em] (x1) at (0,0) {${x_1}$};
\node[circle,draw=black,inner sep=0.0em] (x1a) at (0.5,-0.4) {$~$};
\node[circle,draw=black,inner sep=0.0em] (x1b) at (0.5,+0.4) {$~$};
\node[circle,draw=black,inner sep=0.1em] (x2) at (1,0) {${x_2}$};
\node[circle,draw=black,inner sep=0.0em,fill=black] (x2a) at (1.5,-0.4) {$~$};
\node[circle,draw=black,inner sep=0.0em] (x2b) at (1.5,+0.4) {$~$};
\node[circle,draw=black,inner sep=0.1em] (x3) at (2,0) {${x_3}$};
\node[circle,draw=black,inner sep=0.0em] (x3a) at (2.5,-0.4) {$~$};
\node[circle,draw=black,inner sep=0.0em] (x3b)at (2.5,+0.4) {$~$};
\node[circle,draw=black,inner sep=0.1em] (xm) at (3,0) {${x_m}$};
\node[circle,draw=black,inner sep=0.0em] (xma) at (3.5,-0.4) {$~$};
\node[circle,draw=black,inner sep=0.0em] (xmb) at (3.5,+0.4) {$~$};
\node[circle,draw=black,inner sep=0.0em,scale=0.72] (xm1) at (4,0) {${x_{m+1}}$};
\draw[]
(x1) edge[-{Stealth[scale=0.6]},bend right=20] node[below=1mm,scale=0.67]{$\neg x_1$} (x1a)
(x1a) edge[-{Stealth[scale=0.6]},bend right=20] (x2)
(x2) edge[-{Stealth[scale=0.6]},bend right=20,thick] node[below=1mm,scale=0.67]{$\neg x_2$} (x2a)
(x2a) edge[-{Stealth[scale=0.6]},bend right=20] (x3)
(x3) edge[-{Stealth[scale=0.6]},bend right=20,dotted] node[below=1mm,scale=0.67]{$\neg x_3$} (x3a)
(x3a) edge[-{Stealth[scale=0.6]},bend right=20,dotted] (xm)
(xm) edge[-{Stealth[scale=0.6]},bend right=20] node[below=1mm,scale=0.67]{$\neg x_m$}(xma)
(xma) edge[-{Stealth[scale=0.6]},bend right=20] (xm1);
\draw[]
(x1) edge[-{Stealth[scale=0.6]},bend left=20] node[above=0.5mm,scale=0.67]{$~x_1$} (x1b)
(x1b) edge[-{Stealth[scale=0.6]},bend left=20] (x2)
(x2) edge[-{Stealth[scale=0.6]},bend left=20] node[above=0.5mm,scale=0.67]{$~x_2$} (x2b)
(x2b) edge[-{Stealth[scale=0.6]},bend left=20] (x3)
(x3) edge[-{Stealth[scale=0.6]},bend left=20,dotted] node[above=0.5mm,scale=0.67]{$~x_3$}(x3b)
(x3b) edge[-{Stealth[scale=0.6]},bend left=20,dotted] (xm)
(xm) edge[-{Stealth[scale=0.6]},bend left=20] node[above=0.5mm,scale=0.67]{$~x_m$} (xmb)
(xmb) edge[-{Stealth[scale=0.6]},bend left=20] (xm1);
\node[circle,draw=black,inner sep=0.1em] (c1) at (6,0) {${C_1}$};
\node[circle,draw=black,inner sep=0.0em] (c1a) at (6.5,-0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em] (c1b) at (6.5,+0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em,fill=black] (c1c) at (6.5, 0.0) {$~$};
\node[circle,draw=black,inner sep=0.1em] (c2) at (7,0) {${C_2}$};
\node[circle,draw=black,inner sep=0.0em,fill=black] (c2a) at (7.5,-0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em] (c2b) at (7.5,+0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em] (c2c) at (7.5, 0.0) {$~$};
\node[circle,draw=black,inner sep=0.1em] (c3) at (8,0) {${C_3}$};
\node[circle,draw=black,inner sep=0.0em] (c3a) at (8.5,-0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em] (c3b) at (8.5,+0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em] (c3c) at (8.5, 0.0) {$~$};
\node[circle,draw=black,inner sep=0.1em] (cp) at (9,0) {${C_p}$};
\node[circle,draw=black,inner sep=0.0em] (cpa) at (9.5,-0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em] (cpb) at (9.5,+0.5) {$~$};
\node[circle,draw=black,inner sep=0.0em] (cpc) at (9.5, 0.0) {$~$};
\node[circle,draw=black,inner sep=0.0em,scale=0.75] (cp1) at (10,0) {${C_{p+1}}$};
\draw[]
(xm1) edge[-{Stealth[scale=0.6]}, dashed] node[above,scale=0.67]{$p$} (c1);
\draw[]
(c1) edge[-{Stealth[scale=0.6]},bend right=20] (c1a)
(c1a) edge[-{Stealth[scale=0.6]},bend right=20] (c2)
(c2) edge[-{Stealth[scale=0.6]},bend right=20] (c2a)
(c2a) edge[-{Stealth[scale=0.6]},bend right=20,thick] (c3)
(c3) edge[-{Stealth[scale=0.6]},bend right=20,dotted] (c3a)
(c3a) edge[-{Stealth[scale=0.6]},bend right=20,dotted] (cp)
(cp) edge[-{Stealth[scale=0.6]},bend right=20] (cpa)
(cpa) edge[-{Stealth[scale=0.6]},bend right=20] (cp1);
\draw[]
(c1) edge[-{Stealth[scale=0.6]},bend left=20] (c1b)
(c1b) edge[-{Stealth[scale=0.6]},bend left=20] (c2)
(c2) edge[-{Stealth[scale=0.6]},bend left=20] (c2b)
(c2b) edge[-{Stealth[scale=0.6]},bend left=20] (c3)
(c3) edge[-{Stealth[scale=0.6]},bend left=20,dotted] (c3b)
(c3b) edge[-{Stealth[scale=0.6]},bend left=20,dotted] (cp)
(cp) edge[-{Stealth[scale=0.6]},bend left=20] (cpb)
(cpb) edge[-{Stealth[scale=0.6]},bend left=20] (cp1);
\draw[]
(c1) edge[-{Stealth[scale=0.6]}] (c1c)
(c1c) edge[-{Stealth[scale=0.6]},thick] (c2)
(c2) edge[-{Stealth[scale=0.6]}] (c2c)
(c2c) edge[-{Stealth[scale=0.6]}] (c3)
(c3) edge[-{Stealth[scale=0.6]},dotted] (c3c)
(c3c) edge[-{Stealth[scale=0.6]},dotted] (cp)
(cp) edge[-{Stealth[scale=0.6]}] (cpc)
(cpc) edge[-{Stealth[scale=0.6]}] (cp1);
\draw[]
(-1,0) edge[-{Stealth[scale=0.6]}] node[above,scale=0.75]{$s_i$} (x1)
(cp1) edge[-{Stealth[scale=0.6]}] node[above,scale=0.75]{$s_i^\ast$} (11,0);
\node[rectangle,draw=black,inner sep=0.4em,scale=0.9,thick,text width=3.4cm] (loosen) at (4.5,-1.2) {Loosener from $\neg x_2$ on clauses that contain it.};
\node[scale=0.75] at (1.2,-1.2) {trigger by literal $\neg x_2$};
\node[scale=0.75,text width=3.9cm] at (8.7,-1.3) {consequences on clauses that contain this literal: reduced delay for agent $i$};
\draw[]
(x2a) edge[-{Stealth[scale=0.6]},bend right=15,thick] (loosen)
(loosen) edge[-{Stealth[scale=0.6]},bend right=15,thick] (c1c)
(loosen) edge[-{Stealth[scale=0.6]},bend right=15,thick] (c2a);
\end{tikzpicture}
\caption{Reduction from decision problem 3SAT to \textsc{FRoG/$\mathcal{R}$/Br} under (RO) and (RR): Let a 3SAT instance be defined by a list of $m$ binary variables $x_1,\ldots,x_m$ and a list of $p$ 3-clauses $C_1,\ldots,C_p$. A 3-clause is a disjunction of three literals (e.g. $C_j=x_1\vee\neg x_2\vee\neg x_3$). We reduce it to a \textsc{FRoG/$\mathcal{R}$/Br} instance (digraph, agents, priority and threshold). In the digraph, there is a sequence of $m+1$ nodes $x_1,\ldots,x_m,x_{m+1}$, followed by a sequence of $p+1$ nodes $C_1,\ldots,C_p,C_{p+1}$. Between any nodes $x_k$ and $x_{k+1}$, there are two two-edge paths. (Each path represents a choice of valuation for variable $x_k$: true or false.) From node $x_{m+1}$ to node $C_1$, there is one path with $p$ edges. Between any nodes $C_j$ and $C_{j+1}$, there are three two-edge paths that represent the three literals of clause $C_j$. Agent $i$ aims at traveling from source node $x_1$ to sink node $C_{p+1}$; firstly through $m$ choices of valuation for variables, towards node $x_{m+1}$ and then $C_1$; secondly through one literal per node $C_j$, until she reaches node $C_p$. Traveling the first edge of any choice of valuation/literal for variable $x_k$ during round $r=3(k-2)$ loosens on every clause $C_j$ that contains this literal, the arrival time of an agent with higher priority than $i$, from round $3m+p+2j$ to round $3m+p+2j+1$.
Traveling from vertices $x_1$ to $C_1$ takes constant time $3m+p$. The cost for traveling from vertices $C_j$ to $C_{j+1}$ is two are three, depending on whether a literal was loosened.
The question is whether agent $i$ can travel from vertex $x_1$ to $C_{p+1}$ in less than $\theta=3m+3p$ rounds.}
\label{fig:3SAT}
\end{figure}
\begin{proof}[Lemma \ref{lem:br}, under (RO)]
A path $\pi_i$ with total delay below $\theta$ is a yes-certificate that can be verified in polynomial-time, hence \textsc{FRoG/$\mathcal{R}$/Br} belongs to NP. We show hardness by a reduction from decision problem 3SAT, entirely described in Fig. \ref{fig:3SAT}, and proved below, first under rules (RO).
(yes$\Rightarrow$yes)
Assume a satisfactory instantiation of variables for the 3SAT instance, and let agent $i$ travel from vertices $x_1$ to $x_{m+1}$ accordingly. (It takes two rounds on every trigger and one round on the next edge, hence $3m$ rounds in total.) Since every clause $C_j$ is satisfied by at least one literal (out of three), it means that between $C_j$ and $C_{j+1}$, at least one path has its cost loosened from three to two, path that agent $i$ takes on right time $3m+p+2j$. Therefore her total delay is $3m+3p$.
(yes$\Leftarrow$yes)
Assume a path with total delay no larger than $3m+3p$, hence equal to it. It means that for every clause $C_j$, at least one delay was loosened (on the right time) by a successful choice of path/instantiation between vertices $x_1$ and $x_{m+1}$, which gives us an instantiation that satisfies the 3SAT instance.
\qed
\end{proof}
\begin{proof}[Generalization of Lemma \ref{lem:loosener} and \ref{lem:br} from rule (RO) to rule (RR)]
Both proofs generalize to rule (RR) by using the same construction. For (RR), it suffices to keep the same overall order over agents, and to observe that since agent $i$ is the only one to be strategical\footnote{The others have singleton strategy-sets: a single path from source to sink.}, no simultaneous decisions occur under (RO), hence rules (RO) and (RR) are here strategically equivalent.
\qed
\end{proof}
We are now ready to state the proof of Th. \ref{th:win} under rules (RO) and (RR).
\begin{figure}[t]
\centering
\begin{tikzpicture}
\node[circle,draw=black,inner sep=0.1em] (x1) at (0,0) {${x_1}$};
\node[circle,draw=black,inner sep=0.0em] (x1a) at (0.5,-0.4) {$~$};
\node[circle,draw=black,inner sep=0.0em] (x1b) at (0.5,+0.4) {$~$};
\node[circle,draw=black,inner sep=0.1em] (x2) at (1,0) {${x_2}$};
\node[circle,draw=black,inner sep=0.1em] (x3) at (2,0) {${x_3}$};
\node[circle,draw=black,inner sep=0.0em] (x3a) at (2.5,-0.4) {$~$};
\node[circle,draw=black,inner sep=0.0em] (x3b)at (2.5,+0.4) {$~$};
\node[circle,draw=black,inner sep=0.1em] (xm) at (3,0) {${x_m}$};
\node[circle,draw=black,inner sep=0.0em,scale=0.72] (xm1) at (4,0) {${x_{m+1}}$};
\draw[]
(x1) edge[-{Stealth[scale=0.6]},bend right=20,thick] (x1a)
(x1a) edge[-{Stealth[scale=0.6]},bend right=20] (x2)
(x3) edge[-{Stealth[scale=0.6]},bend right=20,thick] (x3a)
(x3a) edge[-{Stealth[scale=0.6]},bend right=20] (xm);
\draw[]
(x1) edge[-{Stealth[scale=0.6]},bend left=20,thick] (x1b)
(x1b) edge[-{Stealth[scale=0.6]},bend left=20] (x2)
(x3) edge[-{Stealth[scale=0.6]},bend left=20,thick] (x3b)
(x3b) edge[-{Stealth[scale=0.6]},bend left=20] (xm);
\draw[]
(x2) edge[-{Stealth[scale=0.6]}] (x3)
node[circle,draw=black,fill=white,inner sep=-0.1em] (x2a) at (1.35,0) {$~$}
node[circle,draw=black,fill=white,inner sep=-0.1em] (x2b) at (1.55,0) {$~$};
\draw[]
(xm) edge[-{Stealth[scale=0.6]}] (xm1)
node[circle,draw=black,fill=white,inner sep=-0.1em] (xma) at (3.35,0) {$~$}
node[circle,draw=black,fill=white,inner sep=-0.1em] (xmb) at (3.52,0) {$~$};
\node[circle,draw=black,inner sep=0.1em] (y1) at (0,2) {${y_1}$};
\node[circle,draw=black,inner sep=0.1em] (y2) at (1,2) {${y_2}$};
\node[circle,draw=black,inner sep=0.0em] (y2a) at (1.5,+1.6) {$~$};
\node[circle,draw=black,inner sep=0.0em] (y2b) at (1.5,+2.4) {$~$};
\node[circle,draw=black,inner sep=0.1em] (y3) at (2,2) {${y_3}$};
\node[circle,draw=black,inner sep=0.1em] (ym) at (3,2) {${y_m}$};
\node[circle,draw=black,inner sep=0.0em] (yma) at (3.5,+1.6) {$~$};
\node[circle,draw=black,inner sep=0.0em] (ymb) at (3.5,+2.4) {$~$};
\node[circle,draw=black,inner sep=0.0em,scale=0.72] (ym1) at (4,2) {${y_{m+1}}$};
\draw[]
(y2) edge[-{Stealth[scale=0.6]},bend right=20,thick] (y2a)
(y2a) edge[-{Stealth[scale=0.6]},bend right=20] (y3)
(ym) edge[-{Stealth[scale=0.6]},bend right=20,thick] (yma)
(yma) edge[-{Stealth[scale=0.6]},bend right=20] (ym1);
\draw[]
(y2) edge[-{Stealth[scale=0.6]},bend left=20,thick] (y2b)
(y2b) edge[-{Stealth[scale=0.6]},bend left=20] (y3)
(ym) edge[-{Stealth[scale=0.6]},bend left=20,thick] (ymb)
(ymb) edge[-{Stealth[scale=0.6]},bend left=20] (ym1);
\draw[]
(y1) edge[-{Stealth[scale=0.6]}] (y2)
node[circle,draw=black,fill=white,inner sep=-0.1em] (y1a) at (0.35,+2.0) {$~$}
node[circle,draw=black,fill=white,inner sep=-0.1em] (y1b) at (0.55,+2.0) {$~$};
\draw[]
(y3) edge[-{Stealth[scale=0.6]}] (ym)
node[circle,draw=black,fill=white,inner sep=-0.1em] (y3a) at (2.35,+2.0) {$~$}
node[circle,draw=black,fill=white,inner sep=-0.1em] (y3b)at (2.55,+2.0) {$~$};
\begin{scope}[shift={(0,0)}]
\node[circle,draw=black,inner sep=0.1em] (c1) at (5,0) {${C_1}$};
\node[circle,draw=black,inner sep=0.0em] (c1a) at (5.5,-0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (c1b) at (5.5,+0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (c1c) at (5.5, 0.0) {~};
\node[circle,draw=black,inner sep=0.1em] (c2) at (6,0) {${C_2}$};
\node[circle,draw=black,inner sep=0.0em] (c2a) at (6.5,-0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (c2b) at (6.5,+0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (c2c) at (6.5, 0.0) {~};
\node[circle,draw=black,inner sep=0.1em] (c3) at (7,0) {${C_3}$};
\node[circle,draw=black,inner sep=0.0em] (c3a) at (7.5,-0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (c3b) at (7.5,+0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (c3c) at (7.5, 0.0) {~};
\node[circle,draw=black,inner sep=0.1em] (cp) at (8,0) {${C_p}$};
\node[circle,draw=black,inner sep=0.0em] (cpa) at (8.5,-0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (cpb) at (8.5,+0.5) {~};
\node[circle,draw=black,inner sep=0.0em] (cpc) at (8.5, 0.0) {~};
\node[circle,draw=black,inner sep=0.0em,scale=0.75] (cp1) at (9,0) {${C_{p+1}}$};
\draw[]
(xm1) edge[-{Stealth[scale=0.6]}, dotted,thick] node[above,scale=0.67]{$p$} (c1);
\draw[]
(c1) edge[-{Stealth[scale=0.6]},bend right=20] (c1a)
(c1a) edge[-{Stealth[scale=0.6]},bend right=20] (c2)
(c2) edge[-{Stealth[scale=0.6]},bend right=20] (c2a)
(c2a) edge[-{Stealth[scale=0.6]},bend right=20] (c3)
(c3) edge[-{Stealth[scale=0.6]},bend right=20] (c3a)
(c3a) edge[-{Stealth[scale=0.6]},bend right=20] (cp)
(cp) edge[-{Stealth[scale=0.6]},bend right=20] (cpa)
(cpa) edge[-{Stealth[scale=0.6]},bend right=20] (cp1);
\draw[]
(c1) edge[-{Stealth[scale=0.6]},bend left=20] (c1b)
(c1b) edge[-{Stealth[scale=0.6]},bend left=20] (c2)
(c2) edge[-{Stealth[scale=0.6]},bend left=20] (c2b)
(c2b) edge[-{Stealth[scale=0.6]},bend left=20] (c3)
(c3) edge[-{Stealth[scale=0.6]},bend left=20] (c3b)
(c3b) edge[-{Stealth[scale=0.6]},bend left=20] (cp)
(cp) edge[-{Stealth[scale=0.6]},bend left=20] (cpb)
(cpb) edge[-{Stealth[scale=0.6]},bend left=20] (cp1);
\draw[]
(c1) edge[-{Stealth[scale=0.6]}] (c1c)
(c1c) edge[-{Stealth[scale=0.6]}] (c2)
(c2) edge[-{Stealth[scale=0.6]}] (c2c)
(c2c) edge[-{Stealth[scale=0.6]}] (c3)
(c3) edge[-{Stealth[scale=0.6]}] (c3c)
(c3c) edge[-{Stealth[scale=0.6]}] (cp)
(cp) edge[-{Stealth[scale=0.6]}] (cpc)
(cpc) edge[-{Stealth[scale=0.6]}] (cp1);
\end{scope}
\node[rectangle,draw=black,text width=1.5cm,scale=0.9,align=center,dotted] at (-1.5,0) {$\exists$-player};
\node[rectangle,draw=black,text width=1.5cm,scale=0.9,align=center,dotted] at (-1.5,2) {$\forall$-player};
\draw[]
(-0.7,0) edge[-{Stealth[scale=0.6]}] node[above,scale=0.75]{$s_\exists$} (x1)
(cp1) edge[-{Stealth[scale=0.6]}] node[above,scale=0.75]{$s_\exists^\ast$} (10,0.0);
\draw[]
(-0.7,2) edge[-{Stealth[scale=0.6]}] node[above,scale=0.75]{$s_\forall$} (y1)
(ym1) edge[-{Stealth[scale=0.6]}] node[above,scale=0.75]{$s_\forall^\ast$} (4.8,2);
\node[rectangle,dashed,draw=black,below right =-0.4mm and -0.5mm of x1a,scale=0.6](ln1){Loosen $\neg x_1$};
\node[rectangle,dashed,draw=black,below right =-0.4mm and -0.5mm of y2a,scale=0.6](ln2){Loosen $\neg x_2$};
\node[rectangle,dashed,draw=black,below right =-0.4mm and -0.5mm of x3a,scale=0.6](ln3){Loosen $\neg x_3$};
\node[rectangle,dashed,draw=black,below right =-0.4mm and -0.5mm of yma,scale=0.6](lnm){Loosen $\neg x_m$};
\node[rectangle,dashed,draw=black,above right =-0.4mm and -0.5mm of x1b,scale=0.6](lp1){Loosen $x_1$};
\node[rectangle,dashed,draw=black,above right =-0.4mm and -0.5mm of y2b,scale=0.6](lp2){Loosen $x_2$};
\node[rectangle,dashed,draw=black,above right =-0.4mm and -0.5mm of x3b,scale=0.6](lp3){Loosen $x_3$};
\node[rectangle,dashed,draw=black,above right =-0.4mm and -0.5mm of ymb,scale=0.6](lpm){Loosen $x_m$};
\draw[dotted]
(ln1) edge[-{Stealth[scale=0.6]},bend right=18] (cpa)
(lp2) edge[-{Stealth[scale=0.6]},bend left=25] (cpb)
(ln3) edge[-{Stealth[scale=0.6]},bend right=15] (c1a)
(ln3) edge[-{Stealth[scale=0.6]},bend right=15] (c3a)
(lpm) edge[-{Stealth[scale=0.6]},bend left=30] (c1b)
(lpm) edge[-{Stealth[scale=0.6]},bend left=30] (c2b)
(lp1) edge[-{Stealth[scale=0.6]},bend left=15] (c1c)
(ln2) edge[-{Stealth[scale=0.6]},bend left=0] (c2c)
(ln2) edge[-{Stealth[scale=0.6]},bend left=28] (c3c)
(lp3) edge[-{Stealth[scale=0.6]},bend right=8] (c2a)
(lnm) edge[-{Stealth[scale=0.6]},bend left=25] (c3b)
(lnm) edge[-{Stealth[scale=0.6]},bend left=35] (cpc);
\end{tikzpicture}
\caption{Reduction from decision problem QSAT to \textsc{FRoG/$\mathcal{R}$/Win} : Let a QSAT instance be defined by a list of $m$ quantified binary variables $Q_1x_1,\ldots,Q_mx_m$ with $Q_k\in\{\exists,\forall\}$ and a list of $p$ 3-clauses $C_1,\ldots,C_p$ with literals defined on any variable.
E.g.: $\exists x_1\forall x_2\exists x_3\forall x_4
(x_1 \vee \neg x_3\vee x_4)\wedge
(\neg x_2 \vee x_3\vee x_4)\wedge
(\neg x_2 \vee\neg x_3 \vee\neg x_4)\wedge
(\neg x_1\vee x_2\vee \neg x_4)$.
We reduce it to a \textsc{FRoG/$\mathcal{R}$/Win} instance similar to Figure \ref{fig:3SAT}, with the following changes and additions. A new agent who goes from vertices $y_1$ to $y_{m+1}$ is created, in order to model the universal player of the formula. If a binary variable $x_k$ has quantifier $\exists$ (resp. $\forall$), then the same two literal-choosing two-edge paths as in Fig. \ref{fig:3SAT} are created from vertex $x_k$ to $x_{k+1}$ (resp. from vertex $y_k$ to $y_{k+1}$), and a unique path of three edges is created from vertex $y_k$ to $y_{k+1}$ (resp. from vertex $x_k$ to $x_{k+1}$). Similarly to Fig. \ref{fig:3SAT}, when an agent takes a literal-choosing path from $x_k$ to $x_{k+1}$ (or from $y_k$ to $y_{k+1}$), in clauses $C_j$ that contain this literal in a path to $C_{j+1}$, it loosens the delay from three rounds to two, during round $3m+p+2j$. Is there a strategy for the existential player to reach sink $C_{p+1}$ in less than $\theta=3m+3p$, whatever the universal player decides?}\label{fig:winning}
\end{figure}
\begin{proof}[Theorem \ref{th:win}]
We show membership to PSPACE by the following minimax algorithm from combinatorial game theory.
Given a \textsc{FRoG}\ and a threshold $\theta$,
we must decide whether there is a (winning) strategy $\sigma_i$ for agent $i$
that gives her total delay at most $C_i(\sigma_i,\bm{\sigma}_{-i})\leq\theta$
for any adversary strategies $\bm{\sigma}_{-i}$.
To do so, we explore game-tree $\Gamma=(\mathcal{H},\mathcal{Q})$ recursively,
by computing on any subgame $\Gamma[H(r)]$ the best value guaranteed for agent $i$ in $\Gamma[H(r)]$, as follows.
If she is the only agent in $M_{H(r)}$ (to make a decision at this round), then she decides a next edge $e$ that optimizes her total delay according to the values of consecutive subgames $\Gamma((H(r),Q(r+1)))$.
If she does not belong to deciding agents $M_{H(r)}$, then we take the worst value on all consecutive subgames $\Gamma((H(r),Q(r+1)))$.
If there are several agents in $M_{H(r)}$ including her, then a new edge for agent $i$ is evaluated by considering the worst case of what the other agents in $M_{H(r)}$ may decide.
In this optimization process with alternating minima and maxima, agent $i$ reaches her sink in a polynomial number of edges (because that is her interest), whatever strategies the other agents choose, even the weirdest non-finishing ones (for them). Consequently, we only explore a finite subset of subgames and the (interesting part of the) game tree only has polynomial-depth. Therefore, running a depth-first search takes (a long time, but) polynomial-space.
We show hardness for PSPACE by a many-one reduction from decision problem QSAT (also known as TQBF), depicted in Figure \ref{fig:winning}.
A QSAT instance is defined by a list of $m$ quantified binary variables $Q_1x_1,\ldots,Q_mx_m$ with $Q_k\in\{\exists,\forall\}$ and a list of $p$ 3-clauses $C_1,\ldots,C_p$ with literals defined on any variable. It asks whether the following formula is true:
$
Q_1x_1,~Q_2x_2,~\ldots,~Q_mx_m,~
C_1\wedge C_2\wedge\ldots\wedge C_p.
$
It is well known that the validity of a QSAT formula can be interpreted as a zero-sum game between an existential player and a universal player \cite[Proof of Th. 4.1]{PAPADIMITRIOU1991}. The formula is true (resp. false) if and only if the existential (resp. universal) player has a winning strategy for instantiating variables.
Here, the universal player is overall indifferent between all the paths from $y_1$ to $y_{m+1}$, and may decide any strategy on universally quantified variables.
Since no simultaneous actions occur in our construct, it holds for both rules (RO) and (RR), by strategical equivalence.
(yes$\Rightarrow$yes) In the QSAT instance, if the existential player has a winning strategy against the universal player, the existential agent plays it on the path from vertex $x_1$ to $x_{m+1}$, by observing sequentially what the other agent decides from vertex $y_1$ to $y_{m+1}$. Since the formula is satisfied, between any vertex $C_j$ and $C_{j+1}$, there is at least one path loosened on the right round $3m+p+2j$ for the existential agent to travel from vertex $C_1$ to $C_{p+1}$ in $2p$ rounds.
(yes$\Leftarrow$yes) Assume that there is a path strategy for the existential agent from vertex $x_1$ to $x_{m+1}$ such that whatever edges the universal agent decides between $y_1$ and $y_{m+1}$, the delay from vertex $C_1$ to $C_{p+1}$ is $2p$ rounds. Then, between every vertices $C_j,C_{j+1}$, at least one literal-path is loosened. Therefore, the existential player in QSAT has the same winning strategy: the formula is true.\qed
\end{proof}
We now show a deeper negative result on the inapproximability of \textsc{FRoG} s under simultaneous and sequential rules (RO) and (RR).
\begin{theorem}\label{th:approx}
It is PSPACE-hard to approximate the optimization version of \textsc{FRoG/$\mathcal{R}$/Win} within $n^{1-\varepsilon}$ for any $\varepsilon>0$, and within any polynomial of $|V|$, under both simultaneous and sequential tiebreaking rules (RO) and (RR).
\end{theorem}
\begin{proof}[Theorem \ref{th:approx}]
Starting from QSAT, we reuse the same reduction as for Theorem \ref{th:win} (Figure \ref{fig:winning}), and append one last edge after vertex $C_{p+1}$ that the existential agent should cross to reach her goal. A number $M$ of agents enter this last edge on same round $3m+3p+2$, with higher priority than agent $i$. Hence, either agent $i$ succeeds in having total delay $\text{OPT}=3m+3p+1$, either she fails and obtains total delay $\text{OPT}+M$. The resulting problem contains $|V|=\text{poly}(m,p)$ vertices and $n=\text{poly}(m,p)+M$ agents.
If the QSAT formula is a yes-instance, then the optimum for agent $i$ is $3m+3p+1$, and an $n^{1-\varepsilon}$-approximation algorithm should return a solution within $\text{OPT}(\text{poly}(m,p)+M)^{1-\varepsilon}$, which turns out to be strictly less than $\text{OPT}+M$ for $M$ large enough and still polynomial in $(m,p)$. So the only approximate solution becomes the optimum. If the formula is a no, then total delay is the other one: $3m+3p+1+M$. Consequently, $n^{1-\varepsilon}$-approximation is PSPACE-hard.
Similarly, assume a $q(|V|)$-approximation algorithm for any polynomial $q$.
If the QSAT formula is a yes-instance, then it should return a solution within $\text{OPT}q(|V|)$, which is strictly less than $\text{OPT}+M$ for a large enough polynomial $M(m,p)$. Therefore, $q(|V|)$-approximation is also PSPACE-hard.\qed
\end{proof}
\section{The Complexity of Subgame Perfect Equilibrium}
In this section, we settle the computational complexity of subgame perfect equilibrium in \textsc{FRoG} s
under simultaneous actions (RO) and sequential actions (RR).
\begin{theorem}\label{th:exist}
Decision problem \textsc{FRoG/RO/Exist} is PSPACE-complete,
and function problem \textsc{FRoG/RR/Find} is also PSPACE-hard and in FPSPACE.
\end{theorem}
\begin{proof}[Theorem \ref{th:exist}]
We show membership to PSPACE or FPSPACE by the same algorithm,
which recursively explores game-tree $\Gamma=(\mathcal{H},\mathcal{Q})$ as follows.
In the process, we only approve sequential (resp. simultaneous) decision nodes $H(r)$ where (the) agent(s) in $M_{H(r)}$ can play a best-response (resp. pure Nash equilibrium) that leads to a subgame $\Gamma(H(r),Q(r+1))$ that contains an SPE.
In order to only explore a finite subset of subgames, one can cut the sub-trees where an agent went into a dead-end or where an agent's current delay (that is: current round $r$) is already more than $|E|\times|N|$.
Indeed, these subtrees are obviously not in the best individual interest of those agents,
since even a blind strategy can do better whatever the others' strategies.
Consequently, these subtrees cannot contain an SPE.
Because of this cut, the game tree only has polynomial depth.
Therefore, a depth-first search takes polynomial-space (but a lot of time).
For function problem \textsc{FRoG/RR/Find}, we show PSPACE-hardness by turning Figure \ref{fig:winning} into a zero-sum game between the existential and universal agents: It suffices to add one last edge $e$ on both strategical agents' paths (reachable in some constant delay from vertices $C_{p+1}$ and $y_{m+1}$). On this edge, the universal agent (who was overall indifferent) is now delayed by one round if and only if the existential agent travels from vertex $C_1$ to $C_{p+1}$ in $2p$ rounds (formula satisfied) and then collides with her on edge $e$.
Since any winning strategy is always played in an SPE, the induced paths indicate whether the formula is true or false.
For decision problem \textsc{FRoG/RO/Exist}, we proceed from Figure \ref{fig:winning} as for \textsc{FRoG/RR/Find}, with the following addition:
If the universal agent is delayed, then she is synchronized to collide with and cancel out counter-example \cite[Fig. 2]{cao2017arxiv} (e.g. by delaying an agent therein). Therefore, if the formula is a yes (resp. no), the universal agent is delayed, the counter-example is (resp. not) canceled and the \textsc{FRoG/RO/Exist} instance is a yes (resp. no).\qed
\end{proof}
\section{Conclusion}
In this paper, for sequential routing games, under simultaneous and sequential tiebreaking rules (RO) and (RR), we settle the computational complexity of the winning strategy problem as PSPACE-complete. PSPACE-hardness of subgame perfect equilibrium is a consequence of the complexity of winning strategies.
Interestingly, under tiebreaking rule (RE), where there are priorities on incoming edges, rather than on agents, the best-response problem turns out P-time tractable (see Th. \ref{th:br:RE} below). This result contrasts with intractability results Lemma \ref{lem:br} and \cite[Th. 3-5]{ismaili2017} which assume a tie-breaking order on agents. Rule (RE) (first proposed in \cite{cao2017arxiv}) turns out to be a sane model that does not favor agents by their IDs. Hence, it will be worth studying further.
\begin{theorem}\label{th:br:RE}
Best-response problem \textsc{FRoG/RE/Br} is in P.
\end{theorem}
\begin{proof}[Theorem \ref{th:br:RE}]
Let us run an initializing event-based Dijkstra-style algorithm on $G=(V,E)$ with adversary paths $\bm{\pi}_{-i}$.
It provides us with congestions $Q_e(r)$ of every queue $e$ and every round $r$, in polynomial-time.
We now consider augmented graph $\mathcal{G}=(\{s_i\}\cup E,\mathcal{E})$.
Every vertex $v\in V$ is augmented into origin-vertex couple $(u,v)\in E$ (but the source).
In set $\mathcal{E}$, there is an augmented edge from origin-vertex $(u,v)$ (resp. $s_i$) to origin-vertex $(v,w)$ (resp. $(s_i,w)$) if the former's vertex (head) is the later's origin (tail). Costs are \emph{dynamically} defined as follows:
If agent $i$ decides next edge $e=(v,w)$ from origin-vertex $(u,v)$ during round $r$,
then she reaches $w$ on round $r+|Q_e(r+1)|-|P_{(u,v,w)}(r+1)|+1$, where $Q_e(r+1)$ is known from the first algorithm (without agent $i$), and agent set $P_{(u,v,w)}(r+1)\subseteq Q_e(r+1)$ are those entering edge $e=(v,w)$ on same round $r+1$ as agent $i$, but from an origin/edge with lower priority than $(u,v)$ (hence $i$ passes them).
First, let us assume that agent $i$ (whose shortest path we compute) does not incur any queue modification that she causes (as in Fig. \ref{fig:loosener}).
In augmented graph $\mathcal{G}$, despite the fact that costs are dynamic, agent $i$ has no interest in arriving one round later in any waiting list, since it is the maximal rate at which a queue empties. We can then directly run Dijkstra's algorithm on $\mathcal{G}$.
Under tiebreaking rule (RE), any path has same delay for any agent, since delays only depend on when and where they come from, rather than their IDs (like in rules (RO) and (RR)).
Consequently, on a shortest path, agent $i$ can always go at least faster than the chain of queue-modifications that she triggers, by (for instance) taking the same modified path.
The frog passes Mach one and crosses roads without (new) collisions (Easter egg).
Therefore, on shortest paths, she never incurs any queue modification that she caused, as assumed above.
\qed
\end{proof}
\textbf{Acknowledgments}\quad I am grateful to Ilan Nehama for his enthusiasm, to Silvia De Bellis for proofreading and to the anonymous reviewers for their tedious work.
Following recent open practices, reviews will be appended to a preprint.
\bibliographystyle{alphaCapitalization}
|
1,314,259,996,712 | arxiv | \section{Introduction}\label{sec:Intro}
In this article we study the sufficient conditions for the local invertibility of mappings. Our work is motivated by the well-known \emph{inverse function theorem} which states that every continuously differentiable mapping
$$f \in C^{1}(U, \R^n) \quad \text{($U \subset \R^n$ open and connected
set)}$$
is a local $C^1$-diffeomorphism outside the zero set of its Jacobian determinant. We are interested in the conditions under which one can recover the local invertibility when the usual assumptions of the inverse function theorem are not satisfied. This leads us to the following two questions:
\begin{itemize}
\item[(Q1)] How to recover local invertibility when the mappings are less than $C^1$-regular?
\item[(Q2)] How to recover local invertibility near the singular set of the Jacobian determinant of mappings?
\end{itemize}
We study these questions in the class of \emph{quasiregular mappings}, id est, in the class of Sobolev mappings
$$f \in W_{\loc}^{1,n}(U, \R^n) \quad \quad \text{($U \subset \R^n$ open and connected
set with $n \ge 2$)}$$
for which the \emph{operator norm} of the weak differential matrix satisfies the following \emph{distortion inequality}
\begin{align}\label{DistortionInequality}
\norm{Df(x)}^n \le K \det Df(x) \colonequals K J(x; f) \colonequals KJ_f(x) \quad \text{a.e.}
\end{align}
for some constant $K \ge 1$. Homeomorphic quasiregular maps form the well-studied class of \emph{quasiconformal mappings}. For the basic properties and the background of quasiregular and quasiconformal mappings we refer to monographs \cite{Astala-Iwaniec-Martin,IwaniecMartin,Reshetnyak89,Rickman-book, VuorinenBook, VaisalaBook}.
Next we point out that even if the definition of quasiregularity is purely analytical it still harbors a great deal of topological information per se. Especially, the distortion inequality can be applied to provide surprisingly vast amount of information on the invertibility properties of quasiregular mappings. This was first observed by Reshetnyak who originally introduced quasiregular mappings by the name of \emph{mappings of bounded distortion} and discovered their basic properties in a series of papers in 1966--1969. One of the deepest discoveries of these works was that non-constant quasiregular mappings are discrete and open, see e.g. \cite{Reshetnyak89}. This observation connected the study of quasiregular mappings to the earlier studies on \emph{branched coverings} in geometric topology, see e.g \cite{Chernavski1964, Chernavski1965, ChurchHemmingsen1960, ChurchHemmingsen1961, ChurchHemmingsen1963}.
In the critical step of the proof of Reshetnyak's celebrated discreteness and openness theorem one applies non-linear potential theory and non-linear PDEs to transfer analytical data into topological information. This step can be carried out by studying the geometric size of the polar sets of the solutions to the quasilinear elliptic partial differential equation
\begin{align}\label{eq:qrPDE}
-\diver \bigl( \langle G_f^{-1} \nabla u, \nabla u \rangle^{(n-2)/2} G_f^{-1} \nabla u \bigr) = 0 \, ,
\end{align}
where
\begin{displaymath}
G_f^{-1}(x) = \left\{ \begin{array}{ll}
\frac{\cof Df(x)^T \cof Df(x)}{\det Df(x)^{2(n-1)/n}}, & \textrm{if $J_f(x)>0$}\\
\I, & \textrm{otherwise,}
\end{array} \right.
\end{displaymath}
stands for the \emph{inverse dilatation tensor} and $\cof Df(x)$ denotes the \emph{cofactor matrix} of the differential matrix. This way one eventually obtains that
$$\mathcal{H}^1(f^{-1}(y)) = 0 \quad \text{for every } y \in \R^n \, ,$$
which implies that every quasiregular mapping is \emph{light} in the sense that the preimage of every point is totally disconnected. After this Reshetnyak's result follows by showing that quasiregular mappings are sense-preserving and by obtaining that sense-preserving and light maps between connected oriented manifolds are discrete and open, see e.g. \cite{BojarskiIwaniec1983, Heinonen2002,IwaniecMartin, ManfrediVillamor1998} for further details.
Reshetnyak's theorem and its techniques have been studied further by several authors, see e.g. \cite{GoldsteinVodopyanov1976,HeinonenKoskela1993,HenclMaly2002,IwaniecSverak1993,ManfrediVillamor1998,ManfrediVillamor1995,OnninenZhong, Rajala2011}. In this process the developement of \emph{mappings of finite distortion} \cite{HenclKoskelaBook, IwaniecMartin, MRSYBook2009} and the study of their connection to the non-linear elasticity theory of Ball, Antman, and Ciarlet \cite{Ball2,Antman1976,Ci} have been main driving forces. In this context, the generalizations of Reshetnyak's theorem have been applied to investigate \emph{impenetrability of matter} of deformations in non-linear elasticity theory. However, usually these techniques have only been used to obtain discreteness and openness of deformations instead of recovering the actual local invertibility.
In this article we study the local invertibility of quasiregular mappings by utilizing the earlier studies of Onninen and Zhong \cite{OnninenZhong} on Reshetnyak's theorem in order to study \emph{Martio's conjecture} which states that every non-constant quasiregular mapping
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 3$})$$
with an \emph{inner dilatation}
$$K_I(f) \colonequals \esssup_{x \in U} \frac{\norm{\cof Df(x)}^n}{\det Df(x)^{n-1}}$$
less than two is a local homeomorphism. This long-standing unconfirmed conjecture was originally stated by Martio, Rickman, and Väisälä in \cite{MRV-71} and it was motivated by the preliminary work of Martio \cite{Martio1970}, where the conjecture was confirmed when the \emph{branch set}
$$\mathcal{B}_f \colonequals \{ x \in U : f \text{ is not a local homeomorphism at } x \}$$
of a quasiregular map contains a rectifiable curve. From a technical point of view it is usually more natural to study the following generalization of the conjecture from \cite{Tengvall}:
\begin{conjecture}[Strong Martio's conjecture]
The inner dilatation of a non-constant quasiregular mapping
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 3$})$$
satisfies
$$\inf_{x \in \mathcal{B}_f} i(x,f) \le K_I(f) \, ,$$
where and in what follows $i(x,f)$
stands for the local topological index of a point $x \in U$ under the mapping $f$, see \cite[Chapter~I]{Rickman-book}.
\end{conjecture}
We point out for the reader that the standard quasiregular \emph{$m$-to-1 winding mapping}
$$(r,\theta, z) \stackrel{w_m}{\mapsto} (r,m\theta,z) \quad \text{($z \in \R^{n-2}$)} \, ,$$
written here in the cylindrical coordinates, is an extermal for the conjecture. In addition, the holomorphic function
$$f : \mathbb{C} \to \mathbb{C}, \quad f(z) = z^m$$
shows the conjecture to fail in the planar case. In \cite{KLT3} Kauranen, Luisto, and Tengvall verified the conjecture for \emph{mappings of bounded length distortion}, also known as \emph{BLD-mappings}. This class consists of those quasiregular Lipschitz mappings
$$f : U \to f(U) \subset \R^n \quad \text{($U \subset \R^n$ open set with $n \ge 2$)}$$
for which we have
\begin{align}\label{eq:BoundedJacobian}
\det Df(x) > c \quad \text{a.e.}
\end{align}
for some constant $c>0$, see \cite{MartioVaisala}. In \cite{Tengvall} Tengvall relaxed the boundedness condition \eqref{eq:BoundedJacobian} even further by proving the conjecture under certain integrability condition on the reciprocal of the Jacobian determinant. He also offered several alternative proofs for the conjecture in the BLD-class. However, none of the above-mentioned proofs from \cite{KLT3,Tengvall} is self-contained as each one of them heavily relies on the following well-known local modulus of continuity estimate
\begin{align}\label{eq:ModuusOfContinuity}
\abs{f(x)-f(y)} \le C \abs{x-y}^{\bigl( \frac{i(x,f)}{K_I(f)}\bigr)^{\frac{1}{n-1}}} \quad \text{for all } y \in B(x,r)
\end{align}
by Martio \cite{Martio1970} which can be also found from \cite[Theorem~III.4.7]{Rickman-book}. The proof of this estimate requires several layers of preliminary results which makes it technical and rather lengthy. In this article we prove the strong Martio's conjecture for BLD-mappings without any use of the estimate \eqref{eq:ModuusOfContinuity} by providing a rather short and self-contained proof for the following result from \cite{KLT3} which is valid also in the planar case:
\begin{theorem}[Kauranen, Luisto, and Tengvall, 2021]\label{thm:main}
Every non-constant BLD-mapping
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 2$})$$
satisfies $i(x,f) \le K_I(f)$ for every $x \in U$.
\end{theorem}
Finally we highlight the connection of \emph{the generalized Liouville's theorem} to our studies on local invertibility of quasiregular mappings. This rigidity result states that non-planar, non-constant quasiregular mappings with
\begin{align}\label{eq:DilatationOne}
K_I(f) = 1
\end{align}
are restrictions of Möbius transformations. The result generalizes the well-known Liouville's theorem \cite{Capelli1886, Liouville1850,Hartman1947,Hartman1958} on rigidity of non-planar conformal diffeomorphisms. The original proofs of Gehring \cite{Gehring1962} and Reshetnyak \cite{ReshetnyakLiouville1967} for the result are based on the study of regularity properties of the solutions to the non-linear $n$-harmonic equation
\begin{align}\label{eq:nHarmonic}
-\diver \bigl( \abs{\nabla u(x)}^{n-2} \nabla u(x) \bigr) = 0 \, .
\end{align}
This equation can be obtained from \eqref{eq:qrPDE} when
$$G_f^{-1}(x) = \id \quad \text{a.e.}$$
and in the planar case it reduces to the usual Laplace equation. In the earlier-mentioned work \cite{MRV-71} of Martio, Rickman, and Väisälä (see also \cite{Goldstein1971}) generalized Liouville's theorem was applied with a compactness argument to obtain that non-planar, non-constant quasiregular mappings with an inner dilatation close to one are local homeomorphisms. Later a quantitative version of this result was obtained by Rajala \cite{Rajala-MartioResult}.
As by the generalized Liouville's theorem all non-planar, non-constant quasiregular mappings with the property \eqref{eq:DilatationOne} coincide with the identity map up to a conjugation by restrictions of Möbius transformations it is natural to ask whether similar kind of phenomenom occurs also for quasiregular mappings with
\begin{align}\label{InnerAndIndex}
K_I(f) = \inf_{x \in \mathcal{B}_f} i(x,f) \, .
\end{align}
In the light of current knowledge it seems that up to a conjugation by Möbius transformations the only non-planar quasiregular mapping with the property \eqref{InnerAndIndex} is the standard $m$-to-1 winding map. Therefore, we conjecture:
\begin{conjecture}[Rigidity conjecture]
Every non-planar quasiregular mapping with
\begin{align*}
K_I(f) = m_f, \quad \text{where} \quad m_f \colonequals \left\{ \begin{array}{ll}
1, & \textrm{if $\mathcal{B}_f = \emptyset$}\\[0.5em]
\inf_{x \in \mathcal{B}_f} i(x,f), & \textrm{if $\mathcal{B}_f \neq \emptyset$,}
\end{array} \right.
\end{align*}
equals to the standard $m_f$-to-1 winding mapping up to a conjugation by restrictions of Möbius transformations.
\end{conjecture}
\section*{\textbf{Acknowledgments}} The author wishes to thank Jani Onninen for several discussions on his work \cite{OnninenZhong}. He would also like to thank Katrin Fässler and Sebastiano Nicolussi Golo for their comments on this research after author's Jyväskylä Geometric Analysis Seminar presentation in November 2021. Part of the writing process of the article was done while the author was visiting Massey University, New Zealand Institute for Advanced Studies. He would like to thank the institute for its hospitality. The visit was funded by the Mobility Grant 2022 of Faculty of Mathematics and Science of University of Jyväskylä.
\section{Preliminaries}
In this section we recall some basic facts on mappings of bounded length distortion from \cite{MartioVaisala} and on discrete and open maps from \cite{Rickman-book}. If the reader is well-aware of the basic properties and the notation related to these mapping classes, reading this section is not necessary.
\subsection{Preliminary properties for BLD-maps} We recall that mappings of bounded length distortion form a subclass of quasiregular maps. Thus, it follows from Reshetnyak's theorem that these mappings are continuous, sense-preserving, discrete, and open. In addition, by the characterization \cite[Theorem~2.16]{MartioVaisala} of these mappings every BLD-map
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 2$})$$
satisfies the following length distortion bounds
\begin{align}\label{eq:LengthDistortion}
\ell(\gamma)/L \le \ell(f \circ \gamma) \le L \ell(\gamma)
\end{align}
for every path $\gamma$ in $U$ with some length distortion constant $L \ge 1$, where $\ell(\gamma)$ stands for the lenght of a path $\gamma$.
\subsection{Preliminary properies for discrete and open maps} As mappings of bounded lenght distortion form a subclass of discrete and open maps we may apply all the basic results on these maps in order to study BLD-maps further. In this section we recall the definitions and results on discrete and open maps from \cite[Chapter~I]{Rickman-book} that are later needed for the proof of Theorem~\ref{thm:main}. We start by recalling that an open, connected set $D \subset \subset U$ is called a \emph{normal domain} of a continuous, discrete, and open mapping
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 2$})$$
if it satisfies
\begin{align*}
f(\partial D) = \partial f(D).
\end{align*}
If a normal domain $D$ satisfies
$$D \cap f^{-1}(f(x)) = \{ x\} \, ,$$
then it is called a \emph{normal neighborhood} of a point $x \in U$. In addition, for a given point $x \in U$ we denote
$$U(x,f,r) \colonequals \text{``the $x$-component of the preimage $f^{-1}\bigl(B(f(x),r) \bigr)$''}.$$
With the notation introduced above we may recall the following standard lemma from \cite[Lemma~I.4.9]{Rickman-book} which is applied frequently throughout the article:
\begin{lemma}\label{lemma:NormalDomain}
Let
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 2$})$$
be a continuous, discrete, and open mapping. Then for every $x \in U$ there exists a radius $r_x > 0$ for which $U(x,f,r)$ is a normal neighborhood of $x$ such that
$$f\bigl(U(x,f,r) \bigr) = B\bigl(f(x),r\bigr) \quad \text{for every } 0 < r \le r_x.$$
Moreover, we have
$$\diam \bigl(U(x,f,r) \bigr) \to 0 \quad \text{as } r \to 0 \, .$$
\end{lemma}
\subsection{Path lifting} In order to study the compression of BLD-maps later in section~\ref{sec:Compression} we need the following path lifting lemma that follows directly from \cite[Proposition~I.4.10]{Rickman-book} and \cite[Corollary~II.3.4]{Rickman-book}:
\begin{lemma}\label{lemma:PathLifting}
Let
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 2$})$$
be a continuous, discrete, and open mapping and let $D \subset \subset U$ be a normal neighborhood of a point $x_0 \in U$ under the mapping $f$. In addition, denote
$$m \colonequals i(x_0,f) .$$
Then for every path
$$\beta :[a,b) \to f(D)$$
starting at $f(x_0)$ there exist paths
$$\alpha_j : [a,b) \to D \quad (\text{$j=1, \ldots, m$})$$
each of which starts at $x_0$ and such that the following conditions are satisfied:
\begin{itemize}
\item[(i)] $f \circ \alpha_j = \beta$ for each $j=1, \ldots, m$.
\item[(ii)] $\card \{ j :\alpha_j(t) = x\} = i(x,f)$ for $x \in D \cap f^{-1}\bigl(\beta(t) \bigr)$.
\item[(iii)] For the traces of the above-mentioned paths we have the following relation
$$\abs{\alpha_1} \cup \cdots \cup \abs{\alpha_m} = D \cap f^{-1}\bigl(\abs{\beta} \bigr).$$
\end{itemize}
\end{lemma}
\section{Compression of BLD-mappings}\label{sec:Compression}
In this section we shortly discuss the local compression of BLD-maps and show that these maps cannot compress materia too much together in the following sense:
\begin{lemma}\label{lemma:LowerBound}
Every non-constant BLD-mapping
$$f : U \to f(U) \subset \R^n \quad (\text{$U \subset \R^n$ open set with $n \ge 2$})$$
satisfies
$$B\bigl(f(x),r/L \bigr) \subset f\bigl(B(x,r) \bigr) \quad \text{for some $L \ge 1$} \, ,$$
whenever $r>0$ is sufficiently small.
\end{lemma}
\begin{proof}
Fix a point $x \in U$. By Lemma~\ref{lemma:NormalDomain} we may find a radius $r_x > 0$ such that the set $U(x,f,r)$ is a normal neighborhood of $x$ whenever $0 < r \le r_x$. Fix any radius $r > 0$ so small that
$$B(x,r) \subset U(x,r_x,f) \, ,$$
and denote by $\tilde{r} > 0$ the largest radius such that
$$B\bigl(f(x),\tilde{r} \bigr) \subset f\bigl(B(x,r) \bigr).$$
Then to conclude the proof it suffices to show that
\begin{align}\label{ProveThis}
\tilde{r} \ge r/L \quad \text{for some constant $L \ge 1$.}
\end{align}
For this purpose, we obtain that by the openness of $f$ we have
$$\partial f\bigl(B(x,r) \bigr) \subset f\bigl(\partial B(x,r)\bigr),$$
and therefore we may find a point $z \in \partial B(x,r)$ such that
$$f(z) \in \partial B\bigl(f(x),\tilde{r} \bigr).$$
Let us next consider the line-segment
$$I \colonequals [f(x),f(z)].$$
By applying Lemma~\ref{lemma:PathLifting} we may then find a path $\gamma$ in $U(x,f, \tilde{r})$ from $x$ to $z$ such that
$$f \circ \gamma = I \, .$$
However, then by \eqref{eq:LengthDistortion} we get
\begin{align*}
\tilde{r} = \ell(I) = \ell(f \circ \gamma) \ge \ell(\gamma)/L \ge r/L
\end{align*}
for some constant $L \ge 1$. Thus, we have verified \eqref{ProveThis} and the claim follows.
\end{proof}
We point out that Lemma~\ref{lemma:LowerBound} can be actually obtained directly from \cite[Lemma~4.6]{MartioVaisala}. However, for the convenience of the reader and for the self-containedness of this article we have sketched a proof above as well. One should also notice that Lemma~\ref{lemma:LowerBound} is not valid for quasiregular maps in general. This can be demonstrated by the radially symmetric $K$-quasiconformal map
\begin{align*}
f : \R^n \to \R^n, \quad f(x) = \left\{ \begin{array}{ll}
\frac{x}{\abs{x}} \abs{x}^{K^{\frac{1}{n-1}}}, & \textrm{if $x \neq 0$}\\[0.5em]
0, & \textrm{if $x=0$,}
\end{array} \right.
\end{align*}
with a constant $K > 1$. In this context we note for the reader that obtaining Lemma~\ref{lemma:LowerBound} is the only step of our proof for Theorem~\ref{thm:main} where the BLD-assumption is needed as all the other steps are valid for general quasiregular mappings.
\section{Improving the lemma of Onninen and Zhong}
In this section we provide the key lemma for our proof of Theorem~\ref{thm:main}. For this purpose we first recall the result of Onninen and Zhong from \cite{OnninenZhong} according to which for every
$$f \in C^{\infty}(U, \R^n) \quad \text{and} \quad \Psi \in C^1\bigl([0,\infty), [0,\infty) \bigr) \, ,$$
and for every compactly supported test-function $\eta \in C_c^{\infty}\bigl(U,[0,\infty) \bigr)$ we have
\begin{align*}
\bigg{\lvert} \, \int_{U} \eta^n [n \Psi\bigl(\abs{f}^2 \bigr) + 2\abs{f}^2 \Psi'\bigl(\abs{f}^2 \bigr) J_f] \, \bigg{\rvert} \le C \int_{U} \eta^{n-1} \abs{\nabla \eta} \abs{f} \Psi\bigl(\abs{f}^2 \bigr) \norm{Df}^{n-1} \, ,
\end{align*}
where the constant $C>0$ depends on the dimension of the underlying open set $U \subset \R^n$. In what follows, we provide a sharp version of this estimate in Lemma~\ref{lemma:MainEstimate} below. The proof of this refinement is based on the following identity:
\begin{lemma}\label{lemma:BasicLinearAlgebra}
Let $U \subset \R^n$ be an open set with $n \ge 2$ and suppose that
$$f : U \to f(U) \subset \R^n \quad \text{and} \quad \eta : U \to \R \quad $$
are differentiable at a point $x \in U$. Then
\begin{align*}
\sum_{i=1}^n f_i &J(x; f_1, \ldots, f_{i-1}, \eta, f_{i+1}, \ldots, f_n) = \nabla \eta(x)^T \cof Df(x) f(x) \, .
\end{align*}
\end{lemma}
\begin{proof}
Let us denote
$$A_{i,j} \colonequals [\cof Df(x)]_{i,j} \, .$$
Then by direct computation we obtain that
\begin{align}\label{eq:CofactorIdetity}
\nabla \eta(x)^T \cof Df(x) f(x) &= \left[ \begin{array}{ccc}
\partial_1 \eta & \cdots & \partial_n \eta
\end{array} \right] \left[\ \begin{array}{ccc}
A_{1,1} & \cdots & A_{1,n} \\
\vdots & \ddots & \vdots \\
A_{n,1} & \cdots & A_{n,n}
\end{array} \right] \left[ \begin{array}{ccc}
f_1 \\
\vdots \\
f_n
\end{array} \right] \nonumber\\
&= \biggl(\sum_{i=1}^n f_1 \partial_i \eta A_{i,1} \biggr) + \cdots + \biggl(\sum_{i=1}^n f_n \partial_i \eta A_{i,n} \biggr) \\
&= \sum_{i=1}^n \sum_{j=1}^n \partial_i \eta A_{i,j} f_j \, . \nonumber
\end{align}
On the other hand, for each $i=1, \ldots, n$ we have
\begin{align*}
f_i &J(x; f_1, \ldots, f_{i-1}, \eta, f_{i+1}, \ldots, f_n) = \sum_{j=1}^n \partial_j \eta A_{1,j} f_i \, ,
\end{align*}
and by summing over $i$'s we get
\begin{align}\label{eq:JacobianIdentity}
\sum_{i=1}^n f_i &J(x; f_1, \ldots, f_{i-1}, \eta, f_{i+1}, \ldots, f_n) = \sum_{i=1}^n \sum_{j=1}^n \partial_j \eta A_{i,j} f_i \, .
\end{align}
By combining \eqref{eq:CofactorIdetity}--\eqref{eq:JacobianIdentity} we have
\begin{align*}
\sum_{i=1}^n f_i &J(x; f_1, \ldots, f_{i-1}, \eta, f_{i+1}, \ldots, f_n) = \nabla \eta(x)^T \cof Df(x) f(x) \, ,
\end{align*}
which completes the proof.
\end{proof}
\begin{lemma}\label{lemma:MainEstimate}
Let $U \subset \R^n$ be an open set with $n \ge 2$ and suppose that
$$f \in C^{\infty}(U, \R^n) \quad \text{and} \quad \Psi \in C^1\bigl([0,\infty), [0,\infty) \bigr) \, .$$
Then for every test-function $\eta \in C_c^{\infty}(U, [0, \infty))$ we have
\begin{align*}
\bigg{\lvert} \, \int_{U} \eta \Bigl[n \Psi\bigl(\abs{f}^2 \bigr) + 2\abs{f}^2 \Psi'\bigl(\abs{f}^2 \bigr) J_f \Bigr] \, \bigg{\rvert} \le \int_{U} \abs{\nabla \eta} \abs{f} \Psi\bigl(\abs{f}^2 \bigr) \norm{\cof Df} \, .
\end{align*}
\end{lemma}
\begin{proof}
The proof follows the same steps as the proofs of \cite[Lemma~2.1]{OnninenZhong} and \cite[Lemma~3.19]{HenclKoskelaBook}. Indeed, first by fixing $i \in \{ 1, \ldots, n\}$ we obtain by applying Stokes' theorem that
\begin{align}\label{eq:stokes}
\int_{U} J(x; f_1, \ldots, f_{i-1}, \eta \Psi\bigl(\abs{f}^2 \bigr) f_i, f_{i+1}, \ldots, f_n ) \, dx = 0 \, .
\end{align}
On the other hand, the chain rule gives us
\begin{align*}
J(x; f_1, \ldots, f_{i-1}, &\Psi\bigl(\abs{f}^2 \bigr), f_{i+1}, \ldots, f_n) \\
&= \sum_{j=1}^n 2\Psi'\bigl(\abs{f}^2 \bigr) f_j J(x; f_1, \ldots, f_{i-1},f_j,f_{i+1}, \ldots, f_n) \\
&= 2\Psi'\bigl(\abs{f}^2 \bigr) f_i J(x; f_1, \ldots, f_n) \, .
\end{align*}
Thus, by the product rule we obtain
\begin{align*}
&J(x; f_1, \ldots, f_{i-1}, \eta \Psi\bigl(\abs{f}^2 \bigr) f_i, f_{i+1}, \ldots, f_n ) \\
&= \Psi\bigl(\abs{f}^2 \bigr) f_i J(x; f_1, \ldots, f_{i+1},\eta, f_{i+1}, \ldots, f_n) + 2\eta f_i^2 \Psi'\bigl(\abs{f}^2 \bigr) J_f + \eta \Psi\bigl(\abs{f}^2 \bigr) J_f \, .
\end{align*}
By combining this with \eqref{eq:stokes} we get
\begin{align*}
\int_{U} \eta \bigl[ \Psi\bigl(\abs{f}^2 \bigr) + &2\Psi'\bigl(\abs{f}^2 \bigr) f_i^2\bigr] J_f \\
&= -\int_{U} \Psi\bigl(\abs{f}^2 \bigr) f_i J(x; f_1, \ldots, f_{i-1}, \eta, f_{i+1}, \ldots, f_n) \, ,
\end{align*}
and by summing over $i$'s and applying Lemma~\ref{lemma:BasicLinearAlgebra} we have
\begin{align*}
\biggl{\lvert} \int_{U} \eta \Bigl( n&\Psi\bigl(\abs{f}^2 \bigr) + 2\Psi'\bigl(\abs{f}^2 \bigr) \abs{f}^2\Bigr) J_f \biggr{\rvert}\\
&= \biggl{\lvert} \sum_{i=1}^n \int_{U} \Psi\bigl(\abs{f}^2 \bigr) f_i J(x; f_1, \ldots, f_{i-1}, \eta, f_{i+1}, \ldots, f_n) \biggr{\rvert} \\
&= \biggl{\lvert} \int_{U} \Psi\bigl(\abs{f(x)}^2 \bigr) \nabla \eta(x)^T \cof Df(x) f(x) \biggr{\rvert} \\
&= \biggl{\lvert} \int_{U \setminus \{ x : \abs{f(x)} = 0\}} \Psi\bigl(\abs{f(x)}^2 \bigr) \nabla \eta(x)^T \cof Df(x) f(x) \biggr{\rvert} \\
&\le \int_{U \setminus \{ x : \abs{f(x)} = 0\}} \Psi\bigl(\abs{f(x)}^2 \bigr) \abs{\nabla \eta(x)} \, \abs{f(x)} \, \bigg{\lvert} \cof Df(x) \frac{f(x)}{\abs{f(x)}} \bigg{\rvert} \, dx \\
&\le \int_{U} \Psi\bigl(\abs{f(x)}^2 \bigr) \abs{\nabla \eta(x)} \, \abs{f(x)} \, \norm{\cof Df(x)}\, dx \, ,
\end{align*}
which ends the proof.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}
To prove Theorem~\ref{thm:main} we first assume without loss of generality that
$$f(0) = 0 \quad \text{ and } \quad i(0,f) = m.$$
By applying Lemma~\ref{lemma:NormalDomain} we find a radius $r_0>0$ such that $U(0,f,t)$ defines a normal neighborhood of the origin whenever $0 < t \le r_0$. Next, we fix a radius
$$0 < R < \min \{ 1, r_0\} \quad \text{so small that} \quad f\bigl(B(0,R) \bigr) \subset B(0,r_0) ,$$
and assume first that
$$f \in C^{\infty}(U, \R^n).$$
Suppose now that $t \in (0,R)$ and let $\varepsilon >0$. By applying Lemma~\ref{lemma:MainEstimate} with the standard cut-off function
\begin{align*}
\eta_{\varepsilon}(s) = \left\{ \begin{array}{ll}
1, & \textrm{if $0 \le s \le t-\varepsilon$}\\
\frac{t-s}{\varepsilon} \, ,& \textrm{if $t-\varepsilon < s < t$}\\
0, & \textrm{if $s \ge t$,}
\end{array} \right.
\end{align*}
we get for any function $\Psi \in C^1\bigl([0,\infty),[0,\infty) \bigr)$ the following estimate
\begin{align}\label{eq:IntegralDifferation}
\biggl{\lvert}\int_{B_t} \eta_{\varepsilon} \bigl[ n \Psi(\abs{f}^2) +2 \abs{f}^2 \Psi'(\abs{f}^2)\bigr] J_f \bigg{\rvert} &\le \int_{B_t} \abs{\nabla \eta_{\varepsilon}} \abs{f} \Psi(\abs{f}^2) \norm{\cof Df} \nonumber \\
&= \frac{1}{\varepsilon} \int_{B_t \setminus B_{t-\varepsilon}} \abs{f} \Psi(\abs{f}^2) \norm{\cof Df} \, ,
\end{align}
where and in what follows we denote
$$B_t \colonequals B(0,t) \quad \text{for a given radius } t > 0 \, .$$
By letting $\varepsilon \to 0$ we conclude from \eqref{eq:IntegralDifferation} that
\begin{align}\label{eq:BoundaryEstimate}
\biggl{\lvert}\int_{B_t} \bigl[ n \Psi(\abs{f}^2) +2 \abs{f}^2 \Psi'(\abs{f}^2)\bigr] J_f \bigg{\rvert} \le \int_{\partial B_t} \abs{f} \Psi(\abs{f}^2) \norm{\cof Df} \, d \mathcal{H}^{n-1},
\end{align}
see \cite[3.4.4]{EvansGariepyBook}. Next we observe that for the function
\begin{align*}
\Psi(s) = \frac{s^{-\frac{n}{2}}}{\log^{n-1}(1/s)}
\end{align*}
we have
\begin{align*}
\Psi'(s) = -\frac{n}{2\log^{n-1}(1/s) s^{\frac{n}{2} +1}} + \frac{n-1}{\log^n(1/s) s^{\frac{n}{2} +1}} \, .
\end{align*}
By approximating $\Psi$ with $C^1$-functions we conclude from \eqref{eq:BoundaryEstimate} the following estimate
\begin{align}\label{eq:BestEstimate}
2(n-1) \int_{B_t} \frac{J_f}{\abs{f}^n \log^n(1/\abs{f}^2)} &\le \int_{\partial B_t} \frac{\norm{\cof Df}}{\abs{f}^{n-1} \log^{n-1}(1/\abs{f}^2)} \, d \mathcal{H}^{n-1} \, .
\end{align}
Now, the standard approximation argument shows that for almost every radius $t \in (0,R)$ the estimate \eqref{eq:BestEstimate} is valid also for continuous Sobolev mappings
$$f \in W_{\loc}^{1,n}(U, \R^n) \cap C(U,\R^n) \, .$$
Therefore, under the assumptions of the theorem we obtain by applying the estimate \eqref{eq:BestEstimate} together with the inner dilatation inequality
$$\norm{\cof Df(x)}^n \le K_I(f) J_f(x)^{n-1} \colonequals K_I J_f(x)^{n-1} \quad \text{a.e.} \, ,$$
and Hölder's inequality that for almost every $t \in (0,R)$ one has
\begin{align}\label{eq:BasicEstimation}
2(n-1) \int_{B_t} \frac{J_f}{\abs{f}^n \log^n(1/\abs{f}^2)} &\le \int_{\partial B_t} \frac{\norm{\cof Df}}{\abs{f}^{n-1} \log^{n-1}(1/\abs{f}^2)} \nonumber \\
&\le K_I^{\frac{1}{n}} \int_{\partial B_t} \frac{J_f^{\frac{n-1}{n}}}{\abs{f}^{n-1}\log^{n-1}(1/\abs{f}^2)} \nonumber \\
&\le K_I^{\frac{1}{n}} \abs{\partial B_t}^{\frac{1}{n}} \biggl( \int_{\partial B_t} \frac{J_f}{\abs{f}^n \log^{n}(1/\abs{f}^2)} \biggr)^{\frac{n-1}{n}} \\
&= K_I^{\frac{1}{n}} \omega_{n-1}^{\frac{1}{n}} t^{\frac{n-1}{n}} \biggl( \int_{\partial B_t} \frac{J_f}{\abs{f}^n \log^{n}(1/\abs{f}^2)} \biggr)^{\frac{n-1}{n}} \, , \nonumber
\end{align}
where $\omega_{n-1}$ denotes the surface measure of the unit sphere and $\abs{\partial B_t}$ denotes the surface measure of the sphere $\partial B_t$. By considering the increasing, non-negative function
\begin{align*}
\varphi(t) = \int_{B_t} \frac{J_f}{\abs{f}^n \log^n(1/\abs{f}^2)}
\end{align*}
we may rewrite the estimate \eqref{eq:BasicEstimation} as follows
\begin{align*}
\varphi(t) \le C_n t^{\frac{n-1}{n}} \bigl[ \varphi'(t) \bigr]^{\frac{n-1}{n}} \, , \quad \text{where} \quad C_n \colonequals \frac{K_I^{\frac{1}{n}}\omega_{n-1}^{\frac{1}{n}}}{2(n-1)} \, .
\end{align*}
Therefore, we have
\begin{align*}
\frac{1}{C_n^{\frac{n}{n-1}} t} \le \frac{\varphi'(t)}{\varphi(t)^{\frac{n}{n-1}}} = -(n-1) \frac{d}{dt} \, \biggl[ \varphi(t)^{-\frac{1}{n-1}} \biggr] \, ,
\end{align*}
and by integrating both sides over an interval $(r,R)$ we get
\begin{align*}
\frac{1}{C_n^{\frac{n}{n-1}}} \log(R/r) &\le -(n-1) \biggl[ \varphi(R)^{-\frac{1}{n-1}} - \varphi(r)^{-\frac{1}{n-1}} \biggr] \le (n-1) \varphi(r)^{-\frac{1}{n-1}} \, .
\end{align*}
This gives us
\begin{align}\label{eq:UpperMainEstimate}
\varphi(r) \le \frac{C_n^n (n-1)^{n-1}}{\log^{n-1}(R/r)} \, .
\end{align}
Next, we recall that by Lemma~\ref{lemma:LowerBound} we have
\begin{align}\label{eq:ImportantInclusion}
B_{r/L} \subset f(B_r) \quad \text{for all sufficiently small radii $r>0$}.
\end{align}
In addition, for a given set $A \subset \R^n$ we recall the standard notation
$$N(y,f,A) \colonequals \card f^{-1}(y) \cap A \, .$$
Then by applying the change of variables formula \cite[Theorem~A.35]{HenclKoskelaBook} and the inclusion \eqref{eq:ImportantInclusion} we obtain the following lower estimate
\begin{align}\label{eq:LowerMainEstimate1}
\varphi(r) = \int_{B_r} \frac{J_f}{\abs{f}^n \log^n(1/\abs{f}^2)} &= \int_{f(B_r)} \frac{N(y,f,B_r)}{\abs{y}^n \log^{n}(1/\abs{y}^2)} \, dy \nonumber \\
&\ge \int_{B_{r/L}} \frac{N(y,f,U_{r/L})}{\abs{y}^n \log^{n}(1/\abs{y}^2)} \, dy \\
&= i(0,f) \int_{B_{r/L}} \frac{dy}{\abs{y}^n \log^{n}(1/\abs{y}^2)} \nonumber
\end{align}
for all sufficiently small $r>0$, where the last equality follows from the fact that
$$U_{r/L} \colonequals U(0,f,r/L)$$
is a normal neighborhood of the origin and from \cite[Proposition~I.4.10]{Rickman-book}. By a direct calculation we get
\begin{align}\label{eq:LowerMainEstimate2}
\int_{B_{r/L}} \frac{dz}{\abs{y}^n\log^{n}(1/\abs{y}^2)} &= \int_0^{r/L} \int_{\partial B_t} \frac{1}{t^n \log^n(1/t^2)} \nonumber \\
&= \frac{\omega_{n-1}}{2^n} \int_0^{r/L} \frac{dt}{t\log^n(1/t)} \\
&= -\frac{\omega_{n-1}}{2^n} \int_{\infty}^{\log(L/r)} \frac{ds}{s^n} \nonumber \\
&= \frac{\omega_{n-1}}{(n-1)2^n \log^{n-1}\bigl(\frac{L}{r} \bigr)} \nonumber
\end{align}
for all sufficiently small radii $r>0$. Thus, by combining the estimates \eqref{eq:UpperMainEstimate} and \eqref{eq:LowerMainEstimate1}--\eqref{eq:LowerMainEstimate2} we get
\begin{align*}
\frac{i(0,f)\omega_{n-1}}{(n-1)2^n \log^{n-1}\bigl(\frac{L}{r} \bigr)} \le \varphi(r) &\le \frac{C_n^n(n-1)^{n-1}}{\log^{n-1}(R/r)} = \frac{K_I \omega_{n-1}(n-1)^{n-1}}{2^n (n-1)^n\log^{n-1}(R/r)} \, .
\end{align*}
By simplifying both sides we finally obtain
\begin{align*}
i(0,f) \le K_I \biggl( \frac{\log(L/r)}{\log(R/r)} \biggr)^{n-1} \quad \text{for all small } r>0 \, .
\end{align*}
Thus, by letting $r \to 0$ we observe that $i(0,f) \le K_I$ which concludes the proof.
\section{Final remarks}
We close the article with further observations on rigidity and Martio's conjectures. We start by pointing out that in the literature several constructions on quasiregular mappings \cite{BonkHeinonen-Smooth, GehringVaisala1973,KaufmanTysonWu2005}, on mappings of bounded length distortion \cite{MartioVaisala}, and on mappings of finite distortion \cite{GuoHenclTengvall2020} have the conjugated form
\begin{align}\label{eq:ConjugatedFrom}
h \circ w_m \circ g \, ,
\end{align}
where $w_m$ stands for the standard $m$-to-1 winding map, and both $h$ and $g$ are some suitable homeomorphisms. In addition to this, by the results of Church and Hemmingsen \cite[Theorem~4.1]{ChurchHemmingsen1960}, Martio, Rickman, and Väisälä \cite[Lemma~3.20]{MRV-71}, and Luisto and Prywes \cite[Theorem~1.1]{LuistoPrywes2021} quasiregular maps with reasonably regular branch sets or branch set images are indeed topologically equivalent to the standard winding map. Therefore, studying this type of maps is not useful only for the sake of the rigidity and Martio's conjectures, but also for its own right. On the other hand, from the point of view of these conjectures the following question stands out:
\begin{question}\label{question1}
Does the standard $m$-to-1 winding map minimize the inner dilatation for the non-planar quasiregular maps of the conjugated form \eqref{eq:ConjugatedFrom} in the following sense
\begin{align*}
K_I(w_m) = \min \biggl{\{} K_I(h \circ w_m \circ g) : \text{$g$ and $h$ quasiconformal} \biggr{\}} \, ?
\end{align*}
\end{question}
By answering the question above one would solve Martio's conjecture for the conjugated quasiregular mappings. As one may notice, Theorem~\ref{thm:main} already anwers the question in the class of BLD-maps. Therefore, for BLD-maps a natural next step is to study the uniqueness of the winding map by investigating the accuracy of the rigidity conjecture.
We also point out that by the constructions in \cite{BonkHeinonen-Smooth,KaufmanTysonWu2005} there exists even reasonably smooth quasiregular maps with branching. Furthermore, these examples can be even written as \eqref{eq:ConjugatedFrom} for some quasiconformal maps $h$ and $g$. Therefore, by current knowledge studying Martio's conjecture in the class of continuously differentiable quasiregular maps makes perfect sense as well as investigating the following question:
\begin{question}
Let us consider a continuously differentiable non-planar quasiregular map
$$f : \R^n \to \R^n, \quad f = h \circ w_m \circ g \, ,$$
where $g$ and $h$ are quasiconformal maps. Does it follow that $K_I(f) \ge m$?
\end{question}
\newcommand{\etalchar}[1]{$^{#1}$}
\def\cprime{$'$}
|
1,314,259,996,713 | arxiv |
\section{The Environment as a Witness} \label{secDefiningRedundancy}
Previous studies of decoherence have focused on the system's reduced density matrix ($\ensuremath{\rho_{\Sys}}$), and on master equations that describe its evolution. To study information flow into the environment, we require a new paradigm.
We begin with a simple observation: \textbf{information about a system ($\ensuremath{\mathcal{S}}$) is obtained by measuring its environment ($\ensuremath{\mathcal{E}}$)} (see \cite{ZurekRMP03, ZurekPTP93}). Although the standard
theories of quantum measurement (see e.g. Von Neumann \cite{VonNeumannBook55}, etc.) presume a direct measurement on the system, real experiments rely on \emph{indirect} measurements. As you read this paper\xspace, you measure the albedo of the page -- but actually, your eyes are capturing photons from the electromagnetic environment. Information about the page is inferred from assumed correlations between text and photons. A similar argument holds for every physics experiment; the scientist gets information about $\ensuremath{\mathcal{S}}$ by capturing and measuring a \emph{fragment} of $\ensuremath{\mathcal{E}}$.
This motivates us to focus on correlations between $\ensuremath{\mathcal{S}}$ and individual fragments of $\ensuremath{\mathcal{E}}$. In particular, we will seek to determine whether a particular state -- or a particular ensemble of states -- allows an observer who captures a small fragment of $\ensuremath{\mathcal{E}}$ to deduce the system's state. If so, then the system's state is \emph{objectively} recorded.
\subsection{Objectivity}
A property -- e.g., the state of a system -- is objective when many independent observers agree about it. The observers' independence is crucial. When many secondary observers are informed by a single primary observer, then only the primary observer's opinion is objective, \emph{not} necessarily the property which he observed.
\emph{Independent} observers, examining a single quantum system, cannot have agreed on a particular measurement basis beforehand. They will generally measure different observables -- and therefore will not agree afterward. An isolated quantum system's state cannot be objective, because measurements of noncommuting observables invalidate each other.
Classical theory, on the other hand, permits observers to measure a system without disturbing it. Properties of classical systems (e.g., classical states) are thus objective. Each observer can record the state in question without altering it, and afterward all the observers will agree on what they discovered. Of course, observers may obtain \emph{different} information -- e.g., one observer may make a more effective measurement than another -- but not \emph{contradictory} information.
Objectivity provides an excellent criterion for exploring the emergence of classicality through decoherence. A quantum system becomes more classical as its measurable properties become more objective. The use of ``measurable'' is significant. Nothing can make \emph{every} property of a quantum system objective, because some observables are incompatible with others. Two observers can never simultaneously obtain reliable information about incompatible observables (such as position and momentum) of the same system. Decoherence partially solves this problem by destroying all the observables incompatible with a system's \emph{pointer observable}. We are thus motivated to explore (a) how the pointer observable becomes objective, and (b) how decoherence and the emergence of objectivity are related.
\subsection{Technical details and assumptions}
This ``environment as a witness'' paradigm \cite{ZurekRMP03,OllivierPRL04,Ollivier04,ZurekADP00} is ideally suited to exploring objectivity. In order to make \emph{independent} measurements of $\ensuremath{\mathcal{S}}$, multiple observers must partition the environment into fragments. In this paper, we assume that measurements must be made on distinct Hilbert spaces in order to be independent, so we divide the environment into fragments as
\begin{equation}
\ensuremath{\mathcal{E}} = \ensuremath{\mathcal{E}}_A \otimes \ensuremath{\mathcal{E}}_B \otimes \ensuremath{\mathcal{E}}_C \otimes \ldots.
\end{equation}
Several factors limit an observer's ability to obtain information about $\ensuremath{\mathcal{S}}$ by measuring a fragment of the environment ($\ensuremath{\mathcal{E}}_A$). We can make more or less optimistic assumptions about some of these factors, but \textbf{the degree of correlation between $\ensuremath{\mathcal{S}}$ and $\ensuremath{\mathcal{E}}_A$ is clearly a limiting factor}. An observer whose particular fragment is not correlated with $\ensuremath{\mathcal{S}}$ has no way to obtain information about $\ensuremath{\mathcal{S}}$. That fragment of $\ensuremath{\mathcal{E}}$ is irrelevant and, for the purpose of gaining information about $\ensuremath{\mathcal{S}}$, might as well not exist. The absolute prerequisite for demonstrating a property's objectivity is that information about it be recorded in many fragments -- that is, \emph{redundantly}.
We quantify redundancy by counting the number of fragments which can provide sufficient information. The redundancy of information about some property is a natural measure of that property's objectivity \cite{ZurekRMP03}. Classical properties are objective because information about them is recorded with [effectively] infinite redundancy. For instance, if we flip a coin, then its final orientation is recorded by trillions of scattered photons. Thousands of cameras, each capturing a tiny fraction of them, could each provide a record. Redundancy is not dependent on \emph{actual} observers. Instead, it is a statement about what observers \emph{could} do, if they existed.
A pertinent question is ``Why not allow an observer to measure the system itself?'' First, only one observer could be allowed to do so without sacrificing independence. Thus, at most, this would increase redundancy by 1. Furthermore, an observer with access to the central system could measure it in some weird basis, thus destroying its state. Since it's not then clear what the information obtained by the \emph{other} observers would refer to, we regard the system itself as off limits to observers.
\subsection{The overall program}
The work presented here is a natural extension of the decoherence program. However, employing the environment as a communication channel -- not just ``sink'' for information lost to decoherence -- is also in a sense ``beyond decoherence.'' It is the next stage in exploring how classicality emerges from the quantum substrate.
In order to fully understand the role that redundancy and objectivity play in (1) the emergence of classicality, and (2) the destruction of quantum coherence, we'd like to answer the following questions:
\begin{enumerate}
\item Given a state $\ensuremath{\rho_{\Sys\Env}}$ for the system and its environment (the ``universe''), how do we quantify the redundancy of information (about $\ensuremath{\mathcal{S}}$) in $\ensuremath{\mathcal{E}}$?
\item For a particular ``universe,'' what states are \emph{typical} (that is, likely to exist)? Do they display redundancy? If so, how much?
\item What sorts of (a) initial states, and (b) dynamics lead dynamically to redundancy?
\item Do realistic models of decoherence produce the massive redundancy we expect in the classical regime?
\item For complicated systems, with many independent properties, how do we distinguish \emph{what} property a bit of information is about?
\item When information about an observable is redundantly recorded, is information about incompatible observables inaccessible?
\end{enumerate}
The building blocks of this work -- e.g., the reasoning presented in this section -- have been laid in recent years by \cite{ZurekPTRSA98, ZurekADP00, ZurekRMP03, OllivierPRL04, Zurek03}. The first attempt to address items (1) and (3) appeared in \cite{ZurekADP00}, and was refined in \cite{OllivierPRL04}, which also analyzed a particular simple model of decoherence numerically. In the current paper\xspace, we answer (1) and (2) in detail, and consider (3) briefly.
\subsection{Computing redundancy} \label{secQuantitativeR}
To compute the redundancy ($R$) of some information ($\ensuremath{\mathcal{I}}$), we divide the environment into fragments ($\ensuremath{\mathcal{E}} = \ensuremath{\mathcal{E}}_A \otimes \ensuremath{\mathcal{E}}_B \otimes \ldots$), and demand that each fragment supply $\ensuremath{\mathcal{I}}$ independently. The redundancy of $\ensuremath{\mathcal{I}}$ is the number of such fragments into which the environment can be divided. A generalized GHZ state is a good example:
\begin{equation}
\ket\psi_{\ensuremath{\mathcal{S}}\ensuremath{\mathcal{E}}} = \alpha\ket{0}_{\ensuremath{\mathcal{S}}}\ket{00000\ldots 0}_{\ensuremath{\mathcal{E}}} + \beta\ket{1}_{\ensuremath{\mathcal{S}}}\ket{11111\ldots 1}_{\ensuremath{\mathcal{E}}} \label{eqGHZ}
\end{equation}
We can determine the system's state by measuring any sub-environment. Each qubit in $\ensuremath{\mathcal{E}}$ provides all the available information about $\ensuremath{\mathcal{S}}$ (see, however, note \footnote{We must make the \emph{right} measurement -- in this case, one which distinguishes $\ket{0}$ from $\ket{1}$ -- in order to get the information. In this work, the amount of information that one subenvironment has is always \emph{maximized} over all possible measurements.}). To extend this analysis to arbitrary states, we need (a) a metric for information, (b) a protocol for dividing the environment into fragments, and (c) an idea of how much of $\ensuremath{\mathcal{I}}$ is ``available''.
\subsubsection{A metric for information}
\OneSmallPanelFigure[Different locality (tensor product) structures for the ``universe'']{figSubdivision}{EnvSubdivision}
{(Color) \textbf{Three ways to divide up the universe}. The decoherence paradigm divides the universe into a system ($\ensuremath{\mathcal{S}}$) and an environment ($\ensuremath{\mathcal{E}}$) as in \textbf{(a)}. In the environment-as-a-witness paradigm, we further subdivide $\ensuremath{\mathcal{E}}$ into \emph{subenvironments}, as in \textbf{(b)}. No subenvironment can be further subdivided, and it is easier to measure one $\ensuremath{\mathcal{E}}_i$ than to make a joint measurement on several. \emph{Fragments} are constructed, so as to provide enough information to infer the state of $\ensuremath{\mathcal{S}}$, by combining subenvironments as in \textbf{(c)}. Measurements on distinct fragments always commute.}
We use \emph{quantum mutual information} (QMI) as an information metric. QMI is a generalization of the classical mutual information \cite{CoverBook91}. \emph{Quantum} mutual information is defined in terms of the Von Neumann entropy, $H = -\mathrm{Tr}(\rho\log\rho)$, as:
\begin{equation}
\ensuremath{\mathcal{I}}_{A:B} = H_A + H_B - H_{AB}
\end{equation}
This is simple to calculate, provides a reliable measure of correlation between systems, and has been used previously for this purpose \cite{ZurekPRD82,Zurek83,ZurekRMP03}. Unlike classical mutual information, the QMI between system $A$ and system $B$ is not bounded by the entropy of either system. In the presence of entanglement, the QMI can be as large as $H_A + H_B$, which reflects the existence of quantum correlations beyond the classical ones \cite{OllivierPRL02}..
\subsubsection{Dividing $\ensuremath{\mathcal{E}}$ into fragments}
A pre-existing concept of locality, usually expressed as a fixed tensor product structure or as a set of allowable structures, is \emph{fundamental} to redundancy analysis. Allowing an \emph{arbitrary} division of $\ensuremath{\mathcal{E}}$ into fragments would make every state where $\ensuremath{\mathcal{S}}$ is entangled with $\ensuremath{\mathcal{E}}$ (see note \footnote{In this work, we assume that the universe is in a pure state. Any correlation between $\ensuremath{\mathcal{S}}$ and $\ensuremath{\mathcal{E}}$ is due to entanglement. Similar conclusions seem to apply when the environment is initially mixed, but we have not investigated these cases exhaustively.}) equivalent (via re-division of $\ensuremath{\mathcal{E}}$) to a GHZ-like state (Eq. \ref{eqGHZ}). Decoherence would be equivalent to redundancy.
The need for a fixed tensor product structure is familiar; both decoherence and entanglement are meaningless without a fixed division between the system and its environment (\cite{ZurekRMP03,ZurekPTP93}, see e.g. \cite{ZanardiPRL04} for a discussion of a possible
tensor product structures' origins in measurable observables; an explanation that does not refer to
measurements would be needed in the present context). In the environment-as-a-witness paradigm, we divide $\ensuremath{\mathcal{E}}$ into indivisible \emph{subenvironments}:
\begin{equation}
\ensuremath{\mathcal{E}} = \ensuremath{\mathcal{E}}_1 \otimes \ensuremath{\mathcal{E}}_2 \otimes \ensuremath{\mathcal{E}}_3 \otimes \ldots \ensuremath{\mathcal{E}}_{\ensuremath{N_{\mathrm{env}}}}.
\end{equation}
These subenvironments can be rearranged into larger \emph{fragments}. A generic fragment consisting of $m$ subenvironments will be written as $\Envset{m}$. The fragment containing the particular subenvironments $\{\ensuremath{\mathcal{E}}_{i_1},\ensuremath{\mathcal{E}}_{i_2},\ldots\ensuremath{\mathcal{E}}_{i_m}\}$ is denoted $\Envset{i_1,i_2,\ldots i_m}$.
We assume that each observer captures a \emph{random} fragment of $\ensuremath{\mathcal{E}}$. This ensures their strict independence. In essence, we do not allow the observers to caucus over the partition of $\ensuremath{\mathcal{E}}$, dividing it up in an advantageous way.
\subsubsection{How much information is practically available}
The maximum information that could be provided about $\ensuremath{\mathcal{S}}$ is its entropy, $\ensuremath{\Hh_{\Sys}}$. In general, no fragment can provide \emph{all} this information \footnote{The GHZ state in Eq. \ref{eqGHZ} is the exception that proves the rule. Such states are measure-zero in Hilbert space. Perfect C-NOT interactions are required to make them.}. Following the reasoning in \cite{OllivierPRL04}, we demand that each fragment provide some large fraction, $1-\delta$ (where $\delta \ll 1$), of the available information about $\ensuremath{\mathcal{S}}$. The precise magnitude of the \emph{information deficit} $\delta$ should not be important. We denote the redundancy of ``all but $\delta$ of the available information'' by $R_\delta$. That is, when we allow a deficit of $\delta = 0.1$, we are computing $R_{0.1}$ or $R_{10\%}$.
To compute $R_{\delta}$, we start by defining $N_{\delta}$ as \textbf{the number of disjoint fragments $\ensuremath{\mathcal{E}}_i$ such that $\Isen{i} \geq (1-\delta)\ensuremath{\I_{\Sys:\Env}}$}. We might just define $R_{\delta} = N_{\delta}$, except for two caveats.
\begin{enumerate}
\item A large deficit ($\delta$) in the definition of ``sufficient'' information could lead to spurious redundancy. Suppose there exist $N=5$ fragments that provide full information. If $\delta = 0.5$, then we might split each fragment in half to obtain $N_{\delta}=10$ fragments that each provide ``sufficient'' information. To compensate for this, we replace $N_{\delta}$ with $(1-\delta)N_{\delta}$.
\item Because of quantum correlations, $\Isen{i}$ can be as high as $2\ensuremath{\Hh_{\Sys}}$. We allow for this by \emph{assuming} that the information provided by one fragment represents strictly quantum correlations, and throwing this fragment away. This means replacing $(1-\delta)N_{\delta}$ with $(1-\delta)N_{\delta}-1$.
\end{enumerate}
By assuming the worst case, we have obtained a \emph{lower bound} for the true redundancy:
\begin{equation}
R_\delta \geq (1-\delta) N_\delta - 1 \label{eqRedundancy}.
\end{equation}
For small $\delta$, this is fairly tight, as $N_\delta$ is clearly an upper bound. Since our current toolset, subject to the caveats mentioned above, does not permit a more precise determination of $R_\delta$, we report the lower bound throughout. Thus, when we report ``$R_{10\%} = 9$,'' we really mean ``$R_{10\%}$ is at least 9, and not much more.''
\subsection{Identifying qualitative redundancy} \label{secQualitativeR}
\OnePanelFigureNow[Three kinds of partial information plots (PIPs)]{figThreePIPs}{ThreePIPs2}
{(Color) \textbf{Three profiles for partial information plots ($\ensuremath{\mathcal{I}}$ vs. $m$).} \textbf{(a)}: the behavior of \emph{independent} environments. \textbf{(b)}: information is stored \emph{redundantly}. \textbf{(c)}: information is \emph{encoded} in multiple environments.}
The actual \emph{amount} of redundancy is often less important than the qualitative observation that information is stored very redundantly (e.g., $R\gg1$). Whether $R=100$ or $R=1000$, the information in question is certainly objective -- but if $R \sim 1$, then its objectivity is in doubt. We also wish to consider more general questions: e.g., how much does $R_{\delta}$ depend on $\delta$? or \emph{why} does a state display virtually no redundancy?
For these purposes, we plot the amount of information about $\ensuremath{\mathcal{S}}$ supplied by a fragment of size $m$ ($\Isen{\{m\}}$), against $m$. Since there are very many fragments of a given size, we average $\Isen{\{m\}}$ over a representative sample of fragments to obtain $\ensuremath{\overline{\I}}(m)$. The plot of $\ensuremath{\overline{\I}}(m)$, which shows the partial information yielded by a partial environment, is a \emph{partial information plot} (PIP). When the universe is in a pure state (see \cite{RBK04}, and Appendix \ref{appQMIDetails}), the PIP must be anti-symmetric around its center (see Fig. \ref{figThreePIPs}). Together with the observation that $\ensuremath{\overline{\I}}(m)$ must be strictly non-decreasing (capturing \emph{more} of the environment cannot \emph{decrease} the amount of information obtained), this permits the three basic profiles shown in Figure \ref{figThreePIPs}.
Redundancy (see Fig. \ref{figThreePIPs}b) is characterized by a rapid rise of $\ensuremath{\overline{\I}}$ at relatively small $m$, followed by a long ``classical plateau''. In this region, all the easily available information has been obtained. Additional environments confirm what is already known, but provide nothing new. Only by capturing \emph{all} the environments can an observer manipulate quantum correlations. The power to do so is indicated by the sharp rise in $\ensuremath{\overline{\I}}$ at $m \sim \ensuremath{N_{\mathrm{env}}}$.
\section{Information storage in random states} \label{secUniformEnsemble}
Redundant information storage is ubiquitous in the classical world. We might na\"{\i}vely expect that randomly chosen states of a model universe -- e.g., a $\ensuremath{D_{\Sys}}$-dimensional system in contact with a bath of $\ensuremath{N_{\mathrm{env}}}$ $\ensuremath{D_{\Env}}$-dimensional systems -- would display massive redundancy. To test this hypothesis, we compute partial information plots for random states, and average them over the uniform ensemble. This was first done in \cite{RBK04}, for qubits. In this work, we extend the analysis to systems and environments with arbitrary sizes.
\TwoPanelFigure[PIPs for uniform ensembles]{figUniformPIPs}{UniformQubitPIPs}{UniformNitPIPs}
{(Color) \textbf{Partial information plots (PIPs) for the uniform ensemble.} We plot the average information ($\ensuremath{\overline{\I}}$) obtainable from a fragment ($\Envset{m}$), against the fragment's size ($m$). $\ensuremath{\mathcal{I}}(m)$ is averaged over \textbf{all states in the uniform ensemble}. \textbf{(a)}: A qubit system coupled to environments consisting of $\ensuremath{N_{\mathrm{env}}}=2\ldots16$ qubits. \textbf{(b)}: Systems with sizes $\ensuremath{D_{\Sys}}=2\ldots16$ coupled to a 16-qubit environment. \textbf{Discussion:} No significant information is obtained until almost half the subenvironments have been captured. Once $m>\frac{\ensuremath{N_{\mathrm{env}}}}{2}$, virtually all possible information (both quantum and classical) is available. Because more than half the environment is required to obtain useful information, there is no redundant information storage in typical uniformly-distributed states. Instead, the information is \emph{encoded} throughout the environment.}
\subsection{The uniform ensemble}
For any [finite] $D$-dimensional Hilbert space, there exists a unitarily invariant uniform distribution over states, usually referred to as \emph{Haar measure}. We examine the behavior of typical random states by averaging PIPs over this uniform ensemble. This average can be obtained analytically, using a formula for the average entropy of a subspace that was conjectured by Page \cite{PagePRL93}, then proved by Sen \cite{SenPRL96} and others \cite{FoongPRL94,SanchezRuizPRE95}.
Page's formula \cite{PagePRL93,SenPRL96,FoongPRL94,SanchezRuizPRE95} for the mean entropy $\overline{H}(m,n)$ of an $m$-dimensional subsystem of an $mn$-dimensional system (where $m\leq n$) is
\begin{eqnarray}
\overline{H}(m,n) &=& \sum_{k=n+1}^{mn}{\frac{1}{k}} - \frac{m-1}{2n} \\
&=& \Psi(mn) - \Psi(n+1) - \frac{m-1}{2n},
\end{eqnarray}
where the latter expression is given in terms of the \emph{digamma} $\Psi$ function. For a $\ensuremath{D_{\Sys}}$-dimensional system in contact with $\ensuremath{N_{\mathrm{env}}}$ environments of size $\ensuremath{D_{\Env}}$, the average mutual information between the system and $m$ sub-environments is
\begin{eqnarray}
\overline{\Isen{\{m\}}} =&& \overline{H}(\ensuremath{D_{\Sys}},\ensuremath{D_{\Env}}^{\ensuremath{N_{\mathrm{env}}}}) \nonumber \\
&+& \overline{H}(\ensuremath{D_{\Env}}^m,\ensuremath{D_{\Sys}} \ensuremath{D_{\Env}}^{\ensuremath{N_{\mathrm{env}}}-m}) \nonumber \\
&-& \overline{H}(\ensuremath{D_{\Sys}}\ensuremath{D_{\Env}}^m, \ensuremath{D_{\Env}}^{\ensuremath{N_{\mathrm{env}}}-m}) \label{eqHaarMI}.
\end{eqnarray}
\subsection{Partial information plots (PIPs)} \label{secUniformPIPs}
\TwoPanelFigure[Equivalent environments in the uniform ensemble]{figVariousEnvSizePIPs}{VariousEnvSizePIPs}{VariousEnvSizeSPIPs}
{(Color) \textbf{Equivalent enviroments} When the state of the universe is chosen randomly, the environment's Hilbert space dimension determines its information-recording properties. \textbf{(a)}: PIPs for a $16$-d system coupled to several equivalent environments with $D_{\mathrm{total}} = 2^{24}$. The subenvironments are \{2, 4, 8, 16\}-dimensional, and $\ensuremath{N_{\mathrm{env}}}$ is scaled appropriately. The plots are essentially identical -- only the scaling of the $m$-axis changes. \textbf{(b)}: The same data, but with the \emph{captured fraction} of the environment plotted on the independent axis.}
\TwoPanelFigure[Scaled partial information plots (SPIPs) for the uniform ensemble]{figUniformSPIPs}{UniformQubitSPIPs}{UniformNitSPIPs}{(Color) \textbf{Scaled versions (SPIPs) of the plots in Fig. \ref{figUniformPIPs}}. SPIPs are useful for comparing environments with different numbers of subenvironments, and for computing $R_\delta$, the redundancy for a given fraction $1-\delta$ of the total information. To estimate redundancy, simply draw a horizontal line at $\ensuremath{f_{\I}} = \frac{1-\delta}{2}$, and note the value of $\ensuremath{f_{cap}}$ where it intersects the PIP. This provides a good estimate of $1/R_\delta$. It is not a perfect estimate for several reasons; most importantly, the PIP and SPIP plot \emph{the average $\ensuremath{\mathcal{I}}$} obtained from a given-sized fragment of the environment. This is not the same as \emph{the average fragment size} ($\overline{m}$) required to obtain $\ensuremath{\mathcal{I}}$, since we average the same data over different variables. In these plots, of course, no redundancy is evident -- we are looking ahead to the next section.}
Our results (Figs. \ref{figUniformPIPs}-\ref{figUniformSPIPs}) demonstrate that \textbf{typical states from the uniform ensemble do not display redundancy}. Figure \ref{figUniformPIPs}a illustrates typical behavior. As an observer captures successively more subenvironments (increasing $m$), he gains virtually no information about $\ensuremath{\mathcal{S}}$. $\Isen{\{m\}}$ remains close to zero. When approximately $50\%$ of the subenvironments have been captured, the observer begins to gain information. $\ensuremath{\overline{\I}}$ rises rapidly, through $H_s$ and onward nearly to $2H_s$.
Information about $\ensuremath{\mathcal{S}}$ is \emph{encoded} in the environment (as in Fig. \ref{figThreePIPs}c), much as a classical bit can be encoded in the parity of an ancillary bitstring. In the classical example, however, \emph{every} bit of the ancilla must be captured to deduce the encoded bit.
This encoding, or ``anti-redundancy'', is related to quantum error correction \cite{KnillPRA97,GottesmanPRA98,NielsenBook00,ScottPRA04}. In an encoding state, any majority subset of the $\ensuremath{\mathcal{E}}_i$ has nearly-complete information. The recorded information is unaffected by the loss of any minority subset. States with this behavior can be used as a quantum code to protect against bit loss. Our results show that \emph{generic} states -- i.e., states selected randomly from the whole $\ensuremath{\mathcal{S}}\ensuremath{\mathcal{E}}$ Hilbert space -- form a nearly-optimal error-correction code for bit-loss errors. Shannon noted similar behavior for classical codewords \cite{ShannonBook49}.
Figures \ref{figUniformPIPs}b and \ref{figVariousEnvSizePIPs} extend this result to larger systems. The results are consistent; information is still encoded, and only the total amount of encoded information changes.
\subsection{Conclusions}
Our first main result is that \textbf{typical states selected randomly from the uniform ensemble display no redundant information storage}. Instead, they display encoding or anti-redundancy. This is not to say that \emph{all} states are ``antiredundant'', merely that redundant information storage is rare. As $m$ declines from $\frac{\ensuremath{N_{\mathrm{env}}}}{2}$, $\ensuremath{\overline{\I}}(m)$ declines exponentially. For large $\ensuremath{N_{\mathrm{env}}}$, states where information is \emph{not} encoded this way are vanishingly rare. If even a small fixed fraction $\epsilon$ of states displayed the opposite ``redundant'' behavior, then $\ensuremath{\overline{\I}}(m)$ would have to be $O(\epsilon)$ at small $m$. The fact that $\ensuremath{\overline{\I}}(m)$ is exponentially close to zero implies that the fraction of non-``encoding'' states must decline exponentially with $\ensuremath{N_{\mathrm{env}}}$.
The obvious conclusion is that the Universe does not evolve into random states. Our observations of ubiquitous redundancy in the real Universe are inconsistent with the random-state model. This is interesting, but not terribly surprising. There is no good reason to expect that the Universe's state \emph{would} be random -- we are not, for instance, in thermodynamic equilibrium. The interactions of systems with their environments must select states that \emph{are} characterized by greater redundancy. In the next section, we suggest and analyze such an ensemble.
\section{Decoherence and branching states} \label{secBranchingStates}
Decoherence -- the loss of information to the environment -- is a prerequisite for redundancy. The simplest models of decoherence \cite{ZurekPRD81} are essentially identical to those for quantum measurements. A set of pointer states for the system, $\{\ket{n}\}$, are singled out, and the environment ``measures'' which $\ket{n}$ the system is in, by evolving from some initial state ($\ket{\ensuremath{\mathcal{E}}_0}$) into a \emph{conditional} state, $\ket{\ensuremath{\mathcal{E}}_n}$. If $\ensuremath{\rho_{\Sys}}$ is written out in the pointer basis, its diagonal elements ($\rho_{nn}$) remain unchanged. Coherences between different pointer states (e.g., $\rho_{nm}$) are reduced by a \emph{decoherence factor}:
\begin{equation}
\gamma_{nm} \equiv \braket{\ensuremath{\mathcal{E}}_n}{\ensuremath{\mathcal{E}}_m}.
\end{equation}
We presume that (a) the subenvironments are initially unentangled, (b) each subenvironment ``measures'' the same basis of the system, and (c) the state of the universe is pure. In this simple model, the universe is initially in a product state:
\begin{equation}
\ket{\Psi_0} = \ket{\ensuremath{\mathcal{S}}_0} \otimes \ket{\ensuremath{\mathcal{E}}\idx{1}_0} \otimes \ket{\ensuremath{\mathcal{E}}\idx{2}_0} \otimes \ldots \ket{\ensuremath{\mathcal{E}}\idx{\ensuremath{N_{\mathrm{env}}}}_0}.
\end{equation}
The subenvironments do not interact with each other, and the system does not evolve on its own. Letting the system's initial state be $\ket{\ensuremath{\mathcal{S}}_0} = \sum_n{s_n\ket{n}}$, the universe evolves over time into:
\begin{equation}
\ket{\Psi_t}= \sum_n{s_n\ket{n}_\ensuremath{\mathcal{S}}\otimes \ket{\ensuremath{\mathcal{E}}^{(1)}_n}\otimes \ket{\ensuremath{\mathcal{E}}^{(2)}_n}\otimes \ldots\ket{\ensuremath{\mathcal{E}}^{(\ensuremath{N_{\mathrm{env}}})}_n}}, \label{eqBranchingState}
\end{equation}
where $\ket{\ensuremath{\mathcal{E}}^{(j)}_n}$ is the \emph{conditional} state into which the $j$th subenvironment evolves \emph{if} the system is in state $\ket{n}$. Different conditional states of a given subenvironment will \emph{not} generally be orthogonal to one another, except in highly simplified (e.g. C-NOT) models.
\subsection{The branching-state ensemble}
We refer to the states defined by Eq. \ref{eqBranchingState} as \emph{singly-branching states}, or simply as \emph{branching states}. In Everett's many-worlds interpretation \cite{EverettRMP57}, a branching state's wavefunction has $\ensuremath{D_{\Sys}}$ branches. Each branch is perfectly correlated with a particular pointer state of the system. The subenvironments are not entangled with each other, only correlated (classically) via the system. In contrast, a typical random state from the uniform ensemble has $D_{\mathrm{universe}}$ branches, with a new branching at every subsystem.
In dynamical models of decoherence, the universe at a given time will be described by a \emph{particular} branching state that depends on the environment's initial state, and on its dynamics. In this paper\xspace, we sidestep the difficulties of specifying these parameters, by considering the ensemble of \emph{all} branching states. We select the conditional $\ket{\ensuremath{\mathcal{E}}_n^{(j)}}$ at random from each subenvironment's uniform ensemble. Each pointer state of the system is correlated with a randomly chosen product state of all the environments.
The amount of available information is set by the system's initial state (i.e., the $s_n$ coefficients). The eigenvalues of $\ensuremath{\rho_{\Sys}}$ after complete decoherence, which determine its maximum entropy, are $\lambda_n = |s_n|^2$. Since we cannot examine \emph{all} possible states, we focus on maximally ``measurable'' generalized Hadamard states:
\begin{equation}
s_n = \frac{1}{\sqrt{\ensuremath{D_{\Sys}}}}\ \ \ \forall\ n.
\end{equation}
To verify that our results are generally valid, we also treat (briefly) another class of initial states.
By examining the branching-state ensemble, we are \emph{not} conjecturing that the Universe is found exclusively in branching states. Branching states form an interesting and physically well-motivated ensemble to explore. We shall see that, unlike the uniform ensemble, the branching-state ensemble displays redundancy consistent with observations of the physical Universe. Our Universe might well tend to evolve into similar states, but we are not ready to establish such a conjecture. Characterizing the states in which the physical Universe (or a fragment thereof) \emph{is} found is a substantially more ambitious project.
\subsection{Numerical analysis of branching states} \label{secBSNumerics}
We begin our exploration of branching states by examining typical PIPs, for various systems and environments. We average these PIPs over the branching-state ensemble, so there are only three adjustable parameters: $\ensuremath{D_{\Sys}}$, $\ensuremath{D_{\Env}}$, and $\ensuremath{N_{\mathrm{env}}}$. Our results confirm that information is stored redundantly. Next, we examine a quantitative measure of redundancy ($R_{\delta}$), and its dependence on $\ensuremath{D_{\Sys}}$, $\ensuremath{D_{\Env}}$, and $\ensuremath{N_{\mathrm{env}}}$. Finally, we derive some analytical approximations, compare them with numerical data, and discuss the implications of our results.
\subsubsection{Partial information plots} \label{secBSPIPs}
\FourSmallPanelFigureX[PIPs for the branching ensemble]{figBSPIPs}{PIPs_2x2}{PIPs_2x5}{PIPs_5x2}{PIPs_5x5}{(Color) \textbf{PIPs for ensembles of singly-branching states.} The system is initialized in a Hadamard state, and decohered by $\ensuremath{N_{\mathrm{env}}}$ subenvironments. We plot the average information ($\ensuremath{\overline{\I}}$) available from a collection of $m$ subenvironments. \textbf{(a)}: A qubit is decohered by qubits. \textbf{(b)}: A qubit is decohered by $5$-dimensional subenvironments. \textbf{(c)}: A $5$-d system is decohered by qubits. \textbf{(d)}: A $5$-d system is decohered by $5$-d subenvironments. \textbf{Discussion:} As $\ensuremath{N_{\mathrm{env}}}$ is increased from 4 to 12, a ``classical plateau'' appears. This indicates redundant information storage. In the regime $m\ll \ensuremath{N_{\mathrm{env}}}$, the PIP converges to an asymptotic form. When $\ensuremath{\mathcal{S}}$ is larger than $\ensuremath{\mathcal{E}}$ (see \textbf{(c)}), the environment is barely sufficient to decohere the system, and there is no redundancy (see also Fig. \ref{figBSRedundancy}).}
Information is redundant when small fragments yield nearly-complete information -- that is, when the PIP looks like Fig. \ref{figThreePIPs}b. PIPs for branching states (Fig. \ref{figBSPIPs}) show exactly this profile. $\ensuremath{\overline{\I}}(m)$ rises rapidly from $\ensuremath{\overline{\I}}(0)=0$, then approaches $\ensuremath{\Hh_{\Sys}}$ asymptotically to produce a ``classical plateau'' centered at $m = \frac{\ensuremath{N_{\mathrm{env}}}}{2}$.
As $\ensuremath{N_{\mathrm{env}}}$ grows, the interesting regimes at $m \sim 0$ and $m \sim \ensuremath{N_{\mathrm{env}}}$ do not change; the classical plateau simply extends to connect them. The initial bits of information that an observer gains about a system are extremely useful, but eventually a point of diminishing returns is reached, where further information is redundant. The degree of redundancy should therefore scale with $\ensuremath{N_{\mathrm{env}}}$.
\subsubsection{Non-Hadamard states for $\ensuremath{\mathcal{S}}$}
\FourSmallPanelFigureX[PIPs for inhomogeneous branching states]{figGeomPIPs}{GeomPIPs_2}{GeomPIPs_3}{GeomPIPs_4}{GeomPIPs_5}
{(Color) \textbf{PIPs for non-Hadamard states:} $\ensuremath{D_{\Env}} = 2,3,4,5$ in plots \textbf{(a)}, \textbf{(b)}, \textbf{(c)}, \textbf{(d)}, respectively. The system is $16$-dimensional, and initialized in a ``thermal'' state, where $s_n \propto \frac{1}{\sqrt{2^n}}$. The entropy of this density matrix is $\sim2$ bits (as opposed to 4 bits for a $\ensuremath{D_{\Sys}}=16$ Hadamard state). We compare the PIPs for ``thermal'' states with $\ensuremath{D_{\Sys}}=16$ to PIPs for Hadamard $\ensuremath{D_{\Sys}}=4$ states, which also develop $2$ bits of entropy, varying the subenvironments' size. These PIPs confirm that our observations apply to non-Hadamard states, and that $\ensuremath{\Hh_{\Sys}}$ characterizes how information about the system is stored.}
Non-Hadamard states provide a different spectrum of information for $\ensuremath{\mathcal{E}}$ to capture. We consider states defined by
\begin{equation}
s_n \propto \frac{1}{\sqrt{2^n}},
\end{equation}
The post-decoherence spectrum of $\ensuremath{\rho_{\Sys}}$ is non-degenerate -- in fact, it is exactly that of a thermal spin -- i.e., a particle with a Hamiltonian $\ensuremath{\mathbf{{H}}} = \ensuremath{\mathbf{J}}_z$, in equilibrium with a bath at finite temperature. We refer to these states as \emph{``thermal''} branching states (and retain quotation marks to emphasize that our justification of this nomenclature is unphysical).
Our general approach is to assume that the system's maximum entropy determines its informational properties. The entropy of a decohered ``thermal'' state does not increase logarithmically with $\ensuremath{D_{\Sys}}$,
but asymptotes to $\ensuremath{\Hh_{\Sys}}=2$ bits. This is exactly the entropy of a $\ensuremath{D_{\Sys}}=4$ Hadamard state, so in the limit $\ensuremath{D_{\Sys}}\rightarrow\infty$, ``thermal'' states should behave much the same as a $\ensuremath{D_{\Sys}}=4$ Hadamard state.
This conjecture is confirmed in Fig. \ref{figGeomPIPs}, which compares PIPs for ``thermal'' states with $\ensuremath{D_{\Sys}}=16$ to PIPs for Hadamard states with $\ensuremath{D_{\Sys}}=4$. The plots' similarity indicates that $\ensuremath{\Hh_{\Sys}}$ is the major factor in how information about $\ensuremath{\mathcal{S}}$ is recorded. Further numerical results use Hadamard states for specificity's sake.
\subsubsection{How PIPs scale with the composition of $\ensuremath{\mathcal{E}}$}
\TwoPanelFigureNow[Scaled partial information plots (SPIPs), and the two regimes of information gain (linear and exponential)]{figBS_SPIPs}{SPIPs_3x3} {SPIPs_16xD}
{(Color) \textbf{Scaled partial information plots (SPIPS)} compare information storage in different environments. \textbf{(a)}: A qutrit system coupled to $\ensuremath{N_{\mathrm{env}}}=4\ldots128$ qutrit environments. \textbf{(b)}: A qutrit system coupled to nine different environments with the same \emph{information capacity}. \textbf{Discussion:} As $\ensuremath{N_{\mathrm{env}}}$ increases, redundancy (indicated by sharp curvature) grows (plot \textbf{(a)}). If $\ensuremath{N_{\mathrm{env}}}$ and $\ensuremath{D_{\Env}}$ are scaled so that total Hilbert space dimension ($\ensuremath{D_{\Env}}^{\ensuremath{N_{\mathrm{env}}}}$) remains constant, then the SPIP remains unchanged (plot \textbf{(b)}). Plot \textbf{(b)} also illustrates the difference between the regime of \emph{linear} information gain (here, $f_{cap} < 0.04$) and the exponential convergence to the ``classical plateau'' thereafter.}
As the number of subenvironments in $\ensuremath{\mathcal{E}}$ grows, comparing PIPs for different environments becomes difficult. Re-parameterizing the axes, and plotting the \emph{fraction} of $\ensuremath{\mathcal{I}}$ available from a \emph{fraction} of $\ensuremath{\mathcal{E}}$, allows direct comparison of different universes. Scaled PIPs (SPIPs) for environments with $\ensuremath{N_{\mathrm{env}}} = 4\ldots128$ (Fig. \ref{figBS_SPIPs}a) show that the information about $\ensuremath{\mathcal{S}}$ becomes more redundant as $\ensuremath{N_{\mathrm{env}}}$ grows.
Different environments, whose total Hilbert space dimensions are the same, act equivalently (see also Sec. \ref{secUniformPIPs}). We have simulated a 16-dimensional system coupled to nine different, but equivalent, environments (Fig. \ref{figBS_SPIPs}b). Although the \emph{number} and \emph{size} of the subenvironments are varied, the redundancy of the available information depends only on $\ensuremath{\mathcal{E}}$'s total information capacity: $c \equiv \log\left[\mathrm{dim}\left(\mathcal{H}\right)\right]$). Each $\ensuremath{\mathcal{E}}$ in Fig. \ref{figBS_SPIPs}b has $c \simeq 120$ bits, so their SPIPs are essentially identical.
\subsubsection{Redundancy: numerical values} \label{secBSRedundancy}
\FourSmallPanelFigureX[Redundancy in branching states]{figBSRedundancy}{BasicRedundancy_basic}{BasicRedundancy_envsize}{BasicRedundancy_syssize} {BasicRedundancy_fudgefactor}
{(Color) \textbf{Redundancy for an assortment of branching-state ensembles.} \textbf{(a)}: $R_{10\%}$ for a $D$-dimensional system decohered by $D$-dimensional subenvironments. \textbf{(b)}: $R_{10\%}$ for a $5$-dimensional system decohered by $\ensuremath{D_{\Env}}=2\ldots5$-dimensional subenvironments. \textbf{(c)}: $R_{10\%}$ for a $\ensuremath{D_{\Sys}}=2\ldots16$-dimensional system decohered by $4$-dimensional subenvironments. \textbf{(d)}: $R_{\delta}$ for $\delta=0.001\ldots0.25$ and $\ensuremath{D_{\Sys}}=\ensuremath{D_{\Env}}=5$.
\textbf{Discussion:} Each plot shows the ensemble-average of $R_{\delta}$, as a function of $\ensuremath{N_{\mathrm{env}}}$. $R_{\delta}$ increases linearly with the number of environments. $R_{\delta}$ increases with $\ensuremath{D_{\Env}}$, but decreases with $\ensuremath{D_{\Sys}}$. Larger environments store more information, which leads to greater redundancy -- but larger systems have more information to be stored. Information is stored with slightly greater efficiency for large $\ensuremath{D_{\Sys}}$ and $\ensuremath{D_{\Env}}$ (plot \textbf{(a)}). Note that if $\ensuremath{\mathcal{S}}$ is larger than $\ensuremath{\mathcal{E}}$ (e.g., $\ensuremath{D_{\Sys}}=16$ in plot \textbf{(c)}), there may be no redundancy. Finally, $\delta$ affects redundancy (plot \textbf{(d)}) -- but varying $\delta$ by a full order of magnitude (from 2\% to 25\%) changes $R_{\delta}$ by less than 50\%.}
Branching states are natural generalizations of GHZ states, so we expect redundant information storage. Figure \ref{figBSRedundancy} confirms this over a wide range of parameters. The amount of redundancy is proportional to the size of the environment, which agrees with the classical intuition that very large environments should store many copies of information about the system. Larger subenvironments (measured by $\ensuremath{D_{\Env}}$) increase redundancy by storing more information in each subenvironment. Conversely, larger systems have more properties to measure, which in turn require more space for information storage. The total amount of redundancy is reduced for large $\ensuremath{D_{\Sys}}$.
The other important feature of the plots in Fig. \ref{figBSRedundancy} is the relatively weak dependence of $R_{\delta}$ on the information deficit ($\delta$). As we vary $\delta$ from $2\%$ to $25\%$ (a full order of magnitude), $R_{\delta}$ changes by less than a factor of 2. The distinction between classical (massively redundant) and quantum (nonredundant) information is largely independent of $\delta$.
\section{Theoretical analysis of branching states} \label{secBranchingStateTheory}
The numerical analysis in the previous section offers compelling evidence that
\begin{enumerate}
\item Information is stored redundantly in branching states,
\item The amount of redundancy scales with $\ensuremath{N_{\mathrm{env}}}$, and
\item $R_{\delta}$ is relatively insensitive to $\delta$.
\end{enumerate}
In this section, we construct theoretical models for PIPs and redundancy, which confirm these hypotheses.
\subsection{Structural properties of branching states} \label{secBSStructure}
We begin by using the structure inherent to branching states to compute a quantity of fundamental interest,
\begin{equation}
\Isen{\{m\}} = \ensuremath{\Hh_{\Sys}} + \ensuremath{H}_{\Envset{m}} - \ensuremath{H}_{\ensuremath{\mathcal{S}}\Envset{m}},
\end{equation}
the mutual information between the system and a partial environment $\Envset{m}$.
We require the entropies of $\ensuremath{\rho_{\Sys}}$, $\rho_{\Envset{m}}$, and $\rho_{\ensuremath{\mathcal{S}}\Envset{m}}$. Tracing over the rest of the universe is simplified by the structure that Eq. \ref{eqBranchingState} implies. Each relevant density matrix (regardless of its actual dimension) has only $\ensuremath{D_{\Sys}}$ nonzero eigenvalues. That is, the reduced states for $\ensuremath{\mathcal{S}}$, $\Envset{m}$, and $\ensuremath{\mathcal{S}}\Envset{m}$ are all ``virtual qudits'' with $D=\ensuremath{D_{\Sys}}$.
Each $\rho$, when reduced to its $\ensuremath{D_{\Sys}}$-dimensional support, is \emph{spectrally} equivalent to a partially decohered variant of the system's initial state:
\begin{equation}
\proj{\ensuremath{\mathcal{S}}_0} = \sum_{nm}{s_ns^*_m\ketbra{n}{m}}.
\end{equation}
In other words, we can obtain $\ensuremath{\rho_{\Sys}}$, $\rho_{\Envset{m}}$, or $\rho_{\ensuremath{\mathcal{S}}\Envset{m}}$ by taking $\proj{\ensuremath{\mathcal{S}}_0}$ and suppressing the off-diagonal elements according to a specific rule.
To determine this rule, we define (for each subenvironment) a \emph{multiplicative decoherence factor}, $\gamma$:
\begin{equation}
\gamma\idx{k}_{ij} = \braket{{\ensuremath{\mathcal{E}}^{(k)}_j}}{{\ensuremath{\mathcal{E}}^{(k)}_i}},
\end{equation}
and an associated \emph{additive decoherence factor}, $d$:
\begin{equation}
d\idx{k}_{ij} \equiv -\log\gamma^{(k)}_{ij}.
\end{equation}
Now, $\gamma\idx{k}_{ij}$ quantifies how much $\ensuremath{\mathcal{E}}_k$ contributes to decohering $\ket{i}$ from $\ket{j}$. The $\gamma$-factors from different $\ensuremath{\mathcal{E}}_k$ combine multiplicatively; the $d$-factors provide a convenient additive representation. Each relevant density matrix $\rho_X$ (for $X \in \{\ensuremath{\mathcal{S}},\Envset{m},\ensuremath{\mathcal{S}}\Envset{m}\}$) is given by:
\begin{equation}
\braopket{i}{\ensuremath{\rho_{\Sys}}}{j} = (s_is^*_j)e^{-d\idx{X}_{ij}}.
\end{equation}
The $d$-factor for each subsystem is a sum over $d$-factors for the component $\ensuremath{\mathcal{E}}_k$:
\begin{eqnarray}
d^{(\Envset{m})}_{ij} &=& \sum_{k \in \Envset{m}}{d^{(k)}_{ij}} \\
d^{(\ensuremath{\mathcal{S}})}_{ij} &=& \sum_{k \in \ensuremath{\mathcal{E}}}{d^{(k)}_{ij}} \\
d^{(\ensuremath{\mathcal{S}}\Envset{m})}_{ij} &=& \sum_{k \not\in \Envset{m}}{d^{(k)}_{ij}}.
\end{eqnarray}
Thus, each $\rho$ appears to have been decohered by a different subset of $\ensuremath{\mathcal{E}}$:
\begin{itemize}
\item $\ensuremath{\rho_{\Sys}}$ has been decohered by \textbf{every} subenvironment,
\item $\rho_{\ensuremath{\mathcal{S}}\Envset{m}}$ has been decohered by all the subenvironments \emph{not} in $\Envset{m}$,
\item $\rho_{\Envset{m}}$ has been decohered by all the subenvironments in $\Envset{m}$.
\end{itemize}
\textbf{Note}: If the last point seems counter-intuitive, recall that for any bipartite decomposition of $\ket{\Psi}_{AB}$, the reduced $\rho_A$ and $\rho_B$ are spectrally equivalent. Thus $\rho_{\Envset{m}}$ is equal to $\rho_{\ensuremath{\mathcal{S}}\overline{\Envset{m}}}$, where $\overline{\Envset{m}}$ contains all the environments \emph{not} in $\Envset{m}$.
Computing $\Isen{m}$ (in terms of the entropy of these three states) can be done \emph{exactly} via numerical diagonalization. For qubit systems, it can also be done analytically (see \cite{RBK04} for extensive details). For our model, we now derive an approximation for $\ensuremath{H}(\rho)$.
\subsection{Theoretical PIPs: averaging $\ensuremath{\mathcal{I}}(m)$} \label{secBSApproxMI}
As a particular $\rho$ is decohered by more and more subenvironments, its off-diagonal elements decline rapidly toward zero. We will treat the off-diagonal elements of a \emph{partially} decohered state, $\rho = \sum_{ij}{s_is^*_j\gamma_ij\ket{i}\bra{j}}$, as a perturbation around the \emph{fully} decohered state $\rho_0$, which has eigenvalues $\lambda_i = |s_i|^2$ and entropy $\ensuremath{H}_0$.
\subsubsection{Average entropy of partially decohered states}
Let $\rho = \rho_0 + \Delta$, where $\Delta$ is a small off-diagonal perturbation to $\rho_0$, and expand its entropy as $\ensuremath{H}(\rho) \approx \ensuremath{H}(\rho_0) + O(\Delta)$.
An intuitively appealing starting point is the MacLaurin expansion of $\ensuremath{H}(x) = -x\ln(x)$, which yields
\begin{equation}
\ensuremath{H}(\rho_0+\Delta) \approx \ensuremath{H}(\rho_0)- \mathrm{Tr}\left[\Delta(1\!\mathrm{l}-\ln(\rho_0))\right] - \ensuremath{\frac12}\frac{\Delta^2}{\rho_0} +\frac{1}{6}\frac{\Delta^3}{\rho_0^2}\ldots
\label{eqBadEntropyExpansion}
\end{equation}
The first order term in Eq. \ref{eqBadEntropyExpansion} vanishes, because $\Delta$ is purely off-diagonal and $1\!\mathrm{l}-\ln(\rho)$ is purely diagonal. The leading term is thus $\frac{\Delta^2}{2\rho_0}$ -- but the matrix quotient $\frac{\Delta^{k+1}}{\rho_0^k}$ is ill-defined when $\Delta$ and $\rho_0$ do not commute.
A more involved expansion of $\ensuremath{H}(\rho)$ around $\rho=1\!\mathrm{l}$ (see Appendix \ref{appEntropyExpansion}) yields a series for $\ensuremath{H}(\rho_0+\Delta)$. It is equivalent to Eq. \ref{eqBadEntropyExpansion} for scalars, but for matrices it involves (1) expanding $\rho_0^{-k}$ in a power series, and (2) taking a totally symmetric product between $\Delta^{k+1}$ and the resulting power series.
To leading order in $\Delta$,
\begin{equation}
\ensuremath{H}(\rho) \approx \ensuremath{H}(\rho_0) -\frac{\overline{|\gamma|^2}}{2}\left(h(\rho_0)-1\right),
\end{equation}
where $\overline{|\gamma|^2}$ is the average of $|\gamma_{ij}|^2$ over all $i\neq j$, and $h(\rho_0)$ is a nontrivial function,
\begin{equation}
h(\rho_0) = \sum\limits_{n,p=0}^{\infty}{\frac{\mathrm{Tr}\left[\rho_0(1\!\mathrm{l}-\rho_0)^p\right] \mathrm{Tr}\left[\rho_0(1\!\mathrm{l}-\rho_0)^{n}\right]}{n+p+1}}.\label{eqHseries}
\end{equation}
\subsubsection{Effective Hilbert space dimension}
In general, $h(\rho_0)$ cannot be simplified further. However, it is well approximated by the \emph{effective Hilbert space dimension} of $\rho_0$. To see this, we consider the special case where $\rho_0$ has $D$ identical eigenvalues, $\lambda_i = \frac{1}{D}$. When reduced to its support, $\rho_0 = \frac{1\!\mathrm{l}}{D}$. The summation can be done explicitly:
\begin{eqnarray}
h(\rho_0) &=& \sum\limits_{n,p=0}^{\infty}{\frac{\mathrm{Tr}\left[\frac{1\!\mathrm{l}}{D}\left(1\!\mathrm{l}-\frac{1\!\mathrm{l}}{D}\right)^p\right] \mathrm{Tr}\left[\frac{1\!\mathrm{l}}{D}\left(1\!\mathrm{l}-\frac{1\!\mathrm{l}}{D}\right)^{n}\right]}{n+p+1}}\nonumber\\
&=& \sum\limits_{n,p=0}^{\infty}{\frac{\left((1-D^{-1})^p\right) \left((1-D^{-1})^{n}\right)}{n+p+1}}\nonumber\\
&=& \sum\limits_{n,p=0}^{\infty}{\frac{\left(1-D^{-1}\right)^{n+p}}{n+p+1}}\nonumber\\
&=& \sum\limits_{n+p=0}^{\infty}{\left(1-D^{-1}\right)^{n+p}}\nonumber\\
&=& D
\end{eqnarray}
Note that $D$ appeared \emph{only} based on the eigenvalue spectrum of $\rho_0$. In the example above, the $\ensuremath{H}_0 = \ensuremath{H}(\rho_0) = \log(D)$. Since the total range of $\ensuremath{\overline{\I}}(m)$ is proportional to $\ensuremath{H}_0$, a logical generalization is
\begin{eqnarray}
h(\rho_0) &\approx& e^{\ensuremath{H}_0}, \label{eqApprox_h} \\
\ensuremath{H}(\rho) &\approx& \ensuremath{H}(\rho_0) -\frac{\overline{|\gamma|^2}}{2}\left(e^{\ensuremath{H}_0}-1\right). \label{eqApproxEntropy}
\end{eqnarray}
Numerical experimentation, and an analytic calculation in $\ensuremath{D_{\Sys}}=2$, confirm that Eq. \ref{eqApprox_h} is a good approximation everywhere, in addition to being exact for (1) maximally mixed states, and (2) pure states.
\subsubsection{Average decoherence factors}
The $\gamma_{ij}$ depend on the details of $\psi_{\ensuremath{\mathcal{S}}\ensuremath{\mathcal{E}}}$. However, when they are small enough to count as a perturbation on $\rho$, the environment's Hilbert space is very large. The $|\gamma_{ij}|^2$ can then be treated as independent random variables, so $\overline{|\gamma^2|}$ is equal to an average over the entire branching state ensemble:
\begin{eqnarray}
\overline{|\gamma^2|} &=& \overline{\braket{\psi}{\psi'}\braket{\psi'}{\psi}} \nonumber\\
&=& \mathrm{Tr}\left(\overline{\proj{\psi}}\ \overline{\proj{\psi'}}\right) \nonumber\\
&=& \frac{\mathrm{Tr}\left(1\!\mathrm{l}\Id\right)}{\ensuremath{D_{\Env}}^2} \nonumber\\
&=& \ensuremath{D_{\Env}}^{-1}
\end{eqnarray}
This is the mean value of $\overline{|\gamma^2|}$ for a \emph{single} subenvironment. For a collection of $m$ subenvironments, $m$ such $\gamma$ factors are multiplied together, so the mean value of $\overline{|\gamma^2|}$ becomes $\ensuremath{D_{\Env}}^{-m}$.
\subsubsection{The result}
Putting this all together, the average entropy of a $\ensuremath{D_{\Sys}}$-dimensional system decohered by $m$ $\ensuremath{D_{\Env}}$-dimensional environments is
\begin{equation}
\overline{\ensuremath{H}} \simeq \ensuremath{H}_0 - \frac{e^{\ensuremath{H}_0}-1}{2} \ensuremath{D_{\Env}}^{-m}, \label{eqAverageEntropy}
\end{equation}
and the average mutual information between the system and $m$ subenvironments is
\begin{eqnarray}
\ensuremath{\overline{\I}}(m) &\approx& \ensuremath{H}_0 - \frac{e^{\ensuremath{H}_0}-1}{2}\left(\ensuremath{D_{\Env}}^{-m} - \ensuremath{D_{\Env}}^{-(\ensuremath{N_{\mathrm{env}}}-m)}\right) \label{eqAverageMI} \\
&=& \ensuremath{H}_0 + \left(e^{\ensuremath{H}_0}-1\right)\sinh\left[\left(m-\frac{\ensuremath{N_{\mathrm{env}}}}{2}\right)\ln(\ensuremath{D_{\Env}})\right]. \nonumber
\end{eqnarray}
Equation \ref{eqAverageMI} is only a good approximation only near the classical plateau, where $\ensuremath{\overline{\I}} \simeq \ensuremath{H}_0$. Around $m=0$ and $m=\ensuremath{N_{\mathrm{env}}}$, $\ensuremath{\overline{\I}}$ rises linearly, not exponentially. Each subenvironment can provide only $\log_2\ensuremath{D_{\Env}}$ bits of information, so until the information starts to become redundant, we're in a different regime (see Fig. \ref{figBS_SPIPs}b).
Once the information capacity of the captured environments ($m\log\ensuremath{D_{\Env}}$) becomes greater than the amount of information in the system ($\ensuremath{H}_0$), Eq. \ref{eqAverageMI} becomes valid. It describes the slow approach to ``perfect'' information about the system, as $m$ increases. Figure \ref{figBSApprox} compares exact (numerical) results for $\ensuremath{\overline{\I}}(m)$ to the approximation in Eq. \ref{eqAverageMI}.
\SixSmallPanelFigureX[Analytical approximations to branching-state PIPs]
{figBSApprox}{Est_2x2_8}{Est_2x2_32}{Est_4x4_8}{Est_4x4_32}{Est_16x16_8}{Est_16x16_32}
{(Color) \textbf{Numerical PIPs vs. Theory:} We compare the approximation derived in Sec. \ref{secBSApproxMI} with numerics. Error bars on numerics represent typical fluctuations over the branching-state ensemble. \textbf{(a)}: $\ensuremath{D_{\Sys}}=\ensuremath{D_{\Env}}=2$, $\ensuremath{N_{\mathrm{env}}}=8$. \textbf{(b)}: $\ensuremath{D_{\Sys}}=\ensuremath{D_{\Env}}=2$, $\ensuremath{N_{\mathrm{env}}}=32$. \textbf{(c)}: $\ensuremath{D_{\Sys}}=\ensuremath{D_{\Env}}=4$, $\ensuremath{N_{\mathrm{env}}}=8$. \textbf{(d)}: $\ensuremath{D_{\Sys}}=\ensuremath{D_{\Env}}=4$, $\ensuremath{N_{\mathrm{env}}}=32$. \textbf{(e)}: $\ensuremath{D_{\Sys}}=\ensuremath{D_{\Env}}=16$, $\ensuremath{N_{\mathrm{env}}}=8$. \textbf{(f)}: $\ensuremath{D_{\Sys}}=\ensuremath{D_{\Env}}=16$, $\ensuremath{N_{\mathrm{env}}}=32$. \textbf{Discussion:} The approximation is virtually perfect near the classical plateau. For small $m$, the rate of information gain is more nearly linear, and the approximation fails. Although it works well at $m=0$ for $\ensuremath{D_{\Sys}}=4$ (plots \textbf{(b)},\textbf{(e)}), it fails spectacularly near $m=0$ for large $\ensuremath{D_{\Sys}}$ (plots \textbf{(c)},\textbf{(f)})}
\subsection{Theoretical redundancy: averaging $m(\ensuremath{\mathcal{I}})$} \label{secBSSpecR}
Branching states develop when each subenvironment interacts independently with $\ensuremath{\mathcal{S}}$. The data in Section \ref{secBSRedundancy} (esp. Fig. \ref{figBSRedundancy}) confirm that redundancy in branching states is proportional to $\ensuremath{N_{\mathrm{env}}}$. A certain number of subenvironments ($m_{\delta}$) is enough to provide sufficient information.
To capture this scaling, we define \emph{specific redundancy} as
\begin{equation}
r_{\delta} = \lim_{\ensuremath{N_{\mathrm{env}}}\rightarrow\infty} \frac{R_{\delta}}{\ensuremath{N_{\mathrm{env}}}} = \frac{1-\delta}{m_{\delta}} \label{eqSpecR}
\end{equation}
In this section, we use specific redundancy to examine precisely how $\ensuremath{D_{\Sys}}$, $\ensuremath{D_{\Env}}$, and $\delta$ affect information storage in branching states. We derive an approximate formula for $r_{\delta}$, and compare its predictions to numerical data.
In the previous section, we computed the average information yielded by $m$ environments. Now, we compute the average $m$ required to achieve a given $\ensuremath{\mathcal{I}}$.
When $\ensuremath{N_{\mathrm{env}}}$ is large, $\ensuremath{H}_{\ensuremath{\mathcal{S}}\Envset{m}} \simeq \ensuremath{\Hh_{\Sys}} \simeq \ensuremath{H}_0$, so $\Isen{\{m\}} \simeq \ensuremath{H}_{\Envset{m}}$. We take Eq. \ref{eqApproxEntropy},
\begin{equation}
\Isen{\{m\}} \approx \ensuremath{\Hh_{\Sys}} -\ensuremath{\frac12}\overline{|\gamma|^2}\left(e^{\ensuremath{\Hh_{\Sys}}}-1\right),
\end{equation}
as a starting point. For the fragment to provide ``sufficient'' information, $\ensuremath{\mathcal{I}}-\ensuremath{\Hh_{\Sys}}$ must be less than $\delta\ensuremath{\Hh_{\Sys}}$, which requires
\begin{equation}
\frac{\sum_{i\neq j}{|\gamma_{ij}|^2}}{\ensuremath{D_{\Sys}}(\ensuremath{D_{\Sys}}-1)}\left(e^{\ensuremath{\Hh_{\Sys}}}-1\right) \leq 2\delta\ensuremath{\Hh_{\Sys}}.
\end{equation}
Assuming $\rho_0$ is maximally mixed (i.e., $e^{\ensuremath{H}_0}=\ensuremath{D_{\Sys}}$), and replacing the $\gamma_{ij}$ with independent random variables $\gamma_n$, we obtain the following condition on a ``sufficiently large'' fragment:
\begin{equation}
\left[\sum_{n=1}^{\frac{\ensuremath{D_{\Sys}}(\ensuremath{D_{\Sys}}-1)}{2}}{|\gamma_n|^2}\right] \leq \delta\ensuremath{D_{\Sys}}\ensuremath{\Hh_{\Sys}} \label{eqRedundancyCondition}
\end{equation}
The interaction of $\frac12\ensuremath{D_{\Sys}}(\ensuremath{D_{\Sys}}-1)$ independent $\gamma$-factors makes it difficult to solve Eq. \ref{eqRedundancyCondition} rigorously. We begin instead by considering a qubit system, which has only one off-diagonal $\gamma$.
\subsubsection{Specific redundancy for qubit systems}
For a single qubit, there is only one decoherence factor: $d_{01}$, which we'll refer to simply as $d$. Eq. \ref{eqRedundancyCondition} simplifies to:
\begin{equation}
d \geq d_{\delta} \equiv -\frac12\log\left(2\delta\ensuremath{\Hh_{\Sys}}\right) \label{eqQubitRedundancyCondition}
\end{equation}
The increase in $d$ with $m$ can be approximated as a biased random walk, where each step has a mean length ($\overline{d}$) and a variance ($\Delta d$). After $m$ environments are added to the fragment, $d$ obeys a normal distribution ($p_m(d)$), whose mean and variance are $m\overline{d}$ and $\sqrt{m}\Delta d$, respectively. We postpone the calculation of $\overline{d}$ and $\Delta d$ for the moment.
Let $\ensuremath{p_{\mathrm{suff}}}(m)$ be the probability that a fragment consisting of $m$ subenvironments provides sufficient information (i.e., satisfies equation \ref{eqQubitRedundancyCondition}). Then
\begin{equation}
\ensuremath{p_{\mathrm{suff}}}(m) = \int_{d_{\delta}}^{\infty}{p_m(d)d\!d},
\end{equation}
and the probability that $m$ environments are \emph{required} is
\begin{eqnarray}
\ensuremath{p_{\mathrm{req}}}(m) &=& \ensuremath{p_{\mathrm{suff}}}(m) - \ensuremath{p_{\mathrm{suff}}}(m-1) \\
&=& \int_{m-1}^{m}{\pdiff{}{n}\ensuremath{p_{\mathrm{suff}}}(n)d\!n},
\end{eqnarray}
and the expected fragment size ($\overline{m}$) is
\begin{eqnarray}
\overline{m} &=& \sum_{m=0}^{\infty}{m\ \ensuremath{p_{\mathrm{req}}}(m)} \nonumber\\
&=& \sum_{m=0}^{\infty}{m\int_{m-1}^{m}{\pdiff{}{n}\ensuremath{p_{\mathrm{suff}}}(n)d\!n}} \nonumber\\
&\simeq& \int_{0}^{\infty}{\left(m+\frac12\right)\pdiff{}{m}\ensuremath{p_{\mathrm{suff}}}(m)d\!m} \nonumber\\
&=& \frac12 + \int_{0}^{\infty}{m\pdiff{}{m}\ensuremath{p_{\mathrm{suff}}}(m)d\!m} \nonumber\\
&=& \frac12 + \int_{0}^{\infty}{\left(1-\ensuremath{p_{\mathrm{suff}}}(m)\right)d\!m} \nonumber\\
&=& \frac12 + \int_{0}^{\infty}{d\!m\int_{-\infty}^{d_{\delta}}{p_m(d)d\!d}}. \label{eqMeanMIntegral}
\end{eqnarray}
We interchange the order of integration, substitute the appropriate normal distribution for $p_m(d)$, and end up with
\begin{equation}
\overline{m} = \frac{d_{\delta}}{\overline{d}} + \frac{\Delta^2}{2\overline{d}^2}+\frac12. \label{eqMeanMPrelim}
\end{equation}
\subsubsection{Specific redundancy for general $\ensuremath{D_{\Sys}}$}
Whereas Eq. \ref{eqQubitRedundancyCondition} (for qubits) has one $\overline{|\gamma|^2}$ term, Eq. \ref{eqRedundancyCondition} involves a sum of $\frac12\ensuremath{D_{\Sys}}(\ensuremath{D_{\Sys}}+1)$ such terms. Deriving an analyzing a probability distribution for this sum is very difficult, so we take a simpler route. We replace the sum over terms with a single term, $\frac12\ensuremath{D_{\Sys}}(\ensuremath{D_{\Sys}}+1)\cdot\gamma^2$, where $\gamma^2$ represents all the off-diagonal terms. The new condition for sufficient information is:
\begin{eqnarray}
\frac{\ensuremath{D_{\Sys}}(\ensuremath{D_{\Sys}}-1)}{2}{\gamma^2} &\leq& \delta\ensuremath{D_{\Sys}}\ensuremath{\Hh_{\Sys}} \nonumber\\
\gamma^2 &\leq& \frac{2\delta\ensuremath{\Hh_{\Sys}}}{\ensuremath{D_{\Sys}}-1} \nonumber\\
d &\geq& d_{\delta} \equiv -\frac12\log\left(\frac{2\delta\ensuremath{\Hh_{\Sys}}}{\ensuremath{D_{\Sys}}-1}\right).
\end{eqnarray}
$\ensuremath{D_{\Sys}}$ has been incorporated into a redefinition of $d_{\delta}$. Equation \ref{eqMeanMPrelim} is still valid for qubits, but it generalizes to
\begin{equation}
\overline{m} = \frac{\log(\ensuremath{D_{\Sys}}-1)-\log\left(2\delta\ensuremath{\Hh_{\Sys}}\right)}{2\overline{d}} + \frac{\Delta d^2}{2\overline{d}^2} +\frac12. \label{eqMeanM}
\end{equation}
We combine this expression with Eq. \ref{eqSpecR} to obtain a general estimate for specific redundancy:
\begin{equation}
r_{\delta} = \frac{2\overline{d}^2(1-\delta)}{{\Delta}^2 + \overline{d}^2 + \overline{d}\left(\log(\ensuremath{D_{\Sys}}-1)-\log(2\delta\ensuremath{\Hh_{\Sys}})\right)} \label{eqApproxSpecR}
\end{equation}
\subsubsection{Dependence of mean decoherence factor ($d$) on $\ensuremath{D_{\Env}}$}
The computation of $\overline{d}$ and $\Delta d$ in terms of $\ensuremath{D_{\Env}}$ is somewhat tedious. Details can be found in Appendix \ref{appDFactors}, where we calculate:
\begin{eqnarray}
\overline{d} &=& \frac12\left(\Psi(\ensuremath{D_{\Env}})+\ensuremath{\gamma_{\scriptscriptstyle{\mathrm{EM}}}}\right), \label{eqMeanD}\\
\Delta d^2 &=& \frac{\pi^2}{24} - \frac{\Psi_1(\ensuremath{D_{\Env}})}{4}, \label{eqVarianceD}
\end{eqnarray}
in terms of the digamma ($\Psi(n)$) and trigamma ($\Psi_1(n)$ functions \cite{MathworldDigamma,MathworldTrigamma}, and the Euler-Mascheroni constant $\ensuremath{\gamma_{\scriptscriptstyle{\mathrm{EM}}}} = 0.577\ldots$. These functions may not be familiar to all readers, so we present the first few values in Table \ref{tabDValues}.
\begin{table}[hdt!]
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|l||c|c|c|c|c|c|}\hline
$\ensuremath{D_{\Env}}$ & 2 & 3 & 4 & 5 & 6 & 8\\
\hline
$\overline{d}$ & $\frac12$ & $\frac34$ & $\frac{11}{12}$ & $\frac{25}{24}$ & $\frac{137}{120}$ & $\frac{363}{280}$\\
[0.5ex]\hline
$\Delta d$ & $\frac12$ & $\frac{\sqrt{5}}{4}$ & $\frac{7}{12}$ & $\frac{\sqrt{205}}{24}$ & $\frac{\sqrt{5269}}{120}$ & $\frac{\sqrt{266681}}{840}$\\
[0.5ex]\hline
\end{tabular}
\end{center}
\caption[Additive decoherence factors for small environments]{\label{tabDValues}The table shows the first few values of $\overline{d}$ and $\Delta d$, for environments of size $\ensuremath{D_{\Env}} \in [2,3,4,5,6,8]$. See Appendix \ref{appDFactors} for details on the calculation.}
\end{table}
For larger $\ensuremath{D_{\Env}}$, we can safely approximate Eqs. \ref{eqMeanD}-\ref{eqVarianceD} as:
\begin{eqnarray}
\overline{d} &\simeq& \frac12\left(\log(\ensuremath{D_{\Env}})+\ensuremath{\gamma_{\scriptscriptstyle{\mathrm{EM}}}}\right) \\
\Delta d &\simeq& \frac{\pi}{\sqrt{24}}.
\end{eqnarray}
\subsubsection{How good is the estimate?}
\FourPanelFigure[Specific redundancy in branching-state ensembles]{figSpecR}{SpecR_16xN}{SpecR_Nx2}{SpecR_NxN_0.99a.pdf}{SpecR_NxN_0.99b.pdf}
{(Color) \textbf{Specific redundancy} ($r_{\delta} \equiv R_{\delta}/\ensuremath{N_{\mathrm{env}}}$): numerical data (symbols) compared with theory (Eq. \ref{eqApproxSpecR}, solid lines). \textbf{(a)}: $r_{\delta}$ vs. $\delta$, for a $16$-d system coupled to $2,3,4,8$-dimensional subenvironments. \textbf{(b)}: $r_{\delta}$ vs. $\delta$, for $2,3,4,8,16$-d systems coupled to qubit subenvironments. \textbf{(c)}: $r_{1\%}$ vs. $\ensuremath{D_{\Env}}$. \textbf{(d)}: $r_{1\%}$ vs. $\ensuremath{D_{\Sys}}$. \textbf{Discussion:} Theory predicts the overall behavior of redundancy well. It is nearly perfect for $\ensuremath{D_{\Sys}}=2$, but overestimates $r$ for larger systems. As $\delta$ increases, $r_{\delta}$ saturates and even \emph{declines} because of the $(1-\delta)$ prefactor in Eq. \ref{eqRedundancy}. When $\delta$ is large, the theory breaks down (see \textbf{(a)}), because a single subenvironment can provide sufficient information.}
In Figure \ref{figSpecR}, we compare numerical results to the approximation of Eq. \ref{eqApproxSpecR}. The analytical estimate is very good for qubit systems, but loses some fidelity for larger $\ensuremath{D_{\Sys}}$. A more sophisticated treatment of the multiple $\gamma_{ij}$ terms -- each representing an independent observable which the environment must record -- would eliminate this error.
To get an intuitive feel for the dependence of $r_{\delta}$ on its parameters, we consider the regime of large systems, large environments, and small deficit -- i.e., $H_0 \gg 1$, $\overline{d} \sim \frac12 \log(\ensuremath{D_{\Env}})$, $\Delta d \sim \frac{\pi^2}{24}$, and $\delta \ll 1$. In this regime, we can ruthlessly simplify Eq. \ref{eqApproxSpecR} to obtain a simple prediction:
\begin{equation}
r_{\delta} \approx \frac{\log(\ensuremath{D_{\Env}})}{\log(\ensuremath{D_{\Sys}})-\log(\delta)}. \label{eqThumbnailSpecR}
\end{equation}
The plots in Fig. \ref{figEfficiency} show the ratio between numerical $r_\delta$ data and the simple predictions of Eq.\ref{eqThumbnailSpecR}. They confirm that Eq.\ref{eqThumbnailSpecR} is a good rule of thumb.
\FourSmallPanelFigure[``Efficiency'': specific redundancy rescaled by information capacity] {figEfficiency}{Efficiency_16xN}{Efficiency_Nx2}{Efficiency_NxN_0.99a.pdf}{Efficiency_NxN_0.99b.pdf}
{(Color) \textbf{``Efficiency'': specific redundancy rescaled by information capacity.} Equation \ref{eqThumbnailSpecR} provides a simple approximation for redundancy, based on the relative information capacity of the system (with a correction for $\delta$) and its environment. We reproduce the data of Fig. \ref{figSpecR}, but use Eq. \ref{eqApproxSpecR} to rescale specific redundancy. \textbf{Discussion:} Efficiency is consistently near to 1: when the universe is in a random branching state, information about $\ensuremath{\mathcal{S}}$ is efficiently recorded in $\ensuremath{\mathcal{E}}$. Equation \ref{eqApproxSpecR} is accurate for large $\ensuremath{D_{\Sys}}$ and $\ensuremath{D_{\Env}}$ (and small $\delta$). When the system or the subenvironments are small, Eq. \ref{eqApproxSpecR} underestimates information storage efficiency.}
Eq. \ref{eqThumbnailSpecR} can be interpreted as a capsule summary of how redundancy scales in the ``random-state'' model of decoherence.
\begin{enumerate}
\item Redundancy is proportional to $\ensuremath{N_{\mathrm{env}}}$, the number of independent subenvironments. \textbf{More environments produce more redundancy.}
\item Redundancy is proportional to $\bar{d}$, the \emph{mean decoherence factor} of a single subenvironment, which grows as $\log\ensuremath{D_{\Env}}$. \textbf{Larger environments produce more redundancy, in proportion to their information capacity.}
\item Redundancy is (roughly) inversely proportional to $\ensuremath{\Hh_{\Sys}}$, the total information available about the system. \textbf{Larger systems require more space in the environment.}
\item The deficit ($\delta$) appears as a logarithmic addition to $\ensuremath{\Hh_{\Sys}}$. Reducing the amount of ``ignorable'' information is equivalent to making the system bigger. \textbf{Redundancy depends only weakly (logarithmically) on the deficit, $\delta$.}
\end{enumerate}
\section{Conclusions and discussion} \label{secStaticsConclusions}
\OnePanelFigure[Quantum Darwinism in action]{figInfoDivision}{InfoDivision2}
{(Color) \emph{Quantum Darwinism} selects certain observable properties of the system and propagates information about them throughout the environment. The preferred observable[s] become redundant \textbf{at the expense of incompatible observables}. As shown here, PIPs illustrate the results of Quantum Darwinism. Information about $\ensuremath{\mathcal{S}}$ becomes divided into three parts: redundant information ($\ensuremath{\I_{\mathrm{R}}}$), quantum information ($\ensuremath{\I_{\mathrm{Q}}}$), and non-redundant information ($\ensuremath{\I_{\mathrm{NR}}}$). Redundant information is objective, and therefore classical. It can be obtained with relative ease. Quantum information represents the non-preferred observables, marginalized by Quantum Darwinism, which can only be measured by capturing all of $\ensuremath{\mathcal{E}}$. Non-redundant information (determined by the slope of $\ensuremath{\overline{\I}}(m)$ at $m=\frac{\ensuremath{N_{\mathrm{env}}}}{2}$) represents the ambiguous borderline, undifferentiated as yet into classical and quantum fractions. When $\ensuremath{\I_{\mathrm{NR}}}$ is small, the central region of the PIP becomes flat. This ``classical plateau'' indicates that an observer can obtain full information without capturing the entire environment.}
`There is no information without representation': information has to be stored somewhere. To retrieve it, we must measure the systems where it is stored. To understand the properties of information, we look at the properties of this retrieval process. We have focused on the question: \textbf{How easily can information about a system be retrieved from its environment?}
The answer is strongly dependent on how the system became correlated with its environment. Random interactions between $\ensuremath{\mathcal{S}}$ and all of $\ensuremath{\mathcal{E}}$ leave no useful correlations -- to learn about $\ensuremath{\mathcal{S}}$ we must measure most of $\ensuremath{\mathcal{E}}$. However, when localized parts of $\ensuremath{\mathcal{E}}$ interact independently with $\ensuremath{\mathcal{S}}$, an observer can learn about $\ensuremath{\mathcal{S}}$ by measuring a small fragment of $\ensuremath{\mathcal{E}}$. Furthermore, the information that he learns is objective -- another independent observer will arrive at the same conclusions.
This redundant imprinting of selected observables on the environment is \emph{quantum Darwinism}. It leads to objective reality in a quantum Universe. Typical PIPs for branching states (see Fig. \ref{figInfoDivision}) illustrate how different sorts of information are selected or deprecated. The information in $\ensuremath{\mathcal{E}}$ about $\ensuremath{\mathcal{S}}$ divides naturally into three parts.
\begin{equation}
\ensuremath{\I_{\Sys:\Env}} = \ensuremath{\I_{\mathrm{R}}} + \ensuremath{\I_{\mathrm{NR}}} + \ensuremath{\I_{\mathrm{Q}}}.
\end{equation}
The \emph{redundant information} ($\ensuremath{\I_{\mathrm{R}}}$) is classical -- it can be obtained easily, by many independent observers. Its selective proliferation is the essence of quantum Darwinism. Ollivier et. al. showed, in \cite{OllivierPRL04}, that $\ensuremath{\I_{\mathrm{R}}}$ is not only easy to obtain, but difficult to ignore. An observer who succeeds in extracting $\ensuremath{\I_{\mathrm{R}}}$, and continues to probe, finds a ``classical plateau''. Measurements on additional subenvironments increase his knowledge of $\ensuremath{\mathcal{S}}$ only slightly -- mostly, they only confirm what he already knows. Only a \emph{perfect} and \emph{global} measurement of \emph{everything} can reveal more than the redundant information.
Purely \emph{quantum information} ($\ensuremath{\I_{\mathrm{Q}}}$) represents observables that are incompatible with the pointer observable. This is the information that quantum Darwinism selects \emph{against}. It is (a) encoded amongst the environments, much as a classical bit can be encoded in the parity of many ancilla bits; (b) accessible only through a global measurement on \emph{all} of $\ensuremath{\mathcal{E}}$; and (c) easily destroyed when $\ensuremath{\mathcal{E}}$ decoheres.
Finally, \emph{non-redundant information} ($\ensuremath{\I_{\mathrm{NR}}}$) represents a grey area -- the border between the classical and quantum domains. It exists only when the classical plateau in $\ensuremath{\overline{\I}}(m)$ has a nonzero slope. This is why we allow for a deficit ($\delta$) when computing redundancy.
Information storage in randomly selected \emph{arbitrary} states of the model universe is dramatically different from information storage in randomly selected \emph{singly-branching} states. The contrast between these two cases emphasizes the importance of the environment's structure. Overly simple thermodynamic arguments (e.g., maximum entropy in absence of gravity) indicate that the physical Universe should evolve into states that are uniformly distributed. Our results, however, show that objects which display the redundancy characteristic of \emph{our} Universe must have \emph{structured} correlations with their environments.
Decoherence theory emphasizes the role of the environment in the quantum-to-classical transition, but only as a reservoir where unwanted quantum superpositions and correlations can be hidden, out of sight. Even this view -- which now seems somewhat narrow -- has produced important advances in our understanding over the past quarter century. Examples include einselection, the special role of pointer states, and the view of classicality as an emergent phenomenon. Nevertheless, it is clear from our discussion above and from related recent work \cite{OllivierPRL04,Ollivier04}, that ``tracing out $\ensuremath{\mathcal{E}}$'' obscures crucial aspects of the environment's role.
\textbf{The environment is a witness} -- a communication channel through which observers acquire the vast majority (if not all) of their information about the Universe. Surprisingly, this realization has taken more than 75 years since the formulation of quantum mechanics in its present form. It goes against a strong classical tradition of looking for solutions of fundamental problems in {\it isolated} settings. This tradition is incompatible with the role of \emph{states} in quantum theory.
Quantum states, unlike classical states, do not define what ``exists objectively''. They are too malleable -- too easily perturbed and redefined by measurements. Moreover, in quantum mechanics, what is \emph{known} about a system's state is inextricably intertwined with what it \emph{is}. Classical states, in contrast, have existence independently of the knowledge of them. To put it tersely (and in the spirit of complementarity), quantum states play both ontic (describing what is) and epistemic (describing what is known to be) roles\footnote{See \cite{ZurekRMP03} and especially \cite{Zurek04} for further discussion of quantum states' {\it epiontic} nature.}. Thus, for many purposes, it makes no sense to talk about a state of a completely isolated quantum system.
Our Universe is `quantum to the core' (see e.g. Ref. \cite{Schlosshauer05b} for an up-to-date review of the experimental evidence), so the only place to look for objective classicality is within the quantum theory itself. Decoherence has certainly supplied part of the answer: Only \emph{some} of the states in an open system's Hilbert space are stable. Those that are not stable, cannot ``exist objectively''. Even these einselected pointer states, however, are vulnerable to perturbation by an observer who measures directly. Yet, objectivity implies that many different (and initially ignorant) observers can independently find out the state.
The environment-as-a-witness point of view solves this problem by recognizing that we gain essentially all of our information indirectly, from the environmental degrees of freedom (with the possible exception of specific laboratory experiments). As the environment is the ``channel'', and as only a part of it can be intercepted, the obvious question is: \textbf{How is information is deposited in ${\cal E}$? and what kind of information?}
Quantum Darwinism, which we have begun to analyse here and elsewhere \cite{ZurekRMP03,OllivierPRL04,Ollivier04,ZurekADP00}, aims to supply the answer. Our basic conclusion is that the redundancy evident in our Universe is not a generic property of randomly selected states in large multipartite (system plus multi-component environment) Hilbert spaces. However, when states in that Hilbert space are created by the interactions usually invoked in discussions of environment-induced superselection, redundancy appears. Thus, objectivity can arise through the dynamics of decoherence. In that sense, decoherence is the mechanism that delivers quantum Darwinism -- a more complete view of classicality's emergence.
While we have already witnessed the birth of this new point of view, it is still far from mature. In particular, our conclusion about redundancy and the typical structure of entanglement was reached without analyzing dynamics {\it per se}. We have laid the foundation for a full-fledged study of quantum Darwinism by analysing kinematic properties of states, and postponed the study
of evolution in specific models to forthcoming publications \cite{RBK05c,RBK05e}. Moreover, by employing von Neumann entropy, we have focused on the amount of information (rather than on what this information is about). Differences between various definitions of mutual information exist (see ``discord", Ref. \cite{OllivierPRL02}), and are symptomatic of the ``quantumness'' of the underlying correlations. Less ``quantum'' definitions of mutual information, involving conditional information, {\it de facto} presume a measurement. They have also been used \cite{ZurekRMP03,OllivierPRL04,Ollivier04}), along with other tools (\cite{DalvitPRL01, Dalvit05}), to show that the familar pointer observables are the ``fittest'' in the (quantum) Darwinian sense. Studying the dynamics of quantum Darwinism, and the connections with various definitions of information, are the obvious next steps.
\acknowledgments
We thank Harold Ollivier and David Poulin for vigorous discussions.
This research was supported in part by NSA and ARDA.
|
1,314,259,996,714 | arxiv | \section*{Introduction}
The plus construction is a notion which is fundamental for discussing algebras, twists, and more recently monoid definitions in the theory of Feynman categories.
In \cite{KMoManin}, they give a broad definition of the plus construction suitable for what they call unique factorization categories.
Consequently, this incorporates more structures into the theory.
In this paper, we will use techniques from different areas to give explicit examples of unique factorization categories and applications of the plus construction.
We will focus on four fundamental examples of these: the trivial Feynman category $\FF^{triv}$, the category of finite sets $\FinSet$, the category of cospans, and the category of spans.
As described in \cite{feynman,feynmanrep}, the plus constructions of $\FF^{triv}$ and $\FinSet$ are related to monoids and operads respectively.
We will add to this by also considering the nc-plus construction introduced in \cite{KMoManin} and operads with multiplication.
Cospans appear in several seemingly different areas.
One area is algebraic topology where they are used in the study of cobordisms \cite{grandis1,grandis2,grandis3,higher-cobordism-TQFT,steinebrunner}.
Following this line of thought, we will show how Frobenius algebras naturally arise from a combinatorial version of the plus construction.
Another more resent area is applied category theory where they have developed an extensive theory for constructing different cospan categories \cite{decorated-cospans, baez2019structured, courser2020open}.
We will show that the notion of a structured cospan can be used to give colored versions of properads.
In \cite{KMoManin}, they introduce an nc-plus construction as an intermediate definition for the plus construction itself.
We will show the relation between this nc-plus construction and mergers.
In \cite{KMoManin}, they show that the category of spans is a hereditary unique factorization category.
This implies that $\Span^+$ is a Feynman category, so it encodes a sort of operad-like structure.
Despite having some interesting properties, the gadgets corepresented by $\Span^+$ have not been explored to the same extent as the gadgets corepresented by $\Cospan^+$.
To better understand the algebraic significance of this structure, we will briefly survey some of the combinatorial and categorical properties of $\Span$.
We will then end by describing the relation of $\Span$ and $\Span^+$ to bialgebras.
\subsection*{Acknowledgments}
We would like to thank Ralph Kaufmann for his continued support.
We would also like to thank Philip Hackney and Jan Steinebrunner for helpful discussions on this topic.
\section{Plus constructions, Feynman categories, and UFCs}
In \cite{categories-seriously}, Lawvere advocates taking descriptions of mathematical objects as categories ``seriously''.
For example, a group $G$ canonically determines a category $\Sigma G$ with one object and $G$ as a set of morphisms.
Applying this attitude to $\Sigma G$, one recovers many classical ideas such as representations, intertwining operators, induced representations, and Frobenius reciprocity as special cases of different categorical constructions.
Because of this, it is common to blur the distinction between these concepts.
However, it can be useful to keep this distinction to separate the datum describing the structure (the set $G$ with a binary operation and certain properties) from the structure itself (the category $\Sigma G$).
These sorts of distinctions are especially important in the study of operad-like structures.
In \cite{feynman}, Kaufmann and Ward introduced a special type of monoidal category called a Feynman category to encode different ``types'' of operad-like structure.
In their formalism, a strong monoidal functor $\O: \F \to \C$ encodes an operad-like structure of ``type $\F$'' in a category $\C$.
For Feynman categories that come from a plus construction, a strong monoidal functor $\F^+ \to \C$ canonically determines a category $\F_{\O}$ by a so-called indexed enrichment.
This process of index enrichment plays a key role in describing algebras and the theory of twists.
In \cite{KMoManin}, they extended this by showing that the notion of a plus construction applies more broadly to what are called unique factorization categories.
In this section, we will briefly recall the definitions of unique factorization categories and the plus construction.
\subsection{Unique factorization categories}
\begin{df}
\cite{KMoManin}
A (symmetric) monoidal category $(\M,\ot)$ has {\em essentially uniquely factorizable objects}, if there is a groupoid $\V$ of basic objects together with a functor $\imath: \V \to \M$, for which $\imath^\boxtimes$ induces an equivalence.
\begin{equation}
\label{objectcond}
\imath^\boxtimes: \V^\boxtimes\stackrel{\sim}{\to}\Iso(\M)
\end{equation}
A choice of such a pair $(\V,\imath)$ will be called a {\em basis of objects} and its elements will be called {\em irreducibles} or {\em basic objects}.
\end{df}
\begin{definition}
\cite{KMoManin}
Let $\M$ be a symmetric monoidal category equipped with a groupoid $\Indec$ and a functor $\jmath: \Indec \to \Iso(\M\downarrow \M)$.
Using the monoidal structure of $\M$, these induce the following functor.
\begin{equation}
\jmath^{\boxtimes}: \Indec^{\boxtimes} \to \Iso(\M \downarrow \M)
\end{equation}
We say $\M$ has {\em essentially uniquely factorizable morphisms} if this induces an equivalence.
A choice of such a pair $(\Indec,\jmath)$ will be called a {\em basis of morphisms} and its elements will be called {\em irreducibles} or {\em basic morphisms}.
\end{definition}
\begin{definition}
\cite{KMoManin}
Let $\M$ be a symmetric monoidal category with essentially small slice categories, then
\begin{enumerate}
\item We say $\M$ is a \emph{unique factorization category} (UFC) if it has uniquely factorizable morphisms together with a choice of basis $(\M,\Indec,\jmath)$.
\item Moreover, $\M$ is a \emph{Feynman category} if it is equipped with a choice of basic objects $(\V,\imath)$ such that $\P = \Iso(\F\downarrow \V)$ and $\jmath = (id_{\F},id_{\F},\imath)$ is a compatible choice of basic morphisms $\P$ making $\F$ into a unique factorization category.
\end{enumerate}
\end{definition}
\begin{dfprop}
\cite{KMoManin}
A basis for morphisms $(\Indec,\jmath)$ is \emph{hereditary} if for every pair of composable morphisms $(\phi_0,\phi_1)$, with $\phi_1 \circ \phi_0=\phi$, and decomposition into irreducible morphisms
\begin{equation}
\phi_0\simeq \bigotimes_{v\in V}\phi_{0,v}, \quad \phi_1\simeq \bigotimes_{w\in W} \phi_{1,w}, \text{ and }\phi=\bigotimes_{u\in U}\phi_u
\end{equation}
there exists a partition of $V\amalg W=\amalg_{u\in U} P_u$ indexed by $U$, such that for each $u\in U$ there is a decomposition pair $(\phi_{0,u},\phi_{1,u})$ of the $\phi_u$, viz.\ $\phi_{1,u} \phi_{0,u}=\phi_u$, such that
\begin{equation}
\label{eq:hereditarycond}
\phi_{0,u} \simeq \bigotimes_{v\in P_u\cap V}\phi_{0,v} \text{ and } \phi_{1,u} \simeq \bigotimes_{w\in P_u\cap W}\phi_{1,w}
\end{equation}
A unique factorization category is a {\em hereditary UFC} if its basis is hereditary.
\end{dfprop}
\begin{rmk}
Unique factorization categories are versatile structures which admit many descriptions.
In \cite{KMoManin}, these conditions were equivalently formulated as right Ore conditions.
In \cite[Proposition 6.31]{KMoManin}, they also show that a hereditary UFC $\M$ naturally determines an indexing functor $\M \to \Cospan$.
Categories equipped indexing functors that satisfy the appropriate conditions play an important role in the work of Steinebrunner in \cite{steinebrunner} where they go by the name \emph{labeled cospan categories}.
Moreover, Hackney and Beardsley \cite{labeled-cospan-properad} describe a connection between these labeled cospan categories and Segal presheaves of a category of level graphs.
\end{rmk}
\subsection{Plus constructions of categories}
\begin{definition}
\cite{KMoManin}
Given a category $\C$, define $\C^{\boxtimes +}$ so that
\begin{enumerate}
\item The objects are words $\phi_1 \boxtimes \ldots \boxtimes \phi_n$ of morphisms $\phi_i \in \C$.
\item The morphisms are generated by two types of basic morphisms:
\begin{description}
\item[Isomorpisms] are words $(\sigma_1\Downarrow\sigma'_1) \boxtimes \ldots \boxtimes (\sigma_n\Downarrow\sigma'_n)$.
\item[$\gamma$-morphisms] for every composable pair $(\phi_1,\phi_0)$ there is a generator
\begin{equation}
\gamma_{\phi_1,\phi_0}: \phi_1 \boxtimes \phi_0 \to \phi_1 \circ \phi_0
\end{equation}
\end{description}
\item There are several relations including the typical ones like associativity, identities, and interchange as well as some new ones like equivariance with respect to isomorphisms.
See \cite{KMoManin} for the details.
\end{enumerate}
\end{definition}
\begin{definition}
\cite{KMoManin}
In the case where $\C = \M$ is a monoidal category, there is a refinement:
\begin{enumerate}
\item The \emph{nc-plus construction} $\M^{nc +}$ is obtained from $\M^{\boxtimes +}$ by adjoining new generators $\mu_{\phi_0,\phi_1}: \phi_1 \boxtimes \phi_2 \to \phi_1 \ot \phi_2$ and imposing new relations.
\item The \emph{(localized) plus construction} $\M^{loc+}$ is defined to be the localization of $\mu$.
That is, we add the morphism $\mu^{-1}$ and mod out by $\mu \circ \mu^{-1} = \mu^{-1} \circ \mu = \id$.
We will often refer to this simply as \emph{the plus construction} and denote it as $\M^+$.
\end{enumerate}
\end{definition}
\begin{rmk}
In principle, these localizations can be difficult to compute.
However when $\M$ is a hereditary unique factorization category, \cite{KMoManin} defines a plus construction $\M^{+}$ which is a monoidally equivalent to $\M^{+loc}$.
As a consequence of the characterization of the heredity property as a right Ore condition, there is a right roof calculus available making the computations tractable.
Since we only work with hereditary UFCs, we wont make a distinction between $\M^{+loc}$ and $\M^{+}$ and we will refer to both of them as ``the plus construction''.
\end{rmk}
\begin{prop}\cite{KMoManin}
The plus construction of a hereditary unique factorization category is a Feynman category.
\end{prop}
\section{Trivial Feynman category}
Define the \emph{trivial category} $1$ to be the category with a single object $\ast$ and a single identity morphism.
Define the \emph{trivial Feynman category} $\FF^{triv}$ so that $\F = 1^{\boxtimes}$ and $\V = 1$.
In words, the objects of $\FF^{triv}$ are strings $\ast^{\boxtimes n}$ and the morphisms are the commutativity constraints.
\subsection{Monoids}
For a Feynman category $\FF = (\F, \V, \imath)$, a strong monoidal functor $\O: \F \to \C$ is called an $\F$-op in $\C$.
The name is supposed to evoke the idea that $\O$ is an operad-like structure of ``$\F$-type''.
In our particular case, an op of the trivial Feynman category is a strong monoidal functor $\ast^{\boxtimes n} \to \C$.
This is just a choice of an object in $\C$.
Despite the simplicity of their ops, the trivial Feynman category is interesting since a lot theory can be ``boot-strapped'' from this simple Feynman category.
For a full explanation of this idea, we refer the reader to \cite{feynmanrep}.
For our purposes, we single-out the following fact:
\begin{prop}
\cite{feynmanrep}
As a combinatorial object, $(\FF^{triv})^+$ is equivalent to the monoidal category $FS^{>}$ of surjections with ordered fibers.
Moreover $(\FF^{triv})^+$ corepresents monoids as a Feynman category.
\end{prop}
\begin{example}
To understand $(\FF^{triv})^+$ as a combinatorial category, it is helpful to think of it in terms of the diagrams of \cite{feynmanrep}.
In their depiction, an object of $(\FF^{triv})^+$ is equivalent to a $n$-length string of $\id_{\ast}$ and a morphisms is a stacking of these letters.
\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node [style=Black] (0) at (-3.75, -1.25) {};
\node [style=none] (1) at (-3.75, -0.75) {};
\node [style=none] (2) at (-3.75, -1.75) {};
\node [style=Black] (3) at (-3.25, -1.25) {};
\node [style=none] (4) at (-3.25, -0.75) {};
\node [style=none] (5) at (-3.25, -1.75) {};
\node [style=Black] (6) at (-2.75, -1.25) {};
\node [style=none] (7) at (-2.75, -0.75) {};
\node [style=none] (8) at (-2.75, -1.75) {};
\node [style=Black] (9) at (-2.25, -1.25) {};
\node [style=none] (10) at (-2.25, -0.75) {};
\node [style=none] (11) at (-2.25, -1.75) {};
\node [style=Black] (12) at (-1.75, -1.25) {};
\node [style=none] (13) at (-1.75, -0.75) {};
\node [style=none] (14) at (-1.75, -1.75) {};
\node [style=Black] (15) at (-1.25, -1.25) {};
\node [style=none] (16) at (-1.25, -0.75) {};
\node [style=none] (17) at (-1.25, -1.75) {};
\node [style=none] (18) at (-3.75, -2.25) {$1$};
\node [style=none] (19) at (-3.25, -2.25) {$2$};
\node [style=none] (20) at (-2.75, -2.25) {$3$};
\node [style=none] (21) at (-2.25, -2.25) {$4$};
\node [style=none] (22) at (-1.75, -2.25) {$5$};
\node [style=none] (23) at (-1.25, -2.25) {$6$};
\node [style=Black] (24) at (1.25, 0.25) {};
\node [style=Black] (27) at (2.5, 0.25) {};
\node [style=none] (28) at (2.5, 0.75) {};
\node [style=Black] (30) at (0, -0.25) {};
\node [style=none] (31) at (0, 0.25) {};
\node [style=none] (32) at (0, -0.75) {};
\node [style=Black] (33) at (1.25, 0.75) {};
\node [style=none] (34) at (1.25, 1.25) {};
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\node [style=none] (38) at (1.25, -0.75) {};
\node [style=Black] (39) at (2.5, -0.25) {};
\node [style=none] (41) at (2.5, -0.75) {};
\node [style=none] (42) at (1.75, 0.25) {$1$};
\node [style=none] (43) at (3, 0.25) {$2$};
\node [style=none] (44) at (0.5, -0.25) {$3$};
\node [style=none] (45) at (1.75, 0.75) {$4$};
\node [style=none] (46) at (1.75, -0.25) {$5$};
\node [style=none] (47) at (3, -0.25) {$6$};
\node [style=none] (48) at (-0.5, -1.25) {};
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\node [style=none] (58) at (5.25, -1.75) {};
\draw (1.center) to (0);
\draw (0) to (2.center);
\draw (4.center) to (3);
\draw (3) to (5.center);
\draw (7.center) to (6);
\draw (6) to (8.center);
\draw (10.center) to (9);
\draw (9) to (11.center);
\draw (13.center) to (12);
\draw (12) to (14.center);
\draw (16.center) to (15);
\draw (15) to (17.center);
\draw (28.center) to (27);
\draw (31.center) to (30);
\draw (30) to (32.center);
\draw (34.center) to (33);
\draw (36) to (38.center);
\draw (39) to (41.center);
\draw (33) to (24);
\draw (24) to (36);
\draw (27) to (39);
\draw [style=Arrow] (48.center) to (49.center);
\draw (51.center) to (50);
\draw (50) to (52.center);
\draw (54.center) to (53);
\draw (53) to (55.center);
\draw (57.center) to (56);
\draw (56) to (58.center);
\end{tikzpicture}
\end{equation}
\end{example}
\subsection{Graded monoids}
If we acknowledge the importance of the trivial Feynman category for the plus construction, then it is natural to consider the nc-plus construction of this category as well.
First, we observe that the category $(\FF^{triv})^{nc+}$ has a basic object $\id_{\ast^{\boxtimes n}}$ for each natural number $n$, hence any $(\FF^{triv})^{nc+}$-op will naturally involve some sort of a grading.
To fix notation, we denote a graded object by $A = \{A_i\}_{i \in \N}$.
When finite coproducts exist, there is a canonical tensor product of graded objects in $\C$:
\begin{equation}
(A \bullet B)_n = \coprod_{n=p+q} A_p \otimes B_q
\end{equation}
The monoidal unit for this product is then
\begin{equation}
(1_{Gr})_n =
\begin{cases}
1_{\C}, & n=0 \\ 0_{\C}, & n > 0
\end{cases}
\end{equation}
Note that a pointing $\eta: 1_{Gr} \to A$ in graded $\C$-objects amounts to a pointing $1_{\C} \to A_0$ in $\C$.
Hence the unit conditions for $\eta$ say that the following is an isomorphism:
\[ A_n \to 1_{\C} \otimes A_n \to A_0 \otimes A_n \to A_n \]
\begin{prop}
The $(\F^{triv})^{nc+}$-ops in $\C$ are monoid objects in the category of symmetric sequences of $\C$-monoids.
\end{prop}
\begin{proof}
Fix a strong monoidal functor $\O: (\F^{triv})^{nc+} \to \C$.
Define an $\SS_n$-module $M_{\O}(n) = \O(1^{\ot n})$ with the action $\SS_n \to Aut(M_{\O})$.
All together, this forms an $\SS$-module $M_{\O} = \{M_{\O}(n)\}_{n \in \N}$.
The image of $\O$ on the morphism $\gamma: 1^{\ot n} \boxtimes 1^{\ot n} \to 1^{\ot n}$ is a morphism $M_{\O}(n) \otimes M_{\O}(n) \to M_{\O}(n)$ which makes each $M_{\O}(n)$ into a monoid.
The image of $\O$ on the $\mu$-morphism $\mu: 1^{\ot n} \boxtimes 1^{\ot m} \to 1^{\ot (n+m)}$ is a map $M_{\O}(n) \otimes M_{\O}(m) \to M_{\O}(n+m)$.
These assemble into a morphism $M_{\O} \bullet M_{\O} \to M_{\O}$.
Hence we know that $M_{\O} = \{M_{\O}(n)\}_{n \in \N}$ is at least a monoid object in the category of $\N$-graded sets.
The interchange relation for the plus construction implies that the following diagram commutes:
\begin{equation}
\begin{tikzcd}
M_{\O}(n) \otimes M_{\O}(m) \otimes M_{\O}(n) \otimes M_{\O}(m) \arrow[d] \arrow[r] & M_{\O}(n+m) \otimes M_{\O}(n+m) \arrow[d] \\
M_{\O}(n) \otimes M_{\O}(m) \arrow[r] & M_{\O}(n+m)
\end{tikzcd}
\end{equation}
Therefore $M_{\O} \bullet M_{\O} \to M_{\O}$ respects the monoid structure of $M_{\O}$ making it into a monoid objects in the category of symmetric sequences of $\C$-monoids.
\end{proof}
\begin{cor}
Any monoid (an $(\FF^{triv})^+$-op) pulls-back to an monoid object in the category of symmetric monoids (an $(\FF^{triv})^{nc+}$-op).
\end{cor}
\begin{proof}
Abstractly, this is just a pullback of the quotient functor $(\FF^{triv})^{nc+} \to (\FF^{triv})^+$.
Concretely, given a monoid $M$, we define the graded object $A = \{A_i\}_{i \in \N}$ by $A_i = M^{\otimes i}$.
Define $\mu: A_p \otimes A_q \to A_{p+q}$ to be concatenation.
To define $\gamma: A_n \otimes A_n \to A_n$, write the multiplication of $M$ as $m: M \otimes M \to M$ and let $C$ be the commutativity constraint associated to the following permutation:
\begin{equation}
\begin{pmatrix}
1 & 1+n & \ldots & n & n+n \\
1 & 2 & \ldots & 2n-1 & 2n
\end{pmatrix}
\end{equation}
Then $\gamma: A_n \otimes A_n \to A_n$ is the following composition:
\begin{equation*}
A_n \otimes A_n
\overset{C}{\to}
(A \otimes A)^{\otimes n}
\overset{m^{\otimes n}}{\to}
A^{\otimes n} = A_n
\qedhere
\end{equation*}
\end{proof}
\section{Finite Sets}
The category $\FinSet$ of finite sets is a Feynman category where the basic objects are singleton sets and any map $f: X \to Y$ can be factored as a collection of maps $\{f^{-1}(y) \to \{y\}\}_{y \in Y}$.
We will see a connection to operads and make a new connection to operads with multiplication.
We will also point out a structural similarity to Young tableaux which we think is interesting.
\subsection{Operads}
We can think of the plus construction as ascending upwards in some algebraic hierarchy.
For example, we have seen that ``above'' objects ($\FF^{triv}$-ops), there are monoids ($(\FF^{triv})^+$-ops).
Similarly ``above'' commutative monoids ($\FinSet$-ops), there are operads ($\FinSet^+$-ops).
\begin{prop}
\cite{feynmanrep}
The category $\FinSet$ of finite sets corepresents commutative monoids.
\end{prop}
\begin{prop}
\cite{feynmanrep}
Combinatorially, $\FinSet^+$ is equivalent to a Borisov--Manin category of graphs whose objects are rooted corollas and the generating morphisms have level trees as ghost graphs.
Moreover, they corepresent operads as Feynman categories.
\end{prop}
\begin{example}
Let $\phi_n$ denote some $n$-to-$1$ map in $\FinSet$.
Then the morphisms $\gamma_{\phi_2, \phi_1 \amalg \phi_3} \to \phi_4$ in $\FinSet^+$ can be identified with a morphism in a Borisov--Manin category:
\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node [style=White] (0) at (-4.5, 0) {};
\node [style=White] (1) at (-3.5, 0) {};
\node [style=White] (2) at (-2.25, 0) {};
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\node [style=none] (6) at (-2.75, 0.5) {};
\node [style=none] (7) at (-2.25, 0.5) {};
\node [style=none] (8) at (-1.75, 0.5) {};
\node [style=White, inner sep = 0.6mm] (9) at (-0.5, 0.75) {};
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\node [style=none] (28) at (1.5, -0.5) {};
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\draw (3.center) to (0);
\draw (4.center) to (0);
\draw (5.center) to (1);
\draw (6.center) to (2);
\draw (7.center) to (2);
\draw (8.center) to (2);
\draw (14.center) to (10);
\draw (15.center) to (11);
\draw (16.center) to (11);
\draw (17.center) to (11);
\draw (10) to (9);
\draw (11) to (9);
\draw (19.center) to (18);
\draw (20.center) to (18);
\draw (21.center) to (18);
\draw (22.center) to (18);
\draw [style=Arrow] (23.center) to (24.center);
\draw (0) to (25.center);
\draw (1) to (26.center);
\draw (2) to (27.center);
\draw (18) to (28.center);
\draw (9) to (29.center);
\end{tikzpicture}
\end{equation}
The image of $\gamma_{\phi_1 \amalg \phi_3, \phi_2} \to \phi_4$ under a strong monoidal functor $\O: \FinSet^+ \to \C$ is the same thing as an operadic composition $\O(2) \otimes \O(1) \otimes \O(3) \to \O(4)$.
\end{example}
\begin{prop}
\cite{feynmanrep}
Combinatorially, $(\FinSet^{<})^+$ is equivalent to a (decorated) Borisov--Manin category of graphs whose objects are planar rooted corollas and the generating morphisms have level trees as ghost graphs.
Moreover, they corepresent non-symmetric operads as Feynman categories.
\end{prop}
\subsection{Operads with multiplication}
\label{sec:operad-mult}
We will show that we can obtain a Feynman category which corepresents operads with multiplication by using a slight modification on the plus construction.
Rather than shift the focus from a particular example to a general construction, we will simply introduce this modification as an ad hoc construction.
\begin{prop}
Define $\operads_{\pi}$ by starting with $\FinSet^+$ and allowing words morphisms of the form $(\sigma_1 \Downarrow \pi_1) \boxtimes \ldots \boxtimes (\sigma_n\Downarrow \pi_n)$ where $\sigma_i$ are isomorphisms and $\pi_i$ are surjections.
Then $\operads_\pi$ is a Feynman category that corepresents operads with multiplication.
\end{prop}
\begin{proof}
The added actions $(\sigma \Downarrow \pi)$ can be factored as $(\sigma_1 \Downarrow t_1) \otimes \ldots \otimes (\sigma_n \Downarrow t_n)$ where $t_i$ are morphisms with singletons in the target.
The source of $(\sigma_n \Downarrow t_n)$ is an aggregate of corollas and the target is a single corolla.
Hence the new morphisms meet the necessary conditions, so $\operads_{\pi}$ is indeed a Feynman category.
Let $\O: \operads_\pi \to \C$ be a strong monoidal functor.
Since $\operads \simeq \FinSet^+$ is present as a subcategory, we still have the $\SS_n$-actions and operadic compositions:
\begin{equation}
\begin{tikzcd}
\O(\phi_{n_1}) \otimes \ldots \otimes \O(\phi_{n_k}) \otimes \O(\phi_k) \arrow[r, "\gamma"] & \O\left(\phi_{\sum_i n_i} \right)
\end{tikzcd}
\end{equation}
On the other hand, the new actions introduce a multiplication:
\begin{equation}
\begin{tikzcd}
\O(\phi_n) \otimes \O(\phi_m) \arrow[r, "\sim"] & \O(\phi_n \amalg \phi_m) \arrow[r, "\O(Id \Downarrow \pi)"] & \O(\phi_{n+m})
\end{tikzcd}
\end{equation}
Therefore $\O$ is an operad with multiplication.
\end{proof}
\begin{cor}
Pulling back along the inclusion $\iota: \FinSet^+ \to \operads_{\pi}$ forgets the multiplication structure.
\qed
\end{cor}
\begin{example}
For ease of notion, let $F$ denote the skeletal category whose objects are sets $\underline n = \{1, \ldots, n\}$ and whose morphisms are functions.
Then define $\operads_{\pi}^{skel}$ in a similar manner by starting with $F^+$ and allowing words of the form $(\sigma_1 \Downarrow \pi_1) \boxtimes \ldots \boxtimes (\sigma_n\Downarrow \pi_n)$.
Now, given an operad $\O: F^+ \to \Vect$, define a strong monoidal functor $\O^{nc}: \operads_{\pi}^{skel} \to \Vect$ on basic objects by
\begin{equation}
\O^{nc}(n) = \bigoplus_{n = \sum_{i=1}^k n_i}
\O(n_1) \otimes \ldots \otimes \O(n_k)
\end{equation}
\begin{enumerate}
\item The operadic composition in $\O^{nc}$ is given by summing over all possible operadic compositions in $\O$.
\item The multiplication is defined by commutativity of finite colimits and tensors followed by inclusion:
\begin{equation}
\begin{tikzcd}
\left(
\displaystyle \bigoplus_{n = \sum_{i=1}^N n_i}
\O(n_1) \otimes \ldots \otimes \O(n_N)
\right)
\otimes
\left(
\displaystyle \bigoplus_{m = \sum_{j=1}^M m_j}
\O(m_1) \otimes \ldots \otimes \O(m_M)
\right)
\arrow[d, "\sim"] \\
\displaystyle \bigoplus_{n = \sum_{i=1}^N n_i}
\displaystyle \bigoplus_{m = \sum_{j=1}^M m_j}
\O(n_1) \otimes \ldots \otimes \O(n_N)
\otimes
\O(m_1) \otimes \ldots \otimes \O(m_M)
\arrow[d, tail] \\
\displaystyle \bigoplus_{m+n = \sum_{k=1}^P p_k}
\O(p_1) \otimes \ldots \otimes \O(p_P)
\end{tikzcd}
\end{equation}
\end{enumerate}
\end{example}
\subsection{Young tableaux}
We can think of a surjection $p: E \ta B$ as a $B$-indexed partition of a set $E$ by taking fibers $\{ p^{-1}(b) : b \in B \}$.
In the representation theory of symmetric groups, partitions are encoded by Young tableaux/tabloids.
In this section, we look at this structure from the point of view of the plus construction.
To match established conventions, we will use the skeletal category $FS$ whose objects are sets $\underline n = \{1, \ldots, n\}$ and whose morphisms are surjections.
Then we can think of the objects of $FS^{nc+}$ as Young tabloids, see Figure~\ref{fig:tabloid}.
Permuting the fibers leaves the function unchanged.
This corresponds to the fact that the rows are unordered in a tabloid.
Typically, we draw the tableaux so that the width decreases from top to bottom.
However, each object in $FS^{nc+}$ is isomorphic to a tableaux with this property.
Similarly, we can consider ordered surjections, then $(FS^{<})^{nc+}$ corresponds to Young tableaux, see Figure~\ref{fig:tabluex}.
If we think of $\gamma: t \boxtimes a \to t \circ a$ as an operation $a$ acting on a tableaux/tabloid $t$, then the $\gamma$-morphisms correspond to relabeling or adding an entry to a column.
Similarly, the $\mu$-morphisms correspond to adding a new rows.
\begin{figure}
\centering
\begin{subfigure}[c]{0.3\textwidth}
\ytableausetup{boxsize=normal, tabloids}
\ytableaushort{
48,672,53,1
}
\end{subfigure}
\begin{subfigure}[c]{0.4\textwidth}
\begin{tikzpicture}
\node [style=none] (0) at (1.75, 0.5) {$1$};
\node [style=none] (1) at (-0.75, 0.5) {$2$};
\node [style=none] (2) at (0.75, 0.5) {$3$};
\node [style=none] (3) at (-1.75, 0.5) {$4$};
\node [style=none] (4) at (1.25, 0.5) {$5$};
\node [style=none] (5) at (-0.25, 0.5) {$6$};
\node [style=none] (6) at (0.25, 0.5) {$7$};
\node [style=none] (7) at (-1.25, 0.5) {$8$};
\node [style=none] (8) at (-1.5, -0.5) {$1$};
\node [style=none] (9) at (-0.25, -0.5) {$2$};
\node [style=none] (10) at (1, -0.5) {$3$};
\node [style=none] (11) at (1.75, -0.5) {$4$};
\draw (3) to (8);
\draw (7) to (8);
\draw (5) to (9);
\draw (6) to (9);
\draw (1) to (9);
\draw (2) to (10);
\draw (0) to (11);
\draw (4) to (10);
\end{tikzpicture}
\end{subfigure}
\caption{Young tabloids correspond to surjections with \emph{unordered} fibers.}
\label{fig:tabloid}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[c]{0.3\textwidth}
\ytableausetup{notabloids}
\begin{ytableau}
4 & 8 \\
6 & 7 & 2 \\
5 & 3 \\
1
\end{ytableau}
\end{subfigure}
\begin{subfigure}[c]{0.4\textwidth}
\begin{tikzpicture}
\node [style=none] (0) at (1.75, 0.5) {$1$};
\node [style=none] (1) at (-0.75, 0.5) {$6$};
\node [style=none] (2) at (1.25, 0.5) {$3$};
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\node [style=none] (13) at (-1.25, 1) {};
\node [style=none] (14) at (-0.75, 1) {};
\node [style=none] (15) at (0.25, 1) {};
\node [style=none] (16) at (0.75, 1) {};
\node [style=none] (17) at (1.25, 1) {};
\draw (3) to (8);
\draw (7) to (8);
\draw (5) to (9);
\draw (6) to (9);
\draw (1) to (9);
\draw (2) to (10);
\draw (0) to (11);
\draw (4) to (10);
\draw [style=Arrow] (12.center) to (13.center);
\draw [style=Arrow] (14.center) to (15.center);
\draw [style=Arrow] (16.center) to (17.center);
\end{tikzpicture}
\end{subfigure}
\caption{Young tableaux correspond to surjections with \emph{ordered} fibers.}
\label{fig:tabluex}
\end{figure}
The branching rules for representations of the symmetric group are one place where the operations of adding columns and rows appears naturally.
However, there is one important difference.
In this application, these operations occur as linear maps of the form $S(\gamma): S(t) \to S(t \circ a)$ instead of linear maps $S(\gamma): S(t) \otimes S(a) \to S(t \circ a)$.
We will describe the appropriate modification.
Borrowing the idea of Section~\ref{sec:operad-mult}, we can define an appropriate ``branching category'' by allowing additional actions.
We will describe a quick way of doing this using the element category, which is very closely related to the plus construction.
First, define $B(n,m)$ to be the set of surjections $\underline{n} \ta \underline{m}$.
We make this into a functor $B: FS^{op} \times FI \to \Set$ by letting the category $FS$ of finite surjections act by pre-composition and letting the category $FI$ of finite injections act by post-composition.
We can then think of $\el(B)$ as the category whose objects are tabloids and whose morphisms are the branching operations.
\section{Cospans}
The next combinatorial category is Cospans.
This is our first example of a unique factorization category which is not a Feynman category.
We will show the connection to graphs, properads, and Frobenius algebras.
\begin{definition}
Define the category $\Cospan$ so that:
\begin{enumerate}
\item The objects are finite sets.
\item The morphisms are diagrams $S \rightarrow V \leftarrow T$ called \emph{cospans} modulo isomorphisms in the middle:
\begin{equation}
\begin{tikzcd}[row sep = small]
& V \arrow[dd, "\sim"] & \\
S \arrow[ru] \arrow[rd] & & T \arrow[lu] \arrow[ld] \\
& V' &
\end{tikzcd}
\end{equation}
\item Define composition by taking a pushout in the middle of a pair of cospans:
\begin{equation}
\begin{tikzcd}[row sep = small]
&& {V \mathbin{_b\amalg_c} V'} \\
& V && {V'} \\
X && Y && Z
\arrow["a", from=3-1, to=2-2]
\arrow["b"', from=3-3, to=2-2]
\arrow["{i_b}", dashed, from=2-2, to=1-3]
\arrow["{i_c}"', dashed, from=2-4, to=1-3]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-45}, draw=none, from=1-3, to=3-3]
\arrow["c", from=3-3, to=2-4]
\arrow["d"', from=3-5, to=2-4]
\end{tikzcd}
\end{equation}
\end{enumerate}
\end{definition}
\subsection{Zero-to-zero morphisms}
Although they are generally harmless, one has to decide how to handle zero-to-zero morphisms since there are a few reasonable options.
\subsubsection{Lone morphisms}
One option is to accept them and modify the definition of a UFC slightly.
Consider $(\N, +)$ as a discrete monoidal category.
We can augment the definition of a UFC with an injection $L \rightarrowtail \Iso(1_{\M} \downarrow 1_{\M})$.
Using the monoidal structure of $\M$ and the inclusion $\Iso(1_{\M} \downarrow 1_{\M}) \hookrightarrow \Iso(\M \downarrow \M)$, these induce the following functor.
\begin{equation}
\jmath^{\boxtimes} \times l: \Indec^{\boxtimes} \times \N^L \to \Iso(\M \downarrow \M)
\end{equation}
We can think of the image of $L \rightarrowtail \Iso(1_{\M} \downarrow 1_{\M})$ as the irreducible \emph{lone morphisms}.
By the Eckmann--Hilton argument, the only way these lone morphisms can compose is by accumulating.
\begin{example}
\label{ex:cospan-Eckmann-Hilton}
In $\Cospan$, the pushout of $\{*\}\leftarrow \varnothing\rightarrow \{*\}$ is the two--point set $\{*\} \amalg \{*\}$.
\end{example}
\begin{prop}
\cite{KMoManin}
Cospan is a hereditary unique factorization category in this extended sense with $L = \{ \ast \}$ a singleton set.
\end{prop}
\subsubsection{Restrictions}
Perhaps the simplest option is to just prevent the zero-to-zero morphisms from occurring.
\begin{enumerate}
\item Define the \emph{non--unital cospans} ${}'\Cospan$ to be the subcategory where $S=\varnothing$ implies $V=T=\varnothing$
\item Define the \emph{non--co--unital cospans} $\Cospan'$ to be the subcategory where $T=\varnothing$ implies $V=S=\varnothing$.
\item Define ${}'\Cospan'$ to be the subcategory where either $T=\varnothing$ or $S=\varnothing$ implies $S=V=T=\varnothing$.
\end{enumerate}
The connected cospans fail to be a subcategory of $\Cospan$ because of the phenomena described in Example~\ref{ex:cospan-Eckmann-Hilton}.
This is avoided in ${}'\Cospan$, $\Cospan'$, and ${}'\Cospan'$ since $\varnothing$ cannot be both a source and a target for morphisms with $|V|=1$.
This implies that the pushout is of $\{*\} \leftarrow S\rightarrow \{*\}$ is always $\{*\}$.
\subsubsection{Corelations}
Another option is to eliminate them whenever they occur.
This approach is common in applied situations.
\begin{definition}
Define the \emph{(first) category of corelations} $\Corel_{I}$ as follows:
\begin{enumerate}
\item The objects are sets.
\item The morphisms are isomorphism classes of jointly surjective cospans which are cospans $S \rightarrow V \leftarrow T$ such that the induced map $S \amalg T \to V$ is a surjection.
\item In general, a composition of two jointly surjective cospans might compose to some $S \amalg T \to V$ that is not jointly surjective.
However, we can restrict the codomain to get a surjection $S \amalg T \ta V'$.
We take this to be the composition in $\Corel$.
\end{enumerate}
\end{definition}
\begin{definition}
\cite{co-relation}
Let $\mathcal M$ denote the collection of injections.
Define the \emph{(second) category of corelations} $\Corel_{II}$ so that the morphisms are equivalence classes of cospans where two cospans are considered equivalent if there is a zig-zag of morphisms in $\mathcal M$ connecting them:
\begin{equation}
\begin{tikzcd}
S \arrow[r, equals] \arrow[d]
& S \arrow[r, equals] \arrow[d]
& \ldots \arrow[r, equals]
& S \arrow[r, equals] \arrow[d]
& S \arrow[d] \\
V_1 \arrow[r, "\in \mathcal M"]
& V_2
& \ldots \arrow[l, "\in \mathcal M"'] \arrow[r, "\in \mathcal M"]
& V_{n-1}
& V_n \arrow[l, "\in \mathcal M"'] \\
T \arrow[r, equals] \arrow[u]
& T \arrow[r, equals] \arrow[u]
& \ldots \arrow[r, equals]
& T \arrow[r, equals] \arrow[u]
& T \arrow[u]
\end{tikzcd}
\end{equation}
Note that we can always pick a representative without any ``lone vertices'':
\begin{equation}
\begin{tikzcd}
S \arrow[r, "l"] \arrow[d, equal]
& V
& T \arrow[l, "r"'] \arrow[d, equal] \\
S \arrow[r]
& im(l) \cup im(r) \arrow[u, hook]
& T \arrow[l]
\end{tikzcd}
\end{equation}
\end{definition}
\begin{rmk}
As observed in \cite{co-relation}, these definitions are valid in any category $\mathcal C$ with pushouts and a factorization system $(\mathcal E, \mathcal M)$ such that $\mathcal M$ is stable under pushouts.
\end{rmk}
\begin{prop}
\cite{co-relation}
$\Corel_I$ and $\Corel_{II}$ are equivalent categories.
\end{prop}
\subsection{Properads and Frobenius algebras}
To better understand the relation to properads and Frobenius algebras, we will start by considering $\Cospan$ on its own.
\subsubsection{Cospan corepresents special Frobenius algebras}
Note that a strong monoidal functor $A: \Cospan \to \C$ determines an object $A = A(pt)$, a multiplication $\mu = A(\{1, 2\} \rightarrow \{1\} \leftarrow \{1\})$, and a comultiplicaiton $\Delta = A(\{1\} \rightarrow \{1\} \leftarrow \{1,2\})$.
The description of $\Cospan$ as set maps implies that $\mu$ and $\Delta$ are commutative.
The Frobenius $N$-condition is a consequence of composing cospans by pushouts:
\begin{equation}
\begin{tikzcd}[sep = small]
\bullet \arrow[d] & & \bullet \arrow[d] & & \bullet \arrow[rdd] & & \bullet \arrow[ldd] \\
\bullet & & \bullet & & & & \\
\bullet \arrow[u] \arrow[d] & \bullet \arrow[lu] \arrow[rd] & \bullet \arrow[d] \arrow[u] & = & & \bullet & \\
\bullet & & \bullet & & & & \\
\bullet \arrow[u] & & \bullet \arrow[u] & & \bullet \arrow[ruu] & & \bullet \arrow[luu]
\end{tikzcd}
\end{equation}
However, composition by pushout automatically implies that $A \overset{\Delta}{\to} A \otimes A \overset{\mu}{\to} A$ is the identity, see Diagram~\eqref{eq:special-law}.
Hence $\Cospan$-ops have an extra property that is not guaranteed by the usual axioms for a Frobenius algebra.
Because of this extra feature, $\Cospan$ corepresents what are called \emph{special Frobenius algebras} which were first identified by Carboni and Walters in \cite{Carboni-Walters}.
If one uses corelations, one gets the \emph{extra special Frobenius algebras} as described by Coya and Fong in \cite{Extraspecial}.
\begin{equation}
\label{eq:special-law}
\begin{tikzcd}[sep = small]
& \bullet \arrow[d] & & & \bullet \arrow[dd] \\
& \bullet & & & \\
\bullet \arrow[rd] \arrow[ru] & & \bullet \arrow[ld] \arrow[lu] & = & \bullet \\
& \bullet & & & \\
& \bullet \arrow[u] & & & \bullet \arrow[uu]
\end{tikzcd}
\end{equation}
\subsubsection{Genus data}
Considering the equivalence of commutative Frobenius algebras and 2D topological quantum field theories proved by Abrams~\cite{Abrams}, we see that $\Cospan$ would need to be equipped with an extra ``genus datum'' in order to corepresent commutative Frobenius algebras.
To understand the nature of this genus datum, it is useful to recognize the following connection between cospans and the properads of Vallette \cite{Vallette}.
\begin{prop}
\cite{KMoManin}
Combinatorially, $\Cospan^+$ is equivalent to a Borisov--Manin category of graphs where the objects are directed aggregates of corollas and all morphisms except vertex mergers.
Moreover, $\Cospan^+$ corepresents properads as a Feynman category.
\end{prop}
This graphical description is convenient and nicely complements the work of Berger and Kaufmann in \cite{DDecDennis} where they give a categorical and combinatorial account of different structures and operations coming from string topology and adjacent areas.
In particular, they demonstrate that the genus datum can be understood as a strong monoidal functor $\O_{\mathrm{genus}}: (\Graphs, \amalg) \to (\Set, \times)$ that assigns a genus labeling to each vertex.
Specializing to our situation, we pull $\O_{\mathrm{genus}}$ back along $\Cospan^+ \hookrightarrow (\Graphs^{\mathrm{dir}}, \amalg) \ta (\Graphs, \amalg)$ to obtain a strong monoidal functor $\O_{\mathrm{genus}}^*: \Cospan^+ \to (\Set, \times)$.
\subsubsection{Indexed enrichments}
With the extended notion of a plus construction described in \cite{KMoManin}, indexed enrichments of Kaufmann and Ward \cite{feynman} can be adapted to unique factorization categories.
We will describe this briefly here and use it to establish the desired connection to Frobenius algebras.
\begin{definition}
\cite{feynman}
Given a strong monoidal functor $D: \M^+ \to \C$, define the (enriched) category $\M_{\O}$ as follows:
\begin{enumerate}
\item The objects are the same as the original $\M$.
\item The hom $\C$-objects are
\begin{equation}
\Hom_{\M_{\O}}(X, Y) =
\coprod_{\phi \in \Hom_{\M}(X, Y)} D(\phi)
\end{equation}
\item The composition is induced by the gamma-morphisms $D(\phi) \otimes D(\psi) \to D(\phi \circ \psi)$ of the plus construction.
\end{enumerate}
\end{definition}
\begin{example}
The index enrichment of an operad $\O: \FinSet^+ \to \C$ produces a new category $\FinSet_{\O}$ which corepresents $\O$-algebras.
For more examples, we refer the reader to \cite{feynmanrep} where it is used extensively.
\end{example}
\begin{cor}
The category $\Cospan_{\O_{\mathrm{genus}}^*}$ corepresents commutative Frobenius monoids.
\end{cor}
\begin{proof}
With the extra genus datum, the morphisms of $\Cospan_{\O_{\mathrm{genus}}^*}$ agree with the Abrams description of Frobenius algebras as 2D topological quantum field theories.
\end{proof}
\subsection{Structured cospans}
In applied category theory, there are a few constructions that allow one to equip cospans with additional structures such as the decorated cospans of Fong \cite{decorated-cospans} and the structured cospans of Baez and Courser \cite{baez2019structured,courser2020open}.
In this section, we will briefly survey structured cospans and use it to describe colored versions of cospans.
\begin{definition}
We consider the special case of a construction introduced in \cite{courser2020open} applied to cospans.
Given a \emph{foot functor} $F: \Gpd \to \C$, there is a double category $\Cospan(F)$ of \emph{structured cospans}:
\begin{enumerate}
\item The object category is the same as $\Gpd$.
\item Define the horizontal arrows to be cospans with $F$ applied to both feet:
\begin{equation}
\begin{tikzcd}
X & Y && {F(X)} & V & {F(Y)} \\
{X'} & {Y'} && {F(X')} & {V'} & {F(Y')}
\arrow["\sigma"', from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "\tau", from=1-2, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "\shortmid"{marking}, from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "\shortmid"{marking}, from=2-1, to=2-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{\sigma}"', from=1-4, to=2-4]
\arrow["{\tau}", from=1-6, to=2-6]
\arrow[from=2-4, to=2-5]
\arrow[from=2-6, to=2-5]
\arrow[from=1-4, to=1-5]
\arrow[from=1-6, to=1-5]
\arrow["\Phi", from=1-5, to=2-5]
\arrow["\Phi", shorten <=10pt, shorten >=10pt, Rightarrow, from=1, to=2]
\arrow["{:=}"{description}, Rightarrow, draw=none, from=0, to=3]
\end{tikzcd}
\end{equation}
\item $\odot$ is given by taking pushouts.
\end{enumerate}
\end{definition}
\begin{remark}
In \cite{baez2019structured}, the functor $F$ is denoted by $L$ to stand for ``left adjoint'' since the functor often is indeed a left adjoint in their work.
For us, this is generally not the case, so we drop that convention to avoid any confusion.
\end{remark}
\begin{prop}
\cite{courser2020open}
If $\Gpd$ and $\C$ are symmetric monoidal categories and $F: A \to \C$ is a strong symmetric monoidal functor, then the double category $\Cospan(F)$ becomes symmetric monoidal in a canonical way.
\end{prop}
\begin{figure}
\centering
\begin{tikzpicture}
\node [style=Red] (1) at (-4, 3) {};
\node [style=Blue] (2) at (-4, 2) {};
\node [style=Blue] (3) at (-4, 1) {};
\node [style=Red] (5) at (1, 3) {};
\node [style=Red] (6) at (1, 1) {};
\node [style=Red] (8) at (-1.5, 3) {};
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\node [style=Red] (10) at (-1.5, 1) {};
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\node [style=none] (38) at (2, 5) {};
\node [style=none] (39) at (2, 0) {};
\node [style=none] (40) at (-1.5, 4.75) {\large Composition};
\node [style=none] (41) at (4.5, 4.75) {\large Result};
\node [style=Red] (42) at (3, 3) {};
\node [style=Blue] (43) at (3, 2) {};
\node [style=Blue] (44) at (3, 1) {};
\node [style=Red] (45) at (6, 3) {};
\node [style=Red] (46) at (6, 1) {};
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\draw (13) to (8);
\draw (13) to (9);
\draw (9) to (16);
\draw (16) to (5);
\draw (6) to (16);
\draw (1) to (13);
\draw (2) to (14);
\draw (14) to (10);
\draw (3) to (14);
\draw [style=Dash] (18.center) to (21.center);
\draw [style=Dash] (20.center) to (21.center);
\draw [style=Dash] (20.center) to (19.center);
\draw [style=Dash] (19.center) to (18.center);
\draw (38.center) to (39.center);
\draw (8) to (16);
\draw (10) to (16);
\draw (42) to (47);
\draw (47) to (45);
\draw (43) to (47);
\draw (44) to (47);
\draw (46) to (47);
\end{tikzpicture}
\caption{A composition of two morphisms in the colored cospan category. Note that the middle colors are forgotten precisely because composition occurs as a pushout in $\FinSet$.}
\label{fig-colored-cospan}
\end{figure}
\begin{ex}[Cospans with colors]
\label{colored-cospans}
Let $\V$ be a discrete category with two objects $R$ (``red'') and $B$ (``blue'').
There is a canonical strong monoidal functor $clr: \V^{\otimes} \to \FinSet$ which sends an object of length $n$ to the set $\underline{n}$.
By the previous result, this defines a symmetric monoidal category $\Cospan(clr)$.
Composition is illustrated in figure~\ref{fig-colored-cospan}.
\end{ex}
\subsection{Props and mergers}
We always have a localization functor $L_{\M}: \M^{nc+} \to \M^{+}$ for any monoidal category $\M$ which sends a basic object $f_1 \boxtimes \ldots \boxtimes f_n$ to $f_1 \otimes \ldots \otimes f_n$.
If there is another functor $J: \M^{nc+} \to \M^{+}$, we can incorporate these as an extra structure on a plus construction which allows ``mergers'' of morphisms.
\begin{definition}
Given a functor $J: \M^{nc+} \to \M^{+}$, define a new category by starting with $\M^+$ then formally add a generating morphism $B_\phi: L_{\M}(\phi) \to J(\phi)$ for each basic object $\phi \in \M^{nc+}$.
We then add relations making $B_\phi$ natural in the sense that the following diagram commute for each morphism $\Phi$ of $\M^{nc+}$:
\begin{equation}
\begin{tikzcd}
L_{\M}(\phi)
\arrow[d, "B_\phi"']
\arrow[r, "L_{\M}(\Phi)"]
& L_{\M}(\psi)
\arrow[d, "B_\psi"] \\
J(\phi)
\arrow[r, "J(\Phi)"']
& J(\psi)
\end{tikzcd}
\end{equation}
\end{definition}
\begin{example}
In the category of cospans, there is a functor $J: Cospan^{nc+} \to Cospan^{+}$ which sends a basic object $S \rightarrow V \leftarrow T$ to the basic object $S \rightarrow pt \leftarrow T$.
The morphisms $B_{\phi}$ are mergers in the ordinary sense.
The naturality for isomorphisms is just equivariance.
Naturality for $\gamma: \phi_0 \boxtimes \phi_1 \to \phi_0 \circ \phi_1$ is a sort of interchange, see Figure~\ref{fig:B-interchange}.
\end{example}
\begin{figure}
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\node [style=none] (85) at (1, -2.5) {};
\node [style=White, inner sep = 0.6mm] (86) at (0.5, -2.25) {};
\node [style=none] (87) at (0, -2) {};
\node [style=none] (88) at (0.25, -2) {};
\node [style=none] (89) at (0.75, -2) {};
\node [style=none] (90) at (1, -2) {};
\node [style=none] (91) at (0, 2.5) {};
\node [style=White, inner sep = 0.6mm] (92) at (0.5, 2.75) {};
\node [style=White, inner sep = 0.6mm] (93) at (1, 2.75) {};
\node [style=White, inner sep = 0.6mm] (94) at (0, 2.75) {};
\node [style=none] (95) at (0.5, 2.5) {};
\node [style=none] (96) at (0.75, 2.5) {};
\node [style=none] (97) at (1.25, 2.5) {};
\node [style=White, inner sep = 0.6mm] (102) at (0, 3.25) {};
\node [style=White, inner sep = 0.6mm] (103) at (1, 3.25) {};
\node [style=White, inner sep = 0.6mm] (104) at (0.5, 3.25) {};
\node [style=none] (109) at (0, 3.5) {};
\node [style=none] (110) at (0.75, 3.5) {};
\node [style=none] (111) at (1.25, 3.5) {};
\node [style=none] (112) at (0.5, 3.5) {};
\draw [dashed] (6.center) to (7.center);
\draw (16.center) to (0);
\draw (0) to (8.center);
\draw (17.center) to (1);
\draw (1) to (18.center);
\draw (1) to (9.center);
\draw (19.center) to (2);
\draw (2) to (10.center);
\draw (2) to (11.center);
\draw (20.center) to (3);
\draw (3) to (12.center);
\draw (4) to (21.center);
\draw (22.center) to (4);
\draw (4) to (13.center);
\draw (4) to (14.center);
\draw (23.center) to (5);
\draw (5) to (15.center);
\draw [dashed] (30.center) to (31.center);
\draw (41.center) to (25);
\draw (25) to (42.center);
\draw (25) to (33.center);
\draw (28) to (45.center);
\draw (46.center) to (28);
\draw (28) to (37.center);
\draw (28) to (38.center);
\draw (40.center) to (25);
\draw (25) to (32.center);
\draw (25) to (34.center);
\draw (25) to (43.center);
\draw (25) to (35.center);
\draw (44.center) to (28);
\draw (36.center) to (28);
\draw (39.center) to (28);
\draw (47.center) to (28);
\draw [style=Arrow] (48.center) to (49.center);
\draw [style=Arrow] (52.center) to (53.center);
\draw [style=Arrow] (50.center) to (51.center);
\draw (61.center) to (54);
\draw (54) to (57.center);
\draw (55) to (62.center);
\draw (63.center) to (55);
\draw (55) to (58.center);
\draw (55) to (59.center);
\draw (64.center) to (55);
\draw (60.center) to (55);
\draw (66) to (72.center);
\draw (73.center) to (66);
\draw (66) to (68.center);
\draw (66) to (69.center);
\draw (74.center) to (66);
\draw (70.center) to (66);
\draw [style=Arrow] (75.center) to (76.center);
\draw (83.center) to (77);
\draw (77) to (84.center);
\draw (77) to (79.center);
\draw (82.center) to (77);
\draw (77) to (78.center);
\draw (77) to (80.center);
\draw (77) to (85.center);
\draw (77) to (81.center);
\draw (86) to (88.center);
\draw (89.center) to (86);
\draw (87.center) to (86);
\draw (90.center) to (86);
\draw (86) to (82.center);
\draw (86) to (83.center);
\draw (86) to (84.center);
\draw (86) to (85.center);
\draw (91.center) to (94);
\draw (92) to (95.center);
\draw (93) to (96.center);
\draw (93) to (97.center);
\draw (109.center) to (102);
\draw (103) to (110.center);
\draw (111.center) to (103);
\draw (112.center) to (104);
\draw (104) to (92);
\draw (112.center) to (104);
\draw (103) to (93);
\draw (103) to (92);
\draw (71.center) to (66);
\draw (66) to (67.center);
\draw (102) to (94);
\end{tikzpicture}
\caption{The naturality condition of $B$ on the gamma morphisms. The dashed line indicates a $\boxtimes$ that became an $\otimes$ after applying either the functor $L$ or $J$.}
\label{fig:B-interchange}
\end{figure}
\section{Spans}
Similar to cospans, the category $\Span$ is defined so that:
\begin{enumerate}
\item The objects are finite sets.
\item The morphisms are diagrams $S \leftarrow V \rightarrow T$ called \emph{spans} modulo isomorphisms in the middle:
\begin{equation}
\begin{tikzcd}[row sep = small]
& V \arrow[dd, "\sim"] & \\
S \arrow[ru, leftarrow] \arrow[rd, leftarrow]
& & T \arrow[lu, leftarrow] \arrow[ld, leftarrow] \\
& V' &
\end{tikzcd}
\end{equation}
\item Define composition by taking a pushout in the middle of a pair of cospans:
\begin{equation}
\begin{tikzcd}[row sep = small]
&& {V \mathbin{_b\amalg_c} V'} \\
& V && {V'} \\
X && Y && Z
\arrow["a", from=3-1, to=2-2, leftarrow]
\arrow["b"', from=3-3, to=2-2, leftarrow]
\arrow["{i_b}", dashed, from=2-2, to=1-3, leftarrow]
\arrow["{i_c}"', dashed, from=2-4, to=1-3, leftarrow]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-45}, draw=none, from=1-3, to=3-3, leftarrow]
\arrow["c", from=3-3, to=2-4, leftarrow]
\arrow["d"', from=3-5, to=2-4, leftarrow]
\end{tikzcd}
\end{equation}
\end{enumerate}
\begin{prop}
\cite{KMoManin}
$\Span$ is a hereditary unique factorization category.
\end{prop}
\subsection{Relations}
Classically, a relation between two sets $X$ and $Y$ is a subset $R \subseteq X \times Y$.
From a categorical point of view, we can think of a relation as injection $r: R \rightarrowtail X \times Y$, so by the universal property of the product these are a special type of span $X \leftarrow R \rightarrow Y$.
\begin{prop}[well-known]
Composition of this span is the same thing as a composition of the relation.
\end{prop}
\begin{proof}
The pullback $R_0 \rightarrow Y \leftarrow R_1$ can be constructed as $R_0 \amalg R_1/\sim$ where $\sim$ identifies elements that map to the same value in $Y$.
This is the same way the composition of two relations is defined.
\end{proof}
\begin{prop}
The category of relations is a hereditary unique factorization category.
\end{prop}
\begin{proof}
The category $Rel$ is a subcategory of $\Span$.
Hence the result follows straightforwardly from the fact that $\Span$ is a unique factorization category.
\end{proof}
\subsection{Graphical interpretations}
\label{sec:graph-interpretations}
Note that the two middle arrows in a composition of two spans forms a cospan.
We know that cospans factor as $\{V_y \rightarrow \{y\} \leftarrow V'_y\}_{y \in Y}$.
Hence it suffices to look at these basic cospans to understand the composition of spans.
The pullback of a basic cospan is graphically the complete graph between the sets of vertices.
For instance, the pullback of $\{a,b,c\} \rightarrow pt \leftarrow \{1, 2\}$ is the following span:
\begin{equation*}
\begin{tikzcd}[column sep = 0.1cm]
& a &&& b &&& c \\
{(a,1)} && {(a,2)} & {(b,1)} && {(b,2)} & {(c,1)} && {(c,2)} \\
&&& 1 && 2
\arrow[from=2-1, to=1-2]
\arrow[from=2-3, to=1-2]
\arrow[from=2-4, to=1-5]
\arrow[from=2-6, to=1-5]
\arrow[from=2-1, to=3-4]
\arrow[from=2-3, to=3-6]
\arrow[from=2-4, to=3-4]
\arrow[from=2-6, to=3-6]
\arrow[from=2-7, to=1-8]
\arrow[from=2-9, to=1-8]
\arrow[from=2-7, to=3-4]
\arrow[from=2-9, to=3-6]
\end{tikzcd}
\end{equation*}
Another graphical interpretation comes from thinking of a span $V_B \overset{b}{\leftarrow} E \overset{w}{\rightarrow} V_w$ as a black and white graph where the edge $e \in E$ is connected to a white vertex $w(e) \in V_W$ and a black vertex $b(e) \in V_B$.
To define a ``composition'', suppose we have the following two b/w graphs such that $V_b = V'_w$:
\begin{center}
\begin{tabular}{ccc}
$\Gamma = (V_w \leftarrow E \rightarrow V_b)$
& and
& $\Gamma' = (V'_w \leftarrow E' \rightarrow V'_b)$
\end{tabular}
\end{center}
Now define $\Gamma \wedge \Gamma'$ to be the following pullback:
\begin{equation}
\begin{tikzcd}
& & E^{\wedge} = E {}_{\del_b}{\times}{}_{\del'_w} E' \arrow[ld] \arrow[rd] & & \\
& E \arrow[ld] \arrow[rd] & & E' \arrow[ld] \arrow[rd] & \\
V_w & & V_b = V'_w & & V'_b
\end{tikzcd}
\end{equation}
This new graph has $V_w$ as its white vertices and $V'_b$ as its black vertices.
There is an edge between vertices $w \in V_w$ and $b' \in V'_b$ in the new graph if and only if there is a vertex $x \in V_b = V'_w$ with an edge between $x$ and $w$ in the first graph and an edge between $x$ and $b'$ in the second graph.
\begin{ex}
In \eqref{orion} below, we have a simplified version of the constellation Orion with edges labeled by letters and another b/w graph with edges labeled by numbers.
The black and white vertices that are matched are depicted as squares.
\begin{equation}
\label{orion}
\resizebox{0.85\textwidth}{!}{
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node [style=BlackSq] (0) at (-3.5, 2.25) {};
\node [style=BlackSq] (1) at (-4.25, 0) {};
\node [style=BlackSq] (2) at (-2.75, 0) {};
\node [style=BlackSq] (3) at (-3.5, -2.25) {};
\node [style=BlackSq] (4) at (-0.75, 2) {};
\node [style=BlackSq] (5) at (-6.25, 2.25) {};
\node [style=White] (7) at (-4.75, 1.25) {};
\node [style=White] (8) at (-2.25, 1.25) {};
\node [style=White] (9) at (-5, -2) {};
\node [style=White] (10) at (-2, -2) {};
\node [style=White] (11) at (-0.25, 0.5) {};
\node [style=White] (12) at (-1.5, 3) {};
\node [style=White] (14) at (-6.75, 3.5) {};
\node [style=none] (15) at (-6.75, 2.75) {a};
\node [style=none] (16) at (-5.75, 1.5) {b};
\node [style=none] (17) at (-4.75, 0.5) {c};
\node [style=none] (18) at (-5, -1) {d};
\node [style=none] (19) at (-4.5, -2.5) {e};
\node [style=none] (20) at (-2.5, -2.5) {f};
\node [style=none] (21) at (-2, -1) {g};
\node [style=none] (22) at (-2.25, 0.5) {h};
\node [style=none] (23) at (-1.25, 1.25) {i};
\node [style=none] (24) at (0, 1.25) {j};
\node [style=none] (25) at (-0.75, 2.75) {k};
\node [style=none] (26) at (-3, 1.5) {l};
\node [style=none] (27) at (-4, 1.5) {m};
\node [style=none] (30) at (1, 0) {$\bigwedge$};
\node [style=WhiteSq] (31) at (5.25, 2.25) {};
\node [style=WhiteSq] (32) at (4.5, 0) {};
\node [style=WhiteSq] (33) at (6, 0) {};
\node [style=WhiteSq] (34) at (5.25, -2.25) {};
\node [style=WhiteSq] (35) at (8, 2) {};
\node [style=WhiteSq] (36) at (2.5, 2.25) {};
\node [style=Black] (38) at (5.25, 0) {};
\node [style=Black] (39) at (3, 2.75) {};
\node [style=Black] (40) at (2, 1.75) {};
\node [style=Black] (42) at (5.25, 2.75) {};
\node [style=Black] (43) at (5.25, -2.75) {};
\node [style=none] (44) at (4.75, -2.5) {4};
\node [style=none] (45) at (2, 2.25) {2};
\node [style=none] (46) at (2.5, 2.75) {1};
\node [style=none] (47) at (5.75, 2.5) {6};
\node [style=none] (48) at (4.75, -0.5) {3};
\node [style=none] (49) at (5.75, -0.5) {5};
\node [style=none] (50) at (9, 0) {$=$};
\node [style=White] (57) at (12, 1.25) {};
\node [style=White] (58) at (14.5, 1.25) {};
\node [style=White] (59) at (11.75, -2) {};
\node [style=White] (60) at (14.75, -2) {};
\node [style=White] (61) at (16.5, 0.5) {};
\node [style=White] (62) at (15.25, 3) {};
\node [style=White] (63) at (10, 3.5) {};
\node [style=Black] (83) at (13.25, 0) {};
\node [style=Black] (84) at (11, 2.75) {};
\node [style=Black] (85) at (10, 1.75) {};
\node [style=Black] (86) at (13.25, 2.25) {};
\node [style=Black] (87) at (13.25, -2.25) {};
\node [style=none] (88) at (9.5, 2.5) {a2};
\node [style=none] (89) at (11, 1) {b2};
\node [style=none] (90) at (12.25, 0.5) {c3};
\node [style=none] (91) at (12, -1) {3d};
\node [style=none] (92) at (12.5, -2.5) {e4};
\node [style=none] (93) at (14.25, -2.5) {f4};
\node [style=none] (94) at (14.5, -1) {5g};
\node [style=none] (95) at (14.25, 0.5) {5h};
\node [style=none] (96) at (13.75, 1.5) {l6};
\node [style=none] (97) at (12.75, 1.5) {m6};
\node [style=none] (98) at (11.75, 2.25) {b1};
\node [style=none] (99) at (10.75, 3.5) {a1};
\draw (7) to (1);
\draw (1) to (9);
\draw (9) to (3);
\draw (3) to (10);
\draw (10) to (2);
\draw (2) to (8);
\draw (8) to (0);
\draw (0) to (7);
\draw (5) to (7);
\draw (14) to (5);
\draw (8) to (4);
\draw (12) to (4);
\draw (4) to (11);
\draw (32) to (38);
\draw (38) to (33);
\draw (39) to (36);
\draw (36) to (40);
\draw (42) to (31);
\draw (34) to (43);
\draw (63) to (85);
\draw (85) to (57);
\draw (63) to (84);
\draw (84) to (57);
\draw (57) to (86);
\draw (86) to (58);
\draw (57) to (83);
\draw (58) to (83);
\draw (83) to (59);
\draw (59) to (87);
\draw (87) to (60);
\draw (60) to (83);
\end{tikzpicture}
}
\end{equation}
The result of this composition is that the elbow in the arm holding the club gets ``cloned'', Orion's belt gets tightened, and the black vertex that is part of the shield is removed.
\end{ex}
\subsection{Matrices}
Shifting our focus to to an algebraic point of view, we can think of $\Span$ as a categorified version of a matrix.
To see this, note each span $S \leftarrow V \rightarrow T$ corresponds to a map $M: V \to S \times T$ by the universal property of the product.
The map $M$ is determined by its fibers $M_{s,t} = M^{-1}(s,t)$, so $M$ can be understood as a sort of matrix where the coordinates are sets rather than numbers.
This analogy is strengthened by the following fact, which is well-known.
\begin{prop}
Given the maps $M: V \to A \times B$ and $N: U \to B \times C$, let $M \circ N$ denote their composition as a pair of spans.
Then the coordinates of $M \circ N$ are given by a categorified matrix multiplication:
\begin{equation}
(M \circ N)_{a,c}
\cong \coprod_{b \in B} M_{a,b} \times N_{b,c}
\end{equation}
\end{prop}
\begin{proof}
Let $A \leftarrow V \rightarrow B$ be the span for $M$ and $B \leftarrow U \rightarrow C$ be the span for $N$.
Write the pullback as $V \leftarrow W \rightarrow U$.
As shorthand, we will use subscripts for preimages.
For example, $V_a$ is the preimage of $a \in A$ under the map $V \to A$.
We will also use double subscripts for intersections, so $V_{ab} = V_a \cap V_b$.
First, a straightforward pullback argument shows that the 2-by-2 square below is a pullback:
\begin{equation}
\begin{tikzcd}[sep = small]
W_{ac} \arrow[r] \arrow[d] & W_a \arrow[r] \arrow[d] & V_a \arrow[r] \arrow[d] & \{a\} \arrow[d] \\
W_c \arrow[r] \arrow[d] & W \arrow[d] \arrow[r] & V \arrow[r] \arrow[d] & A \\
U_c \arrow[d] \arrow[r] & U \arrow[d] \arrow[r] & B & \\
\{c\} \arrow[r] & C & &
\end{tikzcd}
\end{equation}
Then for each point $b \in B$, the dashed arrow is uniquely determined by the universal property of the pullback:
\begin{equation}
\label{cube}
\begin{tikzcd}[sep=tiny]
V_{ab} \times U_{bc} \arrow[dd] \arrow[rr] \arrow[rd, dashed]
& & V_{ab} \arrow[rd] \arrow[dd] & \\
& W_{ac}
\arrow[rr, crossing over]
& & V_a \arrow[dd] \\
U_{bc} \arrow[rr] \arrow[rd]
& & \{b\} \arrow[rd] & \\
& U_c \arrow[rr] \arrow[from=uu, crossing over] & & B
\end{tikzcd}
\end{equation}
Now consider the diagrams below.
Since (1) and (2) are pullbacks, the rectangle (1,2) is also a pullback.
By commutativity \eqref{cube}, rectangle (1,2) is the same as (3,4).
Since (3,4) and (4) are pullbacks, (3) is also a pullback.
\begin{equation}
\begin{tikzcd}[column sep = small]
{V_{ab} \times U_{bc}} & {V_{ab}} & {V_a} & {V_{ab} \times U_{bc}} & {W_{ac}} & {V_a} \\
{U_{bc}} & {\{b\}} & B & {U_{bc}} & {U_c} & B
\arrow[from=1-2, to=2-2]
\arrow[from=1-2, to=1-3]
\arrow[from=2-2, to=2-3]
\arrow[from=1-3, to=2-3]
\arrow[from=1-1, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow[from=1-1, to=2-1]
\arrow[from=1-4, to=2-4]
\arrow[from=2-4, to=2-5]
\arrow[from=2-5, to=2-6]
\arrow[from=1-6, to=2-6]
\arrow[from=1-5, to=2-5]
\arrow[from=1-5, to=1-6]
\arrow[dashed, from=1-4, to=1-5]
\arrow["{(1)}"{description}, draw=none, from=1-1, to=2-2]
\arrow["{(2)}"{description}, draw=none, from=1-2, to=2-3]
\arrow["{(3)}"{description}, draw=none, from=1-4, to=2-5]
\arrow["{(4)}"{description}, draw=none, from=1-5, to=2-6]
\end{tikzcd}
\end{equation}
Diagram (5) below is a pullback, hence the diagram (3,5) is a pullback for all $b \in B$.
\begin{equation}
\label{eq:3}
\begin{tikzcd}
{V_{ab} \times U_{bc}} & {U_{bc}} & {\{b\}} \\
{W_{ac}} & {U_c} & B
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=2-3]
\arrow[from=1-2, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=2-1, to=2-2]
\arrow[from=1-1, to=1-2]
\arrow["{(5)}"{description}, draw=none, from=1-2, to=2-3]
\arrow[from=1-1, to=2-1]
\arrow["{(3)}"{description}, draw=none, from=1-1, to=2-2]
\end{tikzcd}
\end{equation}
By pullback stability, the following diagram is also a pullback.
\begin{equation}
\begin{tikzcd}
\coprod_{b \in B}V_{ab} \times U_{bc} \arrow[d] \arrow[r] & \coprod_{b \in B}\{b\} \arrow[d] \\
W_{ac} \arrow[r] & B
\end{tikzcd}
\end{equation}
Since pullbacks preserve isomorphisms, we get $W_{ac} \cong \coprod_{b \in B}V_{ab} \times U_{bc}$, as desired.
\end{proof}
\begin{cor}
Thinking about $\Span^{+}$ in terms of matrices:
\begin{center}
\begin{tabular}{rcl}
irreducible objects & are & irreducible matrices \\
$\mu$-morphisms & are & direct sums of block matrices \\
$\gamma$-morphisms & are & compositions of matrices
\end{tabular}
\end{center}
\end{cor}
\subsection{Bialgebras}
It is well-known that $\Span$ is related to commutative bimonoids.
The standard relation $\Delta \circ \mu = (\mu \otimes \mu) \circ (\id \otimes C \otimes \id) \circ (\Delta \otimes \Delta)$ can be understood as a special case of the complete graph picture described in Section~\ref{sec:graph-interpretations}:
\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node [style=WhiteSq] (0) at (-2, 0.25) {};
\node [style=White] (1) at (-2, 0.75) {};
\node [style=none] (2) at (-2.25, 1.25) {};
\node [style=none] (3) at (-1.75, 1.25) {};
\node [style=none] (4) at (-2.25, -0.25) {};
\node [style=none] (5) at (-1.75, -0.25) {};
\node [style=none] (6) at (-2.25, 1.75) {};
\node [style=none] (7) at (-1.75, 1.75) {};
\node [style=none] (8) at (-2.25, -0.75) {};
\node [style=none] (9) at (-1.75, -0.75) {};
\node [style=WhiteSq] (12) at (0.25, 1.25) {};
\node [style=WhiteSq] (13) at (1.25, 1.25) {};
\node [style=White] (14) at (0.25, -0.25) {};
\node [style=White] (15) at (1.25, -0.25) {};
\node [style=none] (16) at (0.25, 1.75) {};
\node [style=none] (17) at (1.25, 1.75) {};
\node [style=none] (18) at (0.25, -0.75) {};
\node [style=none] (19) at (1.25, -0.75) {};
\node [style=none] (20) at (0, 0.5) {};
\node [style=none] (21) at (0.5, 0.5) {};
\node [style=none] (22) at (1, 0.5) {};
\node [style=none] (23) at (1.5, 0.5) {};
\draw (1) to (0);
\draw (2.center) to (1);
\draw (3.center) to (1);
\draw (0) to (4.center);
\draw (0) to (5.center);
\draw (6.center) to (2.center);
\draw (7.center) to (3.center);
\draw (4.center) to (8.center);
\draw (5.center) to (9.center);
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\end{tikzpicture}
\end{equation}
Conversely, we can recover the complete graph picture from this relation by using associativity:
\begin{equation}
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\end{equation}
\begin{example}
We will give a simple example of this.
Let $A$ be a monoid in $\Set$.
We will think of $\Hom(I, A)$ as the set of $I$-tuples of $A$.
There are two basic operations on these tuples:
\begin{enumerate}
\item Given a set map $r: V \to T$, define a map $\mu_r: \Hom(V, A) \to \Hom(T, A)$ so that $c \in \Hom(V, A)$, gets sent to the map
\begin{equation}
\mu_r(c)(t) = \sum_{v \in r^{-1}(t)} c(v)
\end{equation}
This is well-defined because $A$ is commutative.
We will also use the convention that the empty sum results in the unit of $A$.
\item There is automatically a map $l^*: \Hom(S, A) \to \Hom(T, A)$ for each set map $l: S \leftarrow T$ given by precomposition.
\end{enumerate}
Using this, we define a strong monoidal functor $M_A: Span \to \Vect$ by sending the span $S \overset{l}{\leftarrow} V \overset{r}{\rightarrow} T$ to the map
\begin{equation}
\Hom(S, A) \overset{l^*}{\to} \Hom(V,A) \overset{\mu(r)}{\to} \Hom(T,A)
\end{equation}
$M_A$ is essentially the same data as a bimonoid with the same multiplication as $A$ and the diagonal as the comultiplicaiton.
\end{example}
\subsection{Plus construction of Span}
Since $\Span^+$ is a Feynman category, the theory of indexed enrichment carries over implying that there is a natural connection between $\Span^+$ and bialgebras.
\begin{example}
Define $A: Span^+ \to \Set$ so that $A(X \overset{l}{\leftarrow} R \overset{r}{\rightarrow} Y) = OF(l) \times OF(r)$ where $OF(f)$ is the set of orders on the fibers of a function $f$.
This datum composes in the canonical fashion.
For example, suppose we have ordered fibers $\{a < b < c\} \rightarrow pt \leftarrow \{1 < 2\}$, then the composition would produce the following ordered fibers:
\begin{center}
\begin{tabular}{rl}
$\{a\} \leftarrow \{(a,1) < (a,2)\}$
& \\
& $\{(a,1) < (b,1) < (c,1)\} \rightarrow \{1\}$ \\
$\{b\} \leftarrow \{(b,1) < (b,2)\}$
& \\
& $\{(a,2) < (b,2) < (c,2)\} \rightarrow \{2\}$ \\
$\{c\} \leftarrow \{(c,1) < (c,2)\}$ &
\end{tabular}
\end{center}
\end{example}
\begin{prop}
The indexed enrichment $\Span_{A}$ corepresents associative bimonoids.
\end{prop}
\begin{proof}
The only difference between $\Span_{A}$ and $\Span$ is that permuting fibers changes the morphisms of $\Span_{A}$ but keeps the morphisms of $\Span$ the same.
Hence a strong monoidal functor $B: Span_A \to \C$ is the same data as a bimonoid which is not necessarily commutative.
\end{proof}
\subsubsection{\@startsection{subsubsection}{3
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}
\def\index{{\rm {idx}}}
\def\findex{\mathfrak {index}}
\def\gindex{\g\text{-}{\mathrm {index}}}
\def\ndeg{\rm ndeg}
\def\depth{\rm depth}
\def\pure{\rm Pure}
\def\spure{\rm{Plr}}
\def\type{\it type}
\def\gpdhom{\mathcal{H}om_\C}
\def\rin{\rm in}
\def\rout{\rm out}
\def\gcp{\it gcp}
\def\bwb{b\mdash w \mdash b}
\def\mathfrak {M}{\mathfrak {M}} |
1,314,259,996,715 | arxiv | \section{Introduction}
The rapid development of quantum technologies over the last decade has prompted much research
on techniques for checking that the carefully engineered quantum devices are, in fact, functioning
properly \cite{GKK19,EHW20,KR21}.
In the survey of the field presented in \cite{EHW20}, a distinction is drawn between certification (or
verification) and benchmarking. Certification is taken to be confined to the assessment the accuracy
of the output, whereas benchmarking protocols are any more general measures of performance.
Clearly, in this parlance, Shor's factoring algorithm and the associated period-finding algorithm can
be used for verification as the validity of their output is easily checked, but the ancillary properties
mentioned in the abstract and discussed below could serve as the basis for benchmarks that
test circuits as a whole.
In Shor's version of his factoring algorithm \cite{Sh97}, a square-free semiprime $N$ is factored by
first using a quantum computer to determine the order $r$ of an arbitrary element
$b\in (\mathbb{Z}/N\mathbb{Z})^\times$, or, more colloquially, the period of $b^x \Mod{N}$,
and then using a classical computer to calculate the candidate $f_b =\gcd(b^{r/2}-1,N)$
for a non-trivial factor of $N$. (Unless specified otherwise, the notation of \cite{CvD10} is adopted.)
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{TransitionInPrb.png}
\caption{A plot of the approximation $\mathsf{P}_\infty^{(\mathfrak{q})}$ in \eqref{eq:Pq} to the probability of success of the period-finding
algorithm versus the increment $\mathfrak{q}$ of the input register size from its critical size of $\mathfrak{m}_0=\lceil 2\log_2r\rceil$ qubits.
}
\label{fig:figure1}
\end{figure}
It is the dependence on the period $r$ of the probability for success of the period-finding
algorithm which is relevant to benchmarking.
The nature of this dependence can be established by considering the sequence of input register sizes
$\{ \mathfrak{m}_{\mathfrak{q} } \}$, where $\mathfrak{m}_{\mathfrak{q} }$ is the \emph{smallest\/}
positive integer such that
\begin{equation}\label{eq:mqdef}
2^{ \mathfrak{m}_\mathfrak{q} } > 2^\mathfrak{q} r^2 ,
\end{equation}
and $\mathfrak{q}$ is \emph{any\/} integer consistent with the requirement that the input register is large enough
for the quantum Fourier transform of its state to yield information on $r$, i.e. $2^{\mathfrak{m}_\mathfrak{q}}\ge 2r$
from the $k=r-1$ version of the first inequality in \eqref{eq:constraint}, or
$\mathfrak{q}\ge\mathfrak{q}_{\min} = \lceil\log_2 r\rceil- \lfloor 2\log_2 r \rfloor$.
A systematic asymptotic analysis implies that, when $r\gg 1$, a useful
approximation to the probability of finding a divisor of $r$ (or $r$ itself) with an
input register of $\mathfrak{m}_{\mathfrak{q} }$ qubits is
\begin{equation}\label{eq:Pq}
\mathsf{P}^{(\mathfrak{q})}_\infty =
\frac{2}{\pi}\Si(2^\mathfrak{q}\pi) - \Bigl( \frac{\sin 2^{\mathfrak{q}-1}\pi }{2^{\mathfrak{q}/2-1}\pi}\Bigr)^2
\end{equation}
($\Si(x)$ is the sine integral of Eq.~6.2.9 in \cite{DLMF}).
The monotonically increasing sequence
$\{ \mathsf{P}^{(\mathfrak{q})}_\infty \}$ is depicted in Fig.~\ref{fig:figure1}. The interpolating curve is obtained with
the same function used to evaluate the $\mathsf{P}^{(\mathfrak{q})}_\infty$'s. As the derivation of section~\ref{sc:BW}
and \eqref{eq:ro1} make clear, the result in \eqref{eq:Pq} pertains to the determination of the period of
\emph{any\/} function of the integers $\mathbb{Z}$ provided it is one-to-one within a period, and
the period $r$ is not a divisor of $2^{\mathfrak{m}_\mathfrak{q}}$. When $r$ is a power of two (and, hence, a
divisor of $2^{\mathfrak{m}_\mathfrak{q}}$), the probability of success is independent of $\mathfrak{q}$ [see \eqref{eq:ro1}].
The register of $\mathfrak{m}_0=\lceil 2\log_2r\rceil$ qubits
is distinguished by the fact that it is, generically, the smallest for which $\mathsf{P}^{(\mathfrak{q})}_\infty$ exceeds 50\%.
Among the inferences that one can draw from Fig.~\ref{fig:figure1} is that \emph{the magnitude of the
probability of success for the period-finding algorithm amounts to a fingerprint of the critical input register}.
Statistics on the success or failure of runs for $\mathfrak{m}_0$ qubits would permit empirical determination of
the corresponding probability
of success of the period-finding algorithm; unambiguous comparison of the probability found with
$\mathsf{P}^{(\mathfrak{q=0})}_\infty$ should be possible in view of its clear separation in value from the other
$\mathsf{P}^{(\mathfrak{q})}_\infty$'s: this comparison constitutes the first benchmark,
applicable whenever
$r$ is not a divisor of $2^{\mathfrak{m}_0}$ (as is typically the case).
Corrections to $\mathsf{P}_\infty^{(\mathfrak{q})}$, considered in section \ref{sc:asymp}, do not invalidate any
of the observations above. In fact, it is found that, for $\mathfrak{q}\ge0$,
these corrections are negligible (see \eqref{eq:asympq}, \eqref{eq:asymp0} and Table \ref{tb:qc2}).
The clear difference between the success probability for the critical input register and input registers of
other sizes also survives (see Fig.~\ref{fig:figure3}).
Another opportunity for benchmarking is provided by
the choice of the element $b\in(\mathbb{Z}/N\mathbb{Z})^\times$.
Leander has pointed out~\cite{Le02} that if $b$
is selected so that the Jacobi symbol $(b/N)=-1$ (``Choice L''), then,
not only is the order $r$ of $b$ guaranteed to be even, but also the probability that $f_b=\gcd(b^{r/2}-1,N)$
is a non-trivial factor of $N$ is enhanced; for $r$ even, $f_b$ is the desired factor
provided $b^{r/2}\not\equiv-1(\Mod{N})$: the conditional probability
\begin{align}\label{eq:Leander}
\Pr&\left( b^{r/2}\not\equiv-1(\Mod{N}) \;\middle\vert\; (b/N)=-1\right) \nonumber\\
&\hspace*{50pt} = 1 - \frac{1}{2^{1+c_p -c_q}}(1-\delta_{c_p,c_q}) ,
\end{align}
where the positive integer powers $c_p\ge c_q$ are related to the square-free semiprime $N$ by
the parametrisation $N=(2^{c_p} d_p + 1)(2^{c_q} d_q + 1)$,
it being understood that $d_p$ and $d_q$ are, by definition, odd.
The success rate of the prescription for $f_b$ is at least 75\% as compared with 50\% if $b$ is chosen at
random from $(\mathbb{Z}/N\mathbb{Z})^\times$, but this
observation does not do justice to the full implications of \eqref{eq:Leander}.
The crux to building on Leander's criterion for $b$ is to recognise that, in principle,
\emph{it can only fail to generate a factor of $N$ if $c_p>c_q$}. When it is known that $c_p > c_q$,
then $b$ should be a quadratic non-residue modulo $N$ for which
$(b/N)=+1$ (``Choice $\overline{\mathrm{L}}$''); the corresponding probability that $f_b$ is a prime
factor of $N$ is $1-\delta_{c_p,c_q}$: if $c_p > c_q$, $f_b$ \emph{must\/} then be a non-trivial
factor of $N$. (As for choice L, the fact that $b$ is
a quadratic non-residue ensures that its order is even.)
There are two clear-cut complementary bench{\-}mark{\-}ing schemes $\mathcal{A}$ and $\mathcal{B}$:
scheme $\mathcal{A}$ ($\mathcal{B}$) entails determination of the probability that $\overline{\mathrm{L}}$-choices
(L-choices) for $b$ do actually yield a prime factor of $N$ given that $c_p>c_q$ ($c_p=c_q$).
Identification of whether $c_p>c_q$ or $c_p=c_q$ requires successful factorization of $N$, which can be
achieved with a suitable sequence of initial runs,
beginning with a few L-choices, and converting to
$\overline{\mathrm{L}}$-choices if these runs fail.
For both $\mathcal{A}$ and $\mathcal{B}$,
the empirical probability is to be
compared with a predicted value of unity. The overhead placed on classical
computer resources by the selection of appropriate values of $b$ is acceptable:
calculation of the Jacobi symbols is efficient, as is checking that a given $b\in (\mathbb{Z}/N\mathbb{Z})^\times$ is a
quadratic non-residue.
Application of the benchmarking procedures $\mathcal{A}$ and $\mathcal{B}$ should be preceded by determination of
$(-1/N)$ and $(2/N)$. Provided $-1$ and 2 are quadratic non-residues modulo $N$, the values of these two Jacobi symbols
permit one to infer whether $c_p=c_q$ or $c_p>c_q$ for all semiprimes such that $c_q\le 2$ --- see Table \ref{tb:jsi}.
The import of Table \ref{tb:jsi} is that, for these semiprimes, the choice between $\mathcal{A}$ and $\mathcal{B}$ can be made
\emph{before\/} the order-finding algorithm is run.
Blum integers are among the semiprimes to which the results of Table \ref{tb:jsi} apply.
\begin{table}[t]
\begin{onecolumn}
\centering
\begin{tabular}{ccc}
& {\small Interpretation} & Scheme \\ \hline\hline
$(-1/N)=-1$ & $c_p>c_q=1$ & $\mathcal{A}$\\
$(-1/N)=+1$ & $c_p=1=c_q$ & $\mathcal{B}$\\
$(2/N)=-1$ & $c_p>c_q=2$ & $\mathcal{A}$\\
$(2/N)=+1$ & $c_p=2=c_q$ & $\mathcal{B}$ \\ \hline\hline
\end{tabular}
\caption{Choice of benchmarking scheme indicated by the Jacobi symbols $(-1/N)$ and $(2/N)$.
The interpretation of $(-1/N)=+1$ and $(2/N)=+1$ assumes that $-1$ and 2, respectively, are
quadratic non-residues modulo $N$. If $-1\;(2)$ is a quadratic residue, then $c_q\ge 2\;(3)$.}
\label{tb:jsi}
\end{onecolumn}
\end{table}
It remains to justify the assertions made in this introduction. The exact reduction of the success probability
associated with the period-finding algorithm to a form in \eqref{eq:genresult} suitable for controlled approximation
is presented in section \ref{sc:BW}, followed by its asymptotic analysis for large $r$ using the
Euler-Maclaurin summation formula in section \ref{sc:asymp}.
Properties of choices
L and $\overline{\mathrm{L}}$ are proven in section \ref{sc:Choices}, and some closing comments
are made in section \ref{sc:discuss}. Appendices \ref{app:A}, \ref{app:B} and \ref{app:C}
contain technical results required in sections \ref{sc:BW} and \ref{sc:asymp}.
\section{Period-finding: success probability} \label{sc:BW}
After the usual preparatory steps (outlined, for example, in Algorithm 5 of Ref.~\cite{CvD10}),
an $m$-qubit input register is left in the superposition of computational basis states
\begin{equation}\label{eq:superposition}
\tfrac{1}{\sqrt{m_k}} \sum\limits_{l=0}^{m_l-1} |k+l\cdot r\rangle\hspace*{0.025\textwidth}
\left( m_k = 1+ \left\lfloor \tfrac{2^m - 1 - k}{r} \right\rfloor \right),
\end{equation}
where $k\in\mathbb{Z}/r\mathbb{Z}$ is unknown (and unknowable), and the
constraint on $m$ that
\begin{equation}\label{eq:constraint}
2^m \ge r+ 1 + k > r
\end{equation}
guarantees that the superposition contains more than one term and,
hence, that $S_k(x)$ in \eqref{eq:StructureFactor} can manifest dependence on $r$. Since
\begin{equation}
r = r_\mathrm{o}2^{n_r}\hspace*{0.025\textwidth}
\left(r_\mathrm{o}\ \mbox{odd} \right),
\end{equation}
where $n_r$ is a non-negative integer, an implication of \eqref{eq:constraint} found
useful below is that $m>n_r$.
Interpretation of the quantum Fourier transform of the one-dimensional ``array'' of uniformly
spaced ``atoms'' in \eqref{eq:superposition} is possible via its ``structure factor''
\begin{equation}\label{eq:StructureFactor}
S_k(x) = \frac{1}{m_k} \left| \sum\limits_{l=0}^{m_k-1} \left( e^{i 2\pi l}\right)^{r x/ 2^m} \right|^2 ,
\end{equation}
which is related to the conditional probability $P(x|k)$ that the transform is detected in the
state $|x\rangle$ by $P(x|k)=S_k(x)/2^m$.
Central to the standard analysis of $P(x|k)$ is the observation that, for non-zero
integers $x$, $S_k(x)$ is large ($\sim m_k$) when the rational number $rx/2^m$,
which must lie in the closed interval $[r/2^m,r(1-1/2^m)]$, is close to one of the $r-1$
integers $j$ in this interval, i.e.
\begin{equation}
j\in\mathbb{N}_r^+=\{1,2,\ldots,r-1\} .
\end{equation}
Thus, the ``frequencies'' $x$ most likely to be returned by measurement are drawn from the
set of integers $\{x_j\}$ closest to the first $r-1$ members of the harmonic series with fundamental
``frequency'' $2^m/r$:
\begin{equation}\label{eq:LocusMaximum}
\left| x_j - j \frac{2^m}{r}\right| < \tfrac{1}{2}\hspace*{0.025\textwidth}
\left( j \in \mathbb{N}^+_r \right),
\end{equation}
where the inequality is strict
because $2^m\,j/r=2^{m-n_r}\, j/r_\mathrm{o}$ can never be a half-integer.
The solution of \eqref{eq:LocusMaximum} is
\begin{equation}\label{eq:nearestinteger}
x_j =\left\lfloor 2^m \tfrac{j}{r} +\tfrac{1}{2}\right\rfloor
\end{equation}
as inspection of Fig.~\ref{fig:figure2} confirms.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{NI_cropped.png}
\caption{Confirmation that \eqref{eq:nearestinteger}, which is trivially valid if $2^m\tfrac{j}{r}$ is an integer, also
works when $2^m\tfrac{j}{r}$ is a non-integer independent of whether $2^m\tfrac{ j}{r}<x_j$ or $2^m\tfrac{j}{r} >x_j$.}
\label{fig:figure2}
\end{figure}
Equation \eqref{eq:LocusMaximum} acquires additional significance if viewed from the perspective of rational
approximation with continued fractions. Provided the input register size is such that $2^{m/2}>r$, then
one of the finite number of convergents to $x_j/2^m$ coincides with the ratio $j/r$ reduced to lowest terms;
if $m=\mathfrak{m}_0$, then the only observed values of $x$ from which information of this nature can be inferred
are precisely those belonging to the set $\{x_j\}$: the total probability for success of the period-finding algorithm is
\begin{equation}\label{eq:Ptotal}
P_\mathrm{tot} = \sum\limits_{j\in\mathbb{N}^+_r} P(x_j|k) = \frac{1}{2^m} \sum\limits_{j\in\mathbb{N}^+_r} S_k(x_j) .
\end{equation}
More generally, when $m=\mathfrak{m}_\mathfrak{q}$, the restriction on useful values of $x$ reads
\begin{equation}\label{eq:usefulx}
\left| x - j \frac{2^{\mathfrak{m}_\mathfrak{q}}}{r}\right| < 2^{\mathfrak{q}-1} .
\end{equation}
When $\mathfrak{q}\ge 1$, there are $2^\mathfrak{q}-1$ or $2^\mathfrak{q}$ solutions of \eqref{eq:usefulx}
dependent on whether $r_\mathrm{o}$ divides $j$ or not ($j\in \mathbb{N}_r^+$);
if $\mathfrak{q}<0$, then the solutions are a subset of the $x_j$'s.
\subsection{The case $\mathfrak{q}=0$}
Simplification of $P_\mathrm{tot}$ in \eqref{eq:Ptotal} for arbitrary $m \ge \mathfrak{m}_0$ prepares
the ground for analysis of the general case when \eqref{eq:usefulx} applies.
The $j$-fold invocation of the periodicity property $S_k(x) = S_k(x+ 2^m/r)$ permits substitution of $S_k(x_j )$
by $S_k(\Delta_j )$, where
\begin{align}\label{eq:Delta}
\Delta_j &= x_j - 2^m \frac{j}{r} \\
& = \left\lfloor 2^{m-n_r} \tfrac{j}{r_\mathrm{o}} + \tfrac{1}{2}\right\rfloor
- 2^{m - n_r}\tfrac{j}{r_\mathrm{o}} , \nonumber
\end{align}
which, significantly, is periodic in $j$ with period $r_\mathrm{o}$. As a result, the partition of
$\mathbb{N}^+_r$ into congruence classes modulo $r_\mathrm{o}$ serves to identify
the summands $S(\Delta_j )$ for different $j$ in \eqref{eq:Ptotal}
which are identical. For $r_\mathrm{o}>1$, the congruence classes
$\overline{1}_{r_\mathrm{o}},\overline{2}_{r_\mathrm{o}} ,\ldots, \overline{r_\mathrm{o}-1}_{r_\mathrm{o}}$ in $\mathbb{N}^+_r$
all contain $r/r_\mathrm{o}(=2^{n_r})$ elements, but the congruence class $\overline{0}_{r_\mathrm{o}}$ has
only $r/r_\mathrm{o}-1$ elements (because $0\not\in\mathbb{N}^+_r$); for $r_\mathrm{o}=1$, all
$r-1\ (=r/r_\mathrm{o}-1)$ elements of $\mathbb{N}^+_r$ trivially belong to the single congruence
class $\overline{0}_1$: thus,
\begin{equation}\label{eq:reducedPtotal}
P_\mathrm{tot}
= \frac{r}{r_\mathrm{o}} \sum\limits_{j\in\mathbb{Z}/r_\mathrm{o}\mathbb{Z}} P(\Delta_j | k) - P(\Delta_0|k) ,
\end{equation}
where the sum over $j$ now includes $j=0$.
As shown in appendix \ref{app:A}, the properties of least non-negative residues modulo $r_\mathrm{o}$ imply that the $r_\mathrm{o}$ distinct
values of $\Delta_j$ are equal to
$\mathfrak{j}/r_\mathrm{o}$, where the $\mathfrak{j}$'s are the absolute least residues
modulo $r_\mathrm{o}$:
\begin{equation}\label{eq:BZi}
\mathfrak{j}\in\mathfrak{B}[r_\mathrm{o}] = \bigl\{0,\pm 1, \pm 2, \ldots , \pm\lfloor \tfrac{1}{2}r_\mathrm{o} \rfloor \bigr\} .
\end{equation}
The change of summation variable in \eqref{eq:reducedPtotal} from $j$ to $\mathfrak{j}=\mathfrak{j}(j)$
is indicated. The argument $\Delta_j$ is replaced by $\Delta_\mathfrak{j}= \mathfrak{j}/r_\mathrm{o}$
and
\begin{equation}\label{eq:Ptotq0}
P_\mathrm{tot}
= \frac{r}{2^m} \left[\frac{1}{r_\mathrm{o}} \sum\limits_{\mathfrak{j}\in\mathfrak{B}[r_\mathrm{o}] }
S_k(\mathfrak{j}/r_\mathrm{o} ) - \frac{1}{r} S_k(0) \right] ,
\end{equation}
which suffices to analyse the case $m=\mathfrak{m}_0$.
\subsection{Generalization to $\mathfrak{q}<0$}
The result in \eqref{eq:Ptotq0} is easily
adapted to accommodate all cases in which $\mathfrak{q}<0$. Solutions of \eqref{eq:usefulx}
are those members of $\{x_j\}$ for which $|\Delta_j| < 1/2^{1-\mathfrak{q}}$.
In terms of the index $\mathfrak{j}$ introduced in connection with \eqref{eq:BZi}, this inequality
specifies that $|\mathfrak{j}|<r_\mathrm{o}/2^{1-\mathfrak{q}}$ or, as $r_\mathrm{o}$ is odd,
\begin{equation}
|\mathfrak{j}| \le \lfloor 2^{\mathfrak{q}-1}r_\mathrm{o} \rfloor .
\end{equation}
It follows that \eqref{eq:Ptotq0} is replaced by
\begin{equation}\label{eq:Ptotqneg}
P_\mathrm{tot}\!
=\! \frac{r}{2^m}\! \left[\frac{1}{r_\mathrm{o}} \sum\limits_{\mathfrak{j}\in\mathfrak{B}[2^\mathfrak{q}r_\mathrm{o}] }
S_k(\mathfrak{j}/r_\mathrm{o} ) - \frac{1}{r} S_k(0) \right]\! .
\end{equation}
Unfortunately, the generalization to $\mathfrak{q}>0$ is not so immediate.
\subsection{Generalization to $\mathfrak{q}>0$}
If $j\in\overline{0}_{r_\mathrm{o}}$, then \eqref{eq:usefulx} amounts to the inequality
$
\left| x - x_j \right| < 2^{\mathfrak{q}-1},
$
which has the integer solutions $x_j + \kappa$ with
\begin{equation}\label{eq:kappa0}
\kappa \in \mathfrak{S}_0= \{0,\pm 1,\pm 2, \ldots,\pm(2^{\mathfrak{q}-1}-1) \}.
\end{equation}
For members of the remaining congruence classes summed over in
\eqref{eq:Ptotq0}, \eqref{eq:usefulx}
can be recast in terms of the congruence class label $\mathfrak{j}$ of \eqref{eq:BZi} as
$
\left| x - x_j + \mathfrak{j}(j)/r_\mathrm{o} \right| < 2^{\mathfrak{q}-1} .
$
Now, the set of integer solutions $\{x_j+\kappa\}$ depends on $\mathfrak{j}=\mathfrak{j}(j)$ via its sign:
$\kappa$ takes on all values in the set
\begin{equation}\label{eq:kappanonzero}
\mathfrak{S}_\mathfrak{j}=\mathfrak{S}_0\cup\{-(\mathrm{sgn}\,\mathfrak{j}) 2^{\mathfrak{q}-1} \} ,
\end{equation}
where $\mathrm{sgn}\,\mathfrak{j}$ denotes the sign of the non-zero $\mathfrak{j}$ in \eqref{eq:kappanonzero}.
Ostensibly, the summation in Eq, \eqref{eq:Ptotq0} is replaced by
the double summation
\begin{equation}
\sum\limits_{\mathfrak{j}\in\mathfrak{B}[r_\mathrm{o}] } \,
\sum\limits_{\kappa\in\mathfrak{S}_{\mathrm{j}} }
S_k(\kappa+\mathfrak{j}/r_\mathrm{o} ) ,
\end{equation}
but it can be more usefully rewritten in terms of a single summation formally like that in \eqref{eq:Ptotqneg} as
\begin{equation}\label{eq:singlesum}
\sum\limits_{\mathfrak{j}\in\mathfrak{B}[2^\mathfrak{q}r_\mathrm{o}]}\!\! S_k(\mathfrak{j}/r_\mathrm{o})\,
- \epsilon_k ( \mathfrak{q} ) ,
\end{equation}
provided the endpoint ``correction''
\begin{equation}
\epsilon_k ( \mathfrak{q} ) = S_k(2^{\mathfrak{q}-1}) + S_k(-2^{\mathfrak{q}-1})
\end{equation}
is included.
A simpler substitution is that of the coefficient $S_k(0)$ of $1/r$ in \eqref{eq:Ptotq0} by
\begin{equation}\label{eq:coefficientsum}
\sum\limits_{\kappa\in\mathfrak{S}_0 } S_k(\kappa) =
\sum\limits_{\kappa\in\mathfrak{B}[ 2^\mathfrak{q} ]}\!\! S_k(\kappa )\,
- \epsilon_k ( \mathfrak{q} ) .
\end{equation}
Together, \eqref{eq:singlesum} and \eqref{eq:coefficientsum} imply that the generalisation
to positive $\mathfrak{q}$ of \eqref{eq:Ptotq0} is
\begin{align}\label{eq:genresult}
P_\mathrm{tot}^{(\mathfrak{q})}
= \frac{r}{2^{\mathfrak{m}_\mathfrak{q}}} & \Biggl[\frac{1}{r_\mathrm{o}}
\sum\limits_{\mathfrak{j}\in\mathfrak{B}[2^{\mathfrak{q}}r_\mathrm{o}] } \!\!
S_k(\mathfrak{j} / r_\mathrm{o} ) - \frac{1}{r} \sum\limits_{\mathfrak{j}\in\mathfrak{B}[ 2^\mathfrak{q} ]} S_k(\mathfrak{j})
\nonumber \\[1ex]
& \hspace*{0.1\textwidth} - \left(\frac{1}{r_\mathrm{o}} - \frac{1}{r} \right) \epsilon_k ( \mathfrak{q} ) \Biggr] .
\end{align}
The choice of $2^m$ appropriate to \eqref{eq:genresult} (namely, $2^{\mathfrak{m}_\mathfrak{q}}$)
has been made for the
overall multiplicative factor; it has also to be adopted in the expression for $S_k(x)$
in \eqref{eq:StructureFactor}.
With the extended definition of $\epsilon_k (\mathfrak{q})$ in terms of the Heaviside function $H(x)$
(\cite{DLMF}, Eq.~1.16.13) as
\begin{equation}
\epsilon_k (\mathfrak{q}) = H(\mathfrak{q}) \left[ S_k(2^{\mathfrak{q}-1}) + S_k(-2^{\mathfrak{q}-1}) \right] ,
\end{equation}
\eqref{eq:genresult} continues to hold when $\mathfrak{q}\le 0$.
Equation \eqref{eq:genresult} is the goal of this section.
It is a generalization and refinement of results in \cite{BW07}, which, in addition to the features
mentioned in the introduction, includes a more careful treatment of endpoint corrections.
The derivation has invoked only the periodicity of
$S_k(x)$ and is independent of its precise form, which, in \eqref{eq:genresult}, happens to be
\begin{equation}\label{eq:explicitS}
S_k(x) = S_k(0)
\left[ \frac{\sinc(m^{(\mathfrak{q})}_k r x /2^{\mathfrak{m}_\mathfrak{q}})}{ \sinc(
r x / 2^{\mathfrak{m}_\mathfrak{q}} )} \right]^2 ,
\end{equation}
where $S_k(0)=m_k^{(\mathfrak{q})} = 1+ \left\lfloor (2^{\mathfrak{m}_\mathfrak{q}} - 1 - k)/r \right\rfloor $
and $\sinc(x)$ is
the \emph{normalized\/} \mbox{sinc} function [for $x\not=0$, $\sinc(x)=\sin(\pi x)/(\pi x)$].
The endpoint correction term in \eqref{eq:genresult} does not influence the approximate
estimates of $P_\mathrm{tot}^{(\mathfrak{q})}$ considered next. Terms involving non-zero
$\epsilon_k(\mathfrak{q})$ are suppressed by at least three powers of $r$ relative to the dominate large $r$
contribution.
\section{Asymptotic analysis of $P_\mathrm{tot}^{(\mathfrak{q})}$} \label{sc:asymp}
If $r$ is a power of two (i.e.\, $r_\mathrm{o}=1$), then $m_k^{(\mathfrak{q})} = 2^{\mathfrak{m}_\mathfrak{q}} / r$,
$S_k(x)=\delta_{x,0}\, m_k^{(\mathfrak{q})}$ (for integer $x$), and \eqref{eq:genresult} reduces without approximation
to
\begin{equation}\label{eq:ro1}
P_\mathrm{tot}^{(\mathfrak{q})} = 1 - \frac{1}{r} .
\end{equation}
The first summation in \eqref{eq:genresult} is dominant, and the remaining terms are suppressed by one power of $r$
or more. The same pattern is also evident in an expansion of \eqref{eq:genresult} for large $r$ which embraces
the cases in which $r$ is \emph{not\/} a power of two. If the negligible terms of order $1/r$ and higher
are discarded, then the following approximation to $P_\mathrm{tot}^{(\mathfrak{q})}$ emerges:
\begin{equation}\label{eq:domC}
\mathsf{P}^{(\mathfrak{q})}
= \frac{1}{r_\mathrm{o}}
\sum\limits_{\mathfrak{j}\in\mathfrak{B}[2^{\mathfrak{q}}r_\mathrm{o}] } \!\!
\sinc^2 (\mathfrak{j} / r_\mathrm{o} ) .
\end{equation}
The expansions in support of \eqref{eq:domC} are given in appendix \ref{app:B}. Like \eqref{eq:ro1},
\eqref{eq:domC} is independent of $k$, but, when specialized to $r_\mathrm{o}=1$, yields
$\mathsf{P}^{(\mathfrak{q})}=1$.
\begin{table}[t]
\centering
\caption{Sample of dependence of $\mathsf{P}^{(\mathfrak{q})}$ on $r_\mathrm{o}$ for $\mathfrak{q}\ge0$.
The last row contains
the values of $\mathsf{P}^{(\mathfrak{q})}_\infty$ from \eqref{eq:Pq}. }\label{tb:qc2}
\begin{tabular}{llll} \hline
$r_\mathrm{o}\backslash \mathfrak{q} $ & 0 & 1 & 2 \\ \hline
3 & 0.7893 & 0.90326 & 0.949999 \\
5 & 0.7792 & 0.90288 & 0.949946 \\
7 & 0.7765 & 0.902837 & 0.9499411 \\
9 & 0.7754 & 0.902828 & 0.9499400 \\
11 & 0.7748 & 0.902826 & 0.9499396 \\
13 & 0.7745 & 0.9028245 & 0.94993949 \\
15 & 0.7743 & 0.9028240 & 0.94993942 \\ \hline
$\infty$ & 0.7737 & 0.9028233 & 0.94993934 \\ \hline
\end{tabular}
\end{table}
Simple as the expression for $\mathsf{P}^{(\mathfrak{q})}$ is, it would seem of little utility because of its
reliance on the factor $r_\mathrm{o}$ of the unknown (but large) period $r$.
However, the numerical data in Table \ref{tb:qc2} suggests that
$\mathsf{P}^{(\mathfrak{q})}$ is a monotonically decreasing function of $r_\mathrm{o}$ when $\mathfrak{q}\ge0$.
This trend can be confirmed for large $r_\mathrm{o}$ with the aid of the Euler-Maclaurin
summation formula, which, for $\mathfrak{q}\ge1$, implies straightforwardly that
\begin{equation}\label{eq:asympq}
\mathsf{P}^{(\mathfrak{q})} = \frac{2}{\pi}\mathrm{Si}(2^\mathfrak{q}\pi) +
\frac{4}{15\cdot 8^\mathfrak{q}}\frac{1}{r_\mathrm{o}^4} + \ldots\, .
\end{equation}
The calculation is trickier for $\mathfrak{q}=0$; application of the Euler-Maclaurin summation formula
has to be followed by an expansion in inverse powers of $r_\mathrm{o}$ (details are given in
appendix \ref{app:C}): the outcome of these manipulations is
\begin{equation}\label{eq:asymp0}
\mathsf{P}^{(\mathfrak{q}=0)} = \frac{2}{\pi}\left(\mathrm{Si}(\pi) -\frac{2}{\pi} \right)
+ \frac{4}{3 \pi^2 }\frac{1}{r_\mathrm{o}^2} + \ldots\, .
\end{equation}
The results in \eqref{eq:asympq}, \eqref{eq:asymp0} and Table \ref{tb:qc2}
justify the claim that, \emph{provided $1/r$ is negligible, $\mathsf{P}^{(\mathfrak{q})}_\infty$ in \eqref{eq:Pq} is always
an excellent approximation to $P_\mathrm{tot}^{(\mathfrak{q})}$} in \eqref{eq:genresult} when $\mathfrak{q}$ is non-negative.
\begin{table}[t]
\setlength{\tabcolsep}{4pt}
\centering
\caption{Sample of deviation of $\mathsf{P}^{(\mathfrak{q})}$ from $\mathsf{P}^{(\mathfrak{q})}_\infty$ for $\mathfrak{q}<0$.
The scaled deviation $\mathsf{D}^{(\mathfrak{q})}=r_\mathrm{o}\bigl(\mathsf{P}^{(\mathfrak{q})} - \mathsf{P}^{(\mathfrak{q})}_\infty\bigr)$.
The last row contains the values of $\mathsf{D}^{(-2)}$ in the limit $r_\mathrm{o}\rightarrow\infty$.}\label{tb:qc3}
\begin{small}
\begin{tabular}{clclclcl} \hline
$r_\mathrm{o}$ & $\mathsf{D}^{(-2)}$ & $r_\mathrm{o}$ & $\mathsf{D}^{(-2)}$ & $r_\mathrm{o}$ & $\mathsf{D}^{(-2)}$ & $r_\mathrm{o}$ & $\mathsf{D}^{(-2)}$ \\ \hline
& & 3 & 0.263 & 5 & $-$0.229 & 7 & $-$0.720 \\
9 & 0.708 & 11 & 0.243 & 13 & $-$0.234 & 15 & $-$0.716 \\
17 & 0.7098 & 19 & 0.240 & 21 & $-$0.235 & 23 & $-$0.7144 \\
25 & 0.7105 & 27 & 0.239 & 29 & $-$0.236 & 31 & $-$0.7138 \\ \hline
& 0.7122 & & 0.237 & & $-$0.237 & & $-$0.7122 \\ \hline
\end{tabular}
\end{small}
\end{table}
The behaviour of $\mathsf{P}^{(\mathfrak{q})}$ for $\mathfrak{q}<0$ is quite different. Now, the expansion in inverse
powers of $r_\mathrm{o}$, which is obtained along the same lines as \eqref{eq:asymp0}, reads
\begin{align}
\frac{2}{\pi}\mathrm{Si}(2^\mathfrak{q} \pi) & - 2^\mathfrak{q} \sinc^2( 2^{\mathfrak{q}-1}) \label{eq:asympn} \\
& + \sinc^2 (2^{\mathfrak{q}-1})\left( 1- \frac{\nu}{2^{|\mathfrak{q}|}} \right) \frac{1}{r_\mathrm{o}} + \ldots , \nonumber
\end{align}
where $\nu$ is the least non-negative residue of $r_\mathrm{o}$ modulo $2^{|\mathfrak{q}|+1}$. The correction to
the leading term $\mathsf{P}^{(\mathfrak{q})}_\infty$ is of either sign and, being of order $1/r_\mathrm{o}$
(see Table \ref{tb:qc3}),
can be substantial for small $r_\mathrm{o}$ -- see Fig.~\ref{fig:figure3}. Despite the scatter about
$\mathsf{P}^{(\mathfrak{q})}_\infty$ in the values of $\mathsf{P}^{(\mathfrak{q})}$ for $\mathfrak{q}<0$,
none are close to $\mathsf{P}^{(0)}$.
\begin{figure}[b!]
\centering
\includegraphics[width=0.45\textwidth]{Scatter.png}
\caption{Comparison of values of $\mathsf{P}^{(\mathfrak{q})}$ in \eqref{eq:domC} to $\mathsf{P}_\infty^{(\mathfrak{q})}$
for \emph{negative\/} increments $\mathfrak{q}$ of the input register size from its critical size (of $\mathfrak{m}_0$ qubits).
Values for which $r_\mathrm{o}=3\, (5)$ are represented by the symbol
{\color{lightgray}$\blacktriangle\,$}({\color{lightgray}$\blacktriangledown$}); the symbol $\bullet$ is used for all larger values of
$r_\mathrm{o}$ considered ($7\le r_\mathrm{o}\le 31$).
} \label{fig:figure3}
\end{figure}
The leading term
$\mathsf{P}^{(\mathfrak{q})}_\infty$ in all of the above expansions in \eqref{eq:asympq}, \eqref{eq:asymp0} and
\eqref{eq:asympn} is found by evaluation of
\begin{equation}\label{eq:intrep}
\int\limits_{-2^{\mathfrak{q}-1}}^{2^{\mathfrak{q}-1}} \sinc^2(x) dx
\end{equation}
for integer $\mathfrak{q}$. The integral in \eqref{eq:intrep} defines an entire function of $z=2^\mathfrak{q}$.
In the limit $r_\mathrm{o}\rightarrow\infty$, when $\mathsf{P}^{(\mathfrak{q})}\rightarrow\mathsf{P}^{(\mathfrak{q})}_\infty$,
the lower limit on the integers $\mathfrak{q}$ of $\mathfrak{q}_{\min}\rightarrow-\infty$; the values of $2^\mathfrak{q}$
then belong to a set with an accumulation point, and the identity theorem for holomorphic functions
can be invoked to assert that the integral representation in \eqref{eq:intrep} is the unique
analytic continuation of $\mathsf{P}^{(\mathfrak{q})}_\infty$ to all values of $\mathfrak{q}$, real and complex: thus,
the right-hand side of \eqref{eq:Pq}, which is obtained from \eqref{eq:intrep} for arbitrary
$\mathfrak{q}$ by integration by parts, is a natural choice of interpolation function in Fig.~\ref{fig:figure1}.
\section{Foundations for choices L and $\overline{\mathrm{L}}$} \label{sc:Choices}
Insight into good choices of $b$ for the factorization of square-free odd semiprimes
$N=pq$ is gained through the isomorphism
between $(\mathbb{Z}/N\mathbb{Z})^\times$
and the direct product $(\mathbb{Z}/p\mathbb{Z})^\times \times (\mathbb{Z}/q\mathbb{Z})^\times$
of the cyclic groups $(\mathbb{Z}/p\mathbb{Z})^\times$ and $(\mathbb{Z}/q\mathbb{Z})^\times$.
The Jacobi symbol permits some consequences of this isomorphism
to be recast in a manner which does not require any knowledge of the odd primes $p$ and $q$.
\subsection{Use of $(\mathbb{Z}/N\mathbb{Z})^\times\cong(\mathbb{Z}/p\mathbb{Z})^\times
\times (\mathbb{Z}/q\mathbb{Z})^\times$ } \label{sbsc:isomorphism}
The two simultaneous congruence relations
\begin{equation}\label{eq:bcong}
b\equiv b_p(\Mod{p}),\quad b\equiv b_q(\Mod{q})
\end{equation}
establish (via the Chinese Remainder theorem) a bijection between elements
$b\in (\mathbb{Z}/N\mathbb{Z})^\times$ and ordered pairs
$(b_p,b_q)\in (\mathbb{Z}/p\mathbb{Z})^\times \times (\mathbb{Z}/q\mathbb{Z})^\times$,
and there is a one-to-one correspondence between $b^k\,(\Mod{N})$ and $\bigl(b_p^k(\Mod{p}),
b_q^k(\Mod{q})\bigr)$ for any positive integer power $k$. Hence, the order $r$ of $b$ to be
used in $f_b=\gcd(b^{r/2}-1,N)$ is determined by the orders $r_p$ and $r_q$ of $b_p$ and
$b_q$, respectively, via
\begin{equation}\label{eq:rlcm}
r = \lcm (r_p,r_q) ,
\end{equation}
and the \emph{conditions for the failure of $f_b$ as a factor of $N$\/}
find neat expression as a property of $r_p$ and $r_q$:
\emph{the powers of two in the prime factorizations of $r_p$ and $r_q$ are equal\/}
(\cite{CvD10}, Lemma 2).
The appeal of this characterisation is that the distribution of the powers of two associated with
either of the orders $r_p$ or $r_q$ is easily constructed. For odd primes $h=2^c d+1$
($c\ge 1$, $d$ odd), the group $(\mathbb{Z}/h\mathbb{Z})^\times$ comprises elements which
are powers (modulo $h$) of a generator $\mathfrak{g}_h$ of order $h-1=2^c d$. As a result, the element
$\mathfrak{g}_h^k(\Mod{h})$ (where $k=1,2,\ldots,2^c d$) has order
\begin{equation}\label{eq:orderGENERAL}
r_h^{(k)} = \frac{2^c d}{\gcd(k,2^cd)} .
\end{equation}
For the $2^{c-1} d$ \textit{odd\/} values of $k$,
\eqref{eq:orderGENERAL} simplifies to $r_h^{(k)} = 2^c d_k$,
where the odd number
$d_k = d/\gcd(k,d)$; the corresponding result for the equal number of
\textit{even\/} values of $k=2j$ ($ j = 1,2,\ldots,2^{c-1}d$) is
\begin{equation}\label{eq:orderEVEN}
r_h^{(k=2j)} = \frac{2^{c-1} d}{\gcd(j,2^{c-1}d)} ,
\end{equation}
which resembles \eqref{eq:orderGENERAL} with the factor of $2^c$ replaced by $2^{c-1}$:
on the basis of the recursive pattern implied by the similarity of \eqref{eq:orderEVEN} to \eqref{eq:orderGENERAL}, there are
$2^{l-1}d$ elements of $(\mathbb{Z}/h\mathbb{Z})^\times$ with the \textit{even\/} orders
$2^l d_j$ ($ j=1,3,5,\ldots, 2^l d-1$) for each $l\in\{1,2,3,\ldots,c\}$, and only $d$ elements
with the \textit{odd\/} orders $d_j$ ($j=1,2,\ldots,d$).
Altogether, there are $c+1$
different powers of two: $\{0,1,\dots, c-1\}$ for the even choices of $k$ in
\eqref{eq:orderGENERAL}, and exclusively $c$ for the odd choices.
An element $(b_p,b_q)\in(\mathbb{Z}/p\mathbb{Z})^\times\times (\mathbb{Z}/q\mathbb{Z})^\times$ has
the representation $\bigl(\mathfrak{g}_p^{k_p}(\Mod{p}),\mathfrak{g}_q^{k_q}(\Mod{q})\bigr)$ in terms
of generators $\mathfrak{g}_p$ and $\mathfrak{g}_q$ of $(\mathbb{Z}/p\mathbb{Z})^\times$ and
$(\mathbb{Z}/q\mathbb{Z})^\times$, respectively. Consistent with the earlier parametrization of $N$ in
connection with \eqref{eq:Leander}, the primes $p$ and $q$ are taken to be $p=2^{c_p}d_p+1$ and
$q=2^{c_q}d_q+1$, where $d_p, d_q$ are odd and $c_p\ge c_q$.
The results of the previous paragraph can be used to determine the number of pairs $(b_p,b_q)$
for which the powers of two in the prime factorizations of the corresponding orders $r_p$ and $r_q$
are identical. By way of example, whenever both indices $k_p$ and $k_q$ are \emph{odd}, the respective
powers of two are $c_p$ and $c_q$; as there are $\tfrac{1}{2}(p-1)$ odd values of $k_p$ and
$\tfrac{1}{2}(q-1)$ odd values of $k_q$, the corresponding
number of pairs $(b_p,b_q)$ with orders sharing the same power of two is
\begin{equation}
\tfrac{1}{2}(p-1){\times} \tfrac{1}{2}(q-1)\hspace*{0.5pt}\delta_{c_p,c_q}
= 4^{c_q - 1} d_p d_q\hspace*{0.5pt} \delta_{c_p,c_q} .
\end{equation}
Table \ref{tb:qc4} contains a summary of all the findings on the numbers of pairs with such matching powers of two.
No match is possible when $k_p$ is \emph{odd\/} and $k_q$ is \emph{even\/} because $c_q-1<c_p$.
The entry for \emph{even} $k_p$ and $k_q$ excludes the pair $(1,1)$ since it corresponds to a
choice of $b$ ($b=1$) that cannot yield non-trivial factors of $N$ ($f_{b=1}=N$).
\begin{table}[t]\centering
\caption{The numbers of pairs $(b_p,b_q)\in
(\mathbb{Z}/p\mathbb{Z})^\times\times (\mathbb{Z}/q\mathbb{Z})^\times$
for which the powers of two in the prime factorizations of the two related orders $r_p$ and $r_q$
coincide. The sum $S=4^{c_q-1}d_pd_q$. \\[-1em] }\label{tb:qc4}
\begin{small}
\begin{tabular}{|c|cc|} \cline{2-3}
\multicolumn{1}{c|}{} & \rule[-6pt]{0pt}{18pt} $k_q$ \small{even} & $k_q$ \small{odd} \\ \hline
\rule[-6pt]{0pt}{18pt}
\strut $k_p$ \small{even} & $\tfrac{1}{3}S+\tfrac{2}{3} d_pd_q-1$ & $S(1-\delta_{c_p,c_q})$ \\
\rule[-6pt]{0pt}{18pt}
$k_p$ \small{odd} & 0 & $S\hspace*{0.5pt}\delta_{c_p,c_q}$ \\ \hline
\end{tabular}
\end{small}
\end{table}
\subsection{Role of the Jacobi symbol}
Quadratic residues modulo $N$ are those elements of $(\mathbb{Z}/N\mathbb{Z})^\times$ for which
both of the indices $k_p$ and $k_q$ are even.
Table \ref{tb:qc4} exhibits a partition of
$(\mathbb{Z}/N\mathbb{Z})^\times$ into the subgroup comprising
its quadratic residues and the three cosets of this subgroup, all elements of which are quadratic non-residues modulo $N$.
The Jacobi symbol distinguishes two of these cosets from the subgroup of quadratic residues.
For odd primes $h$, it is the Legendre symbol $(a/h)$ which differentiates between quadratic residues and
non-residues modulo $h$: $(a/h)$ is $+1$ ($-1$) if $a$ is a quadratic residue (non-residue), and 0 if
$h$ is a divisor of $a$.
A consequence of Fer{\-}mat's little theorem is that $a^{(h-1)/2}\,(\Mod{h})$
must have one of $+1$, $-1$ or 0 as its least absolute residue. As a result (\cite{HW08}, Theorem 83),
it is possible to express $(a/h)$ in terms of least absolute residues as
\begin{equation}
\left(\frac{a}{h}\right) \equiv a^{(h-1)/2}\,(\Mod{h}) .
\end{equation}
Furthermore, as
the values $+1$ and 0 are inadmissible for any
generator $\mathfrak{g}_h$ of $(\mathbb{Z}/h\mathbb{Z})^\times$
because they are incompatible with the requirement that it be of even order $h-1$, it is necessarily the case
that $\mathfrak{g}_h^{(h-1)/2}\equiv-1 (\Mod{h})$, and
\begin{align}\label{eq:interLC}
\left(\frac{\mathfrak{g}_h^k}{h}\right) &\equiv \bigl(\mathfrak{g}_h^{(h-1)/2}\bigr)^k\,(\Mod{h}) \\
&\equiv (-1)^k\, (\Mod{h}) = (-1)^k \nonumber ,
\end{align}
in accord with the expectation that even (odd) powers of a generator are quadratic residues (non-residues).
For an integer $b$ relatively prime to the square-free semiprime $N=pq$, the Jacobi symbol
\begin{equation}\label{eq:Jdef}
\left(\frac{b}{N}\right) = \left(\frac{b}{p}\right) \left(\frac{b}{q}\right),
\end{equation}
where the right-hand side is the product of the two Legendre symbols $(b/p)$ and $(b/q)$,
which, in view of the congruences in \eqref{eq:bcong}, can be substituted by $(b_p/p)$
and $(b_q/q)$, respectively. Use of the further congruences
$b_p\equiv(\mathfrak{g}_p)^{k_p}\,(\Mod{p})$ and $b_q\equiv (\mathfrak{g}_q)^{k_q}\,(\Mod{q})$
as well as \eqref{eq:interLC} imply finally that
\begin{equation}
\left(\frac{b}{N}\right) = (-1)^{k_p+k_q} ,
\end{equation}
which is the basis for the observation exploited in \cite{Le02} that, when $(b/N)=-1$, $b$
belongs to the union of the ($k_p$ odd, $k_q$ even) and
($k_p$ even, $k_q$ odd) cosets in $(\mathbb{Z}/h\mathbb{Z})^\times$.
According to Table \ref{tb:qc4}, $S(1-\delta_{c_p,c_q})$ of the $\tfrac{1}{2}(p-1){\times} (q-1) = 2^{c_p-c_q+1}S$
members in this union are unsuitable for the purposes of factoring the semiprime $N$. The result for the
conditional probability in \eqref{eq:Leander} follows immediately. Another inference from Table \ref{tb:qc4},
which forms the basis for choice $\overline{\mathrm{L}}$, is that, when it is known $c_p\not=c_q$, then
\emph{all\/} quadratic non-residues $b$ for which $(b/N)=+1$ [i.e. the whole of the ($k_p$ odd, $k_q$ odd) coset]
are good candidates for factoring $N$.
\subsection{Value of $c_q$: special cases}
Specialization
to the square-free semiprime $N=(2^{c_p}d_p+1)(2^{c_q}d_q+1)$ of
the standard formulae for the Jacobi symbols $(-1/N)$ and $(2/N)$ (\cite{DLMF}, \S 27.9)
proves serendipitously fruitful.
To begin with, on the basis of
\begin{align}\label{eq:minus1}
\left(\frac{-1}{N}\right) &= (-1)^{(N-1)/2} \\
&= (-1)^{2^{c_p-1}} (-1)^{2^{c_q-1}} , \nonumber
\end{align}
it is possible to interpret the value of $(-1/N)$ as follows:
if $(-1/N)=-1$, then, without further ado, $c_p>c_q=1$; if, instead, $(-1/N)=+1$, then
$c_p=1=c_q$ when $-1$ is a quadratic \emph{non\/}-residue modulo $N$,
and otherwise it can be deduced that $c_p\ge c_q\ge 2$.
In the latter case, it is appropriate to move onto $(2/N)$. Under the assumption
that
$c_p\ge c_q\ge 2$,
\begin{align}\label{eq:plus2}
\left(\frac{2}{N}\right) &= (-1)^{(N^2-1)/8} \\
&= (-1)^{2^{c_p-2}} (-1)^{2^{c_q-2}} , \nonumber
\end{align}
the implications of which parallel those of \eqref{eq:minus1}: if $(2/N)=-1$, then
$c_p>c_q=2$; if $(2/N)=+1$, then $c_p=2=c_q$ when $+2$ is a quadratic
\emph{non\/}-residue modulo $N$, and, by the exclusion above of other options,
$c_p\ge c_q\ge 3$ when $+2$ is a quadratic residue. All of these findings are summarized in
Table \ref{tb:jsi}.
Larger values of $c_q$ can be identified if it is known that $c_p>c_q$
(because choice L has failed) by the elementary expedient of evaluating
\begin{equation}
s_k = (-1)^{(N-1)/2^k}
\end{equation}
for $k=3,4,\ldots$. As $s_k = (-1)^{2^{c_q-k}}$ for $k\le c_q<c_p$, the sequence of
evaluations is to be terminated when the value $s_k=-1$ is encountered; $c_q$ is the
corresponding value of $k$.
\subsection{Properties of orders}
With the substitutions $r_p=2^{l_p}r_{p\mathrm{o}}$ and $r_q=2^{l_q}r_{q\mathrm{o}}$
($r_{p\mathrm{o}}, r_{q\mathrm{o}}$ odd), \eqref{eq:rlcm} becomes
\begin{equation}
r = 2^{{\max}(l_p,l_q)}\lcm(r_{p\mathrm{o}},r_{q\mathrm{o}}) .
\end{equation}
The properties of the indices $l_p$ and $l_q$ established in subsection \ref{sbsc:isomorphism} imply that,
for choice L,
\begin{equation}
c_q \le \max(l_p,l_q)\le c_p,
\end{equation}
whereas, for choice $\overline{\mathrm{L}}$,
\begin{equation}
\max(l_p,l_q) = c_p.
\end{equation}
For both choices, the order $r$ is even as asserted in the introduction.
Lagrange's theorem for finite groups and the isomorphism
$(\mathbb{Z}/N\mathbb{Z})^\times\cong(\mathbb{Z}/p\mathbb{Z})^\times
\times (\mathbb{Z}/q\mathbb{Z})^\times$ imply that an order $r$ modulo the
square-free semiprime $N$
is a divisor of the value
\begin{equation}
\lambda(N) = \lcm(p-1,q-1)
\end{equation}
of the Carmichael $\lambda$-function.
Thus, substituting for $\lcm(p-1,q-1)$
in terms of $\gcd(p-1,q-1)$,
\begin{equation}\label{eq:rbound}
r \le \frac{(p-1)(q-1)}{\gcd(p-1,q-1)}\le\frac{1}{2^{c_q}} (p-1)(q-1)
\end{equation}
since $\gcd(p-1,q-1)\ge 2^{c_q}$.
In all cases of practical interest, $p+q>2$ and
the right-hand side of \eqref{eq:rbound} can be replaced by
\begin{equation}\label{eq:rmax}
r_{\max} = \tfrac{1}{2^{c_q}} (pq-1) =\tfrac{1}{2^{c_q}} (N-1) .
\end{equation}
Information gleaned from the analysis of the Jacobi symbols $(-1/N)$ and $(2/N)$
or the ad hoc construct $s_k$
can be used to fix a suitable lower limit to $c_q$. The upper bound $r_{\max}$ can be used to
improve on Shor's recommendation that the input quantum
register contain at least $m_{\mbox{\tiny Sh}}=\lceil 2\log_2N\rceil$ qubits.
In terms of $\mathfrak{m}_{\mathfrak{q}}$,
\begin{equation}
m_{\mbox{\tiny Sh}}=\mathfrak{m}_{\mathfrak{q}=2c_q+\Delta} ,
\end{equation}
where $\Delta=\lceil 2\log_2 (N/2^{c_q})\rceil-\lceil 2\log_2r\rceil$ is a non-negative integer.
\section{Discussion} \label{sc:discuss}
As a tool for factoring RSA integers $N$, Shor's algorithm has been displaced by an approach
which computes discrete logarithms~\cite{GE21,GS21}. Nevertheless, the present paper suggests that
there may remain an alternative use for Shor's algorithm as a context for testing the operation of quantum
computers. The benchmarks involving quadratic non-residues (schemes $\mathcal{A}$ and $\mathcal{B}$ above)
derive from structural properties of $(\mathbb{Z}/N\mathbb{Z})^\times$, and group-theoretical considerations
pertinent to other algorithms may also imply similar benchmarks. The benchmark arising from the
period-finding algorithm is fortuitous.
Further studies may yet show that the benchmarks identified in this paper are toothless. However, the findings
on the period-finding algorithm should still be of interest in view of their generality.
According to the results in section \ref{sc:asymp}, the approximation $\mathsf{P}^{(\mathfrak{q})}_\infty$
has the merit of being a lower bound to the probability of success when $\mathfrak{q}\ge0$ and $1/r$ is negligible.
\bibliographystyle{plain}
|
1,314,259,996,716 | arxiv | \section{Introduction}
\label{sec:intro}
Two basic aspects in the study of field \eq/s in mathematical physics
are their \sym/ structure and their \conslaw/ structure.
Geometrically speaking,
symmetries are infinitesimal transformations of the fields
under which all solutions are mapped into solutions.
Symmetries of local form in the fields and partial derivatives
of the fields to a finite order (generalizing classical point symmetries)
are the basis for construction of exact (invariant) solutions
and provide an important connection with separation of variables
in certain cases.
In the situation where a system of field \eq/s has a Lagrangian,
every local \sym/ that leaves the Lagrangian invariant to within a divergence
yields a \conslaw/,
namely a current (density and flux)
whose space-time divergence
(\ie/ time derivative of density plus spatial divergence of flux)
vanishes on all solutions of the field equations,
through Noether's theorem.
Conservation laws given by currents of local form
determine physically important conserved quantities such as
energy, momentum, angular momentum, mass, charge, etc.
which are constants of motion
central to an analysis of the time evolution of the fields.
An infinite hierarchy of local \syms/ and \conslaw/s
of increasing higher order is a hallmark of
complete integrability of field equations.
Nonlocal \syms/ and nonlocal \conslaw/s,
involving other than a local form,
such as essential dependence on potentials or integrals of the fields,
have been less studied
but are also important and useful in the study of field equations ---
for instance, they yield exact solutions and conserved quantities
that are not obtainable from local \syms/ or local \conslaw/s
\cite{AncBlu:1996JMP}.
For the source-free \Meq/ in classical electromagnetic field theory
in 3+1 dimensional Minkowski space,
a complete explicit classification of all local \syms/
and local \conslaw/s in a unified coordinate-invariant form
has been recently carried out by Anco \& Pohjanpelto
\cite{AncPoh:2004,AncPoh:2001}.
The symmetry classification was obtained by
solving the \sym/ determining \eq/s
through the use of spinor techniques
(and properties of Killing spinors).
Since \Meq/ are a non-Lagrangian system,
Noether's theorem cannot be used to find \conslaw/s from \syms/.
A scaling formula that generates all nontrivial conserved currents
from \adjsyms/ was instead used to classify \conslaw/s
through solving the similar \adjsym/ determining equations.
This formula is a variant of a general \conslaw/ formula
that applies to any
PDE system admitting a scaling \sym/ and produces
all local conserved currents with nonzero scaling weight
\cite{Anc:2003JPhysA}.
The local \conslaw/s of \Meq/ comprise
the well-known stress-energy currents and Lipkin's zilch currents
\cite{BesHag,Lip,Kib,Fai,Mor},
and new chiral currents which contain first derivatives of
the electromagnetic field and hence are of one order higher in derivatives
compared to the stress-energy currents.
Associated with these \conslaw/s are corresponding conserved tensors.
The local \syms/ of \Meq/ consist of
infinitesimal scaling and duality-rotation transformations,
infinitesimal Poincar\'e, dilation and conformal transformations
\cite{Bat:1909,Cun:1909,Ibr:1968},
and in addition
infinitesimal chiral transformations
which are, again, of one order higher in derivatives.
Second order \syms/ and conserved currents of chiral type were
first discovered by Fushchich \& Nikitin
\cite{FusNik:1983,FusNik:1987book,FusNik:1992}
several years ago
and possess the striking feature of odd parity
under exchange of electric and magnetic fields,
in contrast to the even parity of the stress-energy and zilch currents
as well as that of the Poincar\'e \syms/ and the dilation/conformal \syms/.
Due to the covariant linear nature of \Meq/,
a hierarchy of higher order \syms/ and currents
arise \cite{Poh:survey}
by repeated replacement of the electromagnetic field
by Poincar\'e and dilation/conformal \sym/ operators
applied to the field or its dual.
The general classification results obtained in
\Ref{AncPoh:2004,AncPoh:2001}
state that no other local \syms/ or
local \conslaw/s exist to all orders,
apart from elementary ones of zeroth order
(which are produced through shifting the electromagnetic field
by any particular solution of \Meq/).
Some nonlocal \syms/ and \conslaw/s for \Meq/ were also derived by
Fushchich \& Nikitin \cite{FusNik:1983,FusNik:1987book},
in a non-covariant manner using Fourier transform methods.
For the reduction of \Meq/ to 2+1 dimensions,
nonlocal \syms/ and \conslaw/s in covariant form
were obtained by Anco \& Bluman \cite{AncBlu:1997JMP}
through the use of electric and magnetic potentials.
The well-known covariant vector potential for \Meq/
in any number of dimensions gives a Lagrangian system with gauge freedom.
Note that this potential arises from the absence of magnetic charges
and currents in free space
and hence we refer to it as the magnetic vector potential.
In Lorentz gauge the magnetic potential system reduces
to the vector wave \eq/.
If electric charges and currents are absent,
as in the source-free \Meq/,
there is an electric \potsys/ analogous
to the magnetic one.
In 3+1 dimensions, the electric and magnetic potentials
are each covariant vector fields that are dual in the sense
that they are exchanged under a \dutr/ on the electromagnetic field.
By comparison, in 2+1 dimensions the electric potential
is a scalar field satisfying the ordinary wave \eq/
and duality is lost.
(In more than three space dimensions, duality is also lost
since the electric potential becomes an antisymmetric tensor field.)
The introduction of these potentials for \Meq/ is an instance of
Bluman's method of \potsys/s
\cite{BluKumRei:1988,Blu:potsys,BluDor:1995}
and Vinogradov's theory of coverings for PDE systems
\cite{Vin:covering,KraVin:covering-1,KraVin:covering-2}.
In these two approaches, nonlocal \syms/ of a given PDE system
are realized as local \syms/ of a \potsys/ (or covering system).
\Potsys/s are characterized by the embedding property
\cite{Blu:potsys} that modulo gauge freedom
their solutions are in one-to-one correspondence with solutions of
the given PDE system.
(Gauge freedom, meaning a local \sym/ that depends on an arbitrary
function of all independent variables,
arises automatically for potentials only
in more than one space dimension.)
It is essential to have a gauge imposed on potentials
in order to obtain nonlocal \syms/,
because as proved by Anco \& Bluman \cite{AncBlu:1997JMP}
when a \potsys/ possesses gauge freedom
then all of its local \syms/ project onto only
local (gauge-invariant) \syms/ of the original PDE system
(provided the system is locally well-posed in the sense that
it is locally solvable \cite{Olv:symmbook}
but has no solutions involving
an arbitrary function of all independent variables).
Furthermore, an extension of their proof shows that
all local \conslaw/s of such a \potsys/
project onto \conscurr/s whose form is necessarily gauge invariant
modulo terms that are trivially divergence-free,
which presents a severe limitation for obtaining nonlocal \conslaw/s
with essential dependence on potentials.
In this paper we use a natural joint \potsys/ with Lorentz gauge
imposed to obtain new nonlocal \syms/ and nonlocal \conslaw/s
of \Meq/ in 3+1 dimensions.
This \potsys/ involves the simultaneous introduction of both
electric and magnetic vector potentials.
Thus, it inherits the electric-magnetic duality
invariance of \Meq/
and similarly is non-Lagrangian.
To our knowledge, there has been no previous systematic investigation of
the \syms/ or \conslaw/s of \Meq/ using these joint potentials.
After setting out some preliminaries in \secref{sec:prelim},
we discuss some important inter-relationships among
\syms/, \conslaw/s, and \adjsyms/ of \Meq/
and its various \potsys/s without gauges
in \secref{sec:cohom}.
These interrelationships come from the \loccohom{$p$} of \Meq/,
i.e. $p$-forms locally constructed from the spacetime coordinates
and the electromagnetic field and its derivatives (to some finite order)
on all solutions.
We show that the cohomology of closed $p$-forms modulo exact $p$-forms
determines mappings between \syms/ and \adjsyms/.
Since \adjsyms/ generate conserved currents
through a scaling formula \cite{AncPoh:2001,Anc:2003JPhysA},
we obtain a correspondence between \conslaw/s and both \syms/ and \adjsyms/.
Most importantly, this leads to an explicit formula that generates
\conscurr/s directly from any \syms/ of \Meq/ or its \jps/
(thus by-passing the absence of a Lagrangian).
We apply these results to the well-known \geom/ \syms/ of \Meq/
and their counterparts for the \potsys/s without gauges.
Our results explicitly demonstrate how these \potsys/s yield only
local \syms/ and local \conscurr/s of \Meq/,
with the exception of one \conscurr/ generated from
the duality-rotation \sym/ of the \jps/.
This current has an essential dependence on the joint potentials
yet is found to be invariant
with respect to the gauge freedom in these potentials
modulo trivially conserved terms.
In \secref{sec:jps-analysis}
we investigate the \jps/ with Lorentz gauge imposed.
A classification of geometric \syms/ is derived
by solving the \sym/ determining equations
using covariant tensorial methods,
from which we obtain new local \syms/
along with corresponding new local \conslaw/s
in terms of the potentials.
As main results, in \secref{sec:nonlocal}
we show that these \syms/ and \conslaw/s are nonlocal
under projection to \Meq/,
and we discuss some of their resulting features.
We make some concluding remarks in \secref{sec:concl}.
\section{Preliminaries}
\label{sec:prelim}
\Meq/ for the electromagnetic field tensor
$\F{}{\mu\nu}(x)=\F{}{[\mu\nu]}(x)$ in \Minksp/
$M\up{4} = (\mathbb{R}^{4},\flat{})$ are given by
\begin{equation}
\der{\mu}\F{\mu\nu}{}(x) = 4\pi \Je{\nu}(x) ,\qquad
\der{\mu}\duF{\mu\nu}{}(x) = 4\pi \Jm{\nu}(x) ,
\end{equation}
with electric and magnetic current sources.
Here
\begin{equation}
\duF{}{\mu\nu} = \frac{1}{2} \vol{\mu\nu\sigma\tau}{} \F{\sigma\tau}{}
\label{dualF}
\end{equation}
is the dual of $\F{}{\mu\nu}$,
$\vol{\mu\nu\sigma\tau}{}$ is the spacetime volume form,
$\x{\mu}{}$ are the standard Minkowski coordinates,
and $\der{\mu} = \parder{}{\x{\mu}{}}$ is the coordinate derivative.
Throughout, indices will be freely lowered or raised using
the spacetime metric $\flat{\mu\nu}$ and its inverse
$\invflat{\mu\nu}$
(with signature $(-+++)$).
When there are no current sources, the field \eq/s
\begin{equation}
\coder{\mu}\F{}{\mu\nu}(x) = 0, \qquad
\coder{\mu}\duF{}{\mu\nu}(x) = 0
\label{ME}
\end{equation}
display invariance under the \dutr/
\begin{equation}
\F{}{\mu\nu} \rightarrow \duF{}{\mu\nu}, \qquad
\duF{}{\mu\nu} \rightarrow -\F{}{\mu\nu}.
\label{ME-duality-F}
\end{equation}
To introduce potential variables in a covariantly natural way,
we rewrite the source-free \Meq/ \eqref{ME} in the equivalent form
\begin{equation}
\der{[\sigma}\F{}{\mu\nu]}(x) = 0, \qquad
\der{[\sigma}\duF{}{\mu\nu]}(x) = 0.
\label{Meq2}
\end{equation}
Since $F(x) = \F{}{\mu\nu}(x) d\x{\mu}{} \wedge d\x{\nu}{}$
is a closed 2-form and \Minksp/ is topologically trivial,
we can conclude by Poincar\'{e}'s Lemma that $F(x)$ is exact, \ie/
\begin{equation}
\F{}{\mu\nu}(x) = \der{[\mu} \A{}{\nu]}(x)
\label{Adefn}
\end{equation}
for some 1-form potential $A(x) = \A{}{\nu}(x) d\x{\nu}{}$.
The standard {\em \mps/} is given by
\begin{equation}
\coder{\mu} \der{[\mu} \A{}{\nu]}(x) = 0
\label{MPS}
\end{equation}
which is a self-adjoint system and thus arises from a Lagrangian.
This system \eqref{MPS} possesses gauge freedom
\begin{equation}
\A{}{\nu}(x) \rightarrow \A{}{\nu}(x) + \der{\nu}\chi(x),
\label{MPS-gauge-sym}
\end{equation}
where $\chi(x)$ is an arbitrary function of $\x{\mu}{}$.
The field \eq/s \eqref{Meq2} also imply that
${*F}(x) = \duF{}{\mu\nu}(x) d\x{\mu}{} \wedge d\x{\nu}{}$
is a closed 2-form.
So again by Poincar\'{e}'s Lemma,
${*F}(x)$ is exact, \ie/
\begin{equation}
\duF{}{\mu\nu}(x) = \der{[\mu} \Ap{}{\nu]}(x) ,
\label{A'defn}
\end{equation}
for some 1-form potential $A'(x) = \Ap{}{\nu}(x) d\x{\nu}{}$
which satisfies an {\em electric \potsys/}
analogous to the magnetic \potsys/ for $A(x)$.
Since by duality
the electric \potsys/ shares all the same properties
as the magnetic \potsys/,
we will omit it in our subsequent discussion and results.
A further natural \potsys/ of \Meq/ \eqref{ME} is obtained by
introducing both electric and magnetic potentials simultaneously.
Since $\F{}{\mu\nu}(x) = \der{[\mu} \A{}{\nu]}(x)$
and $\duF{}{\mu\nu}(x) = \der{[\mu} \Ap{}{\nu]}(x)$
must satisfy the duality relation \eqref{dualF},
we define their joint {\em electric-magnetic \potsys/} to be
\begin{equation}
\der{[\mu} \Ap{}{\nu]}(x) =
\frac{1}{2} \vol{\mu\nu\sigma\tau}{} \coder{\sigma} \A{\tau}{}(x).
\label{JPS}
\end{equation}
The \dutr/ (\ref{ME-duality-F}) on the electromagnetic
field induces a corresponding \dutr/
\begin{equation}
\A{}{\mu} \rightarrow \Ap{}{\mu}, \qquad \Ap{}{\mu} \rightarrow -\A{}{\mu}
\label{JPS-duality}
\end{equation}
on the potentials.
(Note that putting $A'={\rm i} A$ would give the self-dual \Meq/.)
The electric-magnetic \potsys/ \eqref{JPS} admits the gauge freedom
\begin{eqnarray}
\A{}{\nu}(x) \rightarrow \A{}{\nu}(x) + \der{\nu}\chi(x), &&
\Ap{}{\nu}(x) \rightarrow \Ap{}{\nu}(x) + \der{\nu}\chi'(x),
\label{JPS-gaugesym}
\end{eqnarray}
where $\chi(x)$ and $\chi'(x)$ are arbitrary functions of $\x{\mu}{}$.
Unlike the standard \potsys/ \eqref{MPS},
the joint system \eqref{JPS} is not self-adjoint,
and hence it is a non-Lagrangian system.
It is important to note that
via the embedding relations \eqref{Adefn}, \eqref{A'defn},
and the duality relation \eqref{dualF},
the solutions of the \potsys/s \eqrefs{MPS}{JPS}
modulo gauge freedom
are in one-to-one correspondence with the solutions of \Meq/ \eqref{ME}.
Associated with \Meq/ or any of its \potsys/s is
the respective jet space $\Jsp{q}$ of order $0\leq q \leq \infty$
defined as the coordinate manifold such that each point
($q$-jet) in $\Jsp{q}$ is identified with a spacetime point $x$
and the values of the field or potential(s)
and its partial derivatives up to order $q$ at $x$.
Note here that the jet space $\Jsp{0}$ is identified with $M\up{4} \times E$
where $E$ is the vector space of 2-forms for the case of \Meq/,
1-forms for the case of the magnetic \potsys/,
and pairs of 1-forms in the case of the joint \potsys/.
In this setting we use $\D{\mu}$ to denote
the total derivative operator with respect to $\x{\mu}{}$
and write a subscript ``$,\mu$'' for coordinates
corresponding to differentiation by $\D{\mu}$ on the field or potential(s)
in the standard way \cite{AncPoh:2001,Poh:survey}.
For \Meq/ the space of solutions is represented by
the submanifold (solution jet space) $\Rsp{}(F) \subset \Jsp{1}(F)$
whose coordinates quotient out the field equations
on the first-order partial derivatives of the electromagnetic field
in $\Jsp{1}(F)$ \cite{AncPoh:2001,Poh:survey}.
The $q$-prolonged solution jet space $\Rsp{q}(F) \subset \Jsp{q+1}(F)$
of \Meq/ is defined by a similar quotient
with respect to the $q$th-order partial derivatives
of the electromagnetic field equations, $1 \leq q \leq \infty$.
There is an analogous construction of (prolonged)
solution jet spaces $\Rsp{q}(A,A') \subset \Jsp{q+1}(A,A')$
in the case of the joint \potsys/,
and $\Rsp{q}(A) \subset \Jsp{q+2}(A)$
in the case of the magnetic \potsys/ \cite{Poh:survey}.
Note explicit coordinates for $\Rsp{0}(F) :=\Rsp{}(F)$ consist of
$( \x{\mu}{},\F{}{\mu\nu},{\rm trfr} \F{}{\mu(\nu,\sigma)} )$;
likewise
$( \x{\mu}{},\A{}{\nu},\Ap{}{\nu},\F{}{\mu\nu},
\A{}{(\nu,\mu)},\Ap{}{(\nu,\mu)} )$
and
$( \x{\mu}{},\A{}{\nu},\F{}{\mu\nu},\A{}{(\nu,\mu)},
\A{}{(\nu,\mu\sigma)},{\rm trfr} \F{}{\mu(\nu,\sigma)} )$
are coordinates for $\Rsp{0}(A,A'):=\Rsp{}(A,A')$
and $\Rsp{0}(A):=\Rsp{}(A)$,
where ``${\rm trfr}$'' on a tensor stands for its totally trace-free part
with respect to the Minkowski metric.
Similar coordinates can be written down for
the prolonged solution spaces to all orders,
representing those components of
the electromagnetic field, potentials, and their partial derivatives
that are freely specifiable at a spacetime point.
Throughout, we indicate jet space coordinates by writing
(derivatives of) $F,A,A'$ without $(x)$ dependence.
It would be typical to proceed by defining \syms/
as infinitesimal transformations
on the (prolonged) solution jet space $\Rsp{\infty}$.
For our purposes, an equivalent characterization of \syms/
via determining equations \cite{BluAnc:2002book} is better suited.
For \Meq/ or its \potsys/s,
local \syms/ of order $q<\infty$
are characterized by $E$-valued functions
on $\Jsp{q} \subset \Jsp{\infty}$ (in the given coordinates)
whose restriction to $\Rsp{\infty}$
satisfies the linearization of the system equations.
Similarly, local \adjsyms/ of order $q$ are characterized by
$\tilde{E}$-valued functions on $\Jsp{q} \subset \Jsp{\infty}$
whose restriction to $\Rsp{\infty}$
satisfies the adjoint linearization of the system equations,
where $\tilde{E}$ is the vector space of 2-forms
in the case of the joint \potsys/,
1-forms in the case of the magnetic \potsys/,
and pairs of 1-forms in the case of \Meq/.
A \sym/ or \adjsym/ of order $q$ is trivial if it vanishes
when evaluated on $\Rsp{\infty}$;
two \syms/ or \adjsyms/ that differ by a trivial one
are considered to be equivalent.
\mystretch{1.0}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|} \hline
System & Symmetry Equations & Adjoint-symmetry Equations
\\ \hline\hline
$\begin{array}{c} \mbox{\Meq/}\\
\coD{\mu} \F{}{\mu\nu} = 0 \\
\frac{1}{2}\vol{\mu\nu\sigma\tau}{} \coD{\mu} \F{\sigma\tau}{} = 0
\end{array}$ &
$\begin{array}{c}
\coD{\mu} \P{}{\mu\nu} = 0 \\
\frac{1}{2}\vol{\mu\nu\sigma\tau}{}\coD{\mu} \P{\sigma\tau}{} = 0 \\
\end{array}$ &
$\D{[\mu} \Qp{}{\nu]} =
\frac{1}{2} \vol{\mu\nu\sigma\tau}{} \coD{\sigma} \Q{\tau}{}$
\\ \hline
$\begin{array}{c}
\mbox{Magnetic \potsys/}\\
\coD{\mu} \D{[\mu} \A{}{\nu]} = 0
\end{array}$ &
$\coD{\mu} \D{[\mu} \Q{}{\nu]} = 0$ &
$\coD{\mu} \D{[\mu} \Qp{}{\nu]} = 0$
\\ \hline
$\begin{array}{c}
\mbox{Joint \potsys/}\\
\D{[\mu} \Ap{}{\nu]} =
\frac{1}{2} \vol{\mu\nu\sigma\tau}{} \coD{\sigma} \A{\tau}{}
\end{array}$ &
$\D{[\mu} \Qp{}{\nu]} =
\frac{1}{2} \vol{\mu\nu\sigma\tau}{} \coD{\sigma} \Q{\tau}{}$ &
$\begin{array}{c}
\coD{\mu} \P{}{\mu\nu} = 0 \\
\frac{1}{2}\vol{\mu\nu\sigma\tau}{}\coD{\mu} \P{\sigma\tau}{} = 0 \\
\end{array}$
\\ \hline
\end{tabular}
\caption{\Sym/ and \adjsym/ \eq/s for \Meq/ and \potsys/s}
\label{table:sys-sym-adjsym}
\end{center}
\end{table}
\mystretch{1}
Viewed geometrically \cite{AndTor:1996},
a local \sym/ of \Meq/ or its \potsys/s
describes a generalized vector field ${\bf X}$ on $M\up{4} \times E$
whose prolongation ${\rm pr} {\bf X}$ to $\Jsp{\infty}$
is tangent to the solution jet space $\Rsp{\infty}$
and involves no motion on the spacetime coordinates $\x{\mu}{}$.
In particular, corresponding to
$P\down{\mu\nu}$, $Q\down{\mu}$, $(Q\down{\mu},Q'\down{\mu})$
appearing in the \sym/ equations
in \tableref{table:sys-sym-adjsym} we have the generators
\begin{equation}
{\bf X} = \P{}{\mu\nu} \Parder{\F{}{\mu\nu}}, \qquad
{\bf X} = \Q{}{\mu} \Parder{\A{}{\mu}}, \qquad
{\bf X} = \Q{}{\mu} \Parder{\A{}{\mu}} + \Qp{}{\mu} \Parder{\Ap{}{\mu}} .
\end{equation}
This definition can be obviously generalized
to allow motion on $\x{\mu}{}$
but it is well-known that every such \sym/ is equivalent to one
(referred to as its evolutionary form) without any motion on $\x{\mu}{}$
\cite{Olv:symmbook,AndTor:1996}.
In contrast, unlike for \syms/,
there is no obvious geometrical meaning for \adjsyms/
unless the system equations are self-adjoint
(in which case \adjsyms/ coincide with \syms/).
A local conserved current of order $q<\infty$ for \Meq/ or its \potsys/s
is a vector function $\curr{\mu}$ on $\Jsp{q} \subset \Jsp{\infty}$
whose divergence vanishes on the solution jet space, namely
\begin{equation}
\D{\mu} \curr{\mu} = 0 \quad \eqtext{ on $\Rsp{\infty}$} .
\end{equation}
A current is trivial if it is equal to a curl
$\curr{\mu} = \D{\nu}\curl{\mu\nu}{}$
when evaluated on $\Rsp{\infty}$,
where $\curl{\mu\nu}{} = \curl{[\mu\nu]}{}$
is some skew-tensor function on $\Jsp{r}$, $q\leq r<\infty$.
Two currents that differ by a trivial one are considered to be equivalent.
The equivalence class of conserved currents
containing a current $\curr{\mu}$
is called the \conslaw/ associated with $\curr{\mu}$.
A \conslaw/ has order $q$ if the minimum order among all \conscurr/s
in its equivalence class is equal to $q$.
For \Meq/ and its potential systems,
the explicit coordinates
introduced earlier for $\Rsp{}$ and its prolongations
can be used to show that
all nontrivial currents of order $q$ modulo curls
are characterized by multipliers (also called characteristics)
such that the divergence of a current
$\D{\mu} \curr{\mu}$ on $\Jsp{q+1}$
yields, after integration by parts where necessary,
a contracted product of the multipliers and the system equations.
It is well-known from general results
(see for instance \cite{AncBlu:1997PRL})
that multipliers of order $q$ are \adjsyms/
subject to certain conditions on their adjoint-linearization
on $\Jsp{q}$,
and there is a homotopy integral formula to recover a current
(modulo a curl) from its multipliers.
Two main complications arise in dealing with the correspondence
between multipliers and currents for \Meq/.
Firstly,
because it is not a PDE system of Cauchy-Kovalevskaya form,
the simple relation \cite{Olv:symmbook,AncBlu:2002EJAM-2}
that two currents are equivalent
if and only if their multipliers agree on $\Rsp{\infty}$
breaks down
and there exist trivial currents
$\curr{\mu} =\D{\nu}( \F{\mu\nu}{}\chi+\duF{\mu\nu}{}\chi' )$
of order $q+1$ whose multipliers are a class of nontrivial \adjsyms/
$\Q{}{\nu}=\D{\nu}\chi$, $\Qp{}{\nu}=\D{\nu}\chi'$,
which do not vanish on $\Rsp{\infty}$,
for any (non-constant) functions $\chi,\chi'$ on $\Jsp{q}$.
Secondly,
due to the linear nature of \Meq/,
at every order $q>0$ there are other classes of nontrivial \adjsyms/
all of which fail to satisfy the adjoint-linearization conditions
even to within the addition of a trivial \adjsym/.
Similar complications are found to occur for
the magnetic and joint \potsys/s.
However,
an alternative way of generating all nontrivial conserved currents
by-passing these complications
for \Meq/ and its \potsys/s
is provided by a general scaling formula that produces conserved currents
directly from \adjsyms/
\cite{AncPoh:2001,Anc:2003JPhysA}.
This formula is derived from the adjoint relation
between the determining equations for \syms/ and \adjsyms/
of any given PDE system
\cite{AncBlu:1997PRL}.
The resulting \conscurr/ formulas for \Meq/ and its \potsys/s
are displayed in \tableref{table:adjrel-conslaw};
the notation $\mid_{\lambda F}$ is used to denote
a scaling of $F$ and derivatives of $F$ in functions on $\Jsp{q}(F)$
by a parameter $\lambda$,
and likewise for functions on $\Jsp{q}(A)$ or $\Jsp{q}(A,A')$.
\mystretch{1.0}
\begin{table}[h]
\begin{center}
$\begin{array}{|c|c|c|} \hline
\mbox{System} & \mbox{Adjoint-symmetry} &
\mbox{Conserved current formula}\\ \hline\hline
\begin{array}{c} \mbox{\Meq/} \\
\coD{\mu} \F{}{\mu\nu}=0 \\
\coD{\mu} \duF{}{\mu\nu}=0
\end{array} &
\Q{}{\nu}, \Qp{}{\nu} \mbox{ on } \Rsp{q}(F) &
\curr{\mu} = {\displaystyle\int}_0^1 \eval{ \left( \Q{}{\nu} \F{\mu\nu}{}
+ \Qp{}{\nu} \duF{\mu\nu}{} \right)}{\lambda F}
\frac{d\lambda}{\lambda}
\\ \hline
\begin{array}{c} \mbox{Magnetic \potsys/} \\
\coD{\mu} \D{[\mu} \A{}{\nu]}=0
\end{array} &
\Q{}{\nu} \mbox{ on } \Rsp{q}(A) &
\begin{array}{l}
\curr{\mu} =
{\displaystyle\int}_0^1 \eval{ \left( \Q{}{\nu} \coD{[\mu} \A{\nu]}{}
- \coD{[\mu} \Q{\nu]}{} \A{}{\nu} \right) }{\lambda A}
\frac{d\lambda}{\lambda}
\end{array}
\\ \hline
\begin{array}{c} \mbox{Joint \potsys/} \\
\D{[\mu} \Ap{}{\nu]} = {*\D{[\mu} \A{}{\nu]}}
\end{array}
&
\P{}{\mu\nu} \mbox{ on } \Rsp{q}(A,A') &
\curr{\mu} =
{\displaystyle\int}_0^1 \eval{\left( \P{\mu\nu}{} \Ap{}{\nu}
- \duP{\mu\nu}{} \A{}{\nu} \right)}{\lambda A,\lambda A'}
\frac{d\lambda}{\lambda}
\\ \hline
\end{array}$
\caption{Conserved current formulas for \Meq/ and \potsys/s}
\label{table:adjrel-conslaw}
\end{center}
\end{table}
\mystretch{1}
General results in \Ref{Anc:2003JPhysA}
establish a key property of these \conscurr/ formulas.
A direct proof in the case of \Meq/
was given in \Ref{AncPoh:2001}.
\begin{prop}
\label{prop:adjsym-conscur}
For an \adjsym/ that agrees with a multiplier on the solution jet space,
the conserved current scaling formula generates
an equivalent nontrivial current;
otherwise for an \adjsym/ differing from any multiplier
on the solution jet space,
it yields a trivial \conscurr/.
\end{prop}
In the case of the magnetic \potsys/,
which is self-adjoint and thus arises from a Lagrangian,
\adjsyms/ are the same as \syms/,
and multipliers correspond to those \syms/ under which
the Lagrangian of the system is invariant (to within a divergence).
The scaling formula in the Lagrangian case
produces a current equivalent to the one
that comes from Noether's theorem applied to such \syms/.
\section{Cohomology, local \syms/ and local \conslaw/s}
\label{sec:cohom}
In this section we present a unified account
of interrelationships (i.e. mappings) among
local \syms/, local \adjsyms/, and local \conslaw/s of \Meq/,
and the magnetic and joint \potsys/s on \Minksp/.
The cohomology of differential 1-forms and 2-forms on
the solution jet space of these systems together with a
locality-projection theorem for local \syms/ will be the main tools
in the derivation of these results.
We will consistently use $P$ to denote a \horiz/ 2-form,
$Q$ a \horiz/ 1-form, $\chi$ a scalar function (0-form),
e.g. on $J^q(F)$,
\begin{equation}
P = P\down{\mu\nu}[F] d\x{\mu}{} \wedge d\x{\nu}{}, \qquad
Q = Q\down{\mu}[F] d\x{\mu}{}, \qquad
\chi = \chi[F],
\end{equation}
where dependence on jet space coordinates to some finite order
is denoted by $[F]$.
A similar notation will be used for differential forms on $J^q(A)$ and
$J^q(A,A')$ and when these variables carry primes, tildes, etc.
The total differential will be denoted by $D$, e.g.
\begin{equation}
D{P} = \D{[\sigma} P\down{\mu\nu]}[F]
d\x{\sigma}{} \wedge d\x{\mu}{} \wedge d\x{\nu}{}, \qquad
D{Q} = \D{[\mu} Q\down{\nu]}[F]
d\x{\mu}{} \wedge d\x{\nu}{}, \qquad
D{\chi} = \D{\mu} \chi[F]
d\x{\mu}{}.
\end{equation}
In differential form notation,
the system equations and determining equations for \syms/ and \adjsyms/
are summarized in \tableref{table:sym-adjsym-diff-form}.
\mystretch{1.0}
\begin{table}[h]
\begin{center}
$\begin{array}{|c|c|c|} \hline
\mbox{System} & \mbox{\Sym/ Equations} &
\mbox{\Adjsym/ Equations} \\ \hline\hline
\begin{array}{c} D F=0 \\ D{*F}=0 \end{array} &
\begin{array}{c} D P[F]=0 \\ D{*P}[F]=0 \end{array}
& D Q'[F] = *D Q[F] \\ \hline
D{*D A} = 0 & D{*D Q}[A] = 0 &
D{*D Q'}[A] = 0 \\ \hline
D A' = {*D A} & D Q'[A,A'] = *D Q[A,A'] &
\begin{array}{c} D P[A,A']=0 \\ D{*P}[A,A']=0 \end{array} \\ \hline
\end{array}$
\caption{\Sym/ and \adjsym/ \eq/s for \Meq/ and \potsys/s}
\label{table:sym-adjsym-diff-form}
\end{center}
\end{table}
\mystretch{1}
Associated with the magnetic and joint \potsys/s,
there is a natural embedding of the respective jet spaces
$\Jsp{q+1}(A)$ and $\Jsp{q+1}(A,A')$
into the jet space $\Jsp{q}(F)$
under the total differential mapping given by
$F=D{A}=-{*D{A'}}$
and its obvious prolongation $(0 \leq q \leq \infty)$.
This embedding extends to the prolonged solution jet spaces
$\Rsp{\infty} \subset \Jsp{\infty}$
and is linear and one-to-one
modulo the gauge freedom on the potentials,
namely $F=0$ if and only if $A=D\chi$, $A'=D\chi'$,
where $\chi,\chi'$ are any scalar functions on $\Jsp{q}$.
We note the following consequences of gauge freedom
in the \potsys/s here.
The (adjoint-) \sym/ determining equations of the standard \potsys/
have solutions of the form $Q[A] = D\chi[A]$
(i.e. gauge symmetries on $A$)
for an {\it arbitrary} scalar function $\chi$ on $\Rsp{q}(A)$.
Such solutions can be also viewed as representing
a gauge freedom in the form of $Q[A]$.
Similarly, the \sym/ determining equations of the joint \potsys/
have solutions
$Q[A,A']=D\chi[A,A']$, $Q'[A,A']=D\chi'[A,A']$
(i.e. gauge \syms/ on $A,A'$)
for a pair of {\em arbitrary} scalar functions
$\chi,\chi'$ on $\Rsp{q}(A,A')$;
the \adjsym/ determining equations of this system
can be shown to have no gauge freedom in the form of $P[A,A']$
(basically, the latter equations are locally well-posed
as a PDE system for $P$).
By comparison with the corresponding determining equations
for \syms/ and \adjsyms/ of \Meq/,
it follows that $Q[F]=D\chi[F]$, $Q'[F]=D\chi'[F]$
represent gauge freedom in the form of \adjsyms/,
for pairs of {\it arbitrary} scalar functions $\chi,\chi'$
on $\Rsp{q}(F)$,
while there is no gauge freedom in the form
of the \syms/ (since \Meq/ are a locally well-posed PDE system).
The total differential $D$ obviously satisfies $D^2 = 0$
and hence defines a complex of \horiz/ forms on $\Jsp{\infty}$,
which is a generalization of the de Rham complex on \Minksp/
to the jet space setting \cite{And:1992}.
What we will refer to as \horiz/ forms on $\Rsp{q}$ can be defined
more precisely as the pullback to $\Rsp{\infty}$ (via the inclusion map)
of \horiz/ forms on $J^\infty$
whose coefficients depend on the jet coordinates up to a finite order
$q+1$ in the case of \Meq/ and the joint \potsys/;
$q+2$ in the case of the magnetic \potsys/.
For notational convenience, we will also denote the induced
total differential mapping on $\Rsp{\infty}$ by $D$ and
hence obtain the $D$-complex on $\Rsp{\infty}$.
Results concerning the local cohomology of this complex,
i.e. $D$-closed modulo $D$-exact \horiz/ forms,
will underlie our study of the local \syms/
and \adjsyms/ of \Meq/ and its \potsys/s.
It is worth remarking that, in contrast,
off $\Rsp{\infty}$
the local cohomology of the (free) $D$-complex is trivial,
since (see \cite{And:1992})
all $D$-closed \horiz/ $p$-forms on $\Jsp{\infty}$
for $1\leq p<4$ are $D$-exact.
\subsection{Cohomology and locality projection}
We now state the cohomology and locality-projection theorems
and then give their proofs afterwards.
\begin{thm}
{\bf (Local Cohomology)}
1-form cohomology: Let $Q$ be a \horiz/ 1-form on $\Rsp{q}$,
$0 \leq q < \infty$.
If $Q$ is closed, \ie/ $D Q=0$, then
\begin{eqnarray}
{\rm (i)}
& D F=D {*F}=0 & \Longrightarrow \qquad
Q[F] = D\chi[F] ;
\label{ME-1form}\\
{\rm (ii)}
& D{*D A}=0 & \Longrightarrow \qquad
Q[A] = D\chi[A] ;
\label{MPS-1form}\\
{\rm (iii)}
& D A' = *D A & \Longrightarrow \qquad
Q[A,A'] = D\chi[A,A'] .
\label{JPS-1form}
\end{eqnarray}
Thus, the \loccohom{1} is trivial for each of these systems.
2-form cohomology: Let $P$ be a \horiz/ 2-form on $\Rsp{q}$,
$0 \leq q < \infty$.
If $P$ is closed, \ie/ $D P=0$, then
\begin{eqnarray}
{\rm (i)}
& D F = D{*F}=0 & \Longrightarrow \qquad
P[F] = \c{1} F + \c{2} {*F} + D Q[F] ;
\label{ME-2form}\\
{\rm (ii)}
& D{*D A}=0 & \Longrightarrow \qquad
P[A] = c\, {*F} + D Q[A] ,\quad
F= D{A} ;
\label{MPS-2form}\\
{\rm (iii)}
& D A' = *D A & \Longrightarrow \qquad
P[A,A'] = D Q[A,A'] ;
\label{JPS-2form}
\end{eqnarray}
for some constants $c,c_1,c_2$.
Thus, $F$ and $\duF{}{}$ represent the only nontrivial \loccohom{2},
and this cohomology is killed by
the introduction of a corresponding potential.
\end{thm}
We remark that local 1-form and 2-form cohomology has
a field-theoretic interpretation in terms of conserved charges
\cite{Tor:lecturenotes},
describing electromagnetic fluxes through closed loops and surfaces
in spacetime.
(Namely, the integral of $*F(x),F(x)$ over any closed surface
yields the total magnetic and electric charge
enclosed within the surface;
in Minkowski space, these charges vanish
for all smooth solutions $F(x)$ of \Meq/.)
\begin{thm}
{\bf (Locality Projection)}
Let $Q$ be a \horiz/ 1-form on $\Rsp{q}$, $0 \leq q < \infty$.
If the 2-form $*D Q$ is closed, \ie/ $D{*D Q}=0$,
then
\begin{eqnarray}
{\rm (i)}
& D{*D A}=0 & \Longrightarrow \qquad
D Q[A] = P[F] ,\eqtext{ where $F=D A$} ;
\\
{\rm (ii)}
& D A'={*D A} & \Longrightarrow \qquad
D Q[A,A']=P[F] ,\eqtext{ where $F = D A = -{*D A'}$} .
\end{eqnarray}
Thus, any essential dependence on potentials in $Q$ is killed
under total exterior differentiation.
\end{thm}
A proof of the \loccohom{2} theorem for \Meq/
in the case where $P$ is a linear function on $\Rsp{q}(F)$
is given in \Ref{The:MSc}
using tensorial methods.
This proof amounts to showing that
the vanishing of the cohomology equation \eqref{ME-2form}
has no nontrivial linear solutions.
The nonlinear case can be reduced to the linear case
by standard linearization techniques
and the whole computation is especially tractable in spinor form
with the methods used in \Ref{AncPoh:2001}.
The proof for the \potsys/s
and the \loccohom{1} theorem
can be done by the same techniques.
The locality-projection theorem is a consequence of applying
a general result proved by Anco \& Bluman \cite{AncBlu:1997JMP}:
for a locally well-posed PDE system
(i.e. if it is locally solvable \cite{Olv:symmbook}
such that no solutions depend on an arbitrary function of
all independent variables),
the local \syms/ of any \potsys/ with gauge freedom
project onto only local \syms/ under the embedding
of the solution space of the \potsys/ into the solution space
of the given PDE system.
In particular, no projected \syms/ have any essential dependence
on the potentials.
If a \horiz/ 1-form $Q$ on $\Rsp{q}(A)$
satisfying $D{*D Q} =0$ is viewed as
a local \sym/ of the magnetic \potsys/,
then since \Meq/ is a locally well-posed system
we immediately conclude that the projected \sym/ $P=D Q$ for \Meq/
must be a \horiz/ 2-form on $\Rsp{q-1}(F)$.
A similar argument applies to a \horiz/ 1-form $Q$ on $\Rsp{q}(A,A')$.
Because $D{*D Q} =0$ implies that
the \horiz/ 2-form $P'=*D Q$ is closed,
the equation $*D Q=D Q'$
holds for some 1-form $Q'$ on $\Rsp{q}(A,A')$,
by the cohomology equation \eqref{JPS-2form}.
If we then view the pair $(Q,Q')$ as a local \sym/ of the joint \potsys/,
we again conclude
$P=D Q$ must be a 2-form on $\Rsp{q-1}(F)$.
It is crucial that both the cohomology and locality-projection theorems
are formulated on the solution jet spaces $\Rsp{q}$
of finite order $q<\infty$.
Indeed, on the infinite-order solution jet space $\Rsp{\infty}$,
these theorems break down in the following manner.
\begin{prop}
On $\Rsp{\infty}(F)$, the \loccohom{2} becomes formally trivial:
\begin{equation}
F = D A_F \qquad \mbox{and} \qquad \duF{}{} = D A'_F
\end{equation}
where
\begin{eqnarray}
A_F &=&
\sum_{k\geq 0} \frac{(-1)^k}{(k+1)!} x\lrcorner ({\mathscr D}^k F) ,
\label{A-F}\\
A'_F &=&
\sum_{k\geq 0} \frac{(-1)^k}{(k+1)!} x\lrcorner ({\mathscr D}^k {*F}) ,
\label{A'-F}
\end{eqnarray}
and ${\mathscr D}=\x{\sigma}{}\D{\sigma}$
denotes the dilation operator.
Furthermore, these 1-forms satisfy Cronstrom's gauge
\begin{equation}
x \lrcorner A_F = x \lrcorner A'_F = 0 .
\end{equation}
\end{prop}
The proof amounts to an explicit computation and will be omitted.
We note that the formal series \eqrefs{A-F}{A'-F} arise from
integration by parts of the Poincar\'e homotopy formula
for the de Rham cohomology of differential forms on \Minksp/
\cite{Olv:symmbook}.
Thus, the notion of nonlocality or nontrivial cohomology
associated with \Meq/ and its \potsys/s
is meaningful only when we work in finite-order jet spaces.
\subsection{Mappings and duality}
We now give the statements of our main results
which are consequences of the cohomology and
locality-projection theorems.
Throughout this section, it is understood that we work on
finite-order solution jet spaces $\Rsp{q}$, $0\leq q < \infty$.
\begin{thm}
\label{thm:PQdecomp}
The local \syms/ and local \adjsyms/ of \Meq/ and its \potsys/s
have the following decompositions:
\begin{enumerate}
\item[{\rm (i)}] $D F=D{*F}=0 \Longrightarrow$
\begin{eqnarray}
&&
P[F] = c\, F + c'\, {*F} + D\t{Q}[F] ,\quad
{*P}[F] = c\, {*F} -c'\, F + D\t{Q}'[F] ,
\label{ME:Pdecomp}\\
&&
Q[F] = \t{Q}[F] + D\chi[F] ,\quad
Q'[F] = \t{Q}'[F] + D\chi'[F] ;
\label{ME:QQ'decomp}
\end{eqnarray}
\item[{\rm (ii)}] $D{*D A}=0 \Longrightarrow$
\begin{eqnarray}
&&
Q[A] = c\, A + \t{Q}[F] + D\chi[A] ,
\label{PS:Qdecomp}\\
&&
Q'[A] = c\, A + \t{Q}[F] + D\chi[A] ,
\label{PS:Q'decomp}
\end{eqnarray}
with $F=D A$;
\item[{\rm (iii)}] $D A'=*D A \Longrightarrow$
\begin{eqnarray}
&&
Q[A,A'] = c\, A + c'\, A' + \t{Q}[F] + D\chi[A,A'] ,\quad
Q'[A,A'] = c\, A' -c'\, A + \t{Q}'[F] + D\chi'[A,A'] ,
\nonumber\\&&
\label{JPS:QQ'decomp}\\
&&
P[A,A']= c\, F + c'\, {*F} + D\t{Q}[F] ,\quad
{*P}[A,A']= c\, {*F} -c'\, F + D\t{Q}'[F] ,
\label{JPS:Pdecomp}
\end{eqnarray}
with $F=D A, {*F}=D A'$;
\end{enumerate}
for some constants $c,c'$,
where
\begin{equation}
D\t{Q}'[F] =*D\t{Q}[F]
\label{QQ'relation}
\end{equation}
in all cases.
These decompositions
are unique up to addition of arbitrary gradients
$D\chi[F]$ to $\t{Q}[F]$, $D\chi'[F]$ to $\t{Q}'[F]$,
and are stable under the duality invariance
\begin{equation}
P \rightarrow {*P} ,\quad (Q,Q') \rightarrow (Q',-Q)
\label{PQduality}
\end{equation}
of the \sym/ equations and \adjsym/ equations.
\end{thm}
We mention that, moreover,
the 1-forms $\t{Q}[F],\t{Q}'[F]$ here are
canonically related under the duality transformation
\eqref{ME-duality-F} on $F$,
as will be discussed in Proposition~\ref{prop:dualitydecomp} later.
In the \sym/ decompositions
\eqref{ME:Pdecomp}, \eqref{PS:Qdecomp}, \eqref{JPS:QQ'decomp},
the cohomology terms correspond to
infinitesimal scaling and duality-rotation transformations
\begin{eqnarray}
{\bf X}_{\it scal} &=&
F\Parder{F} ,\quad
A\Parder{A} ,\quad
A\Parder{A} + A'\Parder{A'} ,
\label{Xscaling}\\
{\bf X}_{\it dual} &=&
*F\Parder{F} ,\quad
A'\Parder{A} - A\Parder{A'} .
\label{Xduality}
\end{eqnarray}
Note the duality-rotation is not realized as a local \sym/
of the standard \potsys/.
The \sym/-decomposition terms that involve $\chi$ or $\chi'$
correspond to a gauge freedom in the \syms/,
namely the infinitesimal transformations
\begin{equation}
{\bf X}_{\it gauge} =
D\chi\Parder{A} ,\quad
D\chi\Parder{A} + D\chi'\Parder{A'}
\label{Xgauge}
\end{equation}
for arbitrary functions $\chi,\chi'$ on $\Jsp{q}$ are \syms/.
The remaining terms in the \sym/ decompositions
represent gauge-invariant infinitesimal transformations
\begin{equation}
{\bf X} =
D\t{Q}[F] \Parder{F} ,\quad
\t{Q}[F] \Parder{A} ,\quad
\t{Q}[F] \Parder{A} + \t{Q}'[F] \Parder{A'}
\end{equation}
which are themselves \syms/.
Note these terms are well-defined only up to gauge freedom \eqref{Xgauge}
such that $\chi,\chi'$ are functions on $\Jsp{q}(F)$.
These results establish that the vector spaces of \syms/ and \adjsyms/
of each system are a direct sum of
cohomology subspaces
spanned by the separate terms proportional to $c,c'$,
and a complementary subspace
identified with the gauge-invariant non-cohomology terms
involving $\t{Q},\t{Q}'$,
up to gradient terms $D\chi,D\chi'$.
Hereafter we let
$X$ denote a vector space of \syms/,
$Y$ a vector space of \adjsyms/,
and we use superscripts $c,c',0$ to distinguish
the respective subspaces in the direct sum decomposition;
superscripts $\chi,\chi'$ will denote the
vector subspace defined by all $D\chi,D\chi'$ terms
while a tilde will stand for the quotient with respect to this subspace.
Thus we have the following vector space decompositions:
\begin{eqnarray}
&{\rm (i)} &
X_F = X^c_F \oplus X^{c'}_F \oplus X^0_F ,\quad
Y_F = Y^0_F ,
\\
& {\rm (ii)} &
X_A = X^c_A \oplus X^0_A ,\quad
Y_A = Y^c_A \oplus Y^0_A ,
\\
& {\rm (iii)} &
X_{A,A'}
= X^c_{A,A'} \oplus X^{c'}_{A,A'} \oplus X^0_{A,A'} ,\quad
Y_{A,A'} = Y^c_{A,A'} \oplus Y^{c'}_{A,A'} \oplus Y^0_{A,A'} ,
\end{eqnarray}
where the $X^0,Y^0$ subspaces in the case of \horiz/ 1-forms
naturally partition into equivalence classes
modulo gradients
\begin{eqnarray}
& {\rm (iv)} &
\tilde Y^0_F = Y^0_F/ Y^{\chi,\chi'}_F ,\quad
\tilde X^0_A = X^0_A/X^{\chi}_A ,\quad
\tilde Y^0_A = Y^0_A/Y^{\chi}_A ,\quad
\tilde X^0_{A,A'} = X^0_{A,A'}/X^{\chi,\chi'}_{A,A'} .
\label{gaugeinvXY}
\end{eqnarray}
It will be convenient to introduce a linear map $*'$
on \horiz/ 1-forms $\t{Q}[F] \mod{D\chi[F]}$
by the equation
\begin{equation}
D{*'\t{Q}}[F] = *D\t{Q}[F]
\quad\eqtext{ on $\Rsp{q}(F)$, }
\label{Qduality}
\end{equation}
which defines an automorphism of each vector space \eqref{gaugeinvXY}.
Note the relation \eqref{QQ'relation} can be simply written
$\t{Q}' = {*'\t{Q}}$.
Importantly, interrelationships hold between any two decompositions
\eqsref{ME:Pdecomp}{JPS:Pdecomp},
given by linear maps summarized in the following four theorems.
\begin{thm}
{\bf (Self-correspondences related to the systems
$D{F}=D{*F}=0$,
$D{A'}=*D{A}$,
$D{*D{A}}=0$)}
\label{thm:self-mappings}
There is a linear mapping between:
\begin{enumerate}
\item[{\rm (i)}]
local \syms/ on $\Rsp{q}(F)$ and local \adjsyms/ on $\Rsp{q-1}(F)$,
given by
\begin{eqnarray}
&&
P[F] \mod{ F,{*F} } \longleftrightarrow D(\t{Q}[F] \mod{ D\chi[F] } ) ,
\\
&&
*P[F] \mod{ F,{*F} } \longleftrightarrow D( \t{Q}'[F] \mod{ D\chi'[F] } ) ,
\end{eqnarray}
corresponding to the isomorphism of vector spaces
\begin{equation}
X^0_F \cong \tilde Y^0_F .
\end{equation}
\item[{\rm (ii)}]
local \syms/ on $\Rsp{q}(A,A')$ and local \adjsyms/ on $\Rsp{q+1}(A,A')$,
given by
\begin{eqnarray}
&&
D( Q[A,A'] \mod{D\chi[A,A']} ) \longleftrightarrow P[A,A'] ,
\label{JPS-QtoPmap}\\
&&
D( Q'[A,A'] \mod{D\chi'[A,A']} ) \longleftrightarrow *P[A,A'] ,
\label{JPS-Q'toPmap}
\end{eqnarray}
corresponding to the isomorphism of vector spaces
\begin{equation}
X^c_{A,A'} \cong Y^c_{A,A'} ,\quad
X^{c'}_{A,A'} \cong Y^{c'}_{A,A'} ,\quad
\tilde X^0_{A,A'} \cong Y^0_{A,A'} .
\end{equation}
\item[{\rm (iii)}]
local \syms/ on $\Rsp{q}(A)$ and local \adjsyms/ on $\Rsp{q}(A)$,
given by the direct identification
\begin{equation}
Q[A] \longleftrightarrow Q'[A] ,
\label{PS:QtoQ'id}
\end{equation}
namely $X_A = Y_A$,
as well as a dual identification
\begin{equation}
{*'}( Q[A] \mod{A,D\chi[A]} ) \longleftrightarrow Q'[A] \mod{A,D\chi[A]} ,
\label{PS:QtoQ'}
\end{equation}
corresponding to a nontrivial duality (isomorphism) of vector spaces
$\tilde X^0_A \cong \tilde Y^0_A$.
\end{enumerate}
\end{thm}
Note the composition of the maps \eqrefs{PS:QtoQ'id}{PS:QtoQ'}
yields a linear mapping of local (adjoint-) \syms/
modulo $A,D\chi[A]$ on $\Rsp{q}(A)$
into themselves,
corresponding to the vector space automorphism
$*': \tilde X^0_A \longrightarrow \tilde X^0_A$
(and correspondingly $\tilde Y^0_A \longrightarrow \tilde Y^0_A$).
\begin{thm}
{\bf (Correspondences related to the systems
$D{*D A}=0$ and $D A'=*D A$)}
\label{thm:std-joint}
There is a linear mapping between:
\begin{enumerate}
\item[{\rm (i)}]
local (adjoint-) \syms/ on $\Rsp{q}(A)$ and local \syms/ on $\Rsp{q}(A,A')$,
given by
\begin{eqnarray}
Q[A] \mod{D\chi[A]} &\longleftrightarrow& Q[A,A'] \mod{A',D\chi[A,A']} ,
\\
*'Q[A] \mod{D\chi[A]} &\longleftrightarrow& Q'[A,A'] \mod{A, D\chi'[A,A']} ,
\end{eqnarray}
where $*'$ is extended by $*'A := A'$ and linearity
so it is well-defined
on all local (adjoint-) \syms/ $Q[A]$ modulo gradients $D\chi[A]$,
corresponding to the isomorphism of vector spaces
\begin{equation}
X^c_A \cong X^c_{A,A'} ,\quad
\tilde X^0_A \cong \tilde X^0_{A,A'} .
\end{equation}
\item[{\rm (ii)}]
local (adjoint-) \syms/ on $\Rsp{q}(A)$
and local \adjsyms/ on $\Rsp{q+1}(A,A')$,
given by
\begin{equation}
D (Q[A] \mod{D\chi[A]}) \longleftrightarrow P[A,A'] \mod *F ,
\end{equation}
corresponding to the isomorphism of vector spaces
\begin{equation}
X^c_A \cong Y^c_{A,A'} ,\quad
\tilde X^0_A \cong Y^0_{A,A'} .
\end{equation}
\end{enumerate}
\end{thm}
\begin{thm}
{\bf (Correspondences related to the systems
$D F= D{*F}=0$ and $D A'=*D A$)}
\label{thm:ME-joint}
There is a linear mapping between:
\begin{enumerate}
\item[{\rm (i)}]
local \syms/ on $\Rsp{q}(F)$ and local \adjsyms/ on $\Rsp{q+1}(A,A')$,
given by
\begin{equation}
P[F] \longleftrightarrow P[A,A']
\end{equation}
which is an isomorphism of vector spaces
\begin{equation}
X^c_F \cong Y^c_{A,A'} ,\quad
X^{c'}_F \cong Y^{c'}_{A,A'} ,\quad
X^0_F \cong Y^0_{A,A'} .
\end{equation}
\item[{\rm (ii)}]
local \adjsyms/ on $\Rsp{q}(F)$ and local \syms/ on $\Rsp{q+1}(A,A')$,
given by
\begin{eqnarray}
&&
Q[F] \mod{D\chi[F]} \longleftrightarrow Q[A,A'] \mod{A,A',D\chi[A,A']} ,
\\
&&
Q'[F] \mod{D\chi'[F]} \longleftrightarrow Q'[A,A'] \mod{A,A',D\chi'[A,A']} ,
\end{eqnarray}
corresponding to the isomorphism of vector spaces
\begin{equation}
\tilde Y^0_F \cong \tilde X^0_{A,A'} .
\end{equation}
\item[{\rm (iii)}]
local \syms/ on $\Rsp{q+1}(A,A')$ and local \syms/ on $\Rsp{q}(F)$,
given by
\begin{eqnarray}
&&
D( Q[A,A'] \mod{D\chi[A,A']} ) \longleftrightarrow P[F] ,
\\
&&
D( Q'[A,A'] \mod{D\chi'[A,A']} ) \longleftrightarrow *P[F] ,
\end{eqnarray}
corresponding to the isomorphism of vector spaces
\begin{equation}
X^c_{A,A'} \cong X^c_F ,\quad
X^{c'}_{A,A'} \cong X^{c'}_F ,\quad
\tilde X^0_{A,A'} \cong X^0_F .
\end{equation}
\item[{\rm (iv)}]
local \adjsyms/ on $\Rsp{q}(F)$ and local \adjsyms/ on $\Rsp{q+1}(A,A')$,
given by
\begin{eqnarray}
&&
D( Q[F] \mod{D\chi[F] }) \longleftrightarrow P[A,A'] \mod{F,*F} ,
\\
&&
D( Q'[F] \mod{D\chi'[F] }) \longleftrightarrow *P[A,A'] \mod{F,*F} ,
\end{eqnarray}
corresponding to the isomorphism of vector spaces
\begin{equation}
\tilde Y^0_F \cong Y^0_{A,A'} .
\end{equation}
\end{enumerate}
\end{thm}
\begin{thm}
{\bf (Correspondences related to the systems
$D F= D{*F}=0$ and ${D{*D A}}=0$)}
\label{thm:ME-std}
There is a linear mapping between:
\begin{enumerate}
\item[{\rm (i)}]
local \syms/ on $\Rsp{q}(F)$ and local \syms/ on $\Rsp{q+1}(A)$,
given by
\begin{equation}
P[F] \mod{*F} \longleftrightarrow D( Q[A] \mod{D\chi[A]} ) ,
\end{equation}
corresponding to the isomorphism of vector spaces
\begin{equation}
X^c_F \cong X^c_A ,\quad
X^0_F \cong \tilde X^0_A .
\end{equation}
\item[{\rm (ii)}]
local \adjsyms/ of $\Rsp{q}(F)$ and local \syms/ of $\Rsp{q+1}(A)$,
given by
\begin{eqnarray}
Q[F] \mod{D\chi[F]} &\longleftrightarrow & Q[A] \mod{A,D\chi[A]},
\\
Q'[F] \mod{D\chi'[F]} &\longleftrightarrow & *'Q[A] \mod{A,D\chi[A]},
\end{eqnarray}
corresponding to the isomorphism of vector spaces
\begin{equation}
\tilde Y^0_F \cong \tilde X^0_A .
\end{equation}
\end{enumerate}
\end{thm}
\Proof{}
We give the proof of Theorem~\ref{thm:self-mappings} in detail.
The proofs of Theorems \ref{thm:std-joint}--\ref{thm:ME-std}
are readily derived in a similar manner from
the decompositions \eqsref{ME:Pdecomp}{JPS:Pdecomp}
combined with the cohomology equations \eqsref{ME-1form}{JPS-2form}
and Theorem~\ref{thm:self-mappings}.
For part (i) of Theorem~\ref{thm:self-mappings},
by the \sym/ decomposition \eqref{ME:Pdecomp} with $c=c'=0$
(i.e. quotienting out the scaling and duality-rotation terms),
the pair of \horiz/ 1-forms $(\t{Q},\t{Q}')$ satisfy
the \adjsym/ equation on $\Rsp{q}(F)$.
Conversely, in the \adjsym/ decomposition \eqref{ME:QQ'decomp}
we see that the \horiz/ 2-forms $D\t{Q}$, $D\t{Q}'$,
and their duals are closed due to the \adjsym/ equation
and hence they directly satisfy
the \sym/ equations on $\Rsp{q}(F)$.
Part (ii) follows analogously from the
\sym/ and \adjsym/ decompositions \eqrefs{JPS:QQ'decomp}{JPS:Pdecomp},
including the cases $c\neq 0,c'\neq 0$ via the obvious mappings
$F \longleftrightarrow D{A}$, ${*F} \longleftrightarrow D{A'}$.
Finally for part (iii) of the theorem,
we apply the 2-form cohomology equation \eqref{MPS-2form}
to the \sym/ equation
$D{*D{Q}}=0$ for the \horiz/ 1-form $Q$ on $\Rsp{q}(A)$,
which shows that the 2-form $*D(Q+c A)$ is exact
for some constant $c$.
Hence the equation $*D(Q+c A) = D{Q'}$ holds
for some \horiz/ 1-form $Q'$ on $\Rsp{q}(A)$,
and we see that the dual 2-form $*D{Q'}$ is exact
and therefore is closed.
Thus, $Q'$ satisfies the (adjoint-) \sym/ equation $D{*D{Q'}}=0$
on $\Rsp{q}(A)$.
\hfill$\Box$
We now prove the \sym/ and \adjsym/ decompositions
\eqsref{ME:Pdecomp}{JPS:Pdecomp}.
\begin{prop}
\label{prop:MEsym}
Every local \sym/ $P=P[F]$
of order $q$ of $D{F}=D{*F}=0$
has the form \eqref{ME:Pdecomp}
for some constants $c,c'$,
where $(\t{Q}[F],\t{Q}'[F])$ is a local \adjsym/
of order $q-1$ of $D{F}=D{*F}=0$,
and conversely.
\end{prop}
\Proof{}
Since $P[F]$ and ${*P}[F]$ are closed \horiz/ 2-forms on $\Rsp{q}(F)$
for finite $q$
then by the 2-form cohomology equation \eqref{ME-2form} we have
\begin{eqnarray}
P[F] &=& \c{1} F + \c{2} {*F} + D\t{Q}[F],
\label{ME-Pform}\\
{*P}[F] &=& \c{3} F + \c{4} {*F} + D\t{Q}'[F],
\label{ME-*Pform}
\end{eqnarray}
for some constants $\c{1},\c{2},\c{3},\c{4}$
and some \horiz/ 1-forms $\t{Q}[F],\t{Q}'[F]$.
Applying $*$ to \eqref{ME-Pform} and equating it to \eqref{ME-*Pform},
we obtain
\begin{equation}
(\c{1} - \c{4}) {*F} - (\c{2} + \c{3}) F =
D\t{Q}'[F] - *D\t{Q}[F].
\label{Fcohomeq}
\end{equation}
The differential order of the right side is at least one
while the left side is of differential order zero,
hence on $\Rsp{q}(F)$ a descent argument
(similar to the ones used in the classification results
in \Ref{AncPoh:2001,AncPoh:2002})
shows that both sides of \eqref{Fcohomeq} must vanish
and so $D\t{Q}'[F] = *D\t{Q}[F]$.
Since $F$, ${*F}$ are \loccohom{2} elements
(which are linearly independent),
then we must have $\c{1}=\c{4}\equiv c$, $\c{2}=-\c{3}\equiv c'$,
which establishes \eqref{ME:Pdecomp}.
The converse is immediate.
\hfill$\Box$
\begin{prop}
\label{prop:JPSsym}
Every local \sym/ $Q=Q[A,A']$, $Q'=Q'[A,A']$
of order $q$ of $D{A'}={*D{A}}$
has the form \eqref{JPS:QQ'decomp}
for some constants $c,c'$,
and some scalar functions $\chi[A,A'],\chi'[A,A']$,
with $F=D A = -{*D A'}$,
where $(\t{Q}[F], \t{Q}'[F])$ is a local \adjsym/
of order $q-1$ of $D{F} = D{*F}=0$,
and conversely.
\end{prop}
\Proof{}
By the locality-projection theorem, both $D{Q}=P[F]$,
$D{Q'}={*P}[F]$ are local \syms/ of $D{F}=D{*F}=0$.
Using Proposition~\ref{prop:MEsym}
and writing $F=D{A},{*F}=D{A'}$,
we have
\begin{equation}
D( Q[A,A'] - c A - c' A' - \t{Q}[F] ) = 0 ,\quad
D( Q'[A,A'] - c A' + c' A - \t{Q}'[F] ) = 0 ,
\end{equation}
where $(\t{Q}[F]$, $\t{Q}'[F])$ is a local \adjsym/
of $D{F}=D{*F}=0$.
Since the \loccohom{1} on $\Rsp{q}(A,A')$ for finite $q$ is trivial,
we obtain \eqref{JPS:QQ'decomp}.
The converse is immediate.
\hfill$\Box$
\begin{prop}
\label{cor:JPS-adjsym}
Every local \adjsym/ $P=P[A,A']$
of order $q$ of $D{A'}={*D{A}}$
has the form \eqref{JPS:Pdecomp}
for some constants $c,c'$,
with $F=D{A} = -{*D{A'}}$,
where $(\t{Q}[F],\t{Q}'[F])$ is a local \adjsym/
of order $q-2$ of $D F = D{*F}=0$,
and conversely.
\end{prop}
\Proof{}
Since the \loccohom{2} on $\Rsp{q}(A,A')$ for finite $q$ is trivial,
we have $P=D Q$, ${*P}=D Q'$
for some \horiz/ 1-forms $Q=Q[A,A']$, $Q'=Q'[A,A']$.
Hence the pair $(Q,Q')$ is a local \sym/ of $D{A'}=*D{A}$.
By Proposition~\ref{prop:JPSsym},
$D$ applied to \eqref{JPS:QQ'decomp} yields \eqref{JPS:Pdecomp}.
The converse is immediate.
\hfill$\Box$
\begin{prop}
Every local (adjoint-) \sym/ $Q=Q[A]$
of order $q$ of $D{*D{A}}=0$
has the form \eqref{PS:Qdecomp}
for some constant $c$ and some scalar function $\chi[A]$,
with $F = D A$,
where $(\t{Q}[F],\t{Q}'[F])$ is a local \adjsym/
of order $q-1$ of $D F = D{*F}=0$,
and conversely.
\end{prop}
\Proof{}
Regard $Q[A]$ as a \horiz/ 1-form on $\Jsp{q}(A,A') \supset \Jsp{q}(A)$
with no dependence on the coordinates involving $A'$.
Since $D{A'}=*D{A}$ implies $D{*D A}=0$,
the 2-form $*D Q[A]$ is closed
on $\Rsp{q}(A,A')$ whose \loccohom{2} for finite $q$ is trivial.
Hence $*D Q[A] = D Q'[A,A']$
holds for some \horiz/ 1-form $Q'[A,A']$
and thus the pair $(Q,Q')$ is a local \sym/ of $D{A'}={*D{A}}$.
Then by Proposition~\ref{prop:JPSsym}, we have
\begin{equation}
Q[A] = cA + c' A' + \t{Q}[F] + D\chi[A,A'] .
\end{equation}
But since $Q[A]$ is independent of $A'$
we must have $c' = 0$ and $\chi = \chi[A]$.
Consequently, we obtain \eqref{PS:Qdecomp}.
The converse is immediate.
\hfill$\Box$
Finally, we point out the effect of the \dutr/
\begin{equation}
F \rightarrow {*F} ,\quad
(A,A') \rightarrow (A',-A)
\label{FAA'duality}
\end{equation}
on \syms/ and \adjsyms/ of \Meq/ and its \potsys/s,
which stems from
the explicit classification of \adjsyms/
$(Q[F],Q'[F])$ on $\Rsp{\infty}(F)$
given in \Ref{AncPoh:2001}.
This classification shows that
the \horiz/ 1-forms
$Q[F],Q'[F]$, modulo gradients $D\chi[F],D\chi'[F]$,
are {\it linear} functions of the jet space coordinates
and hence have well-defined parity (eigenvalue) decompositions
with respect to the combined \dutr/s \eqrefs{FAA'duality}{PQduality}.
Consider more specifically the transformation \eqref{FAA'duality}
composed with the inverse of the transformation \eqref{PQduality}
\begin{equation}
(\t{Q}[F],\t{Q}'[F]) \rightarrow (-\t{Q}'[*F],\t{Q}[*F])
\label{combined-duality}
\end{equation}
acting on the vector space $\tilde Y^0_F$.
(Here the notation $[*F]$ denotes
the \dutr/ \eqref{FAA'duality}
on the jet space coordinates $[F]$ in $\Jsp{q}(F)$;
likewise $[A',-A]$ will denote
the duality transformation on the coordinates $[A,A']$ in $\Jsp{q}(A,A')$.
Note that the \dutr/ is well-defined on the
corresponding solution jet spaces $\Rsp{q-1}(F)$ and $\Rsp{q-1}(A,A')$.
However, it does not exist on either $\Jsp{q}(A)$ or $\Rsp{q-2}(A)$.)
Due to the linearity of $\t{Q}[F]$ and $\t{Q}'[F]$,
the square of the transformation \eqref{combined-duality} is the identity
and hence its eigenvalues are $\pm 1$.
The eigenspaces of even/odd parity corresponding to these eigenvalues
are then given by
\begin{equation}
\t{Q}_{\pm}[*F] = \pm\t{Q}'_{\pm}[F] ,\quad
\t{Q}'_{\pm}[*F] = \mp\t{Q}_{\pm}[F] \quad\eqtext{ on $\Rsp{q}(F)$. }
\label{QQ'dualityrelation}
\end{equation}
Equivalently, we can write this canonical relation as
\begin{equation}
{*'\t{Q}}_{\pm}[F] = \pm \t{Q}_{\pm}[*F]
\end{equation}
and likewise for $\t{Q}'_{\pm}[F]$,
where $*'$ is the linear map \eqref{Qduality}.
This parity decomposition result \eqref{QQ'dualityrelation}
extends to all solutions of
the \sym/ equations and \adjsym/ equations
on $\Rsp{q}(F)$ and $\Rsp{q}(A,A')$
through the (cohomology and locality-projection)
decompositions in Theorem~\ref{thm:PQdecomp}.
\begin{prop}
\label{prop:dualitydecomp}
The \dutr/ \eqref{FAA'duality}
on the electromagnetic field and its joint potentials
induces a corresponding \dutr/ \eqref{PQduality}
on local \syms/ and local \adjsyms/ via
\begin{eqnarray}
&&
{*P_{\pm}}[F] = \pm {P_{\pm}[*F]} ,\quad
{*P_{\pm}}[A,A'] = \pm {P_{\pm}[A',-A]} ,
\\
&&
Q'_{\pm}[F] \mod{D\chi'[F]} = \pm Q_{\pm}[*F] \mod{D\chi[F]} ,
\\
&&
Q'_{\pm}[A,A'] \mod{D\chi'[A,A']} =
\pm Q_{\pm}[A',-A] \mod{D\chi[A,A']} ,
\end{eqnarray}
where $\pm$ denotes even/odd parity parts
with respect to the combined duality transformations
\begin{eqnarray}
&&
P[F] \rightarrow -{*P[*F]}
\quad\eqtext{ on $X^c_F \oplus X^{c'}_F \oplus X^0_F$, }
\label{combined-dutr1}\\
&&
P[A,A'] \rightarrow -{*P}[A',-A]
\quad\eqtext{ on $Y^c_{A,A'} \oplus Y^{c'}_{A,A'} \oplus Y^0_{A,A'}$, }
\\
&&
(Q[F],Q'[F]) \rightarrow (-Q'[*F],Q[*F])
\quad\eqtext{ on $\tilde Y^0_F$, }
\label{combined-dutr3}\\
&&
(Q[A,A'],Q'[A,A']) \rightarrow (-Q'[A',-A],Q[A',-A])
\quad\eqtext{ on $X^c_{A,A'} \oplus X^{c'}_{A,A'} \oplus \tilde X^0_{A,A'}$. }
\label{combined-dutr4}
\end{eqnarray}
\end{prop}
In addition we remark that a complete and explicit classification of
all local \syms/ and \adjsyms/ of the magnetic and joint \potsys/s
is now available by combining the decompositions in Theorem~\ref{thm:PQdecomp}
with the classification of local \adjsyms/ of \Meq/ from \Ref{AncPoh:2001}.
An illustration of this result applied to
a geometric class of \syms/ and \adjsyms/
will be given later.
\subsection{Conservation law formulas}
There is an important application of the preceding results
to local \conslaw/s of \Meq/ and its \potsys/s.
We begin with a few preliminaries.
In differential form notation
a conserved current of order $q<\infty$
given by a vector function $\curr{\mu}$
corresponds to a \horiz/ 3-form
$*\curr{}\down{\mu\nu\sigma} = \vol{\mu\nu\sigma\tau}{} \curr{\tau}$
on $\Jsp{q}$
that is closed on $\Rsp{\infty}$.
A current is trivial if and only if the corresponding \horiz/ 3-form is exact
(\ie/ $*\curr{}\down{\mu\nu\sigma}
= \D{[\mu} {*\curl{}{\nu\sigma]}}
= \frac{1}{3} \vol{\mu\nu\sigma\tau}{} \D{\alpha}\curl{\alpha\tau}{}$)
on $\Rsp{\infty}$.
Thus, the \loccohom{3} of \Meq/ and its potential systems
describes the nontrivial \conslaw/s of these systems.
(Note this cohomology is clearly quite rich,
in contrast to the local 1-form and 2-form cohomology.
See \Ref{Tor:lecturenotes} for a general discussion of \conslaw/s
of field equations from this perspective.)
The formulas in \tableref{table:adjrel-conslaw}
that generate all conserved currents (modulo curls)
in terms of \adjsyms/ are summarized
in \tableref{table:conscur-diff-form}
in this notation.
\mystretch{1.0}
\begin{table}[h]
\begin{center}
$\begin{array}{|c|c|c|c|} \hline
\mbox{System} & \mbox{Adjoint-symmetry} & \mbox{Conserved 3-form current}
\\ \hline \hline
\begin{array}{c}
\mbox{\Meq/}\\
D F=0 \\ D{*F}=0
\end{array} & D{Q'}[F] =*D{Q}[F] &
*\curr{} =
{\displaystyle\int}_0^1 \eval{\left( Q' \wedge F - Q \wedge *F \right)}{\lambda F}
\frac{d\lambda}{\lambda}
\\ \hline
\begin{array}{c}
\mbox{Magnetic \potsys/}\\
D{*D A}=0
\end{array} & D{*D{Q}}[A] = 0 &
*\curr{} =
{\displaystyle\int}_0^1 \eval{\left( A \wedge *D Q - Q \wedge *D A \right)}{\lambda A}
\frac{d\lambda}{\lambda}
\\ \hline
\begin{array}{c}
\mbox{Joint \potsys/}\\
D A'=*D A
\end{array} & \begin{array}{c} D{P}[A,A'] =0 \\ D{*P}[A,A']=0 \end{array} &
*\curr{} =
{\displaystyle\int}_0^1 \eval{\left( -A \wedge P - A' \wedge {*P} \right)} {\lambda A,\lambda A'}
\frac{d\lambda}{\lambda}
\\ \hline
\end{array}$
\caption{Conserved current formulas in differential form notation}
\label{table:conscur-diff-form}
\end{center}
\end{table}
\mystretch{1}
Now, through the \adjsym/ decompositions in Theorem~\ref{thm:PQdecomp},
the conserved current formulas for the \potsys/s can be simplified
to remove inessential dependence on the potentials.
\begin{prop}
For the \potsys/ $D{*D A}=0$,
all nontrivial local conserved 3-form currents of order $q$
are generated by
\begin{equation}
*\curr{} =
\int_0^1 \eval{\left( \t{Q}'\wedge F - \t{Q} \wedge *F \right)}{\lambda F}
\frac{d\lambda}{\lambda}
\mod{ D{*\curl{}{}} }
\label{MPS-current}
\end{equation}
for $q\geq 1$,
where $\t{Q}[F]$ is the gauge-invariant part of the (adjoint-) \sym/
$Q[A]$ in the decomposition \eqref{PS:Qdecomp}
and $\t{Q}'[F] = {*'\t{Q}}[F]$ is its image under the map \eqref{Qduality},
on $\Rsp{q-1}(F)$.
\end{prop}
\Proof{}
We substitute the decomposition \eqref{PS:Qdecomp}
into the integrand of $*\curr{}$ in \tableref{table:conscur-diff-form}
to obtain
\begin{equation}
A\wedge *D Q - Q\wedge *D A =
A\wedge *D\t{Q} - \t{Q}\wedge *F
+ \chi D\duF{}{} - D(\chi {*F}),
\label{stdPS-integrand}
\end{equation}
where the second last term vanishes on $\Rsp{}(A)$,
and the last term is an exact 3-form which we may drop.
Next we apply the relation
$D{*'\t{Q}}=*D\t{Q}$ to the first term on the right
in \eqref{stdPS-integrand},
where, note, $*'\t{Q}$ is determined only to within a gradient $D\chi'$.
This yields
\begin{equation}
A \wedge *D\t{Q} = A \wedge D{*'}\t{Q}
= F \wedge {*'}\t{Q}- \chi'D{F} + D(\chi' F -A \wedge {*'}\t{Q}),
\end{equation}
where we again drop the last two terms as before.
Hence, the integrand of $*\curr{}$ modulo an exact 3-form on $\Rsp{}(A)$
is simply ${*'\t{Q}} \wedge F - \t{Q} \wedge *F$.
\hfill$\Box$
\begin{prop}
\label{prop:JPScurrentformula}
For the joint \potsys/ $D A' = *D A$,
all nontrivial local conserved 3-form currents of order $q$
are generated by linear combinations of
\begin{equation}
*\curr{} =
\int_0^1 \eval{\left( -\t{Q} \wedge F - \t{Q}' \wedge *F \right)}{\lambda F}
\frac{d\lambda}{\lambda}
\mod{ D{*\curl{}{}} }
\quad \eqtext{ on $\Rsp{q-1}(A,A')$}
\label{JPS-current}
\end{equation}
for $q\geq 1$,
and when $q=1$,
\begin{equation}
*\curr{} = -\frac{1}{2}( A \wedge F + A' \wedge \duF{}{} )
\quad \eqtext{ on $\Rsp{}(A,A')$, }
\label{JPS-dualityrot-current}
\end{equation}
where $\t{Q}[F], \t{Q}'[F]$ are determined
by the gauge-invariant parts of the \adjsym/ $P[A,A']$
and its dual ${*P}[A,A']$ in the decomposition \eqref{JPS:Pdecomp},
satisfying the relation $\t{Q}' = {*'\t{Q}}$.
\end{prop}
\Proof{}
As in the previous proof, we first substitute the decompositions
in \eqref{JPS:Pdecomp}
into the integrand of $*\curr{}$
in \tableref{table:conscur-diff-form},
obtaining
\begin{equation}
-A \wedge P - A' \wedge {*P} = c( -A\wedge F - A' \wedge *F)
+ c' (-A \wedge *F + A' \wedge F) -A \wedge D\t{Q}
- A' \wedge D\t{Q}' .
\end{equation}
Note that on $\Rsp{q-1}(A,A')$ we have
\begin{eqnarray}
A' \wedge F- A \wedge *F &=& A' \wedge D A - A \wedge D A'
= D( A \wedge A') ,
\label{simplify1}\\
-A \wedge D\t{Q} - A' \wedge D\t{Q}' &=& -F \wedge \t{Q}
- *F \wedge \t{Q}'
+D( A \wedge \t{Q} + A' \wedge \t{Q}') .
\label{simplify2}
\end{eqnarray}
The last terms in both \eqrefs{simplify1}{simplify2}
are exact 3-forms which we drop.
Hence,
the integrand of $*\curr{}$ modulo an exact 3-form on $\Rsp{q-1}(A,A')$ is
$c( -A\wedge F - A' \wedge *F) - \t{Q} \wedge F - \t{Q}' \wedge *F$.
\hfill$\Box$
These 3-form formulas \eqref{MPS-current},
\eqref{JPS-current}, \eqref{JPS-dualityrot-current}
now lead to our main results pertaining to \conslaw/s.
First, we see that with the exception of
one conserved current \eqref{JPS-dualityrot-current}
coming from the joint \potsys/,
all other local conserved currents of the two \potsys/s
are equivalent to local conserved currents of \Meq/.
In particular, by the \adjsym/ decompositions in Theorem~\ref{thm:PQdecomp},
the \horiz/ 1-forms $\t{Q}[F]$ and $\t{Q}'[F]$
in the currents \eqrefs{MPS-current}{JPS-current}
can be directly identified with \adjsyms/
$(Q[F],Q'[F])$ of \Meq/
by $Q=\t{Q}[F]$ and $Q'=\t{Q}'[F]$.
There is also a dual identification given by
$Q = \t{Q}'[F]$, $Q' = -\t{Q}[F]$
which reflects the duality invariance \eqref{PQduality}
of the \adjsym/ equation $D Q'=*D Q$ on $\Rsp{q}(F)$.
Note that the gauge freedom in the form of the \adjsyms/
given by
\begin{equation}
Q \rightarrow Q + D\chi[F], \qquad
Q' \rightarrow Q' + D\chi'[F] ,
\end{equation}
for arbitrary functions $\chi,\chi'$ on $\Jsp{q}(F)$
yields only trivial currents,
since
\begin{equation}
D\chi'[F] \wedge F - D\chi[F] \wedge {*F}
= D (\chi'[F] F - \chi[F] {*F})
\quad\eqtext{ on $\Rsp{}(F)$ }
\label{gaugeinvcurr}
\end{equation}
is an exact 3-form.
This result \eqref{gaugeinvcurr} applies as well to
the exceptional current \eqref{JPS-dualityrot-current},
coming from the identification of the \adjsyms/ $Q=A'$, $Q'=-A$
with an essential dependence on the potentials.
Therefore, all \conslaw/s given by the currents
\eqrefs{MPS-current}{JPS-current}
as well as the current
\eqref{JPS-dualityrot-current}
are gauge invariant.
Second,
through the mappings relating local \adjsyms/ of \Meq/
to local \syms/ of \Meq/ and its \potsys/s
in Theorems \ref{thm:ME-joint} and~\ref{thm:ME-std},
we obtain a direct correspondence between local conserved currents
and local \syms/ of each system.
This is especially interesting because \Meq/ and the joint \potsys/
are not self-adjoint systems,
i.e. Noether's theorem relating \conslaw/s and \syms/
through a Lagrangian is inapplicable.
\begin{thm}
\label{thm:MEcurrentformula}
All \syms/ of \Meq/ induced through its \jps/
directly generate conserved currents
via the formula
\begin{equation}
*\curr{}
= \int_0^1 \eval{\left( Q' \wedge F - Q \wedge *F \right)}{\lambda A,\lambda A'}
\frac{d\lambda}{\lambda}
= \int_0^1 \eval{\left( A \wedge {*P} - A' \wedge P \right)}{\lambda A,\lambda A'}
\frac{d\lambda}{\lambda}
\mod{ D{*\curl{}{}} }
\label{MEcurrentformula}
\end{equation}
where
the pair of \horiz/ 1-forms $(Q[A,A'],Q'[A,A'])$ is
identified with any local \sym/ ${\bf X}$ of the joint \potsys/,
and the \horiz/ 2-form $P[A,A']$ is
identified with any corresponding \sym/ of \Meq/,
written in terms of the joint potentials
via the relations $D{Q[A,A']}=P[A,A'],D{Q'}[A,A']={*P}[A,A']$.
For the scaling symmetry \eqref{Xscaling},
the current \eqref{MEcurrentformula} is trivial.
\end{thm}
The proof of this theorem is similar to that of
Proposition~\ref{prop:JPScurrentformula}
and will be omitted.
We note that this scaling formula \eqref{MEcurrentformula}
generates the exceptional current
\eqref{JPS-dualityrot-current}
naturally from the duality-rotation \sym/ \eqref{Xduality}
of the \jps/ and \Meq/.
The implications of the explicit dependence
on the potentials in this current
will be dealt with in more detail in \secref{sec:nonlocal}.
More generally, the scaling formula \eqref{MEcurrentformula}
yields \conscurr/s for any \sym/, {\it local or nonlocal},
${\bf X} = P \Parder{F}$ admitted by \Meq/.
Finally, we emphasize a main aspect of the interrelations
we have derived between the \sym/ structure and the \conslaw/ structure
of \Meq/ and its \potsys/s.
\begin{cor}
No \syms/ of $D{F} =D{*F}=0$
with essential dependence on a potential
arise from projection of local \syms/ of the \potsys/s
$D{*D A}=0$ and $D A' = *D A$
under the mapping $F=D A = -{*D A'}$,
due to their gauge freedom
$A \rightarrow A +D\chi$, $A' \rightarrow A' +D\chi'$.
There is only one \conslaw/ of $D{F} =D{*F}=0$
with explicit dependence on a potential
arising from a projection of the local conserved currents of these \potsys/s,
namely, the duality-rotation current \eqref{JPS-dualityrot-current}.
This exceptional \conslaw/ nevertheless is gauge invariant.
\end{cor}
Thus the locality projection results proven in \Ref{AncBlu:1997JMP}
for \syms/ of well-posed PDE systems
when applied to \Meq/
extend in a natural sense to gauge-invariant \conslaw/s.
Indeed, a gauge-invariance projection result
can be established directly for \conslaw/s of
any locally regular PDE system
by an extension of the proof of locality projection for \syms/,
which we now outline.
By a locally regular PDE system
we mean it and all its prolongations are locally solvable \cite{Olv:symmbook}
and of constant rank \cite{Olv:symmbook},
so all \conslaw/s arise from multipliers.
Consider a conserved 3-form current $*\curr{}$
of any \potsys/ with gauge freedom.
Since conservation of $*\curr{}$ holds for all solutions of the \potsys/,
it must remain conserved under
infinitesimal gauge transformations ${\bf X}_{\it gauge}$ on the potentials.
Hence the 3-form current $*\bar\Psi := {\bf X}_{\it gauge}{*\curr{}}$
is also conserved and consequently projects to a \conscurr/ of
the original PDE system through the embedding property of
the respective solution spaces \cite{Blu:potsys}.
The projected current $*\bar\Psi$ necessarily depends on
an arbitrary function of $x$
and therefore so does its associated multiplier.
Treating this function as an auxiliary dependent variable
and applying the corresponding Euler-Lagrange operator,
we obtain a divergence identity holding on the PDE system.
An integration-by-parts method can then be used to reconstruct
an equivalent current from the divergence identity,
directly resulting in $*\bar\Psi = D{*\curl{}{}}$
holding for all solutions of the system.
Hence we conclude that the 3-form current $*\curr{}$
projects to a gauge-invariant \conslaw/ of the given PDE system,
namely $*\curr{}$ is invariant with respect to the gauge freedom
on the potentials modulo trivially conserved (exact 3-form) terms.
\subsection{Geometric \syms/ and \conslaw/s}
\label{geometricPQsolutions}
To conclude this section, we consider
the geometric \syms/, \adjsyms/, and conserved currents of
\Meq/ and its \potsys/s
in light of our main results.
\begin{defn}
\label{geom-pform}
A \horiz/ $p$-form on $\Jsp{1}$ will be called {\em geometric}
if it is locally constructed from $F$ in the case of \Meq/
and from $A$ or $A'$ in the case of the \potsys/s,
using the Minkowski metric, volume form, spacetime coordinates,
and exterior derivatives.
\end{defn}
To proceed we will make use of the following Lie derivative identities
holding for $p$-forms $\omega$
on any 4-dimensional spacetime manifold:
$\Lie{\xi} \omega = \xi \lrcorner d\omega + d(\xi \lrcorner \omega)$,
and for $p=2$,
$*\Lie{\xi} \omega = \Lie{\xi} {*\omega}$.
These identities extend in an obvious way to a definition of
the Lie derivative on \horiz/ forms
in a jet space setting,
for example,
$\Lie{\xi} \omega[F]
= \xi \lrcorner D\omega[F] + D(\xi \lrcorner \omega[F])$,
and
$*\Lie{\xi} \omega[F]
= \Lie{\xi} {*\omega[F]}$.
Thus, it follows that
any Lie derivatives of $F$, $A$, or $A'$ produce geometric \horiz/ forms
in the sense of Definition~\ref{geom-pform}.
Accordingly,
a \sym/, \adjsym/, or \conscurr/ will be called {\it geometric}
if it is described by a geometric \horiz/ form.
(Higher-order \syms/, \adjsyms/, and \conscurr/s of an analogous form
arise from geometric ones in an explicit manner
by the repeated application of
Lie derivatives on $F,A,A'$,
with respect to general \CKvec/s,
\ie/ general conformal symmetry operators.)
Note the classes of geometric \syms/, \adjsyms/, and \conscurr/s
are each invariant under the duality transformations
\eqrefs{PQduality}{FAA'duality}.
The geometric \syms/ of \Meq/ and its \potsys/s consist of
the general conformal \syms/
\begin{equation}
{\bf X}_{\it conf} =
\Lie{\xi} F\Parder{F} ,\quad
\Lie{\xi} A\Parder{A} ,\quad
\Lie{\xi} A\Parder{A} + \Lie{\xi} A'\Parder{A'} ,
\label{confsymm}
\end{equation}
and their corresponding dual \syms/
\begin{equation}
{\bf X}'_{\it conf} =
\Lie{\xi}{*F}\Parder{F} ,\quad
\Lie{\xi} A'\Parder{A} - \Lie{\xi} A\Parder{A'} ,
\end{equation}
in addition to the obvious scaling and duality-rotation \syms/
\eqrefs{Xscaling}{Xduality}.
Here $\xi$ denotes a general conformal \Kvec/ of the Minkowski metric,
\ie/ $\Lie{\xi}\flat{} = \Omega\flat{}$, $\Omega = \frac{1}{2}{\rm div\,}\xi$,
representing
four translation \syms/,
six \rb/ \syms/,
a dilation \sym/,
and four conformal (inversion) \syms/.
(Note that among these infinitesimal transformations,
${\bf X}_{\it scal}$, ${\bf X}_{\it dual}$, ${\bf X}_{\it conf}$ are of point-type
\cite{BluAnc:2002book}
while ${\bf X}'_{\it conf}$ is not, \ie/ its type is first-order.)
It is well known that the \syms/ ${\bf X}_{\it conf}$
comprise the 15-dimensional conformal Lie algebra $\mathfrak{so}(4,2)$,
while ${\bf X}_{\it conf}$ and ${\bf X}'_{\it conf}$ together form
a 30-dimensional Lie algebra isomorphic to the complexification of $\mathfrak{so}(4,2)$;
the scaling and duality-rotation \syms/ ${\bf X}_{\it scal},{\bf X}_{\it dual}$
commute with both ${\bf X}_{\it conf}$ and ${\bf X}'_{\it conf}$.
Thus we note the Lie algebra of geometric \syms/ of
\Meq/ and its \potsys/s
has the structure $\mathfrak{u}(1)^2 \times \mathfrak{so}(4,2)\otimes\mathbb{C}$.
The \horiz/ 1-forms and 2-forms associated with these \syms/
and the corresponding \adjsyms/ obtained through
the mappings in Theorems~\ref{thm:self-mappings} to~\ref{thm:ME-std}
are summarized in \tableref{table:geometric-forms}.
All these pairs of \horiz/ forms $(P,*P)$ and $(Q,Q')$ have even-parity
under the respective combined \dutr/s
\eqsref{combined-dutr1}{combined-dutr4}.
(We remark that, in contrast, the analogous pairs
associated with the chiral \syms/ \cite{AncPoh:2004,AncPoh:2001}
of \Meq/ possess odd-parity and exhibit a non-geometric form
that involves symmetrized derivatives of $F$.)
\begin{table}[h]
\begin{center}
$\begin{array}{|c|c|c|c|} \hline
Q & Q' & P=D Q & {*P}=D Q' \\ \hline\hline
A & A' & F & {*F} \\ \hline
A' & -A & {*F} & -F \\ \hline
\xi\lrcorner F & \xi \lrcorner {*F} & \Lie{\xi} F & \Lie{\xi} {*F} \\ \hline
\xi \lrcorner {*F} & -\xi \lrcorner F & \Lie{\xi} {*F} & -\Lie{\xi} F \\ \hline
\end{array}$
\caption{1-forms and 2-forms associated with geometric \syms/
and \adjsyms/ for \Meq/ and \potsys/s}
\label{table:geometric-forms}
\end{center}
\end{table}
Substitution of the 1-forms and 2-forms
in \tableref{table:geometric-forms}
into the conserved current formulas
in \tableref{table:conscur-diff-form}
generates four currents, two of which are trivial.
The two other currents generated are the duality-rotation one
\eqref{JPS-dualityrot-current} discussed earlier
and one which is equivalent to the well-known
geometric stress-energy currents of \Meq/
\begin{eqnarray}
*\curr{}
&=& \frac{1}{2} \left( (\xi \lrcorner {*F}) \wedge F - (\xi \lrcorner F) \wedge *F \right)
\quad \eqtext{ on $\Rsp{}(F)$}
\label{ME:stressenergy}
\\
&=& \frac{1}{2} (A \wedge \Lie{\xi}{*F} - A' \wedge \Lie{\xi} F)
\mod{ D{*\curl{}{}} } \eqtext{ on $\Rsp{1}(A,A')$} ,
\nonumber
\end{eqnarray}
where $\xi$ is a general \CKvec/.
\section{Analysis of the \jps/ in Lorentz gauge}
\label{sec:jps-analysis}
In light of the main results obtained in \secref{sec:cohom},
we must investigate the imposition of gauges in order
to obtain new \syms/ or new \conslaw/s
of the \potsys/s for \Meq/.
In the case of the \mps/ \eqref{MPS},
some well-known gauges are shown in \tableref{MPS-common-gauges}.
Note that $\x{\mu}{} =\{ x^0, x^i \}$ denotes time and space
coordinates in \Minksp/.
\mystretch{1.0}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|} \hline
Gauge Name & Description \\ \hline\hline
Lorentz & $\coder{\mu} \A{}{\mu}(x) =0$, \ie/
$\der{0}\A{}{0}(x) = \coder{i} \A{}{i}(x)$ \\ \hline
Coulomb & $\coder{i} \A{}{i}(x) =0$ \\ \hline
Temporal & $\A{}{0}(x) =0$ \\ \hline
Axial & $n\up{i} \A{}{i}(x)=0$, $n^i ={\rm const}$ \\ \hline
Cronstrom & $\x{\mu}{} \A{}{\mu}(x) = 0$, \ie/
$\x{i}{}\A{}{i}(x) = -\x{0}{}\A{}{0}(x)$ \\ \hline
\end{tabular}
\caption{Well-known gauges for the \mps/
\label{MPS-common-gauges}}
\end{center}
\end{table}
\mystretch{1}
Since \Meq/ \eqref{ME} are manifestly covariant and coordinate-independent,
natural gauges to investigate for \sym/ purposes should
also have these properties.
It should be noted that in 2+1 spacetime dimensions,
when temporal gauge or axial gauge is imposed on the \mps/,
no new local geometrical \syms/ arise \cite{AncBlu:1997JMP};
moreover, some of the Poincar\'e and conformal \syms/
are manifestly lost.
A standard covariant gauge choice for the \mps/ \eqref{MPS}
is the Lorentz gauge.
Cronstrom's gauge is also covariant,
but it is explicitly coordinate-dependent
and so we will not consider this gauge here.
With Lorentz gauge imposed on the magnetic potential
we have the augmented system
\begin{equation}
\coder{\mu} \der{[\mu} \A{}{\nu]}(x) = 0, \qquad
\coder{\mu} \A{}{\mu}(x) = 0,
\label{A-MPS}
\end{equation}
which is no longer self-adjoint, and hence is a non-Lagrangian system.
A well-known feature of the \potsys/ \eqref{A-MPS}
compared to the system without Lorentz gauge is that
the conformal \syms/ \eqref{confsymm}
on $\A{}{\mu}(x)$ are lost.
On the other hand, no new geometric \syms/ are gained,
as shown in the linear case in \Ref{The:MSc}.
For the \jps/ \eqref{JPS},
duality between the electric and magnetic potentials
motivates the choice of Lorentz gauge on
both $\A{}{\mu}(x)$ and $\Ap{}{\mu}(x)$.
Hence, we consider the augmented \potsys/
\begin{equation}
\der{[\mu} \Ap{}{\nu]}(x) =
\frac{1}{2} \vol{\mu\nu\sigma\tau}{} \coder{\sigma} \A{\tau}{}(x),
\qquad \coder{\mu} \A{}{\mu}(x) = 0,
\qquad \coder{\mu} \Ap{}{\mu}(x) = 0.
\label{JPS-L}
\end{equation}
Like the \jps/ itself, the augmented system is not self-adjoint
and thus is not a Lagrangian system.
Note, importantly,
the duality invariance \eqref{JPS-duality} on the potentials
is retained.
Due to the imposition of Lorentz gauge,
the \potsys/ \eqref{JPS-L} becomes a locally well-posed PDE system,
with no gauge freedom on the potentials.
As is the case for the standard \potsys/ in Lorentz gauge,
the potentials still admit a residual freedom given by
gradients \eqref{JPS-gaugesym} involving scalar functions $\chi(x),\chi'(x)$,
but in order to preserve Lorentz gauge
these functions are restricted to satisfy the wave equation
$\coder{\mu}\der{\mu}\chi(x)=0$,
$\coder{\mu}\der{\mu}\chi'(x)=0$.
(Hence the transformation \eqref{JPS-gaugesym} no longer
involves {\it arbitrary} functions of $\x{\mu}{}$.)
Modulo this residual freedom,
the solutions of the \potsys/ \eqref{JPS-L} in Lorentz gauge
are in one-to-one correspondence with the solutions of \Meq/ \eqref{ME}.
In this section, we will classify geometric \syms/
and corresponding conserved currents
admitted by this \potsys/ \eqref{JPS-L}.
We will denote the solution jet space of the system
by $\divfrRsp{}(A,A')$.
Note the coordinates of $\divfrRsp{}(A,A')$ are related to
those of $\Rsp{}(A,A')$ by quotienting out
the Lorentz gauge equations
on derivatives of the potentials.
\subsection{Symmetry analysis}
The determining \eq/s for local symmetries
${\bf X} = \Q{}{\mu} \Parder{\A{}{\mu}} + \Qp{}{\mu} \Parder{\Ap{}{\mu}}$
of order $q <\infty$
of the \potsys/ \eqref{JPS-L} are given by
\begin{eqnarray}
&&
\D{[\mu} \Qp{}{\nu]}
= *\D{[\mu} \Q{}{\nu]} ,
\label{JPS-L:Qdeteq1}\\
&&
\coD{\mu} \Q{}{\mu} = 0, \quad
\coD{\mu} \Qp{}{\mu} = 0 ,
\label{JPS-L:Qdeteq2}
\end{eqnarray}
on $\divfrRsp{q}(A,A') \subset \Jsp{q+1}(A,A')$.
These equations retain the duality invariance
\begin{equation}
(Q,Q') \longrightarrow (Q',-Q) .
\label{QQ'duality}
\end{equation}
Note that the loss of gauge freedom in this \potsys/
under Lorentz gauge implies there is no gradient freedom
in the form of \syms/ $\Q{}{\mu},\Qp{}{\mu}$.
We therefore refer to \eqref{JPS-L:Qdeteq2}
as Lorentz gauge equations on $(Q,Q')$.
Anco \& Pohjanpelto have shown that
any local \sym/ of order $0\leq q<\infty$ of \Meq/
is {\it linear} in the field $\F{}{\mu\nu}$ and its derivatives
on $\Rsp{q}(F)$ \cite{AncPoh:2004,AncPoh:2002}.
The same method can be expected to establish an analogous result
for local \syms/ of the \potsys/ \eqref{JPS-L}.
Moreover, the geometric \syms/ of Maxwell's equations and
the joint \potsys/ have even-parity
under the respective combined \dutr/s
\eqrefs{combined-dutr1}{combined-dutr4}.
This motivates a classification using
a linear homogeneous even-parity ansatz for geometric \syms/:
\begin{eqnarray}
&&
\Q{}{\mu}[A,A'] =
\Lie{\xi} \A{}{\mu} + \Lie{\xi'} \Ap{}{\mu}
+ \a{\mu}{\nu}(x) \A{}{\nu} + \ap{\mu}{\nu}(x) \Ap{}{\nu},
\label{JPS-L:XA}\\
&&
\Qp{}{\mu}[A,A'] =
\Lie{\xi} \Ap{}{\mu} - \Lie{\xi'} \A{}{\mu}
+ \a{\mu}{\nu}(x) \Ap{}{\nu} - \ap{\mu}{\nu}(x) \A{}{\nu},
\label{JPS-L:XA'}
\end{eqnarray}
where $\kv{\nu}{}$, $\kvp{\nu}{}$, $\a{\mu}{\nu}$, $\ap{\mu}{\nu}$,
are functions of $x$ to be determined.
Note the even-parity condition characterizing
this class of \syms/ is expressed by
\begin{equation}
\Qp{}{\mu}[A,A'] = \Q{}{\mu}[A',-A] .
\label{JPS-L-symmduality}
\end{equation}
\begin{thm}
\label{thm:JPS-L-sym}
The (even-parity) class of geometric \syms/ ${\bf X}$ of the form
\eqsref{JPS-L:XA}{JPS-L:XA'}
admitted by the \potsys/ \eqref{JPS-L}
consists of:
\begin{enumerate}
\item[{\rm (i)}]
the scaling and duality-rotation transformations
\begin{eqnarray}
&&
\X_{\it scal} =
\A{}{\mu} \Parder{\A{}{\mu}}
+ \Ap{}{\mu} \Parder{\Ap{}{\mu}} ,
\label{JPS-L-Xscaling}\\
&&
\X_{\it dual} =
\Ap{}{\mu} \Parder{\A{}{\mu}}
- \A{}{\mu} \Parder{\Ap{}{\mu}} ,
\label{JPS-L-Xduality}
\end{eqnarray}
\item[{\rm (ii)}]
the \rbvec/ transformation
\begin{equation}
\rbX{}=
\left( \gam{\mu}{\nu} \A{}{\nu} + \dugam{\mu}{\nu} \Ap{}{\nu} \right)
\Parder{\A{}{\mu}}
+ \left( \gam{\mu}{\nu} \Ap{}{\nu} - \dugam{\mu}{\nu} \A{}{\nu} \right)
\Parder{\Ap{}{\mu}} ,
\label{JPS-L-Xrbvec}
\end{equation}
with
\begin{equation}
\gam{}{\mu\nu} = \gam{}{[\mu\nu]}={\rm const} ,
\label{skewmatr}
\end{equation}
\item[{\rm (iii)}]
the \HKV/ transformation and its dual
\begin{eqnarray}
&&
\hX{}=
\Lie{\xi} \A{}{\mu} \Parder{\A{}{\mu}}
+ \Lie{\xi} \Ap{}{\mu} \Parder{\Ap{}{\mu}} ,
\label{JPS-L-XHKvec}\\
&&
\hX{}'=
\Lie{\xi} \Ap{}{\mu} \Parder{\A{}{\mu}}
- \Lie{\xi} \A{}{\mu} \Parder{\Ap{}{\mu}} ,
\label{JPS-L-XHKvecdual}
\end{eqnarray}
with
\begin{equation}
\kv{\mu}{} =
\k{1}{\mu}{} + \k{2}{\mu\nu}{} \x{}{\nu} + \k{3}{}{} \x{\mu}{} ,\quad
\k{1}{\mu}{},\k{2}{\mu\nu}{}=\k{2}{[\mu\nu]}{},\k{3}{}{}={\rm const},
\label{HKV}
\end{equation}
\item[{\rm (iv)}]
the \CKV/ transformation and its dual
\begin{eqnarray}
&& \cX{} =
\left( \wLie{\xi} \A{}{\mu} + \z{\mu}{\nu} \A{}{\nu}
+ \duz{\mu}{\nu} \Ap{}{\nu} \right) \Parder{\A{}{\mu}}
+ \left( \wLie{\xi} \Ap{}{\mu} + \z{\mu}{\nu} \Ap{}{\nu}
- \duz{\mu}{\nu} \A{}{\nu} \right)
\Parder{\Ap{}{\mu}} ,
\label{JPS-L-XCKvec}\\
&& \cX{}' =
\left( \wLie{\xi} \Ap{}{\mu} + \z{\mu}{\nu} \Ap{}{\nu}
- \duz{\mu}{\nu} \A{}{\nu} \right)
\Parder{\A{}{\mu}}
- \left( \wLie{\xi} \A{}{\mu} + \z{\mu}{\nu} \A{}{\nu}
+ \duz{\mu}{\nu} \Ap{}{\nu} \right)
\Parder{\Ap{}{\mu}} ,
\label{JPS-L-XCKvecdual}
\end{eqnarray}
where
$\wLie{\xi} := \Lie{\xi} + \frac{1}{4} \Omega$
and
$\z{}{\mu\nu} := -\frac{1}{2} \coder{[\mu} \kv{\nu]}{}$,
$\Omega:= \frac{1}{2} \der{\mu}\kv{\mu}{}$,
with
\begin{equation}
\kv{\mu}{} =
\k{4}{\sigma}{} \x{}{\sigma} \x{\mu}{}
- \frac{1}{2} \k{4}{\mu}{} \x{\sigma}{} \x{}{\sigma} ,\quad
\k{4}{\mu}{}={\rm const} .
\label{CKV}
\end{equation}
\end{enumerate}
This class is closed under the \dutr/ \eqref{QQ'duality}.
\end{thm}
Before outlining the proof of this theorem,
we first discuss geometrical features of these \syms/
and the structure of their Lie algebra.
The parameters
$\k{1}{\mu}{}$, $\k{2}{\mu\nu}{}$, $\k{3}{}{}$, $\k{4}{\mu}{}$
appearing in the homothetic/conformal \Kvec/s $\kv{\mu}{}$
correspond respectively to
four translations, three rotations and three boosts,
one dilation, and four inversions of $M^4$,
which are conformal isometries determined by
the conformal Killing equation
\begin{equation}
\Lie{\xi} \flat{\mu\nu} = \Omega \flat{\mu\nu} .
\label{conformalisom}
\end{equation}
The isometries \eqref{conformalisom}
generated by translations, \rbs/, and a dilation
comprise the homothetic \Kvec/s \eqref{HKV} on $M^4$
and induce a corresponding Lie derivative action on the potentials,
i.e. infinitesimal Poincar\'e and dilation transformations.
In contrast to the situation
for the standard \potsys/ in Lorentz gauge \eqref{A-MPS},
the infinitesimal conformal transformation
associated with genuine conformal isometries \eqref{conformalisom}
given by inversions on $M^4$
are not lost.
However, the transformation is modified through
combining a weighted Lie derivative
with respect to a \CKvec/ \eqref{CKV}
and an internal rotation on the potentials
via the coefficients $\z{}{\mu\nu}$ and $\duz{}{\mu\nu}$.
The infinitesimal \rbvec/ transformation
with parameters $\gam{}{\mu\nu} = \gam{}{[\mu\nu]}$
is genuinely new as there is no local counterpart of it
in \Meq/ or the standard \potsys/ with or without Lorentz gauge.
Along with the infintesimal scaling and duality-rotation transformations,
these transformations are internal (non-spacetime) \syms/
in the sense that there is no associated motion on spacetime.
Note they comprise three internal rotations and three internal boosts
in addition to the one scaling and one duality-rotation.
We give an alternative representation for the internal \syms/
as follows.
At any point in spacetime,
identify $\mathbb{R}^{4} \times \mathbb{R}^{4} \cong \mathbb{R}^{4} \otimes \mathbb{R}^{2}$,
and introduce basis elements
$e\down{1},e\down{2}$ for $\mathbb{R}^{2}$.
We will identify the jet space coordinates of the potentials
$\A{}{\mu},\Ap{}{\mu}$ with
$\A{}{\mu} \otimes e\down{1}$, $\A{}{\mu} \otimes e\down{2}$.
Let ${\rm id} \in {\rm Hom}(\mathbb{R}^{4})$ be the identity map,
and ${\rm R} \in {\rm Hom}(\mathbb{R}^{2})$ a standard rotation on $\mathbb{R}^{2}$,
\ie/ ${\rm id}(v\down\mu) = v\down{\mu}$,
${\rm R}(e\down{1}) = e\down{2}$,
${\rm R}(e\down{2}) = -e\down{1}$,
for $v\down{\mu} \in \mathbb{R}^{4}$.
Then the duality-rotation is given by the transformation
\begin{equation}
{\rm id} \otimes {\rm R} \in {\rm Hom}(\mathbb{R}^{4} \otimes \mathbb{R}^{2})
\end{equation}
which acts non-trivially only on the $\mathbb{R}^{2}$ factor,
or off-diagonally on the space $\mathbb{R}^{4} \otimes \mathbb{R}^{2}$,
with matrix representation
\begin{equation}
\left( \begin{array}{c|c} 0 & -\mathbb{I} \\ \hline \mathbb{I} & 0 \end{array} \right).
\end{equation}
The \rbsvec/ combine a nontrivial action on
the $\mathbb{R}^{2}$ and $\mathbb{R}^{4}$ factors.
Define ${\rm R}\down{\gamma} \in {\rm Hom}(\mathbb{R}^{4})$
to be the standard \rb/ operator with parameters
$\gam{}{\mu\nu} = \gam{}{[\mu\nu]}$.
Recall that the \rbs/ act infinitesimally
by ${\rm R}\down{\gamma}(v\down{\mu}) = \gam{\mu}{\nu} v\down{\nu}$
for $v\down{\mu} \in \mathbb{R}^{4}$.
Here the constant parameters $\gam{}{\mu\nu}$
geometrically determine a 2-dimensional plane in $\mathbb{R}^{4}$
on which the \rb/ takes place
(namely, the cokernel of ${\rm R}\down{\gamma}$).
With this notation, the internal \rbs/ act via
\begin{equation}
{\rm R}\down{\gamma}\otimes {\rm id}
+ {\rm R}\down{*\gamma}\otimes {\rm R}
\in {\rm Hom}(\mathbb{R}^{4} \otimes \mathbb{R}^{2}).
\label{IRB}
\end{equation}
We then see that the transformation \eqref{IRB}
is a sum of diagonal and off-diagonal
\rbs/ on $\mathbb{R}^{4}\otimes \mathbb{R}^{2}$:
the diagonal action involves a standard \rb/ on $\mathbb{R}^{4}$,
and the plane for the off-diagonal \rb/
is the dual of the plane for the diagonal \rb/.
The geometric \syms/ \eqsref{JPS-L-Xscaling}{JPS-L-XCKvecdual}
comprise a 38-dimensional \sym/ algebra.
To describe its commutator structure,
we first extend the transformation $\cX{}$ and its dual $\cX{}'$
to general \CKvec/s \eqref{conformalisom}.
\begin{prop}
\label{JPS-L-combinedsymm}
For a \HKvec/
$\kv{\mu}{} =
\k{1}{\mu}{} + \k{2}{\mu\nu}{} \x{}{\nu} + \k{3}{}{} \x{\mu}{}$,
\begin{equation}
\cX{} = \hX{} +\rbX{} + \frac{1}{4}\Omega \X_{\it scal} ,\quad
\cX{}' = \hX{}' -\durbX{} + \frac{1}{4}\Omega \X_{\it dual} ,\quad
\gam{}{} = \z{}{}
\end{equation}
is a geometric \sym/ of the \jps/ in Lorentz gauge,
where $\z{}{\mu\nu} = \frac{1}{2} \k{2}{\mu\nu}{}={\rm const}$,
$\Omega=2\k{3}{}{}={\rm const}$
are the (scaled) curl and divergence of $\kv{\mu}{}$.
\end{prop}
Let
$[\gamma_1,\gamma_2]\up{\mu\nu}
= 2 \gamma_1\up{\sigma[\mu} \gamma_2\updown{\nu]}{\sigma}$
denote the commutator of two skew-tensors
$\gamma_1\up{\mu\nu},\gamma_2\up{\mu\nu}$
viewed as matrices,
and
$[\xi_1,\xi_2]\up{\mu}
= \Lie{\xi_1}\xi_2\up{\mu} = -\Lie{\xi_2}\xi_1\up{\mu}$
denote the commutator of two general conformal \Kvec/s
$\xi_1\up{\mu},\xi_2\up{\mu}$.
Note a general \CKvec/ $\xi\up{\mu}$ has a decomposition into
a sum of a \HKvec/ \eqref{HKV} and a \CKvec/ \eqref{CKV}
whose curl and divergence are given by
\begin{equation}
\z{}{\mu\nu} = -\frac{1}{2} \coder{[\mu} \kv{\nu]}{}
= \frac{1}{2} \k{2}{\mu\nu}{} -\k{4}{[\mu}{} \x{\nu]}{} ,\quad
\Omega = \frac{1}{2} \der{\mu} \kv{\mu}{}
= 2\k{3}{}{} + 2\k{4}{\mu}{} \x{}{\mu} ,
\end{equation}
where $\z{}{\mu\nu}$ and $\Omega$ are constant only in the case of \HKvec/.
Recall, any two \CKvec/s \eqref{CKV} commute,
while the commutator of any two \HKvec/s \eqref{HKV} is again a \HKvec/,
which is given by the well-known Poincar\'e Lie algebra
\cite{Bat:1909,Cun:1909,Ibr:1968}
enlarged by the one-dimensional Lie algebra of dilations.
(In particular, dilations commute with all \HKvec/s except translations,
whose commutator is again a translation \ie/
$[x,k_1]\up{\mu} = -\k{1}{\mu}{}$;
the Poincar\'e Lie algebra is the semidirect product of
the Lie algebra $\mathfrak{so}(3,1)$ of rotations/boosts
and the abelian Lie algebra $\mathfrak{u}(1)^4 \cong \mathbb{R}^4$ of translations.)
The commutator of a \CKvec/ \eqref{CKV} and a \HKvec/ \eqref{HKV}
is given by the parameters
\begin{equation}
\kb{1}{\mu}{} =
0 ,\quad
\kb{2}{\mu\nu}{} =
2 \k{4}{[\mu}{} \k{1}{\nu]}{} ,\quad
\kb{3}{}{} =
-\k{4}{}{\nu} \k{1}{\nu}{} ,\quad
\kb{4}{\mu}{} =
- \k{4}{\mu}{} \k{3}{}{} - \k{4}{}{\nu} \k{2}{\nu\mu}{} .
\end{equation}
\begin{thm} {\bf (Geometric symmetry algebra) }
\label{JPS-L-symalg}
The nonzero commutators of the geometric \syms/
\eqsref{JPS-L-Xscaling}{JPS-L-XCKvecdual} of the \jps/ in Lorentz gauge
have the algebraic structure
\begin{eqnarray}
&&
[\rbX{1},\rbX{2}] = 2\rbX{3} ,\quad\mbox{ where }
\gamma_3 = [\gamma_2,\gamma_1] ,
\\
&&
[\cX{1},\cX{2}] = -[\cX{1}',\cX{2}'] = \cX{3} ,\quad
[\cX{1},\cX{2}'] = \cX{3}' ,
\mbox{ where }
\xi_3 = [\xi_2,\xi_1] .
\end{eqnarray}
Moreover,
$\X_{\it scal},\X_{\it dual}$ span a two-dimensional abelian Lie algebra,
$\rbX{}$ spans the six-dimensional \rb/ Lie algebra $\mathfrak{so}(3,1)$,
$\cX{}$ spans the 15-dimensional conformal Lie algebra $\mathfrak{so}(4,2)$;
the Lie algebra spanned by $\cX{},\cX{}'$ is isomorphic to
the complexification of $\mathfrak{so}(4,2)$.
Thus the geometric \syms/ together form a
38-dimensional Lie algebra
$\mathfrak{u}(1)^2 \times \mathfrak{so}(3,1) \times \mathfrak{so}(4,2)\otimes\mathbb{C}$.
\end{thm}
The computation of this \sym/ algebra is straightforward and will be omitted.
To conclude the \sym/ analysis,
we now derive the geometric \sym/ classification
by solving the \sym/ determining equations
\eqrefs{JPS-L:Qdeteq1}{JPS-L:Qdeteq2}.
\Proof{ of Theorem \ref{thm:JPS-L-sym}}
Explicit coordinates for the solution jet space $\divfrRsp{}(A,A')$
are given by
$(\x{\mu}{},\A{}{\nu},\Ap{}{\nu},
\F{}{\mu\nu},{\rm trfr}\A{}{\mu\nu},{\rm trfr}\Ap{}{\mu\nu})$
where
\begin{equation}
\F{}{\mu\nu} = \A{}{[\nu,\mu]}
= -\frac{1}{2} \vol{\mu\nu}{\sigma\tau} \Ap{}{\tau,\sigma}
,\quad
\A{}{\mu\nu} = \A{}{(\nu,\mu)} ,\quad
\Ap{}{\mu\nu} = \Ap{}{(\nu,\mu)} .
\end{equation}
These components represent the linearly independent parts of
the potentials and their first-order derivatives
at a point in spacetime, subject to the system equations \eqref{JPS-L}.
To proceed, we substitute \eqrefs{JPS-L:XA}{JPS-L:XA'}
into the determining equations \eqrefs{JPS-L:Qdeteq1}{JPS-L:Qdeteq2}.
The second-order derivative terms in $A$ and $A'$
are found to vanish on $\divfrRsp{1}(A,A') \subset \Jsp{2}(A,A')$.
For the remaining terms, we extract the coefficients of
the linearly independent coordinates
on $\divfrRsp{}(A,A')$ as follows:
(i) $\A{}{\nu},\Ap{}{\nu}$,
(ii) $\F{}{\mu\nu}$,
(iii) ${\rm trfr}\A{}{\mu\nu},{\rm trfr}\Ap{}{\mu\nu}$.
By setting the coefficients to vanish,
we obtain the equations
\begin{eqnarray}
&&
\der{[\alpha} \a{\beta]\sigma}{} =
\frac{1}{2} \vol{\alpha\beta}{\mu\nu} \der{\mu} \ap{\nu\sigma}{} ,
\label{JPS-L:coeff-eqn-1}\\
&&
\der{\sigma} \ba{}{\sigma\alpha} = 0 ,\quad
\der{\sigma} \bap{}{\sigma\alpha} = 0 ,
\label{JPS-L:coeff-eqn-2}\\
&&
\a{[\alpha}{(\mu} \id{\nu)}{\beta]} + \frac{1}{2} \ap{}{\sigma(\mu}
\vol{}{\nu)}\down{\alpha\beta\sigma} =
\frac{1}{4} \invflat{\mu\nu} (
\a{[\alpha\beta]}{}
- \frac{1}{2} \vol{\alpha\beta\sigma\tau}{} \ap{}{\sigma\tau} ) ,
\label{JPS-L:coeff-eqn-3}\\
&&
\ta{(\alpha\beta)}{} =
\frac{1}{4} \flat{\alpha\beta} \ta{\sigma}{\sigma} , \quad
\tap{(\alpha\beta)}{} =
\frac{1}{4} \flat{\alpha\beta} \tap{\sigma}{\sigma} ,
\label{JPS-L:coeff-eqn-4}\\
&&
\ta{[\alpha}{\sigma} \vol{\beta]\sigma}{\mu\nu}
- \ta{}{\sigma[\mu} \epsilon\up{\nu]}\down{\alpha\beta\sigma}
+ 2(\tap{[\alpha}{[\mu}+ \tap{}{[\mu}\down{[\alpha}) \id{\nu]}{\beta]}
- \tap{\sigma}{\sigma}\id{[\mu}{\alpha} \id{\nu]}{\beta}
=0 ,
\label{JPS-L:coeff-eqn-5}\\
&&
\ap{[\alpha\beta]}{} =
\frac{1}{2} \vol{\alpha\beta\sigma\tau}{} \a{}{\sigma\tau} ,
\label{JPS-L:coeff-eqn-6}
\end{eqnarray}
where, for notational convenience, we have defined
\begin{eqnarray}
&&
\ba{\alpha\beta}{} := \a{\alpha\beta}{} + \der{\alpha} \xi\down{\beta} ,
\quad
\bap{\alpha\beta}{} := \ap{\alpha\beta}{} + \der{\alpha} \xi'\down{\beta} ,
\\
&&
\ta{\alpha\beta}{} := \a{\alpha\beta}{} + 2\der{\alpha} \xi\down{\beta} ,
\quad
\tap{\alpha\beta}{} := \ap{\alpha\beta}{} + 2\der{\alpha} \xi'\down{\beta}.
\end{eqnarray}
From \eqref{JPS-L:coeff-eqn-3},
contracting on $\nu,\beta$,
and then symmetrizing on $\alpha,\mu$,
we find
\begin{equation}
\a{(\alpha\beta)}{} =
\frac{1}{4} \flat{\alpha\beta} \a{\sigma}{\sigma} .
\label{symmproof:1}
\end{equation}
Similarly, multiplication of \eqref{JPS-L:coeff-eqn-3}
by $\vol{}{\alpha\beta\sigma\tau}$ followed by a similar contraction
and symmetrization leads to
\begin{equation}
\ap{(\alpha\beta)}{} =
\frac{1}{4} \flat{\alpha\beta} \ap{\sigma}{\sigma} .
\label{symmproof:2}
\end{equation}
Combining \eqrefs{symmproof:1}{symmproof:2}
with \eqref{JPS-L:coeff-eqn-4}, we obtain
\begin{equation}
\der{(\alpha} \kv{}{\beta)}
=\frac{1}{4} \flat{\alpha\beta} \der{\sigma} \kv{\sigma}{} ,\quad
\der{(\alpha} \kvp{}{\beta)}
= \frac{1}{4} \flat{\alpha\beta} \der{\sigma} \kvp{\sigma}{} .
\label{symmproof:KVeq}
\end{equation}
Through \eqsref{symmproof:1}{symmproof:KVeq}
and \eqref{JPS-L:coeff-eqn-6},
the equations \eqref{JPS-L:coeff-eqn-3}--\eqref{JPS-L:coeff-eqn-5}
reduce to identities.
The differential equation \eqref{symmproof:KVeq}
is equivalent to the conformal Killing equation \eqref{conformalisom}
and hence both $\kv{\mu}{}(x)$ and $\kvp{\mu}{}(x)$
are general \CKvec/s:
\begin{eqnarray}
\kv{\mu}{} &=&
\k{1}{\mu}{} + \k{2}{\mu\nu}{} \x{}{\nu}
+ \k{3}{}{} \x{\mu}{}
+ \k{4}{\nu}{} \x{}{\nu} \x{\mu}{}
- \frac{1}{2} \k{4}{\mu}{} \x{\nu}{} \x{}{\nu} ,
\label{xi-CKV}\\
\kvp{\mu}{} &=&
\kp{1}{\mu}{}
+ \kp{2}{\mu\nu}{} \x{}{\nu}
+ \kp{3}{}{} \x{\mu}{}
+ \kp{4}{\nu}{} \x{}{\nu} \x{\mu}{}
- \frac{1}{2} \kp{4}{\mu}{} \x{\nu}{} \x{}{\nu} ,
\label{xi'-CKV}
\end{eqnarray}
where
$\k{1}{\mu}{}$,
$\k{2}{\mu\nu}{} = \k{2}{[\mu\nu]}{}$,
$\k{3}{}{}$, $\k{4}{\mu}{}$,
and
$\kp{1}{\mu}{}$,
$\kp{2}{\mu\nu}{} = \kp{2}{[\mu\nu]}{}$,
$\kp{3}{}{}$,
$\kp{4}{\mu}{}$ are constants.
We decompose $\a{\nu\sigma}{}$ and $\ap{\nu\sigma}{}$
into their symmetric and antisymmetric parts
using \eqrefs{symmproof:1}{symmproof:2},
with the notation
\begin{equation}
\ah{\nu\sigma}{} := \a{[\nu\sigma]}{} ,\quad
\aph{\nu\sigma}{} := \ap{[\nu\sigma]}{} .
\end{equation}
Elimination of $\aph{\nu\sigma}{}$ in \eqref{JPS-L:coeff-eqn-1}
using \eqref{JPS-L:coeff-eqn-6}
then gives the equation
\begin{equation}
\der{[\alpha} \ah{\beta]\sigma}{}
-\frac{1}{2} \der{\sigma} \ah{\alpha\beta}{}
= \flat{\sigma[\alpha} \coder{\mu} \ah{\beta]\mu}{}
+ \frac{1}{4} \flat{\sigma[\alpha} \der{\beta]} \a{\nu}{\nu}
+\frac{1}{8} \vol{\alpha\beta\mu\sigma}{} \coder{\mu} \ap{\nu}{\nu}.
\label{trproof:1}
\end{equation}
By contracting \eqref{trproof:1} on $\beta,\sigma$,
followed by using \eqrefs{JPS-L:coeff-eqn-2}{xi-CKV},
we obtain
\begin{equation}
\der{\alpha} \a{\sigma}{\sigma}
= \frac{4}{3} \coder{\sigma} \ah{\sigma\alpha}{}
= -\coder{\sigma} \der{\sigma} \xi\down{\alpha} = 2 \k{4}{}{\alpha}.
\label{divpart}
\end{equation}
Consequently, the trace-part of $\a{\alpha\beta}{}$ is
\begin{equation}
\a{\sigma}{\sigma} = 4\lambda + 2\k{4}{\sigma}{} \x{}{\sigma} ,
\label{trproof:2}
\end{equation}
where $\lambda$ is a constant.
Then,
a similar elimination of $\ah{\nu\sigma}{}$ in \eqref{JPS-L:coeff-eqn-1}
leads to the trace-part of $\ap{\alpha\beta}{}$,
\begin{equation}
\ap{\sigma}{\sigma} = 4\lambda' + 2\kp{4}{\sigma}{} \x{}{\sigma} ,
\label{trproof:3}
\end{equation}
where $\lambda'$ is a constant.
Hence \eqref{trproof:1} becomes
\begin{equation}
\der{[\alpha} \ah{\beta]\sigma}{}
- \frac{1}{2} \der{\sigma} \ah{\alpha\beta}{}
=
\k{4}{}{[\alpha} \flat{\beta]\sigma}
- \frac{1}{4} \vol{\alpha\beta\sigma\tau}{} \kp{4}{\tau}{}
\label{asymproof:1}
\end{equation}
through \eqsref{divpart}{trproof:3}.
Antisymmetrization of \eqref{asymproof:1} on $\alpha,\beta,\sigma$
leads to
\begin{equation}
\der{[\alpha} \ah{\beta]\sigma}{}
+ \frac{1}{2} \der{\sigma} \ah{\alpha\beta}{}
= -\frac{3}{4} \vol{\alpha\beta\sigma\tau}{} \kp{4}{\tau}{} .
\label{asymproof:2}
\end{equation}
Combining \eqrefs{asymproof:1}{asymproof:2} we obtain
\begin{equation}
\der{\sigma} \ah{\alpha\beta}{} =
-\k{4}{}{[\alpha} \flat{\beta]\sigma}
- \frac{1}{2} \vol{\alpha\beta\sigma\tau}{} \kp{4}{\tau}{} ,
\label{asymproof:3}
\end{equation}
which yields
\begin{equation}
\ah{\alpha\beta}{} =
\gam{\alpha\beta}{} + \z{\alpha\beta}{} - \duzp{\alpha\beta}{} ,
\label{asymproof:4}
\end{equation}
where $\gam{\alpha\beta}{} = \gam{[\alpha\beta]}{}={\rm const}$,
and
$\z{\alpha\beta}{} =
-\frac{1}{2} \der{[\alpha} \kv{}{\beta]}
= -\k{4}{}{[\alpha} \x{}{\beta]}$,
$\zp{\alpha\beta}{} =
-\frac{1}{2} \der{[\alpha} \kvp{}{\beta]}
= -\kp{4}{}{[\alpha} \x{}{\beta]}$.
Finally, using \eqref{JPS-L:coeff-eqn-6}
we have
\begin{equation}
\aph{\alpha\beta}{} =
\dugam{\alpha\beta}{} + \duz{\alpha\beta}{} + \zp{\alpha\beta}{} .
\label{asymproof:5}
\end{equation}
All equations \eqref{JPS-L:coeff-eqn-1}--\eqref{JPS-L:coeff-eqn-6}
now reduce to identities.
\hfill$\Box$
We remark that if the ansatz \eqrefs{JPS-L:XA}{JPS-L:XA'} is
generalized to include odd parity terms of zeroth order
on $\Jsp{1}(A,A')$,
then the proof goes through and no new \syms/ are obtained.
It would be a natural generalization to next include
odd parity first-order terms,
but we have not investigated the outcome.
\subsection{Conservation law analysis}
Since the solution space of the \jps/ in Lorentz gauge
has a natural embedding into
the solution space of the unaugmented \potsys/,
both the duality-rotation current
\eqref{JPS-dualityrot-current}
connected with duality rotations on the potentials
and the stress-energy currents \eqref{ME:stressenergy}
associated with general conformal \Kvec/s,
discussed in \secref{geometricPQsolutions},
continue to be admitted when Lorentz gauge is imposed.
New conserved currents are suggested by
the appearance of the new local \syms/ admitted under Lorentz gauge.
The \jps/ remains non-self-adjoint in Lorentz
gauge and hence it is a non-Lagrangian system. Consequently, conserved
currents arise through \adjsyms/ via a scaling formula similar
to the one in \tableref{table:adjrel-conslaw}
for the system without Lorentz gauge.
Local \adjsyms/ of system \eqref{JPS-L} consist of
a \horiz/ 2-form $\P{}{\mu\nu}[A,A']$
as in the case without Lorentz gauge,
plus a pair of differential scalar functions $\chi[A,A'],\chi'[A,A']$,
which arise respectively from the adjoint of the \sym/ equations
\eqrefs{JPS-L:Qdeteq1}{JPS-L:Qdeteq2}.
In particular,
the determining equations for local \adjsyms/ of order $q<\infty$
take the form
\begin{eqnarray}
&&
\coD{\mu} \P{}{\mu\nu} + \D{\nu} \chi' =0 ,
\label{JPS-L-Pdeteq1}\\
&&
\coD{\mu} \duP{}{\mu\nu} - \D{\nu} \chi =0 ,
\label{JPS-L-Pdeteq2}
\end{eqnarray}
on $\Rsp{q}(A,A')$.
Note these equations have the duality invariance
\begin{equation}
(\P{}{\mu\nu},\chi,\chi') \rightarrow (\duP{}{\mu\nu},\chi',-\chi) .
\label{JPS-L-Pduality}
\end{equation}
\begin{prop}
All nontrivial local conserved currents of the \potsys/ \eqref{JPS-L}
are generated from local \adjsyms/
by the conserved current formula
\begin{equation}
\curr{\mu} =
\int_0^1 \eval{\left(\P{\mu\nu}{} \Ap{}{\nu}
- \duP{\mu\nu}{} \A{}{\nu} + \chi \A{\mu}{}
+ \chi'\Ap{\mu}{}\right)}{\lambda A, \lambda A'}
\frac{d\lambda}{\lambda} .
\label{JPS-L-currentformula}
\end{equation}
\end{prop}
To apply this result,
rather than solve the \adjsym/ equations
we will now utilize our \sym/ classification results
and derive a mapping from local \syms/ to local \adjsyms/,
extending the mapping \eqsref{JPS-QtoPmap}{JPS-Q'toPmap}
obtained in Theorem \ref{thm:self-mappings} for the \jps/ without gauges.
Since this map \eqsref{JPS-QtoPmap}{JPS-Q'toPmap}
relied only on the adjoint relation between
the respective determining equations
for local \syms/
$\Q{}{\mu}[A,A'],\Qp{}{\mu}[A,A']$
and local \adjsyms/
$\P{}{\mu\nu}[A,A']$
of the unaugmented \potsys/,
it carries over to the \potsys/ in Lorentz gauge
if we project out the functions
$\chi[A,A'],\chi'[A,A']$
associated with the Lorentz gauge equations \eqref{JPS-L:Qdeteq2}.
\begin{thm}
For the joint \potsys/ in Lorentz gauge,
there is a linear mapping from local \syms/ into local \adjsyms/
given by
\begin{equation}
(\Q{}{\nu}, \Qp{}{\nu}) \longrightarrow
(\P{}{\mu\nu}= \D{[\mu} \Q{}{\nu]},
*\P{}{\mu\nu} = \D{[\mu} \Qp{}{\nu]},\chi=0,\chi'=0)
\label{JPS-L:sym-adjsym1}
\end{equation}
as well as a dual mapping
\begin{equation}
(\Q{}{\nu}, \Qp{}{\nu}) \longrightarrow
(\P{}{\mu\nu} = -\D{[\mu}\Qp{}{\nu]},
*\P{}{\mu\nu} = \D{[\mu} \Q{}{\nu]},
\chi=0,\chi'=0)
\label{JPS-L:sym-adjsym2}
\end{equation}
coming from the duality invariance \eqref{JPS-L-Pduality}.
Associated to these correspondences
\eqrefs{JPS-L:sym-adjsym1}{JPS-L:sym-adjsym2}
are respective formulas \eqref{JPS-L-currentformula}
that directly generate local \conscurr/s
\begin{equation}
\curr{\mu} =
\int_0^1 \eval{(\Qp{}{\nu} \F{\mu\nu}{}
- \Q{}{\nu} \duF{\mu\nu}{} )}{\lambda A, \lambda A'}
\frac{d\lambda}{\lambda}
\label{JPS-L:sym-curr1}
\end{equation}
and
\begin{equation}
\curr{\mu} =
\int_0^1 \eval{(\Q{}{\nu} \F{\mu\nu}{}
+ \Qp{}{\nu} \duF{\mu\nu}{} )}{\lambda A, \lambda A'}
\frac{d\lambda}{\lambda}
\label{JPS-L:sym-curr2}
\end{equation}
from local \syms/ of this \potsys/.
\end{thm}
\Proof{}
The \adjsym/ equations \eqrefs{JPS-L-Pdeteq1}{JPS-L-Pdeteq2}
reduce directly to the \sym/ equations
\eqrefs{JPS-L:Qdeteq1}{JPS-L:Qdeteq2}
through substitution of the mappings
\eqrefs{JPS-L:sym-adjsym1}{JPS-L:sym-adjsym2}.
Similarly,
the \conscurr/ formula \eqref{JPS-L-currentformula}
reduces to formulas \eqrefs{JPS-L:sym-curr1}{JPS-L:sym-curr2}
modulo curls, since
$\A{}{\nu} \coD{[\mu}\Qp{\nu]}{}
= \A{}{\nu} {*\coD{[\mu}\Q{\nu]}{}}
= \frac{1}{2} \vol{}{\mu\nu\alpha\beta} \left(
\Q{}{\beta} \D{\nu} \A{}{\alpha}
+ \D{\alpha}( \Q{}{\beta}\A{}{\nu} ) \right)$
and likewise for the other term
$\Ap{}{\nu} \coD{[\mu}\Q{\nu]}{}
= -\Ap{}{\nu} {*\coD{[\mu}\Qp{\nu]}{}}$.
\hfill$\Box$
We remark that a converse for either correspondence
\eqref{JPS-L:sym-adjsym1} or \eqref{JPS-L:sym-adjsym2}
would rely on generalizing the \loccohom{2} theorem to the
\jps/ in Lorentz gauge,
which is unnecessary for the purpose of generating conserved currents.
Of the two \conscurr/ formulas,
the latter one \eqref{JPS-L:sym-curr2} is distinguished
by the property that it generates a trivial current
from the scaling \sym/ \eqref{JPS-L-Xscaling}
and a nontrivial current from
the duality-rotation \sym/ \eqref{JPS-L-Xduality}
(while this correspondence is reversed by the dual formula
\eqref{JPS-L:sym-curr1}).
We emphasize that both formulas generate the same \conscurr/s
when applied to the class of geometric \syms/
\eqsref{JPS-L-Xscaling}{JPS-L-XCKvecdual},
since this class exhibits duality-invariance \eqref{QQ'duality}.
We now list in \tableref{table:JPS-L-sym-curr}
the \conscurr/s that arise through formula \eqref{JPS-L:sym-curr2}
from the geometric \syms/
\eqsref{JPS-L-Xscaling}{JPS-L-XCKvecdual}
classified in Theorem~\ref{thm:JPS-L-sym}.
For comparison,
we write out the stress-energy currents,
\begin{equation}
\curr{\mu} =
\kv{\sigma}{} (
\F{}{\sigma\nu} \F{\mu\nu}{} + \duF{}{\sigma\nu} \duF{\mu\nu}{} )
:= \consT{\mu}{\sigma} \kv{\sigma}{}
\label{hKVcurr}
\end{equation}
where $\kv{\sigma}{}$ is a general \CKvec/
and $\consT{\mu}{\sigma}$ denotes the conserved stress-energy tensor of \Meq/.
\begin{table}[h]
\begin{center}
$\begin{array}{|c|c|c|} \hline
& \mbox{Geometric Symmetry} & \mbox{Conserved current}
\\ \hline\hline
\mbox{scaling} &
\begin{array}{c}
\Q{}{\mu} = \A{}{\mu} \\
\Qp{}{\mu} = \Ap{}{\mu} \end{array} &
\begin{array}{rl}
\scurr{\mu} =&
\frac{1}{2} (\A{}{\nu} \F{\mu\nu}{} + \Ap{}{\nu} \duF{\mu\nu}{} ) \\
= & \D{\nu} \left(
\frac{1}{4} \vol{}{\mu\nu\sigma\tau} \A{}{\sigma} \Ap{}{\tau} \right)
\end{array}
\\ \hline
\mbox{duality-rotation} &
\begin{array}{c}
\Q{}{\mu} = \Ap{}{\mu} \\
\Qp{}{\mu} = -\A{}{\mu} \end{array} &
\dcurr{\mu} =
\frac{1}{2} (\Ap{}{\nu} \F{\mu\nu}{} - \A{}{\nu} \duF{\mu\nu}{} )
\\ \hline
\begin{array}{c} \mbox{internal} \\ \mbox{\rbs/} \\
\gam{\mu\nu}{} \mbox{ given by \eqref{skewmatr} } \end{array} &
\begin{array}{rl}
\Q{}{\mu} &=
\gam{\mu}{\nu} \A{}{\nu} + \dugam{\mu}{\nu} \Ap{}{\nu} \\
\Qp{}{\mu} &=
\gam{\mu}{\nu} \Ap{}{\nu} - \dugam{\mu}{\nu} \A{}{\nu} \end{array} &
\begin{array}{rl}
\rbcurr{\mu} = &
\frac{1}{2} \gam{\nu}{\sigma} (
\A{}{\sigma} \F{\mu\nu}{} + \Ap{}{\sigma} \duF{\mu\nu}{} ) \\
& + \frac{1}{2} \dugam{\nu}{\sigma} (
\Ap{}{\sigma} \F{\mu\nu}{} - \A{}{\sigma} \duF{\mu\nu}{} )
\end{array}
\\ \hline
\begin{array}{c} \mbox{general conformal} \\ \mbox{\Kvec/} \\
\kv{\mu}{}, \z{}{\mu\nu} \mbox{ given by \eqref{CKV} } \end{array} &
\begin{array}{rl}
\Q{}{\mu} &=
\wLie{\xi} \A{}{\mu} + \z{\mu}{\nu} \A{}{\nu} + \duz{\mu}{\nu} \Ap{}{\nu} \\
\Qp{}{\mu} &=
\wLie{\xi} \Ap{}{\mu} + \z{\mu}{\nu} \Ap{}{\nu} - \duz{\mu}{\nu} \A{}{\nu}
\end{array} &
\begin{array}{rl}
\ccurr{\mu}
=&
\frac{1}{2}(\wLie{\xi}\A{}{\nu}) \F{\mu\nu}{}
+ \frac{1}{2}(\wLie{\xi}\Ap{}{\nu}) \duF{\mu\nu}{} \\
& + \frac{1}{2} \z{\nu}{\sigma}( \A{}{\sigma} \F{\mu\nu}{}
+ \Ap{}{\sigma} \duF{\mu\nu}{} ) \\
& + \frac{1}{2} \duz{\nu}{\sigma}( \Ap{}{\sigma} \F{\mu\nu}{}
- \A{}{\sigma} \duF{\mu\nu}{} )
\end{array}
\\ \hline
\begin{array}{c} \mbox{general conformal} \\ \mbox{\Kvec/ dual} \end{array} &
\begin{array}{rl}
\Q{}{\mu} &=
\wLie{\xi} \Ap{}{\mu} + \z{\mu}{\nu} \Ap{}{\nu} -\duz{\mu}{\nu} \A{}{\nu} \\
\Qp{}{\mu} &=
-\wLie{\xi} \A{}{\mu} - \z{\mu}{\nu} \A{}{\nu} -\duz{\mu}{\nu} \Ap{}{\nu}
\end{array} &
\begin{array}{rl}
\cducurr{\mu} =&
\frac{1}{2} (\wLie{\xi}\Ap{}{\nu}) \F{\mu\nu}{}
- \frac{1}{2} (\wLie{\xi}\A{}{\nu}) \duF{\mu\nu}{} \\
& + \frac{1}{2} \z{\nu}{\sigma}( \Ap{}{\sigma} \F{\mu\nu}{}
- \A{}{\sigma} \duF{\mu\nu}{} ) \\
& - \frac{1}{2} \duz{\nu}{\sigma}( \A{}{\sigma} \F{\mu\nu}{}
+ \Ap{}{\sigma} \duF{\mu\nu}{} )
\end{array}
\\ \hline
\begin{array}{c} \mbox{Lie derivative terms} \\
\mbox{in $\Q{}{\mu},\Qp{}{\mu}$} \\ \\
\wLie{\xi} = \Lie{\xi} +\frac{1}{4} \Omega \\
\Omega \mbox{ given by \eqref{CKV} } \end{array} & &
\begin{array}{rl}
& \frac{1}{2} (\Lie{\xi} \A{}{\nu}) \F{\mu\nu}{}
+ \frac{1}{2} (\Lie{\xi} \Ap{}{\nu}) \duF{\mu\nu}{} \\
&= \D{\nu}\left( \frac{1}{2} \kv{\sigma}{} (
\A{}{\sigma} \F{\mu\nu}{} + \Ap{}{\sigma} \duF{\mu\nu}{} ) \right) \\
&\qquad + \consT{\mu}{\sigma}\kv{\sigma}{} \\ \hline
& \frac{1}{2} (\Lie{\xi} \Ap{}{\nu}) \F{\mu\nu}{}
- \frac{1}{2} (\Lie{\xi} \A{}{\nu}) \duF{\mu\nu}{} \\
&= \D{\nu}\left( \frac{1}{2} \kv{\sigma}{} (
\Ap{}{\sigma} \F{\mu\nu}{} - \A{}{\sigma} \duF{\mu\nu}{} ) \right)
\end{array}
\\ \hline
\end{array}$
\caption{Conserved currents derived from geometric symmetries of
the joint \potsys/ in Lorentz gauge}
\label{table:JPS-L-sym-curr}
\end{center}
\end{table}
Trivial currents (i.e. curls) are produced by
both the scaling \sym/ \eqref{JPS-L-Xscaling}
and the \HKV/ dual \syms/ \eqref{JPS-L-XHKvecdual}.
The \HKV/ \syms/ \eqref{JPS-L-XHKvec}
themselves reproduce
the same stress-energy currents (modulo trivial currents)
as admitted by the unaugmented \potsys/.
In contrast the currents associated with
the new \rbvec/ symmetries \eqref{JPS-L-Xrbvec}
as well as the \CKV/ symmetries \eqref{JPS-L-XCKvec}
and their dual symmetries \eqref{JPS-L-XCKvecdual}
are genuinely new local \conscurr/s.
Nontriviality of these currents is established
in \secref{sec:nonlocal},
where we will simplify all inessential dependence on potentials
through the embedding of the solution space of the \potsys/ \eqref{JPS-L}
into the solution space of \Meq/.
Finally, we remark that the
duality-rotation current, \rbvec/ currents,
\CKV/ currents and their dual currents
are related by considering a general \CKvec/
as was noted for the corresponding \syms/
in Proposition~\ref{JPS-L-combinedsymm}.
\begin{prop}
\label{JPS-L-combinedcurr}
For a \HKvec/
$\kv{\mu}{} =
\k{1}{\mu}{} + \k{2}{\mu\nu}{} \x{}{\nu} + \k{3}{}{} \x{\mu}{}$,
\begin{equation}
\ccurr{\mu}
= \consT{\mu}{\sigma}\kv{\sigma}{} +\rbcurr{\mu} \eqtext{ mod curls},\quad
\cducurr{\mu}
= -\rbducurr{\mu} +\frac{1}{4}\Omega \dcurr{\mu} \eqtext{ mod curls},\quad
\gam{}{} = \z{}{}
\end{equation}
is a geometric current of the \jps/ in Lorentz gauge,
where $\z{}{\mu\nu} = \frac{1}{2} \k{2}{\mu\nu}{}={\rm const}$,
$\Omega=2\k{3}{}{}={\rm const}$.
\end{prop}
\section{New nonlocal \syms/ and \conslaw/s of \Meq/}
\label{sec:nonlocal}
In this section
we derive nonlocal symmetries and nonlocal conserved currents of \Meq/
through projection of
the new geometric symmetries and new geometric currents of
the \jps/ in Lorentz gauge found in \secref{sec:jps-analysis}.
The projection is defined on $\Jsp{1}(A,A')$
using the natural embedding of
the solution jet space $\divfrRsp{}(A,A')$ of
the potentials in Lorentz gauge
into the solution jet space of \Meq/ as follows:
We note first that, with the introduction of potentials,
points in $\Jsp{q}(F)$ are identified with
equivalence classes of points in $\Jsp{q+1}(A,A')$
under (prolonged) gauge transformations
$\A{}{\mu} \rightarrow \A{}{\mu} + \D{\mu}\chi$,
$\Ap{}{\mu} \rightarrow \Ap{}{\mu} + \D{\mu}\chi'$,
for any functions $\chi,\chi'$ on $\Jsp{q}(A,A')$.
Any representative in each equivalence class satisfying the equations
\begin{equation}
\coD{\mu}\D{\mu}\chi(A) = \invflat{\mu\nu}\A{}{\mu,\nu} ,\quad
\coD{\mu}\D{\mu}\chi'(A') = \invflat{\mu\nu}\Ap{}{\mu,\nu}
\label{JPS-L:waveeq}
\end{equation}
corresponds to the choice of Lorentz gauge being imposed on the potentials.
(Note $\chi,\chi'$ are unique to within the addition of
any solution of the source-free wave equation.)
Hence we have an embedding of $\divfrRsp{}(A,A')$ into $\Rsp{}(A,A')$
given by the explicit jet space coordinates
$(\x{\mu}{},\tA{}{\nu},\tAp{}{\nu},\F{}{\mu\nu},\tA{}{\mu\nu},\tAp{}{\mu\nu})$
such that
\begin{equation}
\tA{}{\nu} = \A{}{\nu} -\D{\nu}\chi(A) ,\quad
\tAp{}{\nu} = \Ap{}{\nu} -\D{\nu}\chi'(A')
\label{tAA'coords}
\end{equation}
satisfy Lorentz gauge
\begin{equation}
\invflat{\mu\nu} \tA{}{\nu,\mu} = \invflat{\mu\nu} \tAp{}{\nu,\mu} = 0 .
\end{equation}
Note here
\begin{equation}
\F{}{\mu\nu} = \tA{}{[\nu,\mu]} = -{*\tAp{}{[\nu,\mu]}} ,
\end{equation}
and
\begin{eqnarray}
&&
\tA{}{\mu\nu} = {\rm trfr}( \A{}{\mu\nu} -\D{\mu}\D{\nu}\chi(A) )
= \A{}{(\nu,\mu)} -\D{\mu}\D{\nu}\chi(A) ,
\\
&&
\tAp{}{\mu\nu} = {\rm trfr}( \Ap{}{\mu\nu} -\D{\mu}\D{\nu}\chi'(A') )
= \Ap{}{(\nu,\mu)} -\D{\mu}\D{\nu}\chi'(A') ,
\label{dertAA'coords}
\end{eqnarray}
where, we recall,
$\A{}{\mu\nu} = \A{}{(\nu,\mu)}$,
$\Ap{}{\mu\nu} = \Ap{}{(\nu,\mu)}$.
Hereafter, $\Rsp{}(\tilde A,\tilde A') \subset \Jsp{1}(A,A')$
will denote $\divfrRsp{}(A,A') \subset \Jsp{1}(A,A')$
under this embedding of the solution jet spaces.
\subsection{Induced \syms/ of \Meq/}
\begin{defn}
\label{prop:JPS-L-projsymm}
Any local symmetry
${\bf X}\tA{}{\mu} = \Q{}{\mu}[\tilde A,\tilde A']$,
${\bf X}\tAp{}{\mu} = \Qp{}{\mu}[\tilde A,\tilde A']$
of the \jps/ on $\Rsp{}(\tilde A,\tilde A')$
projects to a symmetry
${\bf X}\F{}{\mu\nu} = \D{[\mu}\Q{}{\nu]}[A,A']$
of \Meq/ on $\Rsp{}(A,A')$
via the transformation \eqsref{tAA'coords}{dertAA'coords}.
A projected symmetry is {\em local} in the electromagnetic field
iff
${\bf X}\F{}{\mu\nu}$ has {\it no} essential dependence on the potentials
$\A{}{\nu},\Ap{}{\nu},\A{}{\mu\nu},\Ap{}{\mu\nu}$,
so $\D{[\mu}\Q{}{\nu]}$ is a \horiz/ 2-form on $\Rsp{}(F)$;
and otherwise a projected symmetry ${\bf X}\F{}{\mu\nu}[A,A']$ is {\em nonlocal}.
\end{defn}
We now list in \tableref{JPS-L-ME-sym}
the projected geometric symmetries
obtained from Theorem~\ref{thm:JPS-L-sym}.
The projections are a straightforward computation
using Definition~\ref{prop:JPS-L-projsymm}.
\mystretch{1.0}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|} \hline
Geometric \sym/ type &
\begin{tabular}{c} Induced \sym/ of \\
$\coD{\mu}\F{}{\mu\nu} = 0$,\quad $\coD{\mu}\duF{}{\mu\nu} = 0$
\end{tabular} \\ \hline\hline
\begin{tabular}{c}
scaling \eqref{JPS-L-Xscaling} \\
and duality-rotation \eqref{JPS-L-Xduality}
\end{tabular} &
$\begin{array}{rl}
\X_{\it scal}= &
\F{}{\mu\nu} \parder{\F{}{\mu\nu}} \\
\X_{\it dual}= &
\duF{}{\mu\nu} \parder{\F{}{\mu\nu}}
\end{array}$ \\ \hline
\begin{tabular}{c}
internal \rb/ \eqref{JPS-L-Xrbvec} \\
$\gam{\mu\nu}{}$ given by \eqref{skewmatr} \end{tabular} &
$\rbX{}=
( \tA{\sigma}{[\mu} \gam{\nu]\sigma}{}
+ \tAp{\sigma}{[\mu} \dugam{\nu]\sigma}{} ) \parder{\F{}{\mu\nu}}$
\\ \hline
\begin{tabular}{c}
translation, \rb/, dilation \eqref{JPS-L-XHKvec} \\
and dual \eqref{JPS-L-XHKvecdual} \\
$\kv{\mu}{}$ given by \eqref{HKV} \end{tabular} &
$\begin{array}{rl}
\hX{}= &
(\Lie{\xi} \F{}{\mu\nu}) \parder{\F{}{\mu\nu}} \\
\hX{}'= &
(\Lie{\xi} \duF{}{\mu\nu}) \parder{\F{}{\mu\nu}}
\end{array}$ \\ \hline
\begin{tabular}{c}
conformal \eqref{JPS-L-XCKvec} \\
and dual \eqref{JPS-L-XCKvecdual} \\
$\kv{\mu}{}$, $\z{}{\mu\nu}$, $\Omega$ given by \eqref{CKV} \\
$\wLie{\xi} := \Lie{\xi} + \frac{1}{4} \Omega$
\end{tabular} &
$\begin{array}{rl}
\cX{}= &
( \wLie{\xi} \F{}{\mu\nu}
+\tA{\sigma}{[\mu} \z{\nu]\sigma}{}
+ \tAp{\sigma}{[\mu} \duz{\nu]\sigma}{} \\
& - \tA{\sigma}{}\der{\sigma}\z{\mu\nu}{}
+ \tAp{\sigma}{}\der{\sigma}\duz{\mu\nu}{} )
\parder{\F{}{\mu\nu}} \\
\cX{}'= &
( \wLie{\xi} \duF{}{\mu\nu}
+\tAp{\sigma}{[\mu} \z{\nu]\sigma}{}
- \tA{\sigma}{[\mu} \duz{\nu]\sigma}{} \\
& - \tAp{\sigma}{}\der{\sigma}\z{\mu\nu}{}
- \tA{\sigma}{}\der{\sigma}\duz{\mu\nu}{} ) \parder{\F{}{\mu\nu}}
\end{array}$ \\ \hline
\end{tabular}
\caption{Symmetries of \Meq/ induced by geometric \syms/
of the \jps/ in Lorentz gauge
\label{JPS-L-ME-sym}}
\end{center}
\end{table}
The projected symmetries with essential dependence on the potentials
are simply the \CKV/ \syms/ and their dual \syms/,
and the \rbvec/ \syms/,
which comprise
the new geometric symmetries found for the \jps/ in Lorentz gauge.
Note the Lie derivative term in the \CKV/ \syms/
corresponds itself precisely to
the local conformal \syms/ of \Meq/,
and likewise for the Lie derivative term in the dual \syms/.
By subtraction of these local symmetries
\begin{equation}
{\bf X}_{\it conf} = (\Lie{\xi} \F{}{\mu\nu}) \Parder{\F{}{\mu\nu}} ,\quad
{\bf X}_{\it conf}' = (\Lie{\xi} \duF{}{\mu\nu}) \Parder{\F{}{\mu\nu}} ,
\label{ME:localsymm}
\end{equation}
we are left with new symmetries ${\bf Z}$
of a similar form to the \rbvec/ \syms/,
involving no dependence on derivatives of
the electromagnetic field $\F{}{\mu\nu}$ or its dual,
as given by the following transformations:
\begin{enumerate}
\item[{\rm (i)}]
\rbvec/ transformation
\begin{equation}
\rbtX{} = \Big(
\A{\sigma}{[\mu} \gam{\nu]\sigma}{} + \Ap{\sigma}{[\mu} \dugam{\nu]\sigma}{}
+ \gam{[\mu}{\sigma} \D{\nu]}\D{\sigma}\chi(A)
+ \dugam{[\mu}{\sigma} \D{\nu]}\D{\sigma}\chi'(A')
\Big) \Parder{\F{}{\mu\nu}} ,
\label{ME:rbvecsymm}
\end{equation}
with $\gam{}{\mu\nu} = \gam{}{[\mu\nu]}={\rm const}$,
\item[{\rm (ii)}]
\cvec/ transformation
\begin{eqnarray}
\ctX{} = && \Big(
\frac{1}{4} \Omega \F{}{\mu\nu}
+ \A{\sigma}{[\mu} \z{\nu]\sigma}{} + \Ap{\sigma}{[\mu} \duz{\nu]\sigma}{}
- \A{\sigma}{} \der{\sigma}\z{\mu\nu}{}
+ \Ap{\sigma}{}\der{\sigma}\duz{\mu\nu}{}
\nonumber\\&&
- \D{[\mu}( \z{\nu]}{\sigma} \D{\sigma}\chi(A)
+ \duz{\nu]}{\sigma} \D{\sigma}\chi'(A')
+\frac{1}{3} \coder{\sigma} \z{\nu]\sigma}{} \chi(A) )
\Big) \Parder{\F{}{\mu\nu}} ,
\label{ME:Cvecsymm}
\end{eqnarray}
and dual transformation
\begin{eqnarray}
\ctX{}' = && \Big(
\frac{1}{4} \Omega \duF{}{\mu\nu}
+ \Ap{\sigma}{[\mu} \z{\nu]\sigma}{}
-\A{\sigma}{[\mu} \duz{\nu]\sigma}{}
- \Ap{\sigma}{} \der{\sigma}\z{\mu\nu}{}
- \A{\sigma}{}\der{\sigma}\duz{\mu\nu}{}
\nonumber\\&&
- \D{[\mu}( \z{\nu]}{\sigma} \D{\sigma}\chi'(A')
- \duz{\nu]}{\sigma} \D{\sigma}\chi(A)
+\frac{1}{3} \coder{\sigma} \z{\nu]\sigma}{} \chi'(A') )
\Big) \Parder{\F{}{\mu\nu}} ,
\label{ME:Cvecdualsymm}
\end{eqnarray}
where
$\z{}{\mu\nu} := -\frac{1}{2} \coder{[\mu} \kv{\nu]}{}$
and $\Omega:= \frac{1}{2} \der{\mu} \kv{\mu}{}$,
with
$\kv{\mu}{} =
\k{4}{\sigma}{} \x{}{\sigma} \x{\mu}{}
- \frac{1}{2} \k{4}{\mu}{} \x{\sigma}{} \x{}{\sigma}$,
$\k{4}{\mu}{}={\rm const}$;
$\chi(A),\chi'(A')$ are scalar functions satisfying
the wave equation \eqref{JPS-L:waveeq}.
\end{enumerate}
Their explicit dependence on the potentials means that these symmetries
are nonlocal and nontrivial.
We remark that
the transformations \eqrefs{ME:Cvecsymm}{ME:Cvecdualsymm}
continue to be admitted as \syms/
if the \CKvec/ $\kv{\mu}{}$ is replaced by a \HKvec/ \eqref{HKV}
as seen from Proposition~\ref{JPS-L-combinedsymm}.
Indeed,
in the case of a \rb/ $\kv{\mu}{} = \k{2}{\mu\nu}{} \x{}{\nu}$,
these transformations respectively reduce to
the \rbvec/ \syms/ \eqref{ME:rbvecsymm}
with parameters
$\gam{}{\mu\nu} =
\frac{1}{2} \k{2}{\mu\nu}{},
-\frac{1}{4} \invvol{\mu\nu}{\sigma\tau}\k{2}{\sigma\tau}{}$;
in contrast these transformations yield
a multiple $\frac{1}{2} \k{3}{}{}$ of
the scaling and duality-rotation \syms/
in the case of a dilation $\kv{\mu}{} = \k{3}{}{}\x{\mu}{}$,
and trivial \syms/
in the case of a translation $\kv{\mu}{} =\k{1}{\mu}{}$.
Thus, the nonlocal symmetries we have found for \Meq/
arise from new symmetries of the form \eqrefs{ME:Cvecsymm}{ME:Cvecdualsymm}
for a general \CKvec/
\ie/ $\kv{\mu}{}$ is any generator of
a conformal isometry \eqref{conformalisom} of \Minksp/.
There is a deeper unity between
the \rbvec/ \syms/ and the \cvec/ and dual \syms/.
Consider the transformations \eqrefs{ME:Cvecsymm}{ME:Cvecdualsymm}
using the sum of a \rb/ \Kvec/ and a \CKvec/
\begin{equation}
\kv{\mu}{} =
\k{2}{\mu\nu}{} \x{}{\nu}
+ \k{4}{\sigma}{} \x{}{\sigma} \x{\mu}{}
- \frac{1}{2} \k{4}{\mu}{} \x{\sigma}{} \x{}{\sigma} .
\label{unifyKV}
\end{equation}
We observe that only the (scaled) curl $\z{}{\mu\nu}$ and divergence $\Omega$
of this \Kvec/ enter these transformations,
where $\Omega$ is related to $\z{}{\mu\nu}$ by
\begin{equation}
\z{}{\mu\nu}
= \frac{1}{2} \k{2}{\mu\nu}{} -\k{4}{[\mu}{} \x{\nu]}{} ,\quad
\Omega
= 2\k{4}{\mu}{} \x{}{\mu} = \frac{4}{3} \x{}{\nu} \der{\mu} \z{}{\mu\nu} .
\end{equation}
Moreover, $\z{}{\mu\nu}$ has precisely the form of the dual of
a \KY/ tensor,
namely $\ky{}{\mu\nu}(x) := \duz{}{\mu\nu}$ satisfies the
\KY/ equation \cite{KYtensors:1,KYtensors:2}
\begin{equation}
\coder{(\sigma} \ky{}{\mu)\nu} =0 .
\label{KYeq}
\end{equation}
(More general \KY/ tensors of conformal type
are parameters for local chiral \syms/ of \Meq/
\cite{FusNik:1983,FusNik:1987book,AncPoh:2002,AncPoh:2004}
and first arose in the study of integrals of the geodesic equations
for light rays in the curved Kerr metric;
they are also connected with separation of variables of \Meq/
in that metric.)
Thus the nonlocal internal \syms/
\eqref{ME:rbvecsymm}, \eqref{ME:Cvecsymm}, \eqref{ME:Cvecdualsymm}
have the following geometric form:
\begin{eqnarray}
\tX_{Y}
=&& \Big(
\frac{1}{3} \x{}{\sigma} \der{\tau} \duky{}{\sigma\tau} \F{}{\mu\nu}
-\tA{\sigma}{[\mu} \duky{\nu]\sigma}{}
+ \tAp{\sigma}{[\mu} \ky{\nu]\sigma}{}
+ \tA{}{\sigma} \coder{\sigma}\duky{\mu\nu}{}
+ \tAp{}{\sigma}\coder{\sigma}\ky{\mu\nu}{}
\Big) \Parder{\F{}{\mu\nu}} ,
\nonumber\\
= && \Big(
\frac{1}{3} \x{}{\sigma} \der{\tau} \duky{}{\sigma\tau} \F{}{\mu\nu}
-\A{\sigma}{[\mu} \duky{\nu]\sigma}{}
+ \Ap{\sigma}{[\mu} \ky{\nu]\sigma}{}
+ \A{}{\sigma} \coder{\sigma}\duky{\mu\nu}{}
+ \Ap{}{\sigma}\coder{\sigma}\ky{\mu\nu}{}
\nonumber\\&&
+\D{[\mu}( \duky{\nu]}{\sigma} \D{\sigma}\chi(A)
- \ky{\nu]}{\sigma} \D{\sigma}\chi'(A')
+\frac{1}{3} \coder{\sigma} \duky{\nu]\sigma}{} \chi(A) )
\Big) \Parder{\F{}{\mu\nu}} ,
\label{ME:KYsymm}\\
\tX_{Y}'
= && \Big(
\frac{1}{3} \x{}{\sigma} \der{\tau} \duky{}{\sigma\tau} \duF{}{\mu\nu}
-\tAp{\sigma}{[\mu} \duky{\nu]\sigma}{}
- \tA{\sigma}{[\mu} \ky{\nu]\sigma}{}
+ \tAp{}{\sigma} \coder{\sigma}\duky{\mu\nu}{}
- \tA{}{\sigma}\coder{\sigma}\ky{\mu\nu}{}
\Big) \Parder{\F{}{\mu\nu}} ,
\nonumber\\
= && \Big(
\frac{1}{3} \x{}{\sigma} \der{\tau} \duky{}{\sigma\tau} \duF{}{\mu\nu}
-\Ap{\sigma}{[\mu} \duky{\nu]\sigma}{}
- \A{\sigma}{[\mu} \ky{\nu]\sigma}{}
+ \Ap{}{\sigma} \coder{\sigma}\duky{\mu\nu}{}
- \A{}{\sigma}\coder{\sigma}\ky{\mu\nu}{}
\nonumber\\&&
+\D{[\mu}( \duky{\nu]}{\sigma} \D{\sigma}\chi'(A')
+ \ky{\nu]}{\sigma} \D{\sigma}\chi(A)
+\frac{1}{3} \coder{\sigma} \duky{\nu]\sigma}{} \chi'(A') )
\Big) \Parder{\F{}{\mu\nu}} ,
\label{ME:KYdualsymm}
\end{eqnarray}
where
\begin{equation}
\ky{}{\mu\nu} =
\kb{1}{\mu\nu}{} + \invvol{\mu\nu}{\sigma\tau} \kb{2}{\sigma}{} \x{\tau}{} ,
\quad
\kb{1}{\mu\nu}{} = \kb{1}{[\mu\nu]}{}, \kb{2}{\sigma}{} ={\rm const}
\label{KYT}
\end{equation}
is a \KY/ tensor.
Note that only the set of constant \KY/ tensors is preserved
under duality $\ky{}{\mu\nu} \rightarrow \duky{}{\mu\nu}$.
\begin{thm}
\label{ME-nonlocalsymm}
\Meq/ admits the (nontrivial) nonlocal \syms/
\eqrefs{ME:KYsymm}{ME:KYdualsymm}
depending on an arbitrary \KY/ tensor \eqref{KYT}.
Under the \dutr/ \eqref{JPS-duality} on the potentials,
these \syms/ are interchanged,
and in the case of a constant \KY/ tensor
they are related through directly replacing
this tensor with its dual.
Thus the \syms/ comprise a 14-dimensional vector space.
\end{thm}
The vector space structure of these nonlocal \syms/
has a basis consisting of
six of \rbvec/ type \eqref{ME:rbvecsymm},
four of \cvec/ type \eqref{ME:Cvecsymm}
and four of dual type \eqref{ME:Cvecdualsymm}.
However, their commutator structure is not closed.
Note the \rbvec/ \syms/ themselves comprise a $\mathfrak{so}(3,1)$ Lie algebra,
which follows from projection of the commutator structure
stated in Theorem~\ref{JPS-L-symalg}
for the corresponding local \syms/
on the solution jet space $\divfrRsp{}(A,A')$ of the \jps/ in Lorentz gauge.
In contrast the \cvec/ \syms/ and their duals
do not arise from projection of any local \syms/ of this \potsys/,
and as a consequence,
their commutators do not have the form of
local transformations on $\divfrRsp{}(A,A')$,
\ie/ they necessarily involve a nonlocal dependence on the potentials
and hence there is no natural Lie algebra structure for them.
The same conclusion holds for the commutators of
the \rbvec/ \syms/ with the \cvec/ \syms/ or their duals.
These commutators are still themselves
nonlocal \syms/ of the \potsys/ and hence of \Meq/.
More generally,
there is an enveloping (nonlocal) \sym/ algebra
generated by the span of the \syms/
\eqref{ME:rbvecsymm}, \eqref{ME:Cvecsymm}, \eqref{ME:Cvecdualsymm},
and their repeated commutators.
If instead we consider the respective superpositions
$\ctX{} +{\bf X}_{\it conf}$, $\ctX{}' +{\bf X}'_{\it conf}$,
of the nonlocal \syms/ \eqref{ME:Cvecsymm}, \eqref{ME:Cvecdualsymm}
of \Meq/
and the corresponding local \syms/ \eqref{ME:localsymm}
given by \rb/ \Kvec/s and \CKvec/s,
their collective enveloping algebra collapses to
the Lie algebra of \rbs/ and complexified inversions,
namely the natural semidirect product of
$\mathfrak{so}(3,1)$ with $\mathfrak{u}(1)^4 \otimes \mathbb{C}$,
as follows from Theorem~\ref{JPS-L-symalg}.
\subsection{Induced conservation laws}
\begin{defn}
\label{JPS-L-projcurr}
Any local \conscurr/ $\curr{\mu}[\tilde A,\tilde A']$
on $\Rsp{}(\tilde A,\tilde A')$
directly projects to a \conscurr/
$\curr{\mu}[A,A']$ of \Meq/ on $\Rsp{}(A,A')$
via the transformation \eqsref{tAA'coords}{dertAA'coords}.
A projected current is {\em local} in the electromagnetic field
iff,
up to addition of a curl,
it has {\it no} essential dependence on the potentials
$\A{}{\nu},\Ap{}{\nu},\A{}{\mu\nu},\Ap{}{\mu\nu}$,
so that
$\curr{\mu}[A,A']+ \triv{\nu}{\mu\nu}[A,A']$
for some antisymmetric tensor function $\curl{\mu\nu}{}[A,A']$
is a vector function on $\Rsp{}(F)$;
and otherwise a projected current $\curr{\mu}[A,A']$
is {\em nonlocal}.
\end{defn}
For the geometric \conscurr/s
listed in \tableref{table:JPS-L-sym-curr}
for the \jps/ in Lorentz gauge,
the Lie derivative terms in the general \CKV/ currents
simply project to the corresponding stress-energy currents \eqref{hKVcurr},
while the Lie derivative terms in the dual currents project to a curl.
The remaining internal terms in those currents
as well as the \rbvec/ currents
and duality-rotation current
separately project to new \conscurr/s $\Phi\up{\mu}$ of \Meq/:
\begin{enumerate}
\item[{\rm (i)}]
duality-rotation \conslaw/
\begin{equation}
\dtcurr{\mu}
= \tAp{}{\nu} \F{\mu\nu}{} - \tA{}{\nu} \duF{\mu\nu}{}
= \Ap{}{\nu} \F{\mu\nu}{} - \A{}{\nu} \duF{\mu\nu}{} \eqtext{ mod curls},
\label{ME:dualitycurr}
\end{equation}
\item[{\rm (ii)}]
\rbvec/ \conslaw/
\begin{eqnarray}
\rbtcurr{\mu}
= &&
\gam{\nu}{\sigma}(
\tA{}{\sigma} \F{\mu\nu}{} + \tAp{}{\sigma} \duF{\mu\nu}{} )
+ \dugam{\nu}{\sigma}(
\tAp{}{\sigma} \F{\mu\nu}{} - \tA{}{\sigma} \duF{\mu\nu}{} )
\nonumber\\
= &&
-\frac{1}{2} \gam{}{\nu\sigma}(
\A{\mu}{} \F{}{\nu\sigma} + \Ap{\mu}{} \duF{}{\nu\sigma} )
+ 2 \gam{}{\nu[\sigma} (
\A{}{\sigma} \F{\mu]}{\nu} + \Ap{}{\sigma} \duF{\mu]}{\nu} )
\nonumber\\&&
+\wavecurr{\mu}(\chi(A),\gam{}{\nu\sigma}\F{}{\nu\sigma})
+\wavecurr{\mu}(\chi'(A'),\gam{}{\nu\sigma}\duF{}{\nu\sigma})
\label{ME:rbveccurr}\\&&
\eqtext{ mod curls},
\nonumber
\end{eqnarray}
with $\gam{}{\mu\nu} = \gam{}{[\mu\nu]}={\rm const}$,
\item[{\rm (iii)}]
\cvec/ \conslaw/
\begin{eqnarray}
\ctcurr{\mu}
= &&
\z{\nu}{\sigma}(
\tA{}{\sigma} \F{\mu\nu}{} + \tAp{}{\sigma} \duF{\mu\nu}{} )
+ \duz{\nu}{\sigma}(
\tAp{}{\sigma} \F{\mu\nu}{} - \tA{}{\sigma} \duF{\mu\nu}{} )
+ \frac{1}{4} \Omega ( \tA{}{\nu} \F{\mu\nu}{} + \tAp{}{\nu} \duF{\mu\nu}{} )
\nonumber\\
= &&
-\frac{1}{2} \z{}{\nu\sigma} (
\A{\mu}{} \F{}{\nu\sigma} + \Ap{\mu}{} \duF{}{\nu\sigma} )
+ 2 \z{}{\nu[\sigma} (
\A{}{\sigma} \F{\mu]}{\nu} + \Ap{}{\sigma} \duF{\mu]}{\nu} )
-\frac{1}{8} \vol{}{\mu\sigma\alpha\beta}
\der{\sigma}\Omega\ \A{}{\alpha} \Ap{}{\beta}
\nonumber\\&&
+\wavecurr{\mu}(\chi(A),\z{}{\nu\sigma}\F{}{\nu\sigma})
+\wavecurr{\mu}(\chi'(A'),\z{}{\nu\sigma}\duF{}{\nu\sigma})
\label{ME:Cveccurr}\\&&
\eqtext{ mod curls},
\nonumber
\end{eqnarray}
and dual \conslaw/
\begin{eqnarray}
\ductcurr{\mu}
= &&
\z{\nu}{\sigma}(
\tAp{}{\sigma} \F{\mu\nu}{} - \tA{}{\sigma} \duF{\mu\nu}{} )
- \duz{\nu}{\sigma}(
\tA{}{\sigma} \F{\mu\nu}{} + \tAp{}{\sigma} \duF{\mu\nu}{} )
+ \frac{1}{4} \Omega ( \tAp{}{\nu} \F{\mu\nu}{} - \tA{}{\nu} \duF{\mu\nu}{} )
\nonumber\\
= &&
-\frac{1}{2} \z{}{\nu\sigma} (
\Ap{\mu}{} \F{}{\nu\sigma} - \A{\mu}{} \duF{}{\nu\sigma} )
+ 2 \z{}{\nu[\sigma} (
\Ap{}{\sigma} \F{\mu]}{\nu} - \A{}{\sigma} \duF{\mu]}{\nu} )
\nonumber\\&&
+ \frac{1}{4} \Omega ( \Ap{}{\nu} \F{\mu\nu}{} -\A{}{\nu} \duF{\mu\nu}{} )
-\wavecurr{\mu}(\chi(A),\z{}{\nu\sigma}\duF{}{\nu\sigma})
+\wavecurr{\mu}(\chi'(A'),\z{}{\nu\sigma}\F{}{\nu\sigma})
\label{ME:Cvecdualcurr}\\&&
\eqtext{ mod curls},
\nonumber
\end{eqnarray}
where $\z{}{\mu\nu} := -\frac{1}{2} \coder{[\mu} \kv{\nu]}{}$
and $\Omega:= \frac{1}{2} \der{\mu} \kv{\mu}{}$,
with
$\kv{\mu}{} =
\k{4}{\sigma}{} \x{}{\sigma} \x{\mu}{}
- \frac{1}{2} \k{4}{\mu}{} \x{\sigma}{} \x{}{\sigma}$,
$\k{4}{\mu}{}={\rm const}$;
here $\chi(A),\chi'(A')$ are scalar functions satisfying
the wave equation \eqref{JPS-L:waveeq},
and $\wavecurr{\mu}(f,g) := \frac{1}{2}( g\coD{\mu}f - f\coD{\mu}g )$
is a skew-bilinear vector function depending on any scalar expressions $f,g$.
\end{enumerate}
In writing downs these currents
we have simplified some terms through integration by parts
and multiplied by an overall factor of $2$.
We remark that $\wavecurr{\mu}(f,g)$ has the form of a conserved current
formula for the ordinary scalar wave equation \cite{AncBlu:1996JMP},
taking $f,g$ to be a pair of symmetries.
These terms are not separately conserved in the currents here.
Note that if the \CKvec/ $\kv{\mu}{}$ in the currents \eqref{ME:Cveccurr}
is replaced by a \HKvec/ \eqref{HKV},
we obtain
the \rbvec/ currents \eqref{ME:rbveccurr}
with $\gam{}{\mu\nu} = \frac{1}{2} \k{2}{\mu\nu}{}$
in the case of a \rb/
$\kv{\mu}{} = \k{2}{\mu\nu}{} \x{}{\nu}$,
and trivial currents
in the case of a translation $\kv{\mu}{} =\k{1}{\mu}{}$
or a dilation $\kv{\mu}{} = \x{\mu}{}$.
Similarly, from the dual currents \eqref{ME:Cvecdualcurr}
we obtain
the \rbvec/ current \eqref{ME:rbveccurr}
with $\gam{}{\mu\nu}
= -\frac{1}{4} \invvol{\mu\nu}{\alpha\beta} \k{2}{\alpha\beta}{}$
in the case of a \rb/
$\kv{\mu}{} = \k{2}{\mu\nu}{} \x{}{\nu}$,
the duality-rotation current \eqref{ME:dualitycurr}
in the case of a dilation $\kv{\mu}{} = \x{\mu}{}$,
and a trivial current
in the case of translation $\kv{\mu}{} =\k{1}{\mu}{}$.
Thus, the new currents we have found for \Meq/
come from new \conslaw/s of the form \eqrefs{ME:Cveccurr}{ME:Cvecdualcurr}
that exist for a general \CKvec/,
\ie/ $\kv{\mu}{}$ is any generator of
a conformal isometry \eqref{conformalisom} of \Minksp/.
We now prove these \conslaw/s \eqsref{ME:dualitycurr}{ME:Cvecdualcurr}
are nontrivial and nonlocal.
Firstly, it is useful to define the weight of
a quadratic current $\curr{\mu}$ on $\Jsp{q}(A,A')$
to be the maximum of the weights of all monomial terms in $\curr{\mu}$,
given by counting the total number of derivatives that appear on
the potentials:
\ie/ $\A{}{\mu},\Ap{}{\mu}$ have weight $0$,
$\F{}{\mu\nu}$ has weight $1$,
while $\chi(A),\chi'(A')$ are counted as weight $-1$
through equation \eqref{JPS-L:waveeq}.
The lowest weight nontrivial {\it local} \conscurr/s of \Meq/
as shown in \Ref{AncPoh:2001}
are the stress-energy currents \eqref{ME:stressenergy},
which have weight $2$ on $\Jsp{}(F) \subset \Jsp{1}(A,A')$.
In comparison, the currents \eqsref{ME:dualitycurr}{ME:Cvecdualcurr}
each have weight $1$ and cannot be equivalent consequently
to any nontrivial local current of \Meq/.
This establishes, moreover, that these currents
\eqsref{ME:dualitycurr}{ME:Cvecdualcurr}
are nonlocal.
Thus it remains to show only that they are nontrivial
when restricted to the solution jet space $\Rsp{}(\tilde A,\tilde A')$.
\begin{lem}
Suppose
\begin{equation}
\curr{\mu} =
( \k{1}{\mu}{\alpha\beta\sigma} \tA{\sigma}{}
+ \k{2}{\mu}{\alpha\beta\sigma} \tAp{\sigma}{} )\F{\alpha\beta}{}
\label{currLHS}
\end{equation}
is a \conscurr/ on $\Rsp{}(\tilde A,\tilde A')$.
Then this current is trivial iff
\begin{equation}
\k{2}{\mu\alpha\beta\sigma}{}
= \frac{1}{2} \vol{\nu\tau}{\alpha\beta} \k{1}{\mu\nu\tau\sigma}{}
= k \vol{}{\mu\alpha\beta\sigma} ,\quad
k={\rm const}
\label{trivcondition}
\end{equation}
\end{lem}
\Proof{}
Any trivial current on $\Rsp{}(\tilde A,\tilde A')$ is characterized
on $\Jsp{1}(A,A')$ by having the form
\begin{equation}
\curr{\mu}
= \triv{\nu}{\mu\nu}
+ \bb{}{\mu}{\alpha\beta} (\Fp{\alpha\beta}{} - \duF{\alpha\beta}{}) ,\quad
\Fp{\alpha\beta}{}:= \tAp{[\beta,\alpha]}{} ,\quad
\F{\alpha\beta}{}:= \tA{[\beta,\alpha]}{} ,
\label{currRHS}
\end{equation}
where
\begin{eqnarray}
&&
\curl{\mu\nu}{} =
\T{1}{\mu\nu}{\alpha\beta}(x) \tA{\alpha}{} \tA{\beta}{}
+ \T{2}{\mu\nu}{\alpha\beta}(x) \tAp{\alpha}{} \tAp{\beta}{}
+ \Tp{\mu\nu}{\alpha\beta}(x) \tAp{\alpha}{} \tA{\beta}{} ,
\\
&&
\bb{}{\mu}{\alpha\beta} =
\bb{1}{\mu}{\alpha\beta\sigma}(x) \tA{\sigma}{}
+ \bb{2}{\mu}{\alpha\beta\sigma}(x) \tAp{\sigma}{} ,
\end{eqnarray}
with the coefficients subject to the index symmetries
$\T{i}{\mu\nu}{\alpha\beta} = \T{i}{[\mu\nu]}{(\alpha\beta)}$,
$\Tp{\mu\nu}{\alpha\beta} = \Tp{[\mu\nu]}{\alpha\beta}$,
$\bb{i}{\mu}{\alpha\beta\sigma} = \bb{i}{\mu}{[\alpha\beta]\sigma}$.
To proceed we expand \eqref{currRHS}
and substitute the decompositions
$\tA{\beta,\alpha}{} =
\F{\alpha\beta}{}
+ {\rm trfr} \tA{(\beta,\alpha)}{}$
and
$\tAp{\beta,\alpha}{} =
\Fp{\alpha\beta}{}
+ {\rm trfr} \tAp{(\beta,\alpha)}{}$.
The coefficients of all terms other than
$\F{\alpha\beta}{}\tA{\sigma}{}$ and $\F{\alpha\beta}{}\tAp{\sigma}{}$
cannot match \eqref{currLHS} and must therefore vanish,
which leads to the conditions
\begin{eqnarray}
&&
\T{1}{\mu}{(\nu\alpha)\beta} =
\T{2}{\mu}{(\nu\alpha)\beta} =
\Tp{\mu}{(\nu|\beta|\alpha)} =
\Tp{\mu}{(\nu\alpha)\beta} =
0 ,
\label{conds}\\
&&
2\T{2}{\mu}{[\alpha\beta]\sigma} + \bb{2}{\mu}{\alpha\beta\sigma} = 0 ,
\label{rel1}\\
&&
\Tp{\mu}{[\alpha\beta]\sigma} + \bb{1}{\mu}{\alpha\beta\sigma} = 0 .
\label{rel2}
\end{eqnarray}
The index symmetries imposed by \eqref{conds}
immediately show that
$\T{1}{\mu}{\nu\alpha\beta}$ and $\T{2}{\mu}{\nu\alpha\beta}$ vanish
while $\Tp{\mu}{\nu\alpha\beta}$ must be totally antisymmetric
and hence
\begin{equation}
\Tp{\mu}{\nu\alpha\beta} = k(x) \invvol{\mu}{\nu\alpha\beta}
\label{kcond}
\end{equation}
for some $k(x)$.
It then follows from \eqref{rel1} that
$\bb{2}{\mu}{\alpha\beta\sigma}$ also vanishes.
Finally,
by now equating \eqref{currLHS} to \eqref{currRHS}
and collecting like terms, we obtain
\begin{eqnarray}
\k{1}{\mu}{\alpha\beta\sigma}
= -\frac{1}{2} \vol{\alpha\beta}{\rho\tau} \bb{1}{\mu}{\rho\tau\sigma} ,\quad
\k{2}{\mu}{\rho\tau\sigma}
= - \Tp{\mu}{\rho\tau\sigma} ,\quad
\coder{\rho} \k{2}{\mu}{\rho\tau\sigma} =0 .
\label{kconds}
\end{eqnarray}
Then the relations \eqrefs{rel2}{kconds} establish
the conditions \eqref{trivcondition} stated in the Lemma.
\hfill$\Box$
To apply this Lemma,
we observe the new \conscurr/s \eqsref{ME:dualitycurr}{ME:Cvecdualcurr}
written in terms of $\tA{}{\nu},\tAp{}{\nu},\F{}{\mu\nu}$
have the form \eqref{currLHS} where
\begin{equation}
\k{1}{}{\mu\alpha\beta\sigma} =
\flat{\mu[\alpha} \aa{\beta]\sigma}
- \frac{1}{2} \vol{\mu\alpha\beta}{\nu} \aap{\nu\sigma} ,\quad
\k{2}{}{\mu\alpha\beta\sigma} =
\flat{\mu[\alpha} \aap{\beta]\sigma}
+ \frac{1}{2} \vol{\mu\alpha\beta}{\nu} \aa{\nu\sigma}
\end{equation}
as given by \tableref{ME:currsform}.
The algebraic conditions \eqref{trivcondition} are readily
found to fail for each of these currents.
Therefore our main result is now proven.
\mystretch{1.0}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Conserved current &
$\aa{\mu\nu}$ & $\aap{\mu\nu}$
\\ \hline\hline
Duality-rotation
& $0$ & $\flat{\mu\nu}$
\\ \hline
Internal \rb/
& $\gam{\mu\nu}{}$ & $\dugam{\mu\nu}{}$
\\ \hline
Internal conformal
& $\z{\mu\nu}{} +\frac{1}{4} \Omega \flat{\mu\nu}$
& $\duz{\mu\nu}{}$
\\ \hline
Internal dual conformal
& $-\duz{\mu\nu}{}$
& $\z{\mu\nu}{} + \frac{1}{4} \Omega \flat{\mu\nu}$
\\ \hline
\end{tabular}
\caption{Form of nonlocal currents of \Meq/
\label{ME:currsform}}
\end{center}
\end{table}
\begin{thm}
\label{ME-nonlocalcurr}
\Meq/ admits the (nontrivial) nonlocal \conslaw/s
\eqsref{ME:dualitycurr}{ME:Cvecdualcurr}.
These \conslaw/s span a 15-dimensional vector space
whose basis consists of one of duality-rotation type,
six of \rbvec/ type,
four of \cvec/ type and four of dual type.
\end{thm}
We remark that these nonlocal \conslaw/s also arise directly from
the nonlocal symmetries \eqsref{ME:rbvecsymm}{ME:Cvecdualsymm}
through use of the formula \eqref{MEcurrentformula}
in Theorem~\ref{thm:MEcurrentformula}
that generates \conscurr/s of \Meq/ from symmetries of \Meq/.
As a consequence, firstly,
all the currents \eqsref{ME:dualitycurr}{ME:Cvecdualcurr}
are invariant under the \dutr/ \eqref{JPS-duality}
on the potentials
(with the simultaneous induced transformation \eqref{ME-duality-F}
on the electromagnetic field).
More significantly,
the \cvec/ currents and dual currents
can be unified with the \rbvec/ currents
by the introduction of a \KY/ tensor,
corresponding to the analogous form of the nonlocal \syms/
\eqsref{ME:KYsymm}{ME:KYdualsymm}
stated in Theorem~\ref{ME-nonlocalsymm}.
As we recall, the \KY/ tensor is identified with
the dual of the curl of a \Kvec/ of the form \eqref{unifyKV}
given by the sum of a \rb/ \Kvec/ and a \CKvec/.
\begin{cor}
The nonlocal \conslaw/s
\eqref{ME:rbveccurr}, \eqref{ME:Cveccurr}, \eqref{ME:Cvecdualcurr}
admitted by \Meq/
have the unified form
\begin{eqnarray}
\kycurr{\mu}
= &&
\ky{\nu}{\sigma}(
\tAp{}{\sigma} \F{\mu\nu}{} - \tA{}{\sigma} \duF{\mu\nu}{} )
-\duky{\nu}{\sigma}(
\tA{}{\sigma} \F{\mu\nu}{} + \tAp{}{\sigma} \duF{\mu\nu}{} )
\nonumber\\&&
+ \frac{1}{3} \x{\sigma}{} \coder{\tau}\duky{\sigma\tau}{}
( \tA{}{\nu} \F{\mu\nu}{} + \tAp{}{\nu} \duF{\mu\nu}{} )
\nonumber\\
= &&
\frac{1}{2} \ky{}{\nu\sigma} (
\A{\mu}{} \duF{}{\nu\sigma} - \Ap{\mu}{} \F{}{\nu\sigma} )
- 2 \ky{}{\nu[\sigma} (
\A{}{\sigma} \duF{\mu]}{\nu} - \Ap{}{\sigma} \F{\mu]}{\nu} )
+\frac{1}{6} \coder{\tau}\duky{\tau\nu}{}
\vol{}{\mu\nu\alpha\beta} \A{}{\alpha} \Ap{}{\beta}
\nonumber\\&&
-\wavecurr{\mu}(\chi(A),\ky{}{\nu\sigma}\duF{}{\nu\sigma})
+\wavecurr{\mu}(\chi'(A'),\ky{}{\nu\sigma}\F{}{\nu\sigma})
\label{ME:KYcurr}\\&&
\eqtext{ mod curls},
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\dukycurr{\mu}
= &&
-\ky{\nu}{\sigma}(
\tA{}{\sigma} \F{\mu\nu}{} + \tAp{}{\sigma} \duF{\mu\nu}{} )
-\duky{\nu}{\sigma}(
\tAp{}{\sigma} \F{\mu\nu}{} - \tA{}{\sigma} \duF{\mu\nu}{} )
\nonumber\\&&
+ \frac{1}{3} \x{\sigma}{} \coder{\tau}\duky{\sigma\tau}{}
( \tAp{}{\nu} \F{\mu\nu}{} - \tA{}{\nu} \duF{\mu\nu}{} )
\nonumber\\
= &&
\frac{1}{2} \ky{}{\nu\sigma} (
\A{\mu}{} \F{}{\nu\sigma} + \Ap{\mu}{} \duF{}{\nu\sigma} )
- 2 \ky{}{\nu[\sigma} (
\A{}{\sigma} \F{\mu]}{\nu} + \Ap{}{\sigma} \duF{\mu]}{\nu} )
\nonumber\\&&
- \frac{1}{3} \x{\sigma}{} \coder{\tau}\duky{\sigma\tau}{}
( \A{}{\nu} \duF{\mu\nu}{} - \Ap{}{\nu} \F{\mu\nu}{} )
\nonumber\\&&
-\wavecurr{\mu}(\chi(A),\ky{}{\nu\sigma}\F{}{\nu\sigma})
-\wavecurr{\mu}(\chi'(A'),\ky{}{\nu\sigma}\duF{}{\nu\sigma})
\label{ME:KYdualcurr}\\&&
\eqtext{ mod curls},
\nonumber
\end{eqnarray}
depending on a \KY/ tensor \eqref{KYT}.
Under the \dutr/ \eqref{JPS-duality}, \eqref{ME-duality-F}
on the potentials and electromagnetic field,
the currents \eqrefs{ME:KYcurr}{ME:KYdualcurr} are invariant.
In the case when the \KY/ tensor is constant
they are interchanged under replacing
$\ky{}{\mu\nu}$ with its dual $\duky{}{\mu\nu}$.
Thus these currents span a 14-dimensional vector space
which is duality-invariant.
\end{cor}
\section{Concluding remarks}
\label{sec:concl}
In conclusion we mention a few applications of our main results.
The new nonlocal infinitesimal \syms/ we have obtained for
\Meq/ in Minkowski space are of point-type \cite{BluAnc:2002book}
to within a duality transformation,
when expressed in terms of the joint electric and magnetic potentials
for the electromagnetic field.
(Consequently, they can be realized as genuine point transformations
on the complexified jet space of the \potsys/.)
Thus these symmetries can be used to derive corresponding
new group-invariant solutions of \Meq/
and to generalize physically interesting solutions
(\eg/ plane waves and monopoles)
under the action of the finite symmetry group of transformations.
The associated new nonlocal \conscurr/s we have derived from these \syms/
give rise to constants of motion for the electromagnetic field.
Explicit expressions for them
given by gauge-invariant integrals of the field
can be obtained by extending the methods used in \Ref{AncBlu:1997JMP}
to 3+1 dimensions.
In particular, we expect the resulting constants of motion to be
functionally independent of energy, momentum, angular/boost momentum,
and conformal quantities as well as chiral quantities
that arise as constants of motion from the local \conscurr/s of \Meq/.
For future work,
it is planned to extend our results to classify
all nonlocal \syms/ and nonlocal \conscurr/s of \Meq/
produced via a complete classification
of \syms/ and \conscurr/s of local form
in the potentials and their derivatives to any finite order,
admitted by the \jps/ in Lorentz gauge.
A further extension of obvious interest would be to derive
nonlocal \syms/ and nonlocal \conscurr/s of the electromagnetic field
on curved background spacetimes, for instance
the Schwarzschild and Kerr black hole spacetimes.
This analysis is tractable using the spinor techniques of
\Ref{AncPoh:2003,AncPoh:2001}.
Finally, our methods in this paper readily apply to other
linear physical field equations,
notably the linearized gravity wave equation in \Minksp/
and its spin $s$ generalization.
A classification of all local \syms/ and local \conscurr/s
for the linear spin $s>0$ field equations in \Minksp/
has been carried out in \Ref{AncPoh:2004,AncPoh:2003,AncPoh:2002}.
A systematic investigation of spin $s$ \potsys/s
would be of significant interest.
\acknowledgments
D.T. is supported by NSERC through a Canada Graduate Scholarship.
{
\baselineskip=14pt
|
1,314,259,996,717 | arxiv | \section{Introduction}
Many real world data can be represented as a graph of pairwise relationships. Examples include social networks connections, metabolic networks, protein-protein interaction networks, citations network, recommendations and so on.
Matchmaking algorithms and link prediction algorithms are routinely used in many practical situations to discover biochemical interactions, new contacts, hidden connections between criminals, or to match players in online multiplayers video games and sport tournaments.
As testing a link in biological networks, or discovering connections between criminals can be expansive, link prediction algorithms are useful to focus on the most relevant links.
In social networks or online video games, they can help in finding relevant partners.
These applications raise the following mathematical problem that this paper intends to study.
Suppose that there exists a graph whose nodes represent a set of entities or individuals and whose edges represent successful matches between entities or individuals.
The nodes are known to the statistician while the edges are typically hidden at first.
Matchmaking algorithms make queries on pairs of individuals, trying to discover as many edges as possible.
For biological networks like protein-protein interaction networks, the individuals are proteins, an edge is an interaction between the two proteins and a query is an experiment to test whether the interaction exists.
The goal of matchmaking algorithms is to discover as many edges of the graph as possible while minimizing the number of mismatches.
To stress that the focus lies on discovering graph structures, the problem at hand is called hereafter pair-matching rather than matchmaking.
The pair-matching algorithm is forced to explore the graph as it cannot make queries on edges that have already been observed.
To learn interesting features on unobserved edges from previous observations, it is necessary to make assumptions on the structure of the hidden graph.
This paper considers the arguably simplest situation where the graph has been generated according to an assortative conditional stochastic block model (SBM) \cite{Holland83} with two balanced communities, see
Section~\ref{Sec:Setting} for a formal presentation.
In this model, individuals are grouped into two (unobserved) communities and the probability of successful match (edge) between two individuals is larger if they belong to the same community than to different ones.
In this context, the set of pairs is partitioned into good and bad ones, good pairs contain two individuals from the same community and bad pairs two individuals from different communities.
A pair-matching algorithm samples pairs and should sample as many good pairs as possible.
Of course, the partition into good and bad pairs is unknown.
When the graph is fully observed, communities are recovered using clustering algorithms, which have been extensively studied over the past few years, see for example \cite{AbbeReview2017,MooreReview2017,CanLevinaVershyninICM2018} for recent overviews.
A key parameter in the analysis of clustering algorithms, called here \emph{scaling parameter} $s$, is the ratio
\[
s=\frac{(p-q)^2}{p+q},
\]
where $p$ is the probability of connection within a community and $q$ the probability of connection between communities.
This parameter measures the difficulty of clustering, see Section~\ref{Sec:Setting} for details.
The quality of a pair-matching algorithm is evaluated by the expected number of discovered edges after $T$ queries.
Equivalently, the performance can be measured by the expected number of pairs sampled that do not contain edges, which should be as small as possible, see Section~\ref{Sec:Obj} for details.
This last quantity is proportional to the expected number of bad pairs sampled, which is called \emph{sampling regret} in this paper.
As in practical situations, individuals may not be solicited too many times, we consider algorithms constrained to sample each individual less
than a certain amount of times $B_T$ before $T$ queries have been made.
Our main contribution of the paper is that the sampling regret of any strategy that cannot sample pairs more than once, that is invariant to nodes labelling and which satisfies the above constraint (see Assumptions {\bf (NR)}, {\bf (IL)} and {\bf (SpS)} in Section \ref{Sec:SeqMatch} for details) is larger than
$$T\wedge {\sqrt{T}\vee (T/B_{T})\over s},$$ up to multiplicative constants.
Moreover, a polynomial-time algorithm with sampling regret bounded from above by a constant times ${T\wedge {\sqrt{T}\vee (T/B_{T})\over s}}$ is described and analysed, see Theorem~\ref{thm:contraint}.
These results show that no strategy can achieve sub-linear sampling regret before $T=O(1/s^2)$ pairs have been sampled and that, on the other hand, there exist strategies with sub-linear regret scaling as the optimal rate $(\sqrt{T}\vee (T/B_{T}))/s$ once $T\gtrsim 1/s^2$.
It transpires from this result that the constraint has no substantial effect as long as $B_T\gtrsim \sqrt{T}$.
On the other hand, strong constraints such as $B_T=O(1/s)$ induce unavoidable linear regret.
The following problem, related to matchmaking, has recently attracted attention, in particular in Bradley-Terry models \cite{bradley:terry:1952,zemerlo:1929}.
The task is to infer, from the observation of pairs, a vector of parameters characterizing the strength of players.
Most results considered the case where all the graph is observed, see \cite{hunter:2004,caron:doucet:2012}.
Recent contributions dealing in particular with ranking issues also consider the case of partially observed graphs, see \cite{MR3827087,MR3504618,Jang2016TopKRF} for example and the references therein.
In all cases, the list of observed pairs is given as input to the algorithm evaluating the strength of all players.
The choice of a relevant list of successive observed pairs, independent of the observation of the edges is sometimes called a scheduling problem, see \cite{lecorff:lerasle:vernet:2018}.
Scheduling problems are different from matchmaking problems considered here where the algorithm should choose the observed pairs and can use preliminary observations to make its choice.
For online video games, classical algorithms used to evaluate strength of players are ELO or TRUESKILLS \cite{Trueskills2007,Trueskills2018}.
Matchmaking algorithms such as EOMM \cite{Chen:2017:EEO:3038912.3052559} (used with TRUESKILLS see \cite{Trueskills2018}) are then used to pair players, taking as inputs these estimated strengths.
In this approach, the number of mismatches during the learning phase is not controlled.
It is an important conceptual difference with this paper where the matchmaking problem is considered together with the problem of discovering the strength (communities here).
Here, pair-matching algorithms have to simultaneously explore the graph to evaluate the strength and sample as many ``good" pairs as possible to optimize the number of successful matches.
Closer to our setting is the active ranking literature \cite{NIPS2011_4427, NIPS2015_5903, ActiveRanking2016}, where the goal is to discover adaptively the rank or strength of players with a minimal amount of queries. Contrary to our problem, only the exploration matters in adaptive ranking and no notion of regret is investigated.
Pair-matching algorithms take sequential decisions to explore new pairs exploiting previous observations.
This kind of exploration and exploitation dilemma is typical in multi-armed bandit problems \cite{Thompson33,robbins1952,LaiRobbins85,BURNETAS96}.
In stochastic multi-armed bandit problems, a set of actions, called \emph{arms} is proposed to a player who chooses one of these actions at each time step and receives a payoff.
The payoffs are independent random variables with unknown distribution.
For any arm, payoffs are identically distributed.
The player wants to maximize its total payoff after $T$ queries.
The pair-matching problem introduced above can be seen as a non-standard instance of stochastic multi-armed bandit problems.
In this interpretation, each pair of nodes is an arm and the associated payoff is $1$ if an edge links these nodes and $0$ otherwise.
The payoffs hence follow a Bernoulli distribution with parameter $p$ for good pairs and parameter $q$ for bad pairs.
The unusual feature is that each arm can only be played once, so the pair-matcher must choose a new arm at each time step.
For this reason, optimal strategies differ in spirit from classical strategies in bandit problems, see Section~\ref{sec:bandits} for more details.
On the other hand, useful inequalities are borrowed from the classical bandit literature \cite{KCG16,GMS18} to prove lower bounds.
Forgetting the constraint that a node cannot be sampled more than $B_{T}$ times, the pair-matching bandit problem could be seen as an extreme version of mortal or rotting bandit problems \cite{MortalNIPS2008, RottingNIPS2017,RottingPMLR2019}, where every arm would systematically die or have zero pay-off after the first sampling.
Without additional assumptions, the regret would be inexorably linear in the querying budget $T$.
Here, an important difference with classical mortal or rotting multi-armed bandits is that payoffs are structured by the underlying stochastic block model (SBM).
Stochastic block models have attracted a lot of attention in the recent years, with a focus on the determination of optimal strategies for clustering and for parameter estimation, see \cite{AbbeReview2017,MooreReview2017}.
In this prolific literature, the graph is fully observed and the question is to identify precisely the weakest separation between the probabilities of connection necessary to perfectly or partially recover the communities, or to estimate the parameters of the SBM.
Closer to our setting, the paper \cite{YP14a} investigates the question of recovering communities from a minimal number of observed pairs, sampled sequentially.
In this problem, the question is to assign a community to all nodes after a minimal number $T$ of time steps and try to minimize the number of misclassified nodes.
This is quite different from the minimization of the sampling regret considered here, where we seek to find on a budget as many good pairs as possible and not to classify all nodes.
As discussed in Section~\ref{sec:discuss:step2}, applying the algorithm of \cite{YP14a} would lead to a suboptimal regret in our problem.
The formalization of the pair-matching problem considered in this paper may be restrictive in some applications.
Section~\ref{Sec:KclassesSBM} presents some conjectures that seem reasonable for $K$ classes SBMs.
Other graph structures would also be interesting, such as Bradley-Terry models \cite{bradley:terry:1952,zemerlo:1929} which have been used for sport tournaments \cite{Sir_Red:2009}, chess ranking \cite{joe:1990} and predictions of animal behaviors \cite{whiting:stuartfox:oconnor:firth:bennett:bloomberg:2006}.
Various constraints dealing with first discoveries for example may be interesting depending on the applications: the first match of a node is the most important in some situations\footnote{Richard III in Shakespeare's play offers his ``kingdom for a horse!", he would certainly propose less for a second one!}, and, for the search of a life partner, discovering a match with a node already connected in the observed graph is (for most nodes at least) less interesting than a match with an isolated node.
These constraints naturally induce different versions of the pair-matching problem and raise mathematical questions of interest.
Multiplayer video games suggest the extension to hypergraphs of the pair-matching problem.
Indeed, the value of a player could be evaluated as part of a team and with respect to a possible team of opponents rather than simply as part of a pair.
Finally, in many practical situations, additional information on individuals is available and could be used to improve pair-matching algorithms.
It is clear from our first results that this information is necessary to avoid linear regret in applications such as life partner research.
These extensions are postponed to follow-up works.
This paper should be seen as a first step to formalize and study the important pair-matching problem.
It focuses on a toy example but opens several interesting questions that arise when dealing with natural constraints in practical applications of interest.
The remainder of the paper is decomposed as follows.
Section~\ref{Sec:Set} introduces the formal setting and objectives.
As a warm-up, Section~\ref{sec:Unconstrained} focuses on the case where the algorithms are not constrained to sample nodes more than a certain amount of times.
Section~\ref{sec:contraint} presents the main results where the algorithm are constrained.
Section~\ref{Sec:KclassesSBM} gives conjectures for $K$-classes SBMs.
The proofs of the main results are postponed to the appendix.
Notation:
we write $x_n\lesssim y_n$ and $x_n=O(y_n)$, if there exist numerical constants such that $x_n\leqslant Cy_n$ for all $n\geq n_0$; and we write
$x_{n} \asymp y_{n}$ and $x_n=\Theta(y_n)$, if $x_n=O(y_n)$ and $y_n=O(x_n)$ that is, if there exist numerical constants $c,c'>0$ and $n_0$ such that $c x_{n}\leq y_{n} \leq c' x_{n}$ for all $n\geq n_0$. We denote by $\lceil x\rceil$ (respectively $\lfloor x \rfloor$) the upper (resp. lower) integer part of $x$; by $|A|$ the cardinal of a set $A$; and by $A\Delta B$ the symmetric difference between two sets $A$ and $B$.
\section{Setting and Problem Formalization}\label{Sec:Set}
\subsection{Two-Classes SBM}\label{Sec:Setting}
The $n$ individuals are represented by the set $V=\ac{1,\ldots,n}$. Successful matches are represented by a set of edges $E$ between nodes in $V$: there is a successful match between $a$ and $b$ in $V$ if and only if the pair $\ac{a,b}$ belongs to $E$.
Hereafter, a set of two distinct elements in $V$ is called a \emph{pair} and an element of $E$ is called an \emph{edge}.
The graph $(V,E)$ is conveniently represented by its adjacency matrix $A\in\mathbb{R}^{n\times n}$, with entries $A_{ab}=1$ if $\ac{a,b}\in E$ and $A_{ab}=0$ otherwise.
In the following, any graph $(V,E)$ is identified with its adjacency matrix $(A_{ab})_{a,b \in V}$.
For any pair $e = \{a,b\}$, the notations $A_e$ and $A_{ab}$ are used indifferently.
Since the graph is undirected, the adjacency matrix $A$ is symmetric, and since there is no self-matching (no self-loop in the graph), the diagonal of $A$ is equal to zero.
Individuals are grouped into two (unknown) communities according to their affinity.
To model this situation, the graph $(V,E)$ is random and distributed as a two-classes conditional stochastic block model.
Let $0<q,p<1$, and let $n_1$ denote an integer $n_{1}\geq n-n_{1}\geq 1$.
The collection cSBM$(n_{1},n-n_{1},p,q)$ of two-classes conditional stochastic block model distributions on graphs is defined as follows.
Let $G=\ac{G_{1},G_{2}}$ be a partition of $\ac{1,\ldots,n}$ into two groups, with $|G_{1}|=n_{1}$ and $|G_{2}|=n-n_{1}$. The partition $G$ represents the communities of individuals. Let $\mu_{G}$ denotes the distribution on graphs with nodes $\ac{1,\ldots,n}$, such that the adjacency matrix is symmetric, null on the diagonal and with lower diagonal entries $(A_{ab})_{a < b}$ sampled as independent Bernoulli random variables with $\mu_{G}(A_{ab}=1)=p$ when $a$ and $b$ belong to the same group $G_{i}$, and $\mu_{G}(A_{ab}=1)=q$ when $a$ and $b$ belong to different groups. In other words, two individuals are successfully matched with probability $p$ if they belong to the same community, and with probability $q$ otherwise.
The class cSBM$(n_{1},n-n_{1},p,q)$ is defined as the set of all distributions $\mu_G$ defined above, where $G=\{G_1,G_2\}$ describes the set of partitions of $\{1,\ldots,n\}$ satisfying $|G_1|=n_1$ and $|G_2|=n-n_1$:
\begin{multline*}
\textrm{cSBM}(n_{1},n-n_{1},p,q)\\
=\ac{\mu_{G}: G=\ac{G_{1},G_{2}} \ \textrm{partition satisfying}\ |G_{1}|=n_{1},\ |G_{2}|=n-n_{1}}.
\end{multline*}
In the following, the communities are balanced and successful matches happen with higher probability if individuals belong to the same community.
Formally, $n$ is even and the graph $(V,E)$ has been generated according to a distribution $\mu$ in cSBM$(n/2,n/2,p,q)$, for some unknown parameters $p$ and $q$ such that $0<q<p\leq 1/2$.
As $q<p$, the distribution of $(V,E)$ is called an \emph{assortative} cSBM$(n/2,n/2,p,q)$.
All along the paper, the ratio $p/q$ is also assumed bounded from above.
To sum up, $p$ and $q$ are smaller than $1/2$ and satisfy
\begin{equation}\label{eq:pq}
1<p/q \leq \rho^*.
\end{equation}
Given $p$ and $q$, the following scaling parameter plays a central role
\begin{equation}\label{eq:def:ScalingParam}
s= \frac{(p-q)^2}{p+q}.
\end{equation}
This parameter appears in various results in the literature on SBM.
The following property, proved for example in \cite{YP14b,CRV15, AS15,LZ16,GMZZ17,FC17,GV18}, will be used repeatedly in the paper.
When the graph $(V,E)\sim\text{cSBM}(n_{1},n-n_{1},p,q)$, there exist polynomial-time clustering algorithms that return a partition of $\{1,\ldots,n\}$ such that, with large probability, the proportion of misclassified nodes decreases exponentially:
$$\textrm{Proportion of misclassified nodes} \leq \exp(-c n s), \quad \textrm{when}\ ns\geq c',$$
where $c,c'>0$ are numerical constants.
The rate $ns$ of exponential decay in this result is optimal (up to a constant) when (\ref{eq:pq}) is met.
Hence, the scaling parameter $s$ drives the difficulty of clustering.
To stress the importance of $s$, the following parametrization will be used henceforth
$$p= s \pa{\alpha+\sqrt{\alpha}}/2,\quad q=s \pa{\alpha-\sqrt{\alpha}}/2,$$
with $\alpha=(p+q)^2/(p-q)^2$.
In this parametrization, Assumption (\ref{eq:pq}) is met if and only if $\alpha$ is bounded from below by $(\rho^*+1)^2/ (\rho^*-1)^2$.
Another useful property is that there exist numerical constants $c_{1},c_{2}>0$ such that non-trivial community recovery is possible as soon as $s\geq c_{1}/n$, see \cite{Decelle2011,Mas14,CRV15,AS15,BLM18,FC17,GV18} and perfect community recovery is possible as soon as $s\geq c_{2}\log(n)/n$, see \cite{AS15, CX16, MNS16}.
The reader familiar with SBM literature may be more comfortable with the parametrization $p=a_{n}/n$ and $q=b_{n}/n$ for a SBM distribution with two communities. For a comfortable translation of the results, the following relations between $s$, $\alpha$ and $a_{n}$, $b_{n}$ are provided:
\begin{gather*}
s=\frac{(a_{n}-b_{n})^2}{n(a_{n}+b_{n})}, \qquad \alpha=\frac{(a_{n}+b_{n})^2}{(a_{n}-b_{n})^2},\\
\frac{a_{n}}{b_{n}}=\frac{\alpha+\sqrt{\alpha}}{\alpha-\sqrt{\alpha}}\quad \text{and} \quad a_{n}+b_{n}=n\alpha s.
\end{gather*}
\subsection{Sequential Matching strategies}\label{Sec:SeqMatch}
Denote by $\mathcal{E}$ the set of all pairs of nodes, that is the set of all subsets of $V$ containing two distinct elements.
Heuristically, a sequential matching strategy samples at each time $t$ a new pair $\widehat e_{t}\in \mathcal{E}$, using only past observations $(\widehat e_{1},\ldots,\widehat e_{t-1},A_{\widehat e_{1}},\ldots,A_{\widehat e_{t-1}})$ and an internal randomness of the algorithm.
Formally, let $U_{0}, U_{1},\ldots$ be i.i.d uniform random variables in $[0,1]$, independent of $A$ and representing the sequence of internal randomness for the algorithm.
A sequential matching strategy $\psi$ on $\mathcal{E}$ (shortened \emph{strategy} in the following) is a sequence $\psi = (\psi_{t})_{0\leq t\leq \binom{n}{2}-1}$ of measurable functions $\psi_{t}:\mathcal{E}^{t}\times \ac{0,1}^{t}\times [0,1]^{t+1}\to \mathcal{E}$.
Any sequential matching strategy $\psi$ defines a matching algorithm as follows.
The first pair is sampled as $\widehat e_{1}=\psi_{0}(U_{0})$. Then, at each time $t \geq 0$, the pair $\widehat e_{t+1}$ is defined by
\begin{equation*}
\widehat e_{t+1}=\psi_{t}(\widehat\mathcal{E}_{t}, (A_{e})_{e\in \widehat\mathcal{E}_{t}},U_{0},\ldots,U_t)\quad \textrm{with}\quad \widehat\mathcal{E}_{t}=\ac{\widehat e_{1},\ldots,\widehat e_{t}}.
\end{equation*}
The strategy takes as input the observed graph $(A_{e})_{e\in \widehat\mathcal{E}_{t}}$ and possibly an internal independent randomness $U_t$ to output the new observed pair $\widehat e_{t+1}$.
In the following, strategies are assumed to satisfy the following constraints: a pair can only be sampled once and strategies are invariant to labelling of the nodes.
These constraints can be formalized as follows.
\medskip
\noindent{\bf Non-redundancy (NR).} {\it The strategy $\psi$ samples any pair at most once, that is, for any $0\leq t\leq \binom{n}{2}-1$ and $e_{1},\ldots,e_{t} \in \mathcal{E}$, the map $\psi_{t}$ fulfills $\psi_{t}(\{e_{1},\ldots,e_{t}\},\ldots)\notin \ac{e_{1},\ldots,e_{t}}$. }
\medskip
\noindent
Invariance to labelling requires some notation.
For any pair $e\in\mathcal{E}$ and any strategy $\psi$, let
\begin{equation}\label{def:Ne}
N_{e}(\psi,t):={\bf 1}_{e\in \widehat \mathcal{E}_{t}}
\end{equation}
indicate if the pair $e$ has been sampled or not before time $t$ by the strategy $\psi$.
For any non-redundant strategy $\psi$ (i.e. satisfying {\bf (NR)}), pairs are sampled at most once and the observation of $\{N_{e}(\psi,t): e\in\mathcal{E}\}$ is equivalent to that of
$\widehat\mathcal{E}_{t}$.
Let $\mu$ be a distribution in cSBM$(n/2,n/2,p,q)$ and $\sigma$ be a permutation of $V$.
For any pair $\ac{a,b}\in \mathcal{E}$, let $\sigma(\ac{a,b}):=\ac{\sigma(a),\sigma(b)}$.
Let $\mu^\sigma$ denote the distribution of $(A_{\sigma(e)})_{e\in\mathcal{E}}$, where $(A_{e})_{e \in \mathcal{E}}$ is distributed according to $\mu$. \label{def:mu_sigma}
\medskip
\noindent{\bf Invariance to labelling (IL).} {\it The distribution of the outcomes of the strategy $\psi$ is invariant by permutations of the nodes labels: For any $\mu\in \textrm{cSBM}(n/2,n/2,p,q)$ and any permutation $\sigma$ on $V$,
the distribution of $\big(N_{e}(\psi,t): e\in\mathcal{E}, 1\leq t \leq \binom{n}{2}\big)$ under $\mu^{\sigma}$ is the same as the distribution of $\big(N_{\sigma(e)}(\psi,t): e\in\mathcal{E}, 1\leq t \leq \binom{n}{2}\big)$ under $\mu$.}
\medskip
Besides {\bf (NR)} and {\bf (IL)}, we consider strategies that do not sample a node more than $B$ times before time $T$.
This constraint appears naturally in practical situations.
For example, if the algorithm matches biological entities or individuals, one may not want to query too many times each individual for logistic or acceptability reasons.
To stress that the constraint $B$ typically grows with the time horizon $T$, it is denoted $B_T$.
Formally, for any $a\in V$, let
\begin{equation}
\label{def:Na}
N_{a}(\psi,t)=\sum_{b\in V: b\neq a} N_{\ac{a,b}}(\psi,t)
\end{equation}
denote the number of times the node $a$ has been sampled in a pair $\ac{a,b}$ after $t$ queries.
\medskip
\noindent{\bf Sparse sampling (SpS).} {\it Let $T$ and $B_T$ denote two integers. The strategy $\psi$ is called $B_T$-sparse up to time $T$ if it satisfies}
\begin{equation}
\label{eq:sparse}
\forall a \in V, \quad N_{a}(\psi,T) \leq B_T.
\end{equation}
Since $N_{a}(\psi,T)\leq (n-1)\wedge T$ for all nodes $a$, choosing $B_T\geq (n-1)\wedge T$ corresponds to the unconstrained case.
\subsection{Objectives of the Pair-matcher}\label{Sec:Obj}
Let $\mu \in \text{cSBM}(n/2,n/2,p,q)$ be the distribution of an assortative conditional stochastic block model with associated partition $G=\ac{G_{1},G_{2}}$.
Define
$\mathcal{E}^{good}(\mu)$ (or simply $\mathcal{E}^{good}$) as the set of pairs $\{a,b\}$ with $a$ and $b$ from the same community, and $\mathcal{E}^{bad}(\mu)$ (or simply $\mathcal{E}^{bad}$) as the set of pairs $\{a,b\}$ with $a$ and $b$ from two different communities.
The objective of the pair-matcher is to discover as many edges (i.e. successful matches between individuals) as possible with $T$ queries.
Its strategy $\psi$ should maximize the number of discovered edges, in expectation with respect to the randomness of the SBM and the strategy.
Optimal strategies should therefore sample as many pairs in $\mathcal{E}^{\text{good}}$ as possible.
Formally, consider a time horizon $T$ smaller than $|\mathcal{E}^{\text{good}}|=2\binom{n/2}{2}\sim n^2/4$.
Any strategy $\psi$ has an expected number of discoveries equal to
$$\operatorname{\mathbb{E}}_\mu \cro{\sum_{t=1}^T A_{\widehat e_{t}}} = pT -(p-q) \operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)},$$
where $N^{bad}(\psi,T)=\sum_{e\in \mathcal{E}^{bad}}N_{e}(\psi,T)$ is the number of pairs in $\mathcal{E}^{bad}$ sampled up to time $T$.
Since $p>q$, the maximal expected value of discoveries is achieved by any oracle strategy $\psi^*$ sampling only edges in $\mathcal{E}^{\text{good}}$.
In that case, $N^{bad}(\psi^*,T)=0$ and the maximal expected number of discoveries is equal to $pT$.
The regret of the strategy $\psi$ is defined as the difference between $pT$ and its expected number of discoveries:
$$R_{T}(\psi) = pT-\operatorname{\mathbb{E}}_\mu \cro{\sum_{t=1}^T A_{\widehat e_{t}}} = (p-q) \operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)}.$$
As long as $T\leqslant |\mathcal{E}_{\text{good}}|$, the regret is proportional to the expected number of sampled between-group pairs $\operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)}$.
Therefore, the main results analyse this last quantity rather than the regret.
The expected number of bad sampled pairs $\operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)}$ is called hereafter \emph{sampling-regret}.
\medskip
\noindent{\bf Remark.} \label{page:inv-sampling} Without assumption on $\psi$, the distribution of $N^{bad}(\psi,T)$ may depend on the distribution $\mu$ of the cSBM.
On the other hand, when the strategy $\psi$ fulfils {\bf (IL)}, the distribution of $N^{bad}(\psi,T)$ does not depend on the distribution $\mu$ in cSBM$(n/2,n/2,p,q)$.
Indeed, let $\mu,\mu'$ be two distributions in cSBM$(n/2,n/2,p,q)$.
By definition, there exists a permutation $\sigma$ on $\ac{1,\ldots,n}$ such that $\mu'=\mu^{\sigma}$, where $\mu^{\sigma}$ has been defined page~\pageref{def:mu_sigma}. Since $\mathcal{E}^{bad}(\mu^{\sigma})=\sigma^{-1}(\mathcal{E}^{bad}(\mu))$, it follows from {\bf (IL)} that the distribution under $\mu^{\sigma}$ of $\sum_{e \in \mathcal{E}^{bad}(\mu^{\sigma})} N_{e}(\psi,T)$ is the same as the distribution under $\mu$ of $\sum_{e \in \mathcal{E}^{bad}(\mu)} N_{e}(\psi,T)$.
\subsection{A Special Bandit Problem}
\label{sec:bandits}
The pair-matching problem described above can be interpreted as a non-standard multi-armed bandit problem.
Actually, each pair $\{a,b\}$ can be seen as an arm and the discovery of a successful match as a payoff.
The payoff of the arm $\{a,b\}\in\mathcal{E}$ follows a Bernoulli distribution with parameter $p$ if $\{a,b\}\in\mathcal{E}^{good}$ and with parameter $q$ if $\{a,b\}\in\mathcal{E}^{bad}$.
This bandit problem is non-standard, as arms cannot be sampled more than once and payoffs have a structure inherited from the SBM distribution.
To sum up, the main differences with the standard multi-armed bandit problem are:
\begin{enumerate}
\item
the arms are sampled at most once,
\item at most $B_T$ arms involving a given node can be sampled up to time $T$,
\item the distribution of the payoffs have a hidden structure inherited from the SBM setup.
\end{enumerate}
Compared with the standard multi-armed bandit problem, points $1$ and $2$ make this problem harder, while point $3$ is a strong structural property that gives hope to find regimes with sub-linear regret.
These special features make this problem quite different from classical bandit problems.
In classical bandit problems, optimal strategies have to identify the best arm (or some of the best arms) and each arm is played many times to reach this goal.
Here, half the arms are ``optimal" but one cannot play an arm more than once.
Therefore, instead of identifying one of these, optimal strategies should avoid bad arms,
possibly disregarding a non-negligible proportion of good arms in the process.
The constraint {\bf (SpS)} also induces a specific exploration / exploitation trade-off.
When the community of a node is identified, we wish to pair it with a maximum of nodes of the same community in order to maximise the rewards (exploitation).
Yet, we also need to pair this node to some new nodes in order identify the community of new nodes (exploration).
Since a node can only be paired to $B_{T}$ other nodes, we need to trade-off between these two strategies.
The bandit literature is mainly used to establish our lower bounds which involve inequalities from \cite{GMS18,KCG16}.
\section{Warm-up: Unconstrained Optimal Pair-Matching}\label{sec:Unconstrained}
\subsection{Optimal Rates for Unconstrained Pair-Matching}
As a warm-up, we focus first on the simplest case, where $B_{T}=+\infty$, which amounts to remove the constraint {\bf (SpS)}.
Let $\Psi_{\infty}$ denote the set of strategies $\psi$ fulfilling {\bf (NR)} and {\bf (IL)}. The first main result describes the best sampling-regret that can be achieved by a strategy in $\Psi_{\infty}$, as a function of $s$ and $T$.
\begin{thm}
\label{thm:non-contraint}
Let $T$ and $n$ be positive integers with $T \leq |\mathcal{E}^{\text{good}}|=2\binom{n/2}{2}$. Let $p,q \in [0,1/2]$ be two parameters fulfilling (\ref{eq:pq}) and such tha
$$s \leq {1\over 32(1+\rho^*)},$$
where the scaling parameter $s$ is defined in \eqref{eq:def:ScalingParam}.
Then, for any $\mu \in \text{cSBM}(n/2,n/2,p,q)$,
\begin{equation}
\label{eq:non-contraint}
\inf_{\psi \in \Psi_{\infty}}
\operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)} \geq \frac{1}{32} \cro{\frac{\sqrt{T}}{32 (1+\rho^*) s}\wedge T}.
\end{equation}
Moreover, there exist two numerical constants $c_{1},c_{2}>0$, and a strategy $\psi \in \Psi_\infty$ corresponding to a polynomial-time algorithm described in Section~\ref{sec:algo},
taking $s$ as input,
such that, for any $p,q$ satisfying \eqref{eq:pq}, any $\mu \in \text{cSBM}(n/2,n/2,p,q)$ and any time horizon $1\leq T\leq c_{2}n^2$
$$\operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)} \leq c_{1}\cro{\frac{\sqrt{T}}{s}\wedge T}.$$
\end{thm}
The proof of Theorem \ref{thm:non-contraint} is provided in the appendix. The lower bound is proved in Section \ref{sec:lower} and the upper bound in Section \ref{sec:upper}.
The upper bound derives from a stronger result showing that similar bounds hold with high probability, see Theorem \ref{thm:upper-appendix} for a precise statement.
Theorem~\ref{thm:non-contraint} provides only the upper bound in expectation for clarity.
Theorem~\ref{thm:non-contraint} states that, when (\ref{eq:pq}) holds, for any $\mu \in \text{cSBM}(n/2,n/2,p,q)$ and any time horizon $1\leq T \leq c_{2} n^2$, the optimal sampling-regret
$$\inf_{\psi \in \Psi_{\infty}} \operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)} \asymp {\frac{\sqrt{T}}{s}\wedge T\,},$$
grows linearly with $T$ as long as $T \lesssim 1/s^2$ and becomes sub-linear, of order $\sqrt{T}/s$, when $T \gtrsim 1/s^2$.
This result can be understood intuitively.
As long as communities cannot be recovered better than random, there is no hope of getting better sampling-regret than with purely random sampling of the pairs.
In this regime, the sampling-regret grows linearly with $T$.
To identify when this occurs, consider the situation where pairs are sampled at random among $N$ nodes and $T=\beta N^2/2$ (with $\beta\leqslant 1$).
Then the \emph{observed} edges at time $T$ are approximately distributed as in a SBM with $N$ nodes,
within-group connection probability $p_{\beta}=\beta p$, and between-group connection probability $q_{\beta}=\beta q$.
It follows from \cite{Decelle2011,Mas14,MNS15,BLM18} that weak recovery of the communities is possible if and only if $N(p_{\beta}-q_{\beta})^2\geq 2 (p_{\beta}+q_{\beta})$, which is equivalent to
$\sqrt{\beta T} s \geq \sqrt{2}$ or $T\geq 2/(\beta s^2)$.
Since $\beta\leq 1$ by definition, no information about the communities can be recovered when $T\leq 2/s^2$.
Hence, the sampling-regret is expected to grow linearly with $T$ for $T=O(1/s^2)$. This intuition is confirmed by Eq.~\eqref{eq:non-contraint}.
When $T \gg 1/s^2$, the situation is different.
Classical results, such as \cite{YP14b,CRV15,AS15,LZ16,MNS16,FC17,GV18} among others, ensure that the communities of $N$ nodes can be recovered almost perfectly if $N\gg 1/s$ and all edges between these nodes are observed.
Therefore, when $1/s \ll N =\big(\sqrt{T}/s\big)^{1/2} \ll \sqrt{T}$, one can sample all the edges between $N$ nodes and recover almost perfectly their community with a sampling regret smaller than $N^2=\sqrt{T}/s$.
A recipe in order to get a sublinear regret is the following. If we are able to find the community of $\Theta(\sqrt{T})$ nodes, then we can spend a budget of $T$ queries without further regret by sampling pairs among these $\Theta(\sqrt{T})$ nodes. To do so, we need
to identify the community of $\Theta(\sqrt{T})$ nodes from the $N$ clustered nodes, with a regret smaller than $\sqrt{T}/s$.
Given the $N$ clustered nodes, it is possible to identify the community of a new node with a sampling regret of order $O(1/s)$.
Proceeding recursively, $\Theta(\sqrt{T})$ new nodes can be identified with a sampling-regret of order $O(\sqrt{T}/s)$.
The remaining budget of $T$ queries can then be spent by sampling pairs among these $\Theta(\sqrt{T})$ nodes without further regret if there were no errors in the community assignment.
This informal reasoning suggests that the optimal sampling-regret grows like $\sqrt{T}/s$ when $T\gg 1/s^2$.
Again, this intuition is confirmed by Eq.~(\ref{eq:non-contraint}).
An algorithm achieving the optimal upper bound in Theorem~\ref{thm:non-contraint} and taking as input $s$ and the time horizon $T$ is provided in Section \ref{sec:algo}.
It essentially proceeds as in the informal strategy outlined above, even if some steps have to be refined. In particular, the identification of the community of $\Theta(\sqrt{T})$ nodes has to be conducted with care in order to balance the regret and the community assignment errors.
The dependency of the algorithm of Section \ref{sec:algo} on the time horizon $T$, can be easily dropped out with a classical doubling trick, see Section \ref{sec:upper} in the appendix.
To sum up the discussion: in the early stage where $T=O(1/s^2)$, one cannot do better than random guessing, up to multiplicative constant factors.
In the second stage where $T\geq 1/s^2$, the rate $\sqrt{T}/s$ can be interpreted as follows.
A total of $O(\sqrt{T})$ nodes are involved at time $T$ and, for each of them, $O(1/s)$ observations are necessary to obtain an educated guess of their community.
Finally, Theorem~\ref{thm:non-contraint} can be equivalently stated in terms of the regret $R_{T}(\psi)$:
for any time horizon $1\leq T \leq c_{2} n^2$, the minimal regret satisfies
$$\inf_{\psi \in \Psi_{\infty}} R_{T}(\psi) \asymp \sqrt{\alpha}\pa{\sqrt{T}\wedge (sT)\,},$$
when the assumptions of Theorem~\ref{thm:non-contraint} are met.
\subsection{Algorithm with Specified Horizon \texorpdfstring{$T$}{T}}
\label{sec:algo}
This section presents an algorithm achieving the upper bound in Theorem~\ref{thm:non-contraint}.
This algorithm takes as input the scaling parameter $s$ \emph{and} the time horizon $T$.
This dependency on the time-horizon can be avoided with the classical doubling trick, see Section \ref{sec:upper} in the appendix.
We discuss in Section \ref{sec:SNR} a heuristic for the preliminary estimation of $s$ involving less than $O(1/s^2)$ edges.
When the horizon $T$ is $O(1/s^2)$, any strategy achieves a regret of order $O(T)$.
Hence, without loss of generality, it is assumed in the remaining of the section that $T\geq c_{th}/s^2$ for some numerical constant $c_{th}$.
Moreover, as Theorem~\ref{thm:non-contraint} holds for $T\leq c_2n^2$, it is also assumed that this condition is fulfilled
for a sufficiently small constant $c_2$.
The algorithm proceeds in three steps.
In the first step, a kernel $\mathcal{N}$ of $|\mathcal{N}|=\Theta(\sqrt{T}/\log(s\sqrt{T}))$ vertices is chosen uniformly at random and
each pair within this kernel is sampled with probability $\Theta\big( (\log(s\sqrt{T}))^2/(s\sqrt{T})\big)$.
Hence, an average of $\Theta (\sqrt{T}/s)$ pairs are sampled within this kernel.
A community recovery algorithm is run on this observed graph that outputs two estimated communities with a fraction of misclassified nodes vanishing as $O(\log(s\sqrt{T})/(s\sqrt{T}))$ with high probability.
The second step identifies with high probability $\Theta(\sqrt{T})$ vertices from the same community, say community 1.
To do so, it picks uniformly at random a set $\mathcal{A}_{0}$ of $8 \sqrt{2T}$ vertices outside of the kernel $\mathcal{N}$ (this is possible thanks to the condition $T\leq c_2n^2$) and samples pairs between this set and the estimated community 1 of the kernel.
This set of edges is used to estimate the connectivity between these vertices and community 1.
Vertices with low connectivity, that seem to belong to community 2, are removed online to keep the sampling regret under control. The goal of this screening is
not to classify perfectly the $8 \sqrt{2T}$ picked vertices, but instead to sift out vertices of community 2 with a low sampling regret. In particular, a price to pay to achieve this goal is to
possibly remove a non-negligible proportion of vertices of community 1 from the $8 \sqrt{2T}$ picked vertices.
This second step of the algorithm is crucial for getting the optimal regret rate $O(\sqrt{T}/s)$.
A simplified version of this second step can be connected to a particular $k$ out of $m$ best arms identification problem. This connection is discussed in Section \ref{sec:discuss:step2} below.
The third step samples all pairs $\{a,b\}$ such that $a$ and $b$ belong to the $\Theta(\sqrt{T})$ vertices isolated in the second step of the algorithm, until the remaining budget of $T$ queries is expended.
The pair-matching algorithm calls an external clustering algorithm (generically denoted by {\tt GOODCLUST} in the following).
{\tt GOODCLUST} takes as input a graph $(V,E)$ and outputs a partition $\widehat{G}=(\widehat{G}_1,\widehat{G}_2)$.
We require that {\tt GOODCLUST} fulfills the following recovery property:
There exist numerical constants $c^\text{GC}, c^\text{GC}_{1}>0$ such that, for all $N = N_1+N_2$ and all $\tilde{p}, \tilde{q} \in [0,1]$, if $(V,E)\sim \text{cSBM}(N_1,N_2,\tilde p,\tilde q)$, the proportion of misclassified nodes
\[
\varepsilon_N=\frac{|\widehat{G_1}\Delta G_{1}|+|\widehat{G_2}\Delta G_{2}|}{2N},
\]
with $\Delta$ the symmetric difference, satisfies
\begin{equation}\label{eq:GoodClust}
\varepsilon_N \leq \exp \left( - c^\text{GC}_{1} N \frac{(\widetilde{p}-\widetilde{q})^2}{\widetilde{p}}\right),
\end{equation}
with probability at least $1 - c^\text{GC}/N^3$.
Algorithms achieving this proportion of misclassification can be found e.g. in \cite{GV18}, see also \cite{YP14b,CRV15, AS15,LZ16,GMZZ17,FC17} for similar results.
\noindent { \def1.3{1.3}
\begin{tabular}{|l|}
\hline
\textbf{Unconstrained Algorithm}\label{alg:unconstrained} \\
\hline
\begin{minipage}{0.99\textwidth} \centering
\begin{minipage}{0.97\textwidth}
\medskip
{\bf Inputs:} $s$ scaling parameter, $T$ time horizon, $V$ {set of} nodes.\medskip
{\bf Internal constants:} $c_{\mathcal{O}_{0}}=2\vee (1/c^\text{GC}_{1})$, $C_{k}=2200$ and $C_{I}=4$.
\medskip
\noindent \textbf{Step 1: finding communities in a kernel}
\begin{itemize}[topsep=0pt]
\item[1.] Sample uniformly at random a set $\mathcal{N}\subset V$ of $N = \left\lceil \sqrt{T}/\log(s \sqrt{T}) \right\rceil$ nodes.
\item[2.] Sample each pair of $\mathcal{N}$ with probability $\displaystyle c_{\mathcal{O}_{0}} { \sqrt{T}\over s\binom{N}{2}}$, call $\mathcal{O}_0\subset \mathcal{E}$ the output.
\item[3.] Estimate global connectivity $\tau=(p+q)/2$ by $\displaystyle \hat{\tau} = {1 \over |\mathcal{O}_{0}|} \sum_{e \in \mathcal{O}_{0}} A_{e}$.
\item[4.] Run {\tt GOODCLUST} on the graph with nodes set $\mathcal{N}$ and edges present in $\mathcal{O}_0$. Output, for any $x \in \mathcal{N}$, $\widehat Z_{x}$ the estimated community of $x$. Choose the label $\hat Z=1$ for the largest estimated community.
\end{itemize}
\medskip
\noindent \textbf{Step 2: expanding the communities}
\begin{itemize}[topsep=0pt]
\item[5.] Sample uniformly at random a set $\mathcal{A}_{0}$ of $|\mathcal{A}_{0}|=\left\lceil 8\sqrt{2T} \right\rceil$ nodes in $V\setminus\mathcal{N}$.
\item[6.] Set $k = \left\lceil C_k/s \right\rceil $ and $I = \left\lceil C_I \log (s\sqrt{T}) \right\rceil$
\item[7.] {\bf For} $i=1,\ldots,I$, {\bf do}
\begin{itemize}
\item[(a)] {\bf For} $x\in \mathcal{A}_{i-1}$, sample $k$ nodes $(y_{k(i-1)+a}^x)_{a=1,\ldots,k}$ uniformly at random in $\mathcal{N} \cap \ac{\hat Z=1} \setminus \{y^x_a\}_{a = 1, \dots, k(i-1)}$.
\item[(b)] Sample the pairs $(\{x,y_{k(i-1)+a}^x\})_{a=1,\ldots,k}$ and let $\hat p_{x,i}=\frac{1}{ki} \sum_{a=1}^{ki} A_{xy_{a}^x}$.
\item[(c)] Select $\mathcal{A}_{i}=\ac{x\in \mathcal{A}_{i-1}: \hat p_{x,i} \geq \hat \tau}$.
\item[(d)] {\bf In case}\footnote{with high probability, this undesirable case does not happen} where $\mathcal{A}_i=\emptyset$, {\bf then} set $\mathcal{A}_I=\emptyset$ and {\bf BREAK}.
\end{itemize}
\end{itemize}
\medskip
\noindent \textbf{Step 3: sampling pairs within estimated communities}
\begin{itemize}[topsep=0pt]
\item[8.]
Sample uniformly at random pairs within the set $\mathcal{A}_{I}$ until $T$ pairs have been sampled overall.
If the number of sampled pairs is smaller than $T$ after all pairs in $\mathcal{A}_{I}$ have been sampled$^a$, then sample the remaining pairs at random.
\end{itemize}
\medskip
\noindent \textbf{Output:} $T$ pairs sampled at steps 2., 7.(b) and 8. of the algorithm.
\end{minipage}%
\end{minipage} \\
\hline
\end{tabular} }
\subsection{Community Expansion versus \texorpdfstring{$k$}{k} out of \texorpdfstring{$m$}{m} Best Arm Identification}\label{sec:discuss:step2}
As proved in Lemma~\ref{th_step1} in the Appendix~\ref{sec:upper}, after Step 1, with high probability, we end up with a set of $N$ classified nodes, where at most $O(1/s)$ of them are misclassified, and the empirical connectivity $\hat \tau$ does not deviate from the population one $\tau=(p+q)/2$ by more than $(p-q)/4$. The goal of Step 2 is then to identify $\sqrt{2T}$ new nodes of community 1, with at most $O(1/s)$ misclassified nodes and a regret at most $O(\sqrt{T}/s)$. Let us connect this problem to a $k$ out of $m$ best arms identification problem.
Let us consider a simplified version of the problem of Step 2. Assume that we have identified $N_{1}=N/2$ nodes of community 1 with no error, that we have access to the population connectivity $\tau$ and that among the $M=8\sqrt{2T}$ nodes in $\mathcal{A}_{0}$, half of them are of community 1.
Then, each node $a\in \mathcal{A}_{0}$ can be seen as an arm, and pulling the arm $a$ amounts to query a pair $\{a,b\}$ with $b$ one of the $N_{1}$ nodes of community 1 identified at Step 1.
The mean reward of the arm $a$ is $p$ if it belongs to community 1, and $q$ otherwise.
Hence, a simplified version of the problem in Step 2 amounts to identify $k=\sqrt{2T}$ out of $m=M/2=4\sqrt{2T}$ best arms, with at most $O(1/s)$ errors, and a cumulated regret $O(k/s)$.
We have the additional constraint that an arm can be pulled at most $N_{1}$ times, but we will forget this additional feature in this discussion, for simplicity of the comparison.
The problem of identifying $k$ out of $m$ best arms with a tolerance $\epsilon$ has been investigated in \cite{pmlr-v24-goschin12a, koutofm2019}.
The focus on these papers is on the minimal sample size needed to identify $k$ arms whose expected reward is larger than the $m^{\rm th}$ largest expected reward minus $\epsilon$.
The main results of \cite{koutofm2019} states that, with probability at least $1-k^{-2}$, the algorithm AL-Q-FK can recover with a sample size
$$O\pa{{1\over (p-q)^2}\pa{M\log\pa{ m+1\over m+1-k}+k\log(k)}}$$
$k$ out of the $m$ best arms with a tolerance $\epsilon=(p-q)/2$.
The sampling regret is not considered and it can be as large as the sample size.
In the same setting, the screening algorithm of Step 2 achieves the following performance.
For $m\geq ck\geq c'/s$, with probability at least $1-c''k^{-2}$ a budget of at most $O(ks^{-1}\log(sk))$ queries, and a sampling regret at most $O(k/s)$, the algorithm identifies a set of arms with at least $k$ out of $m$ best arms and at most $O(1/s)$ arms not in the $m$ best ones.
As $s=(p-q)^2/(p+q)$, the sampling regret achieved by the screening algorithm of Step 2 is at least $(p+q)/\log(k)$ times smaller.
We can explain this gain by several reasons.
The $p+q$ improvement comes from the fact that we explicitly take into account the fact that the rewards have a Bernoulli distribution.
The $1/\log(k)$ improvement is obtained by a careful design of the algorithm to keep the regret low, at the price of possibly $O(1/s)$ identification errors.
Specified to the simplified version of the problem in Step 2 depicted above, the AL-Q-FK algorithm would return $k=\sqrt{2T}$ nodes out of the $m=M/2=4\sqrt{2T}$ nodes of community 1 with a sampling regret
$$O\pa{{\sqrt{T}\over (p+q)s}\ \log(\sqrt{T})}.$$
This sampling regret is larger than the $O(\sqrt{T}/s)$ regret needed for our Step 2, so the AL-Q-FK cannot be used as a black-box for Step 2.
We emphasize also that the expansion of the communities in Step 2 is somewhat more complex than the simplified version described above: at Step 1, up to $O(1/s)$ nodes are misclassified, we only have access to the empirical connectivity $\hat \tau$, an arm can only be pulled $N_{1}$ times and the number of best arms is random.
We also emphasize that we cannot use the algorithm of \cite{YP14a} as a black-box to identify $\sqrt{2T}$ nodes of community 1 within $\mathcal{A}_{0}$ with at most $1/s$ errors and with a sampling regret $O(\sqrt{T}/s)$.
Indeed, if we take $\Theta(\sqrt{T})$ nodes and apply the procedure of \cite{YP14a} to classify them with a sampling regret
at most $O(\sqrt{T}/s)$, then a \emph{fixed} proportion of the nodes are misclassified and pairing them together at Step 3 would generate a final regret of order $\Theta(T)$.
In addition, from the lower bounds in \cite{YP14a}, we observe that the above phenomenon occurs, whatever the algorithm, if we try to classify \emph{all} the nodes in $\mathcal{A}_{0}$. To overcome this issue, the algorithm of Step 2 recovers the class for a \emph{fraction} only of the nodes in $\mathcal{A}_{0}$ with a sampling regret at most $O(\sqrt{T}/s)$ and at most $1/s$ errors.
When recovering the class of $\sqrt{2T}$ nodes within $\mathcal{A}_{0}$, we do not sample pairs at random, but we carefully select them in order to avoid as much as possible the sampling of bad pairs.
\section{Constrained Optimal Pair-Matching}
\label{sec:contraint}
\subsection{Main Results}
Let us now consider the general problem, where sparse sampling {\bf (SpS)} is enforced.
The algorithm described in Section \ref{sec:algo} for unconstrained pairs-matching uses extensively the opportunity to make ``localized" queries: At time $T$, a small number of $\Theta(\sqrt{T})$ nodes has been queried a large number of $\Theta(\sqrt{T})$ times, while other nodes have been queried less than $O(\log(s\sqrt{T})^2/s)$ times.
So, the strategy has to be adapted to fulfills {\bf (SpS)}.
For a sparsity bound $B_{T}$, denote by $\Psi_{B_{T},T}$ the set of strategies $\psi$ fulfilling the Non-redundancy {\bf (NR)}, Invariance to labelling {\bf (IL)} and Sparse sampling {\bf (SpS)} properties at time $T$.
\begin{thm}
\label{thm:contraint}
Let $T$ and $n$ be positive integers with $T \leq |\mathcal{E}^{\text{good}}|=2\binom{n/2}{2}$. Let $p,q \in [0,1/2]$ be two parameters fulfilling (\ref{eq:pq}) and such that the parameter $s$, defined in \eqref{eq:def:ScalingParam}, fulfills
\[
s \leq \frac{1}{32(1+\rho^*)}.
\]
Then, for any $\mu \in \text{cSBM}(n/2,n/2,p,q)$,
\begin{equation*}
\inf_{\psi \in \Psi_{B_{T},T}} \operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)} \geq \frac{1}{32} \cro{\frac{\sqrt{T} \vee (T/B_{T})}{32(1+ \rho^*) s}\wedge T}.
\end{equation*}
Conversely, there exist two numerical constants $c_1,c_2>0$ such that, for any time horizon $T$ and constraint $B_T$ satisfying $1 \leq T \leq c_1 n(B_{T}\wedge n)$, there exist a strategy $\psi \in \Psi_{B_{T},T}$ corresponding to a polynomial-time algorithm, described in Section~\ref{sec:algo_contraint},
such that
\begin{equation}\label{eq_upperbound_constrained}
\operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,T)} \leq c_2\cro{\frac{\sqrt{T}\vee (T/B_{T})}{s}\wedge T}.
\end{equation}
\end{thm}
We refer to the appendix for a proof of this theorem.
The lower bound is proved in Section~\ref{sec:lower} and the upper bound in Section~\ref{section::proof:constrainedTHM}.
Compared with Theorem~\ref{thm:non-contraint}, Theorem~\ref{thm:contraint} shows that the sparse sampling constraint {\bf (SpS)} amounts to replace $\sqrt{T}$ by $\sqrt{T}\vee(T/B_{T})$ in the optimal sampling-regret.
In particular, the sparse sampling constraint downgrades optimal rates only when $B_{T}$ is smaller than $\sqrt{T}$.
Actually, a close look at the unconstrained algorithm page \pageref{alg:unconstrained} reveals that, by construction, it satisfies assumption \textbf{(SpS)} with $B_T = 17\sqrt{T}$. So, in the regime where $B_T \geq 17 \sqrt{T}$,
the lower bound
cannot be worse than the upper-bound of the unconstrained setting of Theorem~\ref{thm:non-contraint}.
When $B_{T} \lesssim \sqrt{T}$, the optimal sampling-regret is of order $(T/(B_{T}s))\wedge T$.
This rate can be understood as follows.
If $B_{T}\leq 1/s$, there is not enough observations per node to infer their community better than at random, which induces an unavoidable linear regret.
When $B_{T}\gg 1/s$, to proceed as in Step 3 of the constrained case, one needs to identify a sufficiently large set of nodes of the same community, among which one can sample up to $T$ pairs without adding regret.
As each node can now be paired with at most $B_{T}$ others, this set should be of size $\Theta(T/B_{T})$ instead of $\Theta(\sqrt{T})$ in the unconstrained case.
As the identification of the community of a node requires at least $\Theta(1/s)$ queries, the sampling-regret expected to identify this large set of nodes is $\Theta(T/(B_{T}s))$.
The previous informal discussion suggests to extend the algorithm described in Section~\ref{sec:algo} for the unconstrained case.
This extension, fully described and commented in Section \ref{sec:algo_contraint}, still proceeds in $3$ steps and goes as follows.
The first step of the constrained algorithm is essentially the same as the first step of the unconstrained algorithm, with $\sqrt{T}$ replaced by
$B=(B_{T}\wedge \sqrt{T})/2$.
In this first step, all pairs are sampled among a set of $B/\log(sB)\leq B_T$ nodes, so the constraint cannot be violated.
Then, to keep the sampling-regret under control while not violating the {\bf (SpS)} contraint, the trick is to apply recursively a variant of the screening algorithm in Step~2 and repeat these screenings until a total number of $\Theta(T/(B_{T}\wedge \sqrt{T}))$ nodes are correctly classified, with a small proportion of error.
Finally, one can sample at most $B_T\wedge \sqrt{T}$ pairs for each of these nodes in Step 3 with a controlled regret.
The resulting algorithm extends the unconstrained one of Section~\ref{sec:algo} where $B_T\wedge \sqrt{T}=\sqrt{T}$ and where the screening step is only applied once.
This extension is fully described in Section~\ref{sec:algo_contraint}.
To illustrate the theorem, one can discuss the results with the constraint $B_{T}=T^\gamma$, where $0<\gamma\leq 1/2$.
In this case, the optimal sampling-regret is of order $T\wedge (T^{1-\gamma}/s)$.
It follows that any pair-matching algorithm that is $T^\gamma$-sparse up to time $T$ (besides satisfying {\bf (NR)} and {\bf (IL)}) has linear sampling-regret up to time $s^{-1/\gamma}$.
On the other hand, there exist strategies with optimal sampling-regret of order $T^{1-\gamma}/s$ after time $s^{-1/\gamma}$.
Notice that the sparse sampling property $N_{a}(\psi,T)\leq B_{T}$ only constrains the algorithm at the time horizon $T$.
This time horizon has therefore to be specified beforehand for this constraint to be defined.
In many practical situations, this specification is not reasonable and a more realistic constraint takes the form: $N_{a}(\psi,t)\leq B_{t}$ at any time $t\in \ac{1,\ldots,T}$.
In the case where $B_{t}=\Theta(t^{\gamma}/(\log t)^{\tau})$, the constraint can be enforced using a doubling trick, without enlarging the regret by more than a multiplicative numerical constante.
This doubling trick is discussed in detail in Section~\ref{sec:doubling-contraint}.
\subsection{Algorithm with Sparse Sampling}\label{sec:algo_contraint}
The algorithm described in page \pageref{alg:unconstrained}, that achieves optimal regret in the unconstrained case, identifies first a set of $\Theta(\sqrt{T})$ nodes from one community with $O(1/s)$ misclassified nodes and a regret of order $O(\sqrt{T}/s)$ in Steps 1 and 2.
Then, it pairs these nodes together in Step 3 with a $O(\sqrt{T}/s)$ regret (due to the misclassified nodes).
The algorithm described in this section follows essentially the same steps, identifying first a set of nodes from the same community (with small error) and then sampling pairs among them.
It has to be adapted to fulfill the {\bf (SpS)} constraint.
As the unconstrained algorithm fulfills the {\bf (SpS)} constraint for any $B_{T}\geq 17\sqrt{T}$, it is assumed in the remaining of this section that $B_{T}=O( \sqrt{T})$.
Moreover, as the result holds for $T \leq c_1 n(B_{T}\wedge n)$, this assumption is granted in the remaining of the section.
To respect the constraint {\bf (SpS)}, no node may be sampled in more than $B_{T}$ pairs.
Hence, to perform the last step, the algorithm has to identify $\Theta(T/B_{T})$ nodes from one community.
It should achieve this identification with a sampling-regret smaller than $O(T/(sB_{T}))$ while respecting the {\bf (SpS)} constraint.
To respect the {\bf (SpS)} constraint in the first step of the algorithm, a kernel $\mathcal{N}_{init}$ of cardinality smaller than $B_{T}$ is chosen.
Formally, in points 1. and 2. of Step 1 in the algorithm page \pageref{alg:unconstrained}, $\sqrt{T}$ is replaced by $(B_{T}\wedge \sqrt{T})/2$.
Then, as in the unconstrained case, Step 2 expands the communities in order to identify, with high probability and up to a small error, $\Theta(T/B_{T})$ nodes from one community.
The main difference with the unconstrained case is that this expansion cannot be achieved in a single step of screening.
Actually,
\begin{itemize}
\item[(i)] $\Theta(N/s)$ pairs are required to identify the community of $\Theta(N)$ new nodes.
\item [(ii)] Any node from the kernel $\mathcal{N}_{init}$ cannot be sampled more than $B_{T}$ times.
\end{itemize}
By (ii), one cannot sample more than $O(|\mathcal{N}_{init}|B_{T})$ pairs and by (i), it follows that at most $O(|\mathcal{N}_{init}|B_{T}s)=O(B_{T}^2s)$ nodes can be classified with a single screening step based on $\mathcal{N}_{init}$.
The main idea of the new algorithm is to iterate the screening step, expanding progressively the communities.
Along these iterations, to satisfy the {\bf (SpS)} constraint, the screening has to be conducted with more care than in step 2 of the unconstrained algorithm page \pageref{alg:unconstrained}.
The trick is to apply the \texttt{SCREENING} function described page \pageref{alg:screening}, which compartmentalizes the nodes in order to enforce the condition {\bf (SpS)}.
This iterative process outputs a set of $\Theta(T/B_{T})$ nodes from one community (with a small proportion of error with high probability).
The algorithm finally pairs nodes among this subset while respecting the {\bf (SpS)} constraint in Step 3 of the algorithm.
\newpage
\medskip
\noindent { \def1.3{1.3}
\begin{tabular}{|l|}
\hline
\textbf{Constrained Algorithm}\label{alg:constrained} \\
\hline
\begin{minipage}{0.99\textwidth} \centering
\begin{minipage}{0.97\textwidth}
\medskip
\noindent \textbf{Inputs:} $s$ scaling parameter, $T$ time horizon, $V_{init}$ the set of the $n$ nodes of the whole graph, $B_T$ constraint.
\medskip
\noindent \textbf{Internal constants:} set $c_{\mathcal{O}_0} = 8 \vee (1 / c_1^\text{GC})$ and $B = (B_T \wedge \sqrt{T}) / 2$.
\medskip
\noindent \textbf{Step 1: finding communities in a kernel}
\begin{enumerate}[topsep=0pt]
\item Sample uniformly at random an initial set $\mathcal{N}_{init} \subset V_{init}$ of $N_{init} = \left\lceil \frac{B}{\log(sB)} \right\rceil$ nodes.
\item Sample each pair of $\mathcal{N}_{init}$ with probability $c_{\mathcal{O}_0} \frac{B}{s}/\binom{N_{init}}{2}$, call $\mathcal{O}_0 \subset \mathcal{E}$ the output.
\item Estimate mean connectivity $\tau=\frac{p+q}{2}$ by $\hat{\tau}=\frac{1}{|\mathcal{O}_0|} \sum_{(x,x') \in \mathcal{O}_0} A_{x,x'}.$
\item Run \texttt{GOODCLUST} on the graph $(\mathcal{N}_{init}, \mathcal{O}_{0})$ and output, for any $x \in \mathcal{N}_{init}$, $\widehat{Z}_{x}$ the estimated community of $x$ (with the convention that the largest estimated community is labelled by 1).
\end{enumerate}
\medskip
\noindent \textbf{Step 2: iteratively expanding the communities}
\medskip
\noindent \textbf{Internal constants:} set $N^{(0)} = \left\lceil N_{init}/2 \right\rceil$,
\begin{equation}
t_f = \left\lceil \frac{ \log (\lceil T / B \rceil / N^{(0)}) }{ \log \lfloor \log (sB) \rfloor } \right\rceil
\end{equation}
and for all $t \in \{0, \dots, t_f\}$,
\begin{equation}
N^{(t)} = N^{(0)} \lfloor \log (sB) \rfloor^t \wedge \left\lceil \frac{T}{B} \right\rceil.
\end{equation}
\begin{enumerate}[topsep=0pt, resume]
\item Let $\mathcal{N}^{(0)}$ be a set of $N^{(0)}$ nodes in $\mathcal{N}_{init} \cap \{\widehat{Z} = 1\}$ sampled uniformly at random, and let $V^{(0)} = V_{init} \setminus \mathcal{N}_{init}.$
\item \label{point_screening_algcontraint} \textbf{For} $t = 1, \ldots, t_f $, {\bf set}
\begin{equation}
(\mathcal{N}^{(t)}, V^{(t)}) = \texttt{SCREENING}\pa{
\mathcal{N}^{(t-1)},
N^{(t)},
B,
\hat{\tau},
V^{(t-1)}
}.
\end{equation}
\end{enumerate}
\medskip
\noindent \textbf{Step 3: sampling pairs within estimated communities}
\begin{enumerate}[topsep=0pt, resume]
\item Sample pairs within the set $\mathcal{N}^{(t_f)}$ while respecting the constraint \textbf{(SpS)} with $B_T$, until $T$ pairs have been sampled overall (the sampling method does not matter).
\end{enumerate}
\medskip
\end{minipage}%
\end{minipage} \\
\hline
\end{tabular} }
\noindent { \def1.3{1.3}
\begin{tabular}{|l|}
\hline
\textbf{Function} \texttt{SCREENING}$(\mathcal{N}, N', B, \nu, V) = (\mathcal{N}', V')$ \label{alg:screening} \\
\hline
\begin{minipage}{0.99\textwidth} \centering
\begin{minipage}{0.97\textwidth}
\medskip
\textbf{Inputs:} a reference kernel $\mathcal{N}$ of cardinality $N$, a target number of nodes $N'$, a constraint $B \in \mathbb{R}_+$, a threshold $\nu \in [0,1]$, a set of ``new" nodes $V$.
\medskip
\textbf{Output:} a set of nodes $\mathcal{N}'\subset V$ of cardinality at most $N'$ and the set of nodes $V'\subset V$ that are still ``new" after running \texttt{SCREENING}.
(Most of the nodes of $\mathcal{N}'$ will belong to the most represented community in $\mathcal{N}$.)
\medskip
\textbf{Internal constants:} a number of pairs per step $k = \lceil \frac{C_k}{s} \rceil$ and a number of steps $I = \lceil C_I \log(sB) \rceil$, with $C_k =2500$ and $C_I = 1026$.
\medskip
\begin{enumerate}[topsep=0pt]
\item Sample uniformly at random a set $\mathcal{A}_0$ of $|\mathcal{A}_0| = 4N'$ nodes in $V$.
\item \label{SCREEN_step2} Let $m = \lfloor N / (kI) \rfloor$. Take a uniform partition of $\mathcal{N}$ into $m$ sets $(\mathcal{V}_j)_{1 \leq j \leq m}$ of cardinality $kI$ and one set of cardinality smaller than $kI$.
Likewise, take a uniform partition of $\mathcal{A}_0$ into $m$ sets $(\mathcal{A}_0^{(j)})_{1 \leq j \leq m}$ with cardinality in $\{\lfloor 4N'/m \rfloor, \lceil 4N'/m \rceil \}$.
\item \label{step_4_contraint}
\textbf{For} $j = 1, \dots, m$ \textbf{and} $i = 1, \dots, I$, \textbf{do}
\begin{itemize}
\item[] \textbf{For each} $x \in \mathcal{A}^{(j)}_{i-1}$, \textbf{do}
\begin{itemize}[topsep=0pt]
\item[i.] Sample $k$ nodes $(y_{k(i-1)+a}^x)_{a=1,\ldots,k}$ uniformly at random in $\mathcal{V}_j \setminus \{y_{a}^x\}_{a=1,\ldots,k(i-1)} $.
\item[ii.] Sample pairs $(\{x,y_{k(i-1)+a}^x\})_{a=1,\ldots,k}$ and compute
\begin{equation}
\hat p_{x,i}=\frac{1}{ki} \sum_{a=1}^{ki} A_{xy_{a}^x}.
\end{equation}
\end{itemize}
\item[iii.] Select $\mathcal{A}^{(j)}_{i} = \ac{x \in \mathcal{A}^{(j)}_{i-1}: \hat p_{x,i} \geq \nu}$.
\end{itemize}
\item \label{step_5_contraint} Set $\mathcal{N}'$ a set of $N' $ nodes sampled uniformly at random from $\displaystyle \bigcup _{1 \leq j \leq m}\mathcal{A}^{(j)}_I$.\\*
{\bf In case}\footnote{with high probability, this undesirable case does not happen} $|\bigcup _{1 \leq j \leq m}\mathcal{A}^{(j)}_I|< N'$, {\bf then} sample at random $N'$ nodes in $\mathcal{A}_0$.
\item Set $V' = V \setminus \mathcal{A}_0.$
\end{enumerate}
\textbf{Return} $(\mathcal{N}', V')$.
\medskip
\end{minipage}%
\end{minipage} \\
\hline
\end{tabular} }
\subsection{Screening versus \texorpdfstring{$k$}{k} out of \texorpdfstring{$m$}{m} Best Arms Identification}
Similarly as in Section \ref{sec:discuss:step2}, let us compare the screening step to a $k$ out of $m$ best arms identification problem.
The main additional feature compared to the situation discussed in Section \ref{sec:discuss:step2}, is that an arm $a$ cannot be sampled more than $B$ times.
Hence, a simplified version of the screening problem amounts to identify $k$ out of $m$ best arms with tolerance $\epsilon=(p-q)/2$, with the constraint that each arm cannot be sampled more than $B$ times.
In these simplified setting, the screening function achieves the following performance.
Assume that $M\geq ck$ and $k,B\geq c'/s$.
With probability $1-c(sk)^{-1}$ , with a budget of $O(ks^{-1}\log(s(B\wedge k)))$ queries, and with a sampling regret at most $O(k/s)$, the screening function identifies at least $k$ arms of community 1 with at most $O((k(sB)^{-1})\vee s^{-1})$ errors.
The situation handled by the screening function is actually somewhat more complex than the stylized bandit problem depicted above. Actually, among the initial set of $N$ classified nodes, we have up to $cN/(sB)$ misclassified nodes.
At the same time, we cannot query more than $B$ times any of these classified nodes.
Hence, we need a careful querying policy in order to avoid the misclassified nodes to generate errors, while keeping the {\bf (SpS)} condition enforced.
Fulfilling together these two conditions is the main hurdle in the design and analysis of the screening function.
\subsection{Pathwise Sparse Sampling Algorithm}\label{sec:doubling-contraint}
The algorithm presented above fulfills the sparse sampling condition {\bf (SpS)} at time horizon $T$.
In many practical situations, it is more natural to consider Condition {\bf (SpS)} at all times $t=1,2,\ldots$ rather than only at a predefined time horizon $t=T$.
Formally, Condition {\bf (SpS)} would be replaced by $N_{a}(\psi,t)\leq B_{t}$, for all $t=1,2,\ldots$.
It is possible to modify the previous algorithm to build a strategy $\psi$ such that, when $B_{t}=\Theta(t^{\gamma}\log^{-\tau}(t))$, the sampling regret $\operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,t)}$ fulfills
$$\operatorname{\mathbb{E}}_\mu \cro{N^{bad}(\psi,t)}= O\pa{\frac{\sqrt{t}\vee (t/B_{t})}{s}\wedge t},\quad \textrm{for}\ t=1,2,\ldots.$$
Assume that there exists $\gamma\in (0,1/2]$ and $\tau \in [0,+\infty)$ such that $B_{t}=t^{\gamma}/(\log t)^{\tau}$, so $\sqrt{t}\vee(t/B_{t})=t^{1-\gamma}\log^{\tau}(t)$.
In this case, a pathwise sampling condition can be enforced using the simple doubling trick
For any positive integer $l$, let $t_l = 2^l$.
At each time $t_l$, the new algorithm discards all nodes and pairs previously sampled and starts the algorithm of Section \ref{sec:algo_contraint} with the remaining nodes, time horizon $T=t_{l+1}-t_{l}$ and terminal sparse sampling constraint $N_{a}(\psi,t_{l+1}-t_{l})\leq \min_{t_{l}\leq t\leq t_{l+1}}B_{t}$.
The resulting strategy does not depend on any time horizon and it fulfills the condition $N_{a}(\psi,t)\leq B_{t}$, for all $t=1,2,\ldots$.
Moreover, for any $l$ such that $t_{l}\geq e^{\tau/\gamma}$, $\min_{t_{l}\leq t\leq t_{l+1}}B_{t}=B_{t_{l}}$.
Hence, for any $l$ such that $t_{l}<c_1 n(B_{t}\wedge n)$ and for any $t$ such that $t_{l-1}\leq t \leq t_{l}<c_1 n(B_{t}\wedge n)$,
\begin{align*}
\mathbb{E}\cro{N^{bad}(\psi,t)} &=O\pa{1+\sum_{k=1}^l\frac{\pa{t_k-t_{k-1}}^{1-\gamma}\log^{\tau}(t_{k}-t_{k-1})}{s}\wedge (t_{k}-t_{k-1})}\\
&=O\pa{\pa{\frac{1}{s} \sum_{r=0}^{l-1}2^{r(1-\gamma)} (r\log(2))^{\tau}}\wedge t_{l}}\\
&= O \pa{\frac{t_{l}^{1-\gamma}\log^\tau(t_{l})}{s}\wedge t_{l}}=O\pa{\frac{t^{1-\gamma}\log^\tau(t)}{s} \wedge t}.
\end{align*}
According to Theorem~\ref{thm:contraint}, the sampling-regret of the algorithm derived from the doubling trick is then rate optimal.
\section{Discussion}\label{Sec:KclassesSBM}
The present paper provides the optimal sampling-regret for pair-matching in
the case where $G=(E,V)$ is a conditional SBM with a number of groups $K=2$, where the groups have $n/K$ elements, with intra class probability of connection $p$ and inter-class $q$.
The algorithm depicted p.\pageref{alg:unconstrained} in Section~\ref{sec:algo} runs in polynomial time and has optimal sampling-regret given in Theorem~\ref{thm:non-contraint}, up to a multiplicative constant. Let us discuss the two following questions: How can we estimate the scaling parameter $s$?
How does the rates depend on the number $K$ of groups?
\subsection{A Heuristic to Estimate the Scaling Parameter \texorpdfstring{$s$}{s}}
\label{sec:SNR}
The algorithms described p.\pageref{alg:unconstrained} and p.\pageref{alg:constrained} in Sections \ref{sec:algo} and \ref{sec:algo_contraint} take the scaling parameter $s$ as input.
This parameter is typically unknown in practice and an estimated value $\widehat s$ has to be plugged in the algorithm.
To guarantee a sampling-regret smaller than $O(T\wedge (\sqrt{T}/s))$, the estimator $\widehat s$ should use at most $O(1/s^2)$ edges and satisfy $ \widehat s \asymp s$ with high probability.
The following heuristic builds a possible estimator $\widehat s$.
Pick uniformly at random $N$ nodes in $V$ and sample all $N(N-1)/2$ pairs between these $N$ nodes. When $Ns>2$, $p=a/N$ and $q=b/N$, \cite{MNS15} ensures that, as $N\to\infty$, $a$ and $b$ can be consistently estimated.
Therefore, $Ns=(a-b)^2/(a+b)$ can also be consistently estimated from these $T=N(N-1)/2=O(1/s^2)$ observations.
Yet, this estimator requires $Ns$ larger than $2$ and cannot therefore be used directly when $s$ is unknown.
However, when $p=a/N$ and $q=b/N$ and $N\to\infty$,
it is theoretically possible to detect whether $Ns=(a-b)^2/(a+b)$ is smaller or larger than 2.
To proceed, denote by $\mathcal{B}$ the non-backtracking matrix associated to the graph (see \cite{BLM18} for a definition of the non-backtracking matrix).
Let $\lambda_{1}, \lambda_{2},\ldots$ be the eigenvalues of $\mathcal{B}$ ranked in decreasing order of their moduli.
The main result of \cite{BLM18} shows that, when $p=a/N$ and $q=b/N$, with $a,b>0$ fixed, except on an event of vanishing probability as $N\to \infty$,
\begin{gather*}
|\lambda_{2}|^2<\lambda_{1}\quad \text{when}\quad Ns<2\enspace,\\
|\lambda_{2}|^2>\lambda_{1} \quad \text{when}\quad Ns>2\enspace.
\end{gather*}
In addition, when $Ns>2$, the ratio $2|\lambda_{2}|^2/\lambda_{1}$ consistently estimates $(a-b)^2/(a+b)$.
This result suggests the following recursive algorithm to estimate $s$: start with a set $V_1$ of $2$ nodes $i$ and $j$ picked uniformly at random in $V$. Query the pair $\{i,j\}$ and let $E_1$ denote the set of edges in $E\cap \{i,j\}$.
At each step $k\geqslant 2$, pick at random a set $V_k$ of $2^k$ nodes in $V\setminus \cup_{\ell\leqslant k-1}V_{\ell}$.
Sample all pairs in $V_k$, and denote by $E_k$ the set of edges among these pairs.
Build the non-backtracking matrix $\mathcal{B}_k$ of the graph $(V_k,E_k)$ and compute $\lambda^{(k)}_1$ and $\lambda^{(k)}_2$ the eigenvalues of this matrix with largest moduli.
If $|\lambda^{(k)}_{2}|^2<\lambda^{(k)}_{1}$ iterate. If $|\lambda^{(k)}_{2}|^2>\lambda^{(k)}_{1}$ stop, denote by $\widehat{k}$ the stopping iteration time and $\widehat{N}=\binom{2^{\widehat{k}}}{2}$ the number of edges sampled in the last graph $(V_{\widehat{k}},E_{\widehat{k}})$.
Output $\widehat s=2|\lambda^{(\widehat{k})}_{2}|^2/(\widehat{N}\lambda^{(\widehat{k})}_{1})$.
Assume that $p=a/N$ and $q=b/N$ with $a,b\in \mathbb{R}^+$ fulfilling $(a-b)^2/(a+b)>2$. Let $\Omega_{N}$ denote the event where simultaneously $2\leq \widehat N s \leq 4$ and $s/2\leq \widehat s \leq 2 s$.
Then the results of \cite{BLM18} suggest that the event $\Omega_{N}$ holds with probability tending to $1$ as $N\to \infty$.
In addition, the total number of sampled edges is $\cup_{k=1}^{\widehat{k}} \binom{2^k}2=O(\widehat N^2)=O(1/s^2)$ on this event.
We emphasize yet that the results of \cite{BLM18} only hold in a setting where $p=a/N$, $q=b/N$, with $a,b$ fixed and $N\to \infty$, and we cannot turn them into a theoretical guarantee that $\Omega_{N}$ holds with probability close to $1$.
\subsection{Case with \texorpdfstring{$K>2$}{more than two} Groups}
Let us discuss the case where the number of groups $K$ is larger than $2$, still assuming that all the groups have $n/K$ elements, with intra class probability of connection $p$ and inter-class $q$.
Contrary to $K=2$, we expect in this case an information-computation gap and conjecture the following optimal rates for pair-matching.
\begin{conj}
Define $\Psi_{\infty}^{poly}$ as the intersection of $\Psi_{\infty}$ defined page \pageref{sec:Unconstrained}, with polynomial-time algorithms.
Let
\begin{equation}
\label{eq:sk}
s_{K}= {(p-q)^2\over q+(p-q)/K}.
\end{equation}
Under Assumption (\ref{eq:pq}), without computational constraint:
\begin{equation}\label{eq:conj:expo}
\inf_{\psi\in \Psi_{\infty}} \operatorname{\mathbb{E}}\cro{N^{bad}(\psi,T)} \asymp \pa{\pa{K\log(K)\over s_{K}}^2\vee {K\sqrt{T}\over s_{K}}}\wedge T.
\end{equation}
With polynomial time constraint:
\begin{equation}\label{eq:conj:poly}
\inf_{\psi\in \Psi_{\infty}^{poly}} \operatorname{\mathbb{E}}\cro{N^{bad}(\psi,T)} \asymp \pa{\pa{K^2\over s_{K}}^2\vee {K\sqrt{T}\over s_{K}}}\wedge T.
\end{equation}
\end{conj}
Let us explain the heuristics leading to these rates.\smallskip
For $K=2$, a central tool to design the rate-optimal polynomial-time algorithm p.\pageref{alg:unconstrained} is the existence of polynomial-time algorithms (called {\tt GOODCLUST} p.\pageref{alg:unconstrained}) achieving non trivial classification for a cSBM$(N/2,N/2,p,q)$ when $Ns$ is larger than some constant.
When $K>2$ and the number of nodes $N\to \infty$, for $p$, $q$ scaling as $1/N$, \cite{BLM18,AS15c,LM18} provide polynomial-time algorithms $\text{{\tt GOODCLUST}}^{\text{poly}}_K$
achieving a non trivial classification for
$$Ns_{K}>K^2=:\lambda_{K}^{poly}.$$
Furthermore, it is conjectured \cite{Decelle2011} that there does not exist any polynomial-time algorithm achieving non-trivial classification when $Ns_{K}<K^2$.
The threshold $\lambda^{poly}_{K}$ is known as the Kesten-Stigum (KS) threshold.
The information theoretic threshold $\lambda_{K}^{inf}$ for non-trivial classification is below $\lambda_{K}^{poly}$ for $K\geq 5$.
Actually, \cite{BanksMoore2016} have proved that $\lambda_{K}^{inf}\asymp K\log(K)$ and $\lambda_{K}^{inf}<\lambda_{K}^{poly}$ for $K\geq 5$, so, if the conjecture of \cite{Decelle2011} holds, there is an information-computation gap for $K\geq 5$.
A consequence of the result of \cite{BanksMoore2016} is that there exist algorithms $\text{{\tt GOODCLUST}}^{\text{inf}}_K$, with exponential complexity, achieving non-trivial classification for $Ns_{K}=O(K\log(K))$.
Theorem~\ref{thm:non-contraint} requires that $\text{{\tt GOODCLUST}}$ has more than non-trivial classification, it should have vanishing classification error.
Several papers have established, under Assumption (\ref{eq:pq}), the existence of algorithms $\text{{\tt GOODCLUST}}^{\text{poly}}_K$ and $\text{{\tt GOODCLUST}}^{\text{inf}}_K$ with misclassification proportion smaller than $\exp(-c Ns_{K}/K)$, for some positive constant $c$.
This result is obtained for $Ns_{K}\geq c' \lambda_{K}^{poly}$ for $\text{{\tt GOODCLUST}}^{\text{poly}}_K$, see for example \cite{CRV15,GMZZ17,FC17,GV18} and for $Ns_{K}\gg \lambda_{K}^{inf}$ for $\text{{\tt GOODCLUST}}^{\text{inf}}_K$, see \cite{ZZ16}.
As a consequence, without computational constraint, a linear sampling regret is expected for any algorithm as long as the time horizon satisfies $\sqrt{2T} s_{K}< \lambda_{K}^{inf}$, or equivalently
$$
T< 0.5 (\lambda^{inf}_{K}/s_{K})^2=0.5(K\log K /s_{K})^2.
$$
On the other hand, when $T\gg (K(\log K)^2/s_{K})^2$, one can choose $N$ fulfilling $\lambda^{inf}_{K}/s_{K} \ll N\leq (K\sqrt{T}/s_{K})^{1/2} \ll \sqrt{T}$.
Selecting $N$ nodes uniformly at random and
observing all pairs of these $N$ nodes, {\tt GOODCLUST}$^{inf}_{K}$ classifies correctly the $N$ nodes, but a proportion at most $\exp(-cN s_{K}/K)$ of them.
The sampling-regret for this step does not exceed the number $O(N^2)=O(K\sqrt{T}/s_{K})$ of pairs sampled.
Since $N s_{K}/K \gg \log(K)$, the proportion of misclassified nodes among these $N$ nodes is small and a screening procedure as in Step~2 of the algorithm p.\pageref{alg:unconstrained} can be applied in order to classify correctly $\sqrt{T}$ nodes. As an average of $K/s_{{K}}$ queries is necessary to classify one new node, this step will have a regret scaling as $K\sqrt{T}/s_{K}$. Then, we can pair all nodes of the same group until the budget of $T$ queries is spent. Hence, in the regime where $T\gg (K(\log K)^2/s_{K})^2$,
the final regret should be proportional to $N^2+K\sqrt{T}/s_{{K}}\asymp K\sqrt{T}/s_{{K}}$.
To sum-up the discussion, without computational constraints, one can expect a sampling-regret of order $$\left((K\log(K)/s_{{K}})^2\vee K\sqrt{T}/s_{{K}}\right)\wedge T\,,$$
which is the conjectured rate (\ref{eq:conj:expo}).
Using polynomial time algorithms for clustering, the information-theoretic threshold $\lambda^{inf}_{K}$ should be replaced by the KS-threshold $\lambda^{poly}_{K}$.
Following the same reasoning as before, linear regret is expected as long as
$$
T< 0.5 (\lambda^{poly}_{K}/s_{K})^2=0.5(K^2 /s_{K})^2.
$$
On the other hand, when $\sqrt{T}\gg K^3/s_{K}$,
one can pick $N$ nodes at random with $N$ fulfilling $\lambda^{poly}_{K}/s_{K} \ll N\leq (K\sqrt{T}/s_{K})^{1/2} \ll \sqrt{T}$.
A polynomial time algorithm $\text{{\tt GOODCLUST}}^{\text{poly}}_K$ run with all pairs based on these nodes classifies correctly these $N$ nodes, except for a proportion at most $\exp(-cN s_{K}/K)$ of them.
The sampling-regret associated to this classification step is smaller than $N^2\leq K\sqrt{T}/s_K$.
The screening step classifies correctly $\sqrt{T}$ nodes with a regret $K\sqrt{T}/s_{K}$.
The remaining budget until sampling $T$ pairs is spent by pairing together nodes in a same estimated group.
Ultimately, taking into account the computational constraint, one can expect a sampling-regret of order $((K^2/s_{K})^2\vee K\sqrt{T}/s_{K})\wedge T$, which is the conjectured rate (\ref{eq:conj:poly}).
|
1,314,259,996,718 | arxiv | \section{Introduction}
In the general picture, the gravitational unstable molecular cloud
cores collapse and form the protostars, accretion disks, and
outflows. The first model of a spherical symmetric accretion which
flow towards a central object was made by Bondi~(1952). Years later,
Ulrich~(1976) modified the Bondi's idea by assuming all fluid
particles have a certain angular momentum with negligible pressure
forces at the border of the cloud, so that the collapse problem
analyzed using ballistic trajectories. Since then, many
generalizations of the core collapse have been made (e.g., Shu~1977,
Terebey, Shu \& Cassen~1984, Foster \& Chevalier~1993, Galli \&
Shu~1993, Henriksen, Andr\'{e} \& Bontemps~1997, Fatuzzo, Adams \&
Myers~2004, Mendoza, Tejeda \& Nagel~2009, Bate~2010,
Nejad-Asghar~2010). As a result of rotating collapse, the collisions
at the equatorial plane, between the infalling matters coming from
the northern hemisphere and those coming from the southern one,
cause the formation of strong supersonic shocks. If the shocked gas
cools rapidly, the result is that material accumulates in a thin
structure in the equatorial plane, i.e., accretion disk
(Hartmann~2009).
Observations have revealed not only the existence of the
circumstellar disks around the protostars (e.g., Watson et al.~2007,
Akeson~2008, Quanz et al.~2010), but also the occurrence of jets and
outflows (e.g., Hirth, Mundt \& Solf 1997, Wu et al.~2004, Dunham et
al.~2010). The outflows and jets are ubiquitous together with
accretion disks through the collapsing molecular cloud cores (e.g.,
Furuya, Cesaroni \& Shinnaga~2011). We cannot directly observe the
newborn or very young accretion disks and jets because their
formation places are embedded in a dense infalling envelope. On the
other words, observations cannot directly determine the real
structure and formation history of accretion disks and outflows.
Therefore, theoretical approach and simulations are necessary to
investigate the formation and evolution of them (e.g., Walch et
al.~2009, Machida, Inutsuka \& Matsumoto~2010, Ciardi \&
Hennebelle~2010, Nejad-Asghar~2011). Clearly, consideration of jets
and magnetic fields can affect on formation and evolution of
accretion disks (e.g., Hujeirat~1998, Hujeirat et al.~2003, Machida,
Inutsuka \& Matsumoto~2007), but in this study, we turn our
attention to the phase after formation of of central protostar, and
for simplicity neglect the effect of jets and outflows on
protostellar accretion shocks. In addition to physical evolution of
circumstellar disks, the chemical evolution from the core to the
disk phase is also important (e.g., Ceccarelli, Hollenbach \&
Tielens~1996, Rodgers \& Charnley~2003, Garrod \& Herbst~2006,
Garrod, Weaver \& Herbst 2008, Visser et al.~2009). Here, we assume
that all effects of chemical evolution of shocked gas are simplified
through the appropriate net cooling function.
Interstellar shocks cover a wide range of parameters: velocities of
$1-10^4 \mathrm{km.s^{-1}}$, pre-shock densities of $10^{-2}-10^7
\mathrm{cm^{-3}}$, and post-shock temperatures of $10^2-10^9
\mathrm{K}$. The strength of a shock is indicated by the Mach number
which can range up to $\sim 10^3$, for larger than laboratory shocks
(McKee \& Hollenbach~1980). For an adiabatic strong shock, the jump
in density is limited to a factor of four (for a ratio of specific
heats $\gamma=5/3$), while the temperature of post-shock gas can be
increased proportional to the square of Mach number (e.g., Dyson \&
Williams~1997). As mentioned above, the high velocity of infalling
matters at the equatorial plane of a rotating core collapse, leads
to strong supersonic collisions with great Mach numbers. The high
temperature of the post-shock gas causes the molecule dissociation
and the atom ionization. Neufeld \& Hollenbach (1994) examined the
physical and chemical processes of high-density accretion shocks
which associated with the supersonic infall of material during the
collapse of a molecular cloud core to form a protostar. Since the
rotation and magnetic fields cause the deflection of infalling
matters from the protostar so that a shocked disk gas is formed,
here we study the physical and chemical evolutions of these shocked
circumstellar disks for understanding the formation of structure and
substructures through them.
As a general aspect of star formation, we expect that cooling (as a
result of chemical and physical changes) of the protostellar
accretion shocked gas at the equatorial plane of collapsing core
leads to supply the suitable conditions for grain growth and
formation of proto-planetary entities. The goal of this paper is to
investigate this expectation. For this purpose, the jump shock
(J-shock) structure and the cooling time-scale of post-shock gas are
presented in section~2. In section~3, we use the SPH methodology to
investigate the time evolution of the strong supersonic shocks.
Finally, section~4 is devoted to summary and conclusions.
\section{J-shock structure}
For simplicity, the shock is assumed planar and steady in which the
deceleration is negligible and there is no thermal instability in
the cooling layer. The jump conditions of this shock (J-shock)
relate the quantities at an arbitrary point behind the shock front
to those ahead of it. Conservation of mass, momentum, and energy
across the shock front is given by Rankine-Hugoniot conditions
(e.g., Dyson \& Williams~1997)
\begin{equation}\label{rh1}
\rho_1 v_1=\rho_2 v_2
\end{equation}
\begin{equation}\label{rh2}
\rho_1 v_1^2+ \mathcal{K}_1 \rho_1 T_1 =\rho_2 v_2^2+ \mathcal{K}_2
\rho_2 T_2
\end{equation}
\begin{equation}\label{rh3}
\frac{1}{2}v_1^2 + \frac{\gamma_1}{\gamma_1-1} \mathcal{K}_1 T_1=
\frac{1}{2}v_2^2 + \frac{\gamma_2}{\gamma_2-1} \mathcal{K}_2 T_2 + Q
\end{equation}
where $\gamma$ is the ratio of specific heats, $Q$ is the energy
lost per unit mass during the shock process, and the equation of
state is applied as $p=(k/\mu m_H)\rho T = \mathcal{K}\rho T$. The
jump conditions (\ref{rh1})-(\ref{rh3}) enable one to solve the
J-shock structure.
We would be interested to consider the collision of two gas sheets
with velocities $v_0$ in the rest frame of laboratory. In this
reference frame, the post-shock will be at rest and the pre-shock
velocity is given by $v_1=v_0+v_2$, where $v_2$ is the shock front
velocity. Defining the pre-shock sound speed as $c\equiv
\sqrt{\mathcal{K}_1 T_1}$ and Mach number as $M_0 \equiv v_0/c$, the
equations (\ref{rh1})-(\ref{rh3}) lead to
\begin{equation}\label{temp1}
\mathcal{T} = \frac{1}{\mathcal{R}} + \frac{M_0^2}{\mathcal{R}-1},
\end{equation}
\begin{equation}\label{temp2}
\mathcal{T} = \frac{\gamma_1}{\gamma_2} \frac{\gamma_2 -1}{\gamma_1 - 1} +
\frac{\gamma_2 -1}{2\gamma_2} \frac{\mathcal{R}+1}{\mathcal{R}-1}M_0^2 -
\frac{(\gamma_2 -1)Q}{\gamma_2 c^2},
\end{equation}
where $\mathcal{T} \equiv \mathcal{K}_2 T_2 / \mathcal{K}_1 T_1$ and
$\mathcal{R} \equiv \rho_2 / \rho_1$ are the relative temperature
and density of the post-shock gas, respectively. Eliminating
$\mathcal{T}$ between the equations (\ref{temp1}) and (\ref{temp2}),
gives the square equation
\begin{equation}\label{cubicR}
\mathcal{R}^2 +A_1 \mathcal{R} +A_2 =0,
\end{equation}
with coefficients as follows
\begin{eqnarray}\label{coeff}
\nonumber A_1 &=& -\frac{\frac{\gamma_2+1}{\gamma_2-1}M_0^2
+ \frac{2\gamma_1}{\gamma_1-1}+ \frac{2\gamma_2}{\gamma_2-1}- \frac{2Q}{c^2}}{M_0^2+ \frac{2\gamma_1}{\gamma_1-1}
- \frac{2Q}{c^2}}\\
\nonumber A_2 &=& \frac{\frac{2\gamma_2}{\gamma_2-1}}{M_0^2+ \frac{2\gamma_1}{\gamma_1-1}
- \frac{2Q}{c^2}}.
\end{eqnarray}
In the general case, the solution of equation (\ref{cubicR}) can
directly be found according to four parameters $\gamma_1$,
$\gamma_2$, $Q$ and $M_0$; then the relative temperature
$\mathcal{T}$ is obtained via equations (\ref{temp1}) or
(\ref{temp2}). The adiabatic shock is a special case with $Q=0$,
which in the strong supersonic collision ($M_0\rightarrow \infty$),
the equation (\ref{cubicR}) leads to $\mathcal{R}=4$ with assumption
of $\gamma_2=5/3$. In this case, the relative temperature is
limitless as $\mathcal{T} \approx M_0^2/3$. In fact, this
temperature is the maximum allowed value in a shock process, which
is $T_2^{max} = 3.86 \mu_2 \times 10^5 v_{07}^2 \mathrm{K}$ where
$v_{07} \equiv v_0/100 \mathrm{km.s^{-1}}$ (e.g., Hollenbach \&
McKee~1979). The most important parameter in the strong supersonic
shocks ($M_0>>1$) is the energy lost per unit mass during the shock
process, $Q=(n_2 \Lambda/\mu_2 m_H) t_{dur}$, where $\Lambda$
($\mathrm{erg.cm^{3}.s^{-1}}$) is the cooling function at the
post-shock region with density $n_2$, and $t_{dur}$ is the duration
time of the post-shock gas. Accurate determination of the cooling
time-scale requires specifying the elemental abundance of the
post-shock region, but a simple estimate can be obtained using
$t_{cool}\approx kT_2/(n_2\Lambda)$. Eliminating the $n_2\Lambda$,
we approximately have $Q/c^2\approx T (t_{dur}/t_{cool})$. If the
post-shock gas cools rapidly (i.e., $t_{cool}<<t_{dur}$), its
temperature cannot be grater than the molecular dissociation energy,
while in slow cooling rate (i.e., $t_{cool}>>t_{dur}$), the
temperature may even be grater than $10^4 \mathrm{K}$ causing the
atomic ionization process.
The cooling mechanisms take into account many different processes
that dominate in different ranges of temperature. We apply the
cooling function as outlined in the Figure~1 of Heitsch, Hartmann \&
Burkert~(2008) in which they used a combination of the rates quoted
by Dalgarno and McCray~(1972) and Wolfire et al.~(1995) for $T <
10^4 \mathrm{K}$, and the tabulated curves of Sutherland and
Dopita~(1993) for $T > 10^4 \mathrm{K}$. We can express the
logarithm of cooling function as a piecewise linear function of the
temperature logarithm as follows
\begin{equation}\label{cool}
\Lambda = \Lambda_0 \left( \frac{T}{T_0} \right)^\beta,
\end{equation}
where $\Lambda_0$, $T_0$ and $\beta$ are given in Table~1 for
ionization degree $x_i=0.1$ and metallicities corresponding to the
solar neighborhood. In this way, the cooling time-scale can be
expressed in a piecewise form as
\begin{equation}\label{cooltime}
t_{cool}\approx \frac{kT_0^\beta}{n_2 \Lambda_0} T^{1-\beta},
\end{equation}
which is shown in Fig.~\ref{coolt} for $n_2 \sim 10^4
\mathrm{cm^{-3}}$.
\begin{table}
\begin{center}
\caption{Parameters for the piecewise linear expression of the
cooling function, $\Lambda=\Lambda_0(T/T_0)^\beta$, with ionization
degree $x_i=0.1$ and with metallicities corresponding to the solar
neighborhood.\label{tbl}}
\begin{tabular}{|c|c|c|c|}
\hline\hline $\log T$ & $\Lambda_0 (\mathrm{erg.cm^3.s^{-1}})$ &
$T_0 (\mathrm{K})$ &
$\beta$ \\
\hline
$1.00 \rightarrow 1.75$ & $1.74\times10^{-28}$ & $10$ & $3.59$\\
$1.75 \rightarrow 3.75$ & $8.51\times10^{-26}$ & $56$ & $0.47$\\
$3.75 \rightarrow 4.21$ & $7.41\times10^{-25}$ & $5623$ & $5.11$\\
$4.21 \rightarrow 4.47$ & $1.66\times10^{-22}$ & $16218$ & $-0.42$\\
$4.47 \rightarrow 4.94$ & $1.29\times10^{-22}$ & $29512$ & $1.89$\\
$4.94 \rightarrow 5.35$ & $1.00\times10^{-21}$ & $87096$ & $0.17$ \\
$5.35 \rightarrow 5.68$ & $1.17\times10^{-21}$ & $223872$ & $-2.36$\\
$5.68 \rightarrow 6.17$ & $1.95\times10^{-22}$ & $478630$ & $-0.51$ \\
$6.17 \rightarrow 6.88$ & $1.10\times10^{-22}$ & $1479109$ & $-0.87$\\
$6.88 \rightarrow 7.38$ & $2.63\times10^{-23}$ & $7585778$ & $-0.26$\\
$7.38 \rightarrow 8.00$ & $1.95\times10^{-23}$ & $23988340$ & $0.21$\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{coolt.eps}
\caption{The cooling time-scale as expressed in a piecewise linear
form for ionization degree $x_i=0.1$ and metallicities corresponding
to the solar neighborhood. The dotted curve is the B-spline fit to
this piecewise linear expression. The temperatures in which the
molecules and atoms dissociate/form are depicted as arrows, and the
minimum value of cooling time-scale is nominated as trap.}
\label{coolt}
\end{center}
\end{figure}
When the strong shocking molecular gases proceed, at the first time,
the temperature increases near $T_2^{max}$ so that the molecules
will be dissociated and the atoms may be ionized. In the ionized
region, there is a trap in which $t_{cool}$ reaches to its minimum
value so that the plasma gas cools rapidly to recombine the
electrons and nucleons. Recombination becomes important when the
post-shock plasma cools to $10^4 \mathrm{K}$; a typical $H$ atomic
recombination time-scale is approximately $t_{rec} \approx
1/n_e\alpha_{rec}\approx10^5 /n_e (\mathrm{cm^{-3}}) \mathrm{yr}$,
where $\alpha_{rec}$ is the total recombination coefficient and
$n_e$ is the number density of electron (Osterbrock \& Ferland
2006). If the process of cooling mode continues, the atomic
post-shock gas will approach to the molecular gas. The $H_2$
molecule cannot form in the gas phase because it is not lonely able
to radiate away the excess energy of formation. The grains, which
have been survived in the shock process, can thus transfer this
extra energy of $H_2$ formation to their phonons (Hollenbach \&
Salpeter~1971). Although, the details of this mechanism are rather
complex, a simple estimation of $H_2$ formation time-scale is
$t_{mol} \approx 1/n_H k_m \approx 10^9/n_H^3 (\mathrm{cm^{-3}})
\mathrm{yr}$, where $k_m \approx 3\times 10^{-17}n_H(n_H+2n_{H_2})$
$\mathrm{cm^3.s^{-1}}$ is the hydrogen molecule formation rate on
the grain surfaces and $n_H$ is the hydrogen atomic number density
(Lequeux~2005). For a typical J-shock structure of a protostellar
accretion shocks, with $n_e \approx n_H \approx 10^4
\mathrm{cm^{-3}}$, we have $t_{rec} \approx 10 \mathrm{yr}$ and
$t_{mol} \approx 10^{-3} \mathrm{yr}$, thus the recombination occurs
slowly while the molecule formation occurs very fast. Although, we
find some insights for J-shock structure and its relaxation process,
but the shock process is rather a complex non-equilibrium thermal
and chemical mechanism. Thus, in the next section, we use the SPH
simulation to study the time evolution of processes in the strong
shocking molecular gases.
\section{Investigation by SPH}
We consider the head-on collisions of two molecular gas sheets. An
initial negative velocity is given to the particles with a positive
$x$-coordinate, and a velocity in the opposite direction to those
with a negative $x$-coordinate. Compositions of the sheets are
assumed as global neutral which consists of a mixture of atomic and
molecular hydrogen ($X\approx0.75$), helium ($Y\approx0.25$), and
traces of CO and other rare molecules. The mean molecular weight is
initially $1/\mu = X/2 + Y/4 \approx 0.4375$ for molecular case
($T<10^2 \mathrm{K}$), and in the post-shock region for simplicity
is assumed to be $1/\mu = X + Y/4 \approx 0.8125$ for atomic case
($10^2 \mathrm{K}< T < 10^4 \mathrm{K}$), and $1/\mu = 2X + 3Y/4
\approx 1.6875$ for ionized case ($T> 10^4\mathrm{K}$). The same
procedure is followed for the ratio of specific heats: $\gamma=7/5$
for diatomic molecular gas and $\gamma=5/3$ for atomic and ionized
cases. The chosen physical scales for length and time are $[l]=3.0
\times 10^{14} \mathrm{cm}=20 \mathrm{AU}$ and $[t]=3.0 \times
10^{7} \mathrm{s} = 1 \mathrm{yr}$, respectively, so the velocity
unit is approximately $100~\mathrm{km.s^{-1}}$. The gravitational
constant is set $G= 10^{-12} [l]^{3}.[t]^{-2}.[m]^{-1}$ for which
the calculated mass unit is $[m]=4.5 \times 10^{23} \mathrm{g}$.
Consequently, the physical scales for density and energy are
$[\rho]=1.6\times 10^{-20} \mathrm{g.cm^{-3}}= 10^{4} \mathrm{H.
cm^{-3}}$ and $[e]=4.5\times 10^{30}\mathrm{erg}$, respectively. Two
equal one dimensional molecular sheets with extension $x= 0.1 [l]$
is considered, which have initial uniform density and temperature of
$10^4 \mathrm{cm^{-3}}$ and $\sim 10 \mathrm{K}$, respectively.
The SPH method is well suited to address unbound astrophysical
problems, especially the behavior of gases subjected to compression
(Rosswog~2009). A worthy review of the SPH methodology and its
applications can be found in Monaghan~(2005). In this method, fluid
is represented by $N$ discrete but extended/smoothed particles (i.e.
Lagrangian sample points). The particles are overlapping, so that
all the involved physical quantities can be treated as continuous
functions both in space and time. Overlapping is represented by the
kernel function, $W_{ab} \equiv
W(\textbf{r}_a-\textbf{r}_b,h_{ab})$, where $h_{ab} \equiv
(h_a+h_b)/2$ is the mean smoothing length of two particles $a$ and
$b$. The density is estimated via usual summation over neighboring
particles,
\begin{equation}\label{sphden}
\rho_a=\sum_b m_b W_{ab},
\end{equation}
the acceleration equation in the one-dimensional usual symmetric
form is
\begin{equation}\label{sphacc}
\frac{d v_a}{dt}=-\sum_b m_b (\frac{p_a}{\rho_a^2}+
\frac{p_b}{\rho_b^2}+ \Pi_{ab}) \frac{dW_{ab}}{dx_a},
\end{equation}
and the SPH equivalent of the energy equation is
\begin{eqnarray}\label{sphenergy}
\nonumber \frac{du_a}{d t}=\frac{1}{2} \sum_b m_b
(\frac{p_a}{\rho_a^2} +\frac{p_b}{\rho_b^2} +\Pi_{ab}) v_{ab}\frac{d
W_{ab}}{d x_a}\\ -\frac{\Lambda_0}{(\mu_a m_H)^2} \rho_a \left(
\frac{T_a}{T_0}\right)^\beta,
\end{eqnarray}
where $u_a=\frac{1}{\gamma_a-1}\mathcal{K}_aT_a$ is the thermal
energy per unit mass, and $\Pi_{ab}$ is the artificial viscosity
between particles $a$ and $b$
\begin{equation}\label{av}
\Pi_{ab}=\cases{
\frac{-\alpha^* v_{sig} \mu_{ab}^*
+\beta^* \mu_{ab}^{*2}}{\bar{\rho}_{ab}}, &
if $\textbf{v}_{ab}.\textbf{r}_{ab}<0$,\cr
0 , & otherwise,}
\end{equation}
where $\textbf{v}_{ab}\equiv \textbf{v}_a - \textbf{v}_b$ and
$\textbf{r}_{ab}\equiv \textbf{r}_a - \textbf{r}_b$ are relative
velocity and the distance of particles, $\bar{\rho}_{ab}=
\frac{1}{2}(\rho_a+\rho_b)$ is an average density, $v_{sig} =
(c_a+c_b)/2$ is signal velocity where $c_a$ and $c_b$ are the sound
speed of particles, and $\mu_{ab}^*$ is defined as its usual form
\begin{equation}\label{muab}
\mu_{ab}^*=\frac{\textbf{v}_{ab} \cdot\textbf{r}_{ab}}{h_{ab}}
\frac{1}{r_{ab}^2/h_{ab}^2+\eta^2}
\end{equation}
with $\eta\sim 0.1$.
To prevent unphysical solutions with inter-particle penetration and
unwanted heating, we use $\alpha^*$ and $\beta^*$ in the form of
variables with respect to time,
\begin{equation}\label{alpha}
\frac{d \alpha^*_a}{dt} = - \frac{\alpha^*_a -
\alpha_{\mathrm{min}}} {\tau_a} + z_\alpha \mathcal{S}_a,
\end{equation}
and
\begin{equation}\label{beta}
\frac{d \beta^*_a}{dt} = - \frac{\beta^*_a - \beta_{\mathrm{min}}}
{\tau_a} + z_\beta \mathcal{S}_a,
\end{equation}
respectively, where $\mathcal{S}_a=\max (- \frac{dv_a}{dx_a},
\frac{\dot{\rho}_a}{\rho_a} - \frac{\dot{h}_a}{h_a} -
\frac{v_{\mathrm{sig}}}{r_{ab}}, 0)$ is the restricted source term,
$\tau_a \equiv h_a / (\mathcal{C} v_{sig})$ with $0.1 < \mathcal{C}
< 0.2$ is the decay time-scale, and the parameters $z_\alpha$ and
$z_\beta$ are chosen to regulate the effect of source term so that
the heat production and post-shock oscillations are controlled in
the numerical simulations (Nejad-Asghar, Khesali \& Soltani~2008).
Here, we choose $\alpha_{min}=1$, $\beta_{min}=2$,
$\mathcal{C}=0.2$, $z_\alpha=3$, and $z_\beta=0.01$ for the best
smoothed results of simulations (see, e.g., Fig.~\ref{tempdenpos}).
Clearly, comparison between numerical and analytical results in
Fig.~\ref{tempdenpos} verifies the accuracy and physical consistency
of the employed numerical tool.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{temppos.eps}\\(a)\\
\includegraphics[width=80mm]{denpos.eps}\\(b)\\
\caption{(a) Temperature and (b) density of the adiabatic shock
($\Lambda_0 = 0$) in the head-on collision of two sheets with
initial Mach number $M_0=500$. The solid curves are derived with
variable viscosity (\ref{alpha}) and (\ref{beta}), while the dotted
ones are from the common artificial viscosity of Monaghan~(1989)
with $\alpha^*=1$ and $\beta^*=2$. The analytical results of strong
supersonic adiabatic shocks are depicted by
arrows.}\label{tempdenpos}
\end{center}
\end{figure}
The width of the post-shock region increases by time, until it
reaches to all extensions of the simulated sheets, or the
temperature of center of the shocked region cools less than
$10\mathrm{K}$. For computer experiments which are considered here,
with initial extension $x = 0.1 [l]$, we stop the simulation when
$90\%$ of total SPH particles enter to the post-shock region or the
temperature of center of the shocked region cools less than
$10\mathrm{K}$. In this simulation epoch, the cooling rate affects
on the post-shock temperature as shown in Fig.~\ref{tempmach} for
various Mach numbers. In this figure, the relationship $T_2 =
\frac{\mu_2}{\mu_1} \frac{T_1}{3} M_0^2$, as mentioned for the
strong supersonic adiabatic shocks, is depicted by dash-line. We see
the cooling rate causes to decrease the value of post-shock
temperature. The effect of cooling rate appears more clear in the
Mach numbers which cause to set the post-shock temperature in the
trap region as outlined by Fig.~\ref{tempdenpos}. Since for larger
Mach numbers, the simulation epoch (i.e, the time in which $90\%$ of
SPH particles of our simulation enter to the post-shock region or
the temperature of center of the shocked region cools less than
$10\mathrm{K}$) is shorter, the post-shock temperature reaches
asymptotically to the adiabatic case.
\begin{figure}
\includegraphics[width=80mm]{tempmach.eps} \caption{Post-shock
temperature at the simulation epoch for different initial Mach
numbers. The adiabatic case, $T_2 \propto M_0^2$, is shown by
dash-line.}\label{tempmach}
\end{figure}
In the general case, the high temperature of the post-shock region
lead to splash it into the medium. But, in the accretion shock, the
infall matter confides the shocked gas so that it may cool and end
up with an accretion rotating disk. For finding the relaxation time,
we use the equation (\ref{sphenergy}) with $v_{ab}\approx 0$, thus,
we have
\begin{equation}\label{temprelax}
\frac{1}{\gamma_a-1}\mathcal{K}_a\frac{dT_a}{d t}=
-\frac{\Lambda_0}{(\mu_a m_H)^2} \rho_a \left(
\frac{T_a}{T_0}\right)^\beta.
\end{equation}
The temperature of the accreted shocked gas can be obtained by
integrating the equation (\ref{temprelax}). The result is in a
piecewise change from $T_{a1}$ (at $t_1$) to $T_{a2}$ (at $t_2$) as
follows:
\begin{eqnarray}\label{tempchange}
\nonumber T_{a2}=\{ T_{a1}^{1-\beta}- \frac{(1-\beta)(\gamma_a-1)}{\mathcal{K}_a}
\frac{\Lambda_0}{(\mu_a m_H)^2}\times \\ \frac{\rho_a}{T_0^\beta} (t_2-t_1)
\}^\frac{1}{1-\beta},
\end{eqnarray}
which is shown in Fig.~\ref{temptime}, with assumption of
$\rho_a\approx 4$.
\begin{figure}
\includegraphics[width=80mm]{relaxtemp.eps} \caption{Relaxation of the
temperature of the post-shock gas in the head-on collision of two
sheets.}\label{temptime}
\end{figure}
\section{Summary and conclusions}
Molecular cloud cores are rotating so that in the process of
collapse and protostellar formation, the infalling matters, which
arrive at the equator, collide and dissipate the kinetic energy of
motion perpendicular to the equatorial plane so that an accretion
disk can be formed. The speed of infalling matters at the equator is
so high that the strong supersonic shocks appear, and the
temperature of post-shock gas is increased so causing to dissociate
the molecules and ionize the atoms. In adiabatic strong supersonic
shocks, the density of post-shock region is about fourfold of
initial density, and the temperature is increased proportional to
the square of the Mach number. On the other hand, the cooling
processes of the post-shock gas can decrease the temperature so that
the ionized gases can be recombined to form the atoms and molecules,
if the duration time-scale of the post-shock region is longer than
the cooling time-scale. The suitable cooling function for the
post-shock gas (Table~1) showed that the cooling time-scale has a
minimum which is at the ranges of ionizing temperature
(Fig.~\ref{coolt}). This minimum, which is nominated as a trap,
causes to cool the post-shock gas in a fast rate. Thus, if the
initial speed of colliding matters is so high that the temperature
of post-shock gas settles in this trap region, it will quickly cool
and electrons recombine with nucleons.
For investigating the time evolution of the post-shock gas in the
strong supersonic collisions, which occurs at protostellar accretion
shocks, we used the SPH method with the variable artificial
viscosity to obtain the more smoothed results of simulations
(Fig.~\ref{tempdenpos}). The simulations of strong shocks with great
Mach numbers show that the temperature of the post-shock gas is
quickly increased near to $T_2^{max}$, which is for the adiabatic
case, and gradually decreased by the time according to the cooling
rate. Decrease of temperature of the post-shock gas at the
simulation epoch is shown in Fig.\ref{tempmach} for different values
of the Mach numbers. We see from this figure that the decreasing
rate of temperature is very fast in the trap region as depicted by
Fig.~\ref{coolt}.
The simulations show that the temperature of center of the
post-shock gradually decreases, while in the ridges, it stays about
$T_2^{max}$ because of continuously infall of matters. Temperature
decrease of the central region leads to increase of its density in
an isobaric manner so that an accretion thin disk can be formed.
Thus, over the time, center of the collisional infalling matters
(i.e., equatorial plane) converts to a dense molecular thin disk
with an atomic envelope and ionized gas which comes from strong
shocks of continuous infalling matters. The time in which this
structure occurs, depends on the Mach number that is shown in
Fig.~\ref{temptime}. The temperature range of the trap, which leads
to fast cooling of the post-shock gas, is clearly seen in
Fig.~\ref{temptime}, too. Thus, we see that the cooling processes of
the post-shock gas in the protostellar accretion shocks can lead to
the formation of an accretion thin molecular disk at the equatorial
plane, in a convenient time-scale. This dense molecular thin disk is
appropriate for grain coagulation and formation of proto-planetary
entities.
\section*{Acknowledgments}
This work has been supported by grant of Research and Technology
Deputy of University of Mazandaran.
|
1,314,259,996,719 | arxiv | \section{Introduction}
\begin{figure*}[]
\centering
\includegraphics[height=0.22\textheight,width=0.95\linewidth]{isovec.png}
\caption{Proposed Method. Loss is a weighted combination of skipgram with negative sampling loss (seen left with a reproduction of the familiar image from \citet{mikolov2013efficient} for reader recognizability) and an isomorphism loss (seen right, ours) calculated in relation to a fixed reference space. Gray boxes are two possibilities explored in this work: Proc-L2 (supervised) where $L_{ISO}$ is calculated over given seed translations, and RSIM-U (unsupervised).}
\label{fig:isovec}
\end{figure*}
The task of extracting a translation dictionary from word embedding spaces, called ``bilingual lexicon induction'' (BLI), is a common task in the natural language processing literature.
Bilingual dictionaries are useful in their own right as linguistic resources, and automatically generated dictionaries may be particularly helpful for low-resource languages for which human-curated dictionaries are unavailable.
BLI is also used as an extrinsic evaluation task to assess the quality of cross-lingual spaces.
If a high-quality translation dictionary can be automatically extracted from a shared embedding space, intuition says that the space is high-quality and useful for downstream tasks.
``Mapping-based'' methods are one way to create cross-lingual embedding spaces.
Separately-trained monolingual embeddings are mapped to a shared space by applying a linear transformation to one or both spaces, after which a bilingual lexicon can be extracted via nearest-neighbor search
\cite[e.g.,][]{mikolov2013, lample2018word, artetxe-etal-2018-robust, joulin-etal-2018-loss,patra-etal-2019-bilingual}.
Mapping methods are effective for closely-related languages with embedding spaces trained on high-quality, domain-matched data even without supervision, but critically rely on the ``approximate isomorphism assumption''---that monolingual embedding spaces are geometrically similar.\footnote{In formal mathematicals, “isomorphic” requires two objects to have an invertible correspondence between them. Researchers in NLP loosen the definition to ``geometrically similar'', and consider \textit{degrees} of similarity.
We might say that space X is \textit{more isomorphic} to space Y than is space Z.}
Problematically, researchers have observed that the isomorphism assumption weakens substantially as languages and domains become dissimilar, leading to failure precisely where unsupervised methods might be helpful \cite[e.g.][]{sogaard-etal-2018-limitations,ormazabal-etal-2019-analyzing,glavas-etal-2019-properly, vulic-etal-2019-really,patra-etal-2019-bilingual,marchisio-etal-2020-unsupervised}.
Existing work attributes non-isomorphism to linguistic, algorithmic, data size, or domain differences in training data for source and target languages.
From \citet{sogaard-etal-2018-limitations}, ``the performance of unsupervised BDI [BLI] depends heavily on... language pair, the comparability
of the monolingual corpora, and the parameters of the word embedding algorithms.'' Several authors found that unsupervised machine translation methods suffer under similar data shifts \cite{marchisio-etal-2020-unsupervised, kim2020and,marie-unsup-mt-2020}.
While such factors do result in low isomorphism of spaces trained with traditional methods, we needn't resign ourselves to the mercy of the geometry a training methodology naturally produces.
While multiple works post-process embeddings or map non-linearly, we control similarity explicitly during embedding training by incorporating five global metrics of isomorphism into the skipgram loss function.
Our three supervised and two unsupervised losses gain some control of the relative isomorphism of word embedding spaces, compensating for data mismatch and creating spaces that are linearly mappable where previous methods failed.
\section{Related Work}
\paragraph{Cross-Lingual Word Embeddings}
There is a broad literature on creating cross-lingual word embedding spaces. Two major paradigms are ``mapping-based'' methods which find a linear transformation to map monolingual embedding spaces to a shared space \cite[e.g.,][]{artetxe-etal-2016-learning, artetxe-etal-2017-learning,alvarez-melis-jaakkola-2018-gromov,doval-etal-2018-improving, jawanpuria-etal-2019-learning},
and ``joint-training'' which, as stated in the enlightening survey by \citet{ruder2019survey}, ``minimize the source and target language monolingual losses jointly with the cross-lingual regularization term'' \cite[e.g.][\citet{ruder2019survey} for a review]{luong-etal-2015-bilingual}. \citet{gouws-2015} train skipgram for source and target languages simultaneously, enforcing an L2 loss for known translation. \citet{Wang*2020Cross-lingual} compare and combine joint and mapping approaches.
More recently, researchers have explored massively multilingual language models \cite{devlin-etal-2019-bert, conneau-etal-2020-unsupervised}.
While these have been shown to possess some inherent cross-lingual transfer ability \cite{wu-dredze-2019-beto}, another line of work focuses on improving their cross-lingual representations with explicit cross-lingual signal \cite{wang-etal-2019-cross, liu-etal-2019-investigating, Cao2020Multilingual, kulshreshtha-etal-2020-cross, wu-dredze-2020-explicit}. Recently, \citet{li-etal-2022-improving} combined static and pretrained multilingual embeddings for BLI.
\paragraph{Handling Non-Isomorphism}
\citet{miceli-barone-2016-towards} explore whether comparable corpora induce embedding spaces which are approximately isomorphic.
\citet{ormazabal-etal-2019-analyzing} compare cross-lingual word embeddings induced via mapping methods
and jointly-trained embeddings from \citet{luong-etal-2015-bilingual}, finding that the latter are better in measures of isomorphism and BLI precision.
\citet{nakashole-flauger-2018-characterizing} argue that word embedding spaces are not globally linearly-mappable. Others use non-linear mappings \cite[e.g.][]{mohiuddin-etal-2020-lnmap, glavas-vulic-2020-non} or post-process embeddings after training to improve quality \citep[e.g.][]{peng-etal-2021-cross, faruqui-etal-2015-retrofitting, mu2018allbutthetop}. \citet{eder-etal-2021-anchor} initialize a target embedding space with vectors from a higher-resource source space, then train the low-resource target.
\citet{zhang-etal-2017-earth} minimize earth mover's distance over 50-dimensional pretrained word2vec embeddings. \citet{ormazabal-etal-2021-beyond} learn source embeddings in reference to fixed target embeddings given known or hypothesized translation pairs induced during via self-learning.
\paragraph{Examining \& Exploiting Embedding Geometry} Emerging literature examines geometric properties of embedding spaces. In addition to isomorphism, some examine \textit{isotropy} \cite[e.g.][]{mimno-thompson-2017-strange,mu2018allbutthetop,ethayarajh-2019-contextual, rajaee-pilehvar-2022-isotropy, rudman-etal-2022-isoscore}.
\citet{li-etal-2020-sentence} transform the semantic space from masked language models into an isotropic Gaussian distribution from a non-smooth anisotropic space. \citet{bert_whitening} apply whitening and dimensionality reduction to improve isotropy.
\citet{zhang-etal-2022-effect} inject isotropy into a variational autoencoder, and \citet{ethayarajh-jurafsky-2021-attention} recommend ``adding an anisotropy penalty to the language modelling objective'' as future work.
\section{Background}
We discuss the mathematical background used in our methods.
Throughout, $\bf{X} \in \mathbb{R}^{n \times d}$ and $\bf{Y} \in \mathbb{R}^{m \times d}$ are the source and target word embedding spaces of $d$-dimensional word vectors, respectively. We may assume seed pairs $\{(x_0, y_0), (x_1, y_1),... (x_s, y_s)\}$ are given.
\subsection{The Orthogonal Procrustes Problem}\label{sec:proc}
\citet{schonemann1966generalized} derived the solution to the orthogonal Procrustes problem, whose goal is to find the linear transformation $W$ that solves:
\begin{equation*}
\argmin_{W \in \mathbb{R}^{d \times d}, W^T W=I} ||X W-Y||_F^2
\label{eq:proc-eq}
\end{equation*}
The solution is $W = VU^T$, where $U\Sigma V^T$ is the singular value decomposition of $Y^TX$. If $X$ is a matrix of vectors corresponding to seed words $x_i$ in $\{(x_0, y_0), (x_1, y_1), \ldots, (x_s, y_s)\}$ and $Y$ is a matrix of the corresponding $y_i$, then $W$ is the linear transformation that minimizes the difference between the vector representations of known pairs.
\subsection{Embedding Space Mapping with VecMap}
We use the popular VecMap\footnote{\url{https://github.com/artetxem/vecmap}} toolkit for embedding space mapping, which can be run in supervised, semi-supervised, and unsupervised modes. As of the time of its writing, \citet{glavas-etal-2019-properly} deem VecMap the most robust unsupervised method.
First, source and target word embeddings are unit-normed, mean-centered, and unit-normed again \cite{zhang-etal-2019-girls}. The bilingual lexicon is induced by whitening each space and then solving a variant of the orthogonal Procrustes problem.\footnote{See Appendix \ref{sec:app-white}, \ref{sec:app-ortho} for details} Spaces are reweighted, dewhitened, dimensionality reduced, and translation pairs are extracted via nearest-neighbor search from the mapped embedding spaces. See the original works and implementation for details \cite{artetxe2018generalizing}.
Unsupervised and semi-supervised modes utilize the same framework as supervised mode, but with an iterative self-learning procedure that repeatedly solves the orthogonal Procrustes problem over hypothesized translations. On each iteration, new hypotheses are extracted.
The modes differ only in how they induce the initial hypothesis seed pairs.
In semi-supervised mode, this is a given input seed dictionary. In unsupervised mode, similarity matrices $M_x=XX^T$ and $M_z=ZZ^T$ are created over the first $n$ vocabulary words.\footnote{Default: 4000} Word $z_j$ is the assumed translation of $x_i$ if vector $M_{z_j}$ is most similar to $M_{x_i}$ compared to all others in $M_z$. See \citet{artetxe-etal-2018-robust} for details.
\subsection{Isomorphism Metrics}
In NLP, relative isomorphism is often measured by Relational Similarity, Eigenvector Similarity, and Gromov-Hausdorff Distance. We describe these metrics in detail in this section.
\paragraph{Relational Similarity} Given seed translation pairs, calculate pairwise cosine similarities:
\begin{footnotesize}
\begin{gather*}
\text{cos}(x_0, x_1) \qquad \text{cos}(y_0, y_1) \\
\text{cos}(x_0, x_2) \qquad \text{cos}(y_0, y_2) \\
\text{cos}(x_0, x_3) \qquad \text{cos}(y_0, y_3) \\
\dots \qquad \qquad \dots \\
\text{cos}(x_1, x_0) \qquad \text{cos}(y_1, y_0) \\
\text{cos}(x_1, x_2) \qquad \text{cos}(y_1, y_2) \\
\dots \qquad \qquad \dots \\
\text{cos}(x_s, x_s) \qquad \text{cos}(y_s, y_s)
\end{gather*}
\end{footnotesize}
The Pearson's correlation between the lists of cosine similarities is known as Relational Similarity \cite{vulic-etal-2020-good, zhang-etal-2019-girls}.
\paragraph{Eigenvector Similarity} \cite{sogaard-etal-2018-limitations} measures isomorphism between two spaces based on the Laplacian spectra of their $k$-nearest neighbor ($k$-NN) graphs.
For seeds $\{x_0, x_1, \hdots, x_s\}$ and $\{y_0, y_1, \hdots, y_s\}$, we compute unweighted $k$-NN graphs $G_X$ and $G_Y$, then compute the Graph Laplacians ($L_G$) for both graphs (the degree matrix minus the adjacency matrix: $L_G = D_G - A_G$). We then compute the eigenvalues of $L_{G_X}$ and $L_{G_Y}$, namely $\{\lambda_{L_{G_X}}(i)\}$ and $\{\lambda_{L_{G_Y}}(i)\}$. We select $l = \min(l_X, l_Y)$ where $l_X$ is the maximum $l$ such that the first $l$ eigenvalues of $L_{G_X}$ sum to less than 90\% of the total sum of the eigenvalues. EVS is the sum of squared differences between the partial spectra:
\begin{equation*}
\text{EVS} = \sum_{i=1}^l (\lambda_{L_{G_X}}(i) - \lambda_{L_{G_Y}}(i))^2
\end{equation*}
The Laplacian allows one to decompose a graph function into a sum of weighted Laplacian eigenvectors, which roughly corresponds to a frequency-based decomposition of the function. The weights (eigenvalues) determine the contribution of each eigenvector to the final function. Graphs with similar Laplacian eigenvalues should have similar structure, which is what EVS aims to capture.
\paragraph{Gromov-Hausdorff Distance} is a “worst-case” metric that optimally linearly maps embedding spaces and then calculates the distance between nearest neighbors in a shared space.
\begin{itemize}[]
\item For each $x$ of the source embeddings, find its nearest neighbor $y$ of the target embeddings. Measure the distance.
\item For each $y$ of the target embeddings, find its nearest neighbor $x$ of the source embeddings. Measure the distance.
\item \textit{Hausdorff distance} is the worst of the above.
\item \textit{Gromov-Hausdorff distance} is Hausdorff distance after optimal isometric transformation to minimize distances. As in previous work, since we apply mean-centering to source and target embeddings, we search only over the space of orthogonal transformations \cite{patra-etal-2019-bilingual, vulic-etal-2020-good}. See Figure \ref{fig:gh}.
\end{itemize}
\noindent We follow \citet{chazal-2009-bottleneck} and approximate the Gromov-Hausdorff distance with the Bottleneck distance between the source and target embeddings.
\begin{figure}[]
\centering
\includegraphics[height=0.12\textheight,width=1\linewidth]{gh_dist.png}
\caption{Calculation of Gromov-Hausdorff (GH) Distance: the worst case distance of nearest neighbors in a shared embedding space after optimal orthogonal mapping. The right-most red dots have been orthogonally rotated to turn Hausdorff distance into GH Distance.}
\label{fig:gh}
\end{figure}
\section{Method}
We implement skipgram with negative sampling on GPU using PyTorch and use it to train monolingual embedding spaces for Bengali (bn), Ukranian (uk), Tamil (ta), and English (en).\footnote{Dim: 300, window: 5, negative samples: 10, min\_count: 10, batch size: 16384, LR: 0.001. Adam with linear warmup for 1/4 of batches, then polynomial decay. Run 10 epochs.} Our implementation mirrors the official word2vec\footnote{\url{https://github.com/tmikolov/word2vec}} release closely \cite{mikolov2013efficient}.
We create comparison embedding spaces using the official word2vec release with default hyperparameters and map the resulting spaces from both algorithms with VecMap for BLI. We report precision@1 (P@1) on the development set in Table \ref{tab:baseline}. P@1 is a standard evaluation metric for BLI. Our implementation slightly outperforms word2vec except ta in unsupervised mode.
\begin{table}[htb]
\footnotesize
\begin{center}
\setlength{\tabcolsep}{2.7pt}
\begin{tabular}{@{}c|c|c|c|c||c|c|c||c|c|c@{}}
\toprule
& & \multicolumn{3}{c||}{\bf{bn-en}} & \multicolumn{3}{c||}{\bf{uk-en}} & \multicolumn{3}{c}{\bf{ta-en}}\\
\hline
& & \bf Su & \bf Se & \bf U & \bf Su & \bf Se & \bf U & \bf Su & \bf Se & \bf U \\
\hline
W2V & $\mu$ & 12.6 & 11.9 & 9.0 & 10.8 & 9.7 & 7.7 & 8.4 & 7.3 & 7.1 \\
& $\sigma$ & \itshape 0.3 & \itshape 0.3 & \itshape 5.0 & \itshape 0.5 & \itshape 0.4 & \itshape 4.3 & \itshape 0.5 & \itshape 0.6 & \itshape 0.6 \\
\hline
\hline
Ours & $\mu$ & 13.1 & 12.2 & 10.8 & 12.4 & 11.7 & 10.5 & 9.3 & 8.3 & 1.8 \\
& $\sigma$ & \itshape 1.0 & \itshape 0.4 & \itshape 0.9 & \itshape 0.9 & \itshape 0.5 & \itshape 0.6 & \itshape 0.7 & \itshape 0.9 & \itshape 3.7 \\
\hline
\end{tabular}
\end{center}
\caption{\label{tab:baseline} P@1 of mapped embedding spaces on the development set: our implementation vs. official word2vec release. Shown: Mean ($\mu$) and standard deviation ($\sigma$) over 5 runs in supervised (Su), semi-supervised (Su), and unsupervised (U) modes.}
\end{table}
\begin{table}[htb]
\footnotesize
\begin{center}
\setlength{\tabcolsep}{2.8pt}
\begin{tabular}{@{}ccccc|c|c@{}}
\toprule
& \multicolumn{4}{c|}{{\underline{newscrawl2020}}} & {\underline{Common Crawl}} & {\underline{newscrawl2018-20}} \\
& \textbf{en} & \textbf{bn} & \textbf{ta} & \textbf{uk} & \textbf{en} & \textbf{en} \\
\midrule
& 29.0 & 14.7 & 12.6 & 7.8 & 750 & 2700 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{tab:datasizes} Size of training data (millions of tokens).}
\end{table}
\subsection{Data}
For the main experiments, we train word embeddings on the first 1 million lines from newscrawl2020 for en, bn, and ta \cite{barrault-etal-2020-findings}.\footnote{\url{https://data.statmt.org/news-crawl/}} For uk, we use the entirety of newscrawl2020 ($\sim$427,000 lines). We normalize punctuation, lowercase, remove non-printing characters, and tokenize using standard Moses scripts.\footnote{\url{github.com/moses-smt/mosesdecoder/tree/master/scripts/tokenizer}} Domain mismatch experiments in Section \ref{sec:dom-mismatch} use approximately 33.8 million lines of webcrawl from the English Common Crawl. Larger data experiments in the same section use 93 million lines of English newscrawl2018-2020. The size of the training data in tokens is seen in Table \ref{tab:datasizes}.
We use the publicly available train and test dictionaries from MUSE \cite{lample2018word}.\footnote{\url{https://github.com/facebookresearch/MUSE\#ground-truth-bilingual-dictionaries}} For the development set, we use source words 6501-8000 from the ``full'' set. Train, development, and test sets are non-overlapping. We use all possible training set seed words for our supervised losses, which is 6000-7000 word pairs per language.\footnote{$\sim90\%$ of train set pairs are present in the trained embedding spaces; bn-en: 6859, uk-en: 6476, ta-en: 6019.} We use the test set for evaluating downstream BLI.
\subsection{Integrating Isomorphism Losses}
\label{sec:iso-metrics}
To train the embedding space $\bf{X}$ such that it 1) captures the distributional information via skipgram with negative sampling and 2) is geometrically similar to the reference word embedding space $\bf{Y}$, we propose the objective below.
$\mathcal{L}_{SG}$ is the familiar skipgram with negative sampling loss function and $\mathcal{L}_{ISO}$ is the isomorphism metric loss.
Each $\mathcal{L}_{ISO}$ requires a reference embedding space $\bf{Y}$, trained separately using our base implementation.
\begin{equation*}
\mathcal{L} = (1 - \beta) \, \mathcal{L}_{SG} + \beta\, \mathcal{L}_{ISO}
\end{equation*}
We use English as the reference language because we generally assume that the data quality is higher than the low-resource languages used on the source-side.
$\bf{Y}$ is normalized, mean-centered, and normalized again before use.
On each calculation of $\mathcal{L}_{ISO}$, we perform the same operations on a copy of the current model's word embeddings.
\subsection{Supervised Losses}
\label{sec:sup_losses}
We assume seeds $\{(x_0, y_0), (x_1, y_1),... (x_s, y_s)\}$. $X$ is a matrix of the current model's word embeddings for [$x_0, x_1, ..., x_s$] and $Y$ is the matrix of reference source embeddings for [$y_0, y_1, ..., y_s$].
\paragraph{L2}
We implement L2 distance and normalize over samples. Intuitively, this coaxes translation pairs to have similar vector representations, with the hope that other words in $\bf{X}$ and $\bf{Y}$ will be tugged closer to their translations. \textbf{L2} is easy to implement and understand, and computes quickly.
\[ \frac{1}{|X|}\sum_{i=1}^{s} || x_i - y_i ||_2 \]
\paragraph{Proc-L2} We find $W$ that solves the orthogonal Procrustes problem as in Section \ref{sec:proc}, then minimize L2 distance over the mapped space:
\[ \frac{1}{|X|} \sum_{i=1}^{s} || x_iW - y_i ||_2 \]
\paragraph{Proc-L2+Init}
Same as \textbf{Proc-L2}, except initialize source seed embeddings with the reference translation vectors so that spaces begin with the same representation for known translations.
\paragraph{RSIM} We implement relational similarity over seeds. Higher is better, so we minimize $\mathcal{L}_{ISO} = 1-\text{Pearsons\_Corr}$.
Like \textbf{Proc-L2+Init}, we can also initialize the source space with reference seed embeddings. We call this \textbf{RSIM+Init}.
\subsection{Unsupervised Losses}
\label{sec:unsup_losses}
We use two unsupervised metrics to increase isomorphism when no seed translations are available.
\paragraph{RSIM-U} In this unsupervised variant of \textbf{RSIM}, we calculate pairwise cosine similarities over the first $k$ words in $\bf X$ and $\bf Y$, sort the lists, then calculate Pearson's correlation. As above, $\mathcal{L}_{ISO} = 1-\text{Pearsons\_Corr}$. We use $k=2000$ for efficiency.
\paragraph{EVS-U} We calculate eigenvector similarity over the first 2000 words in $\bf X$ and $\bf Y$.
\subsection{On Differentiability}
Each metric must be differentiable with respect to $\bf X$, a matrix of the model's current word embeddings, to allow isomorphism-based losses to inform parameter updates in $\bf X$.
\textbf{L2} is straightforwardly differentiable, as it is the Frobenius norm of $X-Y$. The same applies for variants \textbf{Proc-L2} and \textbf{Proc-L2+Init}. \textbf{RSIM} is naturally differentiable, seen in the formulation below. For mean-centered cosine similarity vectors\footnote{Mean-centering and cosine similarity are differentiable.} $x_{sim}$ and $y_{sim}$, Pearson's correlation coefficient is:
\begin{align*}
\frac{\sum_i x_{sim}(i) y_{sim}(i)}{\sqrt{\sum_i x_{sim}(i)^2 y_{sim}(i)^2}} = \frac{x_{sim}^Ty_{sim}}{\lVert x_{sim}\rVert \lVert y_{sim}\rVert}
\end{align*}
\textbf{EVS} is not immediately differentiable due to the need for the non-differentiable $k$-NN computation. Instead we modify the graph computation step to use a fully-connected \textit{weighted} graph where the edge weight is the dot product between node vectors.\footnote{$x_i ^T y_j\le 1$, as all vectors are unit-normalized.} With this amended formulation, computing the gradients of Laplacian eigenvalues is possible.
\subsection{$\beta$ and Linear Mapping for BLI}
\label{sec:beta}
Each isomorphism loss may be considered a different method, as each loss may cause the overall framework to behave differently.
Accordingly, we set $\beta$ for each separate loss function based on performance on the development set.\footnote{We try $\beta\in\{0.5, 0.333, 0.2, 0.1, 0.01\}$. For RSIM* and EVS-U, we also try $0.001$. An early L2 run used 0.05, 0.0001.}
After selecting $\beta$, we evaluate and present results only on the test set.
$\beta$s for each method are in Table \ref{tab:beta}.
\begin{table}[htb]
\centering \setlength\tabcolsep{3.5pt}\footnotesize
\begin{tabular}{l|c|cl|cll|c}
\toprule
& L2 & Proc-L2 & +Init & RSIM & +Init & -U & EVS-U \\
\midrule
$\beta$ & 0.1 & 0.333 & 0.2 & 0.01 & 0.001 & 0.1 & 0.333 \\
\bottomrule
\end{tabular}
\caption{$\beta$ parameter for isomorphism losses. Each loss function should be considered a separate method, so $\beta$ is set for each loss based on development set performance. Once $\beta$ is chosen, we evaluate on the test set.}
\label{tab:beta}
\end{table}
VecMap in supervised mode consistently scores higher than semi-supervised mode in all baseline experiments on the development set.
For \textit{IsoVec}, semi-supervised mapping often works best. We thus use VecMap in supervised mode for baselines and semi-supervised mode for \textit{IsoVec} supervised runs.
This sometimes underestimates \textit{IsoVec}'s strength when supervised mapping would have performed better.
For unsupervised experiments and baselines, we map in unsupervised mode. Each is run five times and averaged. \textit{IsoVec} and VecMap use one NVIDIA GeForce GTX 1080Ti GPU.
\section{Experiments \& Results}
We pretrain English embeddings to use as reference space $\bf{Y}$. \textit{IsoVec} trains source space $\bf{X}$.
\subsection{Main Experiments}
\label{sec:main}
For baselines, we train source and target spaces separately for each run using our base implementation.
In experimental conditions, we train the source space with \textit{IsoVec} using each isomorphism loss from Sections \ref{sec:sup_losses} and \ref{sec:unsup_losses}.
In Table \ref{tab:main}, we see that \textit{IsoVec} consistently outperforms the baseline for bn-en and uk-en.
For ta-en, it outperforms with \textbf{Proc-L2+Init} and both unsupervised methods.\footnote{Test set coverage *-to-en; bn: 77\%, uk: 76.8\%, ta: 71.4\%} In terms of training efficiency, L2-based methods perform comparably to the baseline (< 10\% time increase) and RSIM-based methods see a slight time increase ($\sim$10-16\% increase over baseline). EVS-based methods require an expensive eigendecomposition step which causes a $\sim$2.5x time increase over the baseline.
\begin{table}
\centering \setlength\tabcolsep{3.5pt}\footnotesize
\begin{tabular}{l|ll|ll|ll}
\toprule
& \multicolumn{2}{c}{\underline{bn}} & \multicolumn{2}{c}{\underline{uk}} & \multicolumn{2}{c}{\underline{ta}} \\
\midrule
\textbf{Supervised} & $\mu$ & $\sigma$ & $\mu$ & $\sigma$ & $\mu$ & $\sigma$ \\
\hspace{1mm}\textit{Baseline} & \textit{15.2 }& \textit{(0.8) }& \textit{14.4} & \textit{(0.8) }& \textit{11.6} & \textit{(0.4)} \\
\hspace{1mm}L2 & 16.3 & (0.4) & 16.5 & (0.4) & 11.1 & (0.5) \\
\hspace{1mm}Proc-L2 & 16.6 & (0.7) & 16.0 & (0.8) & 10.7 & (0.3) \\
\hspace{5mm}+\textit{Init} & \textbf{16.9} & (0.2) & \textbf{17.1} & (0.6) & \textbf{11.8} & (0.3) \\
\hspace{1mm}RSIM & 16.3 & (0.3) & 15.9 & (0.4) & 10.3 & (0.6) \\
\hspace{5mm}+\textit{Init} & 16.0 & (0.4) & \textbf{17.1} & (0.5) & 11.0 & (0.4) \\
\midrule
\midrule
\textbf{Unsupervised} & & & &&& \\
\hspace{1mm}\textit{Baseline} & 13.2 & (0.6) & 12.6 & (0.5) & 3.2 & (4.4) \\
\hspace{1mm}RSIM-U & \textbf{14.2} & (0.7) & \textbf{14.0} & (0.6) & \textbf{5.4} & (4.9) \\
\hspace{1mm}EVS-U & 13.4 & (0.7) & 13.4 & (0.6) & 5.2 & (4.8) \\
\bottomrule
\end{tabular}
\caption{Main Experiments. Average P@1 ($\mu$) and standard deviation ($\sigma$) over 5 runs of \textit{IsoVec} with isomorphism losses for bn-en, uk-en, ta-en.}
\label{tab:main}
\end{table}
\subsection{Algorithm, Domain, \& Data Mismatch}
\label{sec:dom-mismatch}
\citet{sogaard-etal-2018-limitations} show that mapping methods fail for embeddings trained with different algorithms,
and that BLI performance deteriorates when source and target domains do not match \citep{marchisio-etal-2020-unsupervised}. We test \textit{IsoVec} under algorithm and domain mismatch using the best losses from the main experiments: \textbf{Proc-L2+Init} and \textbf{RSIM-U}. We use $\beta$ as-is from the previous section.
The \textit{IsoVec} base model intends to mirror word2vec closely, but there are likely output differences due to implementation.\footnote{Ours batches on GPU with Adam; word2vec is CPU-only with SGD, no batching.} We map the baseline source embeddings trained in the main experiments to varying en target spaces trained with the official word2vec release, so that algorithms do not match between source and target embedding spaces. We run experiments using the below training data:
\begin{itemize}[]
\item \textit{Algorithm Mismatch}: 1 million lines of en newscrawl2020 (same as main experiments). Shows effect of algorithm mismatch only.
\item \textit{+More Target-Side Data}: 93 million lines of en newscrawl2018-20. Shows effect of target trained with ample in-domain data.
\item \textit{+Domain Mismatch}: 33.8 million lines of en Common Crawl (web-crawl). Shows the effect of different domains in source vs. target.
\end{itemize}
\noindent Table \ref{tab:alg-dom-mismatch-delta} contains baselines for our mismatch experiments and shows the drop in performance compared to Table \ref{tab:main} baselines, where both source and target embedding spaces were trained with the \textit{IsoVec} base model.
This occurs across languages, moderately for supervised baselines, and severely for unsupervised.
The large performance drop given \textit{more} high-quality data of the same domain in unsupervised mode (\textit{+More Target-Side Data}) is surprising given that this target space is \textit{stronger} than the one from only \textit{Algorithm Mismatch}. Perhaps its geometry has changed so considerably because of its additional data and different algorithm that it is too different from the lower-resource source space to be mapped with unsupervised methods.
This should be investigated in future work.
\begin{table*}[htb]
\centering \setlength\tabcolsep{3pt}\footnotesize
\begin{tabular}{l|r@{ }lr|r@{ }lr|r@{ }lr||r@{ }lr|r@{ }lr|r@{ }lr}
\toprule
& \multicolumn{9}{c||}{\underline{Supervised}} & \multicolumn{9}{c}{\underline{Unsupervised}} \\
& \multicolumn{3}{c}{\underline{bn}} & \multicolumn{3}{c}{\underline{uk}} & \multicolumn{3}{c||}{\underline{ta}} & \multicolumn{3}{c}{\underline{bn}} & \multicolumn{3}{c}{\underline{uk}} & \multicolumn{3}{c}{\underline{ta}} \\
& \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\Delta$} & {$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\Delta$} & {$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\Delta$} & {$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\Delta$} & {$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\Delta$} & {$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\Delta$} & {$\sigma$} \\
\midrule
\textit{Main Baseline} & \textit{15.2} & \multicolumn{1}{c}{\textit{-}} & \textit{0.8} & \textit{14.4} & \multicolumn{1}{c}{\textit{-}} & \textit{0.8} & \textit{11.6} & \multicolumn{1}{c}{\textit{-}} & \textit{0.4} & \textit{13.2} & \multicolumn{1}{c}{\textit{-}} & \textit{0.6} & \textit{12.6} & \multicolumn{1}{c}{\textit{-}} & \textit{0.5} & \textit{3.2} & \multicolumn{1}{c}{\textit{-}} & \textit{4.4} \\
\midrule
Algorithm Mism. & 13.6 & \textcolor{red}{\textit{(-1.6)}} & 0.5 & 11.7 & \textcolor{red}{\textit{(-2.7)}} & 0.6 & 9.4 & \textcolor{red}{\textit{(-2.1)}} & 0.5 & 11.9 & \textcolor{red}{\textit{(-1.3)}} & 0.7 & 4.4 & \textcolor{red}{\textit{(-8.2)}} & 6.1 & 2.6 & \textcolor{red}{\textit{(-0.6)}} & 3.6 \\
\hspace{1mm}+More Trg Data & 16.0 & \textcolor{darkgreen}{\textit{(+0.8)}} & 0.7 & 13.5 & \textcolor{red}{\textit{(-0.9)}} & 0.3 & 11.0 & \textcolor{red}{\textit{(-0.6)}} & 0.6 & 5.9 & \textcolor{red}{\textit{(-7.3)}} & 8.0 & 0.0 & \textcolor{red}{\textit{(-12.6)}} & 0.0 & 0.0 & \textcolor{red}{\textit{(-3.2)}} & 0.0 \\
\hspace{1mm}+Domain Mism. & 10.5 & \textcolor{red}{\textit{(-4.7)}} & 0.6 & 10.0 & \textcolor{red}{\textit{(-4.4)}} & 0.4 & 8.4 & \textcolor{red}{\textit{(-3.2)}} & 0.4 & 0.0 & \textcolor{red}{\textit{(-13.2)}} & 0.0 & 0.0 & \textcolor{red}{\textit{(-12.6)}} & 0.0 & 0.0 & \textcolor{red}{\textit{(-3.2)}} & 0.0 \\
\bottomrule
\end{tabular}
\caption{Effect of algorithm and data mismatch in source vs. target embedding spaces. Average P@1 of 5 runs ($\mu$) with $\Delta$ vs. baseline and std.dev. ($\sigma$). Isomorphism losses are \textit{not} used here. Source-side embeddings are trained with our base implementation, target-side with word2vec (algorithm mismatch). \textit{Main Baseline} is from the main experiments, Table \ref{tab:main}. \textit{+More Target-Side Data} (+More Trg Data) uses nearly 100x more data on the target-side than previous experiments. \textit{+Domain Mismatch} uses target embeddings trained on $\sim34$M lines of web crawl.}
\label{tab:alg-dom-mismatch-delta}
\end{table*}
We run \textbf{Proc-L2+Init} and \textbf{RSIM-U} in \textit{Algorithm Mismatch}, \textit{+More Target-Side Data}, and \textit{+Domain Mismatch} conditions as described above. Results are in Table \ref{tab:dom-mismatch}.
In supervised mode, \textit{IsoVec} recovers from algorithm mismatch by 2.7-4.9 points, domain mismatch by 2.5-7.3, and still improves when the target space is trained on $\sim$100x more data.
Whereas \textit{+Domain Mismatch} and \textit{+More Target-Side Data} baselines fail to extract any correct translation pairs in unsupervised mode, \textbf{RSIM-U} method completely recovers in all conditions: equalling or outperforming the main unsupervised baseline from Table \ref{tab:main} which matched on algorithm, domain, \textit{and} data size.\footnote{Dev set P@1 on \textit{+Domain Mismatch} was near zero despite success on test set. We note test uses more common words than dev. \citet{czarnowska-etal-2019-dont} find that BLI performance worsens for rarer words, which may have poorly trained embeddings \citep[also,][]{gong2018frage, sogaard-etal-2018-limitations}.}
\textit{IsoVec} is thus useful for many types of distributional shifts: algorithmic, domain, and amount of data available.
\begin{table*}[htb]
\centering \setlength\tabcolsep{3pt}\footnotesize
\begin{tabular}{l||r@{ }l|r@{ }l|r@{ }l||r@{ }l|r@{ }l|r@{ }l||r@{ }l|r@{ }l|r@{ }l}
\toprule
& \multicolumn{6}{c||}{\underline{Algorithm Mismatch}} & \multicolumn{6}{c||}{\underline{\textit{+ Domain Mismatch}}} & \multicolumn{6}{c}{\underline{\textit{+ More Target-Side Data}}} \\
& \multicolumn{2}{c}{bn} & \multicolumn{2}{c}{uk} & \multicolumn{2}{c}{ta} & \multicolumn{2}{c}{bn} & \multicolumn{2}{c}{uk} & \multicolumn{2}{c}{ta} & \multicolumn{2}{c}{bn} & \multicolumn{2}{c}{uk} & \multicolumn{2}{c}{ta} \\
& \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c||}{$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\sigma$} &
\multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c||}{$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\sigma$} & \multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\sigma$} &
\multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c}{$\sigma$} \\
\midrule
\textbf{Supervised} & & & &&& &&&&&&&&\\
\textit{Baseline} & \textit{13.6} & \textit{(0.5}) & \textit{11.7} & \textit{(0.6}) & \textit{9.4} & \textit{(0.5}) & \textit{10.5} & \textit{(0.6}) & \textit{10.0} & \textit{(0.4}) & \textit{8.4} & \textit{(0.4}) & \textit{16.0} & \textit{(0.7}) & \textit{13.5} & \textit{(0.3}) & \textit{11.0} & \textit{(0.6)} \\
\textbf{Proc-L2+I} (Ours) & \textbf{16.3} & (0.4) & \textbf{16.6} & (2.2) & \textbf{12.1} & (0.7) & \textbf{15.5} & (0.7) & \textbf{17.3} & (0.4) & \textbf{10.9} & (0.5) & \textbf{16.2} & (0.4) & \textbf{17.3} & (0.3) & \textbf{11.4} & (0.3) \\
\midrule
\textbf{Unsupervised} & & & &&& &&&&&&&&\\
\textit{Baseline} & \textit{11.9} & \textit{(0.7)} & \textit{4.4} & \textit{(6.1)} & \textit{2.6} & \textit{(3.6)} & \textit{0.0} & \textit{(0.0)} & \textit{0.0} & \textit{(0.0)} & \textit{0.0 }& \textit{(0.0)} & \textit{5.9} & \textit{(8.0)} & \textit{0.0} & \textit{(0.0)} & \textit{0.0} & \textit{(0.0)} \\
\textbf{RSIM-U} (Ours) & \textbf{13.6} & (0.7) & \textbf{13.4} & (1.2) & \textbf{3.6} & (4.8) & \textbf{13.5} & (0.7) & \textbf{13.1} & (0.8) & \textbf{6.8} & (3.8) & \textbf{14.4} & (0.7) & \textbf{13.6} & (0.6) & \textbf{3.2} & (4.3) \\
\bottomrule
\end{tabular}
\caption{P@1 of \textit{IsoVec} vs. baseline under algorithm and domain mismatch. \textit{Baselines} correspond to Table \ref{tab:alg-dom-mismatch-delta} results. Ours, Supervised: \textbf{Proc-L2+Init}. Ours, Unsupervised: \textbf{RSIM-U}.}
\label{tab:dom-mismatch}
\end{table*}
\subsection{Effect on Isomorphism}
\label{sec:effect-on-iso}
Table \ref{tab:final-iso-scores} (left) shows the effect of \textit{IsoVec} on global isomorphism measures. We measure relational similarity, eigenvector similarity, and Gromov-Hausdorff distance of trained embedding spaces (before mapping) for all main experiments of Section \ref{sec:main} using scripts from \citet{vulic-etal-2020-good}\footnote{\url{https://github.com/cambridgeltl/iso-study/tree/master/scripts}}. We average over experiments. To avoid confusion with the \textit{IsoVec} loss functions, we call the metrics ``RelSim'', ``EigSim'', and ``GH''. The script calculates EigSim ($k=2$) over the first 10,000 embeddings in each space and GH over the first 5000. RelSim is calculated over the first 1000 seed translation pairs.
All supervised methods improved RelSim ($\uparrow$ better). Perhaps surprisingly, initializing the source space with target embeddings (\textbf{+Init}) worsens isomorphism. \textbf{RSIM} is best, directly optimizing for this metric in a supervised manner. RelSim stayed roughly consistent in unsupervised experiments.
All uk-en and ta-en experiments improve GH \cite[$\downarrow$ better]{patra-etal-2019-bilingual}. GH worsened for bn-en despite improved BLI (Table \ref{tab:main}).
EigSim ( $\downarrow$ better) improves across all experiments except uk-en supervised methods, despite improved BLI (notably, initial EigSim for uk was low). \textbf{EVS-U} strongly improves EigSim, optimizing it directly. Table \ref{tab:final-iso-scores} (left) measures the unperturbed geometry of spaces after training and shows that \textit{IsoVec} improves isomorphism in a majority of settings. The same calculation over embeddings \textit{after} mapping with semi-supervised VecMap is in Table \ref{app:final-iso-scores-mapped}.
It is interesting that baseline experiments performed better when mapped in supervised mode while spaces trained with \textit{IsoVec} tended to map better in semi-supervised mode (as mentioned in Section \ref{sec:beta}). This may further indicate that the \textit{IsoVec} spaces have become more geometrically similar.
\begin{table*}[!htb]
\setlength\tabcolsep{3.2pt}\footnotesize
\begin{minipage}{.56\linewidth}
\begin{tabular}{@{}l|ccc|ccc|ccc}
\toprule
& \multicolumn{3}{c}{\underline{Relational Sim. $\uparrow$}} & \multicolumn{3}{c}{\underline{GH Distance $\downarrow$}} & \multicolumn{3}{c}{\underline{Eigenvector Sim. $\downarrow$}} \\
& bn & uk & ta & bn & uk & ta & bn & uk & ta \\
\midrule
\hspace{1mm}\textit{Baseline} & \textit{0.32} & \textit{0.26} & \textit{0.27} & \textit{\textbf{0.34}} & \textit{1.17} & \textit{0.39} & \textit{62.7} & \textit{49.9} & \textit{74.7} \\
\hspace{1mm}\textbf{L2} & 0.36 & 0.34 & 0.33 & 0.43 & 0.76 & \textbf{0.19} & 52.0 & 68.5 & 47.3 \\
\hspace{1mm}\textbf{Proc-L2} & 0.43 & 0.42 & 0.39 & 0.42 & 0.51 & 0.23 & 46.6 & 72.8 & \textbf{37.7} \\
\hspace{5mm}+\textit{\textbf{Init}} & 0.39 & 0.38 & 0.35 & 0.43 & \textbf{0.46} & 0.22 & 52.1 & 77.2 & 41.0 \\
\hspace{1mm}\textbf{RSIM} & \textbf{0.54} & \textbf{0.53} & \textbf{0.47} & 0.37 & 0.56 & 0.20 & 47.2 & 60.4 & 39.8 \\
\hspace{5mm}+\textit{\textbf{Init}} & 0.37 & 0.35 & 0.33 & 0.40 & 0.52 & 0.20 & 48.6 & 65.6 & 44.8 \\
\hspace{1mm}\textbf{RSIM-U} & 0.30 & 0.25 & 0.26 & 0.42 & 0.54 & 0.32 & 56.3 & \textbf{36.2} & 60.1 \\
\hspace{1mm}\textbf{EVS-U} & 0.30 & 0.24 & 0.26 & 0.68 & 0.53 & 0.30 & \textbf{38.7} & 38.3 & 39.5 \\
\bottomrule
\end{tabular}
\end{minipage}%
\begin{minipage}{.44\linewidth}
\;\,
\begin{tabular}{l|c|c|rrr}
\toprule
& & All & bn-en & uk-en & ta-en \\
\midrule
P@1 \hspace{1mm} vs. RelSim & (+) & 0.17 & \textcolor{gray}{\textit{-0.03}} & \textcolor{red}{-0.54} & \textcolor{red}{-0.54} \\
\hspace{8mm} vs. GH & (-) & \textcolor{red}{0.79} & \textcolor{red}{0.28} & -0.23 & \textcolor{red}{0.27} \\
\hspace{8mm} vs. EigSim & (-) & \textcolor{red}{0.57} & \textcolor{red}{0.17} & \textcolor{gray}{\textit{-0.05}} & \textcolor{gray}{\textit{0.05}} \\
\midrule
RelSim \hspace{2mm} vs. GH & (-) & \textcolor{red}{0.09} & -0.45 & -0.24 & \textcolor{gray}{\textit{-0.03}} \\
\hspace{8mm} vs. EigSim & (-) & \textcolor{gray}{\textit{-0.05}} & -0.25 & -0.28 & -0.27 \\
GH \hspace{2.5mm} vs. EigSim & (+) & 0.65 & \textcolor{red}{-0.16} & \textcolor{red}{-0.17} & \textcolor{red}{-0.29} \\
\bottomrule
\end{tabular}
\end{minipage}
\caption{\textbf{Left:} Isomorphism scores of source vs. target embedding space after training, per language, per training method, averaged over 5 runs. Best is \textbf{bold}. \textit{IsoVec} output embeddings vs. base English space.
\textbf{Right:} Pearson's correlation: BLI performance (P@1) of supervised \textit{IsoVec} experiments vs. isomorphism score
and between metrics. (+/-): expected direction of correlation: \textcolor{red}{opposite} or \textit{\textcolor{gray}{weak}} (magnitude <= 0.05).
}
\label{tab:final-iso-scores}
\label{tab:corrs}
\end{table*}
\section{Discussion}
\subsection{The Promise of Geometric Losses}
We have seen that \textit{IsoVec} improves relative isomorphism and downstream BLI from word embedding spaces.
The success of unsupervised methods is particularly encouraging for the use of global isomorphism measures to improve embedding spaces.
Notably, we use only the first 2000 words per space to calculate unsupervised \textit{IsoVec} losses---i.e., we coax these frequent words to have similar representations, regardless of identity.
While there are likely some true translation pairs in the mix, there are almost certainly words this subset of $\bf{X}$ whose translation is not in the first 2000 words of $\bf{Y}$ (and vice-versa)---particularly when source and target corpora are from different domains.
Regardless, \textit{IsoVec} unsupervised methods work.
\subsection{Need for a Sensitive Isomorphism Metric}
Previous authors found that EigSim and GH correlate well with BLI performance \cite{sogaard-etal-2018-limitations, patra-etal-2019-bilingual},
however our results reveal a nuanced story.
In Table \ref{tab:corrs} (right), we correlate the EigSim, RelSim, and GH with BLI P@1 performance over all runs of the main supervised \textit{IsoVec} experiments (\textbf{L2}, \textbf{Proc-L2}, \textbf{Proc-L2+Init}, \textbf{RSIM}, \textbf{RSIM+Init}; 25 data points per calculation).
P@1 should correlate positively with RelSim ($\uparrow$ better for both) and negatively vs. GH/EigSim ($\downarrow$ better for GH/EigSim).
RelSim should correlate negatively with GH/EigSim, and GH positively with EigSim.
In Table \ref{tab:corrs} (right) within language, however, only P@1 vs. GH on uk-en aligns with intuition.
Many correlations are weak (gray, magnitude <= 0.05) or \textit{opposite} of expected; For instance, P@1 should increase with RelSim, but we see the opposite within language pair.
Over languages combined, the relationship is weakly positive. Figure \ref{fig:rsim-v-p1} shows how this is possible.
Samples for Pearson's correlation should be drawn from the same population, and in Table \ref{tab:corrs} (right) we assume that is our \textit{IsoVec} embedding spaces.
Perhaps the assumption is unfair: different \textit{IsoVec} losses might induce different monolingual spaces where specific metrics are indeed predictive of downstream BLI performance, but this may not be visible in the aggregate.
An ideal metric, however, would predict downstream BLI performance regardless of how monolingual spaces were trained; such that we might assess the potential of spaces to align well without having to map them and measure their performance with development or test dictionaries.
In that light, the discrepancies in Table \ref{tab:corrs} (right) highlight the need for a more sensitive metric that works within language and with small differences in BLI performance.\footnote{For instance, the five runs of \textbf{L2} ranged in P@1 from 16.05-16.91, and in RelSim from 0.3606-0.3618.
Maximum and minimum BLI scores differed by only 0.0004 RSIM.
}
We should thus be cautious drawing between- vs. within-language conclusions about isomorphism metrics and downstream BLI.
When isomorphism metrics differ considerably, perhaps BLI performance also differs similarly, as seen in previous work; however if isomorphism scores are poor or too similar, the metrics may not be sensitive enough to be predictive. Future work should investigate these hypotheses and develop isomorphism metrics that are more sensitive. The spectral measures of \citet{dubossarsky-etal-2020-secret} might be examined in these lower-resource contexts, as the authors claim to correlate better with downstream BLI.
All-in-all, though, our main results show that coaxing towards improved isomorphism as measured by the three popular metrics can improve BLI performance even if the scores are not strongly predictive of raw P@1.
\section{Conclusion \& Future Work}
We present \textit{IsoVec}, a new method for training word embeddings which directly injects global measures of embedding space isomorphism into the skipgram loss function.
Our three supervised and two unsupervised isomorphism loss functions successfully improve the mappability of monolingual word embedding spaces, leading to improved ability to induce bilingual lexicons.
\textit{IsoVec} also shows promise under algorithm mismatch, domain mismatch, and data size mismatch between source and target training corpora.
Future work could extend our work to even greater algorithmic mismatches, and in massively multilingual contextualized models.
We release \textit{IsoVec} at \url{https://github.com/kellymarchisio/isovec}.
\section*{Limitations}
As with most methods based on static word embeddings, our work is limited by polysemy.
By using word2vec as a basis, we inherit many of its limitations, many of which are addressed in recent contextualized representation learning work.
Future work might apply our methods to contextualized models.
We also experiment with only English as a target language, limiting our method's universal applicability.
Future work could extend our results to non-English pairs, and also evaluate monolingually if languages will be used separately as recommended by \citet{luong-etal-2015-bilingual}.
|
1,314,259,996,720 | arxiv |
\section{Introduction}
\label{intro}
The interstellar medium is permeated by large-scale magnetic fields \cite{planckxxxiii}. These magnetic fields are observed in molecular clouds and are suggested to have a crucial role in regulating the fragmentation of dense filaments and in channeling filament material into prestellar cores \cite{mckee,crutcher2012}. Observations of linearly-polarized continuum emission from magnetically-aligned dust grains at mm and submm wavelengths are a powerful tool to measure the morphology and structure of the magnetic field lines in star-forming clouds and dense cores \cite{crutcher2012}.
The dust polarization maps provided by the {\it Planck} satellite have revealed a large-scale regular
morphology for the Galactic magnetic field (GMF). The orientation of this magnetic field tends to be parallel to low matter density and perpendicular to high matter density structures \cite{Planck2016}. Using complementary total intensity high angular resolution observations provided by the {\it Herschel} satellite \cite{andre2014,palmeirim2013,Arzouminian2011,Arzouminian2019} it has been shown that galactic filamentary structures are associated with an organized magnetic field topology at scales larger than 0.5 pc.
To probe the role played by the magnetic fields in the star formation mechanism, we need to explore smaller angular scales within the filaments. This could be performed with the polarization channel of NIKA2 \cite{NIKA2-Adam} at 260 GHz (1.15 mm), which will provide an in-depth view of the magnetic fields at the critical scales of 0.01 to 0.1 pc, thanks to an angular resolution of 11.2 arcsec.
In addition, the NIKA2 camera has dual band capability, providing simultaneous observations in total intensity at 150 GHz (2.05 mm) with an angular resolution of 17.5 arcsec.
\section{NIKA2 polarization system and detection strategy}
\label{nika2polsys}
The NIKA2 polarization system consists of a room temperature continuously rotating half-wave plate (HWP) and a fixed polarizer mounted inside the cryostat at base temperature of ~150 mK \cite{NIKA2-Adam}.
The cold polarizer separates with high purity the two polarization onto two different KID arrays (see Figure~1).
Figure~2 shows the configurations in total power observations and polarization mode. In the latter, the HWP is placed in front of the cryostat window. The rotation of the HWP modulates the incident polarization.
As a consequence, the polarization signal is shifted at the fourth harmonic of the HWP rotation frequency $\omega$ and can be recovered by using a demodulation technique, which consists of a numerical lock-in around the 4$\omega$ \cite{ritacco2017} frequency. This detection strategy allows a quasi-simultaneous observation of the three Stokes parameters $I$, $Q$, and $U$ although HWP-induced systematic effects are observed and need to be subtracted. The most important one corresponds to a parasitic signal at all harmonics of the HWP rotation frequency $\omega$. This systematic effect, which is common to all the pixels of the KID array, is called HWP Synchronous Signal (HWPSS) and it is corrected in the data analysis \cite{ritacco2017}.
Furthermore, we also observe in the NIKA2 polarization data intensity to polarization leakage, which constitutes to date the most critical systematic effect. From observations of Uranus, which can be considered as unpolarized, we have estimated a
maximum fractional instrumental polarization varying from 0.45~\% to 1.84~\% in Stokes $U$ and from 1.27~\% to 2.16~\%
in Stokes $Q$ \cite{hamza2019}. A similar effect was observed for the NIKA camera and we developed a procedure to correct for it \cite{ritacco2017}. Although the intensity to polarization leakage in NIKA2 is about a factor 2 lower than in NIKA, it can not be corrected for using the NIKA procedure because it varies more significantly with elevation. Work is in progress to account for this effect and it is the priority of the forthcoming commissioning campaigns. In this paper, we concentrate on extended sources for which the intensity to polarization leakage effect is reduced when averaging across the source (see ~\cite{ritacco2017,ritacco2018}).
\begin{figure}
\begin{center}
\includegraphics[scale=0.3]{Fig1_cryo.png}
\label{fig_cryo}
\caption{Cryostat scheme showing the two NIKA2 channels at 150 GHz (2.05 mm) and 260 GHz (1.15 mm). The two 1mm arrays recover the two orientations of the linear polarization.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{polarization_system_cabin.png}
\label{fig_cabin}
\caption{The two images show the NIKA2 cryostat inside the receiver cabin of the IRAM 30m telescope. The HWP is placed in front of the cryostat entrance when we run observations in polarization mode.}
\end{center}
\end{figure}
\section{NIKA2 first polarization light on the Crab Nebula}
For the NIKA2 commissioning, we concentrated on observations of the Crab Nebula, which were taken in December 2018.
The Crab nebula is a supernova remnant that is generally considered as a good polarization angle calibrator for polarization experiments (\cite{ritacco2018,aumont2019}). The spectral energy distribution in both intensity and polarization of the Crab nebula is well described by a single power law spectrum, as expected from synchrotron emission powered by a single population of relativistic electrons \cite{ritacco2018}. As a consequence, we expect the Crab nebula degree and polarization angle to be constant at radio and millimeter wavelengths. This has been shown by \cite{ritacco2018} using intensity and polarization observations of the Crab nebula from 23 to 217 GHz. However, other emission contributions are expected and they could be investigated using NIKA2 among other experiments. \\
Figure~3 shows the 1.15 mm NIKA2 maps of Stokes $I$, $Q$, and $U$. Polarized vectors, where the polarization intensity SNR is larger than 3$\sigma$, are also overplotted on the intensity map.
The polarization intensity flux integrated over the extension of the Crab nebula is $12.3 \pm 0.1$ Jy (statistical uncertainties only). This value is consistent with the expectation from the polarization intensity SED given in \cite{ritacco2018}. For the Stokes $I$ map we obtain a total flux of 158.04$\pm$0.18 Jy (statistical uncertainties only). The discrepancy between the recovered flux in total intensity and the expected one \cite{ritacco2018} may be due to filtering effects that were not fully taken into account on this preliminary result. In terms of polarization angle, we find (-86.9$\pm$0.1)$^{\circ}$ (statistical uncertainties only), which is only marginally consistent with the averaged value of (-87.7$\pm$0.3)$^{\circ}$ estimated at these frequencies by \cite{ritacco2018}. Although NIKA2 preliminary results are fairly consistent with Planck satellite results given in \cite{ritacco2018}, there are still few systematic effects observed on compact sources that need to be addressed.
Systematic uncertainties due to the HWPSS and to the intensity to polarization leakage are under investigation and will require extra commissioning campaigns, which are planned for early 2020.
At present we still observe significant discrepancy between the polarization angle calibration deduced from compact and extended sources, which is mainly related to the intensity to polarization leakage correction \cite{hamza2019}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{crab_1mm_nika2.png}
\includegraphics[scale=0.45]{crab_q_1mm.png}
\includegraphics[scale=0.45]{crab_u_1mm.png}
\label{fig2}
\caption{NIKA2 Stokes parameter maps of the Crab Nebula at 1.15 mm. From left to right: intensity map with polarization vectors overplotted where the polarization intensity SNR > 3$\sigma$, Stokes Q map, and Stokes U map.}
\end{center}
\end{figure}
\section{Summary and conclusions}
The NIKA2 polarization channel at 260 GHz entered its commissioning phase in Autumn 2017, when the NIKA2 camera was officially offered to the community for total intensity observations. Because of hardware problems and bad weather, first reliable data were only obtained in December 2018.
During this commissioning campaign, we were able to observe well known compact (see \cite{hamza2019}) and extended sources including the Crab nebula. In this paper, we have presented the preliminary NIKA2 polarization maps of the Crab nebula. These maps allowed us to compute the integrated polarization flux, which is consistent with expectations from the polarization SED shown in~\cite{ritacco2018}. Preliminary NIKA2 measurements of the Crab polarization angle indicate fair agreement with expectations although systematic uncertainties are not fully understood to date.
Overall the current analysis is still limited by systematic effects and requires further investigation. Extra commissioning campaigns are planned for early 2020.
\section*{Acknowledgements}
\input{acknowledgements}
|
1,314,259,996,721 | arxiv | \section{Introduction}
On the main-sequence stellar sites the series of reactions that convert hydrogen into helium is known as
the proton-proton chains. It is a key to understand the evolution of the stars.
The cross sections of these charged particle induced reactions are major ingredients to calculate
thermonuclear reaction rates.
They are measured at laboratory energies and are then extrapolated
to thermal energies~\cite{nacre}, because of their smallness at such low energies.
The extrapolation is performed by introducing the astrophysical $S$-factor:
\begin{equation}
\label{eq:sf}
S(E)=\sigma(E)E e^{2\pi \eta(E)},
\end{equation}
where $\sigma(E)$ is the reaction cross section at the incident center-of-mass (c.m.) energy $E$
and $\eta(E)=Z_T Z_P \alpha \sqrt{\frac{\mu c^2}{2E}}$, with $Z_T$, $Z_P$, and $\mu$ denoting the
atomic numbers and the reduced mass of the target and the projectile~\cite{clayton}; $\alpha$ and $c$ are
the fine-structure constant and the speed of light, respectively.
The exponential term in the equation represents the inverse of the Coulomb barrier penetrability.
Since we have factored out the strong energy dependence of $\sigma(E)$ due to the barrier
penetrability, the $S$-factor can be approximated by a smooth polynomial expansion in the absence
of low-energy resonances.
In laboratory experiments, the targets are usually in gas or solid state.
In the low-energy region, the $S$-factors obtained from experiments show
large enhancement to the extrapolation from high energy data for various reactions~\cite{frvr}.
This enhancement is, usually, attributed to the screening by the bound electrons around the target.
In contrast in the stellar nucleosynthesis nuclei are almost fully ionized and
surrounded by the plasma electrons. In deuteron induced reactions
on deuterated metals~\cite{kasagi,raiola1,raiola2} and proton-induced reactions on lithium isotopes
in several forms of lithium chemical compounds~\cite{cruz}, much larger screening enhancements have
been observed with respect to the enhancement by gaseous targets.
The nuclear reactions in such a circumstance are affected by a different
mechanism of the plasma or the conduction electron screening~\cite{shav,ichimaru,kato:014615}.
A similar effect has been discussed in the radioactive decay of a nucleus in a model~\cite{PhysRevLett.74.2824}.
The screening effects in the medium can be dependent on temperature and density of the medium
and we do not consider such effects in this paper.
Hence the screening effect of the bound electrons should be removed from the $S$-factor data
to asses the reaction rate in the stellar site correctly.
The enhancement by the bound electrons
is discussed in terms of a constant potential shift~(screening potential $U_e$).
The upper limit of $U_e$ is obtained, when the adiabatic approximation is fulfilled,
and it is given by the difference
of the binding energies of the target atom and the united atom.~\cite{alr}
On this issue, dynamical effects have been studied
by following the time evolution of the atomic wave function in the
classical allowed region~\cite{skls}.
They solved the time dependent Hartree-Fock equation and evaluated the screening potential.
Their results suggest that the screening potential
approaches the adiabatic limit as the incident energy becomes lower.
The influence of the tunneling phenomenon to this problem has been studied as well~\cite{ktab}.
And, there, the screening potential could go over the above-mentioned adiabatic limit slightly,
only in the case where the electronic wave-function has some excited state components at the classical
turning point of the inter-nuclear motion.
We have examined the problem using molecular dynamics approach with constraints~\cite{pmb,kb-ags},
to see the effect of the fluctuations~\cite{kb-cdf,kb-icfe}.
The obtained average enhancement factors do not exceed the adiabatic
limit. However, there are events that give enhancement factors larger than that in the adiabatic limit.
In this paper, we discuss
the influence of the electron screening effects on the determination
of the bare $S$-factors~($S_b(E)$).
We especially focus on the reactions in the hydrogen burning process and
determine $S_b(E)$
through a fitting of experimental data making use of the polynomial expression and fixing the screening
enhancement in the adiabatic limit.
There are several more sophisticated theoretical models that describe the bare $S$-factor
as the $R$-matrix model~\cite{ad,barker,daacv}, the potential model~\cite{PhysRevC.61.025801},
and the distorted wave Born approximation.
However our aim is
to clarify the effect of the electron shielding on the experimental data of the $S$-factors
rather than to determine the $S$-factor based on a particular theoretical nuclear model.
For this purpose we choose the simplest way: the polynomial expression and try to determine
$S_b(E)$ in a model independent way.
We will
discuss the sensitivity of the $S_b(0)$ values on the choice of the degree of the polynomial and the fitting
range for each particular reaction.
We stress that the use of simple polynomial expressions does not carry physical meaning and hence the derived
$S_b(0)$ values should be taken with caution. To make up this deficit,
we propose another expression in place of the polynomial expression.
The new expression includes an explicit
contribution of the nuclear interaction. Moreover it is based on
a two-step process with
a compound nucleus~(CN) formation and a statistical choice of the exit channel~\cite{weiss,bon87,bb,kb-up}.
Provided that the energy regions of the astrophysical interest is low, we assume that the $l=0$ partial-wave
component is dominant for most of the reactions.
To our knowledge, the statistical model calculation has been used to estimate the $S$-factor of the
radiative proton-capture reactions on Sr isotopes~\cite{PhysRevC.64.065803}. They have used the Hauser-Feshbach
statistical model code and have compared the theoretical results with the experimental data.
Because in the Hauser-Feshbach statistical model code, a global level density, which is based on some models,
is used, one can obtain the absolute value of the $S$-factor. They have found discrepancies between the theoretical
results and the experimental data, especially in the reactions with target isotopes near the neutron
shell-closure. In their paper they did not aim to fit their experimental data.
Electron screening effects have been studied on some specific reactions in the chain,
especially on transfer reactions $^3$He($^3$He,2$p$)$^4$He and $^7$Li($p,\alpha$)$^4$He, which are studied
including extremely low-energy region~\cite{ju98,erag,erag-b}.
However there is no systematical study for all of the reactions including the radiative capture reactions.
We aim to deduce these effects from such
well studied reactions and estimate the screening quantitatively on all the reactions in the chain.
This motivates us to employ the adiabatic limit, rather than leaving the screening potential as a
fitting parameter as it is customary treated in a series of studies~\cite{ju98,barker}.
For comparison we also perform the fitting procedures without enhancement factor and by treating the screening potential
as a parameter.
The obtained $S_b$ at zero incident energy are compared with the results in the NACRE compilation~\cite{nacre},
in which the authors employed the screening potential higher than the adiabatic limit, and
with the results in the $R$-matrix analyses~\cite{daacv}.
This paper is organized as follows,
In Sec.~\ref{sec:efad} we describe the enhancement factor by the bound electrons within the adiabatic limit briefly.
We list up the reactions in PP-chains in Sec.~\ref{sec:ppchain} and explain how we incorporate the enhancement factor
into the fitting procedure. We, especially, give a detailed account on the second fitting procedure based on
the statistical model.
Some reactions are analyzed in this section.
We summarize the paper in Sec.~\ref{sec:sum}.
\section{Enhancement factor in the adiabatic limit}
\label{sec:efad}
To discuss the enhancement quantitatively,
we determine the enhancement factor~\cite{alr}:
\begin{equation}
\label{eq:enh}
f_e=\frac{\sigma(E)}{\sigma_0(E)},
\end{equation}
in terms of the measured cross section $\sigma(E)$ and the bare cross section $\sigma_0(E)$.
If one assumes that the effect of the electron screening can be represented
by the constant shift $U_e$(screening potential) of the potential barrier, the enhancement factor
is approximated by~\cite{alr,skls},
\begin{equation}
\label{eq:scpot}
U_e \sim \frac{E}{\pi\eta(E)} \log \ {f_e}.
\end{equation}
The $U_e$ can be estimated easily in two limiting cases.
In one case the inter-nuclear velocity is much higher than that of electrons
velocity; this limit is called the sudden limit. Within this limit the electron wave function is frozen
during the reaction.
In the opposite case where the inter-nuclear motion is much slower than electrons motion,
the bound electrons follow the motion of the nuclei adiabatically. Within this adiabatic limit
the screening potential is expressed by the difference of the binding energies between
the initial target atom~($BE_{T}$), and the united atom~($BE_{UA}$), which is formed during the reaction.
\begin{equation}
\label{eq:uad}
U_e^{(AD)}=BE_{T}-BE_{UA}
\end{equation}
The screening potential within this limit gives the theoretical upper limit. However, we should stress that these cases deal with
the ideal situation of a ion impinging on an isolated atom. This should be the ideal situation in nuclear physics experiments where
very thin targets are used together with a well collimated mono-energetic beam. However, this situation is not fulfilled if the beam energy is very
low (which is the case of interest in the present paper) or if the atoms are embedded in a medium such as a metal at a given density and temperature.
In the latter case the nuclear process is expected to be influenced by the rearrangement of the electrons in the metal which will give rise to some
peculiarities similar to the studied radioactive decay in a medium~\cite{PhysRevLett.74.2824}.
\section{Bare $S$-factors of PP-chain reactions}
\label{sec:ppchain}
A list of reactions in PP-chains is shown in Table~\ref{tab:a}.
\begin{table*}
\caption{Reactions in PP-chains, their minimum incident energy in the c.m. system measured so far
and the enhancement factor within the adiabatic limit at the minimum energy.}
\label{tab:a}
\begin{center}
\begin{tabular}{lrr}
\hline
reactions & $E_{min}$ (keV) & $f_e^{(AD)}(E_{min})$ \\
\hline
H($p,\beta^+ \nu_e$)D & & \\
D($p,\gamma$)$^3$He & 2.52 & 1.07\\
$^3$He($^3$He,2$p$)$^4$He & 20.76 & 1.22 \\[3pt]
$^3$He($\alpha$,$\gamma$)$^7$Be & 93.$^{\#}$, 127.$^\ast$ & 1.02 \\
$^7$Be($e^-$,$\nu_e$)$^7$Li & &\\
$^7$Li($p,\alpha$)$^4$He & 12.7, 10.$^{\ast\ast}$ & 1.18 \\[3pt]
$^7$Be($p,\gamma$)$^8$B & 115.6 &1.01\\
\hline
\end{tabular}
\end{center}
\vspace*{.6cm}
\noindent
\hspace{2cm} $^{\#}$prompt-$\gamma$ method
\hspace{0.3cm} $^\ast$activation method
\hspace{0.3cm} $^{\ast\ast}$THM
\end{table*}
The first reaction H($p,\beta^+ \nu_e$)D involves the $\beta$-decay and has too small cross section to be measured
experimentally. Its $S$-factor is calculated from first principles~\cite{bahcall}.
We, therefore, concentrate on the other 5 reactions except the electron capture reaction $^7$Be($e^-$,$\nu_e$)$^7$Li.
In the table the minimum incident energies, measured so-far, for each reaction are also shown.
For two transfer reactions, $^3$He($^3$He,2$p$)$^4$He and $^7$Li($p,\alpha$)$^4$He,
the cross sections have been measured already including the low-energy region where the screening enhancement
becomes more than 10\%.
The other three reactions are radiative-capture reactions, which have even smaller cross sections.
The $S$-factor of the reaction $^3$He($\alpha$,$\gamma$)$^7$Be has been re-determined with high precision
recently both by detecting $\gamma$-ray from $^7$Be decay(the activation method)~\cite{bemmerer:122502,gyurky:035805}
and by detecting prompt $\gamma$-ray(the prompt method)~\cite{confortola:065803}.
Its $S$-factor in the low-energy region is extrapolated from high energy data by the $R$-matrix fitting.
The reaction $^7$Be($p,\gamma$)$^8$B involves unstable nuclei. The $S$-factor of this reaction has been
determined by means of the direct capture reaction~\cite{baby:065805,ju03} and the Coulomb dissociation
method~\cite{PhysRevLett.83.2910}.
It has been claimed that there is an inconsistency between the results of the two
methods~\cite{ju03,esbensen:042502}, though the question seems to be resolved by reanalyzing the
data of the Coulomb dissociation method~\cite{schumann:015806}.
If one takes into account modifications due to the nuclear potential and all the contributions from
partial-waves, the bare $S$-factor can be expressed as~\cite{clayton,kb-up}
\begin{eqnarray}
S_b(E)\sim\frac{\pi\hbar^2}{2\mu}\sum_l\Pi_{lf}(E)(2l+1)\exp(W_l), \label{eq:sb0}
\end{eqnarray}
where $\Pi_{lf}(E)$ is the probability to obtain a specified exit channel $f$
from a certain entrance channel $l$ and
\begin{eqnarray}
W_l=\frac{4Z_TZ_Pe^2}{\hbar} \sqrt{\frac{\mu}{2E}}
\left[\sin^{-1}\left(\sqrt{\frac{E}{E_c}}\right) + \sqrt{\frac{E}{E_c}}\sqrt{1-\frac{E}{E_c}} \right] \nonumber \\
- 2\sqrt{\frac{l(l+1)E_l}{E_c}}\left(1-\sqrt{\frac{E}{E_c}}\right), \label{eq:wl}
\end{eqnarray}
with $E_c= Z_TZ_Pe^2/R$ (MeV), $E_l=l(l+1)\hbar^2/(2\mu R^2)$ (MeV fm$^2$), i.e., the heights of the
Coulomb barrier and the centrifugal potential at the nuclear interaction radius $R$.
We assume an empirical formula $R=d\times R_{N_0}$, where $R_{N_0}=1.4 \times (A_T^{1/3}+A_P^{1/3})$ (fm) and
$d$ is a parameter that takes into account the fact that the nuclear potential has a diffuseness;
with $A_T$ and $A_P$ denoting the mass numbers of the target and the projectile nuclei.
We call $d$ radial parameter.
The exponential term in Eq.~(\ref{eq:sb0}) stands for the penetration factor divided by the pure coulomb penetrability.
At zero incident energy limit, Eq.~(\ref{eq:sb0}) reduces to~\cite{kb-up} :
\begin{equation}
\label{eq:sfN3}
S_b(0)\sim \frac{\pi\hbar^2}{2\mu} e^{\frac{4}{\hbar}\sqrt{2\mu Z_TZ_P e^2 R}}\left[\Pi_{0f}(0) +\sum_{l\ge 1}\Pi_{lf}(0)(2l+1)e^{-\frac{2l(l+1)\hbar}{\sqrt{2\mu Z_TZ_P e^2 R}}}\right] .
\end{equation}
The conventional polynomial expression for the bare $S$-factor of non-resonant reactions
is associated to the Taylor expansion of Eq.~(\ref{eq:sb0}).
Most reactions in the PP-chain are non-resonant,
one, therefore, can fit the bare $S$-factor using the enhancement factor:
\begin{eqnarray}
S(E)&=&S_b(E)\cdot f_e; \hspace*{1cm} S_b(E)=S_b(0)+S_1E+S_2E^2+\cdot\cdot\cdot, \label{eq:sbf}
\end{eqnarray}
in an implementation of the nonlinear least-squares
algorithm.
For resonant reactions the energy dependence of the $S$-factor is given by the Breit-Wigner formula~\cite{clayton}
\begin{equation}
\label{eq:bw}
S(E)=\frac{\pi\hbar^2}{2\mu}\frac{\omega\Gamma_1(E)\Gamma_2}{(E-E_r)^2+(\Gamma/2)^2}e^{2\pi\eta(E)},
\end{equation}
where
$\omega=(2J+1)/(2j_1+1)(2j_2+1)$ and $E_r$ are the statistical factor and the resonance energy,
respectively; $\Gamma_1(E), \Gamma_2$ and $\Gamma$ are the entrance and the exit channels partial widths
and the total width. The incident energy dependence of $\Gamma_1(E)$ is given again by the penetration factor
in the case of sub-barrier reactions. One, therefore, can write down
\begin{equation}
\label{eq:bw2}
S(E)\sim\frac{\pi\hbar^2}{2\mu}\frac{c_r e^{W_l}\Pi_{lf}(E)}{(E-E_r)^2+(\Gamma/2)^2},
\end{equation}
where $c_r=\omega\Gamma_2 \frac{3\hbar}{R}\sqrt{\frac{2E_c}{\mu}}$~(MeV$^2$), but we determine $c_r$ from the fitting procedure.
In the case where there are more than one data sets
we weight the data depending on its standard error.
As one can easily imagine, higher order terms are important to fit the experimental
data in the high incident energy region. In this paper
we limit the fitting energy range less than 1 MeV and
choose the degree of the polynomial to obtain convergence of the $S_b(0)$ value within the statistical errors.
We will discuss the sensitivity of $S_b(0)$ on the degree of the polynomial and the fitting
range for each particular reaction.
Alternatively, the experimental data are fitted using Eq.~(\ref{eq:sb0}) directly,
instead of the polynomial expression. For this purpose we derive the incident-energy dependence
of $\Pi_{lf}(E)$ in Eq.~(\ref{eq:sb0}).
According to Weisskopf model~\cite{weiss,bon87,kb-up}, the probability to obtain a certain exit
channel $f$ after the CN formation is proportional to
\begin{equation}
\label{eq:pif}
\pi_{f} \propto g_f (T_{k_f}^2+2m_fT_{k_f})\exp\left(-\frac{T_{k_f}\sqrt{a}}{\sqrt{Q_{\rm{CN}}}}\right) \sigma_f^{abs},
\end{equation}
where $g_f, T_{k_f}$, and $m_f$ are the number of states for the spin, the kinetic energy, and the mass, respectively,
of the lightest reaction product in the exit channel;
$Q_{\rm{CN}}, \sigma_f^{abs}$, and $a$ are the $Q$-value for the CN formation,
the cross section of the inverse process, and
the level density parameter $a\sim A_{\rm{CN}}/8.0$ (MeV$^{-1}$)\cite{weiss}; with $A_{\rm{CN}}$ denoting the mass number of
the CN.
The absolute value of $\Pi_{lf}(E)$ is, in principle, given by
\begin{equation}
\label{eq:pifab}
\Pi_{lf}(E)=\frac{\pi_f}{\sum_f \pi_{f}},
\end{equation}
where the sum in the denominator is taken over all possible exit channels.
We, however, avoid calculating the sum and determine $\Pi_{lf}(0)$ from fitting procedures of the experimental data.
Instead, we scale the incident energy dependence of $\Pi_{lf}(E)$
\begin{equation}
\label{eq:pi}
\Pi_{lf}(E)=\Pi_{lf}(0)\frac{T_{k_f}(E)^2+2m_iT_{k_f}(E)}{T_{k_f}(0)^2+2m_iT_{k_f}(0)}\exp\left(\frac{T_{k_f}(0)\sqrt{a}}{\sqrt{Q_{\rm{CN}}}}-\frac{T_{k_f}(E)\sqrt{a}}{\sqrt{E+Q_{\rm{CN}}}}\right),
\end{equation}
where $T_{k_f}(E)=\frac{A_{\rm{CN}}-A_f}{A_{\rm{CN}}}(E+Q)$, with $Q$, and $A_f$ denoting the reaction $Q$-value, and
the mass number of the lightest reaction product.
Eq.~(\ref{eq:wl}) is obviously valid only at the incident energy lower than the Coulomb barrier.
In the fitting procedure using Eq.~(\ref{eq:sb0}),
$\Pi_{lf}(0)$~($l=0,1,2,\cdots$) and the radial parameter $d$ are treated
as fitting parameters.
We assume that the nuclear interaction radius can differ for each partial-wave:
$R_l=d_l\times R_{N_0}$.
The fitting procedures are performed
only in the energy region below the barrier. The sum of the partial-waves is taken up to the order with which
the fit converges, mostly $l=0, 1, 2$ are sufficient.
We anticipate that the description of the reaction mechanism through the CN formation works well,
in particular, in the reactions with the $l=$0 partial-wave in the entrance channel.~\cite{kb-up}
\subsection{$^3$\rm{He}($^3$\rm{He},2$p$)$^4$\rm{He}}
\label{sec:3he3he}
The $S$-factor of the reaction $^3$He($^3$He,2$p$)$^4$He from several measurements
are shown with error bars in Fig.~\ref{fig:3he3he}.
At the minimum incident energy, which has been reached in an experiment by the LUNA collaboration~\cite{ju98},
the screening enhancement is estimated to be more than 20\% of the adiabatic approximation.
In the NACRE~\cite{nacre}compilation, the authors used the screening potential $U_e=$ 330~(eV)
and a quadratic polynomial
to obtain the $S$-factor. The fitting parameters in~\cite{nacre} are shown in the first row of table 2.
\begin{figure}
\includegraphics[height=.7\textheight]{3he3he.ps}
\caption{The $S$-factor for the reaction $^3$He($^3$He,2$p$)$^4$He as a function of
the incident C.M. energy. The experimental points are
from~\cite{ba67}(Backer67), from~\cite{dw71}(Dwarakanath71), from~\cite{kr87}(Krauss87) and
from~\cite{ju98}(Junker98).
The solid curves represent the $S$-factors multiplied by the enhancement factor within the adiabatic limit
from our fitting. The dashed curves show the corresponding bare $S$-factor.
In the top and bottom panels the fitting is performed using the polynomial expression and using
Eq.(\ref{eq:sb0}), respectively.}
\label{fig:3he3he}
\end{figure}
For the reaction $^3$He($^3$He,2$p$)$^4$He, the adiabatic screening potential is obtained
under the following considerations.
In the target medium $^3$He projectiles are likely to be $^3$He$^+$ or $^3$He charge neutral state.
If we consider $^3$He neutral projectiles, the adiabatic screening potential is
$U_e^{(AD)}=$ 246.8~(eV)~\cite{ju98}.
\begin{equation}
f_e^{(AD)}(^3He)=e^{\pi\eta(E)\frac{U_e^{(AD)}}{E}}.
\end{equation}
For $^3$He$^+$ projectiles the adiabatic screening potential is calculated
taking into account the charge symmetry of the system~\cite{lichten}.
$U_e^{(AD)1}=$ 255.5~(eV) and $U_e^{(AD)2}=$ 122.2~(eV) in the cases where the system ends up with
$^6$Be$^+$(1s)$^2$(2s) state and $^6$Be$^+$(1s)(2p)$^2$ state respectively.
The corresponding enhancement factor within the adiabatic limit is written as~\cite{ktab}
\begin{equation}
\label{eq:fad2}
f_e^{(AD)}(^3He^+)=\frac{1}{2}\left(\exp\left[\pi\eta(E)\frac{U_e^{(AD)1}}{E} \right]
+\exp\left[\pi\eta(E)\frac{U_e^{(AD)2}}{E} \right]\right).
\end{equation}
The results of the polynomial fitting using a quadratic polynomial are shown in Table~\ref{tab:3he3he}
together with fitting parameters in~\cite{nacre}.
The obtained fitting parameters from the fit without enhancement factor are shown in the 4th row.
The parameters in the second row are for $^3$He neutral projectile and ones in the third row
are for $^3$He$^+$ projectile.
The corresponding curve for $^3$He neutral projectile case is shown in the top panel of Fig.~\ref{fig:3he3he}
together with the experimental points.
Notice that the adiabatic limit gives a smaller $S_b(0)$ but within the standard
error of the one obtained by the NACRE collaboration.
If we fit the same data by varying the screening potential $U_e$, we obtain $U_e=371 \pm 46$~(eV)
and $S_b(0)=$ 5.06$\pm$0.09~(MeVb) , with $\chi^2_{\nu}=0.7$. The deduced $U_e$ is higher than the adiabatic limit
but the obtained $S_b(0)$ is very close to the one in the adiabatic limit.
Fixing the fitting range from the lowest experimental data 0.0208~MeV to 1~MeV,
we obtained $S_b(0)$=5.32 $\pm$0.08 (MeVb) using a cubic polynomial with $\chi_{\nu}^2$=0.7.
This $S_b(0)$ coincides with the result using a quadratic polynomial~(the second row in Tab.~\ref{tab:3he3he}).
Limiting the fitting range from 0.0208~(MeV) to 1~(MeV), $S_b(0)$ is insensitive to the choice of the
degree of the polynomial.
Assuming the quadratic polynomial and the adiabatic enhancement factor,
the value $S_b(0)$ varies from 5.23$\pm$0.06~(MeVb) to 5.26$\pm$0.3~(MeVb), as one changes the
upper-limit of the fitting from 1~(MeV) to 0.1~(MeV) but fixing the lower limit 0.0208~(MeV).
On the other hand $S_b(0)$ varies from 5.23~(MeVb) to 5.17~(MeVb), as one changes the lower limit from 0.02~(MeV) to 0.2~(MeV).
Thus the $S_b(0)$ obtained by using the polynomial expression is not much sensitive to the choice of both the lower and the upper limit.
\begin{table}
\caption{Fitting parameters of the reaction $^3$He($^3$He,2$p$)$^4$He using polynomial expression.
The first row is from~\cite{nacre}. The second and the third rows are obtained by using the screening
potential in the adiabatic limit. The last row is obtained by assuming without enhancement.}
\label{tab:3he3he}
\begin{tabular}{llllrl}
\hline
& $S_{b}(0)$(MeVb) & $S_{1}$(b) & $S_{2}$(MeV$^{-1}$b) & $U_e$(eV) & $\chi^2_{\nu}$ \\
\hline
& 5.18 & -2.22 & 0.80 & 330 & \\
$f_e^{(AD)}(^3$He) & 5.23 $\pm$ 0.06 & -3.1 $\pm$ 0.5 & 1.6 $\pm$ 0.5 & 246.8 & 0.7 \\
$f_e^{(AD)}(^3$He$^+)$ & 5.32 $\pm$ 0.06 & -3.5 $\pm$ 0.5 & 1.9 $\pm$ 0.6 & & 0.8 \\
& 5.56 $\pm$ 0.07 & -4.7 $\pm$ 0.6 & 3.0 $\pm$ 0.7 & 0 & 1.1 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Fitting parameters and the $S_{b}(0)$ of the reaction $^3$He($^3$He,2$p$)$^4$He using Eq.~(\ref{eq:sb0}).
The first row is obtained by using the screening potential in the adiabatic limit,
the second row without enhancement, and the last row is obtained by treating $U_e$ as a fitting parameter.}
\label{tab:3he3he-2}
\begin{tabular}{lclcllc}
\hline
$\Pi_{0f}(0)$ & $\Pi_{1f}(0)$ & $d_0$ & $d_1$ & $U_e$(eV) & $\chi_{\nu}^2$ & $S_{b}(0)$(MeVb) \\
\hline
0.19 & 7.4$\times 10^{-8}$ & 0.61 & 9.9 & 246.8 & 0.68 & 5.4 \\
0.17 & 2.2$\times 10^{-8}$ & 0.63 &13. & 0 & 0.73 & 6.5 \\
0.18 & 1.0$\times 10^{-7}$ & 0.60 & 9.1 & 299$\pm$102 & 0.68 & 5.2 \\
\hline
\end{tabular}
\end{table}
The fitting procedures using Eq.~(\ref{eq:sb0}) with the screening potentials $U_e=$~246.8, 0~(eV)
and treating $U_e$ as a fitting parameter
give zero energy $S$-factors 5.4, 6.5, 5.2 (MeVb), respectively, and they are shown in Table~\ref{tab:3he3he-2}.
We have used the $l=$0 and 1 partial-waves.
The fitting procedure with $U_e=$ 0 gives
$\chi^2_{\nu}$ slightly larger than the other two cases.
If we take into account the enhancement factor,
the radial parameters $d_{0}$ and $d_{1}$ for two cases are about 0.6 and 9., respectively,
and $\Pi_{1f}(0)$ is much smaller than $\Pi_{0f}(0)$.
The former implies that the effective radius of the $l=$1 partial-wave component is larger than that of the
$l=$ 0 component. And the latter implies that the $l=$ 0 component gives a dominant contribution to the
$S$-factor. The $l=$1 component plays a major role in the higher energy region.
The resulting $S_b(0)$ are
in agreement with the extrapolations using quadratic polynomials for the corresponding screening
potentials.
The curve obtained by this fitting procedure for the adiabatic enhancement is shown in the bottom
panel in Fig.~\ref{fig:7lip2}.
If we fit the same data by varying the screening potential $U_e$, we obtain $U_e=299 \pm 102$~(eV)
and $S_b(0)=$5.2~(MeVb), with $\chi^2_{\nu}=0.68$. This $\chi^2_{\nu}$ is the same as
in the case where we used the adiabatic screening potential.
\subsection{$^7$\rm{Li}($p,\alpha$)$^4$\rm{He}}
\label{sec:7lip}
\begin{figure}
\includegraphics[height=.80\textheight]{7lip-st.ps}
\caption{$S$-factor for the reaction $^7$Li($p,\alpha$)$^4$He as a function of the incident c.m. energy.
In the top panel experimental points are taken
from~\cite{cjmsf}(Cassagnou62), from~\cite{rk86}(Rolfs86), from~\cite{erag,erag-b}(Engstler92).
The solid curve represents our fit by polynomial expression with the adiabatic enhancement.
The dashed curve corresponds to the bare $S$-factor.
In the bottom panel the data from~\cite{la01}(Lattuada01) are shown, together with other experimental
data in the top panel.
The solid curve represents our fit by using Eq.(\ref{eq:sb0}), instead of
the polynomial expression, and with treating the screening potential as a parameter.}
\label{fig:7lip2}
\end{figure}
In Fig.~\ref{fig:7lip2} experimental data of the $S$-factor of the reaction
$^7$Li($p,\alpha$)$^4$He from several direct measurements
are shown with error bars.
We performed the fit of the data
in the incident energy region from 0.01~MeV to 1~MeV
using a cubic polynomial without enhancement factor.
The obtained fitting parameters are shown in the third row of Table~\ref{tab:7liph}.
This fitting procedure without enhancement factor is quite sensitive to the choice
of both the upper and the lower limits of the fitting range.
\begin{table}
\caption{Fitting parameters of the reaction $^7$Li($p,\alpha$)$^4$He.
The three rows are obtained by using a cubic polynomial
and from~\cite{nacre}~(the first row), with the adiabatic approximation(the second row),
and without enhancement~(the third row).}
\label{tab:7liph}
\begin{tabular}{llllrl}
\hline
$S_{b}(0)$(MeVb) & $S_{1}$(b) & $S_{2}$(MeV$^{-1}$b) & $S_{3}$(MeV$^{-2}$b) & $U_e$(eV) & $\chi^2_{\nu}$\\
\hline
0.0593 & 0.193 & -0.355 & 0.236 & 300 & \\
0.0620 $\pm$ 0.0006 & 0.15 $\pm$ 0.01 & -0.24 $\pm$ 0.03 & 0.14 $\pm$ 0.03 & 175 & 0.41 \\
0.0673 $\pm$ 0.0008 & 0.10 $\pm$ 0.01 & -0.13 $\pm$ 0.04 & 0.08 $\pm$ 0.03 & 0 & 0.62 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Fitting parameters and the $S_{b}(0)$ of the reaction $^7$Li($p,\alpha$)$^4$He
using Eq.~(\ref{eq:sb0}).
The first row is obtained by using the screening potential 300 eV, the second row in the adiabatic limit,
the third row without enhancement, and the last row is obtained by treating $U_e$ as a fitting parameter.}
\label{tab:7lip-2}
\begin{tabular}{llrlc}
\hline
$\Pi_{1f}(0)$ & $d_{1}$ & $U_e$~(eV) & $\chi_{\nu}^2$ & $S_{b}(0)$(MeVb) \\
\hline
7.1$\times$10$^{-5}$ & 4.7 & 300 & 1.4 & 0.033 \\
6.0$\times$10$^{-5}$ & 4.9 & 175 & 2.1 & 0.035 \\
5.0$\times$10$^{-5}$ & 5.1 & 0 & 3.5 & 0.038 \\
9.6$\times$10$^{-5}$ & 4.3 & 495 $\pm$ 41 & 1.0 & 0.031 \\
\hline
\end{tabular}
\end{table}
The experimental data from~\cite{erag,erag-b} show the enhancement of the $S$-factor in the low-energy region.
In the compilation NACRE~\cite{nacre}, authors used the screening potential $U_e=$ 300~(eV), which is larger than
the adiabatic limit, and a cubic polynomial to obtain the $S$-factor. The fitting parameters
in~\cite{nacre} are shown in the first row.
However in the experiments~\cite{erag,erag-b} LiF solid targets and deuteron projectiles as well as
deuterium molecular gas targets and Li projectiles are utilized. In the case of LiF target, which is
an ionic crystal similar to NaCl and a large band gap insulator, one can approximate the electronic
structure of the target $^7$Li state by the $^7$Li$^+$ with
only two innermost electrons. Thus one expects the screening potential in the adiabatic
limit $U^{(AD)}_e$ = 371.8-198.2$\sim$174~(eV).
The best fit of the experimental data in the form of Eq.~(\ref{eq:sbf}) and $U_e^{(AD)}=$ 175~(eV)~\cite{kb-icfe} gives
the fitting parameters in the second row of Table~\ref{tab:7liph} with the reduced $\chi^2=0.46$.
The corresponding $S$-factor is shown with the dashed curve in Fig.~\ref{fig:7lip2}.
If we fit the same data by varying the screening potential $U_e$, we obtain $U_e=195\pm 28$~(eV)
and $S_b(0)=$0.061$\pm$0.001~(MeVb), with $\chi^2_{\nu}=0.41$. This is, practically, consistent with the result
(the second row of Table~\ref{tab:7liph}) in the adiabatic limit.
$S_b(0)$ in the adiabatic limit is lower than the value obtained by the fit assuming no screening enhancement
but higher than the value obtained in~\cite{nacre} where the authors used $U_e=$ 300~(eV).
We have checked
the sensitivity of $S_b(0)$ on the degree of the polynomial and the fitting range.
Fixing the fitting range from the lowest experimental data 0.011MeV to 1MeV,
we obtained $S_b(0)$=0.0620$\pm$0.0006~(MeVb) and 0.0617$\pm$0.0009~(MeVb), using a cubic polynomial
and using a quartic polynomial, respectively. Both have $\chi_{\nu}^2$=0.41.
Hence we choose a cubic polynomial for the following fit.
In addition to this, assuming the adiabatic enhancement factor,
the value $S_b(0)$ varies from 0.0620~(MeVb) to 0.0632~(MeVb), as one changes the fitting upper limit from 1~(MeV) to 0.5~(MeV)
with fixing the lower limit 0.011~MeV.
On the other hand $S_b(0)$ varies from 0.0620~(MeVb) to 0.578~(MeVb), as one changes the lower limit from 0.01~(MeV) to 0.04~(MeV).
$S_b(0)$ obtained by using the polynomial expression is more sensitive to the choice of the lower limit
than to the choice of the upper limit.
The fitting procedures of the same data but using Eq.~(\ref{eq:sb0}) are performed.
We have used only the $l=$1 partial-wave,
because $l$ must be odd to obtain positive-parity state of $^8$Be~\cite{barker2000,daacv}.
Using only the $l=$1 component, our fitting procedure gives a steeper incident
energy dependence than the case where $l=$0 is used. In passing we mention briefly that the fitting procedure
of the reaction $^6$Li($p$,$\alpha$)$^3$He in the next subsection for a comparison.
With the screening potential $U_e=$300, 175, 0~(eV)
we obtain zero energy $S$-factors 0.033, 0.035, 0.038~(MeVb), respectively, as they are shown
in Table~\ref{tab:7lip-2}. They are all considerably smaller than the results of polynomial fitting
and $\chi_{\nu}^2$ of all cases are larger than the results of polynomial fitting.
The radial parameters $d_{1}$ for all three cases are considerably larger than 1.
This fact suggests that the interaction radius is 5 times larger than the empirical formula for the
$l=$1 partial-wave.
Again, if we fit the same data by varying the screening potential $U_e$, we obtain $U_e=495 \pm 41$~(eV).
The fitting parameters of this procedure are
shown in the last row in Table~\ref{tab:7lip-2}.
The curve obtained by this fitting procedure is shown in the bottom
panel in Fig.~\ref{fig:7lip2}.
The extracted bare $S$-factor data by THM are especially shown with the closed squares~\cite{la01} but
these data are not included in the fitting procedure. Nevertheless the obtained bare $S$-factor curve follows
the data by THM, which is thought to give the bare $S$-factor.
In Ref.~\cite{daacv} the $R$-matrix fitting for higher energy region~($E > 40$~keV) has been used to determine the $S$-factor.
They obtained the zero-energy $S$-factor 0.067 $\pm$ 0.004~(MeVb) and the screening potential
$U_e=100\pm25$~(eV), which is less than that within the adiabatic limit.
The $S$-factor at zero energy from our results using polynomials $S_{b}(0)$=0.065 $\pm$ 0.005~(MeVb)
and 0.0620 $\pm$ 0.0006~(MeVb) are in agreement with this result from the $R$-matrix fitting.
\subsection{$^6$\rm{Li}($p$,$\alpha$)$^3$\rm{He}}
\label{sec:6lip}
In contrast to the reaction $^7$Li($p$,$\alpha$)$^4$He, the reaction $^6$Li($p$,$\alpha$)$^3$He does not have
the restriction on the incident partial-wave. For a comparison, we show the fitting curves of this reaction
in Fig.~\ref{fig:6lip}, where the top panel shows the two curves from a fitting procedure using a polynomial
and the bottom panel shows the curves obtained by using Eq.~(\ref{eq:sb0}), although this
reaction is not included in the PP-chains,
The fitting procedures of the experimental data using Eq.~(\ref{eq:sb0}) are performed.
Using the $l=$0 partial-wave alone,
we obtain
$S_b(0)=$~3.3~(MeV), with $\chi^2_{\nu}$ =2.5 with fixing the screening potential
$U_e$ = 175~(eV) in the adiabatic limit.
We obtain $U_e$ = 466 $\pm$ 31~(eV) and $S_b(0)=$~3.1~(MeV), with $\chi^2_{\nu}$ =1.4
by treating $U_e$ as a fitting parameter.
The fitting parameters are shown in Table~\ref{tab:6lip-2}.
For this reaction the radial parameter $d_{0}$ for all three cases are about unity.
Although the screening potentials are different, the difference
does not affect either the $S_b(0)$ or the fitting parameters, $\Pi_{0f}(0)$ and $d_{0}$.
Only the $\chi^2_{\nu}$ for $U_e$ = 466 $\pm$ 31~(eV) is much smaller than the others.
The obtained screening potential from the latter procedure is in agreement with one in the
reaction $^7$Li($p$,$\alpha$)$^4$He.
This fact supports the isotopic independence of the electron screening.
\begin{figure}
\includegraphics[height=.7\textheight]{6lip.ps}
\caption{$S$-factor for the reaction $^6$Li($p$,$\alpha$)$^3$He as a function of the incident c.m. energy.
The experimental points are from~\cite{ma56}(Marion56), from~\cite{ge66}(Gemeinhardt66),
from~\cite{sp71}(Spinka71), from~\cite{el79}(Elwin79),
from~\cite{sh79}(Shinozuka79), from~\cite{kw89}(Kwon89) and from~\cite{erag}(Engstler92).
In the top panel the solid curve represents our fit by polynomial expression with the adiabatic enhancement.
The dashed curve corresponds to the bare $S$-factor.
In the bottom panel the solid curve represents our fit by using Eq.~(\ref{eq:sb0}), instead of
the polynomial expression, and with treating the screening potential as a parameter.}
\label{fig:6lip}
\end{figure}
\begin{table}
\caption{Fitting parameters and the $S_{b}(0)$ of the reaction $^6$Li($p,\alpha$)$^3$He by using fitting procedure Eq.~(\ref{eq:sb0}). The first row is obtained by using the screening potential in the adiabatic limit,
the second row corresponds to the zero screening potential,
the last row is obtained by treating $U_e$ as a fitting parameter.}
\label{tab:6lip-2}
\begin{tabular}{llrlc}
\hline
$\Pi_{0f}(0)$ & $d_{0}$ & $U_e$~(eV) & $\chi_{\nu}^2$ & $S_{b}(0)$(MeVb) \\
\hline
0.11 & 1.2 & 175 & 2.5 & 3.3 \\
0.11 & 1.2 & 0 & 3.8 & 3.5 \\
0.12 & 1.1 & 466 $\pm$ 31 & 1.4 & 3.1 \\
\hline
\end{tabular}
\end{table}
From the tree results of all the transfer reactions considered,
$^3$He($^3$He,2$p$)$^4$He, $^7$Li($p,\alpha$)$^4$He, and $^6$Li($p$,$\alpha$)$^3$He,
one can say that the enhancement by the screening is crucial, in the sense that the fitting
procedure without enhancement gives $\chi_{\nu}^2$ larger than the others.
However the obtained $S_b(0)$ is insensitive to the magnitude of the screening potential.
\subsection{\rm{D}($p$,$\gamma$)$^3$\rm{He}}
\label{sec:}
The $S$-factor data of the reaction D($p$,$\gamma$)$^3$He from several measurements
are shown with error bars in Fig.~\ref{fig:dp}.
We performed the fitting procedure of the data
in the incident energy region from 0.0025~(MeV) to 1~(MeV)
using a quadratic polynomial without enhancement factor.
The obtained fitting parameters are shown in the second row of Table~\ref{tab:Dp}.
In~\cite{nacre} the same polynomial degree has been used but the low-energy data by~\cite{ca02}
were not available at that time.
At the minimum incident energy in Ref.~\cite{ca02}
the screening enhancement is estimated to be 7\% at utmost.
This enhancement within the adiabatic limit is,
again, estimated by using a linear combination of the even and odd states of the electronic wave function,
reflecting the charge symmetry of the system as Eq.~(\ref{eq:fad2})
where $U_e^{(AD)1}$ and $U_e^{(AD)2}$ are replaced by 40.8~eV and 0.0~eV, respectively.
\begin{figure}
\includegraphics[height=.7\textheight]{dp.ps}
\caption{$S$-factor for the reaction D($p$,$\gamma$)$^3$He as a function of the incident c.m. energy.
The experimental points are from~\cite{gr62}(Griffiths62), from~\cite{wa63}(Warren63),
from~\cite{be64}(Berman64), from~\cite{wo67}(Wolfli67), from~\cite{sc95}(Schmid95) and
from~\cite{ca02}(Casella02).
In the top panel the solid curve represents our fit by polynomial expression with the adiabatic enhancement.
The dashed curve corresponds to the bare $S$-factor.
In the bottom panel the curves are same with ones in the top panel, but using Eq.~(\ref{eq:sb0}) instead of
the polynomial expression.}
\label{fig:dp}
\end{figure}
\begin{table}
\caption{Fitting parameters of the reaction D($p,\gamma$)$^3$He.
The three rows are obtained by using a quadratic polynomial
and from~\cite{nacre}(the first row), without enhancement~(the second row) and with the
adiabatic approximation~(the third row).}
\label{tab:Dp}
\begin{tabular}{lllrl}
\hline
$S_{b}(0)$~(eVb) &$S_{1}$~(b) & $S_{2}$~(eV$^{-1}$b) & $U_e$~(eV) & $\chi_{\nu}^2$ \\
\hline
0.20$\pm$0.07 & 5.60$\pm$2.00 & 3.10$\pm$1.10 & 0.0 & (NACRE) \\
0.261$\pm$0.006 & 1.3$\pm$0.2 & 12.0$\pm$1.0 & 0.0 & 2.7 \\
0.256$\pm$0.006 & 1.4$\pm$0.2 & 11.8$\pm$1.0 & 40.8,0.0 & 3.9 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Fitting parameters and the $S_{b}(0)$ of the reaction D($p,\gamma$)$^3$He by using the
fitting procedure Eq.~(\ref{eq:sb0}).
The first and the second rows are obtained by the fitting without enhancement, and by using
the screening potential $U_e$ in the adiabatic limit, respectively.}
\label{tab:dp-2}
\begin{tabular}{llllrlc}
\hline
$\Pi_{0f}(0)$ & $\Pi_{1f}(0)$ & $d_0$ & $d_1$ & $U_e$~(eV) & $\chi_{\nu}^2$ & $S_{b}(0)$~(eVb) \\
\hline
2.5$\times$10$^{-8}$ & 3.4$\times$10$^{-8}$ & 3.4 & 4.0 & 0.0 & 2.7 & 0.25 \\
2.3$\times$10$^{-8}$ & 3.6$\times$10$^{-8}$ & 3.7 & 3.9 & 40.8, 0.0 & 2.8 & 0.25 \\
\hline
\end{tabular}
\end{table}
The fitting parameters obtained using a quadratic polynomial with the adiabatic enhancement are shown in the
third row in Table~\ref{tab:Dp}
and it is shown with the dashed curve in the top panel in Fig.~\ref{fig:dp}.
Because the enhancement is less than 7\% even at the lowest measured incident energy,
it changes insignificantly the zero-energy $S$-factor:
$S(0)$=0.261 $\pm$ 0.006~(eVb) obtained by neglecting
the enhancement differs only slightly from
the bare $S$-factor at zero-energy $S_{b}(0)$=~0.256 $\pm$ 0.006~(eVb) from our fitting procedure.
$S_{b}(0)$=~0.256 $\pm$ 0.006~(eVb) is
slightly higher than
the result from the $R$-matrix fit $S_{b}(0)$=~0.223$\pm$0.010~(eVb) in Ref.~\cite{daacv}, which is obtained
as a sum of M1 and E1 contributions.
Limiting the fitting range from 0.0025~MeV to 1~MeV, $S_b(0)$ is insensitive to the choice of the
degree of the polynomial. However
the $S_b(0)$ obtained using polynomials is quite sensitive to the choice
of both the upper and the lower limits of the fitting range.
We performed the fitting procedures using Eq.~(\ref{eq:sb0}).
This fitting procedure without enhancement and with the adiabatic enhancement factor
lead the fitting parameters and the $S_{b}(0)$ in Table~\ref{tab:dp-2}.
We have used the $l=$~0 and 1 states.
The obtained $S_{b}(0)$ are essentially the same for two cases and are in agreement with the extrapolations
using quadratic polynomials.
The radial parameters $d_{0}$ and $d_{1}$ for both cases are larger than one.
This can be interpreted
because the effective radius of deuteron is larger than the one given by the empirical formula.
The curve obtained by this fitting procedure for the adiabatic enhancement is shown in the bottom
panel in Fig.~\ref{fig:dp}.
\subsection{$^3$\rm{He}($\alpha,\gamma$)$^7$\rm{Be}}
\label{sec:3he4he}
The $S$-factor of the reaction $^3$He($\alpha$,$\gamma$)$^7$Be has been investigated
recently both by the activation~\cite{bemmerer:122502,gyurky:035805} and
the prompt methods~\cite{confortola:065803}.
The latter confirmed that there is no discrepancy between the obtained $S$-factors by
two different methods.
They have discussed the electron screening enhancement factor in the adiabatic limit
in~\cite{gyurky:035805}, but the $S$-factor data has not been corrected by the effect.
The $S$-factor of the reaction $^3$He($\alpha,\gamma$)$^7$Be from several measurements
are shown with error bars in Fig.~\ref{fig:3he4he}.
The fitting parameters in~\cite{nacre} are shown in the first row of Table~\ref{tab:3he4he}.
The screening potential for the reaction $^3$He($\alpha,\gamma$)$^7$Be is estimated in the same way
with the reaction $^3$He($^3$He,2$p$)$^4$He.
The estimated enhancement at the minimum incident energy within the adiabatic limit is 2\% at utmost.
\begin{figure}
\includegraphics[height=.7\textheight]{3he4he.ps}
\caption{$S$-factor for the reaction $^3$He($\alpha,\gamma$)$^7$Be as a function of the incident c.m. energy.
The experimental points are
from~\cite{pa63}(Parker63), from~\cite{kr82}(Kraewinkel82), from~\cite{os82}(Osborne82), from~\cite{hi88}(Hilgemeier88),
from~\cite{bemmerer:122502}(Bemmerer06) and from~\cite{confortola:065803}(Confortola07).
In the top panel the solid curve represents our fit by polynomial expression with the adiabatic enhancement.
The dashed curve corresponds to the bare $S$-factor.
In the bottom panel the curves are same with ones in the top panel, but using Eq.~(\ref{eq:sb0}) instead of
the polynomial expression.}
\label{fig:3he4he}
\end{figure}
We performed the fit of the data
in the incident energy region from 0.1072~MeV to 1~MeV
using quadratic polynomials with the adiabatic enhancement factor.
The obtained fitting parameters are shown in the second row of Table~\ref{tab:3he4he}.
We have performed the same fit but without enhancement factor and
obtained the same $S_b(0)$ as the one with the adiabatic enhancement.
The obtained $S_b(0)$ coincides with the one in~\cite{nacre} within the error.
However $S_b(0)$ is rather sensitive to the choice of the fitting range, especially to the choice
of the lower limit.
$S_b(0)$ is insensitive to the choice of the degree of polynomial in the selected fitting range.
In the top panel in Fig.~\ref{fig:3he4he} we have shown the results of the fitting using the adiabatic
enhancement factor.
The $S$-factor at zero-energy $S_{b}(0)$=0.49 $\pm$ 0.01~(keVb) from our procedure is in agreement with
the result from the $R$-matrix fitting 0.51$\pm$0.04~(keVb) in Ref.~\cite{daacv}.
\begin{table}
\caption{Fitting parameters of the reaction $^3$He($\alpha,\gamma$)$^7$Be.
The two rows are obtained by using polynomial expression
and from~\cite{nacre}~(the first row), with the adiabatic approximation~(the second row).}
\label{tab:3he4he}
\begin{tabular}{lllrl}
\hline
$S_{b}(0)$ (keVb) & $S_{1}$ (b) & $S_{2}$ (keV$^{-1}$b) & $U_e$ (eV) & $\chi_{\nu}^2$\\
\hline
0.54 $\pm$ 0.09 & -0.52 & -0.52 & 0.0 & (NACRE)\\
0.50 $\pm$ 0.01 & -0.35 $\pm$ 0.04 & 0.13 $\pm$ 0.03 & 246.8 & 0.0014\\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Fitting parameters and the $S_{b}(0)$ of the reaction $^3$He($\alpha,\gamma$)$^7$Be
by using the fitting procedure Eq.~(\ref{eq:sb0}).
The first and the second rows are obtained by the fitting without enhancement, and by using
the screening potential $U_e$ in the adiabatic limit, respectively.}
\label{tab:3he4he-2}
\begin{tabular}{llllrlc}
\hline
$\Pi_{0f}(0)$ & $\Pi_{2f}(0)$ & $d_0$ & $d_2 $ & $U_e$~(eV) & $\chi_{\nu}^2$ & $S_{b}(0)$~(keVb) \\
\hline
5.1$\times$10$^{-7}$ & 4.9$\times$10$^{-7}$ & 4.0 & 2.5 & 0.0 & 2.3 & 0.51 \\
5.3$\times$10$^{-7}$ & 4.9$\times$10$^{-7}$ & 4.0 & 2.5 & 246.8 & 2.3 & 0.50 \\
\hline
\end{tabular}
\end{table}
The fitting parameters from the fitting procedures with Eq.~(\ref{eq:sb0}) are shown
in Table~\ref{tab:3he4he-2}. We have used the $l=$0, 2 partial-wave contributions.
This procedure using the enhancement factors with $U_e^{(AD)}=$ 246.8, 0.0~(eV) gives zero
energy $S$-factors 0.50 and 0.51~(keVb), respectively. Both are in agreement with the result
from the polynomial fitting procedure.
The radial parameters $d_{0}$ and $d_{2}$ for both cases are larger than one.
The curve obtained by this fitting procedure with the adiabatic enhancement is shown in the bottom
panel in Fig.~\ref{fig:3he4he}.
The obtained zero energy $S$-factor is smaller than $S(0)=0.560\pm 0.017$~(keV) in~\cite{confortola:065803}.
This is because their result is obtained by a normalization to their data.
\subsection{$^7$\rm{Be}($p$,$\gamma$)$^8$\rm{B}}
\label{sec:7bep}
The reaction $^7$Be($p$,$\gamma$)$^8$B is a key process to produce the high energy solar neutrinos through
the $\beta$-decay of $^8$B. The $S$-factor of this reaction is investigated intensively by many groups by means
of the direct capture~(DC)
reaction~\cite{baby:065805,ju03}, the indirect Coulomb dissociation~(CD) method~\cite{PhysRevLett.83.2910,schumann:015806}
and the asymptotic normalization coefficients~(ANCs)~\cite{trache:062801}.
It was claimed that the experimental data of the $S$-factor by CD experiments gives a steeper energy dependence than that by
DC experiments in the low-energy region and the lower zero-energy $S$-factor as an average~\cite{ju03}.
Recently this inconsistency has been resolved by reanalyzing data by the Coulomb dissociation method~\cite{schumann:015806}.
\begin{figure}
\includegraphics[height=.7\textheight]{7bep8bg.ps}
\caption{$S$-factor for the reaction $^7$Be($p$,$\gamma$)$^8$B as a function of the incident c.m. energy.
The experimental points are from~\cite{ju03}(Junghans03) and from~\cite{schumann:015806}(Schuemann06).
In the top panel the solid curve represents our fit by polynomial expression with the adiabatic enhancement.
The dashed curve corresponds to the bare $S$-factor.
In the bottom panel the curves are same with ones in the top panel, but using Eq.~(\ref{eq:sb0}) instead of
the polynomial expression.}
\label{fig:7bep}
\end{figure}
For the purpose of the extrapolation of zero-energy $S$-factor,
both the DC and the CD experiments above mentioned use the microscopic cluster model~\cite{desc} and
give the zero-energy $S$-factor 22.1 $\pm$ 0.6(expt.) $\pm$ 0.6(theor.)~(eVb)
[$21.4 \pm 0.5$(expt.) $\pm$ 0.6(theor.)~(eVb) as a mean of all modern direct measurements]
and 20.6 $\pm 0.8$~(stat.) $\pm$ 1.2~(syst.) (eVb), respectively.
Let us remind you that the purpose of this paper is not a precise
determination of the $S$-factor but to see the effect of the electron screening on the
determination of the $S$-factor.
So that we rather use the consistent approaches with the other reactions than employ a special
treatment for this reaction.
However we, at least, need to include the resonances to analyzing the DC data~\cite{ju03}.
For this purpose we use the Breit-Wigner formula Eq.~(\ref{eq:bw}).
Moreover it is well known that this reaction has a low-energy bound state in the $^8$B~\cite{PhysRevC.58.3711, PhysRevC.61.045801}. To take into account this state, we use
\begin{eqnarray}
S_b(E)= \frac{S_{-1}}{E_B+E}+S_0+S_1E, \label{eq:sbf2w}
\end{eqnarray}
where $E_B=0.1375$ (MeV)~\cite{PhysRevC.58.3711}, in place of the polynomial expression.
We make a special mention of $^7$Be metallic target being used in the experiment~\cite{ju03}.
The experimental data of the $S$-factor in Ref.~\cite{ju03} is fitted with
the Breit-Wigner single-level resonance formula for 1$^+$ and 3$^+$ resonances plus Eq.~(\ref{eq:sbf2w}).
We use the resonance parameters in Ref.~\cite{ju03} and
obtained $S_b(0)=$ 20.8~(eVb) with $\chi_{\nu}^2=$0.3 from the fitting procedure
without enhancement factor.
Assuming the adiabatic enhancement,
the screening enhancement factor is of the order of 1\% at the minimum incident energy of the
experiment in~\cite{ju03}. By making use of the enhancement factor with the adiabatic screening
potential $U_e^{(AD)}=$ 222.0~(eV), we obtain $S_b(0)=$ 20.5~(eVb) with $\chi_{\nu}^2=$0.3.
The corresponding bare $S$-factor is shown with the dashed curve in the top panel in Fig.~\ref{fig:7bep}.
The difference between two $S_b(0)$ using different screening potentials is less than 2\%.
Considering the isotopic independence of the electron screening,
we, tentatively, use the screening potential obtained by the measurement of the reaction
$^9$Be($p,\alpha$)$^6$Li: $U_e$=900$\pm$50~(eV)~\cite{zahnow}.
The fitting procedure gives $S_b(0)=$ 19.7~(eVb) with $\chi_{\nu}^2=$ 0.3.
This is 4\% smaller than the former two results.
We summarize the fitting parameters in above fitting procedures in Table~\ref{tab:7bep}.
\begin{table}
\caption{Fitting parameters and $S_b(0)$ of the reaction $^7$Be($p$,$\gamma$)$^8$B by using
the polynomial Eq.~(\ref{eq:sbf2w}). The four rows are obtained from~\cite{nacre}(the first row),
without enhancement~(the second row), with the adiabatic approximation~(the third row),
and with the screening potential $U_e$= 900~(eV)~\cite{zahnow}~(the fourth row).}
\begin{tabular}{llllrll}
\hline
{$S_{-1}$~(eV$^2$b) } & {$S_0$~(eVb) } & {$S_{1}$~(b) } & {$S_{2}$ } & $U_e$~(eV) & $\chi^2_{\nu}$ & $S_b(0)$\\
\hline
& 21. $\pm$ 2. & 18. & 38. & 0.0 & (NACRE) & 21. $\pm$ 2. \\
0.6 $\pm$ 0.2 & 15.7 $\pm$ 0.6 & 8.3 $\pm$ 0.4 & & 0.0 & 0.3 & 20.8 \\
0.7 $\pm$ 0.2 & 15.7 $\pm$ 0.6 & 8.3 $\pm$ 0.4 & & 222.0 & 0.3 & 20.5 \\
0.7 $\pm$ 0.2 & 16.0 $\pm$ 0.6 & 8.1 $\pm$ 0.4 & & 900.0 & 0.3 & 19.7 \\
\hline
\end{tabular}
\label{tab:7bep}
\end{table}
\begin{table}
\caption{Fitting parameters and the $S_{b}(0)$ of the reaction $^7$Be($p,\gamma$)$^8$B by using the
fitting procedure Eq.~(\ref{eq:sb0}) with resonant terms.
The three rows are obtained by the fitting without enhancement~(the first), and by using
the screening potential $U_e$ in the adiabatic limit~(the second), and
with the screening potential $U_e$= 900~(eV)~(the third), respectively.}
\label{tab:7bep-2}
\begin{tabular}{llllrlc}
\hline
$\Pi_{0f}(0)$ & $\Pi_{2f}(0)$ & $d_0$ & $d_2$ & $U_e$~(eV) & $\chi_{\nu}^2$ & $S_0$~(eVb) \\
\hline
2.2$\times$10$^{-8}$ & 2.5$\times$10$^{-4}$ & 0.9 & 0.2 & 0.0 & 0.4 & 22.4 \\
2.2$\times$10$^{-8}$ & 2.3$\times$10$^{-4}$ & 0.9 & 0.2 & 222. & 0.4 & 22.3 \\
2.2$\times$10$^{-8}$ & 2.0$\times$10$^{-4}$ & 0.9 & 0.2 & 900. & 0.5 & 22.1 \\
\hline
\end{tabular}
\end{table}
We perform the fitting procedures of the experimental data in Ref.~\cite{ju03}
using Eq.~(\ref{eq:sb0}) including 1$^+$ and 3$^+$
resonances plus another resonance with negative energy -0.1375~(MeV), which corresponds
to the pole term in Eq.~(\ref{eq:sbf2w}).
\begin{eqnarray}
\label{eq:fres}
S_b(E)&=&\frac{\pi\hbar^2}{2\mu}(\Pi_{0f}(E)e^{W_0}+5\Pi_{2f}(E)e^{W_2}) \nonumber \\
&+&\frac{c_{r1}e^{W_0}\Pi_{0f}(E)}{(E+0.1375)^2}
+\frac{c_{r2}e^{W_0}\Pi_{0f}(E)}{(E-0.630)^2+\Gamma^2/4}
+\frac{c_{r3}e^{W_2}\Pi_{2f}(E)}{(E-2.183)^2+\Gamma^2/4}
\end{eqnarray}
where $c_{r1}, c_{r2}$ and $c_{r3}$ are scaling factors in Eq.~(\ref{eq:bw2}).
Our fitting procedures using different screening potentials give common
$c_{r1}=5.\times 10^{-7}$ (MeV$^2$), $c_{r2}=2.\times 10^{-9} $(MeV$^2$) and
$c_{r1}=2.\times 10^{-6}$ (MeV$^2$).
In the bottom panel of Fig.~\ref{fig:7bep}
the thin dotted and dot-dashed curves show the contributions from resonances, including
the negative-energy resonance, and the non-resonant part, respectively.
We have used the $l=$~0 and 2 partial-waves.
The fitting procedure using Eq.~(\ref{eq:fres})
without enhancement, with the adiabatic enhancement factor, and with $U_e$= 900~(eV)
lead the fitting parameters and the $S_{b}(0)$ in Table~\ref{tab:7bep-2}.
Despite of the difference of the utilized screening potentials, neither the obtained zero-energy
$S$-factor nor $\chi_{\nu}^2$ does have much differences.
The values $\Pi_{2f}(0)$ are much larger than $\Pi_{0f}(0)$, i.e.,
the $d$-wave contribution is dominant in the energy region investigated.
The radial parameter $d_{0}$ is smaller than one, and $d_{1}$ is smaller than $d_{0}$ for
all three cases.
The curve obtained by this fitting procedure for the adiabatic enhancement is shown in the bottom
panel in Fig.~\ref{fig:7bep}.
The obtained $S_b(0)$ using different screening potentials are in accordance with the result with
the microscopic cluster model~\cite{desc} $S_b$(0)=22.1 $\pm$ 0.6(expt.) $\pm$ 0.6(theor.)~(eVb) in Ref.~\cite{ju03} within the error-bar.
From the tree results of all the radiative capture reactions considered,
D($p$,$\gamma$)$^3$He, $^3$He($\alpha,\gamma$)$^7$Be, and $^7$Be($p,\gamma$)$^8$B,
one can say that the obtained $S_b(0)$ is insensitive if
the screening enhancement, within the adiabatic approximation, is taken into account or not.
For all the reactions investigated in this paper the fitting procedure with the polynomial
expression is more sensitive to the difference of the screening potential than the fitting procedure
with Eq.~(\ref{eq:sb0}).
\section{Conclusions}
\label{sec:sum}
We discussed the bound electron screening corrections to the bare $S$-factors of the reactions in PP-chains.
Our approach is based on fitting procedures of the experimental data.
For this purpose we employed two different fitting procedures: one is the conventional polynomial expressions
and the other includes explicitly the contribution of the nuclear interaction and based on the statistical model
to describe exit channels.
The later fitting procedure works especially well for the reactions that have a dominant $s$-wave entrance channel component.
We have applied different types of screening enhancements:
in the adiabatic limit, determined through a fit and larger than adiabatic
screening potentials, as well.
From the tree results of all the transfer reactions considered,
$^3$He($^3$He,2$p$)$^4$He, $^7$Li($p,\alpha$)$^4$He, and $^6$Li($p$,$\alpha$)$^3$He,
the enhancement by the screening is crucial, in the sense that the fitting
procedure without enhancement gives $\chi_{\nu}^2$ larger than the others in which the screening
enhancement is taken into account. However
the obtained $S_b(0)$ is insensitive to the magnitude of the screening potential.
Especially for the radiative capture reactions D($p$,$\gamma$)$^3$He
and $^7$Be($p,\gamma$)$^8$B,
the screening correction within the adiabatic approximation has been considered for the first time.
However the results
suggest that the obtained $S_b(0)$ is insensitive whether the screening enhancement, within the adiabatic
approximation, is taken into account or not.
Making a comparative study of the bare $S$-factors obtained by two-ways of fitting procedures
using different screening enhancement factors, we found that
all $S_b(E)$ coincide within the standard errors.
$S_b(E)$ is, practically, insensitive to the magnitude of the screening potential.
\bigskip
The authors acknowledge Prof. S. Kubono for the suggestion of the problem and valuable comments.
One of us~(S. K.) thanks Dr. H. Costantini and Dr. R. G. Pizzone for stimulating discussions and
for providing us experimental data.
|
1,314,259,996,722 | arxiv | \section{Introduction}
Some years ago the Yang-Mills gradient flow was introduced to the Lattice Field
Theory community by M.L\"uscher~\cite{Luscher:2010iy} as an additional tool for
studying the dynamics of gauge theories non-perturbatively.
Since then the number of applications of the YM gradient flow to
probe non-perturbative aspects of non-Abelian gauge theories using old and new
ideas steadily increases --- as can be seen in the various contributions to
this conference for instance.
So far most of them are influenced by~\cite{Luscher:2010iy} and the all-order
proof in perturbation theory given in~\cite{Luscher:2011bx} which tells us that
the gauge field that is generated by the flow equations
\begin{subequations}
\begin{align}\label{eq:DEQ}
\dfrac{ {\rm d}B_\mu(x,t)}{{\rm d}t} &= D_\nu G_{\nu\mu}(x,t) \;, & t&> 0 \;,\\
G_{\mu\nu}(x,t) &= \partial_\mu B_\nu - \partial_\nu B_\mu + [B_\mu,B_\nu] \,, & D_\nu &= \partial_\nu + [B_{\nu},\star] \;, \\[0.2em]
B_\mu(x,t)|_{t=0} &= A_\mu(x) \,, \label{eq:IC}
\end{align}
\end{subequations}
does not require renormalization. Here $A_{\mu}(x)$ is the fundamental gauge
field of the underlying theory and the flow field $B_{\mu}(x,t)$ is the
solution to \eqref{eq:DEQ} subject to the initial condition~\eqref{eq:IC}.
On the lattice the gradient flow is also referred to as Wilson flow if the
Wilson plaquette gauge action has been used to define the flow equations in
terms of parallel transporters~\cite{Luscher:2009eq,Luscher:2010iy}. In that
case it has rigorously been shown that the action is a monotonically decreasing
function of $t$ and the flow thus represents a smoothing operation of the
initial gauge field.
\section{Definition of a new coupling}
Here we are especially interested in one of the applications that were already
mentioned in~\cite{Luscher:2010iy}, namely to use the energy density
\begin{align}
\left\langle E(t) \right\rangle &\equiv \frac{1}{4} \left\langle {G}_{\mu\nu}(t){G}_{\mu\nu}(t) \right\rangle
= \dfrac{3(N^2-1)}{2(8\pi t)^{2}} \times \gbsq_{\rm MS}(\mu) \Big\{ 1 + c_1 \gbsq_{\rm MS} + \Or\big(\gbar_{\rm MS}^4\big) \Big\}
\end{align}
at positive flow time to define a running coupling. This perturbative expansion
of the energy density in terms of a renormalized coupling is given for 4
Euclidean space-time dimensions in infinite volume for gauge group $SU(N)$ at
scale $\mu=1/\sqrt{8t}$, where $\sqrt{8t}$ is the mean-square radius over which
the gauge field is effectively smoothed. This obviously can serve as a
definition of a non-perturbatively renormalized coupling after switching from
dimensional regularisation to the lattice as a regulator, leading to the
gradient flow coupling
\begin{align}
\left\langle t^2 E(t) \right\rangle &\equiv \mathcal{N} \times \gbsq_{\rm GF}(\mu) \;,
& \mu&=1/\sqrt{8t} = 1\big/ cL \;.
\end{align}
The energy density as a gauge invariant quantity scales $\propto t^{-2}$
and is a renormalized quantity at positive flow time. The normalization
constant $\mathcal{N}$ has to be chosen such that $\gbsq_{\rm GF} = g_0^2 +
\Or(g_0^4)$ holds and can be computed by expanding the energy density in the
bare gauge coupling as defined through the field strength tensor
$G_{\mu\nu}(t)$. In a finite volume one has the additional length scale $L$,
the physical size of a hypercube with volume $V=L^4$, which in a finite size
scaling procedure usually sets the renormalization scale in some well-defined
way. It is necessary to fix the factor $c=\sqrt{8t}/L$ that defines the
effective smoothing range in terms of the physical extent.
In a finite volume, boundary conditions of the field variables become important
and the energy density has in fact already been used to define a running coupling
with periodic boundary conditions~\cite{Fodor:2012td}. Unfortunately, this
leads to a definition of the coupling that is non-analytic in $g_0^2$ and thus
has a non-universal 2--loop coefficient in the QCD $\beta$-function. It
considerably complicates the perturbative computations beyond tree-level that
are needed to safely relate the $\Lambda$-parameter of this scheme to
$\Lambda_{\overline{\rm MS}}$.
As it is known for a long time, this behaviour can be avoided from the start by either
using twisted boundary conditions or the Schr\"odinger functional as
finite-volume renormalization scheme~\cite{Luscher:1992an} where Dirichlet
boundary conditions are imposed at the time boundaries. One of the authors was
also working on a computation of the gradient flow coupling using twisted
boundary conditions and presented its results for the full step-scaling
function in $SU(2)$ pure gauge theory~\cite{Alberto:Lat13} that shows a
promising accuracy towards a new computation of the running coupling and the
Lambda parameter in QCD.
\section{The gradient flow coupling in the SF}
The Schr\"odinger functional (SF) is the Euclidean propagation kernel of some
field configuration at Euclidean time $x_0=0$ to $x_0=T$ where $T$ is the
extent of the finite-volume world in time direction. In the spatial direction,
gauge fields have periodic boundary conditions with period $L$ while Dirichlet
boundary conditions are imposed to the spatial components of the gauge fields
at $x_0=0,T$. Accordingly time translation invariance is lost and all physical
observables explicitly depend on the Euclidean time. In our setup we are only
considering vanishing boundary fields. This means that for a spatial Fourier
transformed flow field, $\tilde B_{\mu}(\mathbf{p},x_0,t)$, one would impose
\begin{align}
\forall\mathbf{p}: && \tilde B_k(\mathbf{p}, x_0, t)\big|_{x_0=0,T} &=0 \;,
\intertext{while the boundary condition of the time component emerges through the gauge fixing procedure,}
\mathbf{p} \ne 0 : && \partial_0 \tilde B_0(\mathbf{p}, x_0, t)\big|_{x_0=0,T} &=0 \;, \\
\mathbf{p} = 0 : && \tilde B_0(\mathbf{0}, x_0, t)\big|_{x_0=0\hphantom{,T}} &=0 \;,
& \partial_0 \tilde B_0(\mathbf{0}, x_0, t)\big|_{x_0=T} &=0 \;. \label{eq:cont-B0-p=0}
\end{align}
This is best seen by starting from the lattice formulation and taking the
continuum limit. For additional details we have to refer the interested reader
to our paper~\cite{Fritzsch:2013je} and references therein.
As mentioned earlier the normalization factor $\mathcal{N}$ is obtained by
expanding the flow field and thus the energy density in terms of the bare
coupling. At $t>0$ this reads
\begin{align}
B_{\mu} &= \sum_{n=1}^{\infty} B_{\mu,n} g_0^{n} \;, &
\langle E(t,x_0) \rangle &= \sum_{n=0}^{\infty} \mathcal E_n(t,x_0) \;,~\text{with}\quad\mathcal E_n=\Or\big(g_0^{2+n}\big) \;,\\
&& {\cMagenta\text{LO:}}\quad
\mathcal E_0(t,x_0) &= \frac{g_0^2}{2}\langle
\partial_{\mu}B_{\nu,1}^a\partial_{\mu}B_{\nu,1}^a -
\partial_{\mu}B_{\nu,1}^a\partial_{\nu}B_{\mu,1}^a
\rangle \;, \label{eq:E0}
\end{align}
and only the leading order (LO) will contribute to $\mathcal{N}$. Inserting
the expansion of $B_{\mu}$ into the flow equation reads to leading order
\begin{align}
\dfrac{{\rm d}}{{\rm d}t} \tilde B_{\mu,1}(\mathbf p, x_0, t) &= (-\mathbf p^2 + \partial_0^2)
\tilde B_{\mu,1}(\mathbf p, x_0, t) \;,&
\tilde B_{\mu,1}(\mathbf p, x_0, t)\big|_{t=0} &= \tilde A_{\mu}(\mathbf p, x_0) \;,
\end{align}
which evidently is a heat equation. Solutions to these kind of equations
have been known for a long time and can be written in terms of heat kernels.
Those that enter our observable can be written as
\begin{align}
\tilde B_{\mu,1}(\vecp,x_0,t) &= {\rm e}^{-\vecp^2 t} \int^T_0\!\!{\rm d}x_0^\prime\, K^{D,N}(x_0,x_0^\prime,t) \, \tilde A_{\mu}(\vecp,x_0^\prime,t)
\;,\qquad (\mathbf{p}\ne 0)
\end{align}
where $K^{D}$, $K^{N}$ are heat kernels consistent with {\it Dirichlet}
$(\mu=1,2,3)$ and {\it Neumann} $(\mu=0)$ boundary conditions, respectively,
at $x_0=0,T$. Inserting these into eq.~\eqref{eq:E0}
leads to partial derivatives of the heat kernels convoluted with the standard
gluon propagator w.r.t. the fundamental gauge fields $A_{\mu}$. Since the
latter is known analytically everything can be written down in closed form and
the normalization factor be read-off. In general the continuum normalization
factor $\mathcal{N}$ depends on the parameter $c$, the time-slice used in
the evaluation of $E(x_0,t)$ and the geometry that is applied, i.e., the ratio
$T/L$. It is most advantageous to work at $x_0=T/2$ with $T=L$. Useful choices
for $c$ will be discussed below.
To ease notation and the discussion of the overall computation we worked in the
continuum but the same procedure is carried over to the lattice formulation.
There the corresponding norm $\hat{\mathcal{N}}$ will explicitly depend on the
lattice size $L/a$ and implicitly on further details as the lattice action
and the actual discretisation of the energy density that has been used.
In~\cite{Fritzsch:2013je} we have done this computation for the Wilson plaquette
gauge action and the clover definition of the energy density. Our results have
also been checked by using some dedicated small coupling simulations that
show the correct asymptotic behaviour.
\section{Non-perturbative results}
To immediately learn more about the general non-perturbative behaviour of the
gradient flow coupling in the SF as defined earlier we decided to compute the
corresponding observable $E(x_0,t)$ for different values of the smearing ratio
$c$ on existing $\nf=2$ gauge field configurations for lattice sizes
$L/a=6,8,10,12,16$ with $T=L$. These have been set up along a trajectory in the
space of bare couplings at which the traditional SF coupling and PCAC quark
mass are fixed to
\begin{align} \label{eq:LCP}
{\rm \cRedDark LCP:}\qquad
\overline{g}^{2}_{\rm SF}(L_1) &\equiv u = 4.484 \qquad\text{and}\qquad m(L_1) = 0 \;,
\end{align}
corresponding to a physical box size of $L_1\sim 0.4\,\fm$. Our observable of
interest thus reads
\begin{align}
\Omega(u;c,a/L) &=
\left[
\hat{\mathcal{N}}^{-1}(c,T/L,x_0/T,a/L)\cdot t^2\big\langle
E(T/2,t)\big\rangle
\right]_{\raisebox{0em}{\scriptsize$t=c^2L^2/8$}}^{\raisebox{0em}{\scriptsize{\cRedDark \text{LCP}}}}
\end{align}
and we show some selected data in figure~\ref{fig:CL}. There the by far
dominating part of the overall error budget ($\gtrsim 85\%$) comes from
propagating the uncertainty of setting up a line of constant physics (LCP)
according to~\eqref{eq:LCP}. The continuum limit is taken without the coarsest
lattice $L/a=6$. We observe the following: (a) below $c=0.3$ we see deviations
from the naive leading scaling expectation $(a^{-2})$ that gets stronger and
stronger with decreasing smearing range, (b) in the range $0.3\le c \le 0.5$
relative cutoff effects stay below $10\%$ and decrease with increasing $c$, (c)
the uncertainty increases with increasing smoothing range.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{plots/CLextrap2}
\vskip-.75em
\caption{Exemplary results of $\Omega(u;c,a/L)$ with continuum limits $\omega(u;c)$ and its dependence on $c$.}
\vskip.25em
\label{fig:CL}
\end{figure}
\pagebreak
\vskip1em
\noindent{\bf The relative variance}
\vskip0.5em
In order to judge how accurately the continuum limit can be reached with the SF
gradient flow coupling compared to the traditional SF coupling, we also
computed the relative variance that for any observable $\mathcal{O}$ is given by
\begin{align}
\mathcal{V_O} &= \dfrac{\langle \mathcal O^2 \rangle - \langle \mathcal O \rangle^2}{\langle \mathcal O \rangle^2} \;.
\end{align}
Since numerical prefactors cancel one has $\mathcal{V}_{\gbsq_{\rm
GF}}=\mathcal{V}_E$. Note that $\mathcal{V_O}$ is a genuine observable that
tells us about the statistical accuracy that can be achieved for an observables
$\mathcal{O}$ when the continuum limit is approached in contrast to the
integrated autocorrelation time $\tau_{\mathcal{{\rm int},\mathcal{O}}}$ that
tells us how the underlying algorithm performs for this observable.
\begin{figure}[t]
\centering
\includegraphics[height=0.2775\textwidth]{plots/RNS_v3}
\raisebox{0.38em}{\includegraphics[height=0.300\textwidth]{plots/VarianceComparison}}
\vskip-.75em
\caption{Relative variance of the SF gradient flow coupling compared to that of the traditional SF coupling.}
\label{fig:relvar}
\end{figure}
The results for our five lattices are shown in the left panel of
figure~\ref{fig:relvar}. As we have noted earlier one wants to avoid ---or is
unable--- to take the continuum limit for $c\ll 0.3$ due to large cutoff
effects and it seems that for say $c>0.2$ the variances $\mathcal{V}_E(a/L)$
fall onto a universal curve. A similar study has been done for the SF
coupling in the case of pure $SU(2)$ gauge theory~\cite{deDivitiis:1994yz},
showing that the variance of the SF coupling diverges towards the continuum
limit. In the right panel of figure~\ref{fig:relvar} we plot their results from
table~1 together with our result for the SF gradient flow coupling at $c=0.3$.
Note that both are obtained along a line of constant physics defined through a
large but slightly different SF coupling and that the overall behaviour of the
SF coupling is the same in QCD with dynamical flavours as studied in the case
of the gradient flow coupling.
\vskip1em
\noindent{\bf The integrated autocorrelation time of $\gbsq_{\rm GF}$}
\vskip0.5em
Our statistical errors always include an estimate of the integrated
autocorrelation time using the $\Gamma$-method~\cite{Madras:1988ei} and we
observe a scaling of
\begin{align}
\left. \tau_{{\rm int},\gbsq_{\rm GF}} \right|_{c=0.3} &\approx (L/a)^2 \times 0.05\,{\rm MDU}
\end{align}
that agrees with the scaling behaviour that is naively expected for a Hybrid
Monte Carlo simulation ($\propto a^{-2}$). At finite flow time also the
topological charge can be measured much more accurately and one could ask how
the different topological sectors couple to $\gbsq_{\rm GF}$, especially in
physically larger lattices, i.e., in the low energy regime of QCD. This question
is directly related to critical slowing down and was studied in more detail
in~\cite{Felix:Lat13}.
\vskip1em
\noindent{\bf Scaling of the numerical Wilson flow integrator scheme}
\vskip0.5em
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{plots/IntegratorScaling}
\caption{Scaling behaviour of standard Runge-Kutta integrator (RK3)~\cite{Luscher:2010iy} versus
adaptive step-size integrator (RK23)~\cite{Fritzsch:2013je} for an equivalent setup integrated up
to $c_{\rm max}=0.5$}
\label{fig:intscal}
\end{figure}
In order to integrate the associated flow equations, a first-order differential
equation in the gauge group, the Euler or any Runge-Kutta scheme can be used.
In~\cite{Fritzsch:2013je} we
extend the originally proposed Runge-Kutta scheme (RK3) with fixed
step-size~\cite{Luscher:2010iy} by nesting a 2nd order scheme to define an adaptive
step size scheme (RK23). Due to the smoothing property of the flow the step size
is safely increased with flow time. For
simulations that we are currently performing with lattices up to $L/a=64$ we
have collected the run time for measurements of Wilson flow observables up to a
fixed flow time and identical setup. The results are plotted in
figure~\ref{fig:intscal} and it can be easily inferred that for the largest
lattice a speed-up factor of $\sim 10$ is seen. Already on a $L/a=32$ lattice,
which may typically be used for the finest resolution in a step-scaling
procedure, a significant speed-up is achieved.
\section{Conclusions \& Outlook}
We have perturbatively computed the continuum and lattice behaviour of the energy
density at positive gradient flow time in the Schr\"odinger functional with
vanishing boundary fields~\cite{Fritzsch:2013je}. This allows us to define a new
finite-volume renormalization scheme.
From our studies we see that for wisely chosen flow parameter
$(0.25\lesssim c \lesssim 0.5)$ a controlled continuum limit can be taken.
Furthermore, we find strong numerical evidence that the new coupling
can be computed with high accuracy. We also observe that the variance of the
coupling is independent of $L/a$ which will improve continuum determinations
of observables such as the $\Lambda$-parameter. This new non-perturbative coupling
may also be very useful in the search for a conformal window in beyond the standard
model theories.
However, due to the highly improved statistical accuracy there are still many
corners to explore. For instance, in the past perturbatively computed boundary
improvement terms like $c_{t}$ and $\tilde{c}_t$ were not affecting the results
or error budget. We are currently investigating such issues with the tree-level
improved L\"uscher-Weisz gauge action.
\section*{Acknowledgments}
\small
This work is supported in part by the Deutsche Forschungsgemeinschaft under SFB/TR~9.
We gratefully acknowledge the computer resources provided by the John von Neumann
Institute for Computing as well as at HLRN and at DESY, Zeuthen.
|
1,314,259,996,723 | arxiv | \section{Introduction}
Let $f:M\to M$ be a diffeomorphism on some compact Riemannian manifold $M$.
An invariant probability $\mu$ of $f$ is a \emph{physical measure} if the set of points $z\in M$ for which
\begin{equation}\label{eq.basin}
\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}\to \mu \text{(in the weak$^*$
sense)}
\end{equation}
has positive volume. This set is denoted by ${\mathcal B}(\mu)$ and called the \emph{basin} of $\mu$.
In the present paper, we investigate the existence of existence and finiteness of physical measures in the
setting of partially hyperbolic diffeomorphisms. More precisely, we assume that there exists an
splitting $TM = E^u \oplus E^{cs}$ of the tangent bundle that is invariant under the tangent map $Df$ and
satisfies
\begin{equation}\label{eq.dfpartially}
\|(Df\mid_{E^{u}_x})^{-1})\| < 1 \text{ and } \|(Df\mid_{E^{u}_x})^{-1})\| \|Df \mid_{E^{cs}_x}\| < 1 \text{ at every } x \in M.
\end{equation}
In other words, the \emph{unstable bundle} $E^u$ is uniformly expanding, and it \emph{dominates}
the \emph{center-stable bundle} $E^{cs}$.
A program
for investigating the physical measures of partially hyperbolic diffeomorphisms was initiated by Alves, Bonatti, Viana
in \cite{BoV00,ABV00}. Their starting point was the observation that physical measures must be Gibbs $u$-states,
a notion they borrowed from Pesin, Sinai~\cite{PS82}. Not all Gibbs $u$-states are physical measures, but that is
easily seen to be the case for those Gibbs $u$-states whose center stable Lyapunov exponents are all negative.
Bonatti, Viana~\cite{BoV00} introduced the notion of \emph{mostly contracting center}, and proved that under this condition
there are finitely many ergodic Gibbs $u$-states, all with negative center Lyapunov exponents, and they are the
physical measures. Moreover, the union of their basins is a full volume set.
Further results on physical measures of diffeomorphims with mostly contracting center have been
obtained by Dolgopyat~\cite{Dol00}, Castro~\cite{Cas02}, Burns, Dolgopyat, Pesin~\cite{BDP02},
De Simoi, Liverani~\cite{DSL16}, Dolgopyat, Viana, Yang~\cite{DVY16} and others.
Here we take a different, although related viewpoint. One key new observation (Theorem~\ref{t.limit} below) is that,
for any $C^1$ diffeomorphism, every physical measure $\mu$ must be \emph{volume non-expanding} in the sense that
$$
\int_M \log|\det Df| \, d\mu \le 0.
$$
Thus, going back to the partially hyperbolic setting, we may restrict our analysis to volume non-expanding Gibbs
$u$-states. We say that the diffeomorphism $f$ is \emph{partially volume expanding} if
$$
\left|\det Df(x) \mid_{H}\right| > 1
$$
for any codimension-one subspace $H$ of $T_xM$ that contains $E_x^u$.
Being partially volume expanding is clearly a $C^1$ open property (the corresponding statement for mostly contracting
cente is more subtle, and was proven by Andersson~\cite{An10} and Yang~\cite{Yang-partial}).
Moreover, it is not difficult to find examples. For instance, let $S^1$ be the circle and $D$ be the $2$-dimensional disk.
Then consider an embedding $f_0: M \to M$ of the solid torus $M=S^1\times D$ of the form
\begin{equation}\label{eq_solenoid}
f_0(\theta,x)=\left(k\theta \mod 1, h_\theta(x)\right)
\end{equation}
where $k \ge 3$ is an integer larger and $h_\theta(x)$ is such that $\|Dh_\theta(x)\|$ and $\|Dh_\theta(x)^{-1}\|$
are both strictly less than $k$ at every point.
As we will check later, the first condition implies that $f_0$ is partially hyperbolic, and the second one ensures
that it is partially volume expanding.
\begin{main}\label{main.global}
Any partially volume expanding $C^{1+}$ diffeomorphism admits finitely many physical measures, the union of whose basins is
a full volume subset of the ambient manifold.
\end{main}
Several other results on the physical measures of partially hyperbolic maps have been obtained, especially in the setting
of maps \emph{mostly expanding cente} that was introduced by Alves, Bonatii, Viana~\cite{ABV00}.
Andersson, V\'asquez~\cite{AnV18} use a slightly stronger definition, which they prove is $C^2$ open.
The latter was improved by Yang~\cite{Yang-partial}, who proved $C^1$ openness. Bifurcation properties of the
physical measures have been studied by Andersson, V\'asquez~\cite{AnV} and Yang~\cite{Yang-expanding}
The main novelty of our present results, with respect to those more studied cases, is perhaps that the
notion of partial volume expansion requires no assumptions on the signs of the Lyapunov exponents:
it allows us to focus on the more significant Gibbs $u$-states, with negative center exponents,
and disregard all the other ones.
This comes with a price: while the conclusion of Theorem~\ref{t.limit}
remains true for nearby maps, just because the assumptions are $C^1$ open, we have no control on how the
number of physical measures unfolds under perturbation. This is in contrast with the case of diffeomorphisms
with mostly contracting center, where a very precise bifurcation theory for physical measures exists
(see \cite{DVY16,HYY,Yang-partial}) that describes the number and supports of physical measures for all the
perturbations of an initial map.
We call \emph{$u$-codimension} of a partially hyperbolic diffeomorphism the dimension of its center sub-bundle.
It is clear from the definition that any partially hyperbolic diffeomorphism with $u$-codimension 1 is
partially volume expanding. The theorem that follows provides a new way to establish the property of mostly
contracting center. In particular, it implies that the generalized solenoid $f_0:M \to M$ presented above
has mostly contracting center, which was not known previously.
\begin{main}\label{main.relation}
Every partially volume expanding $C^{1+}$ diffeomorphism that has $u$-codimension less than or equal to 2
has mostly contracting center.
\end{main}
In particular, any partially hyperbolic diffeomorphism with $u$-codimension 1 has mostly contracting center.
This does not seem to have been pointed out before.
It is easy to find examples (e.g. Smale solenoids) of $u$-codimension 2 partially hyperbolic diffeomorphisms
that have mostly contracting center, and yet fail to be partially volume expanding.
Although we ere not able to produce a counterexample, we believe that Theorem~\ref{main.relation}
does not extend to $u$-codimension 3.
In Section~\ref{local} we extend these ideas to a semi-local setting, namely to
partially hyperbolic attracting sets of embeddings $M\to{\rm int}(M)$ of compact
manifolds with boundary, including the generalized solenoid $f_0:M \to M$ in
\eqref{eq_solenoid}, and its perturbations. Note that our conditions on $f_0$
are a lot more flexible than in the usual construction of the Smale
solenoid~\cite{Shu87,Sma67}: in particular, they include many non-hyperbolic examples. In fact, it is clear that the same ideas can be applied to more general solenoids,
including the natural extensions of expanding maps on (possibly branched)
manifolds studied by Williams~\cite{Wil74}, Bonatti, Pumari\~{n}o, Viana~\cite{BPV00}
and Du~\cite{Du13}. We are grateful to Bin Yu for pointing this out to us.
\section{Preliminaries}
In this section, we collect a number of classical notations and facts that are useful for our arguments.
We start by reinterpreting the partial volume expansion assumption.
\subsection{Partial volume expansion}
Let $f:M\to M$ be a $C^1$ diffeomorphism and $\mu$ be an invariant probability measure.
For $\mu$-almost every $x$, let $l=l(x)\ge 1$ and $\lambda_1(x) > \cdots > \lambda_l(x)$ be the Lyapunov exponents and
$$
T_xM = E_x^1 \oplus \cdots \oplus E_x^l
$$
be the Oseledets splitting, given by the Oseledets multiplicative ergodic theorem (Oseledets~\cite{Ose68},
see \cite[Theorem~4.2]{LLE}).
Denote $\Delta(x) = \sum_{i=1}^l \lambda_i(x) \dim E_x^i$. The Oseledets theorem also gives that
\begin{equation}\label{eq.Delta}
\lim_n \frac 1n \log \left|\det Df^n(x)\right| = \Delta(x)
\end{equation}
for $\mu$-almost every $x$. Similarly, for any hyperplane $H \subset T_x M$ there exists $i$ such that
\begin{equation}\label{eq.used1}
\lim_n \frac 1n \log \left|\det Df^n(x) _{\mid H}\right| = \Delta(x)-\lambda_i(x).
\end{equation}
It is equally clear that, for any given $i\in\{i, \dots, l\}$ a hyperplane $H$ such that this identity holds.
Now take $f\in{\rm Diff}^1(M)$ to admit a partially hyperbolic $T_x M = E^u \oplus E^{cs}$.
Then there exists $k = k(x) \in\{1, \dots, l-1\}$ such that
$$
E^u(x)=E^1_x\oplus \cdots \oplus E^k_x
\text{ and }
E^{cs}(x)=E^{k+1}_x \oplus \cdots \oplus E^l_x.
$$
We refer to $\lambda_{k+1}(x), \dots, \lambda_l(x)$ as the \emph{center Lyapunov exponents} at $x$.
The partial volume expansion assumption means that $\log\left|\det Df(x)_{\mid H} \right|$ is positive
for any $x\in M$ and any hyperplane $H$ containing the unstable subspace $E_x^u$.
Since the set of such pairs $(x,H)$ is compact (keep in mind that $x \mapsto E_x^u$ is continuous),
we get that there exists $c>0$ such that
\begin{equation}\label{eq.used3}
\log\left|\det Df(x)_{\mid H} \right| \ge c
\text{ for any $x\in M$ and any hyperplane $H\supset E_x^u$.}
\end{equation}
As the set of such pairs $(x,H)$ is also invariant under iteration, it follows that $c$ is a lower
bound for the limit in \eqref{eq.used1}. Thus, our assumption implies that
\begin{equation}\label{eq.used2}
\Delta(x) - \lambda_i(x) \ge c \text{ for $\mu$-almost every $x\in M$ and any $i\in\{k+1, \dots, l\}$}
\end{equation}
\subsection{Gibbs $u$-states}
Following Pesin and Sinai~\cite{PS82} and Bonatti and Viana~\cite{BoV00} (see also~\cite[Chapter 11]{Beyond}),
we call \emph{Gibbs $u$-state} any invariant probability measure whose conditional probabilities
along strong unstable leaves are absolutely continuous with respect to the Lebesgue measure on the leaves.
It is shown in \cite[Section 11.2]{Beyond} (see also Dolgopyat~\cite{Dol04a}) that every physical measure
of a $C^{1+}$ partially hyperbolic diffeomorphism is a Gibbs $u$-state. A partial converse is true:
any ergodic Gibbs $u$-state whose center Lyapunov exponents are all negative is a physical measure.
Let ${\rm Gibb}^u(f)$ denote the space of all Gibbs $u$-states. By an \emph{unstable disk} we mean any
embedded disk contained in some unstable leaf of $f$.
By \emph{empirical measures} of a point $x\in M$ we mean any accumulation point of the sequence
${n}^{-1}\sum_{i=0}^{n-1}\delta_{f^i(x)}$.
The proofs of the following basic properties can also be found in~\cite[Section 11.2]{Beyond}:
\begin{proposition}\label{p.Gibbsustates}
Let $f$ be a $C^{1+}$ partially hyperbolic diffeomorphism. Then
\begin{itemize}
\item[(1)] ${\rm Gibb}^u(f)$ is non-empty, weak$^*$ compact and convex. Ergodic components of Gibbs $u$-states are Gibbs u-states.
\item[(2)] The support of every Gibbs $u$-state is ${\mathcal F}^u$-saturated, that is, it consists of entire unstable leaves.
\item[(3)] For Lebesgue almost every point $x$ in any unstable disk, every empirical measure $\nu_x$ is a Gibbs $u$-state.
\item[(4)] For every unstable disk $D^u$, any weak$^*$ limit of the sequence of measures
$$
\frac{1}{n}\sum_{i=1}^n (f^i)_* (\operatorname{vol}_{D^u})
$$
is a Gibbs $u$-state.
\end{itemize}
\end{proposition}
\begin{remark}\label{r.Gibbsustates}
Since the unstable foliation is absolutely continuous, part (3) of the proposition implies that there is a full volume
subset of points $x\in M$ for which every empirical measure is a Gibbs $u$-state.
\end{remark}
\section{A physical property}\label{physical}
We use the expression \emph{physical property} to refer to any property that holds on a positive
volume measure subset of the ambient manifold for any diffeomorphism.
The physical property is \emph{full} if it holds on a full volume subset.
The main result of this section is the following full physical property for $C^1$ diffeomorphism:
\begin{theorem}\label{t.limit}
Let $U\subseteq M$ be a compact subset of a manifold and $f: U\to {\rm int}(U)$ be an embedding.
Then there is a full volume measure subset $\Gamma$ of $U$ such that
\begin{equation}\label{eq.limit}
\limsup_n \frac 1n \log|Df^n(x)| \le 0 \text{ and } \int_M \log|\det Df| \, d\nu_x \le 0
\end{equation}
for any empirical measure $\nu_x$ of every $x\in \Gamma$.
\end{theorem}
Note that the left hand side of \eqref{eq.limit} is the average, with respect to $\nu_x$,
of the sum of all Lyapunov exponents.
For volume preserving maps, the statement is contained in the Oseledets theorem.
\begin{proof}
Let ${\mathcal P}$ denote the space of probabilities of $M$ and, for each $r\ge 0$,
$$
{\mathcal K}_r=\left\{\mu\in{\mathcal P}: \int_M \log|\det Df| \, d\mu \leq r\right\}.
$$
We are going to show that for each $r>0$ there is a full volume subset $\Gamma_r$ such that,
for every $x\in\Gamma_r$,
$$
\limsup_n \frac 1n \log|Df^n(x)| \le r
$$
and every empirical measure $\nu_x $ of every $x$ belongs to ${\mathcal K}_r$.
Clearly, we may choose $r\mapsto \Gamma_r$ to be monotone.
Then $\Gamma=\cap_{r>0} \Gamma_r$ is still a full volume subset satisfying all
the conditions in the statement.
Observe that ${\mathcal K}_r$ is weak$^*$-compact and convex. For fixed $r>0$, denote
$$
{\mathcal I}_{n,r}=\left\{x\in M: \frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x)}\in {\mathcal K}_r\right\}.
$$
The definition means that $|\det Df^n(z) \geq e^{nr}$ for every $z\in {\mathcal I}_{n,r}^c$. Then
$$
\operatorname{vol}(M) \geq
\int_{{\mathcal I}{n,r}^c} |\det Df^n| d\operatorname{vol}(x)
\geq e^{nr} \operatorname{vol}({\mathcal I}_{n,r}^c).
$$
Consequently, $\operatorname{vol}({\mathcal I}_{n,r}^c)\leq \operatorname{vol}(M) e^{-nr}$ which, by Borel--Cantelli, implies that
$$
\operatorname{vol}\left(\bigcap_n\bigcup_{k\geq n} {\mathcal I}_{k,r}^c\right)=0.
$$
Now, the definition also means that if $x$ is in $\bigcup_n\bigcap_{i\geq n} {\mathcal I}_{i,r}$
then there exists $n\ge 1$ such that
$$
\frac 1n \sum_{i=0}^{k-1}\delta_{f^i(x)} \in {\mathcal K}_r
\text{ for every } k \ge n.
$$
Then, by compactness, every empirical measure of $\nu_x$ is also in ${\mathcal K}_r$. This shows that
we may take $\Gamma_r$ to be $\bigcup_n\bigcap_{i\geq n} {\mathcal I}_{i,r}$.
\end{proof}
The following immediate consequence of the theorem was used previously in \cite{Yan17}:
\begin{remark}
Suppose that $f:M\to M$ is such that $\int \log|\det Df| \, d\mu \geq 0$ for any invariant probability $\mu$.
Then there is a full Lebesgue measure subset $\Gamma$ of $M$ such that
$$
\int \log|\det Df| \, d\nu_x = 0
$$
for any $x\in\Gamma$ and any accumulation point $\nu_x$ of ${n}^{-1}\sum_{i=0}^{n-1} \delta_{f^i(x)}$.
In particular, if there exists a unique invariant probability $\mu$ for which $\int \log|\det Df| \, d\mu = 0$,
and then $\mu$ must be a physical measure (because it must coincide with any $\nu_x$).
\end{remark}
\section{Proof of Theorem~\ref{main.global}}\label{s.Theorem A}
Throughout this section we take $f$ to be a $C^{1+}$ partially volume expanding diffeomorphism.
Let $c>0$ be as in \eqref{eq.used3}.
\begin{lemma}\label{l.simple_fact1}
If $\tilde\mu$ is an ergodic Gibbs $u$-state such that
$$
\int_M \log|\det Df| \, d\tilde\mu \le 0
$$
then $\tilde\mu$ is a physical measure and all its center Lyapunov exponents are less than $-c$.
\end{lemma}
\begin{proof}
Using the Birkhoff ergodic theorem and inequality \eqref{eq.used2},
$$
0 \ge \int_M \log| \det Df| \, d\tilde\mu
= \int_M \Delta \, d\tilde\mu
\ge c + \int_M \hat\lambda \, d\tilde\mu
$$
where $\hat\lambda$ denotes the largest center exponent. Thus, all the center Lyapunov exponents
are less than $-c$. By \cite[Section 11.2]{Beyond}, the fact that the center exponents are negative
ensures that $\tilde\mu$ is a physical measure.
\end{proof}
\begin{corollary}\label{c.physicalmeasure}
There is a full volume measure subset $\Gamma$ of $M$ such that every empirical measure $\nu_x$ of any $x\in\Gamma$
has an ergodic component $\mu_x$ which is a Gibbs $u$-state with $\int \log|\det Df| \, d\mu_x \leq 0$ and whose
center Lyapunov are all smaller than $-c$. In particular, $\mu_x$ is a physical measure.
\end{corollary}
\begin{proof}
By Remark~\ref{r.Gibbsustates} and Theorem~\ref{t.limit}, there is a full volume subset $\Gamma$ of $M$ such that
every empirical measure $\nu_x$ of every $x\in\Gamma$ is a Gibbs $u$-state and satisfies $\int_M \log| \det Df| \, d\nu_x \leq 0$. By part (1) of Proposition~\ref{p.Gibbsustates}, this $\mu_x$ is a Gibbs $u$-state.
Using Lemma~\ref{l.simple_fact1} with $\tilde\mu=\mu_x$, we get the claim of the proposition.
\end{proof}
The ergodic decomposition theorem (see \cite[Chapter~5]{FET}) states given any invariant probability
measure $\mu$, there exists a probability measure $\Phi_\mu$ on the space of $f$-invariant probability
measures on $M$, giving full weight to the subset of ergodic probability measures, and such that
$\mu(E) = \int_{\mathcal P} \eta(E) \,d\Phi_\mu(\eta)$ for every measurable set $E\subset M$.
The ergodic probability measures in the support of $\Phi_\mu$ are called \emph{ergodic components} of $\mu$.
If $\mu$ is a Gibbs $u$-state, the support of $\Phi_\mu$ is contained in the space of Gibbs $u$-states
(recall Proposition~\ref{p.Gibbsustates}).
\begin{lemma}\label{l.simple_fact2}
If $\tilde\mu$ is an ergodic Gibbs $u$-state whose center Lyapunov exponents are all negative then
it is an isolated point in the set of ergodic Gibbs $u$-states.
Consequently, $\tilde\mu$ is an ergodic component of some Gibbs $u$-state $\mu$ then
$\Phi_\mu(\{\tilde\mu\})>0$.
\end{lemma}
\begin{proof}
By now, the argument is quite standard (see for instance \cite[Lemma~2.9]{BoV00}).
Let $(\mu_j)_j$ be any sequence of ergodic Gibbs $u$-states accumulating on $\tilde\mu$.
Since $\tilde\mu$ is a hyperbolic measure, almost every point has a (Pesin) local stable manifold.
Since the stable lamination is absolutely continuous, and $\operatorname{supp}\tilde\mu$ is accumulated by the
supports of the $\mu_j$, local stable manifolds of $\tilde\mu$-typical points contain $\mu_j$-typical
points. That implies that $\mu_j=\tilde\mu$ for every large $j$. This proves that $\tilde\mu$ is isolated.
The last claim is a consequence, since $\Phi_\mu$ gives full weight to the set of ergodic Gibbs $u$-states.
\end{proof}
We are ready to prove that under our assumptions $f:M\to M$ has only finitely many physical measures,
and that their basins cover a full volume set. That is done in the next couple of lemmas.
\begin{lemma}\label{l.finite}
The diffeomorphism $f$ has finitely many ergodic Gibbs $u$-states with negative center exponents.
\end{lemma}
\begin{proof}
Suppose that $f$ admits infinitely many distinct ergodic Gibbs $u$-states $\mu_j$ whose center exponents
are all negative.
By compactness of the space of Gibbs $u$-states (part (1) of Proposition~\ref{p.Gibbsustates}),
we may assume that the sequence $\mu_j$ converges in the weak-* topology to some Gibbs $u$-state $\mu$.
As pointed out before, every $\mu_j$ is a physical measure. Thus,
$$
\lim \frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^i(x_n)}=\mu_j
$$
for every $x$ in some positive volume set. Take $x$ in the intersection of such a set with the full volume
set $\Gamma$ in Theorem~\ref{t.limit}, so that
$$
\int \log|\det Df| \, d\mu_j \leq 0.
$$
Making $j \to \infty$, we get that
$$
\int \log|\det Df| \, d\mu \leq 0.
$$
So, by part (1) of Proposition~\ref{p.Gibbsustates}, there is an ergodic component $\tilde{\mu}$ of $\mu$
which is a Gibbs $u$-state of $f$ such that
$$
\int \log|\det Df| \, d\tilde{\mu}\leq 0,
$$
and it is no restriction to assume that the support of $\tilde{\mu}$ is contained in the support of $\mu$.
Thus, using \eqref{eq.used2},
$$
0 \ge \int_M \log| \det Df| \, d\tilde\mu
= \int_M \Delta \, d\tilde\mu
\ge c + \int_M \hat\lambda \, d\tilde\mu
$$
By Lemma~\ref{l.simple_fact1}, it follows that $\tilde\mu$ is a physical measure and its
center Lyapunov exponents are negative.
Now the same argument as in Lemma~\ref{l.simple_fact2} proves that $\mu_j=\tilde\mu$
for every large $j$, a contradiction.
\end{proof}
\begin{lemma}\label{l.fullvolume}
The union of the basins of the physical measures $\mu_1,\cdots, \mu_m$ in Lemma~\ref{l.finite}
is a full volume subset of $M$.
\end{lemma}
\begin{proof}
Assume that the complement ${\mathcal C}$ of the union has positive volume.
Let ${\mathcal C}_0$ be the intersection of ${\mathcal C}$ with the full volume set $\Gamma$ in Corollary~\ref{c.physicalmeasure}
and with the full volume set of points $x\in M$ for which every empirical measure is a Gibbs $u$-state
(recall Remark~\ref{r.Gibbsustates}).
Since the unstable foliation is absolutely continuous, there exists some unstable disk that intersects ${\mathcal C}_0$
at a positive volume subset $I$. We may take $I$ to be compact.
By part (4) of Proposition~\ref{p.Gibbsustates}, any accumulation point $\mu$ of the sequence
$$
\mu_n=\frac{1}{n}\sum_{j=0}^{n-1}f_*^j(\operatorname{vol}_{I})
$$
($\operatorname{vol}_I$ denotes the normalized restriction of the
volume measure to $I$)
is a Gibbs $u$-state. Then, for some subsequence $(n_k)_k$ we have
$$
\begin{aligned}
\int \log|\det Df| \, d\mu
& = \lim_k \int \log|\det Df| \, d\mu_{n_k}
= \lim_k \int_I \frac{1}{n_k} \log|\det Df^{n_k}| \,d\operatorname{vol} \\
& \le \int_I \limsup_k \frac{1}{n_k} \log|\det Df^{n_k}| \,d\operatorname{vol}
\le 0,
\end{aligned}
$$
by Theorem~\ref{t.limit}.
Using part (1) of Proposition~\ref{p.Gibbsustates}, it follows that there is an
ergodic component $\tilde\mu$ of $\mu$ which is a Gibbs $u$-state of $f$ and satisfies
$$
\int \log \det(T_xf)\, d\tilde{\mu}(x)\leq 0.
$$
In particular, $\operatorname{supp}\tilde\mu$ is contained in $\operatorname{supp}\mu$.
By Lemma~\ref{l.simple_fact1}, $\tilde\mu$ is a physical measure and its center
Lyapunov exponents are negative.
By Lemma~\ref{l.simple_fact2}, it follows that $a=\Phi_\mu(\{\tilde\mu\})$ is positive.
Let $D$ be an unstable disk such that (volume) almost all points in $D$ are $\tilde\mu$-typical, in the sense of Pesin: in particular, they admit local Pesin stable manifolds.
Then let $B$ be a compact positive volume subset of $D$ (the volume will be denoted as $b$) such that
\begin{itemize}
\item[(i)] the size of the local stable manifolds of points in $B$ is uniformly bounded from below;
\item[(ii)] the holonomy maps of the stable lamination through $B$ are
uniformly absolutely continuous.
\end{itemize}
The second condition means, more precisely, that there exists $K>1$ such that the
projection ${\mathcal H}^s_{D_1,D_2}: D_1 \to D_2$ along the stable laminae of $L$
between any two nearby small unstable disks $D_1$ and $D_2$ has a Jacobian that
is bounded above by $K$ and bounded below by $1/K$.
Existence of such a structure follows from
classical Pesin theory, as we observed
in our previous paper~\cite{ViY13}, where we called
$cs$-block the union of the laminae through a set
$B$ with these properties.
Indeed, (i) and (ii) together with the assumption that $L$ consists of $\tilde\mu$-typical points and has positive volume inside $D$, imply that the $cs$-block ${\mathcal B}$ has positive $\tilde\mu$-measure and positive volume in the ambient manifold, and is contained in the basin of $\tilde\mu$.
We claim that there exists $n\ge 1$ and a positive
volume subset $I_0$ of $I$ such that every point of $f^n(I_0)$ is contained in ${\mathcal B}$. This yields a contradiction, since $I$ is assumed to be in the complement of the basins of all physical measures.
So, we are left to prove the claim. This is analogous to Proposition~6.9 in~\cite{ViY13},
so we only sketch the arguments.
Let $b>0$ denote the volume of $B$ inside $D$. We say that an unstable disk $L$ crosses the $cs$-block if it
intersects every lamina once. Then $\operatorname{vol}(L \cap {\mathcal B})\ge b/K$. Let ${\mathcal U}$ be a small open neighborhood of ${\mathcal B}$.
Obviously, $\operatorname{vol}(L \cap {\mathcal U})\ge b/K$ and $\tilde\mu({\mathcal U})\ge \tilde\mu({\mathcal B})>0$.
Let $c=a\tilde\mu({\mathcal U})$, where $a=\Phi_\mu(\{\tilde\mu\})>0$. Then $\mu({\mathcal U})\ge c >0$.
Next, for each small $\delta>0$ let $J_\delta$ denote the $\delta$-neighborhood of $I_0$ inside the corresponding
unstable leaf which contains $I_0$. Define
$$
\mu^\delta_n=\frac{1}{n}\sum_{j=0}^{n-1}f_*^j(\operatorname{vol}_{J_\delta})
$$
and let $\mu^\delta$ be an accumulation point along the same subsequence as for $\mu$.
It is clear that $\mu^\delta\to\mu$ as $\delta\to 0$, and so we may assume that $\mu^\delta({\mathcal U})\ge c/2$.
Then, there exist $n$ arbitrarily large such that $f^{-n}({\mathcal U})$ intersects $J_\delta$ on a subset with relative
measure larger than $c/4$.
By considering disjoint disks with bounded diameter inside $f^{n}(J_\delta)$ that cross the $cs$-block,
and using bounded distortion of the volume measure under backwards iterates along such disks, we conclude
that the relative volume of $J_\delta \cap f^{-n}({\mathcal B})$ inside $J_\delta$ is bounded below by some constant $r>0$
that depends only on $c$ and on the distortion bound. In particular, $r$ is independent of $\delta$.
On the other hand, the relative volume of $I_0$ inside $J_\delta$ goes to $1$ as $\delta$ goes to $0$.
Thus making $\delta\to 0$, we conclude that the relative volume of $I_0 \cap f^{-n}({\mathcal B})$ inside $I_0$
is still bounded below by $r$. This implies the claim, obviously.
\end{proof}
Theorem~\ref{main.global} follows immediately from Lemma~\ref{l.finite} and Lemma~\ref{l.fullvolume}.
\section{Proof of Theorem~\ref{main.relation}}
Recall (from~\cite{BoV00,An10}) that a partially hyperbolic diffeomorphism $f$ has
\emph{mostly contracting center} if the center Lyapunov exponents of every ergodic Gibbs $u$-state
are all negative.
Let $\mu$ be any ergodic Gibbs $u$-state of $f$.
The entropy $h_\mu(f)$ of $\mu$ is larger than or equal to its conditional entropy along the unstable foliation.
Moreover, Ledrappier~\cite{Led84a} implies that for Gibbs $u$-states the latter is equal to
$\int \log|\det Df\mid_{E^u}| \, d\mu$, which is obviously positive. Thus,
\begin{equation}\label{eq.entropyofGibbsu}
h_\mu(f) \geq \int \log|\det Df\mid_{E^u}| d\mu > 0.
\end{equation}
That implies that the smallest center Lyapunov exponent of $\mu$ is strictly negative:
otherwise, all the exponents would be non-negative which, by the Ruelle inequality~\cite{Rue78}
applied to the inverse $f^{-1}$, would imply that $h_\mu(f)=0$.
In particular, this proves the theorem when the $u$-codimension is 1.
In the $u$-codimension 2 case, let $\lambda^c_1(\mu) \geq \lambda^c_2(\mu)$ be the center Lyapunov exponents.
We already know that $\lambda_2^c(\mu)<0$, so are left to proving that the same holds for $\lambda_1^c(\mu)$.
Suppose otherwise: $\lambda_1^c(\mu) \ge 0$.
Since $-\lambda_2^c(\mu)$ is the unique positive exponent of $\mu$ for $f^{-1}$,
the Ruelle inequality gives that $h_\mu(f)\leq -\lambda^c_2(\mu)$.
Combining this with \eqref{eq.entropyofGibbsu}, we get that
\begin{equation}\label{eq.entropyofGibbsu2}
\int \log |Df\mid_{E^u}| d\mu+\lambda^c_2(\mu) \leq 0.
\end{equation}
By the Oseledets theorem (see \cite[Chapter 4]{LLE}) the integral on the left hand side coincides with
the sum of all Lyapunov exponents along the unstable sub-bundle $E^u$.
Thus the left hand side of \eqref{eq.entropyofGibbsu2} is equal to the sum of all Lyapunov exponents
except for $\lambda_1^c(\mu)$, and so the inequality directly contradicts \eqref{eq.used2}.
This contradiction proves the theorem.
\section{Partially volume expanding attracting sets}\label{local}
As a first step, let us state a variation of Theorem~\ref{main.global} for attracting sets of
diffeomorphisms. Then we will check that it applies to the generalized solenoid on the solid
torus $M=S^1\times D$
$$
f_0: M \to M, \quad f_0(\theta,x)=\left(k\theta \mod 1, h_\theta(x)\right)
$$
presented in the Introduction. Since $\|Dh_\theta(x)\|$ and $\|Dh_\theta(x)^{-1}\|$ are taken
to be strictly less than $k$ at every point, we may find $a<1$ such that
\begin{equation}\label{eq_ak}
\frac{1}{ak} \|v\| \le \|Dh_\theta(x) v\| \le a k \|v\|
\text{ for any } v \in T_x D \text{ and } (\theta,x) \in M.
\end{equation}
Let us point out that the dynamics of this embedding can be very complicated, for
instance, it may exhibit infinitely many coexisting periodic repellers (see~\cite{BLY13}).
\subsection{A semi-local finiteness theorem}
Let $f:M\to {\rm int}(M)$ be a $C^{1+}$ embedding of a compact manifold with boundary to its interior.
By the \emph{attracting} set we mean the maximal invariant set
$\Lambda_f = \cap_{n=0}^\infty f^n(M)$.
Observe that the map $f\mapsto\Lambda_f$ is upper semi-continuous with respect to the uniform
topology on $f$ and the Hausdorff topology on the space of compact subsets of $M$.
We call $f$ \emph{partially hyperbolic} if $\Lambda_f$ is a partially hyperbolic set.
A useful equivalent condition (see~\cite{Yoc95}) is that there exist two continuous
families ${\mathcal C}^{u}$ and ${\mathcal C}^{cs}$ of closed cones in the tangent space satisfying:
\begin{itemize}
\item[(a)] ${\mathcal C}^{u}(x)\cap {\mathcal C}^{cs}(x)=\{0\}$ for any $x\in M$;
\item[(b)] $Df(x)({\mathcal C}^u(x))\subset {\mathcal C}^u(f(x))$ and $Df(x)({\mathcal C}^{cs}(x))\supset {\mathcal C}^{cs}(f(x))$
for any $x\in M$;
\item[(c)] $\|Df(x)u\|>\|u\|$ for any non-zero vector $u \in {\mathcal C}^u(x)$ and any $x$ close to $\Lambda_f$.
\end{itemize}
Similarly, we call $f$ \emph{partially volume expanding} if $\Lambda_f$ is a partially volume
expanding set, meaning that $\log\left|\det Df(x)_{\mid H} \right|$ is positive for any
$x\in\Lambda_f$ and any hyperplane $H$ of the tangent space $T_x M$ containing the unstable
subspace $E_x^u$. Using the upper semi-continuity of $\Lambda_f$ one can easily that this is a
stable property, that is, it remains true for every $C^1$ perturbation of $f$.
Moreover, the consequences \eqref{eq.used3} and \eqref{eq.used2} still hold in this context.
\begin{theorem}\label{main.local}
Any partially volume expanding $C^{1+}$ embedding $f:M\to{\rm int}(M)$ admits finitely many physical
measures, the union of whose basins is a full volume subset of the ambient manifold.
\end{theorem}
The proof is virtually identical to that of Theorem~\ref{main.global}.
Section~\ref{physical} was already formulated in the language of embeddings, and so it applies
immediately to the present context.
Gibbs $u$-states on the invariant set $\Lambda_f$ are defined in precisely the same way as in the
previous global setting. We denote by ${\rm Gibb}^u(f)$ the space of Gibbs $u$-states of $\Lambda_f$.
Proposition~\ref{p.Gibbsustates} remains valid here, as long as we define a $u$-disk to mean any
disk embedded in ${\rm int}(M)$ whose tangent space is contained in the unstable cone at every point.
With that same convention, the arguments in Section~\ref{s.Theorem A} also extend immediately to
this setting. Thus one gets Theorem~\ref{main.local}.
\subsection{A partially volume expanding solenoid}
The final next couple of lemmas assert that generalized solenoid $f_0:M \to M$ is partially
hyperbolic and partially volume expanding. Thus Theorem~\ref{main.local} applies to it and all its
perturbations.
\begin{lemma}\label{l.solenoidpartiallyhyperbolic}
$f_0$ is a partially hyperbolic diffeomorphism.
\end{lemma}
\begin{proof}
Take $a$ as in \eqref{eq_ak} and
$$
K=\max\left\{\left\|\frac{\partial h_\theta}{\partial \theta}(\theta,x)\right\|: (\theta,x)\in M \right\},
$$
and then define
\begin{equation*}
\begin{aligned}
{\mathcal C}^u(\theta,x)
& = \left\{(u,v)\in T_{(\theta,x)} M: \|v\| \leq \frac{2K}{k(1-a)}\|u\|\right\}\\
{\mathcal C}^{cs}(\theta,x)
& = \left\{(u,v)\in T_{(\theta,x)} M: \|v\| \geq \frac{3K}{k(1-a)}\|u\|\right\}.
\end{aligned}
\end{equation*}
It is straightforward to check that these two cone fields satisfy the conditions in the definition above.
\end{proof}
Observe that the two-dimensional vertical sub-bundle $\{0\} \times T_x D$ is invariant under $Df$.
Indeed, it coincides with the center-stable sub-bundle $E^{cs}_{(\theta,x)}$ of $f$.
\begin{lemma}\label{l.solenoidweakcontraction}
There is an integer $N \ge 1$ such that $f_0^N$ is partially volume expanding.
\end{lemma}
\begin{proof}
Let $H$ be any 2-plane that contains the unstable subspace $E^u_{(\theta,x)}$.
Of course, $H$ must intersect the vertical 2-plane $\{0\} \times T_x D$ along some direction $F$.
The iterates of vectors along $E^u_{(\theta,x)}$ remain inside the unstable cone
${\mathcal C}^u$ and so their angle to the horizontal direction is uniformly bounded from
$\pi/2$. In particular, their growth rate under iteration is equal to $k$.
For the vectors along $F$, the growth rate is given by the vertical derivative.
In particular, using \eqref{eq_ak}, it is bounded below by $1/(ak)$.
Since the angle between the iterates of $E^u_{(\theta,x)}$ and $F$ are uniformly bounded from zero,
it follows that the rate of growth of the determinant along the 2-plane $H$ is
bounded below by $1/a>1$. Thus, there exists $N \ge 1$ such that
$|\det Df^N(\theta,x)|_{H}$ is strictly
bigger than $1$ at every point of $\Lambda_f$. This proves the claim.
\end{proof}
|
1,314,259,996,724 | arxiv | \section{Introduction}
Many modern scientific problems involve solving high-dimensional statistical problems where the sample size is small relative to the ambient dimension of the underlying parameter to be estimated. Over the past few decades there has been a large amount of work on solving such problems by imposing low-dimensional structure on the parameter of interest. In particular sparsity, low-rankness and other low-dimensional subspace assumptions have been studied extensively both in terms of the development of fast algorithms and theoretical guarantees. See, e.g., \cite{BuhlmannVDGBook} and \cite{HastieTibshiraniWainwrightBook}, for an overview. Most of the prior work has focussed on scenarios in which the parameter of interest is a vector or matrix. Increasingly common in practice, however, the parameter or object to be estimated naturally has a higher order tensor structure. Examples include hyper-spectral image analysis \citep{LiLi10}, multi-energy computed tomography \citep{Semerci14}, radar signal processing \citep{SidNion10}, audio classification \citep{Mesgarani06} and text mining \citep{CohenCollins12} among numerous others. It is much less clear how the low dimensional structures inherent to these problems can be effectively accounted for. The main purpose of this article is to fill in this void and provide a general and unifying framework for doing so.
Consider a general tensor regression problem where covariate tensors $X^{(i)}\in \mathbb{R}^{d_1\times\cdots\times d_M}$ and response tensors $Y^{(i)}\in \mathbb{R}^{d_{M+1}\times\cdots\times d_N}$ are related through:
\begin{equation}
\label{eq:model}
Y^{(i)}=\langle X^{(i)}, T\rangle+\epsilon^{(i)}, \qquad i=1,2,\ldots,n.
\end{equation}
Here $T\in \mathbb{R}^{d_1\times \cdots \times d_N}$ is an unknown parameter of interest, and $\epsilon^{(i)}$s are independent and identically distributed noise tensors whose entries are independent and identically distributed centered normal random variables. Further, for simplicity we assume the covariates $(X^{(i)})_{i=1}^n$ are Gaussian, but with fairly general dependence assumptions. The notation $\langle\cdot,\cdot \rangle$ will refer throughout this paper to the standard inner product taken over appropriate Euclidean spaces. Hence, for $A\in \mathbb{R}^{d_1\times\cdots\times d_M}$ and $B\in \mathbb{R}^{d_1\times\cdots\times d_N}$:
$$\langle A , B \rangle=\sum_{j_1 = 1}^{d_1}\cdots\sum_{j_M = 1}^{d_M}A_{j_1,\ldots,j_M}B_{j_1,\ldots,j_M}\in \mathbb{R}$$
is the usual inner product if $M=N$; and if $M<N$, then $\langle A , B \rangle\in \mathbb{R}^{d_{M+1}\times\cdots\times d_N}$ such that its $(j_{M+1},\ldots, j_N)$ entry is given by
$$\left(\langle A , B \rangle\right)_{j_{M+1},\ldots, j_N}=\sum_{j_1 = 1}^{d_1}\cdots\sum_{j_M = 1}^{d_M}{A_{j_1,\ldots,j_M}B_{j_1,\ldots,j_M,j_{M+1},\ldots,j_N}}.$$
The goal of tensor regression is to estimate the coefficient tensor $T$ based on observations $\{(X^{(i)},Y^{(i)}): 1\le i\le n\}$. In addition to the canonical example of tensor regression with $Y$ a scalar response (i.e., $M=N$), many other commonly encountered regression problems are also special cases of the general tensor regression model (\ref{eq:model}). Multi-response regression \citep[see, e.g.,][]{AndersonStat}, vector autoregressive model \citep[see, e.g.,][]{Lut06}, and pairwise interaction tensor model \citep[see, e.g.,][]{RendleSchmidt10} are some of the notable examples. In this article, we provide a general treatment to these seemingly different problems.
Our main focus here is on situations where the dimensionality $d_k$s are large when compared with the sample size $n$. In many practical settings, the true regression coefficient tensor $T$ may have certain types of low-dimensional structure. Because of the high ambient dimension of a regression coefficient tensor, it is essential to account for such a low-dimensional structure when estimating it. Sparsity and low-rankness are the most common examples of such low dimensional structures. In the case of tensors, sparsity could occur at the entry-wise level, fiber-wise level, or slice-wise level, depending on the context and leading to different interpretations. There are also multiple ways in which low-rankness may be present when it comes to higher order tensors, either at the original tensor level or at the \emph{matricized} tensor level.
In this article, we consider a general class of convex regularization techniques to exploit either type of low-dimensional structure. In particular, we consider the standard convex regularization framework:
\begin{equation}
\label{EqnGeneral}
\widehat{T} \in \argmin_{A\in \mathbb{R}^{d_1\times \cdots\times d_N}}\left\{\frac{1}{2n} \sum_{i=1}^{n} \|Y^{(i)} - \langle A, X^{(i)} \rangle \|_{\rm F}^2 + \lambda \mathcal{R}(A)\right\},
\end{equation}
where the regularizer $\mathcal{R}(\cdot)$ is a norm on $\mathbb{R}^{d_1\times \cdots\times d_N}$, and $\lambda>0$ is a tuning parameter. Hereafter, for a tensor $A$, $\|A\|_{\rm F}=\langle A, A\rangle^{1/2}$. We derive general risk bounds for a family of so-called \emph{weakly decomposable} regularizers under fairly general dependence structure among the covariates. These general upper bounds apply to a number of concrete statistical inference problems including the aforementioned multi-response regression, high-dimensional vector auto-regressive models, low-rank tensor models, and pairwise interaction tensors where we show that they are typically optimal in the minimax sense.
In developing these general results, we make several contributions to a fast growing literature on high dimensional tensor estimation. First of all, we provide a principled approach to exploit the low dimensional structure in these problems. In doing so, we extend the notion of decomposability originally introduced by \cite{Neg10} for vector and matrix models to \emph{weak decomposability} which allows us to handle more delicate tensor models such as the nuclear norm regularization for low-rank tensor models. Moreover, we provide, for the regularized least squared estimate given by (\ref{EqnGeneral}), a general risk bound under an easily interpretable condition on the design tensor. The risk bound we derive is presented in terms of merely two geometric quantities, the \emph{Gaussian width} which depends on the choice of regularization and the \emph{intrinsic dimension} of the subspace that the tensor $T$ lies in. Finally, our general results lead to novel upper bounds for several important regression problems involving high-dimensional tensors: multi-response regression, multi-variate auto-regressive models and pairwise interaction models, for which we also prove that the resulting estimates are minimiax rate optimal with appropriate choices of regularizers.
The remainder of the paper is organized as follows: In Section~\ref{SecProbSetup} we introduce the general framework of using weakly decomposable regularizers for exploiting low-dimensional structures in high dimensional tensor regression. In Section~\ref{SecBounds} we present a general upper bound for weakly decomposable regularizers and discuss specific risk bounds for commonly used sparsity or low-rankness regularizers for tensors. In Section~\ref{SecExamples} we apply our general result to three specific statistical problems, namely, multi-response regression, multivariate autoregressive model, and the pairwise interaction model. We show that in each of the three examples appropriately chosen weakly decomposable regularizers leads to minimax optimal estimation of the unknown parameters. The proofs are presented in Section~\ref{SecProofs}.
\section{Methodology}
\label{SecProbSetup}
Recall the regularized least-squares objective:
\begin{equation*}
\widehat{T}=\argmin_{A\in \mathbb{R}^{d_1\times \cdots\times d_N}}\left\{\frac{1}{2n} \sum_{i=1}^{n} \|Y^{(i)} - \langle A, X^{(i)} \rangle \|_{\rm F}^2 + \lambda \mathcal{R}(A)\right\}.
\end{equation*}
For brevity, we assume implicitly hereafter that the minimizer on its left hand side is uniquely defined. Our development here actually applies to the more general case where $\widehat{T}$ can be taken as an arbitrary element from the set of the minimizers. Of particularly interest here is the so-called \emph{weakly decomposable} convex regularizers, extending a similar concept introduced by \cite{Neg10} for vectors and matrices.
Let $\mathcal{A}$ be an arbitrary linear subspace of $\mathbb{R}^{d_1 \times \cdots \times d_N}$ and $\mathcal{A}^{\perp}$ its orthogonal complement:
$$\mathcal{A}^{\perp} := \{A \in \mathbb{R}^{d_1 \times \cdots\times d_N}\;| \; \langle A, B \rangle = 0\; \mbox{for all}\; B \in \mathcal{A}\}.$$
We call a regularizer $\mathcal{R}(\cdot)$ weakly decomposable with respect to a pair $(\mathcal{A}, \mathcal{B})$ where $\mathcal{B}\subseteq \mathcal{A}$ if there exist a constant $0<c_\mathcal{R}\le 1$ such that for any $A \in \mathcal{A}^\perp$ and $B \in \mathcal{B}$,
\begin{equation}
\label{eq:decomcond}
\mathcal{R}(A + B)\ge \mathcal{R}(A) + c_{\mathcal{R}}\mathcal{R}(B).
\end{equation}
In particular, if (\ref{eq:decomcond}) holds for any $B\in \mathcal{B}=\mathcal{A}$, we say $\mathcal{R}(\cdot)$ is weakly decomposable with respect to $\mathcal{A}$. Because $\mathcal{R}$ is a norm, by triangular inequality, we also have
$$
\mathcal{R}(A + B) \le \mathcal{R}(A) + \mathcal{R}(B).
$$
Many of the commonly used regularizers for tensors are weakly decomposable, or decomposable for short. When $c_\mathcal{R}=1$, our definition of decomposability naturally extends from similar notion for vectors ($N=1$) and matrices ($N=2$) introduced by \cite{Neg10}. We also allow for more general choices of $c_\mathcal{R}$ here to ensure a wider applicability. For example as we shall see the popular tensor nuclear norm regularizer is decomposable with respect to appropriate linear subspaces with $c_\mathcal{R}=1/2$, but not decomposable if $c_\mathcal{R}=1$.
We now described a catalogue of commonly used regularizers for tensors and argue that they are all decomposable with respect to appropriately chosen subspaces of $\mathbb{R}^{d_1\times \cdots\times d_N}$. To fix ideas, we shall focus in what follows on estimating a third-order tensor $T$, that is $N=3$, although our discussion can be straightforwardly extended to higher-order tensors.
\subsection{Sparsity Regularizers}
An obvious way to encourage entry-wise sparsity is to impose the vector $\ell_1$ penalty on the entries of $A$:
\begin{equation}
\label{eq:lasso}
\mathcal{R}(A):=\sum_{j_1=1}^{d_1}\sum_{j_2=1}^{d_2}\sum_{j_3=1}^{d_3} |A_{j_1j_2j_3}|,
\end{equation}
following the same idea as the Lasso for linear regression \citep[see, e.g.,][]{Tibshirani96}. This is a canonical example of decomposable regularizers. For any fixed $I\subset [d_1]\times[d_2]\times[d_3]$ where $[d]=\{1,2,\ldots,d\}$, write
\begin{equation}
\label{eq:defA1}
\mathcal{A}(I)=\mathcal{B}(I)=\left\{A\in \mathbb{R}^{d_1 \times d_2\times d_3}: A_{j_1j_2j_3}=0 {\rm \ for\ all\ } (j_1,j_2,j_3)\notin I\right\}.
\end{equation}
It is clear that
$$
\mathcal{A}^\perp(I)=\left\{A\in \mathbb{R}^{d_1 \times d_2\times d_3}: A_{j_1j_2j_3}=0 {\rm \ for\ all\ } (j_1,j_2,j_3)\in I\right\},
$$
and $\mathcal{R}(A)$ defined by (\ref{eq:lasso}) is decomposable with respect to $\mathcal{A}$ with $c_{\mathcal{R}}=1$.
In many applications, sparsity arises with a more structured fashion for tensors. For example, a fiber or a slice of a tensor is likely to be zero simultaneously. Mode-$1$ fibers of a tensor $A\in \mathcal{R}^{d_1\times d_2\times d_3}$ are the collection of $d_1$-dimensional vectors
$$\left\{A_{\cdot j_2j_3}=(A_{1j_2j_3},\ldots,A_{d_1j_2j_3})^\top: 1\le j_2\le d_2, 1\le j_3\le d_3\right\}.$$
Mode-$2$ and -$3$ fibers can be defined in the same fashion. To fix ideas, we focus on mode-$1$ fibers. Sparsity among mode-$1$ fibers can be exploited using the group-based $\ell_1$ regularizer:
\begin{equation}
\label{eq:groupLasso}
\mathcal{R}(A) = \sum_{j_2=1}^{d_2}\sum_{j_3=1}^{d_3} \|A_{\cdot j_2j_3}\|_{\ell_2},
\end{equation}
similar to the group Lasso \citep[see, e.g.,][]{YuaLi06}, where $\|\cdot\|_{\ell_2}$ stands for the usual vector $\ell_2$ norm. Similar to the vector $\ell_1$ regularizer, the group $\ell_1$-based regularizer is also decomposable. For any fixed $I\subset [d_2]\times[d_3]$, write
\begin{equation}
\label{eq:defA2}
\mathcal{A}(I)=\mathcal{B}(I)=\left\{A\in \mathbb{R}^{d_1 \times d_2\times d_3}: A_{j_1j_2j_3}=0 {\rm \ for\ all\ } (j_2,j_3)\notin I\right\}.
\end{equation}
It is clear that
$$
\mathcal{A}^\perp(I)=\left\{A\in \mathbb{R}^{d_1 \times d_2\times d_3}: A_{j_1j_2j_3}=0 {\rm \ for\ all\ } (j_2,j_3)\in I\right\},
$$
and $\mathcal{R}(A)$ defined by (\ref{eq:groupLasso}) is decomposable with respect to $\mathcal{A}$ with $c_{\mathcal{R}}=1$. Note that in defining the regularizer in (\ref{eq:groupLasso}), instead of vector $\ell_2$ norm, other $\ell_q$ ($q>1$) norms could also be used. See, e.g., \cite{Turlach05}.
Sparsity could also occur at the slice level. The $(1,2)$ slices of a tensor $A\in \mathbb{R}^{d_1\times d_2\times d_3}$ are the collection of $d_1\times d_2$ matrices
$$\left\{A_{\cdot \cdot j_3}=(A_{j_1j_2j_3})_{1\le j_1\le d_1,1\le j_2\le d_2}: 1\le j_3\le d_3\right\}.$$
Let $\|\cdot\|$ be an arbitrary norm on $d_1\times d_2$ matrices. Then the following group regularizer can be considered:
\begin{equation}
\label{eq:groupLassoMat}
\mathcal{R}(A)=\sum_{j_3=1}^{d_3}\|A_{\cdot \cdot j_3}\|.
\end{equation}
Typical examples of the matrix norm that can be used in (\ref{eq:groupLassoMat}) include Frobenius norm and nuclear norm among others. In the case when $\|\cdot\|_{\rm F}$ is used, $\mathcal{R}(\cdot)$ is again a decomposable regularizer with respect to
\begin{equation}
\label{eq:defA3}
\mathcal{A}(I)=\mathcal{B}(I)=\left\{A\in \mathbb{R}^{d_1 \times d_2\times d_3}: A_{j_1j_2j_3}=0 {\rm \ for\ all\ } j_3\notin I\right\}.
\end{equation}
for any $I\subset[d_3]$.
Now consider the case when we use the matrix nuclear norm $\|\cdot\|_\ast$ in (\ref{eq:groupLassoMat}). Let $P_{1j}$ and $P_{2j}$, $j=1,\ldots, d_3$ be two sequences of projection matrices on $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$ respectively. Let
\begin{equation}
\label{eq:defA4}
\mathcal{A}(P_{1j},P_{2j}: 1\le j\le d_3)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: P_{1j}^\perp A_{\cdot\cdot j}P_{2j}^\perp=0, j=1,\ldots, d_3\right\},
\end{equation}
and
\begin{equation}
\label{eq:defB4}
\mathcal{B}(P_{1j},P_{2j}: 1\le j\le d_3)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: A_{\cdot\cdot j}=P_{1j}A_{\cdot\cdot j}P_{2j}, j=1,\ldots, d_3\right\}.
\end{equation}
By pinching inequality \citep[see, e.g.,][]{Bhatia97}, it can be derived that $\mathcal{R}(\cdot)$ is decomposable with respect to $\mathcal{A}(P_{1j},P_{2j}: 1\le j\le d_3)$ and $\mathcal{B}(P_{1j},P_{2j}: 1\le j\le d_3)$.
\subsection{Low-rankness Regularizers}
In addition to sparsity, one may also consider tensors with low-rank. There are multiple notions of rank for higher-order tensors. See, e.g., \cite{KoldarBader}, for a recent review. In particular, the so-called CP rank is defined as the smallest number $r$ of rank-one tensors needed to represent a tensor $A\in \mathbb{R}^{d_1\times d_2\times d_3}$:
\begin{equation}
\label{eq:cpdecomp}
A=\sum_{k=1}^r u_k\otimes v_k\otimes w_k
\end{equation}
where $u_k\in \mathbb{R}^{d_1}$, $v_k\in \mathbb{R}^{d_2}$ and $w_k\in \mathbb{R}^{d_3}$. To encourage a low rank estimate, we can consider the nuclear norm regularization. Following \cite{YuanZhang14}, we define the nuclear norm of $A$ through its dual norm. More specifically, let the spectral norm of $A$ be given by
$$
\|A\|_{s}=\max_{\|u\|_{\ell_2},\|v\|_{\ell_2},\|w\|_{\ell_2}\le 1}\langle A, u\otimes v\otimes w\rangle.
$$
Then its nuclear norm is defined as
$$
\|A\|_\ast=\max_{\|B\|_s\le 1}\langle A, B\rangle.
$$
We shall then consider the regularizer:
\begin{equation}
\label{eq:nuclear}
\mathcal{R}(A) = \|A\|_\ast.
\end{equation}
We now show this is also a weakly decomposable regularizer.
Let $P_k$ be a projection matrix in $\mathbb{R}^{d_k}$. Define
$$
(P_1\otimes P_2\otimes P_3) A=\sum_{k=1}^r P_1u_k\otimes P_2v_k\otimes P_3w_k.
$$
Write
$$
Q=P_1\otimes P_2\otimes P_3+P_1^\perp\otimes P_2\otimes P_3+P_1\otimes P_2^\perp\otimes P_3+P_1\otimes P_2\otimes P_3^\perp,
$$
and
$$Q^\perp =P_1^\perp\otimes P_2^\perp\otimes P_3^\perp+P_1^\perp\otimes P_2^\perp\otimes P_3+P_1\otimes P_2^\perp\otimes P_3^\perp+P_1^\perp\otimes P_2\otimes P_3^\perp,$$
where $P_k^\perp=I-P_k$.
\blems
\label{le:pinch}
For any $A\in \mathbb{R}^{d_1\times d_2\times d_3}$ and projection matrices $P_k$ in $\mathbb{R}^{d_k}$, $k=1,2,3$, we have
$$
\|A\|_{\ast}\ge \|(P_1\otimes P_2\otimes P_3) A\|_\ast+{1\over 2}\|Q^\perp A\|_\ast.
$$
\elems
Lemma \ref{le:pinch} is a direct consequence from the characterization of sub-differential for tensor nuclear norm given by \cite{YuanZhang14}, and can be viewed as a tensor version of the pinching inequality for matrices.
Write
\begin{equation}
\label{eq:defA5}
\mathcal{A}(P_1,P_2,P_3)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: Q A=A\right\},
\end{equation}
and
\begin{equation}
\label{eq:defB5}
\mathcal{B}(P_1,P_2,P_3)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: (P_1\otimes P_2\otimes P_3) A=A\right\}.
\end{equation}
By Lemma \ref{le:pinch}, $\mathcal{R}(\cdot)$ defined by (\ref{eq:nuclear}) is weakly decomposable with respect to $\mathcal{A}(P_1,P_2,P_3)$ and $\mathcal{B}(P_1,P_2,P_3)$ with $c_{\mathcal{R}}=1/2$. We note that a counterexample is also given by \cite{YuanZhang14} which shows that, for the tensor nuclear norm, we cannot take $c_\mathcal{R}=1$.
Another popular way to define tensor rank is through the so-called Tucker decomposition. Recall that the Tucker decomposition of a tensor $A\in \mathbb{R}^{d_1\times d_2\times d_3}$ is of the form:
\begin{equation}
\label{eq:tucker}
A_{j_1j_2j_3}=\sum_{k_1=1}^{r_1}\sum_{k_2=1}^{r_2}\sum_{k_3=1}^{r_3} S_{k_1k_2k_3}U_{j_1k_1}V_{j_2k_2}W_{j_3k_3}
\end{equation}
so that $U$, $V$ and $W$ are orthogonal matrices, and the so-called core tensor $S=(S_{k_1k_2k_3})_{k_1,k_2,k_3}$ is such that any two slices of $S$ are orthogonal. The triplet $(r_1,r_2,r_3)$ are referred to as the Tucker ranks of $A$. It is not hard to see that if (\ref{eq:cpdecomp}) holds, then the Tucker ranks $(r_1,r_2,r_3)$ can be equivalently interpreted as the dimensionality of the linear spaces spanned by $\{u_k: 1\le k\le r\}$, $\{v_k: 1\le k\le r\}$, and $\{w_k: 1\le k\le r\}$ respectively. The following relationship holds between CP rank and Tucker ranks:
$$
\max\{r_1,r_2,r_3\}\le r\le \min\{r_1r_2,r_2r_3,r_1r_3\}.
$$
A convenient way to encourage low Tucker ranks in a tensor is through matricization. Let $\mathcal{M}_1(\cdot)$ denote the mode-$1$ matricization of a tensor. That is $\mathcal{M}_1(A)$ is a $d_1\times (d_2d_3)$ matrix whose column vectors are the the mode-$1$ fibers of $A\in \mathbb{R}^{d_1\times d_2\times d_3}$. $\mathcal{M}_2(\cdot)$ and $\mathcal{M}_3(\cdot)$ can also be defined in the same fashion. It is clear
$$
{\rm rank}(\mathcal{M}_k(A))=r_k(A).
$$
A natural way to encourage low-rankness is therefore through nuclear norm regularization:
\begin{equation}
\label{eq:matricize}
\mathcal{R}(A) = {1\over 3}\sum_{k=1}^3 \|\mathcal{M}_k(A)\|_{\ast}.
\end{equation}
By the pinching inequality for matrices, $\mathcal{R}(\cdot)$ defined by (\ref{eq:matricize}) is also decomposable with respect to $\mathcal{A}(P_1,P_2,P_3)$ and $\mathcal{B}(P_1,P_2,P_3)$ with $c_{\mathcal{R}}=1$.
\section{Risk Bounds for Decomposable Regularizers}
\label{SecBounds}
We now establish risk bounds for general decomposable regularizers. In particular, our bounds are given in terms of the \emph{Gaussian width} of a suitable set of tensors. Recall that the Gaussian width of a set $S \subset \mathbb{R}^{d_1 \times d_2 \times...\times d_N}$ is given by
\begin{equation*}
w_G(S) := \mathbb{E}\left(\sup_{A \in S} \langle A, G \rangle \right),
\end{equation*}
where $G \in \mathbb{R}^{d_1 \times d_2 \times ... \times d_N}$ is a tensor whose entries are independent $\mathcal{N}(0,1)$ random variables.
See, e.g., \cite{Gordon88}.
Note that the Gaussian width is a geometric measure of the volume of the set $S$ and can be related to other volumetric characterizations \citep[see, e.g.,][]{Pisier89}.We also define the unit ball for the norm-regularizer $\mathcal{R}(.)$ as follows:
$$\mathbb{B}_{\mathcal{R}}(1) := \{A \in \mathbb{R}^{d_1 \times d_2 \times...\times d_N}\;|\; \mathcal{R}(A) \leq 1 \}.$$ We impose the mild assumption that $\|A\|_{\rm F} \leq \mathcal{R}(A)$ which ensures that the regularizer $\mathcal{R}(\cdot)$ encourages low-dimensional structure.
Now we define a quantity that relates the size of the norm $\mathcal{R}(A)$ to the Frobenius norm $\|A\|_{\rm F}$ over the the low-dimensional subspace $\mathcal{A}$. Following \cite{Neg10}, for a subspace $\mathcal{A}$ of $\mathbb{R}^{d_1\times \cdots\times d_N}$, define its compatibility constant $s(\mathcal{A})$ as
\begin{equation*}
s(\mathcal{A}) := \sup_{A \in \mathcal{A}/\{0\}} \frac{\mathcal{R}^2(A)}{\|A\|_{\rm F}^2},
\end{equation*}
which can be interpreted as a notion of intrinsic dimensionality of $\mathcal{A}$.
Now we turn our attention to the covariate tensor. Denote by $X^{(i)}={\rm vec}(X^{(i)})$ the vectorized covariate from the $i$th sample. With slight abuse of notation, write
$$X={\rm vec}((X^{(1)})^\top,\ldots, (X^{(n)})^\top) \in \mathbb{R}^{n.d_1d_2\cdots d_M}$$
the concatenated covariates from all $n$ samples. For convenience let $D_M = d_1d_2\cdots d_M$. Further for brevity we assume a Gaussian design so that
$$X \sim \mathcal{N}(0,\Sigma)$$ where $$\Sigma = {\rm cov}(X, X) \in \mathbb{R}^{nD_M \times nD_M}.$$
With more technical work our results may be extended beyond Gaussian designs. We note that we do not require that the sample tensors $X^{(i)}$ be independent.
We shall assume that $\Sigma$ has bounded eigenvalues which we later verify for a number of statistical examples. Let $\lambda_{\min}(\cdot)$ and $\lambda_{\max}(\cdot)$ represent the smallest and largest eigenvalues of a matrix, respectively. In what follows, we shall assume that
\begin{equation}
\label{AssCov}
c_{\ell}^2 \leq \lambda_{\min}(\Sigma) \leq \lambda_{\max}(\Sigma) \leq c_u^2,
\end{equation}
for some constants $0< c_\ell\le c_u<\infty$.
Note that in particular if all covariates $\{X^{(i)}: i=1,\ldots,n\}$ are independent and identically distributed, then $\Sigma$ has a block diagonal structure, and (\ref{AssCov}) boils down to similar conditions on ${\rm cov}(X^{(i)},X^{(i)})$. However (\ref{AssCov}) is more general and applicable to settings in which the $X^{(i)}$'s may be dependent such as time-series models, which we shall discuss in further detail in Section~\ref{SecExamples}.
We are now in position to state our main result on the risk bounds in terms of both Frobenius norm $\|\cdot\|_{\rm F}$ and the empirical norm $\|\cdot\|_n$ where for a tensor $A\in \mathbb{R}^{d_1\times \cdots \times d_N}$, which we define as:
$$
\|A\|_n^2 := {1\over n}\sum_{i=1}^n \|\langle A, X^{(i)}\rangle\|_{F}^2.
$$
\btheos
\label{ThmUpper}
Suppose that (\ref{eq:model}) holds for a tensor $T$ from a linear subspace $\mathcal{A}_0\subset \mathbb{R}^{d_1\times \cdots\times d_N}$ where (\ref{AssCov}) holds. Let $\widehat{T}$ be defined by~\eqref{EqnGeneral} where the regularizer $\mathcal{R}(\cdot)$ is decomposable with respect to $\mathcal{A}$ and $\mathcal{A}_0$ for some linear subspace $\mathcal{A}\supseteq\mathcal{A}_0$. If
\begin{equation}
\lambda \geq \frac{2 c_u (3+c_\mathcal{R})}{c_\mathcal{R}\sqrt{n}}w_G[\mathbb{B}_{\mathcal{R}}(1)],
\end{equation}
then there exists a constant $c>0$ such that with probability at least $1 - \exp\{-c w_G^2[\mathbb{B}_{\mathcal{R}}(1)]\}$,
\begin{equation}
\label{eq:risk}
\max\left\{\|\widehat{T}-T\|_n^2, \|\widehat{T}-T\|_{\rm F}^2\right\} \leq \frac{6(1+c_\mathcal{R})}{3+c_\mathcal{R}}\frac{9 c_u^2}{c_{\ell}^2} s(\mathcal{A}) \lambda^2,
\end{equation}
when $n$ is sufficiently large, assuming that the right hand side converges to zero as $n$ increases.
\etheos
As stated in Theorem \ref{ThmUpper}, our upper bound boils down to bounding two quantities, $s(\mathcal{A})$ and $w_G[\mathbb{B}_{\mathcal{R}}(1)]$ which are both purely geometric quantities. To provide some intuition, $w_G[\mathbb{B}_{\mathcal{R}}(1)]$ captures how large the $\mathcal{R}(\cdot)$ norm is relative to the $\|\cdot\|_{\rm F}$ norm and $s(\mathcal{A})$ captures the low dimension of the subspace $\mathcal{A}$.
Note that $w_G[\mathbb{B}_{\mathcal{R}}(1)]$ can be expressed as expectation of the \emph{dual norm} of $G$. According to $\mathcal{R}$ \citep[see, e.g.,][for details]{Rockafellar}, the dual norm $\mathcal{R}^*(\cdot)$ is given by:
\begin{equation*}
\mathcal{R}^*(B) := \sup_{A \in \mathbb{B}_{\mathcal{R}}(1)} \langle A, B \rangle,
\end{equation*}
where the supremum is taken over tensors of the same dimensions as $B$.
It is straightforward to see that $w_G[\mathbb{B}_{\mathcal{R}}(1)] = \mathbb{E}[\mathcal{R}^*(G)]$.
Now we develop upper bounds on both quantities in different scenarios. As in the previous section, we shall focus on third order tensor in the rest of the section for the ease of exposition.
\subsection{Sparsity regularizers}
We first consider sparsity regularizers described in the previous section.
\subsubsection{Entry-wise and fiber-wise sparsity}
Recall that vectorized $\ell_1$ regularizer:
$$
\mathcal{R}_1(A)=\sum_{j_1=1}^{d_1}\sum_{j_2=1}^{d_2}\sum_{j_3=1}^{d_3}|A_{j_1j_2j_3}|,
$$
could be used to exploit entry-wise sparsity. Clearly,
$$
\mathcal{R}^\ast_1(A)=\max_{j_1,j_2,j_3}|A_{j_1j_2j_3}|.
$$
It can then be shown that:
\blems
\label{LemSparsity}
There exists a constant $0 < c < \infty$ such that
\begin{equation}
\label{eq:lassog}
w_G[\mathbb{B}_{\mathcal{R}_1}(1)] \leq c\sqrt{\log(d_1d_2d_3)}.
\end{equation}
\elems
Let
$$
\Theta_1(s)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: \sum_{j_1=1}^{d_1}\sum_{j_2=1}^{d_2}\sum_{j_3=1}^{d_3}{\mathbb I}(A_{j_1j_2j_3}\neq 0)\le s\right\}.
$$
For an arbitrary $A\in \Theta_1(s)$, write
$$
I(A)=\left\{(j_1,j_2,j_3)\in [d_1]\times [d_2]\times [d_3]: A_{j_1j_2j_3}\neq 0\right\}.
$$
Then $\mathcal{R}_1(\cdot)$ is decomposable with respect to $\mathcal{A}(I(A))$ as defined by (\ref{eq:defA1}). It is easy to verify that for any $A\in \Theta_1(s)$,
\begin{equation}
\label{eq:lassos}
s_1(\mathcal{A}(I))=\sup_{B \in \mathcal{A}(I(A))/\{0\}} \frac{\mathcal{R}_1^2(B)}{\|B\|_{\rm F}^2}\le s.
\end{equation}
In light of (\ref{eq:lassos}) and (\ref{eq:lassog}), Theorem \ref{ThmUpper} implies that
$$
\sup_{T\in \Theta_1(s)}\max\left\{\|\widehat{T}_1-T\|_n^2, \|\widehat{T}_1-T\|_{\rm F}^2\right\} \lesssim {s\log(d_1d_2d_3)\over n},
$$
with high probability by taking
$$
\lambda\asymp \sqrt{\log(d_1d_2d_3)\over n},
$$
where $\widehat{T}_1$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) when using regularizer $\mathcal{R}_1(\cdot)$.
A similar argument can also be applied to fiber-wise sparsity. To fix ideas, we consider here only sparsity among mode-1 fibers. In this case, we use a group Lasso type of regularizer:
$$
\mathcal{R}_2(A)=\sum_{j_2=1}^{d_2}\sum_{j_3=1}^{d_3}\|A_{\cdot j_2j_3}\|_{\ell_2}.
$$
Then
$$
\mathcal{R}^\ast_2(A)=\max_{j_2,j_3}\|A_{\cdot j_2j_3}\|_{\ell_2}.
$$
\blems
\label{LemSparsityfiber}
There exists a constant $0 < c < \infty$ such that
\begin{equation}
\label{eq:glassog}
w_G[\mathbb{B}_{\mathcal{R}_2}(1)]\leq c \sqrt{ \max\{d_1,\log(d_2d_3)\}}.
\end{equation}
\elems
Let
$$
\Theta_2(s)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: \sum_{j_2=1}^{d_2}\sum_{j_3=1}^{d_3}{\mathbb I}(A_{\cdot j_2j_3}\neq \mathbf{0})\le s\right\}.
$$
Similar to the previous case, for an arbitrary $A\in \Theta_1(s)$, write
$$
I(A)=\{(j_2,j_3)\in [d_2]\times [d_3]: A_{\cdot j_2j_3}\neq \mathbf{0}\}.
$$
Then $\mathcal{R}_2(\cdot)$ is decomposable with respect to $\mathcal{A}(I(A))$ as defined by (\ref{eq:defA2}). It is easy to verify that for any $A\in \Theta_2(s)$,
\begin{equation}
\label{eq:glassos}
s_2(\mathcal{A}(I))=\sup_{B \in \mathcal{A}(I(A))/\{0\}} \frac{\mathcal{R}_2^2(B)}{\|B\|_{\rm F}^2}\le s.
\end{equation}
In light of (\ref{eq:glassos}) and (\ref{eq:glassog}), Theorem \ref{ThmUpper} implies that
$$
\sup_{T\in \Theta_2(s)}\max\left\{\|\widehat{T}_2-T\|_n^2, \|\widehat{T}_2-T\|_{\rm F}^2\right\} \lesssim {s\max\{d_1,\log(d_2d_3)\}\over n},
$$
with high probability by taking
$$
\lambda\asymp \sqrt{\max\{d_1,\log(d_2d_3)\}\over n},
$$
where $\widehat{T}_2$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) when using regularizer $\mathcal{R}_2(\cdot)$.
Comparing with the rates for entry-wise and fiber-wise sparsity regularization, we can see the benefit of using group Lasso type of regularizer $\mathcal{R}_2$ when sparsity is likely to occur at the fiber level. More specifically, consider the case when there are a total of $s_1$ nonzero entries from $s_2$ nonzero fibers. If an entry-wise $\ell_1$ regularization is applied, we can achieve the risk bound:
$$
\|\widehat{T}_1-T\|_{\rm F}^2\lesssim {s_1\log(d_1d_2d_3)\over n}.
$$
On the other hand, if fiber-wise group $\ell_1$ regularization is applied, then the risk bound becomes:
$$
\|\widehat{T}_2-T\|_{\rm F}^2\lesssim {s_2\max\{d_1,\log(d_2d_3)\}\over n}.
$$
When nonzero entries are clustered in fibers, we may expect $s_1\approx s_2d_1$. In this case, $\widehat{T}_2$ enjoys performance superior to that of $\widehat{T}_1$.
\subsubsection{Slice-wise sparsity and low-rank structure}
Now we consider slice-wise sparsity and low-rank structure. Again, to fix ideas, we consider here only sparsity among $(1,2)$ slices. As discussed in the previous section, two specific types of regularizers could be employed:
$$
\mathcal{R}_3(A)=\sum_{j_3=1}^{d_3}\|A_{\cdot \cdot j_3}\|_{\rm F},
$$
and
$$
\mathcal{R}_4(A)=\sum_{j_3=1}^{d_3}\|A_{\cdot \cdot j_3}\|_\ast,
$$
where recall that $\|.\|_{\ast}$ denotes the nuclear norm of a matrix, that is the sum of all singular values.
Note that
$$
\mathcal{R}^\ast_3(A)=\max_{1\le j_3\le d_3}\|A_{\cdot \cdot j_3}\|_{\rm F}.
$$
Then we have the following result:
\blems
\label{LemSparsityslice}
There exists a constant $0 < c < \infty$ such that
\begin{equation}
\label{eq:sglassog}
w_G[\mathbb{B}_{\mathcal{R}_3}(1)] \leq c \sqrt{ \max\{d_1d_2,\log(d_3)\}}.
\end{equation}
\elems
Let
$$
\Theta_3(s)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: \sum_{j_3=1}^{d_3}{\mathbb I}(A_{\cdot \cdot j_3}\neq \mathbf{0})\le s\right\}.
$$
For an arbitrary $A\in \Theta_1(s)$, write
$$
I(A)=\{j_3\in [d_3]: A_{\cdot \cdot j_3}\neq \mathbf{0}\}.
$$
Then $\mathcal{R}_3(\cdot)$ is decomposable with respect to $\mathcal{A}(I(A))$ as defined by (\ref{eq:defA3}). It is easy to verify that for any $A\in \Theta_3(s)$,
\begin{equation}
\label{eq:sglassos}
s_3(\mathcal{A}(I(A)))=\sup_{B \in \mathcal{A}(I(A))/\{0\}} \frac{\mathcal{R}_3^2(B)}{\|B\|_{\rm F}^2}\le s.
\end{equation}
Based on (\ref{eq:sglassos}) and (\ref{eq:sglassog}), Theorem \ref{ThmUpper} implies that
$$
\sup_{T\in \Theta_3(s)}\max\left\{\|\widehat{T}_3-T\|_n^2, \|\widehat{T}_3-T\|_{\rm F}^2\right\} \lesssim {s\max\{d_1d_2,\log(d_3)\}\over n},
$$
with high probability by taking
$$
\lambda\asymp \sqrt{\max\{d_1d_2,\log(d_3)\}\over n},
$$
where $\widehat{T}_3$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) when using regularizer $\mathcal{R}_3(\cdot)$.
Alternatively, for $\mathcal{R}_4(\cdot)$,
$$
\mathcal{R}^\ast_4(A)=\max_{j_3}\|A_{\cdot \cdot j_3}\|_{s},
$$
we have the following:
\blems
\label{LemSparsityslice2}
There exists a constant $0 < c < \infty$ such that
\begin{equation}
\label{eq:sglassog2}
w_G[\mathbb{B}_{\mathcal{R}_4}(1)] \leq c \sqrt{ \max\{d_1,d_2,\log(d_3)\}}.
\end{equation}
\elems
Now consider
$$
\Theta_4(r)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: \sum_{j_3=1}^{d_3}{\rm rank}(A_{\cdot \cdot j_3})\le r\right\}.
$$
For an arbitrary $A\in \Theta_4(r)$, denote by $P_{1j}$ and $P_{2j}$ the projection onto the row and column space of $A_{\cdot \cdot j}$ respectively. It is clear that $A\in \mathcal{B}(P_{1j}, P_{2j}: 1\le j\le d_3)$ as defined by (\ref{eq:defB4}). In addition, recall that $\mathcal{R}_4$ is decomposable with respect to $\mathcal{B}(P_{1j}, P_{2j}: 1\le j\le d_3)$ and $\mathcal{A}(P_{1j}, P_{2j}: 1\le j\le d_3)$ as defined by (\ref{eq:defA4}). It is not hard to see that for any $A\in \Theta_4(r)$, $\mathcal{A}(P_{1j}, P_{2j}: 1\le j\le d_3)\subset \Theta_4(2r)$, from which we can derive that:
\blems
\label{le:sglasso2}
For any $A\in \Theta_4(r)$,
\begin{equation}
\label{eq:sglassos2}
s_4(\mathcal{A}(P_{1j}, P_{2j}: 1\le j\le d_3))\le\sup_{B\in \mathcal{A}/\{0\}} \frac{\mathcal{R}_4^2(B)}{\|B\|_{\rm F}^2}\le 2r.
\end{equation}
\elems
In light of (\ref{eq:sglassos2}) and (\ref{eq:sglassog2}), Theorem \ref{ThmUpper} implies that
$$
\sup_{T\in \Theta_4(r)}\max\left\{\|\widehat{T}_4-T\|_n^2, \|\widehat{T}_4-T\|_{\rm F}^2\right\} \lesssim {r\max\{d_1,d_2,\log(d_3)\}\over n},
$$
with high probability by taking
$$
\lambda\asymp \sqrt{\max\{d_1,d_2,\log(d_3)\}\over n},
$$
where $\widehat{T}_4$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) when using regularizer $\mathcal{R}_4(\cdot)$.
Comparing with the rates for estimates with regularizers $\mathcal{R}_3$ and $\mathcal{R}_4$, we can see the benefit of using $\mathcal{R}_4$ when the nonzero slices are likely to be of low-rank. In particular, consider the case when there are $s_1$ nonzero slices and each nonzero slice has rank up to $r$. Then applying $\mathcal{R}_3$ leads to risk bound:
$$
\|\widehat{T}_{3}-T\|_{\rm F}^2\lesssim {s_1\max\{d_1d_2,\log(d_3)\}\over n},
$$
whereas applying $\mathcal{R}_4$ leads to:
$$
\|\widehat{T}_{4}-T\|_{\rm F}^2\lesssim {s_1 r\max\{d_1,d_2,\log(d_3)\}\over n}.
$$
It is clear that $\widehat{T}_{4}$ is a better estimator when $r\ll d_1=d_2=d_3$.
\subsection{Low-rankness regularizers}
We now consider regularizers that encourages low rank estimates. We begin with the tensor nuclear norm regularization:
$$
\mathcal{R}_5(A)=\|A\|_\ast.
$$
Recall that $\mathcal{R}^\ast_5(A)=\|A\|_s$.
\blems
\label{LemNuclearNorm}
There exists a constant $0 < c < \infty$ such that
\begin{equation}
\label{eq:glassog}
w_G[\mathbb{B}_{\mathcal{R}_5}(1)] \leq c \sqrt{(d_1 + d_2 + d_3)}.
\end{equation}
\elems
Now let
$$
\Theta_5(r)=\left\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: \max\{r_1(A), r_2(A), r_3(A)\} \le r\right\}.
$$
For an arbitrary $A\in \Theta_5(r)$, denote by $P_1$, $P_2$, $P_3$ the projection onto the linear space spanned by the mode-1, -2 and -3 fibers respectively. As we argued in the previous section, $\mathcal{R}_5(\cdot)$ is weakly decomposable with respect to $\mathcal{A}(P_1, P_2,P_3)$ and $\mathcal{B}(P_1, P_2,P_3)$, and $A\in \mathcal{B}(P_1, P_2,P_3)$ where $\mathcal{A}(P_1, P_2,P_3)$ and $\mathcal{B}(P_1, P_2,P_3)$ are defined by (\ref{eq:defA5}) and (\ref{eq:defB5}) respectively. It can also be shown that
\blems
For any $A\in \Theta_5(r)$,
\label{LemNuclearNorm1}
$$s_5(\mathcal{A}(P_1, P_2,P_3))=\sup_{B \in \mathcal{A}(P_1, P_2,P_3)/\{0\}} \frac{\mathcal{R}_5^2(B)}{\|B\|_{\rm F}^2}\le r^2.$$
\elems
Lemmas~\ref{LemNuclearNorm} and \ref{LemNuclearNorm1} show that
$$
\sup_{T\in \Theta_5(r)}\max\left\{\|\widehat{T}_5-T\|_n^2, \|\widehat{T}_5-T\|_{\rm F}^2\right\} \lesssim {r^2(d_1+d_2+d_3)\over n},
$$
with high probability by taking
$$
\lambda\asymp \sqrt{d_1+d_2+d_3\over n},
$$
where $\widehat{T}_5$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) when using regularizer $\mathcal{R}_5(\cdot)$.
Next we consider the low-rankness regularization via matricization:
$$
\mathcal{R}_6(A)={1\over 3}\left(\|\mathcal{M}_1(A)\|_\ast+\|\mathcal{M}_2(A)\|_\ast+\|\mathcal{M}_3(A)\|_\ast\right).
$$
It is not hard to see that
$$
\mathcal{R}_6^\ast(A)=3\max\left\{\|\mathcal{M}_1(A)\|_s, \|\mathcal{M}_2(A)\|_s, \|\mathcal{M}_3(A)\|_s\right\}.
$$
\blems
\label{LemLowRank}
There exists a constant $0 < c < \infty$ such that
\begin{equation}
\label{eq:glassog}
w_G[\mathbb{B}_{\mathcal{R}_6}(1)] \leq c \sqrt{\max\{d_1d_2,d_2d_3,d_1d_3\}}.
\end{equation}
\elems
On the other hand,
\blems
\label{LemLowRank1}
For any $A\in \Theta_5(r)$,
$$s_6(\mathcal{A}(P_1,P_2,P_3))=\sup_{B \in \mathcal{A}(P_1,P_2,P_3)/\{0\}} \frac{\mathcal{R}_6^2(B)}{\|B\|_{\rm F}^2}\le r.$$
\elems
Lemmas \ref{LemLowRank} and \ref{LemLowRank1} suggest that
$$
\sup_{T\in \Theta_5(r)}\max\left\{\|\widehat{T}_6-T\|_n^2, \|\widehat{T}_6-T\|_{\rm F}^2\right\} \lesssim {r\max\{d_1d_2, d_2d_3, d_1d_3\}\over n},
$$
with high probability by taking
$$
\lambda\asymp \sqrt{\max\{d_1d_2, d_2d_3, d_1d_3\}\over n}.
$$
where $\widehat{T}_6$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) when using regularizer $\mathcal{R}_6(\cdot)$.
Comparing with the rates for estimates with regularizers $\mathcal{R}_5$ and $\mathcal{R}_6$, we can see the benefit of using $\mathcal{R}_5$. For any $T\in \Theta_5(r)$, If we apply regularizer $\mathcal{R}_5$, then
$$
\|\widehat{T}_{5}-T\|_{\rm F}^2\lesssim {r^2(d_1+d_2+d_3)\over n}.
$$
This is to be compared with the risk bound for matricized regularization:
$$
\|\widehat{T}_{6}-T\|_{\rm F}^2\lesssim {r\max\{d_1d_2,d_2d_3,d_1d_3\}\over n}.
$$
Obviously $\widehat{T}_{5}$ always outperform $\widehat{T}_6$ since $r\le \min\{d_1,d_2,d_3\}$. The advantage of $\widehat{T}_5$ is typically rather significant since in general $r\ll \min\{d_1,d_2,d_3\}$. On the other hand, $\widehat{T}_6$ is more amenable for computation.
Both upper bounds on Frobenius error on $\widehat{T}_{5}$ and $\widehat{T}_{6}$ are novel results and complement the existing results on tensor completion~\cite{GandRect11, MuGoldfarb14} and \cite{YuanZhang14}.
\section{Specific Statistical Problems}
\label{SecExamples}
In this section, we apply our results to several concrete examples where we are attempting to estimate a tensor under certain sparse or low rank constraints, and show that the regularized least squares estimate $\widehat{T}$ is typically minimiax rate optimal with appropriate choices of regularizers.
\subsection{Multi-Response regression with large $p$}
The first example we consider is the multi-response regression model:
\begin{equation*}
Y_{k}^{(i)} = \sum_{j=1}^p \sum_{\ell=1}^m {X^{(i)}_{j \ell} T_{j \ell k}} + \epsilon^{(i)}_{k},
\end{equation*}
where $1 \leq i \leq n$ represents the index for each sample, $1 \leq k \leq m$ represents the index for each response and $1 \leq j \leq p$ represents the index for each feature. For the multi-response regression problem we have $N = 3$, $M = 2$, $d_1 = d_2 = m$ which represents the total number of responses and $d_3 = p$, which represent the total number of parameters.
Since we are in the setting where $p$ is large but only a small number $s$ are relevant, we define the subspace:
\begin{equation*}
\mathcal{T}_1 = \left\{A \in \mathbb{R}^{m \times m \times p}\;|\; \sum_{j=1}^p \mathbb{I}({\|A_{\cdot\cdot j}\|_{\rm F}} \neq 0) \leq s \right\}.
\end{equation*}
Furthermore for each $i$ we assume $X^{(i)} \in \mathbb{R}^{m \times p}$ where each entry of $X^{(i)}$, $[X^{(i)}]_{k,j}$, corresponds to the $j^{th}$ feature for the $k^{th}$ response. For simplicity, we assume the $X^{(i)}$'s are independent Gaussian with covariance $\widetilde{\Sigma} \in \mathbb{R}^{mp \times mp}$. The penalty function we are considering is:
\begin{equation}
\label{eq:multipen1}
\mathcal{R}(A) = \sum_{j = 1}^{p} \| A_{\cdot\cdot j} \|_{\rm F},
\end{equation}
and the corresponding dual function applied to the i.i.d. Gaussian tensor $G$ is:
\begin{equation*}
\mathcal{R}^*(G) = \max_{1 \leq j \leq p} \| G_{..j}\|_{\rm F}.
\end{equation*}
\btheos
\label{ThmUpperMultiReg}
Under the multi-response regression model with $T\in \mathcal{T}_1$ and independent Gaussian design where $c_\ell^2 \leq \lambda_{min}(\widetilde{\Sigma}) \leq \lambda_{max}(\widetilde{\Sigma}) \leq c_u^2$, if
$$\lambda \geq 3 c_u \sqrt{\frac{\max\{m^2,\log p\}}{n}},$$
such that $\sqrt{s} \lambda$ converges to zero as $n$ increases, then there exist some constants $c_1,c_2>0$ such that with probability at least $1 - p^{-c_1}$
\begin{equation*}
\max\left\{\|\widehat{T} - T\|_n^2, \|\widehat{T} - T\|_{\rm F}^2\right\} \leq \frac{c_2c_u^2}{c_{\ell}^2} s \lambda^2,
\end{equation*}
when $n$ is sufficiently large, where $\widehat{T}$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) with regularizer given by (\ref{eq:multipen1}). In addition,
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_1} \|\widetilde{T} - T\|_{\rm F}^2 \geq \frac{c_3 s \max\{m^2, \log p/s\}}{c_u^{2} n},
\end{equation*}
for some constant $c_3>0$, with probability at least $1/2$, where the minimum is taken over all estimators $\widetilde{T}$ based on data $\{(X^{(i)},Y^{(i)}): 1\le i\le n\}$.
\etheos
Theorem \ref{ThmUpperMultiReg} shows that when taking
$$
\lambda\asymp \sqrt{\frac{\max\{m^2,\log p\}}{n}},
$$
the regularized least squares estimate defined by (\ref{EqnGeneral}) with regularizer given by (\ref{eq:multipen1}) achieves minimax optimal rate of convergence over the parameter space $\mathcal{T}_1$.
Alternatively, there are settings where the effect of covariates on the multiple tasks may be of low rank structure. In such a situation, we may consider
\begin{equation*}
\mathcal{T}_2 = \left\{A \in \mathbb{R}^{m \times m \times p}\;|\; \sum_{j=1}^p {\rm rank}({A_{..j}}) \leq r\right\}.
\end{equation*}
An appropriate penalty function in this case is:
\begin{equation}
\label{eq:multipen2}
\mathcal{R}(A) = \sum_{j = 1}^{p} \| A_{..j} \|_{\ast},
\end{equation}
and the corresponding dual function applied to $G$ is:
\begin{equation*}
\mathcal{R}^*(G) = \max_{1 \leq j \leq p} \| G_{..j}\|_{s}.
\end{equation*}
\btheos
\label{ThmUpperMultiReg1}
Under the multi-response regression model with $T\in \mathcal{T}_2$ and independent Gaussian design where $c_\ell^2 \leq \lambda_{min}(\widetilde{\Sigma}) \leq \lambda_{max}(\widetilde{\Sigma}) \leq c_u^2$, if
$$\lambda \geq 3 c_u\sqrt{\frac{\max\{m,\log p\}}{n}},$$
such that $\sqrt{r} \lambda$ converges to zero as $n$ increases, then there exist some constants $c_1,c_2>0$ such that with probability at least $1 - p^{-c_1}$,
\begin{equation*}
\max\left\{\|\widehat{T} - T\|_n^2, \|\widehat{T} - T\|_{\rm F}^2\right\} \leq \frac{c_2 c_u^2}{c_{\ell}^2} r \lambda^2
\end{equation*}
when $n$ is sufficiently large, where $\widehat{T}$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) with regularizer given by (\ref{eq:multipen2}). In addition,
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_2} \|\widetilde{T} - T\|_{\rm F}^2 \geq \frac{c_3 r \max\{m, \log (p/r)\}}{c_u^{2} n},
\end{equation*}
for some constant $c_3>0$, with probability at least $1/2$, where the minimum is taken over all estimators $\widetilde{T}$ based on data $\{(X^{(i)},Y^{(i)}): 1\le i\le n\}$.
\etheos
Again Theorem \ref{ThmUpperMultiReg1} shows that by taking
$$
\lambda\asymp \sqrt{\frac{\max\{m,\log p\}}{n}},
$$
the regularized least squares estimate defined by (\ref{EqnGeneral}) with regularizer given by (\ref{eq:multipen2}) achieves minimax optimal rate of convergence over the parameter space $\mathcal{T}_2$. Comparing with optimal rates for estimating a tensor from $\mathcal{T}_1$, one can see the benefit and importance to take advantage of the extra low rankness if the true coefficient tensor is indeed from $\mathcal{T}_2$.
\subsection{Multivariate Sparse Auto-regressive Models}
Now we consider the setting of vector auto-regressive models. In this case, our generative model is:
\begin{equation}
\label{EqnVAR}
X^{(t+p)} = \sum_{j=1}^p {A_j X^{(t+p-j)}} + \epsilon^{(t)},
\end{equation}
where $1 \leq t \leq n$ represents the time index, $1 \leq j \leq p$ represents the lag index, $\{X^{(t)}\}_{t=0}^{n+p}$ is an $m$-dimensional vector, $\epsilon^{(t)} \sim \mathcal{N}(0, I_{m \times m})$ represents the additive noise. Note that the parameter tensor $T$ is an $m \times m \times p$ tensor so that $T_{\cdot\cdot j}=A_j$, and $T_{k\ell j}$ represents the co-efficient of the $k^{th}$ variable on the $\ell^{th}$ variable at lag $j$. This model is studied by \cite{BasuMichail15} where $p$ is relatively small (to avoid introducing long-range dependence) and $m$ is large. Our main results allow more general structure and regularization schemes than those considered in \cite{BasuMichail15}.
Since we assume the number of series $m$ is large, and there are $m^2$ possible interactions between the series we assume there are only $s \ll m^2$ interactions in total.
\begin{equation}
\label{EqnVARClass}
\mathcal{T}_3 = \left\{A \in \mathbb{R}^{m \times m \times p}\;|\; \sum_{k=1}^m \sum_{\ell=1}^m \mathbb{I}({A_{k\ell\cdot}} \neq \mathbf{0}) \leq s \right\}.
\end{equation}
The penalty function we are considering is:
\begin{equation}
\label{eq:varpen}
\mathcal{R}(A) = \sum_{k = 1}^{m} \sum_{\ell=1}^m \| A_{k\ell\cdot}\|_{\ell_2},
\end{equation}
and the corresponding dual function applied to $G$ is:
\begin{equation*}
\mathcal{R}^*(G) = \max_{1 \leq k,\ell \leq m} \| G_{k,\ell,.}\|_{\ell_2}.
\end{equation*}
The challenge in this setting is that the $X$'s are highly dependent and we use the results developed in \cite{BasuMichail15} to prove that (\ref{AssCov}) is satisfied.
Prior to presenting the main results, we introduce concepts developed in \cite{BasuMichail15} that play a role in determining the constants $c_{u}^2$ and $c_{\ell}^2$ which relate to the stability of the auto-regressive processes. A $p$-variate Gaussian time series is defined by its auto-covariance matrix function
$$\Gamma_X(h) = \mbox{Cov}(X^{(t)}, X^{(t+h)}),$$
for all $t, h \in \mathbb{Z}$. Further, we define the spectral density function:
\begin{equation*}
f_X(\theta) := \frac{1}{2 \pi} \sum_{\ell = -\infty}^{\infty} {\Gamma_X(\ell) e^{-i \ell \theta}},\;\; \theta \in [-\pi, \pi].
\end{equation*}
To ensure the spectral density is bounded, we make the following assumption:
\begin{equation*}
\mathcal{M}(f_X) := \esssup_{\theta} \Lambda_{\max}(f_X(\theta)) < \infty.
\end{equation*}
Further, we define the matrix polynomial
$$\mathcal{A}(z) = I_{m \times m} - \sum_{j=1}^p {A_j z^j}$$
where $\{A_j\}_{j=1}^p$ denote the back-shift matrices, and $z$ represents any point on the complex plane. Note that for a stable, invertible AR($p$) process,
\begin{equation*}
f_X(\theta) = \frac{1}{2 \pi} \mathcal{A}^{-1}(e^{-i \theta}) \overline{\mathcal{A}^{-1}(e^{-i \theta})}.
\end{equation*}
We also define the lower extremum of the spectral density:
\begin{equation*}
m(f_X) := \essinf_{\theta} \Lambda_{\min}(f_X(\theta)).
\end{equation*}
Note that $m(f_X)$ and $\mathcal{M}(f_X)$ satisfy the following bounds:
\begin{equation*}
m(f_X) \geq \frac{1}{2 \pi \mu_{\max}(\mathcal{A})},\qquad {\rm and}\qquad \mathcal{M}(f_X) \leq \frac{1}{2 \pi \mu_{\min}(\mathcal{A})},
\end{equation*}
where
$$\mu_{\min}(\mathcal{A}) := \min_{|z| = 1} {\Lambda_{\min}(\overline{\mathcal{A}(z)}\mathcal{A}(z)) }$$
and
$$\mu_{\max}(\mathcal{A}) := \max_{|z| = 1} {\Lambda_{\max}(\overline{\mathcal{A}(z)}\mathcal{A}(z)) }.$$
From a straightforward calculation, we have that for any fixed $\Delta$:
\begin{equation}
\label{EqnCondVAR}
\frac{1}{\mu_{\max}} \|\Delta\|_{\rm F}^2 \leq \mathbb{E}\left[\|\Delta\|_n^2\right] \leq \frac{1}{\mu_{\min}} \|\Delta\|_2^2.
\end{equation}
Hence $c_u^2 = 1/\mu_{\min}$ and $c_{\ell}^2 = 1/\mu_{\max}$. Now we state our main result for auto-regressive models.
\btheos
\label{ThmUpperVAR}
Under the vector auto-regressive model defined by~\eqref{EqnVAR} with $T\in \mathcal{T}_3$, if
$$\lambda \geq 3 \sqrt{\frac{\max\{p,2\log m\}}{n\mu_{\min}}},$$
such that $\sqrt{s}\lambda$ converges to zero as $n$ increases, then there exist some constants $c_1,c_2>0$ such that with probability at least $1-m^{-c_1}$,
\begin{equation*}
\max\left\{\|\widehat{T} - T\|_n^2, \|\widehat{T} - T\|_{\rm F}^2\right\} \leq \frac{c_2 \mu_{\max}}{\mu_{\min}}s \lambda^2,
\end{equation*}
when $n$ is sufficiently large, where $\widehat{T}$ is the regularized least squares estimators defined by (\ref{EqnGeneral}) with regularizer given by (\ref{eq:varpen}). In addition,
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_3} \|\widetilde{T} - T\|_{\rm F}^2 \geq c_3\mu_{\min} \frac{s \max\{p, \log (m/\sqrt{s})\}}{n},
\end{equation*}
for some constant $c_3>0$, with probability at least $1/2$, where the minimum is taken over all estimators $\widetilde{T}$ based on data $\{X^{(t)}: t=0,\ldots,n+p\}$.
\etheos
Theorem \ref{ThmUpperVAR} provides, to our best knowledge, the only lower bound result for multivariate time series, and the upper bound is different from Proposition 4.1 in \cite{BasuMichail15} since we impose sparsity only on the large $m$ directions and not over the $p$ lags. Our framework also extends to any low-dimensional structure on the tensor $A$ defined by matricization, whereas \cite{BasuMichail15} impose sparsity through vectorization. Note that Proposition 4.1 in \cite{BasuMichail15} follows directly from Lemma~\ref{LemSparsity} with $d_1 = p$ and $d_2 = d_3 = m$.
\subsection{Pairwise interaction tensor models}
Finally, we consider the tensor regression (\ref{eq:model}) where $T$ follows a pairwise interaction model. More specifically, $(X^{(i)},Y^{(i)})$, $i=1,2,\ldots, n$ are independent copies of a random couple $X\in \mathbb{R}^{d_1\times d_2\times d_3}$ and $Y\in \mathbb{R}$ such that
$$
Y=\langle X, T\rangle +\epsilon
$$
and
$$
T_{j_1j_2j_3}=A^{(12)}_{j_1j_2}+A^{(13)}_{j_1j_3}+A^{(23)}_{j_2j_3}.
$$
Here $A^{(k_1,k_2)}\in \mathbb{R}^{d_{k_1}\times d_{k_2}}$ such that
$$
A^{(k_1,k_2)}\mathbf{1} =\mathbf{0}, \qquad {\rm and}\qquad (A^{(k_1,k_2)})^\top\mathbf{1} =\mathbf{0}.
$$
The pairwise interaction was used originally by \cite{RendleSchmidt09,RendleSchmidt10} for personalized tag recommendation, and later analyzed in \cite{ChenLyu13}. \cite{Hoff2014} briefly introduced a single index additive model (amongst other tensor models) which is a sub-class of the pairwise interaction model. The regularizer we consider is:
\begin{equation}
\label{eq:pairwise0}
\mathcal{R}(A)= \|A^{(12)}\|_\ast+\|A^{(13)}\|_\ast+\|A^{(23)}\|_\ast.
\end{equation}
It is not hard to see that $\mathcal{R}$ defined above is decomposable with respect to $\mathcal{A}(P_1,P_2,P_3)$ for any projection matrices.
Let
\begin{eqnarray*}
\mathcal{T}_4=\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: A_{j_1j_2j_3}=A^{(12)}_{j_1j_2}+A^{(13)}_{j_1j_3}+A^{(23)}_{j_2j_3}, A^{(k_1,k_2)}\in \mathbb{R}^{d_{k_1}\times d_{k_2}},\\
A^{(k_1,k_2)}\mathbf{1} =\mathbf{0}, \qquad {\rm and}\qquad (A^{(k_1,k_2)})^\top\mathbf{1} =\mathbf{0}\\
\max_{k_1,k_2} {\rm rank}(A^{(k_1,k_2)})\le r\}.
\end{eqnarray*}
For simplicity, we assume i.i.d. Gaussian design so $c_\ell^2 = c_u^2 = 1$.
\btheos
\label{th:pairwise}
Under the pairwise interaction model with $T\in \mathcal{T}_4$, if
$$
\lambda \geq 3\sqrt{\max\{d_1,d_2,d_3\}\over n},
$$
such that $\sqrt{r}\lambda$ converges to zero as $n$ increases, then there exist constants $c_1,c_2>0$ such that with probability at least $1-\min\{d_1,d_2,d_3\}^{-c_1}$,
\begin{equation*}
\max\left\{\|\widehat{T} - T\|_n^2, \|\widehat{T} - T\|_{\rm F}^2\right\} \leq c_2 r \lambda^2,
\end{equation*}
when $n$ is sufficiently large, where $\widehat{T}$ is the regularized least squares estimate defined by (\ref{EqnGeneral}) with regularizer given by (\ref{eq:pairwise0}). In addition,
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_4} \|\widetilde{T} - T\|_{\rm F}^2 \geq \frac{c_3r\max\{d_1,d_2,d_3\}}{n},
\end{equation*}
for some constant $c_3>0$, with probability at least $1/2$, where the minimum is taken over all estimate $\widetilde{T}$ based on data $\{(X^{(i)},Y^{(i)}): 1\le i\le n\}$.
\etheos
As in the other settings, Theorem \ref{th:pairwise} establishes the minimax optimality of the regularized least squares estimate (\ref{EqnGeneral}) when using an appropriate convex decomposable regularizer.
\section{Proofs}
\label{SecProofs}
In this section, we present the proofs to our main results.
\subsection{Proof of Theorem~\ref{ThmUpper}}
The initial steps exploit weak decomposability and are similar to those from \cite{Neg10}. After the initial steps, we use properties of Gaussian random variables and suprema of Gaussian processes to derive our general upper bound. Throughout $\mathcal{R}(A)$ refers to the weakly decomposable regularizer over the tensor $A$. For a tensor $A$, we shall write $A_0$ and $A^\perp$ as its projections onto $\mathcal{A}_0$ and $\mathcal{A}^\perp$ with respect to the Frobenius norm, respectively.
Since $\widehat{T}$ is the empirical minimizer,
\begin{eqnarray*}
\frac{1}{2n} \sum_{i=1}^n \| Y^{(i)}- \langle X^{(i)} , \widehat{T} \rangle \|_{\rm F}^2 + \lambda \mathcal{R}( \widehat{T}) & \leq & \frac{1}{2n} \sum_{i=1}^n \| Y^{(i)}- \langle X^{(i)} , T \rangle \|_{\rm F}^2 + \lambda \mathcal{R}(T).
\end{eqnarray*}
Substituting $Y^{(i)} = \langle X^{(i)} , T \rangle + \epsilon^{(i)}$ and $\Delta = \widehat{T} - T$,
\begin{eqnarray*}
\frac{1}{2n}\sum_{i=1}^n \| \langle X^{(i)}, \Delta\rangle \|_{\rm F}^2 & \leq & \frac{1}{n} \left|\sum_{i=1}^{n} \langle \epsilon^{(i)} \otimes X^{(i)}, \Delta \rangle\right| + \lambda (\mathcal{R}(T) - \mathcal{R}(\widehat{T}))\\
& \le & \mathcal{R}^*\left(\frac{1}{n} \sum_{i=1}^n {\epsilon^{(i)} \otimes X^{(i)}} \right) \mathcal{R}(\Delta) + \lambda (\mathcal{R}(T) - \mathcal{R}(\widehat{T}_0) - c_\mathcal{R}\mathcal{R}(\widehat{T}^\perp))\\
& \leq & \mathcal{R}^*\left(\frac{1}{n} \sum_{i=1}^n {\epsilon^{(i)} \otimes X^{(i)}} \right) \mathcal{R}(\Delta) + \lambda (\mathcal{R}(\Delta_0) - c_\mathcal{R}\mathcal{R}(\Delta^\perp)),
\end{eqnarray*}
where the second inequality follows from the decomposability and the last one follows from triangular inequality.
Let $G \in \mathbb{R}^{d_1 \times d_2 \times ...\times d_N}$ be an tensor where each entry is i.i.d. $\mathcal{N}(0,1)$. Recall the definition of Gaussian width:
\begin{equation*}
w_G[\mathbb{B}_\mathcal{R}(1)] = \mathbb{E}[\mathcal{R}^*(G)].
\end{equation*}
For simplicity let $$\eta_\mathcal{R} = \frac{3+c_\mathcal{R}}{2c_\mathcal{R}}$$
and recall that $\lambda \geq {2 c_u \eta_{\mathcal{R}}}n^{-1/2} \mathbb{E}[\mathcal{R}^*(G)]$. We have the following Lemma:
\blems
\label{LemGauss}
If $\lambda\geq {2 c_u \eta_{\mathcal{R}}}n^{-1/2} \mathbb{E}[\mathcal{R}^*(G)]$, then
\begin{equation*}
\lambda \geq \eta_{\mathcal{R}} \mathcal{R}^*\left(\frac{1}{n} \sum_{i=1}^n {\epsilon^{(i)} \otimes X^{(i)}} \right),
\end{equation*}
with probability at least $1 - \exp\{-{\eta_{\mathcal{R}}^2 \mathbb{E}[\mathcal{R}^*(G)]^2}/{4}\}$
\elems
The proof relies on Gaussian comparison inequalities and concentration inequalities
\begin{proof}[Proof of Lemma~\ref{LemGauss}]
Recall that we have set:
$$\lambda \geq \frac{2 \eta_\mathcal{R} c_u }{\sqrt{n}} \mathbb{E}[\mathcal{R}^*(G)].$$
First we show that $\lambda \geq {2 c_u \eta_{\mathcal{R}}}n^{-1/2} \mathcal{R}^*(G)$ with high probability using concentration of Lipschitz functions for Gaussian random variables (see Theorem~\ref{ThmConcGaussLip} in Appendix~\ref{AppTailBounds}). First we prove that $f(G) = \mathcal{R}^*(G) = \sup_{A \in \mathbb{B}_\mathcal{R}(1)}\langle G, A \rangle$ is a $1$-Lipschitz function in terms of $G$. In particular note that:
\begin{eqnarray*}
f(G) - f(G') = \sup_{A:\mathcal{R}(A) \leq 1} \langle G, A \rangle - \sup_{A: \mathcal{R}(A) \leq 1} \langle G', A \rangle.
\end{eqnarray*}
Let $\widetilde{A} := \argmax_{A: \mathcal{R}(A) \leq 1} \langle G, A \rangle$. Then
\begin{eqnarray*}
\sup_{A:\mathcal{R}(A) \leq 1} \langle G, A \rangle - \sup_{A: \mathcal{R}(A) \leq 1} \langle G', A \rangle & = & \langle G, \widetilde{A} \rangle - \sup_{\mathcal{R}(A) \leq 1} \langle G', A \rangle\\
& \leq & \langle G, \widetilde{A} \rangle - \langle G', \widetilde{A} \rangle\\
& \leq & \langle G - G', \widetilde{A} \rangle \\
& \leq & \sup_{A:\mathcal{R}(A) \leq 1} \langle G - G', A \rangle \\
& \leq & \sup_{A:\|A\|_{\rm F} \leq 1} \langle G - G', A \rangle \\
& \leq & \|G - G'\|_{\rm F},
\end{eqnarray*}
where recall that $\|A\|_F \leq \mathcal{R}(A)$ which implies the second last inequality. Therefore $f(G)$ is a $1$-Lipschitz function with respect to the Frobenius norm. Therefore, by applying Theorem~\ref{ThmConcGaussLip} in Appendix~\ref{AppTailBounds},
\begin{equation*}
\mathbb{P}\left\{\left|\sup_{A \in \mathbb{B}_\mathcal{R}(1)}\langle G, A \rangle - \mathbb{E}[\sup_{A \in \mathbb{B}_\mathcal{R}(1)}\langle G, A \rangle]\right| > w_G(\mathbb{B}_\mathcal{R}(1)) \right\} \leq 2 \exp\left(-\frac{1}{2} w_G^2[\mathbb{B}_\mathcal{R}(1)]\right).
\end{equation*}
Therefore $$\lambda \geq \frac{\eta_\mathcal{R} c_u }{\sqrt{n}} \mathcal{R}^*(G)$$ with probability at least $1 - 2\exp\{-w_G^2[\mathbb{B}_\mathcal{R}(1)]/2\}$.
To complete the proof, we use a Gaussian comparison inequality between the supremum of the process ${c_u }n^{-1/2} \langle G, A \rangle$ and $n^{-1} \sum_{i=1}^n \langle \epsilon^{(i)} \otimes X^{(i)}, A\rangle$ over the set $\mathbb{B}_\mathcal{R}(1)$. Recall that:
\begin{equation*}
\mathcal{R}^*\left(\frac{1}{n} \sum_{i=1}^n {\epsilon^{(i)} \otimes X^{(i)}} \right) = \sup_{A\in \mathbb{B}_\mathcal{R}(1)}\left\langle A, \frac{1}{n} \sum_{i=1}^n {\epsilon^{(i)} \otimes X^{(i)}} \right\rangle.
\end{equation*}
Recall that each $\epsilon^{(i)} \in \mathbb{R}^{d_{M+1} \times d_{M+2} \times ... \times d_N}$ is an i.i.d. standard Gaussian tensor and $\mbox{vec}(X) \in \mathbb{R}^{nd_1d_2\cdots d_M}$ is a Gaussian vector covariance $\Sigma \in \mathbb{R}^{(nD_M) \times (nD_M)}$. Further let $\{w^{(i)}: i=1,\ldots,n\}$ be i.i.d. standard normal Gaussian tensors where $w^{(i)} \in \mathbb{R}^{d_{1} \times d_{2} \times ... \times d_M}$. Assuming (\ref{AssCov}) and using a standard Gaussian comparison inequality due to Lemma~\ref{LemAnderson} in Appendix~\ref{AppTailBounds} proven earlier in~\cite{Anderson55}, we get
\begin{equation*}
\mathbb{P}\left\{\sup_{A: \mathcal{R}(A) \leq 1}\frac{1}{n} \sum_{i=1}^n {\langle \epsilon^{(i)} \otimes X^{(i)}, A\rangle} > x \right\} \leq \mathbb{P}\left\{\sup_{A: \mathcal{R}(A) \leq 1}\frac{1}{n} \sum_{i=1}^n {\langle \epsilon^{(i)} \otimes w^{(i)}, A\rangle} > \frac{x}{c_u} \right\},
\end{equation*}
since
$${\rm Cov}({\rm vec}(X)) = \Sigma \preceq c_u^2 I_{(nD_M) \times (nD_M)}.$$
Now we apply Slepian's lemma \citep{Slepian62} to complete the proof. For completeness, Slepian's lemma is included in Appendix~\ref{AppTailBounds}. Clearly for any $A$,
\begin{equation*}
\frac{1}{n}\sum_{i=1}^n {\langle \mathbb{E}[\epsilon^{(i)} \otimes w^{(i)}], A\rangle} = 0.
\end{equation*}
Further a simple calculation shows that for any $A$,
\begin{equation*}
\mbox{Var}\left(\frac{1}{n} \sum_{i=1}^n {\langle [\epsilon^{(i)} \otimes w^{(i)}], A\rangle}\right) = \frac{\|A\|_{\rm F}^2}{n},
\end{equation*}
where we have exploited independence between across all samples and fibers, and the fact that $\epsilon$ and $w$ are independent. Further, for all $A, A',$
\begin{equation*}
\mbox{Var}\left(\frac{1}{n} \sum_{i=1}^n {\langle [\epsilon^{(i)} \otimes w^{(i)}], A - A'\rangle}\right) = \frac{\|A- A'\|_{\rm F}^2}{n}.
\end{equation*}
Now let $G \in \mathbb{R}^{d_1 \times d_2 \times ... \times d_N}$ be an i.i.d. standard normal tensor and define the zero-mean Gaussian process,
\begin{equation*}
\frac{1}{\sqrt{n}}\langle G, A\rangle,
\end{equation*}
for any $A\in \mathbb{B}_\mathcal{R}(1)$. It is straightforward to show that,
\begin{equation*}
\mbox{Var}\left(\frac{1}{\sqrt{n}}\langle G, A\rangle\right) = \frac{\|A\|_{\rm F}^2}{n},
\end{equation*}
and
\begin{equation*}
\mbox{Var}\left(\frac{1}{\sqrt{n}}\langle G, A - A'\rangle\right) = \frac{\|A-A'\|_{\rm F}^2}{n},
\end{equation*}
for all $A, A'$. Therefore, directly applying Slepian's lemma (Lemma~\ref{LemSlepian} in Appendix~\ref{AppTailBounds}),
\begin{equation*}
\mathbb{P}\left\{\sup_{\mathcal{R}(A)\leq 1}\frac{1}{n} \sum_{i=1}^n {\langle \epsilon^{(i)} \otimes w^{(i)}, A\rangle} > x \right\} \leq \mathbb{P}\left\{\sup_{\mathcal{R}(A)\leq 1}\frac{1}{\sqrt{n}} \langle G, A\rangle > x \right\},
\end{equation*}
for all $x > 0$. Substituting $x$ by $x/c_u$ means that
\begin{equation*}
\mathbb{P}\left\{\mathcal{R}^*\left(\frac{1}{n} \sum_{i=1}^n {\langle \epsilon^{(i)} \otimes w^{(i)}, A \rangle}\right) > x \right\} \leq \mathbb{P}\left\{ \frac{c_u}{\sqrt{n}} \mathcal{R}^*(G) > x \right\},
\end{equation*}
for any $x > 0$. This completes the proof.
\end{proof}
\vskip 25pt
In light of Lemma~\ref{LemGauss}, for the remainder of the proof, we can condition on the event that
$$\lambda \geq \eta_{\mathcal{R}} \mathcal{R}^*\left(\frac{1}{n} \sum_{i=1}^n {\epsilon^{(i)} \otimes X^{(i)}} \right).$$
Under this event,
\begin{eqnarray*}
\frac{1}{2n}\sum_{i=1}^n \| \langle X^{(i)}, \Delta \rangle \|_{\rm F}^2 & \leq & \frac{1}{\eta_\mathcal{R}}\lambda \mathcal{R}(\Delta) + \lambda (\mathcal{R}(\Delta_0) - c_\mathcal{R}\mathcal{R}(\Delta^\perp))\\
&\leq & \left(1+\frac{1}{\eta_\mathcal{R}}\right)\lambda \mathcal{R}(\Delta_0) - \left(c_\mathcal{R} - \frac{1}{\eta_\mathcal{R}}\right) \lambda \mathcal{R}(\Delta^\perp).
\end{eqnarray*}
Since
$$\frac{1}{2n}\sum_{i=1}^n \| \langle \Delta, X^{(i)} \rangle \|_{\rm F}^2 \geq 0,$$
we get
\begin{equation*}
\mathcal{R}(\Delta^\perp) \leq \frac{3}{c_\mathcal{R}} \mathcal{R}(\Delta_{0}).
\end{equation*}
Hence we define the cone
$$\mathcal{C} = \left\{ \Delta\;|\; \mathcal{R}(\Delta^\perp) \leq 3c_\mathcal{R}^{-1} \mathcal{R}(\Delta_{0}) \right\},$$
and know that $\Delta \in \mathcal{C}$. Hence
$$
\frac{1}{2n}\sum_{i=1}^n \| \langle X^{(i)},\Delta \rangle \|_{\rm F}^2\leq \frac{3(1+c_\mathcal{R})}{3+c_\mathcal{R}} \lambda \mathcal{R}(\Delta_{0})\leq \frac{3(1+c_\mathcal{R})}{3+c_\mathcal{R}} \sqrt{s(\mathcal{A})} \lambda \|\Delta\|_{\rm F}.
$$
Recall that
$$\frac{1}{n}\sum_{i=1}^n \| \langle X^{(i)}, \Delta\rangle \|_{\rm F}^2 = \|\Delta\|_n^2.$$
Thus,
\begin{eqnarray*}
\|\Delta\|_n^2 & \leq & \frac{6(1+c_\mathcal{R})}{3+c_\mathcal{R}}\sqrt{s(\mathcal{A})} \lambda \|\Delta\|_{\rm F}.
\end{eqnarray*}
For convenience, in the remainder of this proof let $$\delta_n:= \frac{6(1+c_\mathcal{R})}{3+c_\mathcal{R}}\sqrt{s(\mathcal{A})} \lambda.$$
Now we split into three cases. (i) If $\|\Delta\|_n \geq \| \Delta \|_{\rm F}$, then $$\max\{\| \Delta\|_n, \| \Delta\|_{\rm F}\} \leq \delta_n.$$
On the other hand if (ii) $\|\Delta\|_n \leq \| \Delta \|_{\rm F}$ and $\|\Delta\|_{\rm F} \leq \frac{c_u}{c_{\ell}} \delta_n$, then
$$\max\{\| \Delta\|_n, \| \Delta\|_{\rm F}\} \leq \frac{c_u}{c_{\ell}} \delta_n.$$
Hence the only case we need to consider is (iii) $\|\Delta\|_n \leq \| \Delta \|_{\rm F}$ and $\|\Delta\|_{\rm F} \geq c_uc_{\ell}^{-1} \delta_n$. Now we follow a similar proof technique to the proof for Theorem 1 in \cite{RasWaiYu12}.
Let us define the following set:
\begin{eqnarray*}
\mathcal{C}(\delta_n) := \left\{ \Delta \in \mathbb{R}^{d_1 \times d_2 \times\cdots\times d_N}\;|\;\mathcal{R}(\Delta^\perp) \leq 3 c_\mathcal{R}^{-1} \mathcal{R}(\Delta_{0}),\;\|\Delta\|_n \leq \|\Delta\|_{\rm F} \right\}.
\end{eqnarray*}
Further, let us define the event:
\begin{equation*}
\mathcal{E}(\delta_n) := \left\{ \|\Delta\|_n^2 \geq \frac{1}{4}\|\Delta\|_{\rm F}^2 \;|\; \Delta \in \mathcal{C}(\delta_n),\; \|\Delta\|_{\rm F} \geq \frac{c_u}{c_\ell}\delta_n \right\}.
\end{equation*}
Let us define the alternative event:
\begin{equation*}
\mathcal{E}'(\delta_n) := \{ \|\Delta\|_{n}^2 \geq \frac{1}{4} \|\Delta\|_{\rm F}^2 \;|\; \Delta \in \mathcal{C}(\delta_n),\; \|\Delta\|_{\rm F} = \frac{c_u}{c_\ell} \delta_n \}.
\end{equation*}
We claim that it suffices to show that $\mathcal{E}'(\delta_n)$ holds with probability at least $1 - \exp(-c n )$ for some constant $c > 0$. In particular, given an arbitrary non-zero $\Delta \in \mathcal{C}(\delta_n)$, consider the re-scaled tensor
$$\widetilde{\Delta} = \frac{c_u\delta_n}{c_\ell} \frac{\Delta}{\|\Delta\|_{\rm F}}.$$
Since $\Delta \in \mathcal{C}(\delta_n)$ and $\mathcal{C}(\delta_n)$ is star-shaped, we have $\widetilde{\Delta} \in \mathcal{C}(\delta_n)$ and $\|\widetilde{\Delta}\|_{\rm F} = c_uc_\ell^{-1}\delta_n$ by construction. Consequently, it is sufficient to prove that $\mathcal{E}'(\delta_n)$ holds with high probability.
\blems
\label{LemEmpLowerBound}
Under the assumption that for any $c' > 0$, there exists an $n$ such that $\sqrt{s} \lambda \leq c'$, there exists a $\tilde{c} > 0$ such that
\begin{equation*}
\mathbb{P}\big(\mathcal{E}'(\delta_n)\big) \geq 1 - \exp(-\tilde{c}n).
\end{equation*}
\elems
\begin{proof}[Proof of Lemma~\ref{LemEmpLowerBound}]
Denote by $D_N=d_1d_2\cdots d_N$ and $D_M=d_1d_2\cdots d_M$. Now we define the random variable
$$Z_n(\mathcal{C}(\delta_n)) = \sup_{\Delta \in \mathcal{C}(\delta_n)} \left\{\frac{c_u^2}{c_\ell^2}\delta_n^2 - \frac{1}{n} \sum_{i=1}^n \|\langle \Delta, X^{(i)} \rangle\|_{F}^2\right\},$$
then it suffices to show that
$$Z_n(\mathcal{C}(\delta_n)) \leq \frac{c_u^2\delta_n^2}{2 c_\ell^2}.$$
Recall that the norm
$$\|\Delta\|_n^2 = \frac{1}{n} \sum_{i=1}^n \|\langle \Delta, X^{(i)} \rangle\|_{F}^2.$$
Let $N_{\rm pr}(\epsilon; \mathcal{C}(\delta_n); \|\cdot\|_{n})$ denote the proper covering number of $\mathcal{C}(\delta_n)$ in $\|\cdot\|_{n}$ norm. Now let $\Delta^{1}, \Delta^{2},\ldots,\Delta^{\mathcal{N}}$, be a minimal $c_u \delta_n/(8 c_\ell)$-proper covering of $\mathcal{C}(\delta_n)$, so that for all $\Delta \in \mathcal{C}(\delta_n)$, there exists a $k$ such that
$$\|\Delta^{k} - \Delta\|_n \leq \frac{c_u \delta_n}{8 c_\ell},$$
and
$$\mathcal{N} = N_{\rm pr}\left(\frac{c_u \delta_n}{8 c_\ell}; \mathcal{C}(\delta_n); \|\cdot\|_n\right).$$
Note that
\begin{eqnarray*}
\frac{c_u^2 \delta_n^2}{c_\ell^2} - \frac{1}{n} \sum_{i=1}^n \|\langle \Delta, X^{(i)} \rangle\|_{F}^2 &=& \left(\frac{c_u^2 \delta_n^2}{c_\ell^2} - \frac{1}{n} \sum_{i=1}^n \|\langle \Delta^{k}, X^{(i)} \rangle\|_{F}^2\right) \\
&&+ \left(\frac{1}{n} \sum_{i=1}^n \|\langle \Delta^{k}, X^{(i)} \rangle\|_{F}^2 -\frac{1}{n} \sum_{i=1}^n \|\langle \Delta, X^{(i)} \rangle\|_{F}^2\right).
\end{eqnarray*}
By the Cauchy-Schwarz inequality, we have
\begin{eqnarray*}
&&\frac{1}{n} \sum_{i=1}^n \|\langle\Delta^{k}, X^{(i)} \rangle\|_{\rm F}^2 - \|\langle \Delta, X^{(i)} \rangle\|_{\rm F}^2 \\
& = & \frac{1}{n} \sum_{i=1}^n \biggr\langle \langle \Delta^{k} - \Delta, X^{(i)}\rangle, \langle \Delta^{(k)} + \Delta, X^{(i)}\rangle \biggr\rangle\\
& \leq & \left(\frac{1}{n} \sum_{i=1}^n \|\langle \Delta^{k} - \Delta, X^{(i)} \rangle\|_{\rm F}^2\right)^{1/2} \left({\frac{1}{n} \sum_{i=1}^n \|\langle \Delta^{k} + \Delta, X^{(i)} \rangle\|_{\rm F}^2}\right)^{1/2}\\
& = & \|\Delta^{k} - \Delta\|_n \left(\frac{1}{n} \sum_{i=1}^n \|\langle \Delta^{k} + \Delta, X^{(i)} \rangle\|_{\rm F}^2 \right)^{1/2}.
\end{eqnarray*}
By our choice of covering, $\|\Delta^{k} - \Delta\|_n \leq {c_u \delta_n}/{8c_\ell}$. On the other hand, we have
\begin{equation*}
\left({\frac{1}{n} \sum_{i=1}^n \|\langle \Delta^{k} + \Delta, X^{(i)} \rangle\|_{\rm F}^2}\right) \leq \left({2\|\Delta^{k}\|_n^2 + 2\|\Delta\|_n^2 }\right)^{1/2} \leq \sqrt{4c_u^2\delta_n^2/c_\ell^2} = 2 \frac{c_u \delta_n}{c_\ell}.
\end{equation*}
Overall, we have established the upper bound
$$\frac{1}{n} \|\langle \Delta^{k}, X^{(i)} \rangle\|_{\rm F}^2 - \|\langle \Delta, X^{(i)} \rangle\|_{\rm F}^2 \leq \frac{c_u^2\delta_n^2}{4 c_\ell^2}.$$
Hence we have:
\begin{equation*}
Z_n(\mathcal{C}(\delta_n)) \leq \max_{1\le k\le \mathcal{N}} \left\{\frac{c_u^2 \delta_n^2}{c_\ell^2} - \frac{1}{n} \sum_{i=1}^n \|\langle \Delta^{k}, X^{(i)} \rangle\|_{\rm F}^2 \right\} + \frac{c_u^2 \delta_n^2}{4 c_\ell^2}.
\end{equation*}
Now we use (\ref{AssCov}) combined with the Hanson-Wright inequality \citep{HanWri71} to prove that for any $\Delta^{(k)}$ in our covering set,
\begin{equation*}
\mathbb{P}\left\{\frac{c_u^2}{c_\ell^2}\delta_n^2 - \|\Delta^{k}\|_{n}^2 > \frac{c_u^2 \delta_n^2}{4 c_\ell^2}\right\} \leq \exp(-c n ),
\end{equation*}
for some constant $c>0$. Recall that
$$\Sigma = \mathbb{E}[\mbox{vec}(X)\mbox{vec}(X)^\top] \in \mathbb{R}^{(nD_M) \times (nD_M)}.$$
Further, recall $[M] = \{1,2,...,M\}$ and define an extension of the standard matricization
$$\widetilde{\Delta}:= \mathcal{M}_{[M]}(\Delta) \in \mathbb{R}^{D_M \times D_N/D_M}$$
which groups together the first $M$ modes. Further we define the matrix $Q \in \mathbb{R}^{(nD_M) \times (nD_M)}$ such that
$$Q_{r\ell, sm} = \mathbb{I}(r=s) \langle \widetilde{\Delta}_{\ell}, \widetilde{\Delta}_m \rangle$$
where $1 \leq r,s \leq n$ and $1 \leq \ell,m \leq D_M$ and $\widetilde{\Delta}_{\ell}, \widetilde{\Delta}_m \in \mathbb{R}^n$. Simple algebra shows that
\begin{equation*}
\|\Delta\|_n^2 = \frac{1}{n} Z^\top Q^{1/2} \Sigma Q^{1/2} Z.
\end{equation*}
for some $Z \in \mathbb{R}^{nD_M}$ such that
$$Z \sim \mathcal{N}(0, I_{(nD_M) \times (nD_M)}).$$
Note that
\begin{equation*}
\mathbb{E}[\|\Delta\|_n^2] = \frac{1}{n} \mathbb{E}[Z^\top Q^{1/2} \Sigma Q^{1/2} Z]
\geq \frac{c_\ell^2}{n} \mathbb{E}[Z^\top QZ],
\end{equation*}
using (\ref{AssCov}). Furthermore,
\begin{equation*}
\frac{c_\ell^2}{n} \mathbb{E}[Z^\top QZ] = c_\ell^2 \|\widetilde{\Delta}\|_{\rm F}^2 = c_u^2 \delta_n^2.
\end{equation*}
Now we apply the Hanson-Wright inequality \citep[see, e.g.,][]{HanWri71} to get
\begin{equation*}
\mathbb{P}\left\{\frac{c_u^2}{c_\ell^2} \delta_n^2 - \|\Delta\|_n^2 > \frac{c_u^2}{c_\ell^2}\delta_n^2 \zeta \right\} \leq 2 \exp \left(-c \min\left\{\frac{n^2 \zeta^2 \delta_n^4}{\|Q^{1/2} \Sigma Q^{1/2}\|_{\rm F}^2}, \frac{n \zeta \delta_n^2}{\|Q^{1/2} \Sigma Q^{1/2}\|_{s}} \right\} \right).
\end{equation*}
First we upper bound $\|Q^{1/2} \Sigma Q^{1/2}\|_{\rm F}^2$. If (\ref{AssCov}) holds, then
$$\|Q^{1/2} \Sigma Q^{1/2}\|_{\rm F}^2 \leq c_u^2 \|Q\|_{\rm F}^2.$$
Furthermore,
\begin{eqnarray*}
\|Q\|_{\rm F}^2 & = & \sum_{s=1}^n \sum_{r=1}^n \mathbb{I}(r=s) \sum_{\ell = 1}^{D_M} \sum_{m=1}^{D_M}{\langle \widetilde{\Delta}_\ell, \widetilde{\Delta}_m \rangle^2}\\
& = & \sum_{r=1}^n \sum_{\ell = 1}^{D_M} \sum_{m=1}^{D_M}{\langle \widetilde{\Delta}_\ell, \widetilde{\Delta}_m \rangle^2} \\
& = & n \sum_{\ell = 1}^{D_M} \sum_{m=1}^{D_M}{\langle \widetilde{\Delta}_\ell, \widetilde{\Delta}_m \rangle^2}\\
& \leq & n \sum_{\ell = 1}^{D_M}\|\widetilde{\Delta}_\ell\|_{\ell_2}^2 \sum_{m=1}^{D_M}{\|\widetilde{\Delta}_m\|_{\ell_2}^2} \\
& = & \frac{c_u^2}{c_\ell^2}n\delta_n^4.
\end{eqnarray*}
Thus,
$$
\|Q^{1/2} \Sigma Q^{1/2}\|_{\rm F}^2\le \frac{c_u^4}{c_\ell^2}n\delta_n^4.
$$
Next we upper bound $\|Q^{1/2} \Sigma Q^{1/2}\|_{s}$. If (\ref{AssCov}) holds, then
$$\|Q^{1/2} \Sigma Q^{1/2}\|_{s} \leq c_u \|Q\|_{s}.$$
Let $v \in \mathbb{R}^{nD_M}$ such that $\|v\|_{\ell_2}^2 = 1$. Then
\begin{eqnarray*}
v^\top Q v & = & \sum_{s=1}^n \sum_{r=1}^n \sum_{\ell = 1}^{D_M} \sum_{m=1}^{D_M}{ v_{r \ell} \langle \widetilde{\Delta}_\ell, \widetilde{\Delta}_m \rangle v_{s m} \mathbb{I}(r=s)}\\
& = & \sum_{r=1}^n \sum_{\ell = 1}^{D_M} \sum_{m=1}^{D_M}{\langle \widetilde{\Delta}_\ell, \widetilde{\Delta}_m \rangle} v_{r \ell} v_{r m} \\
& = & \sum_{\ell = 1}^{D_M} \sum_{m=1}^{D_M}{\langle \widetilde{\Delta}_\ell, \widetilde{\Delta}_m \rangle \sum_{r=1}^n v_{r\ell} v_{rm}}\\
& \leq & \|v\|_{\ell_2}^2 \|\Delta\|_{\rm F}^2 \\
& = & \frac{c_u^2}{c_\ell^2}\delta_n^2.
\end{eqnarray*}
This implies that
$$
\|Q^{1/2} \Sigma Q^{1/2}\|_{s}\le \frac{c_u^3}{c_\ell^2}\delta_n^2.
$$
Hence, applying the Hanson-Wright inequality yields:
\begin{equation*}
\mathbb{P}\left\{\frac{c_u^2}{c_\ell^2} \delta_n^2 - \|\Delta\|_n^2 > \frac{c_u^2}{c_\ell^2}\delta_n^2 \zeta \right\} \leq 2 \exp\left(-\frac{c c_\ell^2 }{c_u^2} \min\{n \zeta^2, n \zeta\} \right).
\end{equation*}
Setting $\zeta = 1/4$ yields
\begin{equation*}
\mathbb{P}\left\{\frac{c_u^2}{c_\ell^2} \delta_n^2 - \|\Delta\|_n^2 > \frac{c_u^2}{4 c_\ell^2}\delta_n^2 \right\} \leq 2 \exp \left(-\frac{c c_\ell^2 n}{16c_u^2} \right).
\end{equation*}
Next using the union bound, we have
\begin{equation*}
\mathbb{P}\left\{\max_{s=1,2,\ldots,\mathcal{N}} \left\{\frac{c_u^2}{c_\ell^2}\delta_n^2 - \|\Delta^{(s)}\|_n^2\right\} > \frac{\delta_n^2}{4}\right\} \leq \exp\left(\log N_{\rm pr}\left(\frac{c_u \delta_n}{8 c_\ell }, \mathcal{C}(\delta_n), \|\cdot\|_n\right)-c n \right).
\end{equation*}
It remains to bound $\log N_{\rm pr}( c_u\delta_n/(8 c_\ell), \mathcal{C}(\delta_n), \|\cdot\|_n)$. Since the proper covering entropy is upper bounded by the standard covering entropy so that
$$\log N_{\rm pr}\left(\frac{c_u \delta_n}{8 c_\ell}, \mathcal{C}(\delta_n), \|\cdot\|_n\right)\le \log N\left(\frac{c_u \delta_n}{16 c_\ell}, \mathcal{C}(\delta_n), \|\cdot\|_n\right),$$
it suffices to upper bound $\log N(c_u \delta_n/(16 c_\ell), \mathcal{C}(\delta_n), \|\cdot\|_n)$. Viewing the samples $X$ as fixed, let us define the zero-mean Gaussian process $\{ W_{\Delta} \}_{\Delta \in \mathcal{B}}$ via
$$W_{\Delta} = \frac{1}{\sqrt{n}} \sum_{i=1}^n \langle \epsilon^{(i)} \otimes X^{(i)}, \Delta \rangle$$
where $\{ \epsilon^{(i)}: i=1,\ldots,n \}$ are i.i.d. standard Gaussian random variables. By construction, we have
$$\mbox{var}[(W_{\Delta} - W_{\Delta'})] = \|\Delta - \Delta'\|_n^2.$$
By the Sudakov minoration \citep[see, e.g.,][]{LedTal91}, for all $\eta > 0$ we have
$$\eta \sqrt{\log N(\eta, \mathcal{C}(\delta_n), \|\cdot\|_n)} \leq 4 \mathbb{E}_{\epsilon} \left(\sup_{\Delta \in \mathcal{C}(\delta_n)} W_{\Delta}\right).$$
Setting $\eta = c_u \delta_n/(16 c_\ell)$, we obtain the upper bound:
\begin{equation*}
\sqrt{\frac{1}{n}\log N\left(\frac{c_u \delta_n}{16 c_\ell}, \mathcal{C}(\delta_n), \|.\|_n\right)} \leq \frac{64 c_\ell}{c_u \delta_n} \mathbb{E}_{\epsilon} \left(\sup_{\Delta \in \mathcal{C}(\delta_n)} \frac{1}{n} \sum_{i=1}^n \langle \epsilon^{(i)} \otimes X^{(i)}, \Delta \rangle \right).
\end{equation*}
The final step is to upper bound the Gaussian complexity
$$\mathbb{E}_{\epsilon} \left(\sup_{\Delta \in \mathcal{C}(\delta_n)} \frac{1}{n} \sum_{i=1}^n \langle \epsilon^{(i)} \otimes X^{(i)}, \Delta \rangle \right).$$
Clearly,
$$
\frac{1}{n} \sum_{i=1}^n \langle \epsilon^{(i)} \otimes X^{(i)}, \Delta \rangle \leq \mathcal{R}^*\left(\frac{1}{n} \sum_{i=1}^n \epsilon^{(i)} \otimes X^{(i)} \right) \mathcal{R}(\Delta) \leq \frac{\lambda}{\eta_{\mathcal{R}}} \mathcal{R}(\Delta).
$$
by the definition of $\lambda$ and our earlier argument. Since $\Delta \in \mathcal{C}(\delta_n)$,
$$
\frac{\lambda}{\eta_{\mathcal{R}}} \mathcal{R}(\Delta) \leq \frac{\lambda(1 + 3c^{-1}_{\mathcal{R}})}{\eta_{\mathcal{R}}} \mathcal{R}(\Delta_0) \leq \frac{\lambda(1 + 3c^{-1}_{\mathcal{R}})}{\eta_{\mathcal{R}}} \sqrt{s(\mathcal{A})}\|\Delta_0\|_F \leq \frac{c_u(1 + 3c^{-1}_{\mathcal{R}})}{c_\ell \eta_{\mathcal{R}}}\delta_n \sqrt{s(\mathcal{A})}\lambda.
$$
Therefore,
$$\mathbb{E}_{\epsilon} \left(\sup_{\Delta \in \mathcal{C}(\delta_n)} \frac{1}{n} \sum_{i=1}^n \langle \epsilon^{(i)} \otimes X^{(i)}, \Delta \rangle \right) \leq \frac{c_u(1 + 3c^{-1}_{\mathcal{R}})}{c_\ell \eta_{\mathcal{R}}}\delta_n \sqrt{s(\mathcal{A})}\lambda,$$
and
\begin{equation*}
\sqrt{\frac{1}{n}\log N(\frac{c_u \delta_n}{16 c_\ell}, \mathcal{C}(\delta_n), \|.\|_n)} \leq 64 \frac{(1 + 3c^{-1}_{\mathcal{R}})}{\eta_{\mathcal{R}}} \sqrt{s(\mathcal{A})}\lambda.
\end{equation*}
Hence
$$\sqrt{\log N(\frac{c_u \delta_n}{16c_\ell}, \mathcal{C}(\delta_n), \|.\|_n)} \leq 64 \frac{(1 + 3c^{-1}_{\mathcal{R}})}{\eta_{\mathcal{R}}} \sqrt{n} \sqrt{s(\mathcal{A})}\lambda$$
and
$$
\mathbb{P}\left\{\max_{s=1,2,\ldots,N} \left\{\frac{c_u^2}{c_\ell^2}\delta_n^2 - \|\Delta^{(s)}\|_n^2\right\} > \frac{c_u^2 \delta_n^2}{4 c_\ell^2}\right\} \leq \exp(64^2 c n s(\mathcal{A}) \lambda^2 -c n )\leq \exp(- \tilde{c} n )
$$
where the finally inequality holds since $s(\mathcal{A})\lambda^2$ converges to $0$ so if we choose $n$ to be sufficiently large.
\end{proof}
\vskip 25pt
Finally we return to the main proof. On the event $\mathcal{E}(\delta_n)$, it now follows easily that,
\begin{equation*}
\max\{\|\Delta\|_2^2, \|\Delta\|_n^2\} \leq \frac{\eta_{\mathcal{R}} c_u^2}{c_\ell^2} s(\mathcal{A})\lambda^2.
\end{equation*}
This completes the proof for Theorem~\ref{ThmUpper}.
\subsection{Proof of other results in Section~\ref{SecBounds}}
In this section we present proofs for the other main results from Section \ref{SecBounds}, deferring the more technical parts to the appendix.
\begin{proof}[Proof of Lemmas ~\ref{LemSparsity}, \ref{LemSparsityfiber} and \ref{LemSparsityslice}]
We prove these three lemmas together since the proofs follow a very similar argument. First let $S \subset \{1,2,3\}$ denote the directions in which sparsity is applied and $D_S = \prod_{k \in S}{d_k}$ denote the total dimension in all these directions. For example, in Lemma~\ref{LemSparsity} $S = \{1,2,3\}$ and $D_S = d_1d_2d_3$, for Lemma~\ref{LemSparsityfiber}, $S = \{2,3\}$ and $D_S = d_2d_3$ and for Lemma~\ref{LemSparsityslice}, $S =\{1\}$ and $D_S = d_1$. Recall $N = \{1,2,3\}$ and $D_N = d_1d_2d_3$.
Note that $\mathcal{R}^{*}(G)$ can be represented by the variational form:
\begin{equation*}
\mathcal{R}^{*}(G) = \sup_{\|\mbox{vec}(u)\|_{\ell_1} \leq 1, \|v\|_{\rm F} \leq 1} \langle G, u \otimes v \rangle,
\end{equation*}
where $u \in \mathbb{R}^{d_{S_1} \times ...\times d_{S_{|S|}}}$ and $v \in \mathbb{R}^{d_{S^c_1} \times ...\times d_{S^c_{N-|S|}}}$. Now we express the supremum of this Gaussian process as:
\begin{equation*}
\sup_{(u,v) \in V} \mbox{vec}(u)^\top \mathcal{M}_S(G) \mbox{vec}(v),
\end{equation*}
where recall $\mathcal{M}_S$ is the matricization involving either slice or fiber $S$. The remainder of the proof follows from Lemma~\ref{LemSupGauss1} in Appendix~\ref{AppSupGauss}.
\fpro
\vskip 25pt
\begin{proof}[Proof of Lemma~\ref{LemSparsityslice2}]
Recall that
\begin{equation*}
\mathcal{R}^{*}(G) := \max_{1 \leq j_3 \leq d_3} \left\|G_{..j_3}\right\|_s.
\end{equation*}
For each $1 \leq j_3 \leq d_3$, Lemma~\ref{LemGaussTens} in Appendix~\ref{AppSupGauss} with $N = 2$ satisfies the concentration inequality
\begin{equation*}
\mathbb{E}[\|G_{..j_3}\|_s] \leq \sqrt{6(d_1 + d_2)}.
\end{equation*}
Applying standard bounds on the maximum of functions of independent Gaussian random variables,
\begin{equation*}
\mathbb{E}[\max_{1 \leq j_3 \leq d_3}\|G..j_3\|_s] \leq \sqrt{6(d_1 + d_2 + \log d_3)}.
\end{equation*}
This completes the proof.
\end{proof}
\vskip 25pt
\spro[Proof of Lemma \ref{le:sglasso2}]
Using the standard nuclear norm upper bound for a matrix in terms of rank and Frobenius norm:
\begin{eqnarray*}
\mathcal{R}_4^2(A) & = & \left(\sum_{j_3=1}^{d_3}\|A_{\cdot \cdot j_3}\|_\ast\right)^2 \\
& \leq & \left(\sum_{j_3=1}^{d_3}\sqrt{{\rm rank}(A_{\cdot \cdot j_3})}\|A_{\cdot \cdot j_3}\|_{\rm F}\right)^2\\
& \leq & \sum_{j_3=1}^{d_3}{\rm rank}(A_{\cdot \cdot j_3}) \sum_{j_3=1}^{d_3}\|A_{\cdot \cdot j_3}\|_{\rm F}^2 = \sum_{j_3=1}^{d_3}{\rm rank}(A_{\cdot \cdot j_3}) \|A\|_{\rm F}^2,
\end{eqnarray*}
where the final inequality follows from the Cauchy-Schwarz inequality. Finally, note that for any $A \in \Theta_4(r)/\{0\}$,
$$\sum_{j_3=1}^{d_3}{\rm rank}(A_{\cdot \cdot j_3})\le r,$$
which completes the proof.
\fpro
\vskip 25pt
\begin{proof}[Proof of Lemma \ref{LemNuclearNorm}]
Note that $\mathcal{R}^*(G) = \|G\|_s$, we can directly apply Lemma~\ref{LemGaussTens} with $N=3$ from Appendix~\ref{AppSupGauss}.
\end{proof}
\vskip 25pt
\spro[Proof of Lemma \ref{LemNuclearNorm1}]
From Tucker decomposition (\ref{eq:tucker}), it is clear that for any $A\in \Theta_5(r)$, we can find sets of vectors $\{u_k: k=1,\ldots, r^2\}$, $\{v_k: k=1,\ldots, r^2\}$ and $\{w_k: k=1,\ldots, r^2\}$ such that
$$
A= \sum_{k=1}^{r^2}{u_k \otimes v_k \otimes w_k},
$$
and in addition,
$$
u_{k}^\top u_{k'}=(v_{k}^\top v_{k'})(w_{k}^\top w_{k'})=0
$$
for any $k\neq k'$. It is not hard to see that
$$
\|A\|_{\rm F}^2=\sum_{k=1}^{r^2}\left(\|u_k\|_{\ell_2}^2\|v_k\|_{\ell_2}^2\|w_k\|_{\ell_2}^2\right).
$$
On the other hand, as shown by \cite{YuanZhang14},
$$
\|A\|_{\ast}=\sum_{k=1}^{r^2}\left(\|u_k\|_{\ell_2}\|v_k\|_{\ell_2}\|w_k\|_{\ell_2}\right).
$$
The claim then follows from an application of Cauchy-Schwartz inequality.
\fpro
\vskip 25pt
\begin{proof}[Proof of Lemma~\ref{LemLowRank}]
Recall that we are considering the regularizer
$$
\mathcal{R}_6^\ast(A)=3\max\left\{\|\mathcal{M}_1(A)\|_s, \|\mathcal{M}_2(A)\|_s, \|\mathcal{M}_3(A)\|_s\right\},
$$
and our goal is to upper bound
$$
\mathcal{R}^*_6(G)=3\max_{1\le k\le 3}\|\mathcal{M}_k(G)\|_s.
$$
Once again apply Lemma~\ref{LemGaussTens} in Appendix~\ref{AppSupGauss} with $N=2$ for each matricization implies
$$
\mathbb{E}[\mathcal{R}^*_6(G)] \leq 4 \max(\sqrt{d_1},\sqrt{d_2},\sqrt{d_3}).
$$
\end{proof}
\vskip 25pt
\spro[Proof of Lemma \ref{LemLowRank1}]
It is not hard to see that
\begin{eqnarray*}
\mathcal{R}_6(A)^2 & = & {1\over 9}\left(\|\mathcal{M}_1(A)\|_\ast+\|\mathcal{M}_2(A)\|_\ast+\|\mathcal{M}_3(A)\|_\ast\right)^2 \\
& \leq & {1\over 9}(\sqrt{r_1} + \sqrt{r_2} + \sqrt{r_3})^2 \|A\|_{\rm F}^2\\
& \leq & \max\{r_1(A), r_2(A), r_3(A)\}\|A\|_{\rm F}^2,
\end{eqnarray*}
which completes the proof.
\fpro
\subsection{Proof of results in Section~\ref{SecExamples}}
In this section we prove the results in Section~\ref{SecExamples}. First we provide a general minimax lower result that we apply to our main results. Let $\mathcal{T} \subset \mathbb{R}^{d_1 \times d_2 \times\cdots\times d_N}$ be an arbitrary subspace of order-$N$ tensors.
\btheos
\label{ThmLower}
Assume that (\ref{AssCov}) holds and there exists a finite set $\{A^1, A^2,\ldots,A^{m} \} \in \mathcal{T}$ of tensors such that $\log m \geq 128 n \delta^2$, such that
\begin{equation*}
n c_u^{-2} \delta^2 \leq \|A^{\ell_1} - A^{\ell_2}\|_{\rm F}^2 \leq 8 n c_u^{-2} \delta^2 ,
\end{equation*}
for all $\ell_1\neq \ell_2 \in [m]$ and all $\delta > 0$. Then
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \delta^2,
\end{equation*}
with probability at least $1/2$ for some $c >0$.
\etheos
\spro
We use standard information-theoretic techniques developed in \cite{IbrHas81} and extended in ~\cite{YanBar99}. Let $\{A^1, A^2,\ldots,A^m \}$ be a set such that
$$\|A^{\ell_1} - A^{\ell_2}\|_{\rm F}^2 \geq n c_u^{-2} \delta^2$$
for all $\ell_1 \neq \ell_2$, and let $\widetilde{m}$ be a random variable uniformly distributed over the index set $[m] = \{1,2,\ldots,m\}$.
Now we use a standard argument which allows us to provide a minimax lower bound in terms of the probability of error in a multiple hypothesis testing problem \citep[see, e.g.,][]{YanBar99,Yu} then yields the lower bound (write out steps here).
\begin{eqnarray*}
\inf_{\widetilde{T}} \sup_{T \in \mathcal{T}} \mathbb{P}\left\{\|\widetilde{T} - T\|_{\rm F}^2 \geq \frac{c_u^{-2} \delta^2}{2} \right\} \geq \inf_{\widetilde{T}} \mathbb{P}(\widetilde{T} \neq A^{\widetilde{m}})
\end{eqnarray*}
where the infimum is taken over all estimators $\widetilde{T}$ that are measurable functions of $X$ and $Y$.
Let $X = \{ X^{(i)}: i=1,\ldots,n\}$, $Y = \{ Y^{(i)}: i=1,\ldots,n \}$ and $E = \{ \epsilon^{(i)}: i=1,\ldots,n \}$. Using Fano's inequality \citep[see, e.g.,][]{Cover}, for any estimator $\widetilde{T}$, we have:
\begin{equation*}
\mathbb{P}[\widetilde{T} \neq A^{\widetilde{m}} | X ] \geq 1 - \frac{I_X(A^{\widetilde{m}}; Y)+\log 2}{\log m}.
\end{equation*}
Taking expectations over $X$ on both sides, we have
\begin{equation*}
\mathbb{P}[\widetilde{T} \neq A^{\widetilde{m}}] \geq 1 - \frac{\mathbb{E}_X[I_X(A^{\widetilde{m}}; Y)]+\log 2}{\log m}.
\end{equation*}
For $\ell = 1,2,\ldots,m$, let $\mathbb{Q}^{\ell}$ denote the condition distribution of $Y$ conditioned on $X$ and the event $\{T = A^{\ell}\}$, and $D_{\rm KL}(\mathbb{Q}^{\ell_1} ||\mathbb{Q}^{\ell_2} )$ denote the Kullback-Leibler divergence between $\mathbb{Q}^{\ell_1}$ and $\mathbb{Q}^{\ell_2}$. From the convexity of mutual information \citep[see, e.g.,][]{Cover}, we have the upper bound
$$I_X(T; Y) \leq \frac{1}{{m \choose 2}} \sum_{\ell_1,\ell_2 = 1}^m {D_{\rm KL}(\mathbb{Q}^{\ell_1} ||\mathbb{Q}^{\ell_2} )}.$$
Given our linear Gaussian observation model~\eqref{eq:model},
\begin{equation*}
D_{\rm KL}(\mathbb{Q}^{\ell_1} ||\mathbb{Q}^{\ell_2} ) = \frac{1}{2} \sum_{i=1}^n \left(\langle A^{\ell_1}, X^{(i)} \rangle - \langle A^{\ell_2}, X^{(i)} \rangle \right)^2 = \frac{n\| A^{\ell_1} - A^{\ell_2}\|_n^2}{2}.
\end{equation*}
Further if (\ref{AssCov}) holds, then
\begin{equation*}
\mathbb{E}_X[I_X(T; Y)] \leq \frac{n}{2{m \choose 2} } \sum_{\ell_1 \neq \ell_2} {\mathbb{E}_X[\| A^{\ell_1} - A^{\ell_2}\|_n^2] } \leq c_u^2\frac{n}{2{m \choose 2} } \sum_{\ell_1 \neq \ell_2} {\| A^{\ell_1} - A^{\ell_2}\|_{\rm F}^2 }.
\end{equation*}
Based on our construction, there exists a set $\{A^1, A^2,\ldots,A^{m} \}$ where each $A^\ell \in \mathcal{T}$ such that $\log m \geq C n \delta^2$ and
$$c_u^{-1} \delta \leq \|A^{\ell_1} - A^{\ell_2} \|_{\rm F} \leq 8 c_u^{-1} \delta$$
for all $\ell_1 \neq \ell_2 \in \{1,2,\ldots,m\}$.
If (\ref{AssCov}) holds, then
$$\mathbb{E}_X\left(\| A^{\ell_1} - A^{\ell_2}\|_n^2\right) \leq c_u^2\| A^{\ell_1} - A^{\ell_2}\|_{\rm F}^2$$
and we can conclude that
\begin{equation*}
\mathbb{E}_X[I_X(T; Y)] \leq 32 c_u^2 n \delta^2,
\end{equation*}
and from the earlier bound due to Fano's inequality, for and $\delta > 0$ such that
$$\frac{32 c_u^2 n \delta^2 + \log 2}{\log m} \leq \frac{1}{2},$$
we are guaranteed that
$$\mathbb{P}\left\{\widetilde{T} \neq A^{\widetilde{m}}\right\} \geq \frac{1}{2}.$$
The proof is now completed because $\log m \geq 128 n \delta^2$ and $32 n \delta^2 \geq \log 2$.
\fpro
\vskip 25pt
\spro[Proof of Theorem \ref{ThmUpperMultiReg}]
The proof for the upper bound follows directly from Lemma~\ref{LemSparsityslice} with $d_1 = d_2 = m$ and $d_3 = p$ and noting that the overall covariance $\Sigma \in \mathbb{R}^{(nD_M) \times (nD_M)}$ is block-structured with blocks $\widetilde{\Sigma}$ since each of the samples is independent. Hence
$$c_\ell^2 \leq \lambda_{\min}(\Sigma) \leq \lambda_{\max}(\Sigma) \leq c_u^2.$$
To prove the lower bound, we use Theorem~\ref{ThmLower} and construct a suitable packing set for $\mathcal{T}_1$. The way we construct this packing is to construct two separate packing sets and select the set with the higher packing number using a similar argument to that used in \cite{RasWaiYu12} which also uses two separate packing sets. The first packing set we consider involves selecting the $s$-dimensional slice $A_{..S}$ where $A \subset[j_3]$ and $S = \{1,2,...,s\}$. Consider vectorizing each slice so $v = \mbox{vec}(A_{..S}) \in \mathbb{R}^{sm^2}$. Hence in order to apply Theorem~\ref{ThmLower}, we define the set $\mathcal{T}$ to be slices which is isomorphic to the vector space $\mathbb{R}^{sm^2}$. Using Lemma~\ref{LemFullHuperCube} in Appendix~\ref{SecHypercube}, there exists a packing set $\{v^1, v^2,...,v^N\} \in \mathbb{R}^{sm^2}$ such that $\log N \geq c s m^2$ and for all $v^{\ell_1}, v^{\ell_2}$ where $\ell_1 \neq \ell_2 $,
$$\frac{\delta^2}{4} \leq \|v^{\ell_1} - v^{\ell_2}\|_{\rm F}^2 \leq \delta^2$$
for any $\delta > 0$. If we choose $\delta = c\sqrt{s}m/\sqrt{n}$, then Theorem~\ref{ThmLower} implies the lower bound
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_1} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2}\frac{sm^2}{n},
\end{equation*}
with probability greater than $1/2$.
The second packing set we construct is for the slice $A_{11\cdot} \in \mathbb{R}^p$. Since in the third direction only $s$ of the $p$ co-ordinates are non-zero, the packing number for any slice is analogous to the packing number for $s$-sparse vectors with ambient dimension $p$. Letting $v = A_{11\cdot}$, we need to construct a packing set for
$$\{v \in \mathbb{R}^p\;|\;\|v\|_{\ell_0} \leq s \}.$$
Using Lemma~\ref{LemSparseHuperCube} in Appendix~\ref{SecHypercube}, there exists a discrete set $\{v^1, v^2,...,v^N\}$ such that $\log N \geq cs \log(p/s)$ for some $c >0$ and $$\frac{\delta^2}{8} \leq \|v^k - v^{\ell}\|_2^2 \leq \delta^2$$
for $k \neq \ell$ for any $\delta > 0$. Setting $\delta^2 = s n^{-1}\log (p/s)$,
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_1} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{s \log(p/s)}{n},
\end{equation*}
with probability greater than $1/2$.
Taking a maximum over lower bounds involving both packing sets completes the proof of the lower bound in Theorem~\ref{ThmUpperMultiReg}.
\fpro
\vskip 25pt
\spro[Proof of Theorem \ref{ThmUpperMultiReg1}]
The upper bound follows directly from Lemma~\ref{LemSparsityslice2} with $d_1 = d_2 = m$ and $d_3 = p$ and noting that the overall covariance $\Sigma \in \mathbb{R}^{(nD_M) \times (nD_M)}$ is block-structured with blocks $\widetilde{\Sigma}$ since each of the samples is independent.
To prove the lower bound, we use Theorem~\ref{ThmLower} and construct a suitable packing set for $\mathcal{T}_2$. Once again we construct two separate packings and choose the set that leads to the larger minimax lower bound. For our first packing set, we construct a packing a long once slice. Let us assume $A = (A_{\cdot\cdot1},...,A_{\cdot\cdot p})$, where $\mbox{rank}(A_{\cdot\cdot1}) \leq r$ and
$$A_{\cdot\cdot 2} =\cdots= A_{\cdot\cdot p} = 0.$$
If we let $A_{\cdot\cdot 1} = M$ where $M \in \mathbb{R}^{m \times m}$ then $A = (M, 0,..,0) \in \mathbb{R}^{m \times m \times p}$. Using Lemma~\ref{LemLowRankHypercube} in Appendix~\ref{SecHypercube}, there exists a set $\{A^1, A^2,...,A^{N} \}$ such that $\log N \geq c r m$ and
$$\frac{\delta^2}{4} \leq \|A^{\ell_1} - A^{\ell_2}\|_{\rm F}^2 \leq \delta^2$$
for all $\ell_1 \neq \ell_2$ and any $\delta > 0$. Here we set $\delta = \sqrt{rm/n}$. Therefore using Theorem~\ref{ThmLower}
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_2} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{rm}{n},
\end{equation*}
with probability greater than $1/2$.
The second packing set for $\mathcal{T}_2$ involves a packing in the space of singular values since
$$\sum_{j=1}^p {\mbox{rank}(A_{\cdot\cdot j})} \leq r.$$
Let $\{\sigma_{jk}: k=1,\ldots,m\}$ be the singular values of the matrix $A_{\cdot\cdot j}$. Under our rank constraint, we have
$$\sum_{j=1}^p \sum_{k=1}^m {\mathbb{I}(\sigma_{jk} \neq 0)} \leq s.$$
Let $v \in \mathbb{R}^{mp}$ where
$$v = \mbox{vec}((\sigma_{jk})_{1 \leq j \leq p, 1 \leq k \leq m}).$$
Note that
$$\sum_{j=1}^p \sum_{k=1}^m {\mathbb{I}(\sigma_{jk} \neq 0)} \leq r$$
implies $\|v\|_{\ell_0} \leq r$. Using Lemma~\ref{LemSparseHuperCube}, there exists a set $\{v^1, v^2,...,v^N \}$, such that $\log N \geq c r \log(mp/r)$ and for all $\ell_1\neq \ell_2$,
$$\frac{\delta^2}{4} \leq \|v^{\ell_1}- v^{\ell_2}\|_2^2 \leq \delta^2$$
for any $\delta > 0$. If we set $\delta^2 = rn^{-1}\log(mp/r)$. Therefore using Theorem~\ref{ThmLower},
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_2} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{r \log(mp/r)}{n},
\end{equation*}
with probability greater than $1/2$. Hence taking a maximum over both bounds,
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_2} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{r \max\{m, \log(p/r), \log m \}}{n} = cc_u^{-2} \frac{r \max\{m, \log(p/r)\}}{n},
\end{equation*}
with probability greater than $1/2$.
\fpro
\vskip 25pt
\spro[Proof of Theorem \ref{ThmUpperVAR}]
The upper bound with
$$\lambda \ge 3 \sqrt{\frac{\max\{p,2\log m\}}{\mu_{\min} n}}$$
follows directly from Lemma \ref{LemSparsityfiber} with $d_1=p$ and $d_2 = d_3 = m$ and (\ref{AssCov}) is satisfied with $c_u^2 = 1/\mu_{\min}$ and $c_{\ell}^2 =1/\mu_{\max}$ according to (\ref{EqnCondVAR}).
To prove the lower bound is similar to the proof for the lower bound in Theorem~\ref{ThmUpperMultiReg}. Once again we use Theorem~\ref{ThmLower} and construct a two suitable packing sets for $\mathcal{T}_3$. The first packing set we consider involves selecting an arbitrary subspace
$$\widetilde{\mathcal{T}} := \{ A = (A_{j_1,j_2,j_3})_{j_1, j_2, j_3}\;|\; 1 \leq j_1 \leq \sqrt{s},\; 1 \leq j_2 \leq \sqrt{s},\; 1 \leq j_3 \leq p \}.$$
Now if we let $v = \mbox{vec}(A)$, then $v$ comes from an $sp$-dimensional vector space for any $A\in \widetilde{\mathcal{T}}$. Using Lemma~\ref{LemFullHuperCube} in Appendix~\ref{SecHypercube}, there exists a packing set $\{v^1, v^2,...,v^N\} \in \mathbb{R}^{sp}$ such that $\log N \geq c s p$ and for all $v^{\ell_1}, v^{\ell_2}$ where $\ell_1 \neq \ell_2$,
$$\frac{\delta^2}{4} \leq \|v^{\ell_1} - v^{\ell_2}\|_{\rm F}^2 \leq \delta^2$$
for any $\delta > 0$. If we choose $\delta = \sqrt{sp/n}$, then Theorem~\ref{ThmLower} implies the lower bound
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_3} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{sp}{n},
\end{equation*}
with probability greater than $1/2$. Further $c_u^2 = 1/\mu_{\min}$.
For the second packing set we construct is for the slice $A_{1,j_2,j_3}$ for any $1 \leq j_2, j_3 \leq m$. Since in the second and third direction only $s$ of the co-ordinates are non-zero, we consider the vector space
$$\{v \in \mathbb{R}^{m^2}\;|\; \|v\|_{\ell_0} \leq s\}.$$
Once again using the standard standard hypercube construction in Lemma~\ref{LemSparseHuperCube} in Appendix~\ref{SecHypercube}, there exists a discrete set $\{v^1, v^2,...,v^N\}$ such that $\log N \geq cs \log(m^2/s)$ for some $c >0$ and
$$\frac{\delta^2}{8} \leq \|v^{\ell_1} - v^{\ell_2}\|_2^2 \leq \delta^2$$
for $\ell_1 \neq \ell_2$ for any $\delta > 0$. Setting $\delta = sn^{-1} \log (m^2/s)$ yields
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_3} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{s \log(m/\sqrt{s})}{n},
\end{equation*}
with probability greater than $1/2$. Taking a maximum over lower bounds involving both packing sets completes the proof of of our lower bound.
\fpro
\vskip 25pt
\spro[Proof of Theorem \ref{th:pairwise}]
The upper bound follows from a slight modification of the statement in Lemma~\ref{LemSparsityslice2}. In particular since $\mathcal{R}(A)= \|A^{(12)}\|_\ast+\|A^{(13)}\|_\ast+\|A^{(23)}\|_\ast$, the dual norm is
\begin{equation}
\label{eq:pairwise}
\mathcal{R}^*(A)=\max_{1\le k_1<k_2\le 3}\|A^{(k_1 k_2)}\|_s.
\end{equation}
Hence, following the same technique as used in Lemma~\ref{LemSparsityslice2}
\begin{equation}
\label{eq:pairwise}
\mathbb{E}[\mathcal{R}^*(G)] \leq c\max_{1\le k_1<k_2\le 3}\sqrt{\frac{\max\{d_{k_1}, d_{k_2}\}}{n}} = c\sqrt{\frac{\max\{d_1, d_2, d_3\}}{n}}.
\end{equation}
It is also straightforward to see that $s(\mathcal{T}_4) \leq r$.
To prove the lower bound, we construct three packing sets and select the one with the largest packing number. Recall that
\begin{eqnarray*}
\mathcal{T}_4=\{A\in \mathbb{R}^{d_1\times d_2\times d_3}: A_{j_1j_2j_3}=A^{(12)}_{j_1j_2}+A^{(13)}_{j_1j_3}+A^{(23)}_{j_2j_3}, A^{(k_1,k_2)}\in \mathbb{R}^{d_{k_1}\times d_{k_2}},\\
A^{(k_1,k_2)}\mathbf{1} =\mathbf{0}, \qquad {\rm and}\qquad (A^{(k_1,k_2)})^\top\mathbf{1} =\mathbf{0}\\
\max_{k_1,k_2} {\rm rank}(A^{(k_1,k_2)})\le r\}.
\end{eqnarray*}
Therefore our three packings are for $A^{(12)} \in \mathbb{R}^{d_1 \times d_2}$, $A^{(13)} \in \mathbb{R}^{d_1 \times d_3}$, and $A^{(23)} \in \mathbb{R}^{d_2 \times d_3}$ assuming each has rank $r$. We focus on packing in $A^{(12)} \in \mathbb{R}^{d_1 \times d_2}$ since the approach is similar in the other two cases. Using Lemma~\ref{LemGaussTens} from Appendix~\ref{AppSupGauss} in combination with Theorem~\ref{ThmLower},
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_4} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{r \min\{d_1, d_2\} }{n},
\end{equation*}
with probability greater than $1/2$. Repeating this process for packings in $A^{(13)} \in \mathbb{R}^{d_1 \times d_3}$, and $A^{(23)} \in \mathbb{R}^{d_2 \times d_3}$ assuming each has rank $r$ and taking a maximum over all three bounds yields the overall minimax lower bound
\begin{equation*}
\min_{\widetilde{T}} \max_{T \in \mathcal{T}_4} \| \widetilde{T} - T \|_{\rm F}^2 \geq cc_u^{-2} \frac{r \max\{d_1, d_2, d_3\}}{n},
\end{equation*}
with probability greater than $1/2$.
\fpro
\bibliographystyle{plainnat}
|
1,314,259,996,725 | arxiv | \section{Introduction and statement of the main results} \label{morse-sec:cr}
The purpose of this paper is to establish analogues of the holomorphic
Morse inequalities of Demailly for CR manifolds.
Demailly \cite{De:85} proved remarkable asymptotic Morse inequalities
for the ${\overline\partial}$ complex constructed over the line bundle
$L^k$ as $k\to\infty$, where $L$ is a holomorphic
hermitian line bundle. Shortly after, Bismut \cite{B87c} gave a heat
equation proof of Demailly's inequalities, which involves probability
theory. Later Demailly \cite{De:91} and Bouche \cite{Bou:96} replaced
the probability technique by a classical heat kernel argument. The book \cite{MM07} introduced an argument
based on the asymptotic of the heat kernel
of the Kodaira Laplacian by using the rescaling of the coordinates and functional analytic
techniques inspired by Bismut-Lebeau \cite[\S 11]{BL91} (see also Bismut-Vasserot \cite{BVa89}). A different approach was introduced by Berndtsson \cite{Bern:02} and developed by Berman
\cite{Be04,Be05}; they work with the Bergman kernel and use the mean value estimate for eigensections
of the Kodaira Laplacian.
The idea of all these proofs is localization of the analytic objects (eigenfunctions, kernels) and scaling techniques.
See also Fu-Jacobowitz \cite{FJ07} for related results on domains of finite type.
Inspired by Bismut's paper, Getzler~\cite{Ge89} gave an expression involving local data for the large $k$ limit of the trace of heat kernel of the $\overline\partial_b$-Laplacian on $L^k$, where $L$ is a CR line bundle
over a CR strongly pseudoconvex manifold.
But Getzler didn't infer from this asymptotics Morse inequalities for the $\overline\partial_b$-complex.
In this paper we introduce a method that
produces Morse inequalities with computable bounds for the growth of the
$\overline\partial_b$ coholmology and allows also more general CR manifolds.
Our approach is related to the techniques of Berman~\cite{Be04} and Shaw-Wang~\cite{SW05}.
In a project developed jointly with R. Ponge \cite{Pon04,P05} we use the heat kernel asymptotics and Heisenberg calculus to prove holomorphic Morse inequalities for a line bundle endowed with the CR Chern connection.
This method predicts similar results and applications as of the present paper.
For a complex manifold with boundary, the
$\overline\partial_b$-cohomology of the boundary is linked to the
$\overline\partial$-cohomology of the interior, cf. Kohn-Rossi
\cite{KR65}, Andreotti-Hill \cite{AH72a,AH72b}.
Stephen S.T. Yau \cite{YauSST81} exhibited the relation between the
$\overline\partial_b$-cohomology of the boundary of a strictly
pesudoconvex Stein analytic space with isolated singularities and
invariants of the singular points.
Holomorphic Morse inequalites for manifolds with boundary were obtained
by Berman \cite{Be05} and in \cite{Ma04,MarTC} (cf.\,also \cite[Ch.\,3]{MM07}). The bounds in the Morse
inequalities appearing in this paper are similar to the boundary terms
in Berman's result \cite{Be05}. For the relation between the boundary
and interior cohomology of high tensor powers $L^k$ see also \cite{Ma96}.
On the other hand, the $\overline\partial_b$-complex on an abstract CR
manifold has important consequences for the embedability and deformation
of the CR-structure, see the embedding theorem of Boutet de Monvel
\cite{BdM1:74b} for strictly pseudoconvex CR manifolds and the paper of
Epstein-Henkin \cite{EH00}.
In this paper we will study the large $k$ behavior of the Szeg\"{o}
kernel function $\mathit\Pi^{(q)}_k(x)$, which is the restriction to the
diagonal of the integral kernel of the projection $\mathit\Pi^{(q)}_k$ on the
harmonic $(0,q)$-forms with values in $L^k$. The Szeg\"{o} kernel for
functions on a strictly pseudoconvex CR manifold was studied by
Boutet de Monvel \cite{BdM1:74a} and Boutet de Monvel-Sj\"ostrand
\cite{BouSj76} and has important applications in complex analysis and
geometry.
\subsection{Terminology and Notations}
Let $(X, \Lambda^{1, 0}T(X))$ be a compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$.
Fix a smooth Hermitian metric $(\ |\ )$ on $\mathbb C T(X)$ so that $\Lambda^{1, 0}T(X)$
is orthogonal to $\Lambda^{0, 1}T(X)$ and $(u\ |\ v)$ is real if $u$, $v$ are real tangent vectors,
where $\Lambda^{0, 1}T(X)=\overline{\Lambda^{1, 0}T(X)}$ and $\mathbb C T(X)$ is the complexified tangent bundle.
Then there is a real non-vanishing vector field $T$ on $X$ which is pointwise orthogonal to
$\Lambda^{1, 0}T(X)\oplus\Lambda^{0, 1}T(X)$. The Hermitian metric $(\ |\ )$ on $\mathbb C T(X)$ induces,
by duality, a Hermitian metric on the complexified cotangent bundle $\mathbb C T^*(X)$ that we shall also denote by $(\ |\ )$.
Let $\Lambda^{0, q}T^*(X)$ be the bundle of $(0, q)$ forms of $X$.
The Hermitian metric $(\ |\ )$ on $\mathbb C T^*(X)$ induces a Hermitian metric on
$\Lambda^{0, q}T^*(X)$ also denoted by $(\ |\ )$. Let $D\subset X$ be an open set. Let $\Omega^{0,q}(D)$ denote the space of smooth sections of $\Lambda^{0,q}T^*(X)$ over $D$. Similarly, if $E$ is a vector bundle over $D$, then we let $\Omega^{0,q}(D, E)$
denote the space of smooth sections of $\Lambda^{0,q}T^*(X)\otimes E$ over $D$. Let $\Omega^{0,q}_c(D, E)$ be the subspace of
$\Omega^{0,q}(D, E)$ whose elements have compact support in $D$.
If $w\in\Lambda^{0,1}T^*_z(X)$, let
$w^{\wedge, *}: \Lambda^{0,q+1}T^*_z(X)\rightarrow \Lambda^{0,q}T^*_z(X),\ q\geqslant0$,
be the adjoint of the left exterior multiplication
$w^\wedge: \Lambda^{0,q}T^*_z(X)\rightarrow \Lambda^{0,q+1}T^*_z(X)$, $w^\wedge u:=w\wedge u$:
\begin{equation} \label{s1-e1}
(w^\wedge u\ |\ v)=(w\wedge u\ |\ v)=(u\ |\ w^{\wedge, *}v),
\end{equation}
for all $u\in\Lambda^{0,q}T^*_z(X)$, $v\in\Lambda^{0,q+1}T^*_z(X)$.
Notice that $w^{\wedge, *}$ depends anti-linearly on $w$.
Let $\Lambda^{1, 0}T^*(X)$ denote the bundle with fiber $\Lambda^{1, 0}T^*_z(X):=\overline{\Lambda^{0, 1}T^*_z(X)}$
at $z\in X$. Locally we can choose an orthonormal frame $\omega_1(z),\ldots,\omega_{n-1}(z)$
for the bundle $\Lambda^{1,0}T^*_z(X)$, then $\overline\omega_1(z),\ldots,\overline\omega_{n-1}(z)$
is an orthonormal frame for the bundle $\Lambda^{0,1}T^*_z(X)$. The real $(2n-2)$ form
$\omega=i^{n-1}\omega_1\wedge\overline\omega_1\wedge\cdots\wedge\omega_{n-1}\wedge\overline\omega_{n-1}$
is independent of the choice of the orthonormal frame. Thus $\omega$ is globally
defined. Locally there is a real $1$-form $\omega_0(z)$ of length one which is orthogonal to
$\Lambda^{1,0}T^*_z(X)\oplus\Lambda^{0,1}T^*_z(X)$. The form $\omega_0(z)$ is unique up to the choice of sign.
Since $X$ is orientable, there is a nowhere vanishing $(2n-1)$ form $Q$ on $X$.
Thus, $\omega_0$ can be specified uniquely by requiring that $\omega\wedge\omega_0=fQ$,
where $f$ is a positive function. Therefore $\omega_0$, so chosen, is globally defined. We call $\omega_0$
the uniquely determined global real $1$-form.
We choose a vector field $T$ so that $\norm{T}=1$, $\langle T, \omega_0\rangle=-1$.
Therefore $T$ is uniquely determined. We call $T$ the uniquely determined global real vector field. We have the
pointwise orthogonal decompositions:
\begin{equation} \label{s1-e3}\begin{split}
\mathbb C T^*(X)&=\Lambda^{1,0}T^*(X)\oplus \Lambda^{0,1}T^*(X)\oplus\set{\lambda\omega_0;\,
\lambda\in\mathbb C}, \\
\mathbb C T(X)&=\Lambda^{1,0}T(X)\oplus \Lambda^{0,1}T(X)\oplus\set{\lambda T;\,\lambda\in\mathbb C}.
\end{split}\end{equation}
In the sequel we will denote by $\langle\cdot,\cdot\rangle$ both scalar products as well as the duality bracket between vector fields and forms.
\begin{defn} \label{s1-d2}
For $p\in X$, the \emph{Levi form} $\mathcal{L}_p$ is the Hermitian quadratic form on $\Lambda^{1,0}T_p(X)$ defined as follows. For any $U,\ V\in \Lambda^{1,0}T_p(X)$, pick $\mathcal{U},\mathcal{V}\in
C^\infty(X;\, \Lambda^{1,0}T(X))$ such that
$\mathcal{U}(p)=U$, $\mathcal{V}(p)=V$. Set
$\mathcal{L}_p(U,\overline V)=\frac{1}{2i}\big\langle\big[\mathcal{U}\ ,\overline{\mathcal{V}}\,\big](p)\ ,\omega_0(p)\big\rangle$\,,
where $\big[\mathcal{U}\ ,\overline{\mathcal{V}}\,\big]=\mathcal{U}\ \overline{\mathcal{V}}-\overline{\mathcal{V}}\ \mathcal{U}$ denotes the commutator of $\mathcal{U}$ and $\overline{\mathcal{V}}$.
Recall that $\mathcal{L}_p$ does not depend of the choices of $\mathcal{U}$ and $\mathcal{V}$.
Given $q\in\{0,\ldots,n-1\}$, the Levi form is said to satisfy \emph{condition $Y(q)$} at $p\in X$, if $\mathcal{L}_p$ has $\max{(q+1, n-q)}$ eigenvalues of the same sign or $\min{(q+1,n-q)}$ pairs of eigenvalues with opposite signs.
\end{defn}
\begin{ass} \label{s1-a1}
Throughout the paper we assume that the Levi form is non-degene-rate and condition $Y(q)$ holds at each point of $X$.
\end{ass}
\subsection{CR complex line bundles, Semi-classical $\overline\partial_b\text{-Complex}$ and $\Box_b$}
Let
\begin{equation} \label{s1-e5}
\overline\partial_b:\Omega^{0,q}(X)\rightarrow\Omega^{0,q+1}(X)
\end{equation}
be the tangential Cauchy-Riemann operator. We say that $u\in C^\infty(X)$ is CR if
$\overline\partial_b u=0$.
\begin{defn} \label{s1-d1}
Let $L$ be a complex line bundle over $X$. We say that $L$ is a CR complex line bundle over $X$
if its transition functions are CR.
\end{defn}
From now on, we let $(L,h^L)$ be a CR Hermitian line bundle over $X$, where
the Hermitian fiber metric on $L$ is denoted by $h^L$. We will denote by
$\phi$ the local weights of the Hermitian metric. More precisely, if
$s$ is a local trivializing
section of $L$ on an open subset $D\subset X$, then the pointwise
norm of $s$ is
\begin{equation} \label{s1-e6}
\abs{s(x)}^2_{h^L}=e^{-\phi(x)},\quad\phi\in C^\infty(D; \mathbb R).
\end{equation}
Let $L^k$, $k>0$, be the $k$-th tensor power of the line bundle $L$. The Hermitian fiber metric on $L$ induces a Hermitian
fiber metric on $L^k$ that we shall denote by $h^{L^k}$. If $s$ is a local trivializing section
of $L$ then $s^k$ is a local trivializing section of $L^k$. For $f\in\Omega^{0,q}(X, L^k)$, we denote the poinwise norm $\abs{f(x)}^2:=\abs{f(x)}^2_{h^{L^k}}$. We write $\overline\partial_{b,k}$ to denote the tangential
Cauchy-Riemann operator with values in $L^k$:
\begin{equation} \label{s1-e7}
\overline\partial_{b,k}:\Omega^{0,q}(X, L^k)\rightarrow\Omega^{0,q+1}(X, L^k)\,,\quad \overline\partial_{b,k}(s^ku):=s^k\overline\partial_bu,
\end{equation}
where $s$ is a local trivialization of $L$ on an open subset $D\subset X$
and $u\in\Omega^{0,q}(D)$.
We obtain a $\overline\partial_{b,k}$-complex $(\Omega^{0,\bullet}(X, L^k),\overline\partial_{b,k})$ with cohomology
\begin{equation}\label{db-cohom}
H^{\bullet}_b(X,L^k):=\ker\overline\partial_{b,k}/\operatorname{Im} \overline\partial_{b,k}.
\end{equation}
We denote by $dm=dm(x)$ the volume form on $X$ induced by the fixed Hermitian metric $(\ |\ )$ on $\mathbb C T(X)$. Then we get natural global $L^2$ inner products $(\ |\ )_k$, $(\ |\ )$
on $\Omega^{0,q}(X, L^k)$ and $\Omega^{0,q}(X)$ respectively. Let
\begin{equation} \label{s1-e8}
\overline{\partial}^*_{b,k}:\Omega^{0,q+1}(X, L^k)\rightarrow\Omega^{0,q}(X, L^k)
\end{equation}
be the formal adjoint of $\overline\partial_{b,k}$ with respect to $(\ |\ )_k$. The \emph{Kohn Laplacian} with values in $L^k$ is given by
\begin{equation} \label{s1-e9}
\Box_{b,k}^{(q)}=\overline{\partial}^*_{b,k}\overline\partial_{b,k}+\overline\partial_{b,k}\overline{\partial}^*_{b,k}:
\Omega^{0,q}(X, L^k)\rightarrow\Omega^{0,q}(X, L^k).
\end{equation}
We extend $\Box_{b,k}^{(q)}$ to the $L^2$ space by
$\Box_{b,k}^{(q)}:{\rm Dom\,}\Box_{b,k}^{(q)}\subset L^2_{(0,q)}(X, L^k)\rightarrow L^2_{(0,q)}(X, L^k)$,
where
${\rm Dom\,}\Box^{(q)}_{b,k}:=\{u\in L^2_{(0,q)}(X, L^k);
\Box^{(q)}_{b,k}u\in L^2_{(0,q)}(X, L^k)\}$.
Consider the space of harmonic forms
\begin{equation} \label{s1-e12}
\cali{H}_b^q(X, L^k):=\Ker\Box^{(q)}_{b,k}\,.
\end{equation}
By \cite[7.6-7.8]{Ko65}, \cite[5.4.11-12]{FK72}, \cite[Props.\,8.4.8-9]{CS01}, condition $Y(q)$ implies that
$\Box^{(q)}_{b,k}$ is hypoelliptic, has compact resolvent and the strong Hodge decomposition holds. Hence
\begin{equation} \label{s2-e11}
\dim\cali{H}_b^q(X, L^k)<\infty\,,\quad \cali{H}_b^q(X, L^k)\subset\Omega^{0,q}(X, L^k)\,,\quad \cali{H}_b^q(X, L^k)\cong H_b^q(X, L^k) \,.
\end{equation}
Let $f_j(x)\in\Omega^{0,q}(X, L^k)$, $j=1,\ldots,N$, be an orthonormal frame for the space
$\cali{H}_b^q(X, L^k)$. The \emph{Szeg\"{o} kernel function} is defined by
\begin{equation} \label{s2-e1}
\mathit\Pi^{(q)}_k(x)=\sum^N_{j=1}\abs{f_j(x)}_{h^{L^k}}^2=:\sum^N_{j=1}\abs{f_j(x)}^2\,.
\end{equation}
It is easy to see that $\mathit\Pi^{(q)}_k(x)$ is independent of the choice of orthonormal frame and
\begin{equation} \label{s1-e13}
\dim \cali{H}_b^q(X, L^k)=\int_X\!\mathit\Pi^{(q)}_k(x)dm(x).
\end{equation}
\subsection{The main results}
We will express the bound of the Szeg\"o kernel with the help of the following Hermitian form.
\begin{defn} \label{s1-d3}
Let $s$ be a local trivializing section of $L$ and $\phi$ the corresponding local weight as in \eqref{s1-e6}.
The Hermitian quadratic form $M^\phi_p$ on $\Lambda^{1,0}T_p(X)$, $p\in D$, defined by
\begin{equation} \label{s1-e14}
M^\phi_p(U, \overline V)=\frac{1}{2}\Big\langle U\wedge\overline V, d\big(\overline\partial_b\phi-\partial_b\phi\big)(p)\Big\rangle,\ \ U, V\in\Lambda^{1,0}T_p(X),
\end{equation}
where $d$ is the usual exterior derivative and $\overline{\partial_b\phi}=\overline\partial_b\overline\phi$.
\end{defn}
In Proposition \ref{s7-p1} we show that in the embedded case $M^\phi_p$ is the restriction of the Chern curvature of the holomorphic extension of $L$.
But in the abstract case the definition of $M^\phi_p$ depends on the choice of local trivializations.
However, set
\begin{equation}\label{s1-e15}
\begin{split}
\mathbb R_{\phi(p),\,q}=\big\{s\in\mathbb R;\, \text{$M^\phi_p+s\mathcal{L}_p$ has exactly $q$ negative eigenvalues} \\
\text{and $n-1-q$ positive eigenvalues}\big\}\,,
\end{split}
\end{equation}
where the eigenvalues of the Hermitian quadratic form $M^\phi_p+s\mathcal{L}_p$, $s\in\mathbb R$,
are calculated with respect to $(\ |\ )$.
It turns out (see Proposition \ref{s5-p1}) that the function
\begin{equation} \label{s1-e16-1}
X\to\mathbb R\,,\quad x\mapsto\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds
\end{equation}
does not depend on the choice of $\phi$, where $\det(M^\phi_x+s\mathcal{L}_x)$ is the product of all the eigenvalues of $M^\phi_x+s\mathcal{L}_x$. Thus, the function \eqref{s1-e16-1}
is well-defined.
Since $M^\phi_x$ and $\mathcal{L}_x$ are continuous functions of $x\in X$, we conclude that the function \eqref{s1-e16-1}
is continuous.
Now, the main result of this work is the following
\begin{thm} \label{t-main1}
There is a constant $C_0>0$ independent of $k$, such that
\begin{equation} \label{s1-e18}
k^{-n}\mathit\Pi^{(q)}_k(x)\leqslant C_0\,,\quad\text{for all $x\in X$.}
\end{equation}
Furthermore, we have
\begin{equation} \label{s1-e19}
\limsup_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_k(x)\leqslant (2\pi)^{-n}\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds\,,\quad\text{for all $x\in X$.}
\end{equation}
\end{thm}
\noindent
From \eqref{s1-e13}, Theorem~\ref{t-main1} and Fatou's lemma, we get weak Morse inequalities on CR manifolds.
\begin{thm} \label{t-main2}
We have for $k\to\infty$
\begin{equation} \label{s1-e191}
\dim H^q_b(X, L^k)\leqslant \frac{k^{n}}{(2\pi)^{n}}\int_X\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds\,dm(x)+o(k^n)\,.
\end{equation}
\end{thm}
From the classical work of Kohn \cite[Th.\,7.6]{Ko65}, \cite[Th.\,5.4.11--12]{FK72}, \cite[Cor.\,8.4.7--8]{CS01}, we know that if $Y(q)$ holds, then $\Box^{(q)}_{b,k}$ has a discrete
spectrum, each eigenvalues occurs with finite multiplicity and all eigenforms are smooth. For $\lambda\in\mathbb R$, let
$\cali{H}^q_{b,\,\leqslant \lambda}(X, L^k)$ denote the spectral space spanned by the eigenforms of $\Box^{(q)}_{b,k}$ whose eigenvalues are bounded by
$\lambda$. We denote by $\mathit\Pi^{(q)}_{k,\,\leqslant \lambda}$ the restriction to the diagonal of the integral kernel of the orthogonal projector on $\cali{H}^q_{b,\leqslant \lambda}(X,L^k)$ and call it the Szeg\"{o} kernel function of the space $\cali{H}^q_{b,\leqslant \lambda}(X,L^k)$
Then
$\mathit\Pi^{(q)}_{k,\,\leqslant \lambda}(x)=\sum^M_{j=1}\abs{g_j(x)}^2$,
where $g_j(x)\in\Omega^{0,q}(X, L^k)$, $j=1,\ldots,M$, is any orthonormal frame for the space
$\cali{H}_{b,\,\leqslant\lambda}^q(X, L^k)$. The following is one of the main results.
\begin{thm} \label{t-main3}
For any sequence $\nu_k>0$ with $\nu_k\To0$, as $k\rightarrow\infty$, there is a constant $C'_0$ independent of $k$,
such that
\begin{equation} \label{s1-e18-1}
k^{-n}\mathit\Pi^{(q)}_{k,\,\leqslant k\nu_k}(x)\leqslant C'_0
\end{equation}
for all $x\in X$. Moreover, There is a sequence $\mu_k>0$, $\mu_k\To0$, as $k\rightarrow\infty$, such that for any sequence
$\nu_k>0$ with $\lim_{k\rightarrow\infty}\frac{\mu_k}{\nu_k}=0$, we have
\begin{equation} \label{s1-e19-1}
\lim_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_{k,\,\leqslant k\nu_k}(x)=(2\pi)^{-n}\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds,
\end{equation}
for all $x\in X$.
\end{thm}
By integrating \eqref{s1-e19-1} we obtain the following semi-classical Weyl law:
\begin{thm} \label{t-main4}
There is a sequence $\mu_k>0$, $\mu_k\To0$, as $k\rightarrow\infty$, such that for any sequence
$\nu_k>0$ with $\lim_{k\rightarrow\infty}\frac{\mu_k}{\nu_k}=0$, we have
\[{\rm dim\,}\cali{H}^q_{b,\,\leqslant k\nu_k}(X, L^k)=\frac{k^{n}}{(2\pi)^{n}}\int_X\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds\,dm(x)+o(k^n).\]
\end{thm}
\noindent
From Theorem~\ref{t-main4} and the linear algebra argument from Demailly~\cite{De:85} and~ \cite{Ma96}, we obtain strong Morse inequalities on CR manifolds (see \S6):
\begin{thm} \label{t-main5}
If $Y(j)$ holds, for all $j=0,1,\ldots,q$, then as $k\to\infty$
\begin{equation*}
\begin{split}
\sum^q_{j=0}(-1)^{q-j}&{\rm dim\,}H^j_b(X, L^k)\\
&\leqslant \frac{k^{n}}{(2\pi)^{n}}\sum^q_{j=0}(-1)^{q-j}\int_X\int_{\mathbb R_{\phi(x),j}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds\,dm(x)+o(k^n).
\end{split}
\end{equation*}
If $Y(j)$ holds, for all $j=q,q+1,\ldots,n-1$, then as $k\to\infty$
\begin{equation*}
\begin{split}
\sum^{n-1}_{j=q}(-1)^{q-j}&{\rm dim\,}H^j_b(X, L^k)\\
&\leqslant \frac{k^{n}}{(2\pi)^{n}}\sum^{n-1}_{j=q}(-1)^{q-j}\int\limits_X\int\limits_{\mathbb R_{\phi(x),j}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds\,dm(x)+o(k^n).
\end{split}
\end{equation*}
\end{thm}
In section 6.1, we will state our main results in the embedded case, that is, when $X$ is a real hypersurface of a complex manifold $M$ and the bundle $L$ is the restriction of a holomorphic line bundle over $M$.
In this case the form $M^\phi_p$ is the restriction to $\Lambda^{1,0}T_p(X)$ of the curvature form $R^L$.
To wit, we deduce from the weak Morse inequalities (Theorem~\ref{t-main1}):
\begin{cor} \label{t-main6}
Let $M$ be a complex manifold of dimension $n$ and let $D=\{p\in
M:r(p)<0\}$ be a strongly pseudoconvex compact domain with smooth
definition function $r:M\to\mathbb R$ which is strictly plurisubharmonic in
a neighbourhood of $X=\partial D$. Let $(L,h^L)$ be a Hermitian
holomorphic line bundle whose curvature is proportional to the Levi form
of $D$ on $X$, i.e.\! there exists a smooth function $\lambda:X\to\mathbb R$
such that $R^L=\lambda \mathcal{L}_r$ on the holomorphic tangent bundle of $X$. Then
$\dim H^q_b(X,L^k)=o(k^n)$ as $k\to\infty$ for all $1\leqslant q\leqslant n-2$.
\end{cor}
\begin{ex}\label{t-main7}
Let $N$ be a compact complex manifold of dimension $n$ and $(E,h^E)$ be
a positive line bundle on $N$.
Let $D=\{v\in E^*;|v|_{h^{E^*}}<1\}$ be the Grauert tube, set $X=\partial D$ and let
$\pi:X\to N$ be the canonical projection. Then we can apply Corollary \ref{t-main6} and we obtain that the $\overline\partial_b$-cohomology of the CR line bundle $L:=\pi^*E$ satisfies $\dim H^q_b(X,L^k)=o(k^{n+1})$ as $k\to\infty$ for all
$1\leqslant q\leqslant n-1$.
\end{ex}
To exemplify the use of the strong Morse inequalities on CR manifolds, we formulate a condition to guarantee that a CR line bundle is big in the embedded
cases.
\begin{thm} \label{th-app}
Let $M$ be a relatively compact open subset with $C^\infty$ boundary $X$ of a
complex manifold $M'$ of dimension $n$. Furthermore, let $L$ be a Hermitian holomorphic
line bundle over $M'$ with positive curvature $R^L$. We assume that the Levi form of $X$ is non-degenerate and has at least
two negative and two positive eigenvalues. Let $\lambda_1(x),\ldots,\lambda_{n-1}(x)$ be the eigenvalues of the Levi form with
respect to $R^L$. Assume that $\lambda_j<0$, $j=1,\ldots,n_-$, $\lambda_j>0$, $j=n_-+1,\ldots,n-1$, $\abs{\lambda_1}\leqslant \abs{\lambda_2}\leqslant \cdots\leqslant \abs{\lambda_{n-}}$, $\abs{\lambda_{n_-+1}}\leqslant \cdots\leqslant \abs{\lambda_{n-1}}$.
If $\lambda_1(x)=\lambda_2(x)$, $\lambda_{n_-+1}(x)=\lambda_{n_-+2}(x)$, at each point of $X$, then there is a positive constant $c$, such that $\dim H^0_b(X, L^k)\geqslant ck^n$.
\end{thm}
In \S6, we will give examples which satisfy the assumptions of Theorem~\ref{th-app}.
Keeping in mind the notion of $q$-pseudoconvexity and pseudoconcavity of
Andreotti-Grauert \cite{AG:62} we introduce the following.
\begin{defn}
A complex manifold $M$ with $\dim_\mathbb C M=n$ is called a
$(n-2)$-convex--concave strip if there exists a smooth proper map
$\rho:M\to\mathbb R$ whose Levi form $\partial\overline\partial\rho$ has at least three
negative and three positive eigenvalues on $M$. The function $\rho$ is
called an exhaustion function.
\end{defn}
\begin{thm} \label{th-strip}
Let $M$ be a $(n-2)$-convex--concave strip with exhaustion function
$\rho$. Let $a\in\mathbb R$ be a regular value of $\rho$ and set
$X:=\{\rho=a\}$. Assume that there exists a holomorphic line bundle
$L\to M$ whose curvature form
$R^L$ is positive on $X$ and the Levi form
$\partial\overline\partial\rho|_X$ satisfies the assumptions of Theorem~\ref{th-app}. Then the line bundle
$L$ is big. Therefore, the transcendence degree of the meromorphic
function field $\mathcal{K}_M$ equals $n=\dim_\mathbb C M$ and the
Kodaira map $\Phi_k:M\cdots\longrightarrow\mathbb{P}(H^0(M,L^k)^*)$ is
an immersion
outside a proper analytic set.
\end{thm}
\subsection{Sketch of the proof of Theorem \ref{t-main1}}
To simplify the exposition we consider only the case $q=0$, i.e.\,we show how to pointwise estimate the function $\limsup\limits_{k\rightarrow\infty}\mathit\Pi^{(0)}_k$.
It is easy to see that for all $x\in X$ we have
\[
\mathit\Pi^{(0)}_k(x)=S^{(0)}_k(x):=\sup_{\alpha\in H^0_b(X,L^k),\norm{\alpha}=1}\abs{\alpha(x)}^2\,,
\]
where $S^{(0)}_k(x)$ is called the extremal function.
For a given point $p\in X$, by definition,
there is a sequence $u_k\in H^0_b(X, L^k)$, $\norm{u_k}=1$, such that
$\limsup_{k\rightarrow\infty}k^{-n}S^{(0)}_k(p)=\lim_{k\rightarrow\infty}k^{-n}\abs{u_k(p)}^2$.
Near $p$, take local coordinates $(x, \theta)=(z, \theta)=(x_1,\ldots,x_{2n-2},\theta)$, $z_j=x_{2j-1}+ix_{2j}$, $j=1,\ldots,n-1$, $(x(p), \theta(p))=0$, such that
$\frac{\partial}{\partial z_j}=\frac{1}{2}(\frac{\partial}{\partial x_{2j-1}}-\frac{\partial}{\partial x_{2j}})$, $j=1,\ldots,n-1$, is an orthonormal basis for $\Lambda^{1,0}T_p(X)$ and
the Levi form and local weight are given by $\mathcal{L}_p=\sum ^{n-1}_{j=1}\lambda_jd z_j\otimes d\overline z_j$ and
\[\phi=\beta\theta+\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t+R(z)
+O(\abs{z}\abs{\theta})+O(\abs{\theta}^2)+O(\abs{(z, \theta)}^3),\]
where $R(z)=O(\abs{z}^2)$, $\frac{\partial}{\partial\overline z_j}R=0$, $j=1,\ldots,n-1$.
Let $F_k(z, \theta):=(\frac{z}{\sqrt{k}}, \frac{\theta}{k})$ be the scaling map. For $r>0$, let
$D_r=\set{(z, \theta)=(x, \theta);\, \abs{x_j}<r, \abs{\theta}<r, j=1,\ldots,2n-2}$.
Now, we consider the restriction of $u_k$ to the domain $F_k(D_{\log k})$.
The function $\alpha_k:=k^{-\frac{n}{2}}F^*_k(e^{-kR}u_k)\in C^\infty(D_{\log k})$, satisfies $\limsup_{k\rightarrow\infty}k^{-n}S^{(0)}(p)=\lim_{k\rightarrow\infty}\abs{\alpha_k(0)}^2$, where $F^*_kf\in C^\infty(D_{\log k})$ denotes the scaled function
$f\big(\frac{x}{\sqrt{k}}, \frac{\theta}{k}\big)$, $f\in C^\infty(F_k(D_{\log k}))$. Moreover, $\alpha_k$ is harmonic with respect to
the scaled Kohn Laplacian $\Box^{(0)}_{s,(k)}$ (cf.\,\eqref{s3-e8-9-1}). The point is that $\Box^{(0)}_{s,(k)}$ converges in some sense to the model Laplacian $\Box^{(0)}_{b,H_n}$ on $H_n:=\mathbb C^{n-1}\times\mathbb R$ (cf.\,\eqref{s3-e221}).
In fact, $\Box^{(0)}_{b,H_n}$ is the Kohn Laplacian defined with respect to the CR structure $U_{j,H_n}:=\frac{\partial}{\partial z_j}-\frac{1}{\sqrt{2}}i\lambda_j\overline z_j\frac{\partial}{\partial\theta}$, $j=1,\ldots,n-1$, and the weight $e^{-\psi_0}$, $\psi_0=\beta\theta+\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t$.
Since $Y(q)$ holds, $\Box^{(0)}_{s,(k)}$ is hypoelliptic with loss of one derivative. Thus, the standard techniques for partial differential operators
(Rellich's theorem and Sobolev embedding theorem) yield a subsequence $\alpha_{k_j}$ converging uniformly with all the derivatives
on any compact subset of $H_n$ to a smooth function $\alpha$, which is harmonic with respect to $\Box^{(0)}_{b,H_n}$. This implies
that
\[
\limsup_{k\rightarrow\infty}k^{-n}S^{(0)}_k(p)=\abs{\alpha(0)}^2\leqslant S^{(0)}_{H_n}(0):=
\sup_{\Box^{(0)}_{b,H_n}f=0, \norm{f}_{\psi_0}=1}\abs{f(0)}^2\,.
\]
Computing the extremal function in the model case explicitly (see \S4) finishes the proof of \eqref{s1-e19}.
This paper is organized as follows. In \S2 we first introduce the extremal function and we relate it to the Szeg\"o kernel function.
Then we introduce the scaled Kohn Laplacian $\Box^{(q)}_{s,(k)}$ and prove the rough upper-bound for the Szeg\"o kernel function \eqref{s1-e18} (cf.\,Theorem \ref{s3-t1}). Moreover, by comparing the scaled operator $\Box^{(q)}_{s,(k)}$ to the Kohn Laplacian $\Box^{(q)}_{b,H_n}$ on the Heisenberg group we estimate in Theorem \ref{s3-t2} the Szeg\"o kernel function on $X$ in terms of the extremal function on the Heisenberg group.
The latter is computed explicitely in \S3. In \S4 we use this information in order to prove the local Morse inequalities \eqref{s1-e19} and by integration the weak Morse inequalities \eqref{s1-e191}.
In \S5 we analyse the spectral function of $\Box^{(q)}_{b,(k)}$ and deduce the semi-classical Weyl law, thus proving Theorems \ref{t-main3}--\ref{t-main5}. In \S6 we specialize the previous results to the case of an embedded
CR manifold and prove Theorems \ref{th-app} and \ref{th-strip}. Moreover, we exemplify our results in two concrete examples, one of a Grauert tube over the torus and the other of a quotient of the Heisenberg group.
\section{The estimates of the Szeg\"{o} kernel function $\mathit\Pi^{(q)}_k$} \label{morse-sec:estimates}
\subsection{The Szeg\"{o} kernel function $\mathit\Pi^{(q)}_k(x)$ and the extremal function $S^{(q)}_{k,J}(x)$} \label{morse-sec:szego}
We first introduce some notations. For $p\in X$, we can choose an orthonormal frame
$e_1(y),\ldots,e_{n-1}(y)$
for $\Lambda^{0,1}T^*_y(X)$ varying smoothly with $y$ in a neighborhood $U$ of $p$. For a multiindex $J=(j_1,\ldots,j_q)\in\{1,\ldots,n-1\}^q$ we write $\abs{J}=q$. We say that $J$ is strictly increasing if $1\leqslant j_1<j_2<\cdots<j_q\leqslant n-1$.
Then $(e^J(y):=e_{j_1}\wedge\cdots\wedge e_{j_q})_{1\leqslant j_1<j_2<\cdots<j_q\leqslant n-1}$
is an orthonormal frame for $\Lambda^{0,q}T^*_y(X)$. For $f\in\Omega^{0,q}(X, L^k)$, we may write $f|_U=\sum'_{\abs{J}=q} f_Je^J$, with $f_J=\langle f,e^J\rangle\in C^\infty(U;\, L^k)$, where $\sum'$ means that the summation is performed only over strictly increasing multiindices. We call $f_J$ the component of $f$ along $e^J$. It will be clear from the context what frame is being used. The \emph{extremal function} $S^{(q)}_{k,J}$ along the direction $e^J$ is defined by
\begin{equation} \label{s2-e2}
S^{(q)}_{k,J}(y)=\sup_{\alpha\in\,\cali{H}_b^q(X, L^k),\,\norm{\alpha}=1}\abs{\alpha_J(y)}^2\,.
\end{equation}
\begin{lem} \label{s2-l1}
For a given local orthonormal frame $\{e^J(y); \text{$\abs{J}=q$,\,$J$ strictly increasing}\}$ of $\Lambda^{0,q}T^*(X)|_U$, $U\subset X$ open,
we have that $\mathit\Pi^{(q)}_k(y)=\sum_{\abs{J}=q}'S^{(q)}_{k,J}(y)$.
\end{lem}
\begin{proof}
Let $(f_j)_{j=1,\ldots,N}$ be an orthonormal frame for the space
$\cali{H}_b^q(X, L^k)$.
On $U$ we write $\mathit\Pi^{(q)}_k(y)=\sum'_{\abs{J}=q}\mathit\Pi^{(q)}_{k,J}(y)$, where $\mathit\Pi^{(q)}_{k,J}(y):=\sum_j\abs{f_{j,J}(y)}^2$.
It is easy to see that
$\mathit\Pi^{(q)}_{k,J}(y)$ is independent of the choice of the orthonormal frame $(f_j)$. Take
$\alpha\in\cali{H}^q_b(X, L^k)$ of unit norm. Since $\alpha$ is contained in an orthonormal base, obviously $|\alpha_J(y)|^2\leqslant \mathit\Pi^{(q)}_{k,J}(y)$. Thus,
\begin{equation} \label{s2-e4}
S^{(q)}_{k,J}(y)\leqslant \mathit\Pi^{(q)}_{k,J}(y)\,,\quad\text{for all strictly increasing $J$, $\abs{J}=q$.}
\end{equation}
Fix a point $p\in U$ and a strictly incresing $J$, $\abs{J}=q$. For simplicity, we may assume that $\phi(p)=0$. Put
\[
\textstyle
u(y)=\Big(\sum^N_{j=1}\abs{f_{j,J}(p)}^2\Big)^{-1/2}\cdot\sum^N_{j=1}\overline{f_{j,J}(p)}f_j(y)\,.
\]
We can check that $u\in H^q_b(X, L^k)$ and $\norm{u}=1$. Hence, $|u_{J}(p)|^2\leqslant S^{(q)}_{k,J}(p)$, therefore
$\mathit\Pi^{(q)}_{k,J_0}(p)=\sum^N_{j=1}\abs{f_{j,J}(p)}^2=|u_{J}(p)|^2\leqslant S^{(q)}_{k,J}(p)$.
By \eqref{s2-e4}, $\mathit\Pi^{(q)}_{k,J}=S^{(q)}_{k,J}$ for all strictly increasing $J$ with $\abs{J}=q$
and the lemma follows.
\end{proof}
\subsection{The scaling technique}
For a given point $p\in X$, let $U_1(y),\ldots, U_{n-1}(y)$
be an orthonormal frame of $\Lambda^{1, 0}T_y(X)$ varying smoothly with $y$ in a neighborhood of\, $p$,
for which the Levi form is diagonalized at $p$. Furthermore, let $s$ be a local trivializing section of $L$ on an open neighborhood of $p$ and $\abs{s}^ 2=e^{-\phi}$. We take local coordinates
$(x, \theta)=(z, \theta)=(x_1,\ldots,x_{2n-2},\theta)$, $z_j=x_{2j-1}+ix_{2j}$, $j=1,\ldots,n-1$,
defined on an open set $D$ of $p$ such that $\omega_0(p)=\sqrt{2}d\theta$, $(x(p), \theta(p))=0$,
$(\frac{\partial}{\partial x_j}(p)\ |\ \frac{\partial}{\partial x_t}(p))=2\delta_{j,t}$, $(\frac{\partial}{\partial x_j}(p)\ |\ \frac{\partial}{\partial\theta}(p))=0$,
$(\frac{\partial}{\partial\theta}(p)\ |\ \frac{\partial}{\partial\theta}(p))=2$, $j, t=1,\ldots,2n-1$,
and
\begin{equation} \label{s1-e20}
U_j=\frac{\partial}{\partial z_j}-\frac{1}{\sqrt{2}}i\lambda_j\overline z_j\frac{\partial}{\partial\theta}-
\frac{1}{\sqrt{2}}c_j\theta\frac{\partial}{\partial\theta}+O(\abs{(z, \theta)}^2),\ \ j=1,\ldots,n-1,
\end{equation}
and
\begin{equation} \label{s1-e21}
\begin{split}
\phi=&\sum^{n-1}_{j=1}(\alpha_j z_j+\overline\alpha_j\overline z_j)+\beta\theta+\sum^{n-1}_{j,t=1}(a_{j,t}z_jz_t+\overline a_{j,t}\overline z_j\overline z_t)
+\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t\\
&+O(\abs{z}\abs{\theta})+O(\abs{\theta}^2)+O(\abs{(z, \theta)}^3),
\end{split}
\end{equation}
where
$\beta\in\mathbb R, c_j, \alpha_j, a_{j,t}, \mu_{j,t}\in\mathbb C$, $\delta_{j,t}=1$ if $j=t$, $\delta_{j,t}=0$ if $j\neq t$, $\frac{\partial}{\partial z_j}=\frac{1}{2}(\frac{\partial}{\partial x_{2j-1}}-i\frac{\partial}{\partial x_{2j}})$, for $j,t=1,\ldots,n-1$ and $\lambda_j$, $j=1,\ldots,n-1$ are the eigenvalues of\, $\mathcal{L}_p$.
This is always possible, see \cite[p.\,157--160]{BG88}.
In this section, we work with this local coordinates and we identify $D$ with some open set in $\mathbb R^{2n-1}$. Put
\begin{gather}
R(z, \theta)=\sum^{n-1}_{j=1}\alpha_j z_j+\sum^{n-1}_{j,t=1}a_{j,t}z_jz_t\,,\label{s1-e22}\\
\phi_0=\phi-R(z, \theta)-\overline{R(z, \theta)} =\beta\theta+\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t+O(\abs{z}\abs{\theta})+O(\abs{\theta}^2)+O(\abs{(z, \theta)}^3)\,.\label{s1-e23}
\end{gather}
Let $(\ |\ )_{k\phi}$ and $(\ |\ )_{k\phi_0}$ be the inner products on the space
$\Omega^{0,q}_c(D)$ defined as follows:
$(f\ |\ g)_{k\phi}=\int_D\!(f\ |\ g)e^{-k\phi}dm$, $(f\ |\ g)_{k\phi_0}=\int_D\!(f\ |\ g)e^{-k\phi_0}dm$,
where $f, g\in\Omega^{0,q}_c(D)$. Put
\[
\begin{split}
L^2_{(0,q)}(D, k\phi)=\Big\{u\in\mathscr{D}^{\,\prime}(D;\, \Lambda^{0,q}T^*(X));\, \int_{D}\!
\abs{u}^2e^{-k\phi}dm<\infty\Big\}\,,\\
L^2_{(0,q)}(D, k\phi_0)=\Big\{u\in\mathscr D'(D;\, \Lambda^{0,q}T^*(X));\, \int_D\!
\abs{u}^2e^{-k\phi_0}dm<\infty\Big\}\,,
\end{split}
\]
where $\mathscr{D}^{\,\prime}(D, \Lambda^{0,q}T^*(X))$ denotes the space of distribution sections of $D$ over $\Lambda^{0,q}T^*(X)$.
We extend the inner products $(\ |\ )_{k\phi}$ and $(\ |\ )_{k\phi_0}$ to the spaces
$L^2_{(0,q)}(D, k\phi)$ and $L^2_{(0,q)}(D, k\phi_0)$ respectively.
We have the unitary identification
\begin{equation} \label{s1-e27}
\left\{\begin{aligned}
L^2_{(0,q)}(D, k\phi_0)&\leftrightarrow L^2_{(0,q)}(D, k\phi) \\
u&\rightarrow \widetilde u=e^{kR}u, \\
u=e^{-kR}\widetilde u&\leftarrow \widetilde u.
\end{aligned}
\right.
\end{equation}
Let
$\overline\partial^{*,k\phi}_b:\Omega^{0,q+1}(D)\rightarrow\Omega^{0,q}(D)$
be the formal adjoint of $\overline\partial_b$ with respect to $( \ |\ )_{k\phi}$. Put
$\Box^{(q)}_{b,k\phi}=\overline\partial_b\overline\partial^{*,k\phi}_b+\overline\partial^{*,k\phi}_b\overline\partial_b:\Omega^{0,q}(D)\rightarrow\Omega^{0,q}(D)$.
Let $u\in\Omega^{0,q}(D, L^k)$.
On $D$, we write $u=s^k\hat u$, $\hat u\in\Omega^{0,q}(D)$. We have $\Box^{(q)}_{b,k}u=s^k\Box^{(q)}_{b,k\phi}\hat u$.
In this section, we identify $u$ with $\hat u$ and $\Box^{(q)}_{b,k}$ with $\Box^{(q)}_{b, k\phi}$. Note that $\abs{u(0)}^2=\abs{\hat u(0)}^2e^{-k\phi(0)}=\abs{\hat u(0)}^2$.
In the sequel we denote by $\alpha\wedge$ the operator of left exterior multiplication with a form $\alpha$.
The adjoint of this operator is denoted by $\alpha^{\wedge,*}$.
If $u\in\Omega^{0,q}(D)\cap L^2_{(0,q)}(D, k\phi_0)$, using \eqref{s1-e27}, we have
$\overline\partial_b\widetilde u=\widetilde{\overline\partial_s u}=e^{kR}\overline\partial_su$,
where
\begin{equation} \label{s3-e2-5}
\overline\partial_s=\overline\partial_b+k(\overline\partial_bR)\wedge\;.
\end{equation}
Let $(e_j(z, \theta))_{j=1,\ldots,n-1}$ denote the basis of $\Lambda^{0,1}T^*_{(z,\theta)}(X)$,
dual to $(\overline U_j(z,\theta))_{j=1,\ldots,n-1}$. Then
$\overline\partial_b=\sum^{n-1}_{j=1}\bigr(e_j\wedge\overline U_j+\overline\partial_b e_j\wedge e_j^{\wedge,*}\bigr)$. Note that $e_j^{\wedge,*}=i_{\overline{U}_j}$, the interior product with $\overline{U}_j$.
Thus,
\begin{equation} \label{s3-e4}
\overline\partial_s=\sum^{n-1}_{j=1}e_j\wedge\Bigr(\overline U_j+k(\overline U_jR)\Bigr)
+\sum^{n-1}_{j=1}(\overline\partial_b e_j)\wedge e_j^{\wedge,*}
\end{equation}
and correspondingly
\begin{equation} \label{s3-e5}
\overline\partial^*_s=\sum^{n-1}_{j=1}e_j^{\wedge,*}\Bigr(\overline U^{*,k\phi_0}_j+k(U_j\overline R)\Bigr)
+\sum^{n-1}_{j=1}e_j\wedge(\overline\partial_b e_j)^{\wedge,*},
\end{equation}
where $\overline\partial^{*,k\phi}_b\widetilde u=e^{kR}\overline\partial^*_su$ and $\overline U^{*,k\phi_0}_j$ is the formal adjoint of $\overline U_j$ with respect to $(\ |\ )_{k\phi_0}$, $j=1,\ldots,n-1$. We can check that
\begin{equation} \label{s3-e6}
\overline U_j^{*,k\phi_0}=-U_j+k(U_j\phi_0)+s_j(z, \theta),
\end{equation}
where $s_j\in C^\infty(D)$, $s_j$ is independent of $k$, $j=1,\ldots,n-1$. Put
\begin{equation} \label{s3-e7}
\Box^{(q)}_s=\overline\partial_s\overline\partial^*_s+\overline\partial^*_s\overline\partial_s:\Omega^{0,q}(D)\rightarrow\Omega^{0,q}(D).
\end{equation}
We have
\begin{equation} \label{s3-e7-0}
\widetilde{\Box^{(q)}_s u}=e^{kR}\Box^{(q)}_su=\Box^{(q)}_{b,k\phi}\widetilde u.
\end{equation}
\begin{prop}[{\cite[Prop.\,2.9]{Hsiao08}}] \label{s3-p0}
We have
\begin{equation} \label{s3-e7-1}
\begin{split}
\Box^{(q)}_s &=\overline\partial_s\overline\partial^*_s+\overline\partial^*_s\overline\partial_s \\
&=\sum^{n-1}_{j=1}\bigr(\overline U^{*,k\phi_0}_j+k(U_j\overline R)\bigr)\bigr(\overline U_j+k(\overline U_jR)\bigr)\\
&+ \sum^{n-1}_{j,t=1}e_j\wedge e^{\wedge, *}_t\big[\overline U_j+k(\overline U_j R)\ ,\overline U^{*,k\phi_0}_t+k(U_t\overline R)\big] \\
&\quad +\epsilon(\overline U+k(\overline U R))+\epsilon(\overline U^{*,k\phi_0}+k(U\overline R))+f(z, \theta),
\end{split}
\end{equation}
where $\epsilon(\overline U+k(\overline U R))$ denotes remainder terms of the form $\sum a_j(z, \theta)\bigr(\overline U_j+k(\overline U_j R)\bigr)$ with $a_j$ smooth, matrix-valued and independent of $k$, for all $j$, and similarly for $\epsilon(\overline U^{*,k\phi_0}+k(\overline U R))$ and $f(z, \theta)\in C^\infty$ independent of $k$.
\end{prop}
We recall some notations we used before for the convenience of the reader. For $r>0$, let
$D_r=\set{(z, \theta)=(x, \theta)\in\mathbb R^{2n-1};\, \abs{x_j}<r,\ \abs{\theta}<r,\ j=1,\ldots,2n-2}$.
Let $F_k$ be the scaling map:
$F_k(z, \theta)=(\frac{z}{\sqrt{k}}, \frac{\theta}{k})$.
From now on, we assume that $k$ is large enough so that $F_k(D_{\log k})\subset D$.
On $D_{\log k}$, the scaled bundle $F^*_k\Lambda^{0,q}T^*(X)$ is the bundle with fiber
$F^*_k\Lambda^{0,q}T^*_{(z_0, \theta_0)}(X):=\set{\sum'_{\abs{J}=q}a_Je^J(\frac{z_0}{\sqrt{k}},\frac{\theta_0}{k});\, a_J\in\mathbb C,\abs{J}=q}$ at $(z_0, \theta_0)\in D_{\log k}$.
We take the Hermitian metric $(\ |\ )_{F^*_k}$ on $F^*_k\Lambda^{0,q}T^*(X)$ so that at each point $(z_0, \theta_0)\in D_{\log k}$\,,
\[
\Bigr\{e^J\big(\tfrac{z}{\sqrt{k}}\;,\tfrac{\theta}{k}\big)\,; \text{$\abs{J}=q$, $J$ strictly increasing}\Bigl\}\,,
\]
is an orthonormal basis for $F^*_k\Lambda^{0,q}T^*_{(z_0,\theta_0)}(X)$. For $r>0$, let $F^*_k\Omega^{0,q}(D_r)$
denote the space of smooth sections of $F^*_k\Lambda^{0,q}T^*(X)$ over $D_r$. Let $F^*_k\Omega^{0,q}_c(D_r)$ be the subspace of
$F^*_k\Omega^{0,q}(D_r)$ whose elements have compact support in $D_r$.
Given $f\in\Omega^{0,q}(F_k(D_{\log k}))$ we write
$f=\sum'_{\abs{J}=q}f_Je^J$.
We define the scaled form $F_k^*f\in F^*_k\Omega^{0,q}(D_{\log k})$ by:
\[
F_k^*f=\sideset{}{'}\sum_{\abs{J}=q}f_J\Big(\frac{z}{\sqrt{k}}\;, \frac{\theta}{k}\Big)e^J\Big(\frac{z}{\sqrt{k}}\;,\frac{\theta}{k}\Big)\,.
\]
Let $P $ be a partial differential operator of order one on $F_k(D_{\log k})$ with $C^\infty$ coefficients. We write
$P=a(z, \theta)\frac{\partial}{\partial\theta}+\sum^{2n-2}_{j=1}a_j(z, \theta)\frac{\partial}{\partial x_j}$, $a, a_j\in C^\infty(F_k(D_{\log k}))$, $j=1,\ldots,2n-2$. The partial diffferential operator
$P_{(k)}$ on $D_{\log k}$ is given by
\begin{equation} \label{s3-e8-4}
P_{(k)}=\sqrt{k}F^*_ka\frac{\partial}{\partial\theta}+\sum^{2n-2}_{j=1}F^*_ka_j\frac{\partial}{\partial x_j}
=\sqrt{k}a\Big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\Big)\frac{\partial}{\partial\theta}+\sum^{2n-2}_{j=1}a_j\Big(\frac{z}{\sqrt{k}}, \frac{\theta}{k}\Big)\frac{\partial}{\partial x_j}.
\end{equation}
Let $f\in C^\infty(F_k(D_{\log k}))$. We can check that
\begin{equation} \label{s3-e8-5}
P_{(k)}(F^*_kf)=\frac{1}{\sqrt{k}}F^*_k(Pf).
\end{equation}
The scaled differential operator $\overline\partial_{s,(k)}:F^*_k\Omega^{0,q}(D_{\log k})\rightarrow F^*_k\Omega^{0,q+1}(D_{\log k})$ is given by (compare $\overline\partial_s$, see \eqref{s3-e4})
\begin{equation} \label{s3-e8-6}
\begin{split}
\overline\partial_{s,(k)}=&\sum^{n-1}_{j=1}e_j\Big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\Big)\wedge\Bigr(\overline U_{j(k)}+\sqrt{k}F^*_k(\overline U_jR)\Bigr)\\
&+\sum^{n-1}_{j=1}\frac{1}{\sqrt{k}}(\overline\partial_b e_j)\Big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\Big)\wedge e_j\Big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\Big)^{\wedge,*}.
\end{split}
\end{equation}
From \eqref{s3-e4} and \eqref{s3-e8-5}, we can check that if $f\in\Omega^{0,q}(F_k(D_{\log k}))$, then
\begin{equation} \label{s3-e8-7}
\overline\partial_{s,(k)}F^*_kf=\frac{1}{\sqrt{k}}F^*_k(\overline\partial_sf).
\end{equation}
Let $(\ |\ )_{kF^*_k\phi_0}$ be the inner product on the space $F^*_k\Omega^{0,q}_c(D_{\log k})$
defined as follows: $(f\ |\ g)_{kF^*_k\phi_0}=\int_{D_{\log k}}\!(f\ |\ g)_{F^*_k}e^{-kF^*_k\phi_0}(F^*_km)(z, \theta)dv(z)dv(\theta)$,
where $mdv(z)dv(\theta)$ is the volume form, $dv(z)=2^{n-1}dx_1\cdots dx_{2n-2}$, $dv(\theta)=\sqrt{2}d\theta$. Note that $m(0,0)=1$. Let $\overline\partial^*_{s,(k)}:F^*_k\Omega^{0,q+1}(D_{\log k})\rightarrow F^*_k\Omega^{0,q}(D_{\log k})$ be the formal adjoint of $\overline\partial_{s,(k)}$ with respect to $(\ |\ )_{kF^*_k\phi_0}$. Then, we can check that (compare $\overline\partial^*_s$, see \eqref{s3-e5} and \eqref{s3-e6})
\begin{equation} \label{s3-e8-8}
\begin{split}
\overline\partial^*_{s,(k)}=&\sum^{n-1}_{j=1}e_j\Big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\Big)^{\wedge,*}\Bigr(-U_{j(k)}+\sqrt{k}F^*_k( U_j\overline R)+\sqrt{k}F^*_k(U_j\phi_0)+\frac{1}{\sqrt{k}}F^*_ks_j\Bigr)\\
&+\sum^{n-1}_{j=1}\frac{1}{\sqrt{k}}\,e_j\Big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\Big)\wedge(\overline\partial_b e_j)\Big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\Big)^{\wedge,*},
\end{split}
\end{equation}
where $s_j\in C^\infty(D_{\log k})$, $j=1,\ldots,n-1$, are independent of $k$.
We also have
\begin{equation} \label{s3-e8-9}
\overline\partial^*_{s,(k)}F^*_kf=\frac{1}{\sqrt{k}}F^*_k(\overline\partial^*_sf),\,\quad f\in\Omega^{0,q+1}(F_k(D_{\log k}))\,.
\end{equation}
We define now the \emph{scaled Kohn Laplacian}:
\begin{equation} \label{s3-e8-9-1}
\Box^{(q)}_{s,(k)}:=\overline\partial^*_{s,(k)}\overline\partial_{s,(k)}+\overline\partial_{s,(k)}\overline\partial^*_{(s,(k)}:F^*_k\Omega^{0,q}(D_{\log k})\rightarrow F^*_k\Omega^{0,q}(D_{\log k}).
\end{equation}
From \eqref{s3-e8-7} and \eqref{s3-e8-9}, we see that if $f\in\Omega^{0,q}(F_k(D_{\log k}))$, then
\begin{equation} \label{s3-e9}
(\Box^{(q)}_{s,(k)})F^*_kf=\frac{1}{k}F^*_k(\Box^{(q)}_sf).
\end{equation}
From \eqref{s1-e20} and \eqref{s1-e22}, we can check that
\begin{equation} \label{s3-e10}
\overline U_{j(k)}+\sqrt{k}F^*_k(\overline U_j R)=\frac{\partial}{\partial\overline z_j}+\frac{1}{\sqrt{2}}i\lambda_jz_j\frac{\partial}{\partial \theta}+\epsilon_kZ_{j,k}\,,\quad j=1,\ldots,n-1,
\end{equation}
on $D_{\log k}$, where $\epsilon_k$ is a sequence tending to zero with $k\rightarrow\infty$ and $Z_{j,k}$ is a first order differential operator and all the derivatives of the coefficients of $Z_{j,k}$ are uniformly bounded in $k$ on $D_{\log k}$, $j=1,\ldots,n-1$. Similarly, from \eqref{s1-e22} and \eqref{s1-e23}, we can check that
\begin{equation} \label{s3-e11}
\begin{split}
&-U_{t(k)}+\sqrt{k}F^*_k( U_t\overline R)+\sqrt{k}F^*_k(U_t\phi_0)+\frac{1}{\sqrt{k}}F^*_ks_t\\
&=-\frac{\partial}{\partial z_t}+\frac{1}{\sqrt{2}}i\lambda_t\overline z_t\frac{\partial}{\partial\theta}-\frac{1}{\sqrt{2}}i\lambda_t\overline z_t\beta+\sum^{n-1}_{j=1}\mu_{j,\,t}\,\overline{z}_j+\delta_kV_{t,\,k}\,,\quad t=1,\ldots,n-1,
\end{split}
\end{equation}
on $D_{\log k}$, where $\delta_k$ is a sequence tending to zero with $k\rightarrow\infty$ and $V_{j,k}$ is a first order differential operator and all the derivatives of the coefficients of $V_{j,k}$ are uniformly bounded in $k$ on $D_{\log k}$, $j=1,\ldots,n-1$.
From \eqref{s3-e10}, \eqref{s3-e11} and \eqref{s3-e8-6}, \eqref{s3-e8-8}, \eqref{s3-e8-9-1}, it is straightforward to see that
\begin{prop} \label{s3-p1}
We have that
\begin{equation*}
\begin{split}
\Box^{(q)}_{s,(k)}&=\sum^{n-1}_{j=1}\Bigr[\Bigr(-\frac{\partial}{\partial z_j}+\frac{i}{\sqrt{2}}\lambda_j\overline z_j\frac{\partial}{\partial \theta}-\frac{i}{\sqrt{2}}\lambda_j\overline z_j\beta+\sum^{n-1}_{t=1}\mu_{t,\,j}\,\overline z_t\Bigr)\Bigr(\frac{\partial}{\partial\overline z_j}+\frac{i}{\sqrt{2}}\lambda_jz_j\frac{\partial}{\partial\theta}\Bigr)\Bigr]\\
&
+\sum^{n-1}_{j,\,t=1}e_j\Big(\frac{z}{\sqrt{k}}\,, \frac{\theta}{k}\Big)\wedge e_t\Big(\frac{z}{\sqrt{k}}\,,\frac{\theta}{k}\Big)^{\wedge,*}\Bigr(\Bigr(\mu_{j,\,t}-\frac{i}{\sqrt{2}}\lambda_j\delta_{j,\,t}\beta\Bigr)+\sqrt{2}i\lambda_j\delta_{j,\,t}\frac{\partial}{\partial \theta}\Bigr)+\varepsilon_kP_k,
\end{split}
\end{equation*}
on $D_{\log k}$, where $\varepsilon_k$ is a sequence tending to zero with $k\rightarrow\infty$ and $P_k$ is a second order differential operator and all the derivatives of the coefficients of $P_k$ are uniformly bounded in $k$ on $D_{\log k}$.
\end{prop}
Let $D\subset D_{\log k}$ be an open set and let $W^s_{kF^*_k\phi_0}(D;\, F^*_k\Lambda^{0, q}T^*(X))$,
$s\in\mathbb N_0:=\mathbb N\cup\set{0}$, denote the Sobolev space of order $s$ of sections of $F^*_k\Lambda^{0,q}T^*(X)$
over $D$ with respect to the weight $e^{-kF^*_k\phi_0}$. The Sobolev norm on this space is given by
\begin{equation} \label{s1-e37}
\norm{u}^2_{kF^*_k\phi_0,s,D}
=\sum_{\substack{\alpha\in\mathbb{N}^{2n-1}_0,\;\abs{\alpha}\leqslant s\;\\{\abs{J}=q}}}
\int_{D}\!\abs{\partial^\alpha_{x,\theta}u_J}^2e^{-kF^*_k\phi_0}(F^*_km)(z, \theta)dv(z)dv(\theta),
\end{equation}
where
$u=\sum'_{\abs{J}=q}u_Je^J\big(\frac{z}{\sqrt{k}},\frac{\theta}{k}\big)\in W^s_{kF^*_k\phi_0}
(D;\, F^*_k\Lambda^{0,q}T^*(X))$ and $m$ is the volume form.
If $s=0$, we write $\norm{\cdot}_{kF^*_k\phi_0,D}$ to denote $\norm{\cdot}_{kF^*_k\phi_0,0,D}$.
We need the following
\begin{prop} \label{s3-p2}
For every $r>0$ with $D_{2r}\subset D_{\log k}$ and $s\in\mathbb N\cup\{0\}$, there is a constant $C_{r,s}>0$
independent of $k$, such that
\begin{equation} \label{s3-e12}
\norm{u}^2_{kF^*_k\phi_0,s+1,D_{r}}\leqslant C_{r,s}\Bigr(\norm{u}^2_{kF^*_k\phi_0,D_{2r}}+\big\|\Box^{(q)}_{s,(k)}u\big\|^2_{kF^*_k\phi_0,s,D_{2r}}\Bigl)\,,\;
u\in F^*_k\Omega^{0,q}(D_{\log k})\,.
\end{equation}
\end{prop}
\begin{proof}
Since $Y(q)$ holds, we see from the classical work of Kohn (\cite[Th.\,7.7]{Ko65}, \cite[Prop.\,5.4.10]{FK72}, \cite[Th.\,8.4.3]{CS01}), that
$\Box^{(q)}_{s,(k)}$ is hypoelliptic with loss of one derivative and we have \eqref{s3-e12}.
Since all the derivatives of the coefficients of the operator $\Box^{(q)}_{s,(k)}$ are uniformly bounded in $k$,
if we go through the proof of~\cite[pp.\,193--199]{CS01} (see also Remark~\ref{s3-r1} below), it is
straightforward to see that $C_{r,s}$ can be taken to be independent of $k$.
\end{proof}
\begin{rem} \label{s3-r1}
Put
\begin{equation*}
\begin{split}
A=\{&\mbox{all the coefficients of $\Box^{(q)}_{s,(k)}$, $\overline\partial_{s,(k)}$, $\overline\partial^*_{s,(k)}$,
$[\overline U_{j(k)}\ ,U_{t(k)}]$, $\overline U_{j(k)}$, $U_{t(k)}$},\\
&j,t=1,\ldots,n-1,\ \mbox{and of $kF^*_k\phi_0$, $F^*_km$}\}
\end{split}
\end{equation*}
and $B=\set{\mbox{all the eigenvalues of $\mathcal{L}_p$}}$.
From the proof of Kohn, we see that for $r>0$, $s\in\mathbb{N}_0$, there exist a semi-norm $P$ on $C^\infty(D_{2r})$ and a strictly positive continuous function $F:\mathbb R\rightarrow\mathbb R_+$ such that
\begin{equation} \label{s3-e12-1}
\norm{u}^2_{kF^*_k\phi_0,s+1,D_{r}}\leqslant \Bigr(\textstyle\sum\limits_{f\in A}F(P(f))+\sum\limits_{\lambda\in B}F(\lambda)\Bigr) \Bigr(\big\|u\big\|^2_{kF^*_k\phi_0,D_{2r}}+\big\|\Box^{(q)}_{s,(k)}u\big\|^2_{kF^*_k\phi_0,s,D_{2r}}\Bigr),
\end{equation}
where $u\in F^*_k\Omega^{0,q}(D_{\log k})$.
Roughly speaking, the constant $C_{r,s}$ in \eqref{s3-e12} depends continuously on the eigenvalues of $\mathcal{L}_p$ and the elements of $A$ in $C^\infty(D_{2r})$ topology. (See also the proof of \cite[Lemma\,4.1]{SW05}.)
\end{rem}
\begin{lem} \label{s3-l1}
Let $\alpha_k\in F^*_k\Omega^{0,q}(D_{\log k})$ with $\Box^{(q)}_{s,(k)}\alpha_k=0$ and $\norm{\alpha_k}_{kF^*_k\phi_0,D_{\log k}}\leqslant 1$. Then, there is a constant $C$, $C$ is independent of $k$, such that
$\abs{\alpha_k(0)}^2\leqslant C$.
\end{lem}
\begin{proof}
Fix $r>0$, $r$ small and let $\chi\in C^\infty_0(D_{r})$, $\chi=1$ on $D_{\frac{r}{2}}$. Identify $\alpha_k$ with a form in $\mathbb R^{2n-1}$ by extending with zero. Then
\begin{align*}
\abs{\chi(0)\alpha_k(0)}&=\abs{\int_{\mathbb R^{2n-1}}\!\widehat{\chi\alpha_k}(\xi)d\xi}
=\abs{\int_{\mathbb R^{2n-1}}\!(1+\abs{\xi}^2)^{-\frac{n}{2}}(1+\abs{\xi}^2)^{\frac{n}{2}}\widehat{\chi\alpha_k}(\xi)d\xi}\\
&\quad\leqslant \Bigl(\int_{\mathbb R^{2n-1}}\!(1+\abs{\xi}^2)^{-n}d\xi\Bigr)^{\frac{1}{2}}\Bigl(\int_{\mathbb R^{2n-1}}\!
(1+\abs{\xi}^2)^{n}\abs{\widehat{\chi\alpha_k}(\xi)}^2d\xi\Bigr)^{\frac{1}{2}}\\
&\quad\leqslant \widetilde c\norm{\alpha_k}_{kF^*_k\phi_0,n,D_r},
\end{align*}
where $\widehat{\chi\alpha_k}$ denotes the Fourier transform of $\chi\alpha_k$.
Proposition~\ref{s3-p2} implies that there exists $C'>0$ independent of $k$ such that $\norm{\alpha_k}_{kF^*_k\phi_0,n,D_r}\leqslant C'$. The lemma follows.
\end{proof}
Now, we can prove the first part of Theorem \ref{t-main1} (estimate \eqref{s1-e18}).
\begin{thm} \label{s3-t1}
There is a constant $C_0$, $C_0$ is independent of $k$, such that
\begin{equation} \label{s3-e14}
k^{-n}\mathit\Pi^{(q)}_k(x_0)\leqslant C_0
\end{equation}
for all $x_0\in X$.
\end{thm}
\begin{proof}
Let $u_k\in H^q_b(X, L^k)$, $\norm{u_k}=1$. Set
$\alpha_k:=k^{-\frac{n}{2}}F^*_k(e^{-kR}u_k)\in F^*_k\Omega^{0,q}(D_{\log k})$.
We recall that $R$ is given by \eqref{s1-e22}. (See also \eqref{s1-e27}.) We check that
$\norm{\alpha_k}_{kF^*_k\phi_0,D_{\log k}}\leqslant 1$.
Using \eqref{s3-e9} and \eqref{s3-e7-0}, it is not difficult to see that $\Box^{(q)}_{s,(k)}\alpha_k=0$ on $D_{\log k}$. From this and Lemma~\ref{s3-l1}, we see that $\abs{\alpha_k(0)}^2=k^{-n}\abs{u_k(0)}^2\leqslant C(0)$,
$C(0)$ is independent of $k$.
Let $\widetilde x_0$ be another point of $X$ near $0$. We can repeat the procedure above and get
$k^{-n}\abs{u_k(\widetilde x_0)}^2\leqslant C(\widetilde x_0)$,
with $C(\widetilde x_0)$ independent of $k$.
In view of Remark~\ref{s3-r1}, we see that the constant $C(\widetilde x_0)$ can be taken to be uniformly bounded in some neighborhood of $0$. Since $X$ is compact, we get that there is a constant $C'_0$ independent of $k$, such that
$k^{-n}\abs{u_k(x_0)}^2\leqslant C'_0$
for all $x_0\in X$. Thus, for a local orthonormal frame $\{e^J;\text{$\abs{J}=q$, $J$ strictly increasing}\}$ we have $k^{-n}S^{(q)}_{k,J}(x_0)\leqslant C'_0$ (see \eqref{s2-e2} for the definition of $S^{(q)}_{k,J}$). From this and Lemma~\ref{s2-l1}, the theorem follows.
\end{proof}
\subsection{The Heisenberg group $H_n$}
We pause and introduce some notations. We identify $\mathbb R^{2n-1}$ with the Heisenberg group $H_n:=\mathbb C^{n-1}\times\mathbb R$. We also write $(z, \theta)$ to denote the coordinates of $H_n$, $z=(z_1,\ldots,z_{n-1})\in\mathbb C^{n-1}$, $z_j=x_{2j-1}+ix_{2j}$, $j=1,\ldots,n-1$, and $\theta\in\mathbb R$. Then
\[
\begin{split}
&\Big\{U_{j,H_n}=\frac{\partial}{\partial z_j}-\frac{1}{\sqrt{2}}i\lambda_j\overline z_j\frac{\partial}{\partial\theta}\,;\, j=1,\ldots,n-1\Big\}\,\\
&\Big\{U_{j,H_n}, \overline{U}_{j,H_n}, T=-\frac{1}{\sqrt{2}}\frac{\partial}{\partial\theta}\, ;\,j=1,\ldots,n-1\Big\}
\end{split}
\]
are orthonormal bases for the bundles $\Lambda^{1,0}T(H_n)$ and $\mathbb C T(H_n)$, respectively. Then
\[
\Big\{dz_j, d\overline z_j,\omega_0=\sqrt{2}d\theta+\sum^{n-1}_{j=1}(i\lambda_j\overline z_jdz_j-i\lambda_jz_jd\overline z_j)\,;j=1,\ldots,n-1\Big\}
\]
is the basis of $\mathbb C T^*(H_n)$ which is dual to $\{U_{j,H_n},\overline U_{j,H_n}, -T;j=1,\ldots,n-1\}$.
We take the Hermitian metric $(\ |\ )$ on $\Lambda^{0,q}T^*(H_n)$ such that $\{d\overline z^J: \text{$\abs{J}=q$, $J$ strictly increasing}\}$ is an orthonormal basis of $\Lambda^{0,q}T^*(H_n)$. The Cauchy-Riemann operator $\overline\partial_{b,H_n}$ on $H_n$ is given by
\begin{equation} \label{s3-e16}
\overline\partial_{b,H_n}=\sum^{n-1}_{j=1}d\overline z_j\wedge\overline U_{j,H_n}:\Omega^{0,q}(H_n)\rightarrow\Omega^{0,q+1}(H_n).
\end{equation}
Put
$\psi_0(z, \theta)=\beta\theta+\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t\in C^\infty(H_n;\, \mathbb R)$,
where $\beta$ and $\mu_{j,t}$, $j,t=1,\ldots,n-1$, are as in \eqref{s1-e21}. Note that
\begin{equation} \label{s3-e17-1}
\sup_{(z, \theta)\in D_{\log k}}\abs{kF^*_k\phi_0-\psi_0}\To0,\ \ \mbox{as $k\rightarrow\infty$}.
\end{equation}
Let $(\ |\ )_{\psi_0}$
be the inner product on $\Omega^{0,q}_c(H_n)$ defined as follows:
\[
(f\ |\ g)_{\psi_0}=\int_{H_n}\!(f\ |\ g)e^{-\psi_0}dv(z)dv(\theta)\,, \quad f, g\in\Omega^{0,q}_c(H_n)\,,
\]
where $dv(z)=2^{n-1}dx_1dx_2\cdots dx_{2n-2}$, $dv(\theta)=\sqrt{2}d\theta$. Let $\overline\partial^{*,\psi_0}_{b,H_n}:\Omega^{0,q+1}(H_n)\rightarrow\Omega^{0,q}(H_n)$
be the formal adjoint of $\overline\partial_{b,H_n}$ with respect to $(\ |\ )_{\psi_0}$. We have
\begin{equation} \label{s3-e21}
\overline\partial^{*,\psi_0}_{b,H_n}=\sum^{n-1}_{t=1}d\overline z_t^{\wedge,*}\circ\overline U_{t,H_n}^{*,\psi_0}:\Omega^{0,q+1}(H_n)\rightarrow \Omega^{0,q}(H_n),
\end{equation}
where
\begin{equation} \label{s3-e22}
\overline U_{t,H_n}^{*,\psi_0}=-U_{t,H_n}+U_{t,H_n}\psi_0=-U_{t,H_n}+\sum^{n-1}_{j=1}\mu_{j,t}\overline z_j-\frac{1}{\sqrt{2}}i\lambda_t\overline z_t\beta\,.
\end{equation}
The Kohn Laplacian on $H_n$ is given by
\begin{equation}\label{s3-e221}
\Box^{(q)}_{b,H_n}=\overline\partial_{b,H_n}\overline\partial^{*,\psi_0}_{b,H_n}+\overline\partial^{*,\psi_0}_{b,H_n}\overline\partial_{b,H_n}:\Omega^{0,q}(H_n)\rightarrow\Omega^{0,q}(H_n)\,.
\end{equation}
From \eqref{s3-e16}, \eqref{s3-e21} and \eqref{s3-e22}, we can check that
\begin{equation} \label{s3-e23}
\begin{split}
&\Box^{(q)}_{b,H_n}\\
&=\sum^{n-1}_{j=1}\overline U^{*,\psi_0}_{j,H_n}\overline U_{j,H_n}
+\sum^{n-1}_{j,t=1}d\overline z_j\wedge d\overline z^{\wedge, *}_t\Bigr[\Bigr(\mu_{j,t}-\frac{i}{\sqrt{2}}\lambda_j\delta_{j,t}\beta\Bigr)+i\sqrt{2}\lambda_j\delta_{j,t}\frac{\partial}{\partial\theta}\Bigr]\\
&=\sum^{n-1}_{j=1}\Bigr[\Bigr(-\frac{\partial}{\partial z_j}+\frac{i}{\sqrt{2}}\lambda_j\overline z_j\frac{\partial}{\partial\theta}+\sum^{n-1}_{t=1}\mu_{t,j}\overline z_t-\frac{1}{\sqrt{2}}i\lambda_j\overline z_j\beta\Bigr)\Bigr(\frac{\partial}{\partial\overline z_j}+\frac{i}{\sqrt{2}}\lambda_jz_j\frac{\partial}{\partial\theta}\Bigr)\Bigr]\\
&\quad+\sum^{n-1}_{j,t=1}d\overline z_j\wedge d\overline z^{\wedge, *}_t\Bigr[\Bigr(\mu_{j,t}-\frac{i}{\sqrt{2}}\lambda_j\delta_{j,t}\beta\Bigr)+i\sqrt{2}\lambda_j\delta_{j,t}\frac{\partial}{\partial\theta}\Bigr].
\end{split}
\end{equation}
\subsection{The estimates of the Szeg\"{o} kernel function $\mathit\Pi^{(q)}_k$}
We need the following
\begin{prop} \label{s3-p3}
For each $k$, pick an element $\alpha_k\in F^*_k\Omega^{0,q}(D_{\log k})$ with
$\Box^{(q)}_{s,(k)}\alpha_k=0$
and $\norm{\alpha_k}_{kF^*_k\phi_0,D_{\log k}}\leqslant 1$. Identify $\alpha_k$ with a form in $H_n$ by extending it with zero and write $\alpha_k=\sum'_{\abs{J}=q}\alpha_{k,J}e^J(\frac{z}{\sqrt{k}},\frac{\theta}{k})$. Then there is a subsequence $\set{\alpha_{k_j}}$ of $\set{\alpha_k}$ such that for each strictly increasing multiindex $J$, $\abs{J}=q$, $\alpha_{k_j,\,J}$ converges uniformly with all its derivatives on any compact subset of $H_n$ to a smooth function $\alpha_J$. Furthermore, if we put $\alpha=\sum'_{\abs{J}=q}\alpha_Jd\overline z^J$, then
$\Box^{(q)}_{b,H_n}\alpha=0$.
\end{prop}
\begin{proof}
Fix a strictly increasing multiindex $J$, $\abs{J}=q$,
and $r>0$. From \eqref{s3-e12} and Remark~\ref{s3-r1}, we see that for
all $s>0$, there is a constant $C_{r,s}$, $C_{r,s}$ is independent of $k$, such that
$\norm{\alpha_{k,J}}_{s,D_r}\leqslant C_{r,s}$
for all $k$. Rellich 's compactness theorem \cite[p.\,281]{Yo80} yields a subsequence of $\set{\alpha_{k,J}}$, which converges in all Sobolev spaces $W^s(D_r)$ for $s>0$. From the Sobolev embedding theorem \cite[p.\,170]{Yo80}, we see that the sequence converges in all $C^l(D_r)$, $l\geqslant0$, $l\in\mathbb Z$, locally unformly. Choosing a diagonal sequence, with respect to a sequence of $D_r$ exhausting $H_n$, we get a subsequence $\set{\alpha_{k_j,J}}$ of $\set{\alpha_{k,J}}$ such that $\alpha_{k_j,J}$ converges uniformly with all derivatives on any compact subset of $H_n$ to a smooth function $\alpha_J$.
Let $J'$ be another strictly increasing multiindex, $\abs{J'}=q$. We can repeat the procedure above and get a subsequce $\set{\alpha_{k_{j_s},J'}}$ of $\set{\alpha_{k_j,J'}}$ such that $\alpha_{k_{j_s},J'}$ converges uniformly with all derivatives on any compact subset of $H_n$ to a smooth function $\alpha_{J'}$. Continuing in this way, we get the first statement of the proposition.
Now, we prove the second statement of the proposition. Let $P=(p_1,\ldots,p_q)$, $R=(r_1,\ldots,r_q)$ be multiindices, $\abs{P}=\abs{R}=q$. Define
\[\varepsilon^P_R=\left\{ \begin{array}{ll}
&0,\ \ \mbox{if $\set{p_1,\ldots,p_q}\neq\set{r_1,\ldots,r_q}$}, \\
&\mbox{the sign of permutation $\left(
\begin{array}[c]{c}
P \\
R
\end{array}\right)$
},\ \ \mbox{if $\set{p_1,\ldots,p_q}=\set{r_1,\ldots,r_q}$}.
\end{array}\right.\]
For $j, t=1,\ldots,n-1$, define
\[\sigma^{jtP}_{R}=\left\{ \begin{array}{ll}
&0,\ \ \mbox{if $d\overline z_j\wedge d\overline z_t^{\wedge,*}(d\overline z^P)=0$}, \\
&\varepsilon^Q_R,\ \ \mbox{if $d\overline z_j\wedge d\overline z_t^{\wedge,*}(d\overline z^P)=d\overline z^Q$, $\abs{Q}=q$}.
\end{array}\right.\]
We may assume that $\alpha_{k,J}$ converges uniformly with all derivatives on any compact subset of $H_n$ to a smooth function $\alpha_J$, for all strictly increasing $J$, $\abs{J}=q$. Since $\Box^{(q)}_{s,(k)}\alpha_k=0$, from the explicit formula of $\Box^{(q)}_{s,(k)}$
(see Prop. \ref{s3-p1}), it is not difficult to see that for all strictly increasing $J$, $\abs{J}=q$, we have
\begin{equation} \label{s3-e24}
\begin{split}
\sum^{n-1}_{j=1}\overline U^{*,\psi_0}_{j,H_n}\overline U_{j,H_n}\alpha_{k,J}=-\sideset{}{'}\sum_{\substack{\abs{P}=q,\\1\,\leqslant\, \, j\,,\,t\,\leqslant \, n-1}}
\sigma^{jtP}_{J}
\Bigr[\Bigr(\mu_{j,t}-\frac{i}{\sqrt{2}}\lambda_j\delta_{j,t}\beta\Bigr)+&\sqrt{2}i\lambda_j\delta_{j,t}\frac{\partial}{\partial\theta}\Bigr]\alpha_{k,P}\\
&+\epsilon_kP_{k,J}\alpha_k
\end{split}
\end{equation}
on $D_{\log k}$, where $\epsilon_k$ is a sequence tending to zero with $k\rightarrow\infty$ and $P_{k,J}$ is a second order differential operator and all the derivatives of the coefficients of $P_{k,J}$ are uniformly bounded in $k$ on $D_{\log k}$. by letting $k\rightarrow\infty$ in \eqref{s3-e24} we get
\begin{equation} \label{s3-e25}
\sum^{n-1}_{j=1}\overline U^{*,\psi_0}_{j,H_n}\overline U_{j,H_n}\alpha_{k,J}=-\sideset{}{'}\sum_{\substack{\abs{P}=q,\\1\,\leqslant\, \, j\,,\,t\,\leqslant \, n-1}}
\sigma^{jtP}_{J}
\Bigr[\Bigr(\mu_{j,t}-\frac{i}{\sqrt{2}}\lambda_j\delta_{j,t}\beta\Bigr)+\sqrt{2}i\lambda_j\delta_{j,t}\frac{\partial}{\partial\theta}\Bigr]\alpha_{k,P}
\end{equation}
on $H_n$, for all strictly increasing $J$, $\abs{J}=q$. From this and the explicit formula of $\Box^{(q)}_{b,H_n}$ (see \eqref{s3-e23}),
we conclude that $\Box^{(q)}_{b,H_n}\alpha=0$. The proposition follows.
\end{proof}
Now, we can prove the main result of this section.
In analogy to \eqref{s2-e2} we define the extremal functions $S^{(q)}_{J,H_n}$ on the Heisenberg group along the direction $d\overline z^J$ is defined by
\begin{equation} \label{s2-e271}
S^{(q)}_{J,H_n}(0)=\sup\big\{\abs{\alpha_J(0)}^2;\Box^{(q)}_{b,H_n}\alpha=0, \norm{\alpha}_{\psi_0}=1\big\}\,.
\end{equation}
where $\alpha=\sum'_{\abs{J}=q}\alpha_Jd\overline z^J$.
\begin{thm} \label{s3-t2}
We have
\[
\limsup_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_k(0)\leqslant \sideset{}{^\prime}\sum_{\abs{J}=q}S^{(q)}_{J,H_n}(0)\,.
\]
\end{thm}
\begin{proof}
Fix a strictly increasing $J$, $\abs{J}=q$. We claim that
\begin{equation} \label{s3-e28}
\limsup_{k\rightarrow\infty}k^{-n}S^{(q)}_{k,J}(0)\leqslant S^{(q)}_{J,H_n}(0).
\end{equation}
By definition, there are $\alpha_{k_j}\in H^q_b(X, L^{k_j})$, $\norm{\alpha_{k_j}}=1$, $j=1,2,\ldots$, $k_1<k_2<\cdots$,
such that $\lim_{j\rightarrow\infty}k_j^{-n}\abs{\alpha_{k_j,J}(0)}^2=\limsup_{k\rightarrow\infty}k^{-n}S^{(q)}_{k,J}(0)$, where $\alpha_{k_j,J}$ is the component
of $\alpha_{k_j}$ in the direction of $e_J$. On $D_{\log k_j}$, put
$\beta_{k_j}=k_j^{-\frac{n}{2}}F^*_{k_j}(e^{-k_jR}\alpha_{k_j})\in F^*_{k_j}\Omega^{0,q}(D_{\log k_j})$.
It is easy to see that $\norm{\beta_{k_j}}_{k_jF^*_{k_j}\phi_0,D_{\log k_j}}\leqslant 1$ and $\Box^{(q)}_{s,(k_j)}\beta_{k_j}=0$ on $D_{\log k_j}$. In view of Proposition~\ref{s3-p3}, we see that
there is a subsequence $\set{\beta_{k_{j_s}}}$ of $\set{\beta_{k_j}}$ such that for each $J$, $\beta_{k_{j_s},J}$ converges uniformly with all derivatives on any compact subset of $H_n$ to a smooth function $\beta_J$. Furthermore, if we put
$\beta=\sum'_{\abs{P}=q}\beta_Pd\overline z^P$,
then $\Box^{(q)}_{b,H_n}\beta=0$. From \eqref{s3-e17-1}, we can check that $\norm{\beta}_{\psi_0}\leqslant 1$. Thus,
\begin{equation} \label{s3-e29}
|\beta_J(0)|^2\leqslant \frac{|\beta_J(0)|^2}{\norm{\beta}_{\psi_0}^2}\leqslant S^{(q)}_{J,H_n}(0).
\end{equation}
Note that $\lim_{j\rightarrow\infty}k_j^{-n}\abs{\alpha_{k_j,J}(0)}^2=\lim_{s\rightarrow\infty}|\beta_{k_{j_s},J}(0)|^2=|\beta_J(0)|^2$.
From this and \eqref{s3-e29}, we get \eqref{s3-e28}. From Lemma~\ref{s2-l1}, the theorem follows.
\end{proof}
\section{The Szeg\"{o} kernel function on the Heisenberg group $H_n$}
In this section, we will use the same notations as in section 3.2.
The main goal of this section is to compute $\sum'_{\abs{J}=q}S^{(q)}_{J,H_n}(0)$.
\subsection{The partial Fourier transform}
Let $u(z, \theta)\in\Omega^{0,q}(H_n)$ with $\norm{u}_{\psi_0}=1$
and $\Box^{(q)}_{b,H_n}u=0$. Put
$v(z, \theta)=u(z, \theta)e^{-\frac{\beta}{2}\theta}$
and set $\Phi_0=\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t$.
We have \[\int_{H_n}\!\abs{v(z,\theta)}^2e^{-\Phi_0(z)}dv(z)dv(\theta)=1\,.\]
Put $L^2_{(0,q)}(H_n, \Phi_0)=\{u\in\mathscr{D}^{\,\prime}(H_n;\, \Lambda^{0,q}T^*(H_n));\, \int_{H_n}\abs{u}^2e^{-\Phi_0}dv(z)dv(\theta)<\infty\}$.
Choose $\chi(\theta)\in C^\infty_0(\mathbb R)$ so that $\chi(\theta)=1$ when $\abs{\theta}<1$ and $\chi(\theta)=0$ when $\abs{\theta}>2$ and set $\chi_j(\theta)=\chi(\theta/j)$, $j\in\mathbb{N}$. Let
\begin{equation} \label{s4-e11}
\hat v_j(z, \eta)=\int_{\mathbb R}\!v(z,\theta)\chi_j(\theta)e^{-i\theta\eta}dv(\theta)\in\Omega^{0,q}(H_n),\ j=1,2,\ldots.
\end{equation}
From Parseval's formula, we have
\begin{align*}
&\int_{H_n}\!\abs{\hat v_j(z,\eta)-\hat v_t(z,\eta)}^2e^{-\Phi_0(z)}dv(\eta)dv(z)\\
&=2\pi\int_{H_n}\!\abs{v(z,\theta)}^2\abs{\chi_j(\theta)-\chi_t(\theta)}^2e^{-\Phi_0(z)}dv(\theta)dv(z)\To0,\ j,t\rightarrow\infty.
\end{align*}
Thus, there is $\hat v(z, \eta)\in L^2_{(0,q)}(H_n, \Phi_0)$ such tht $\hat v_j(z,\eta)\rightarrow\hat v(z, \eta)$ in $L^2_{(0,q)}(H_n, \Phi_0)$. We call $\hat v(z, \eta)$ the Fourier transform of $v(z, \theta)$ with respect to $\theta$. Formally,
\begin{equation} \label{s4-e12}
\hat v(z, \eta)=\int_{H_n}\! e^{-i\theta\eta}v(z,\theta)dv(\theta).
\end{equation}
Moreover, we have
\begin{equation} \label{s4-e12-1}
\begin{split}
&\int_{H_n}\!\abs{\hat v(z, \eta)}^2e^{-\Phi_0(z)}dv(z)dv(\eta)=\lim_{j\rightarrow\infty}\int_{H_n}\!\abs{\hat v_j(z, \eta)}^2e^{-\Phi_0(z)}dv(z)dv(\eta)\\
&=2\pi\lim_{j\rightarrow\infty}\int_{H_n}\!\abs{u(z, \theta)e^{-\frac{\beta}{2}\theta}\chi_j(\theta)}^2e^{-\Phi_0(z)}dv(z)dv(\theta)\\
&=2\pi\int_{H_n}\!\abs{u(z, \theta)}^2e^{-\psi_0(z,\theta)}dv(z)dv(\theta)=2\pi<\infty.
\end{split}
\end{equation}
From Fubini's theorem,
$\int_{\mathbb C^{n-1}}\!\abs{\hat v(z,\eta)}^2e^{-\Phi_0(z)}dv(z)<\infty$
for almost all $\eta\in\mathbb R$. More precisely, there is a negligeable set $A_0\subset\mathbb R$
such that $\int_{\mathbb C^{n-1}}\!\abs{\hat v(z,\eta)}^2e^{-\Phi_0(z)}dv(z)<\infty$,
for every $ \eta\notin A_0$.
Let $s\in L^2_{(0,q)}(H_n, \Phi_0)$.
Assume that $\int\!\abs{s(z, \eta)}^2dv(\eta)<\infty$ and $\int\!\abs{s(z, \eta)}dv(\eta)<\infty$ for all $z\in\mathbb C^{n-1}$. Then, from Parseval's formula, we can check that
\begin{equation} \label{s4-e12-2}
\begin{split}
&\iint\!(\hat v(z, \eta)\ |\ s(z, \eta))e^{-\Phi_0(z)}dv(\eta)dv(z)\\
&=\iint\!(u(z, \theta)e^{-\frac{\beta}{2}\theta}\ |\ \int\! e^{i\theta\eta}s(z, \eta)dv(\eta))e^{-\Phi_0(z)}dv(\theta)dv(z).
\end{split}
\end{equation}
We pause and introduce some notations. Fix $\eta\in\mathbb R$, put
\begin{equation} \label{s4-e13}
\Phi_\eta=-\sqrt{2}\eta\sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2+\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t\in C^\infty(\mathbb C^{n-1};\, \mathbb R).
\end{equation}
We take the Hermitian metric $(\ |\ )$ on the bundle $\Lambda^{0,q}T^*(\mathbb C^{n-1})$ of $(0, q)$ forms of $\mathbb C^{n-1}$ so that $\{d\overline z^J; \text{$\abs{J}=q$, $J$ strictly increasing}\}$ is an orthonormal basis. We also let $\Omega^{0,q}(\mathbb C^{n-1})$ denote the space of smooth sections of $\Lambda^{0,q}T^*(\mathbb C^{n-1})$ over $\mathbb C^{n-1}$. Let $\Omega^{0,q}_c(\mathbb C^{n-1})$ be the subspace of $\Omega^{0,q}(\mathbb C^{n-1})$ whose elements have compact support in $\mathbb C^{n-1}$ and let $(\ |\ )_{\Phi_\eta}$ be the inner product on $\Omega^{0,q}_c(\mathbb C^{n-1})$ defined by
$(f\ |\ g)_{\Phi_\eta}=\int_{\mathbb C^{n-1}}\!(f\ |\ g)e^{-\Phi_\eta(z)}dv(z)$,
$f, g\in\Omega^{0,q}_c(\mathbb C^{n-1})$. Let
\begin{equation} \label{s4-e14}
\Box^{(q)}_{\Phi_\eta}=\overline{\partial}^{*,\Phi_\eta}\overline\partial+\overline\partial\,\overline{\partial}^{*,\Phi_\eta}:\Omega^{0,q}(\mathbb C^{n-1})\rightarrow\Omega^{0,q}(\mathbb C^{n-1})
\end{equation}
be the complex Laplacian with respect to $(\ |\ )_{\Phi_\eta}$, where $\overline\partial^{*,\Phi_\eta}$ is the formal adjoint of $\overline\partial$ with respect to $(\ |\ )_{\Phi_\eta}$. We can check that
\begin{equation} \label{s4-e15}
\begin{split}
\Box^{(q)}_{\Phi_\eta}&=\sum^{n-1}_{t=1}\Bigr(-\frac{\partial}{\partial z_t}-\sqrt{2}\lambda_t\overline z_t\eta+\sum^{n-1}_{j=1}\mu_{j,t}\overline z_j\Bigr)\frac{\partial}{\partial\overline z_t}\\
&+\sum^{n-1}_{j,t=1}d\overline z_j\wedge d\overline z_t^{\wedge,*}\Bigr(\mu_{j,t}-\sqrt{2}\lambda_j\eta\delta_{j,t}\Bigr).
\end{split}
\end{equation}
Now, we return to our situation. We identify $\Lambda^{0,q}T^*(\mathbb C^{n-1})$ with $\Lambda^{0,q}T^*(H_n)$. Set
\begin{equation}\label{s4-e161}
\alpha(z, \eta)=\hat v(z, \eta)\exp\Bigl[\Bigl(\frac{i\beta}{2\sqrt{2}}-\frac{1}{\sqrt{2}}\eta\Bigl) \sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2\Bigr]\,.
\end{equation}
We remind that $\hat v(z, \eta)$ is given by \eqref{s4-e12}.
\begin{thm} \label{s4-t1}
For almost all $\eta\in\mathbb R$, we have
$\int_{\mathbb C^{n-1}}\!\abs{\alpha(z, \eta)}^2e^{-\Phi_\eta(z)}dv(z)<\infty$
and
\begin{equation} \label{s4-e17}
\Box^{(q)}_{\Phi_\eta}\alpha(z, \eta)=0
\end{equation}
in the sense of distributions.
Thus $\alpha(z,\eta)\in\Omega^{0,q}(\mathbb C^{n-1})$
for almost all $\eta\in\mathbb R$.
\end{thm}
\begin{proof}
Let $A_0\subset\mathbb R$ be as in the discussion after \eqref{s4-e12-1}.
Thus, for all $\eta\notin A_0$,
\[\int_{\mathbb C^{n-1}}\!\abs{\hat v(z,\eta)}^2e^{-\Phi_0(z)}dv(z)=\int_{\mathbb C^{n-1}}\!\abs{\alpha(z, \eta)}^2e^{-\Phi_\eta(z)}dv(z)<\infty\,.\]
We only need to prove the second statement of the theorem. Let
$f(z)\in\Omega^{0,q}_c(\mathbb C^{n-1})$.
Put
$h(\eta)=\int_{\mathbb C^{n-1}}\!\big(\alpha(z,\eta)\ \big|\ \Box^{(q)}_{\Phi_\eta}f(z)\big)e^{-\Phi_\eta(z)}dv(z)$
if $\eta\notin A_0$, $h(\eta)=0$ if $\eta\in A_0$. We can check that
\begin{equation} \label{s4-e18-1}
\abs{h(\eta)}^2\leqslant \int_{\mathbb C^{n-1}}\!\abs{\alpha(z,\eta)}^2e^{-\Phi_\eta(z)}dv(z)\int_{\mathbb C^{n-1}}\!\big\vert\Box^{(q)}_{\Phi_\eta}f\big\vert^2e^{-\Phi_\eta(z)}dv(z).
\end{equation}
For $R>0$, put $\varphi_R(\eta)=\mathds{1}_{[-R,R]}(\eta)\overline h(\eta)$,
where $\mathds{1}_{[-R,R]}(\eta)=1$ if $-R\leqslant \eta\leqslant R$, $\mathds{1}_{[-R,R]}(\eta)=0$ if $\eta<-R$ or $\eta>R$. From \eqref{s4-e18-1}, we have
\begin{equation} \label{s4-e18-2}
\begin{split}
&\int\!\abs{\varphi_R(\eta)}^2dv(\eta)=\int^R_{-R}\!\abs{h(\eta)}^2dv(\eta)\\
&\leqslant C\iint\!\abs{\alpha(z,\eta)}^2e^{-\Phi_\eta(z)}dv(\eta)dv(z)=C\iint\!\abs{\hat v(z, \eta)}^2e^{-\Phi_0(z)}dv(\eta)dv(z)<\infty\,,
\end{split}
\end{equation}
where $C>0$. Thus, $\varphi_R(\eta)\in L^2(\mathbb R)\cap L^1(\mathbb R)$. Set
$\lambda\abs{z}^2:=\sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2$.
We have
\begin{equation} \label{s4-e19}
\begin{split}
&\int_{\mathbb R}\!h(\eta)\varphi_R(\eta)dv(\eta)=\int^R_{-R}\!\abs{h(\eta)}^2dv(\eta)\\
&=\iint\!(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}f(z))e^{-\Phi_\eta(z)}\varphi_R(\eta)dv(\eta)dv(z)\\
&=\iint\!\Big(\hat v(z,\eta)\ \big|\ e^{-\big(\frac{i\beta}{2\sqrt{2}}-\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2}\Box^{(q)}_{\Phi_\eta}f(z)\overline\varphi_R(\eta)\Big)e^{-\Phi_0(z)}dv(\eta)dv(z)\\
&\stackrel{\eqref{s4-e12-2}}{=}\iint\!\Big(u(z,\theta)\ \big|\ \int\!
e^{i\theta\eta}
e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,\Box^{(q)}_{\Phi_\eta}(f\overline\varphi_R)dv(\eta)\Big)e^{-\psi_0(z, \theta)}dv(\theta)dv(z).
\end{split}
\end{equation}
From Lemma~\ref{s4-l1} below, we know that
\begin{equation} \label{s4-e22}
\begin{split}
&\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,\Box^{(q)}_{\Phi_\eta}(f(z)\overline\varphi_R(\eta))dv(\eta)\\
&=\Box^{(q)}_{b,H_n}\Bigr(\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,f(z)\overline\varphi_R(\eta)dv(\eta)\Bigr).
\end{split}
\end{equation}
Put
\begin{equation} \label{s4-e23}
S(z,\theta)=\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,f(z)\overline\varphi_R(\eta)dv(\eta).
\end{equation}
From \eqref{s4-e22} and \eqref{s4-e23}, \eqref{s4-e19} becomes
\begin{equation} \label{s4-e24}
\int^R_{-R}\!\abs{h(\eta)}^2dv(\eta)
=\iint\!\big(u(z,\theta)\ \big|\ \Box^{(q)}_{b,H_n}S(z,\theta)\big)e^{-\psi_0(z,\theta)}dv(\theta)dv(z).
\end{equation}
Choose $\chi(\theta)\in C^\infty_0(\mathbb R)$ so that $\chi(\theta)=1$ when $\abs{\theta}<1$ and $\chi(\theta)=0$ when $\abs{\theta}>2$. Then,
\begin{equation} \label{s4-e25}
\begin{split}
\int^R_{-R}\!\abs{h(\eta)}^2dv(\eta)
&=\lim_{j\rightarrow\infty}\iint\!(u(z,\theta)\ |\ \chi(\tfrac{\theta}{j})\Box^{(q)}_{b,H_n}S(z,\theta))e^{-\psi_0(z,\theta)}dv(\theta)dv(z)\\
&=\lim_{j\rightarrow\infty}
\Bigr(\iint\!(\Box^{(q)}_{b,H_n}u(z,\theta)\ |\ \chi(\tfrac{\theta}{j})S(z,\theta))e^{-\psi_0(z,\theta)}dv(\theta)dv(z)\\
&+
\iint\!(u(z,\theta)\ |\ [\chi(\tfrac{\theta}{j})\ ,\Box^{(q)}_{b,H_n}]S(z,\theta))e^{-\psi_0(z,\theta)}dv(\theta)dv(z)\Bigr)\\
&=\lim_{j\rightarrow\infty}\iint\!(u(z,\theta)\ |\ [\chi(\tfrac{\theta}{j})\ ,\Box^{(q)}_{b,H_n}]S(z,\theta))e^{-\psi_0(z,\theta)}dv(\theta)dv(z).
\end{split}
\end{equation}
We can check that $[\chi(\frac{\theta}{j})\ ,\Box^{(q)}_{b,H_n}]$ is a first order partial differential operator and all the coefficients of $[\chi(\frac{\theta}{j})\ ,\Box^{(q)}_{b,H_n}]$ converge to $0$ as $j\rightarrow\infty$ uniformly in $\theta$ and locally uniformly in $z$. Moreover, from Parseval's formula, \eqref{s4-e18-2} and \eqref{s4-e23}, we can check that
\begin{align*}
&\sum_{\abs{\alpha}\leqslant 1}\int\!\abs{\partial^\alpha_{x,\theta}S}^2e^{-\psi_0}dv(\theta)dv(z)\\
&\leqslant C\sum_{\abs{\alpha}\leqslant 1}\iint\!(1+\abs{z}+\abs{\eta}+\abs{z}\abs{\eta})^2\abs{\partial^\alpha_xf}^2\abs{\varphi_R(\eta)}^2e^{-\Phi_\eta(z)}dv(z)dv(\eta)\\
&\leqslant \widetilde C\int\!\abs{\varphi_R(\eta)}^2dv(\eta)<\infty,
\end{align*}
where $C>0$, $\widetilde C>0$. Thus,
\[\lim_{j\rightarrow\infty}\iint\!(u(z,\theta)\ |\ [\chi(\tfrac{\theta}{j})\ ,\Box^{(q)}_{b,H_n}]S(z,\theta))e^{-\psi_0(z,\theta)}dv(\theta)dv(z)=0.\]
From this and \eqref{s4-e25}, we conclude that
$\int^R_{-R}\!\abs{h(\eta)}^2dv(\eta)=0$.
Letting $R\rightarrow\infty$, we get $h(\eta)=0$ almost everywhere. We have proved that for a given $f(z)\in\Omega^{0,q}_c(\mathbb C^{n-1})$, $\int_{\mathbb C^{n-1}}\!(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}f(z))e^{-\Phi_\eta(z)}dv(z)=0$ almost everywhere.
Let $W^2_{(0,q)}(\mathbb C^{n-1}):=\{u\in\mathscr D'(\mathbb C^{n-1};\, \Lambda^{0,q}T^*(\mathbb C^{n-1}));
\sum_{\abs{\alpha}\leqslant 2}\int_{\mathbb C^{n-1}}\!\abs{\partial^\alpha_xu}^2dv(z)<\infty\}$.
Since $W^2_{(0,q)}(\mathbb C^{n-1})$
is separable and $\Omega^{0,q}_c(\mathbb C^{n-1})$ is dense in $W^2_{(0,q)}(\mathbb C^{n-1})$,
we can find $f_j\in\Omega^{0,q}_c(\mathbb C^{n-1})$, $j=1,2,\ldots$, such that
$\set{f_1,f_2,\ldots}$
is a dense subset of $W^2_{(0,q)}(\mathbb C^{n-1})$. Moreover, we can take $\set{f_1,f_2,\ldots}$ so
that for all $g\in\Omega^{0,q}_c(\mathbb C^{n-1})$ with
${\rm supp\,}g\subset B_r:=\set{z\in\mathbb C^{n-1};\, \abs{z}<r}$, $r>0$,
we can find $f_{j_1}, f_{j_2},\ldots$, ${\rm supp\,}f_{j_t}\subset B_r$, $t=1,2,\ldots$, such that
$f_{j_t}\rightarrow g$ for $t\rightarrow\infty$ in $W^2_{(0,q)}(\mathbb C^{n-1})$.
Now, for each $j$, we can repeat the method above and find a measurable set $A_j\supset A_0$, $\abs{A_j}=0$ ($A_0$ is as in the beginning of the proof), such that
$(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}f_j(z))_{\Phi_\eta}=0$ for all $\eta\notin A_j$.
Put $A=\bigcup_jA_j$. Then, $\abs{A}=0$ and for all $\eta\notin A$,
$(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}f_j(z))_{\Phi_\eta}=0$ for all $j$.
Let $g\in\Omega^{0,q}_c(\mathbb C^{n-1})$ with
${\rm supp\,}g\subset B_r$. From the discussion above,
we can find $f_{j_1}, f_{j_2},\ldots$, ${\rm supp\,}f_{j_t}\subset B_r$, $t=1,2,\ldots$, such that
$f_{j_t}\rightarrow g$ in $W^2_{(0,q)}(\mathbb C^{n-1})$, $t\rightarrow\infty$.
Then, for $\eta\notin A$,
\begin{align*}
(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}g)_{\Phi_\eta}&=(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}(g-f_{j_t})))_{\Phi_\eta}
+(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}f_{j_t})_{\Phi_\eta}\\
&=(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}(g-f_{j_t})))_{\Phi_\eta}.
\end{align*}
Now,
\begin{equation}
\begin{split}
\abs{(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}(g-f_{j_t})))_{\Phi_\eta}}&=\abs{\int_{B_r}\!(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}(g-f_{j_t}))e^{-\Phi_\eta(z)}dv(z)}\\
&\leqslant C\sum_{\abs{\alpha}
\leqslant 2}\int\!\abs{\partial^\alpha_x(g-f_{j_t})}^2dv(z)\rightarrow 0, \quad t\rightarrow\infty.
\end{split}
\end{equation}
Thus, for $\eta\notin A$,
$(\alpha(z,\eta)\ |\ \Box^{(q)}_{\Phi_\eta}g)_{\Phi_\eta}=0$
for all $g\in\Omega^{0,q}_c(\mathbb C^{n-1})$. The theorem follows.
\end{proof}
\begin{lem} \label{s4-l1}
Let $f\in\Omega^{0,q}_c(\mathbb C^{n-1})$. Let $\varphi(\eta)\in L^2(\mathbb R)$ with compact support. Then, we have
\begin{equation*}
\begin{split}
&\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,
\Box^{(q)}_{\Phi_\eta}f(z)\varphi(\eta)dv(\eta)\\
&=\Box^{(q)}_{b,H_n}\Bigr(\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,f(z)\varphi(\eta)dv(\eta)\Bigr),
\end{split}
\end{equation*}
where $\lambda\abs{z}^2=\sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2$.
\end{lem}
\begin{proof}
For any $g\in\Omega^{0,q}_c(\mathbb C^{n-1})$, we can check that
\begin{equation} \label{s4-e26}
\begin{split}
&\overline U_{t,H_n}\Bigr(\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,g(z)\varphi(\eta)dv(\eta)\Bigr)\\
&=\Bigr(\frac{\partial}{\partial\overline z_t}+\frac{1}{\sqrt{2}}i\lambda_tz_t\frac{\partial}{\partial\theta}\Bigr)
\Bigr(\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,g(z)\varphi(\eta)dv(\eta)\Bigr)\\
&=\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,\frac{\partial g}{\partial \overline z_t}\varphi(\eta)dv(\eta),
\end{split}
\end{equation}
where $t=1,\ldots,n-1$,
\begin{equation} \label{s4-e27}
\begin{split}
\overline U^{*,\psi_0}_{t,H_n}&\Bigr(\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,g(z)\varphi(\eta)dv(\eta)\Bigr)\\
=&\Bigr(-\frac{\partial}{\partial z_t}+\frac{1}{\sqrt{2}}i\lambda_t\overline z_t\frac{\partial}{\partial\theta}+\sum^{n-1}_{j=1}\mu_{j,t}\overline z_j-\frac{1}{\sqrt{2}}i\lambda_t\overline z_t\beta\Bigr)\\
&\Bigr(\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,g(z)\varphi(\eta)dv(\eta)\Bigr)\\
=&\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,(-\frac{\partial g}{\partial z_t}+\sum^{n-1}_{j=1}\mu_{j,t}\overline z_jg-\sqrt{2}\lambda_t\overline z_t\eta g)\varphi(\eta)dv(\eta),
\end{split}
\end{equation}
where $t=1,\ldots,n-1$, and
\begin{align} \label{s4-e28}
&\Bigr(\mu_{j,t}-\frac{1}{\sqrt{2}}i\lambda_j\delta_{j,t}\beta+\sqrt{2}i\lambda_j\delta_{j,t}\frac{\partial}{\partial\theta}\Bigr)\circ\Bigr(\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,g\varphi(\eta)dv(\eta)\Bigr)\nonumber\\
&=\int\! e^{i\theta\eta}e^{-\big(\frac{i\beta}{2\sqrt{2}}+\frac{1}{\sqrt{2}}\eta\big)\lambda\abs{z}^2
+\frac{\beta}{2}\theta}\,(\mu_{j,t}g-\sqrt{2}\eta\lambda_j\delta_{j,t}g)\varphi(\eta)dv(\eta),
\end{align}
where $j,t=1,\ldots,n-1$. From \eqref{s4-e26}, \eqref{s4-e27}, \eqref{s4-e28} and the explicit formulas for
$\Box^{(q)}_{b,H_n}$ and $\Box^{(q)}_{\Phi_\eta}$ (see \eqref{s3-e23} and \eqref{s4-e15}),
the lemma follows.
\end{proof}
\subsection{Estimates for the extremal function on the Heisenberg group
We will use the same notations as before. For $\eta\in\mathbb R$, put
\begin{equation*}
L^2_{(0,q)}(\mathbb C^{n-1}, \Phi_\eta)
=\Big\{u\in\mathscr{D}^{\,\prime}(\mathbb C^{n-1};\, \Lambda^{0,q}T^*(\mathbb C^{n-1}));\, \int_{\mathbb C^{n-1}}\!\abs{u}^2e^{-\Phi_\eta(z)}dv(z)<\infty\Big\}.
\end{equation*}
Let $B^{(q)}_{\Phi_\eta}:L^2_{(0,q)}(\mathbb C^{n-1}, \Phi_\eta)\rightarrow\Ker\Box^{(q)}_{\Phi_\eta}$
be the Bergman projection, i.e.~the orthogonal projection onto $\Ker\Box^{(q)}_{\Phi_\eta}$ with respect to $(\ |\ )_{\Phi_\eta}$. Let
$(B^{(q)}_{\Phi_\eta})^*$ be the adjoint of $B^{(q)}_{\Phi_\eta}$ with respect to $(\ |\ )_{\Phi_\eta}$.
We have
$B^{(q)}_{\Phi_\eta}=(B^{(q)}_{\Phi_\eta})^*=(B^{(q)}_{\Phi_\eta})^2$.
Let
\begin{equation*}
\begin{split}
B^{(q)}_{\Phi_\eta}(z,w)\in C^\infty(\mathbb C^{n-1}\times\mathbb C^{n-1};\, \mathscr L(\Lambda^{0,q}T^*_w(\mathbb C^{n-1}),\Lambda^{0,q}T^*_z(\mathbb C^{n-1})))\\
(B^{(q)}_{\Phi_\eta}u)(z)=\int_{\mathbb C^{n-1}}\!B^{(q)}_{\Phi_\eta}(z,w)u(w)e^{-\Phi_\eta(w)}dv(w)\,,\quad
u\in L^2_{(0,q)}(\mathbb C^{n-1}, \Phi_\eta)
\end{split}
\end{equation*}
be the distribution kernel of $B^{(q)}_{\Phi_\eta}$ with respect to $(\ |\ )_{\Phi_\eta}$.
We take the Hermitian metrix $(\ |\ )$ on $\Lambda^{1,0}T_z(\mathbb C^{n-1})$, $z\in\mathbb C^{n-1}$, so that
$\frac{\partial}{\partial z_j}$, $j=1,\ldots,n-1$, is an orthonormal basis.
Let
\begin{equation} \label{s4-e29-1}
M_{\Phi_\eta}:\Lambda^{1,0}T_z(\mathbb C^{n-1})\rightarrow\Lambda^{1,0}T_z(\mathbb C^{n-1})\,,\quad z\in\mathbb C^{n-1}
\end{equation}
be the linear map defined by
$(M_{\Phi_\eta} U\ |\ V)=\langle\partial\overline\partial\Phi_\eta, U\wedge\overline V\rangle$,
$U, V\in\Lambda^{1,0}T_z(\mathbb C^{n-1})$. Put
\begin{equation} \label{s4-e30}
\begin{split}
\mathbb R_q=&\{\eta\in\mathbb R;\, \mbox{$M_{\Phi_\eta}$ has exactly $q$ negative eigenvalues}\\
&\quad\mbox{and $n-1-q$ positive eigenvalues}\}.
\end{split}
\end{equation}
The following is essentially well-known (see Berman~\cite{Be04}).
\begin{thm} \label{s4-t2}
If $\eta\notin \mathbb R_q$, then $B^{(q)}_{\Phi_\eta}(z,z)=0$,
for all $z\in\mathbb C^{n-1}$.
If $\eta\in \mathbb R_q$, let $Z_1(\eta),\ldots,Z_{n-1}(\eta)$
be an orthonormal frame of $\Lambda^{1,0}T_z(\mathbb C^{n-1})$, for which $M_{\Phi_\eta}$ is diagonal. We assume that
$M_{\Phi_\eta}Z_j(\eta)=\nu_j(\eta)Z_j(\eta)$ for $j=1,\ldots,n-1$,
with $\nu_j(\eta)<0$ for $j=1,\ldots,q$ and $\nu_j(\eta)>0$ for $j=q+1,\ldots,n-1$.
Let $(T_j(\eta))_{j=1,\ldots,\,n-1}$,
denote the basis of $\Lambda^{0,1}T^*_z(\mathbb C^{n-1})$, which is dual to $(\overline Z_j(\eta))_{j=1,\ldots,\,n-1}$. Then,
\begin{equation} \label{s4-e32}
B^{(q)}_{\Phi_\eta}(z,z)=e^{\Phi_\eta(z)}(2\pi)^{-n+1}\abs{\nu_1(\eta)}\cdots\abs{\nu_{n-1}(\eta)}\prod^q_{j=1}T_j(\eta)\wedge (T_j(\eta)\wedge)^*.
\end{equation}
In particular,
\begin{equation} \label{s4-e33}
\begin{split}
{\rm Tr\,}B^{(q)}_{\Phi_\eta}(z,z):&=\sideset{}{'}\sum_{\abs{J}=q}(B^{(q)}_{\Phi_\eta}(z,z)d\overline z^J\ |\ d\overline z^J)\\
&=e^{\Phi_\eta(z)}(2\pi)^{-n+1}\abs{\nu_1(\eta)}\cdots\abs{\nu_{n-1}(\eta)}\mathds{1}_{\mathbb R_q}(\eta)\\
&=e^{\Phi_\eta(z)}(2\pi)^{-n+1}\abs{\det M_{\Phi_\eta}}\mathds{1}_{\mathbb R_q}(\eta),
\end{split}
\end{equation}
where $\mathds{1}_{\mathbb R_q}(\eta)$ is the characteristic function of $\mathbb R_q$.
\end{thm}
\begin{rem} \label{s4-r1}
We recall that
$\Phi_\eta=-\sqrt{2}\eta\sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2+\sum^{n-1}_{j,t=1}\mu_{j,t}\overline z_jz_t$.
Since the Levi form is non-degenerate and $Y(q)$ holds, we conclude that $\mathbb R_q\subset[-R, R]$ for some $R>0$.
\end{rem}
We return to our situation.
Let $u(z, \theta)\in\Omega^{0,q}(H_n)$, $\norm{u}_{\psi_0}=1$, $\Box^{(q)}_{b,H_n}u=0$. As before,
let $\hat v(z, \eta)$ be the Fourier transform of $u(z, \theta)e^{-\frac{\beta}{2}\theta}$ with respect to $\theta$. From Theorem~\ref{s4-t1}, we know that for $\alpha$ defined in \eqref{s4-e161} we have
\begin{equation} \label{s4-e34}
\begin{split}
\alpha(z,\eta
\in\Ker\Box^{(q)}_{\Phi_\eta}\cap L^2_{(0,q)}(\mathbb C^{n-1}, \Phi_\eta)\cap \Omega^{0,q}(\mathbb C^{n-1})
\end{split}
\end{equation}
for almost all $\eta\in\mathbb R$. Thus, $\alpha(z, \eta)=\int_{\mathbb C^{n-1}}\!B^{(q)}_{\Phi_\eta}(z, w)\alpha(w,\eta)e^{-\Phi_\eta(w)}dv(w)$ for almost all $\eta\in\mathbb R$. Put
$\hat v(z, \eta)=\sum'_{\abs{J}=q}\hat v_J(z, \eta)d\overline z^J$.
We have the following
\begin{lem} \label{s4-l2}
For $J$ strictly increasing, $\abs{J}=q$, $z\in\mathbb C^{n-1}$, we have that
\begin{equation} \label{s4-e37}
\abs{\hat v_J(z, \eta)}^2\leqslant e^{\sqrt{2}\eta\sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2}(B^{(q)}_{\Phi_\eta}(z, z)d\overline z^J\ |\ d\overline z^J)\int_{\mathbb C^{n-1}}\!\abs{\hat v(w, \eta)}^2e^{-\Phi_0(w)}dv(w)
\end{equation}
for almost all $\eta\in\mathbb R$.
\end{lem}
\begin{proof}
Let $\varphi\in C^\infty_0(\mathbb C^{n-1})$ such that $\int_{\mathbb C^{n-1}}\!\varphi(z)dv(z)=1$, $\varphi\geqslant0$, $\varphi(z)=0$ if $\abs{z}>1$. Put $f_j(z)=j^{2n-2}\varphi(jz)e^{\Phi_\eta(z)}$, $j=1,2,\ldots$.
Then, $\int_{\mathbb C^{n-1}}\!f_j(z)e^{-\Phi_\eta(z)}dv(z)=1$ and $f_j(z)\rightarrow \delta_0$ in the sense of distributions with
respect to $(\ |\ )_{\Phi_\eta}$, that is,
$(h(z)\ |\ f_j(z))_{\Phi_\eta}\rightarrow h(0)$, $j\rightarrow\infty$,
for all $h\in C^\infty(\mathbb C^{n-1})$. Thus, for almost all $\eta\in\mathbb R$,
\begin{equation} \label{s4-e38}
\begin{split}
&\abs{e^{-\frac{\eta}{\sqrt{2}}\sum^{n-1}_{j=1}\lambda_j\abs{z_{0,j}}^2}\hat v_J(z_0, \eta)}=\abs{\alpha_J(z_0, \eta)}=\lim_{j\rightarrow\infty}\abs{(\alpha(z, \eta)\ |\ f_j(z-z_0)d\overline z^J)_{\Phi_\eta}}\\
&=\lim_{j\rightarrow\infty}\abs{(B^{(q)}_{\Phi_\eta}\alpha\ |\ f_j(z-z_0)d\overline z^J)_{\Phi_\eta}}=\lim_{j\rightarrow\infty}\abs{(\alpha\ |\ B^{(q)}_{\Phi_\eta}(f_j(z-z_0)d\overline z^J))_{\Phi_\eta}},
\end{split}
\end{equation}
for all $z_0=(z_{0,1},z_{0,2},\ldots,z_{0,n-1})\in\mathbb C^{n-1}$. Now,
\begin{equation} \label{s4-e39}
\abs{(\alpha(z, \eta)\ |\ B^{(q)}_{\Phi_\eta}(f_j(z-z_0)d\overline z^J))_{\Phi_\eta}}^2\leqslant \norm{\alpha}^2_{\Phi_\eta}\norm{B^{(q)}_{\Phi_\eta}(f_j(z-z_0)d\overline z^J)}^2_{\Phi_\eta}
\end{equation}
and
\begin{equation} \label{s4-e40-0}
\begin{split}
\norm{\alpha}^2_{\Phi_\eta}&\norm{B^{(q)}_{\Phi_\eta}(f_j(z-z_0)d\overline z^J)}^2_{\Phi_\eta}
=\norm{\hat v}^2_{\Phi_0}\norm{B^{(q)}_{\Phi_\eta}(f_j(z-z_0)d\overline z^J)}^2_{\Phi_\eta}\\
&=\norm{\hat v}^2_{\Phi_0}(B^{(q)}_{\Phi_\eta}(f_j(z-z_0)d\overline z^J)\ |\ B^{(q)}_{\Phi_\eta}(f_j(z-z_0)d\overline z^J))_{\Phi_\eta}\\
&\longrightarrow\norm{\hat v}^2_{\Phi_0}(B^{(q)}_{\Phi_\eta}(z_0,z_0)d\overline z^J\ |\ d\overline z^J),\ \ j\rightarrow\infty.
\end{split}
\end{equation}
From \eqref{s4-e38}, \eqref{s4-e39} and \eqref{s4-e40-0}, we get for all $z_0\in\mathbb C^{n-1}$,
\[\abs{e^{-\frac{\eta}{\sqrt{2}}\sum^{n-1}_{j=1}\lambda_j\abs{z_{0,j}}^2}\hat v_J(z_0, \eta)}^2\leqslant \norm{\hat v}^2_{\Phi_0}(B^{(q)}_{\Phi_\eta}(z_0,z_0)d\overline z^J\ |\ d\overline z^J)\]
for almost all $\eta\in\mathbb R$. The lemma follows.
\end{proof}
Put $u(z, \theta)=\sum'_{\abs{J}=q}u_J(z, \theta)d\overline z^J$.
We have the following
\begin{prop} \label{s4-p1}
For $\abs{J}=q$, $J$ is strictly increasing, we have
\begin{equation} \label{s4-e40}
\abs{u_J(0, 0)}^2\leqslant \frac{1}{2\pi}\int_{\mathbb R}\!(B_{\Phi_\eta}(0,0)d\overline z^J\ |\ d\overline z^J)dv(\eta).
\end{equation}
\end{prop}
\begin{proof}
Let $\chi\in C^\infty_0(\mathbb R)$, $\int_\mathbb R\!\chi dv(\theta)=1$, $\chi\geqslant0$ and $\chi_\varepsilon\in C^\infty_0(\mathbb R)$, $\chi_\varepsilon(\theta)=\frac{1}{\varepsilon}\chi(\frac{\theta}{\varepsilon})$. Then,
$\chi_\varepsilon\rightarrow\delta_0$, $\varepsilon\To0^+$
in the sense of distributions.
Let $\hat\chi_\varepsilon$
be the Fourier transform of $\chi_\varepsilon$. We can check that
$\abs{\hat\chi_\varepsilon(\eta)}\leqslant 1$ for all $\eta\in\mathbb R$, $\hat\chi_\varepsilon(\eta)=\hat\chi(\varepsilon\eta)$ and
$\lim_{\varepsilon\To0}\hat\chi_\varepsilon(\eta)=\lim_{\varepsilon\To0}\hat\chi(\varepsilon\eta)=\hat\chi(0)=1$.
Let $\varphi(z)$ be as in the proof of Lemma~\ref{s4-l2}. Put
$g_j(z)=j^{2n-2}\varphi(jz)e^{\Phi_0(z)}$, $j=1,2,\ldots$.
Then, for $J$ is strictly increasing, $\abs{J}=q$, we have
\begin{equation} \label{s4-e41}
u_J(0,0)=\lim_{j\rightarrow\infty}\lim_{\varepsilon\To0^+}\int_{H_n}\!(u(z,\theta)e^{-\frac{\beta}{2}\theta}\ |\ \chi_\varepsilon(\theta)g_j(z)d\overline z^J)e^{-\Phi_0(z)}dv(z)dv(\theta).
\end{equation}
From \eqref{s4-e12-2}, we see that
\begin{equation} \label{s4-e42}
\begin{split}
&\iint\!\big(u(z,\theta)e^{-\frac{\beta}{2}\theta}\ \big|\ \chi_\varepsilon(\theta)g_j(z)d\overline z^J\big)e^{-\Phi_0(z)}dv(z)dv(\theta)\\
&=\frac{1}{2\pi}\iint\!\big(\hat v(z, \eta)\ \big|\ \hat\chi_\varepsilon(\eta)g_j(z)d\overline z^J\big)e^{-\Phi_0(z)}dv(\eta)dv(z).
\end{split}
\end{equation}
From \eqref{s4-e37} and Theorem~\ref{s4-t2}, we see that
\begin{equation*}
\abs{\hat v_J(z, \eta)}^2
\leqslant e^{\sqrt{2}\eta\sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2}(B_{\Phi_\eta}(z, z)d\overline z^J\ |\ d\overline z^J)\mathds{1}_{\mathbb R_q}(\eta)\int_{\mathbb C^{n-1}}\abs{\hat v(w, \eta)}^2e^{-\Phi_0(w)}dv(w),
\end{equation*}
for almost all $\eta\in\mathbb R$.
Thus, for fixed $j$,
$\iint\!\abs{(\hat v\ |\ g_jd\overline z^J)}e^{-\Phi_0(z)}dv(\eta)dv(z)<\infty$.
From this and Lebesque dominated convergence theorem, we conclude that
\begin{equation} \label{s4-e42-1}
\begin{split}
\lim_{\varepsilon\To0^+}&\iint\!\big(\hat v(z, \eta)\ \big|\ \hat\chi_\varepsilon(\eta)g_j(z)d\overline z^J\big)e^{-\Phi_0(z)}dv(\eta)dv(z)\\
&=\iint\!\big(\hat v(z, \eta)\ \big|\ g_j(z)d\overline z^J\big)e^{-\Phi_0(z)}dv(\eta)dv(z).
\end{split}
\end{equation}
From \eqref{s4-e42} and \eqref{s4-e42-1}, \eqref{s4-e41} becomes
\begin{equation} \label{s4-e42-2}
u_J(0, 0)=\lim_{j\rightarrow\infty}\frac{1}{2\pi}\iint\!(\hat v(z, \eta)\ |\ g_j(z)d\overline z^J)e^{-\Phi_0(z)}dv(\eta)dv(z).
\end{equation}
Put $f_j(\eta)=\frac{1}{2\pi}\int\!(\hat v(z, \eta)\ |\ g_j(z)d\overline z^J)e^{-\Phi_0(z)}dv(z)$.
Since $\hat v(z, \eta)\in\Omega^{0,q}(\mathbb C^{n-1})$ for almost all $\eta$, we have
$\lim_{j\rightarrow\infty}f_j(\eta)=\frac{1}{2\pi}\hat v_J(0, \eta)$
almost everywhere. Now,
\begin{equation} \label{s4-e42-3}
\begin{split}
\abs{f_j(\eta)}&=\frac{1}{2\pi}\abs{\int\!\big(\hat v(z, \eta)\ \big|\ g_j(z)d\overline z^J\big)e^{-\Phi_0(z)}dv(z)}\\
&=\frac{1}{2\pi}\abs{\int_{\abs{z}\leqslant \frac{1}{j}}\!
\big(\hat v(z, \eta)\ \big|\ j^{2n-2}\varphi(jz)d\overline z^J\big)dv(z)}\\
&\leqslant \frac{1}{2\pi}\Bigr(\int_{\abs{z}\leqslant \frac{1}{j}}\!
\abs{\hat v(z, \eta)}^2e^{-\Phi_0(z)}j^{2n-2}dv(z)
\Bigr)^{\frac{1}{2}}
\Bigr(\int_{\abs{z}\leqslant \frac{1}{j}}\!\abs{\varphi(jz)}^2e^{\Phi_0(z)}j^{2n-2}dv(z)\Bigr)^{\frac{1}{2}}\\
&\leqslant C_1\Bigr(\int_{\abs{z}\leqslant 1}\!
\abs{\hat v(\tfrac{z}{j}, \eta)}^2e^{-\Phi_0(z/j)}dv(z)\Bigr)^{\frac{1}{2}}\\
&\leqslant C_2\Bigr(\int_{\abs{z}\leqslant 1}\!e^{\sqrt{2}\eta\sum^{n-1}_{t=1}\lambda_t\abs{\frac{z_t}{j}}^2}\abs{{\rm Tr\,}B_{\Phi_\eta}(\tfrac{z}{j},\tfrac{z}{j})}\mathds{1}_{\mathbb R_q}(\eta)dv(z)\Bigr)^{\frac{1}{2}}\\
&\times\Bigr(\int_{\mathbb C^{n-1}}\abs{\hat v(w, \eta)}^2e^{-\Phi_0(w)}dv(w)\Bigr)^{\frac{1}{2}}\ \ (\mbox{here we used \eqref{s4-e37} and Theorem~\ref{s4-t2}})\\
&\leqslant C_3\Bigr(\int_{\mathbb C^{n-1}}\abs{\hat v(w, \eta)}^2e^{-\Phi_0(w)}dv(w)\Bigr)^{\frac{1}{2}}\mathds{1}_{\mathbb R_q}(\eta),
\end{split}
\end{equation}
where $C_1, C_2, C_3$ are positive constants. From this and Lebesque dominated convergence theorem, we conclude that
\[
u_J(0, 0)=\lim_{j\rightarrow\infty}\int f_j(\eta)dv(\eta)=\int\lim_{j\rightarrow\infty}f_j(\eta)dv(\eta)
=\frac{1}{2\pi}\int\hat v_J(0, \eta)dv(\eta)\,.
\]
Thus,
\begin{equation} \label{s4-e48}
\abs{u_J(0,0)}\leqslant \frac{1}{2\pi}\int\!\abs{\hat v_J(0,\eta)}dv(\eta).
\end{equation}
Since $\iint\!\abs{\hat v(w, \eta)}^2e^{-\Phi_0(w)}dv(\eta)dv(w)=2\pi$ we obtain from Lemma~\ref{s4-l2} that
\begin{equation} \label{s4-e49}
\begin{split}
\abs{\int\!\abs{\hat v_J(0,\eta)}dv(\eta)}^2&\leqslant 2\pi\int\!\frac{\abs{\hat v_J(0,\eta)}^2}{\int\!\abs{\hat v(w, \eta)}^2e^{-\Phi_0(w)}dv(w))}dv(\eta)\\
&\leqslant 2\pi\int\!(B^{(q)}_{\Phi_\eta}(0,0)d\overline z^J\ |\ d\overline z^J)dv(\eta).
\end{split}
\end{equation}
From \eqref{s4-e49} and \eqref{s4-e48}, the proposition follows.
\end{proof}
From Proposition~\ref{s4-p1}, we know that for all
$u(z, \theta)=\sum'_{\abs{J}=q}u_J(z, \theta)d\overline z^J\in\Omega^{0,q}(H_n)$,
$\norm{u}_{\psi_0}=1$, $\Box^{(q)}_{b,H_n}u=0$, we have
\[\abs{u_J(0,0)}^2\leqslant \frac{1}{2\pi}\int\!(B^{(q)}_{\Phi_\eta}(0,0)d\overline z^J\ |\ d\overline z^J)dv(\eta).\]
Thus, $S^{(q)}_{J,H_n}(0)\leqslant \frac{1}{2\pi}\int\!(B^{(q)}_{\Phi_\eta}(0,0)d\overline z^J\ |\ d\overline z^J)dv(\eta)$
for all strictly increasing $J$, $\abs{J}=q$.
Hence $\sum'_{\abs{J}=q}S^{(q)}_{J,H_n}(0)\leqslant \frac{1}{2\pi}\int\!{\rm Tr\,}B^{(q)}_{\Phi_\eta}(0,0)dv(\eta)$.
From this and Theorem~\ref{s4-t2}, we get
\begin{thm} \label{s4-t3}
We have
$\sum'_{\abs{J}=q}S^{(q)}_{J,H_n}(0)\leqslant (2\pi)^{-n}\int_{\mathbb R_q}\!\abs{\det M_{\Phi_\eta}}dv(\eta)$,
where $M_{\Phi_\eta}$ is as in \eqref{s4-e29-1} and $\mathbb R_q$ is as in \eqref{s4-e30}.
\end{thm}
\subsection{The Szeg\"{o} kernel function on the Heisenberg group}
In the rest of this section, we calculate the extremal function for the Heisenberg group (see Theorem \ref{s4-t4}).
For $\eta\in\mathbb R$, we can find $z_j(\eta)=\sum^{n-1}_{t=1}a_{j,t}(\eta)z_t$, $j=1,\ldots,n-1$,
such that $\Phi_\eta=\sum^{n-1}_{j=1}\nu_j(\eta)\abs{z_j(\eta)}^2$,
where $\nu_1(\eta),\ldots,\nu_{n-1}(\eta)$, are the eigenvalues of $M_{\Phi_\eta}$, $a_{j,t}(\eta)\in\mathbb C$, $j, t=1,\ldots,n-1$. If $\eta\in \mathbb R_q$, we assume that
$\nu_1(\eta)<0,\ldots,\nu_q(\eta)<0,\nu_{q+1}(\eta)>0,\ldots,\nu_{n-1}(\eta)>0$.
The following is essentially well-known (see~\cite{Be04}).
\begin{prop} \label{s4-p2}
Put
\begin{equation} \label{s4-e53}
\alpha(z, \eta)=C_0\abs{\det M_{\Phi_\eta}}\mathds{1}_{\mathbb R_q}(\eta)e^{\nu_1(\eta)\abs{z_1(\eta)}^2+\cdots+\nu_q(\eta)\abs{z_q(\eta)}^2}
d\overline{z_1(\eta)}\wedge\cdots\wedge d\overline{z_q(\eta)},
\end{equation}
where
$C_0=(2\pi)^{1-\frac{n}{2}}\Bigr(\int_{\mathbb R_q}\abs{\det M_{\Phi_\eta}}dv(\eta)\Bigr)^{-\frac{1}{2}}$.
Then, $\Box^{(q)}_{\Phi_\eta}\alpha(z, \eta)=0$
and
\begin{equation} \label{s4-e55}
\int_{\mathbb C^{n-1}}(1+\abs{z}^2)^{m'}\abs{\partial^m_x\alpha(z, \eta)}^2e^{-\Phi_\eta(z)}dv(z)<\infty
\end{equation}
and the value $\int_{\mathbb C^{n-1}}(1+\abs{z}^2)^{m'}\abs{\partial^m_x\alpha(z, \eta)}^2e^{-\Phi_\eta(z)}dv(z)$ can be bounded by some positive continuous function of the eigenvalues of $M_{\Phi_\eta}$, $\eta\in \mathbb R_q$,
for all $m\in\mathbb N_0^{2n-2}$, $m'\in\mathbb N_0$. Moreover, we have
\begin{equation} \label{s4-e56}
\int_{\mathbb C^{n-1}}\abs{\alpha(z, \eta)}^2e^{-\Phi_\eta(z)}dv(z)=2\pi\Bigr(\int_{\mathbb R_q}\abs{\det M_{\Phi_\eta}}dv(\eta)\Bigr)^{-1}\abs{\det M_{\Phi_\eta}}\mathds{1}_{\mathbb R_q}(\eta).
\end{equation}
\end{prop}
Set
\begin{equation} \label{s4-e57}
u(z,\theta)=\frac{1}{2\pi}\int e^{i\theta\eta+\frac{\beta\theta}{2}+\big(\frac{\eta}{\sqrt{2}}-
\frac{i\beta}{2\sqrt{2}}\big)\lambda\abs{z}^2}\alpha(z, \eta)\,dv(\eta)\in\Omega^{0,q}(H_n),
\end{equation}
where $\alpha(z, \eta)$ is as in \eqref{s4-e53} and $\lambda\abs{z}^2:=\sum^{n-1}_{j=1}\lambda_j\abs{z_j}^2$.
\begin{prop} \label{s4-p3}
We have that
\begin{equation} \label{s4-e58}
\Box^{(q)}_{b,H_n}u=0,
\end{equation}
\begin{equation} \label{s4-e59}
\norm{u}_{\psi_0}=1
\end{equation}
and
\begin{equation} \label{s4-e60}
\abs{u(0,0)}^2=(2\pi)^{-n}\int_{\mathbb R_q}\abs{\det M_{\Phi_\eta}}dv(\eta).
\end{equation}
Moreover, we have
\begin{equation} \label{s4-e61}
\int_{H_n}\abs{\partial^m_x\partial^{m'}_\theta u(z, \theta)}^2e^{-\psi_0(z, \theta)}dv(z)dv(\theta)<\infty
\end{equation}
and the value $\int_{H_n}\abs{\partial^m_x\partial^{m'}_\theta u(z, \theta)}^2e^{-\psi_0(z, \theta)}dv(z)dv(\theta)$ can be bounded by some positive continuous function of the eigenvalues of $M_{\Phi_\eta}$, $\eta\in \mathbb R_q$, $\beta$ and $\lambda_j$, $j=1,\ldots,n-1$, for all $m\in\mathbb N_0^{2n-2}$, $m'\in\mathbb N_0$.
\end{prop}
\begin{proof}
In view of the proof of Lemma~\ref{s4-l1}, we see that
\[\Box^{(q)}_{b,H_n}u(z, \theta)=\frac{1}{2\pi}\int e^{i\theta\eta+\frac{\beta\theta}{2}+\big(\frac{\eta}{\sqrt{2}}-
\frac{i\beta}{2\sqrt{2}}\big)}(\Box^{(q)}_{\Phi_\eta}\alpha)(z, \eta)dv(\eta)=0.\]
We get \eqref{s4-e58}. Now,
\begin{equation} \label{s4-e62}
\begin{split}
&\int\abs{u(z, \theta)}^2e^{-\psi_0(z, \theta)}dv(z)dv(\theta)\\
&\quad=\frac{1}{(2\pi)^2}\int\abs{\int e^{i\theta\eta+\frac{\beta\theta}{2}+\big(\frac{\eta}{\sqrt{2}}-
\frac{i\beta}{2\sqrt{2}}\big)}\alpha(z, \eta)dv(\eta)}^2e^{-\beta\theta-\Phi_0(\eta)}dv(\theta)dv(z)\\
&\quad=\frac{1}{(2\pi)^2}\int\abs{\int e^{i\theta\eta+\frac{\eta}{\sqrt{2}}\lambda\abs{z}^2}\alpha(z, \eta)dv(\eta)}^2dv(\theta)e^{-\Phi_0(z)}dv(z).
\end{split}
\end{equation}
From Parseval's formula, we have
\begin{equation} \label{s4-e63}
\begin{split}
&\frac{1}{(2\pi)^2}\int\abs{\int e^{i\theta\eta+\frac{\eta}{\sqrt{2}}\lambda\abs{z}^2}\alpha(z, \eta)dv(\eta)}^2dv(\theta)=\frac{1}{2\pi}\int e^{\sqrt{2}\eta\lambda\abs{z}^2}\abs{\alpha(z, \eta)}^2dv(\eta).
\end{split}
\end{equation}
In view of \eqref{s4-e63}, \eqref{s4-e62} becomes
\[\int\abs{u(z, \theta)}^2e^{-\psi_0(z, \theta)}dv(z)dv(\theta)=\frac{1}{2\pi}\iint\abs{\alpha(z, \eta)}^2e^{-\Phi_\eta(z)}dv(z)dv(\eta).\]
From \eqref{s4-e56}, we can check that
$\frac{1}{2\pi}\iint\abs{\alpha(z, \eta)}^2e^{-\Phi_\eta(z)}dv(z)dv(\eta)=1$ so we infer
\eqref{s4-e59}.
We obtain \eqref{s4-e60} from the following
\begin{equation*}
\begin{split}
\abs{u(0,0)}^2&=\frac{1}{(2\pi)^2}\abs{\int\alpha(0, \eta)dv(\eta)}^2=\frac{1}{(2\pi)^2}C^2_0\Bigr(\int_{\mathbb R_q}\abs{\det M_{\Phi_\eta}}dv(\eta)\Bigr)^2\\
&=(2\pi)^{-n}\int_{\mathbb R_q}\abs{\det M_{\Phi_\eta}}dv(\eta).
\end{split}
\end{equation*}
Finally, from \eqref{s4-e55}, \eqref{s4-e57}, Parseval's formula and the statement after \eqref{s4-e55}, we get \eqref{s4-e61} and the last statement of this proposition.
\end{proof}
From Proposition~\ref{s4-p3} and Theorem~\ref{s4-t3}, we get the main result of this section:
\begin{thm} \label{s4-t4}
We have
$\sum'_{\abs{J}=q}S^{(q)}_{J,H_n}(0)=(2\pi)^{-n}\int_{\mathbb R_q}\!\abs{\det M_{\Phi_\eta}}dv(\eta)$,
where $M_{\Phi_\eta}$ is as in \eqref{s4-e29-1} and $\mathbb R_q$ is as in \eqref{s4-e30}.
\end{thm}
\section{Szeg\"{o} kernel asymptotics and weak Morse inequalities on CR manifolds}
In this section we first study the properties of the Hermitian form $M^\phi_p$ introduced in Definition \ref{s1-d3}, especially its dependence of local trivializations. We then prove \eqref{s1-e19}, i.e. the second part of Theorem \ref{t-main1} (cf. Theorem \ref{s5-t1}). Finally, we prove Theorem \ref{t-main2}.
Let $s$ be a local trivializing section of $L$ on an open subset
$D\subset X$. Let $\phi$ be the weight of the Hermitian metric $h^L$ relative to $s$, that is, the point-wise norm of $s$ is $\abs{s(x)}^2=e^{-\phi(x)}$, $\phi\in C^\infty(D;\, \mathbb R)$.
Until further notice, we work on $D$.
Recall that
$M^\phi_p$, $p\in D$, is the Hermitian quadratic form on $\Lambda^{1,0}T_p(X)$ defined by
\[
M^\phi_p(U, \overline V)=\frac{1}{2}\Big\langle U\wedge\overline V, d(\overline\partial_b\phi-\partial_b\phi)(p)\Big\rangle\,,\quad U, V\in\Lambda^{1,0}T_p(X)\,.
\]
\begin{lem} \label{s5-l1}
For any $U, V\in\Lambda^{1,0}T_p(X)$, pick $\mathcal{U}, \mathcal{V}\in C^\infty(D;\, \Lambda^{1,0}T(X))$ that satisfy $\mathcal{U}(p)=U$,
$\mathcal{V}(p)=V$. Then,
\begin{equation} \label{s5-e2}
M^\phi_p(U, \overline V)=-\frac{1}{2}\big\langle\big[\,\mathcal{U}, \overline{\mathcal{V}}\,\big](p), \overline\partial_b\phi(p)-\partial_b\phi(p)\big\rangle
+\frac{1}{2}\big(\mathcal{U}\overline{\mathcal{V}}+\overline{\mathcal{V}}\mathcal{U}\big)\phi(p).
\end{equation}
\end{lem}
\begin{proof}
Recall that for a $1$-form $g$ and vectorfields $V_1$, $V_2$ we have
\begin{equation} \label{s5-e3}
\langle V_1\wedge V_2, dg\rangle=V_1(\langle V_2, g\rangle)-V_2(\langle V_1, g\rangle)-\langle[V_1, V_2], g\rangle,
\end{equation}
Taking $V_1=\mathcal{U}$, $V_2=\overline{\mathcal{V}}$ and $g=\overline\partial_b\phi-\partial_b\phi$ in
\eqref{s5-e3}, we get
\begin{equation} \label{s5-e4}
\begin{split}
&\big\langle\mathcal{U}\wedge\overline{\mathcal{V}}, d(\overline\partial_b\phi-\partial_b\phi)\big\rangle\\
&\quad=\mathcal{U}\big(\big\langle\overline{\mathcal{V}},\overline\partial_b\phi-\partial_b\phi\big\rangle)-\overline{\mathcal{V}}\big(\big\langle\mathcal{U},\overline\partial_b\phi-\partial_b\phi\big\rangle\big)-\big\langle\big[\mathcal{U}, \overline{\mathcal{V}}\,\big], \overline\partial_b\phi-\partial_b\phi\big\rangle.
\end{split}
\end{equation}
Note that $\langle\overline{\mathcal{V}}, \overline\partial_b\phi-\partial_b\phi\rangle=\langle\overline{\mathcal{V}},\overline\partial_b\phi\rangle=\overline{\mathcal{V}}\phi$
and $\langle\mathcal{U}, \overline\partial_b\phi-\partial_b\phi\rangle=\langle\mathcal{U}, -\partial_b\phi\rangle=-\mathcal{U}\phi$.
From this observation, \eqref{s5-e4} becomes
$\langle\mathcal{U}\wedge\overline{\mathcal{V}}, d(\overline\partial_b\phi-\partial_b\phi)\rangle=(\mathcal{U}\overline{\mathcal{V}}+\overline{\mathcal{V}}\mathcal{U})\phi-\langle[\,\mathcal{U}, \overline{\mathcal{V}}\,], \overline\partial_b\phi-\partial_b\phi\rangle$.
The lemma follows.
\end{proof}
The definition of $M^\phi_p$ depends on the choice of local trivializations.
Let $\widetilde D$ be another local trivialization with
$D\cap\widetilde D\neq\emptyset$. Let $\widetilde s$ be a local trivializing section of $L$ on the open subset $\widetilde D$ and the point-wise norm of $\widetilde s$ is $\abs{\widetilde s(x)}^2=e^{-\widetilde\phi(x)}$, $\widetilde\phi\in C^\infty(\widetilde D; \mathbb R)$.
Since $\widetilde s=gs$ on $D\cap\widetilde D$, for some non-zero CR function $g$, we
can check that
\begin{equation} \label{s5-e6}
\widetilde\phi=\phi-2\log{\abs{g}}
\end{equation}
on $D\cap\widetilde D$. Moreover, we have the following
\begin{prop} \label{s5-p1}
For $p\in D\cap\widetilde D$, we have
\begin{equation} \label{s5-e7}
M^\phi_p=M^{\widetilde\phi}_p+\Big(\,\frac{Tg}{g}-\frac{T\,\overline g}{\overline g}\,\Bigr)(p)\mathcal{L}_p\,.
\end{equation}
\end{prop}
\begin{proof}
From \eqref{s5-e6}, we can check that $\overline\partial_b\widetilde\phi=\overline\partial_b\phi-\frac{\overline\partial_b\overline g}{\overline g}$ and $\partial_b\widetilde\phi=\partial_b\phi-\frac{\partial_bg}{g}$ on $D\cap\widetilde D$.
From above, we have
\begin{equation} \label{s5-e9}
\langle[U, \overline V\,], \overline\partial_b\phi-\partial_b\phi\rangle=\langle[U, \overline V\,], \overline\partial_b\widetilde\phi-\partial_b\widetilde\phi\rangle+
\Big\langle[U, \overline V\,], \frac{\overline\partial_b\overline g}{\overline g}-\frac{\partial_bg}{g}\Big\rangle,
\end{equation}
where $U, V\in C^\infty(D\cap\widetilde D;\, \Lambda^{1,0}T(X))$.
From \eqref{s5-e6}, we have
\begin{equation} \label{s5-e10}
\begin{split}
(U\overline V+\overline VU)\phi&=(U\overline V+\overline VU)(\widetilde\phi+2\log{\abs{g}}) \\
&=(U\overline V+\overline VU)\widetilde\phi+\frac{\overline VUg}{g}+\frac{U\overline V\overline g}{\overline g}\ \ (\mbox{since $\overline Vg=0$, $U\overline g=0$}) \\
&=(U\overline V+\overline VU)\widetilde\phi-\frac{[U, \overline V\,]g}{g}+\frac{[U, \overline V\,]\overline g}{\overline g}.
\end{split}
\end{equation}
From \eqref{s5-e9}, \eqref{s5-e10} and \eqref{s5-e2}, we see that
\begin{equation} \label{s5-e11}
\begin{split}
M^\phi_p(U(p), \overline V(p))=&M^{\widetilde\phi}_p(U(p), \overline V(p))-\Big\langle[U, \overline V\,](p), \frac{1}{2}\frac{\overline\partial_b\overline g}{\overline g}(p)-\frac{1}{2}\frac{\partial_bg}{g}(p)\Big\rangle\\
&-\frac{1}{2}\frac{[U, \overline V\,]g}{g}(p)+\frac{1}{2}\frac{[U, \overline V\,]\overline g}{\overline g}(p).
\end{split}
\end{equation}
We write $[U, \overline V\,]=Z+\overline W+\alpha(x)T$, where $Z, W\in C^\infty(D\cap\widetilde D;\, \Lambda^{1,0}T(X))$ and
$\alpha(x)\in C^\infty(D\cap\widetilde D;\, \mathbb C)$. We can check that
$\alpha(p)=-2i\mathcal{L}_p(U(p), \overline V(p))$.
Since $\overline Wg=0$ and $Zg=\langle Z, \partial_bg\rangle=\langle[U, \overline V\,], \partial_bg\rangle$, we have
\begin{equation} \label{s5-e12}
[U, \overline V\,]g(p)=Zg(p)+\alpha(p)Tg(p)=\langle[U, \overline V\,](p), \partial_b g(p)\rangle-2i\mathcal{L}_p(U(p), \overline V(p))Tg(p).
\end{equation}
Similarly, we have
\begin{equation} \label{s5-e13}
[U, \overline V\,]\overline g(p)=\langle[U, \overline V\,](p), \overline\partial_b\overline g(p)\rangle-2i\mathcal{L}_p(U(p), \overline V(p))T\overline g(p).
\end{equation}
Combining \eqref{s5-e12}, \eqref{s5-e13} with \eqref{s5-e11}, we get
\[M^\phi_p(U(p), \overline V(p))=M^{\widetilde\phi}_p(U(p), \overline V(p))+i\mathcal{L}_p(U(p), \overline V(p))\Bigl(\frac{Tg}{g}-\frac{T\overline g}{\overline g}\Bigr)(p).\]
The proposition follows.
\end{proof}
\noindent
Recall that $\mathbb R_{\phi(p),q}$ was defined in \eqref{s1-e15}. From \eqref{s5-e7}, we see that
\begin{equation} \label{s5-e14}
\begin{split}
\mathbb R_{\widetilde\phi(p),q}&=\mathbb R_{\phi(p),q}+i\big(\tfrac{Tg}{g}-\tfrac{T\overline g}{\overline g}\big)(p)\\
&=\big\{s+i\big(\tfrac{Tg}{g}-\tfrac{T\overline g}{\overline g}\big);\, s\in\mathbb R_{\phi(p),q}\big\}.
\end{split}
\end{equation}
Since $Y(q)$ holds, $\mathbb R_{\phi(p),q}\subset[-R, R]$ for some $R>0$.
From \eqref{s5-e7} and \eqref{s5-e14}, we see that the function
$x\rightarrow\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds$
does not depend on the choice of $\phi$, where $\det(M^\phi_x+s\mathcal{L}_x)$ is the product of all the eigenvalues of $M^\phi_x+s\mathcal{L}_x$. Thus, the function $x\rightarrow\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds$ is well-defined.
Since $M^\phi_x$ and $\mathcal{L}_x$ are continuous functions of $x$, we conclude that
$x\rightarrow\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds$ is a continuous function of $x$.
\begin{rem} \label{s5-r3}
We take local coordinates
$(x, \theta)=(z, \theta)=(x_1,\ldots,x_{2n-2}, \theta)$, $z_j=x_{2j-1}+ix_{2j}$, $j=1,\ldots,n-1$, as in \eqref{s1-e20} and \eqref{s1-e21}
defined on some neighborhood of $p$. Then, it is straight forward to see that
$\mathcal{L}_p=\sum^{n-1}_{j=1}\lambda_jdz_j\otimes d\overline z_j$
and $M^\phi_p=\sum^{n-1}_{j,t=1}\mu_{j,t}dz_t\otimes d\overline z_j$.
Thus,
\begin{equation} \label{s5-e18-1}
\int_{\mathbb R_{\phi(p),q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds=
\int_{\mathbb R_{\phi(p),q}}\abs{\det\left(\mu_{j,t}+s\delta_{j,t}\lambda_j\right)^{n-1}_{j,t=1}}ds
\end{equation}
and
\begin{equation} \label{s5-e19}
\begin{split}
&\mathbb R_{\phi(p),q}=\big\{s\in\mathbb R;\, \mbox{the matrix $\left(\mu_{j,\,t}+s\delta_{j,\,t}\lambda_j\right)^{n-1}_{j,\,t=1}$ has $q$ negative eigenvalues} \\
&\quad\mbox{and $n-1-q$ positive eigenvalues}\big\}.
\end{split}
\end{equation}
\end{rem}
We prove now the precise bound \eqref{s1-e19} which is one of the main result of this work.
\begin{thm} \label{s5-t1}
We have for all $p\in X$
\[\limsup_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_k(p)\leqslant (2\pi)^{-n}\int_{\mathbb R_{\phi(p),\,q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds\, .
\]
\end{thm}
\begin{proof}
For $p\in X$, let $(x, \theta)=(z, \theta)=(x_1,\ldots,x_{2n-1})$, $z_j=x_{2j-1}+ix_{2j}$, $j=1,\ldots,n-1$, be the coordinate as in \eqref{s1-e20} and
\eqref{s1-e21} defined on some neighborhood of $p$. From Theorem~\ref{s3-t2}, we have that
$\limsup_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_k(0)\leqslant \sum'_{\abs{J}=q}S^{(q)}_{J,H_n}(0)$.
From Theorem~\ref{s4-t4}, we know that
$\sum'_{\abs{J}=q}S^{(q)}_{J,H_n}(0)=(2\pi)^{-n}\int_{\mathbb R_q}\!\abs{\det M_{\Phi_\eta}}dv(\eta)$,
where $M_{\Phi_\eta}$ is as in \eqref{s4-e29-1} and $\mathbb R_q$ is as in \eqref{s4-e30}. Thus,
\begin{equation} \label{s5-e21}
\limsup_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_k(0)\leqslant (2\pi)^{-n}\int_{\mathbb R_q}\!\abs{\det M_{\Phi_\eta}}dv(\eta).
\end{equation}
From \eqref{s4-e29-1}, \eqref{s4-e30} and the definition of $\Phi_\eta$ (see \eqref{s4-e13}), we see that
\begin{equation} \label{s5-e22}
\det M_{\Phi_\eta}=\det\left(\mu_{j,\,t}-\sqrt{2}\eta\lambda_j\delta_{j,\,t}\right)^{n-1}_{j,\,t=1}
\end{equation}
and
\begin{equation} \label{s5-e23}
\begin{split}
&\mathbb R_q=\big\{\eta\in\mathbb R;\, \mbox{the matrix $\left(\mu_{j,\,t}-\sqrt{2}\eta\delta_{j,\,t}\lambda_j\right)^{n-1}_{j,\,t=1}$ has $q$ negative eigenvalues} \\
&\quad\mbox{and $n-1-q$ positive eigenvalues}\big\}.
\end{split}
\end{equation}
Note that $dv(\eta)=\sqrt{2}d\eta$. From this and \eqref{s5-e22}, \eqref{s5-e23}, \eqref{s5-e18-1}, \eqref{s5-e19}, it is easy to see that
$\int_{\mathbb R_q}\!\abs{\det M_{\Phi_\eta}}dv(\eta)=\int_{\mathbb R_{\phi(p),q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds$.
From this and \eqref{s5-e21}, the theorem follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t-main2}]
By \eqref{s2-e11}-\eqref{s1-e13} we have $\dim H^q_b(X,L^k)=\int_X\!\mathit\Pi^{(q)}_k(x)dm(x)$.
In view of Theorem~\ref{s3-t1}, $\sup_k k^{-n}\mathit\Pi^{(q)}_k(\cdot)$ is integrable on $X$.
Thus, we can apply Fatou's lemma and we get by using Theorem \ref{s5-t1}:
\begin{equation*}
\begin{split}
&\limsup_{k\rightarrow\infty}\,k^{-n}\dim H^q_b(X,L^k)\leqslant \int_X\!\limsup_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_k(x)dm(x)\\
&\quad\leqslant (2\pi)^{-n}\int_X\Bigr(\int_{\mathbb R_{\phi(x),q}}\abs{\det(M^\phi_x+s\mathcal{L}_x)}ds\Bigr)dm(x).
\end{split}
\end{equation*}
The theorem follows.
\end{proof}
\section{Strong Morse inequalities on CR manifolds}
In this section, we will establish the strong Morse inequalities on CR manifolds. We first recall some well-known facts.
We know \cite[Th.\,7.6]{Ko65}, \cite[Th.\,5.4.11--12]{FK72}, \cite[Cor.\,8.4.7--8]{CS01} that if $Y(q)$ holds, then $\Box^{(q)}_{b,k}$ has a discrete
spectrum, each eigenvalues occurs with finite multiplicity and all eigenforms are smooth. For $\lambda\in\mathbb R$, let
$\cali{H}^q_{b,\,\leqslant\lambda}(X, L^k)$ denote the space spanned by the eigenforms of $\Box^{(q)}_{b,k}$ whose eigenvalues are bounded by
$\lambda$ and denote by $\mathit\Pi^{(q)}_{k,\,\leqslant \lambda}$ the Szeg\"{o} kernel function of the space $\cali{H}^q_{b,\,\leqslant \lambda}(X,L^k)$.
Similarly, let $\cali{H}^q_{b,>\lambda}(X, L^k)$ denote the space spanned by the eigenforms of $\Box^{(q)}_{b,k}$ whose eigenvalues are
$>\lambda$.
Let $Q_b$ be the
Hermitian form on $\Omega^{0,q}(X, L^k)$ defined for $u, v\in\Omega^{0,q}(X, L^k)$ by
\[
Q_b(u, v)=(\overline\partial_{b,k}u\ |\ \overline\partial_{b,k}v)_k+(\overline\partial^*_{b,k}u\ |\ \overline\partial^*_{b,k}v)_k+(u\ |\ v)_k=(\Box^{(q)}_{b,k}u\ |\ v)_k+(u\ |\ v)_k\,.
\]
Let $\overline{\Omega^{0,q}(X, L^k)}$ be the completion
of $\Omega^{0,q}(X, L^k)$ under $Q_b$ in $L^2_{(0,q)}(X, L^k)$.
For $\lambda>0$, we have the orthogonal spectral decomposition with
respect to $Q_b$:
\begin{equation} \label{s6-e1}
\overline{\Omega^{0,q}(X, L^k)}=\cali{H}^q_{b,\,\leqslant \lambda}(X, L^k)\oplus\overline{\cali{H}^q_{b,\,>\lambda}(X, L^k)},
\end{equation}
where $\overline{\cali{H}^q_{b,\,>\lambda}(X, L^k)}$ is the completion of $\cali{H}^q_{b,\,>\lambda}(X, L^k)$ under $Q_b$ in $L^2_{(0,q)}(X, L^k)$.
Let
$u\in\overline{\cali{H}^q_{b,\,>\lambda}(X, L^k)}\cap\Omega^{0,q}(X, L^k)$. There are $f_j\in \cali{H}^q_{b,\,>\lambda}(X, L^k)$, $j=1,2,\ldots$, such that $Q_b(f_j-u)\rightarrow 0$, as $j\rightarrow\infty$.
From this, we can check that $(\Box^{(q)}_{b,k}f_j\ |\ f_j)_k\rightarrow(\Box^{(q)}_{b,k}u\ |\ u)_k$, as $j\rightarrow\infty$, and
\begin{equation} \label{s6-e2}
\norm{u}^2=\lim_{j\rightarrow\infty}\norm{f_j}^2=\lim_{j\rightarrow\infty}(f_j\ |\ f_j)_k
\leqslant \lim_{j\rightarrow\infty}\frac{1}{\lambda}\big(\Box^{(q)}_{b,k}f_j\ \big|\ f_j\big)_k=\frac{1}{\lambda}\big(\Box^{(q)}_{b,k}u\ \big|\ u\big)_k\,.
\end{equation}
We return to our situation. We will use the same notations as in section 3. For a given point $p\in X$, let $s$ be a local trivializing section of $L$ on an open neighborhood of $p$ and $\abs{s}^ 2=e^{-\phi}$. Let $(x, \theta)=(z, \theta)=(x_1,\ldots,x_{2n-2},\theta)$, $z_j=x_{2j-1}+ix_{2j}$,
$j=1,\ldots,n-1$, be the local coordinates as in \eqref{s1-e20} and \eqref{s1-e21} defined on an open set $D$ of $p$.
Note that $(x(p), \theta(p))=0$. We identify $D$ with some open set of $H_n$.
Let $u(z, \theta)=\sum'_{\abs{J}=q}u_J(z, \theta)d\overline z^J\in\Omega^{0,q}(H_n)$ be as in \eqref{s4-e57} and Proposition~\ref{s4-p3}.
From \eqref{s4-e61} and the statement after \eqref{s4-e61}, we know that the value
\[\int\abs{\partial^m_x\partial^{m'}_\theta u}^2e^{-\psi_0}dv(z)dv(\theta)\]
is finite and can be bounded by some positive continuous function of the eigenvalues of $M_{\Phi_\eta}$, $\eta\in\mathbb R_q$, $\beta$ and $\lambda_j$, $j=1,\ldots,n-1$, for all $m'\in\mathbb N_0$, $m\in\mathbb N_0^{2n-2}$. Since $X$ is compact, we deduce that for every
$m\in\mathbb N_0^{2n-2}$, $m'\in\mathbb N_0$, we can find $M_{m,m'}>0$ independent of the point $p$, such that
\begin{equation} \label{s6-e3}
\int_{H_n}\abs{\partial^m_x\partial^{m'}_\theta u}^2e^{-\psi_0}dv(\theta)dv(z)< M_{m,m'}.
\end{equation}
Set
$\beta_k(z, \theta)=\chi_k(\sqrt{k}z, k\theta)\sum'_{\abs{J}=q}u_J(\sqrt{k}z, k\theta)e^J(z, \theta)\in\Omega^{0,q}(D)$. Here $\chi$ is a smooth function, $0\leqslant \chi\leqslant 1$, supported on $D_1$
which equals one on $D_{\frac{1}{2}}$ and
$\chi_k(z, \theta)=\chi(\frac{z}{\log k}, \frac{\theta}{\sqrt{k}\log k})$.
We remind that $(e_j)_{j=1,\ldots,\,n-1}$ denotes the basis of $\Lambda^{0,1}T^*(X)$,
which is dual to $(\overline U_j)_{j=1,\ldots,\,n-1}$, where $(U_j)_{j=1,\ldots,\,n-1}$ are as in \eqref{s1-e20}. We notice that for $k$ large,
${\rm Supp\,}\beta_k\subset D_{\frac{\log k}{\sqrt{k}}}$. From Proposition~\ref{s3-p1} and \eqref{s3-e23}, we have
\begin{equation} \label{s6-e5}
(\Box^{(q)}_{s,(k)})(F^*_k\beta_k)=\Box^{(q)}_{b,H_n}\bigr(\chi_k(z, \theta)u(z, \theta)\bigr)+\varepsilon_kP_k(F^*_k\beta_k),
\end{equation}
where $\varepsilon_k$ is a sequence tending to zero with $k\rightarrow\infty$ and $P_k$ is a second order differential operator and all
the derivatives of the coefficients of $P_k$ are uniform bounded in $k$. Note that
$\Box^{(q)}_{b,H_n}u=0$
and
$\sup_{(z, \theta)\in D_{\log k}}\abs{kF^*_k\phi_0-\psi_0}\To0$ as $k\rightarrow\infty$ ($\phi_0$ is as in \eqref{s1-e23}).
From this, \eqref{s6-e5} and \eqref{s6-e3}, we deduce that there is a sequence $\delta_k>0$, independent of the point $p$ and
tending to zero such that
\begin{equation} \label{s6-e6}
\big\|\Box^{(q)}_{s,(k)}(F^*_k\beta_k)\big\|_{kF^*_k\phi_0}\leqslant \delta_k.
\end{equation}
Similarly, we have for all $m\in\mathbb N$
\begin{equation} \label{s6-e7}
\big\|(\Box^{(q)}_{s,(k)})^m(F^*_k\beta_k)\big\|_{kF^*_k\phi_0}\To0\ \ \mbox{as $k\rightarrow\infty$}\,.
\end{equation}
Now define $\alpha_k\in\Omega^{0,q}(X, L^k)$ by
\begin{equation} \label{s6-e8}
\alpha_k(z, \theta)=s^kk^{\frac{n}{2}}e^{kR}\beta_k(z, \theta)\ \ (\mbox{$R(z, \theta)$ is as in \eqref{s1-e22}}).
\end{equation}
We can check that
\begin{equation} \label{s6-e9}
k^{-n}\abs{\alpha_k(0, 0)}^2=\abs{\beta_k(0, 0)}^2=\abs{u(0, 0)}^2=(2\pi)^{-n}\int_{\mathbb R_{\phi(p),q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds
\end{equation}
for all $k$, and
\begin{equation} \label{s6-e10}
\begin{split}
\norm{\alpha_k}^2&=\int k^ne^{k(R+\overline R)}\abs{\beta_k}^2e^{-k\phi}m(z, \theta)dv(z)dv(\theta)\\
&=\int k^ne^{-k\phi_0}\abs{\beta_k}^2m(z, \theta)dv(z)dv(\theta)\\
&=\int e^{-kF^*_k\phi_0}\abs{\chi_k(z, \theta)}^2\abs{u(z, \theta)}^2
m\big(\tfrac{z}{\sqrt{k}}, \tfrac{\theta}{k}\big)
dv(z)dv(\theta)\\
&\rightarrow\int\abs{u}^2e^{-\psi_0(z, \theta)}dv(z)dv(\theta)=1,\ \ \mbox{as $k\rightarrow\infty$},
\end{split}
\end{equation}
where $m(z, \theta)dv(z)dv(\theta)$ is the volume form. Note that $m(0, 0)=1$. Moreover, we have
\begin{equation} \label{s6-e11}
\begin{split}
&(\tfrac{1}{k}\Box^{(q)}_{b,k}\alpha_k\ |\ \alpha_k)_k\\
&\quad=\int k^n(\tfrac{1}{k}\Box^{(q)}_s\beta_k\ |\ \beta_k)e^{-k\phi_0}m(z, \theta)dv(z)dv(\theta)\ \ (\text{by \eqref{s3-e7-0}})\\
&\quad=\int(\tfrac{1}{k}F^*_k(\Box^{(q)}_s\beta_k)\ |\ F^*_k\beta_k)_{F^*_k}e^{-kF^*_k\phi_0}(F^*_km)dv(z)dv(\theta)\\
&\quad=\int((\Box^{(q)}_{s,(k)})F^*_k\beta_k\ |\ F^*_k\beta_k)_{F^*_k}e^{-kF^*_k\phi_0}(F^*_km)dv(z)dv(\theta)\ \ (\text{by \eqref{s3-e9}}).
\end{split}
\end{equation}
From \eqref{s6-e6} and the fact that $\norm{F^*_k\beta_k}_{kF^*_k\phi_0}\leqslant 1$, we deduce that
there is a sequence $\mu_k>0$, independent of the point $p$ and
tending to zero such that
\begin{equation} \label{s6-e12}
\big(\tfrac{1}{k}\Box^{(q)}_{b,k}\alpha_k\ \big|\ \alpha_k\big)_k\leqslant \mu_k.
\end{equation}
Similarly, from \eqref{s6-e7}, we can repeat the procedure above with minor changes and get
\begin{equation} \label{s6-e13}
\big\|\big(\tfrac{1}{k}\Box^{(q)}_{b,k}\big)^m\alpha_k\big\|\To0\ \ \mbox{as $k\rightarrow\infty$},
\end{equation}
for all $m\in\mathbb N$. Now, we can prove
\begin{prop} \label{s6-p1}
Let $\nu_k>0$ be any sequence with
$\lim_{k\rightarrow\infty}\frac{\mu_k}{\nu_k}=0$,
where $\mu_k$ is as in \eqref{s6-e12}. Then, $\liminf_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_{k,\,\leqslant k\nu_k}(0)\geqslant(2\pi)^{-n}\int_{\mathbb R_{\phi(p),q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds$.
\end{prop}
\begin{proof}
Let $\alpha_k$ be as in \eqref{s6-e8}. From \eqref{s6-e1}, we have
$\alpha_k=\alpha^{1}_k+\alpha^2_k$,
where $\alpha^1_k\in\cali{H}^q_{b,\le k\nu_k}(X, L^k)$, $\alpha^2_k\in\overline{\cali{H}^q_{b,>k\nu_k}(X, L^k)}$. From \eqref{s6-e2}, we have
\[\norm{\alpha^2_k}^2\leqslant \frac{1}{k\nu_k}\big(\Box^{(q)}_{b,k}\alpha^2_k\ \big|\ \alpha^2_k\big)_k\leqslant \frac{1}{k\nu_k}\big(\Box^{(q)}_{b,k}\alpha_k\ \big|\ \alpha_k\big)_k\leqslant \frac{\mu_k}{\nu_k}\rightarrow 0,\]
as $k\rightarrow\infty$.
Thus, $\lim_{k\rightarrow\infty}\norm{\alpha^2_k}=0$. Since $\alpha_k\To1$ as $k\rightarrow\infty$, we get
\begin{equation} \label{s6-e15}
\lim_{k\rightarrow\infty}\norm{\alpha^1_k}=1.
\end{equation}
Now, we claim that
\begin{equation} \label{s6-e16}
\lim_{k\rightarrow\infty}k^{-n}\abs{\alpha^2_k(0)}^2=0.
\end{equation}
On $D$, we write $\alpha^2_k=s^kk^{\frac{n}{2}}e^{kR}\beta^2_k$,
$\beta^2_k\in\Omega^{0,q}(D)$. From \eqref{s3-e12} and the proof of Lemma~\ref{s3-l1}, we see that
\begin{equation} \label{s6-e17}
\abs{F^*_k\beta^2_k(0)}^2\leqslant C_{n-1,r}\Bigr(\norm{F^*_k\beta^2_k}^2_{kF^*_k\phi_0,D_r}+\norm{(\Box^{(q)}_{s,(k)})F^*_k\beta^2_k}^2_{kF^*_k\phi_0,n-1,D_r}\Bigr),
\end{equation}
for some $r>0$. Now, we have
\begin{equation} \label{s6-e18}
\norm{F^*_k\beta^2_k}^2_{kF^*_k\phi_0,D_r}\leqslant \norm{\alpha^2_k}^2\To0,\ \ \mbox{as $k\rightarrow\infty$}.
\end{equation}
Moreover, from \eqref{s3-e12} and using induction, we get
\begin{equation} \label{s6-e19}
\norm{(\Box^{(q)}_{s,(k)}F^*_k\beta^2_k}^2_{kF^*_k\phi_0,n-1,D_r}\leqslant C'\sum^n_{m=1}\norm{(\Box^{(q)}_{s,(k)})^mF^*_k\beta^2_k}^2_{kF^*_k\phi_0,D_{r'}}\,,
\end{equation}
for some $r'>0$, where $C'>0$ is independent of $k$. We can check that for all $m\in\mathbb N$,
\begin{equation} \label{s6-e20}
\Big\|(\Box^{(q)}_{s,(k)})^mF^*_k\beta^2_k\Big\|^2_{kF^*_k\phi_0,D_{r'}}\leqslant \Big\|(\tfrac{1}{k}\Box^{(q)}_{b,k})^m\alpha^2_k\Big\|^2
\leqslant \Big\|(\tfrac{1}{k}\Box^{(q)}_{b,k})^m\alpha_k\Big\|^2\To0\ \ \text{as $k\rightarrow\infty$}.
\end{equation}
Here we used \eqref{s6-e13}.
Combining \eqref{s6-e20}, \eqref{s6-e19}, \eqref{s6-e18} with \eqref{s6-e17}, we get
\[\lim_{k\rightarrow\infty}\abs{F^*_k\beta^2_k(0)}^2=\lim_{k\rightarrow\infty}\abs{\beta^2_k(0)}^2=
\lim_{k\rightarrow\infty}k^{-n}\abs{\alpha^2_k(0)}^2=0.\]
Hence \eqref{s6-e16} follows. From this and \eqref{s6-e9}, we conclude
\begin{equation} \label{s6-e21}
\lim_{k\rightarrow\infty}k^{-n}\abs{\alpha^1_k(0)}^2=(2\pi)^{-n}\int_{\mathbb R_{\phi(p),q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds.
\end{equation}
Now,
\[
k^{-n}\mathit\Pi^{(q)}_{k,\,\leqslant k\nu_k}(0)\geqslant k^{-n}\frac{\abs{\alpha^1_k(0)}^2}{\norm{\alpha^1_k}^2}\rightarrow(2\pi)^{-n}\int_{\mathbb R_{\phi(p),q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds,\ \ \mbox{as $k\rightarrow\infty$}.
\]
The proposition follows.
\end{proof}
\begin{prop} \label{s6-p2}
Let $\nu_k>0$ be any sequence with $\nu_k\To0$, as $k\rightarrow\infty$. Then,
\[\limsup_{k\rightarrow\infty}k^{-n}\mathit\Pi^{(q)}_{k,\,\leqslant k\nu_k}(0)\leqslant (2\pi)^{-n}\int_{\mathbb R_{\phi(p),q}}\abs{\det(M^\phi_p+s\mathcal{L}_p)}ds.\]
\end{prop}
\begin{proof}
The proof is a simple modification of the proof of Theorem~\ref{s5-t1} and in what follows these modifications will be presented.
Let $\alpha_k\in\cali{H}^q_{b,\leqslant k\nu_k}(X, L^k)$ with $\norm{\alpha_k}=1$. On $D$, we write
$\alpha_k=s^kk^{\frac{n}{2}}e^{kR}\beta_k$,
$\beta_k\in\Omega^{0,q}(D)$. From \eqref{s3-e12} and using induction, we get
\begin{equation} \label{s6-e22}
\norm{F^*_k\beta_k}^2_{kF^*_k\phi_0,s+1,D_r}
\leqslant C_{r,s}\Bigr(\norm{F^*_k\beta_k}^2_{kF^*_k\phi_0,D_{2r}}+\sum^{s+1}_{m=1}
\norm{(\Box^{(q)}_{s,(k)})^mF^*_k\beta_k}^2_{kF^*_k\phi_0,D_{2r}}\,\Bigr).
\end{equation}
We can check that
$\big\|(\Box^{(q)}_{s,(k)})^mF^*_k\beta_k\big\|^2_{kF^*_k\phi_0,D_{2r}}\leqslant \big\|(\frac{1}{k}\Box^{(q)}_{b,k})^m\alpha_k\big\|
\leqslant \nu_k^m\To0$.
Thus, the conclusion of Proposition~\ref{s3-p3} is still valid and the rest of the argument goes through word by word.
\end{proof}
\begin{proof}[Proof of Theorems \ref{t-main3}, \ref{t-main4} and \ref{t-main5}]
We can repeat the proof of Theorem~\ref{s3-t1} and conclude that
for any sequence $(\nu_k)$ with $\nu_k\To0$, as $k\rightarrow\infty$,
there is a constant $C_0$ independent of $k$, such that
$k^{-n}\mathit\Pi^{(q)}_{k,\,\leqslant k\nu_k}(x_0)\leqslant C_0$ for all $x_0\in X$.
From this, Proposition~\ref{s6-p1} and Proposition~\ref{s6-p2} and the fact that the sequence $(\mu_k)$ in \eqref{s6-e12} is independent of the point $p$, we get Theorem \ref{t-main3}. By integrating Theorem \ref{t-main3} we obtain Theorem \ref{t-main4}.
By applying the algebraic Morse inequalities \cite[Lemma\,3.2.12]{MM07} to the $\overline\partial_{b,k}$-complex \eqref{s1-e7}
we deduce in view of Theorem \ref{t-main4} the strong Morse inequalities of Theorem \ref{t-main5}.
\end{proof}
\section{Examples}
In this section, some examples are collected. The aim is to illustrate the main results in some simple situations. First, we state
our main results in the embedded case.
\subsection{The main results in the embedded cases}
Let $M$ be a relatively compact open subset with $C^\infty$ boundary $X$ of a
complex manifold $M'$ of dimension $n$ with a smooth Hermitian metric $(\ |\ )$. Furthermore, let $L$ be a Hermitian holomorphic
line bundle over $M'$ with fiber metric $\phi$. If $s(x)$ is a local trivializing
section of $L$ on an open subset $D\subset M'$, then $\abs{s(x)}^2=e^{-\phi}$. If we restict $L$ on the boundary $X$, then $L$
is a CR line bundle over the CR manifold $X$. For $p\in X$, let $M^\phi_p$ be as in Definition~\ref{s1-d3}.
\begin{prop} \label{s7-p1}
For $U, V\in\Lambda^{1,0}T_p(X)$, we have
$M^\phi_p(U, \overline V)=\big\langle\partial\overline\partial\phi(p), U\wedge\overline V\big\rangle$.
\end{prop}
\begin{proof}
Let $r\in C^\infty(X;\ \mathbb R)$ be a defining function of $X$. For $U, V\in\Lambda^{1,0}T_p(X)$, pick $\mathcal{U}, \mathcal{V}\in C^\infty(M';\, \Lambda^{1,0}T(M'))$ that satisfy $\mathcal{U}(p)=U$,
$\mathcal{V}(p)=V$ and $\mathcal{U}(r)=\mathcal{V}(r)=0$ in a neighborhood of $p$ in $M'$. From \eqref{s5-e3}, we have
\begin{equation} \label{s7-e2}
\begin{split}
2\big\langle\mathcal{U}\wedge\overline{\mathcal{V}}, \partial\overline\partial\phi\big\rangle
&=\big\langle\mathcal{U}\wedge\overline{\mathcal{V}}, d(\overline\partial\phi-\partial\phi)\big\rangle\\
&=\mathcal{U}\big(\big\langle\overline{\mathcal{V}},\overline\partial\phi-\partial\phi\big\rangle\big)-\overline{\mathcal{V}}\big(\big\langle\mathcal{U},\overline\partial\phi-\partial\phi\big\rangle\big)-\big\langle\big[\mathcal{U}, \overline{\mathcal{V}}\,\big], \overline\partial\phi-\partial\phi\big\rangle.
\end{split}
\end{equation}
Note that
$\langle\overline{\mathcal{V}}, \overline\partial\phi-\partial\phi\rangle=\langle\overline{\mathcal{V}},\overline\partial\phi\rangle=\overline{\mathcal{V}}\phi$
and $\langle\mathcal{U}, \overline\partial\phi-\partial\phi\rangle=\langle\mathcal{U}, -\partial\phi\rangle=-\mathcal{U}\phi$.
From this observation, \eqref{s7-e2} becomes
\begin{equation} \label{s7-e3}
2\big\langle\mathcal{U}\wedge\overline{\mathcal{V}}, \partial\overline\partial\phi\big\rangle=\big(\mathcal{U}\overline{\mathcal{V}}+\overline{\mathcal{V}}\mathcal{U}\big)\phi-\big\langle\big[\mathcal{U}, \overline{\mathcal{V}}\,\big], \overline\partial\phi-\partial\phi\big\rangle.
\end{equation}
Since $\mathcal{U}(r)=\mathcal{V}(r)=0$ in a neighborhood of $p$ in $M'$, we have
\[\big(\mathcal{U}\overline{\mathcal{V}}+\overline{\mathcal{V}}\mathcal{U}\big)\phi(p)=\big(\mathcal{U}|_X\overline{\mathcal{V}}|_X+\overline{\mathcal{V}}|_X\mathcal{U}|_X\big)\phi|_X(p)\] and
\[\big\langle\big[\mathcal{U}, \overline{\mathcal{V}}\,\big], \overline\partial\phi-\partial\phi\big\rangle(p)=
\big\langle\big[\mathcal{U}|_X, \overline{\mathcal{V}}|_X\big], \overline\partial_b\phi|_X-\partial_b\phi|_X\big\rangle(p),\]
where $\mathcal{U}|_X$ is the restriction to $X$ of $\mathcal{U}$ and similarly for $\overline{\mathcal{V}}$ and $\phi$.
From this observation and \eqref{s7-e3}, we conclude that
\begin{equation} \label{s7-e4}
2\big\langle\mathcal{U}\wedge\overline{\mathcal{V}}, \partial\overline\partial\phi\big\rangle(p)=\big(\mathcal{U}|_X\overline{\mathcal{V}}|_X+\overline{\mathcal{V}}|_X\mathcal{U}|_X\big)\phi|_X(p)-\big\langle\big[\mathcal{U}|_X, \overline{\mathcal{V}}|_X\big], \overline\partial_b\phi|_X-\partial_b\phi|_X\big\rangle(p).
\end{equation}
From \eqref{s7-e4} and Lemma~\ref{s5-l1}, the proposition follows.
\end{proof}
We denote by $R^L_X$ the restriction of $R^L$ to $\Lambda^{1,0}T(X)$.
As before, let $\mathcal{L}_p$ be the Levi form of $X$ at $p\in X$. We define the set $\mathbb R_{\phi(p),q}$ as in
\eqref{s1-e15}. Set
\begin{equation}
I^q(X,L):=\int_X\int_{\mathbb R_{\phi(x),q}}\!\abs{\det\bigl(R^L_X+s\mathcal{L}_x\bigr)}ds\,dm(x)\,.
\end{equation}
Now, we can reformulate Theorem~\ref{t-main2} and Theorem~\ref{t-main5}:
\begin{thm} \label{s7-t1}
If the Levi form is non-degenerate and
condition $Y(q)$ holds, then
\begin{equation*}
\dim H^q_b(X, L^k)
\leqslant k^n(2\pi)^{-n} I^q(X,L)+o(k^n),
\end{equation*}
If condition $Y(j)$ holds, for all $j=0,1,\ldots,q$, then
\begin{equation}\label{s7-e5}
\sum^q_{j=0}(-1)^{q-j}{\rm dim\,}H^j_b(X, L^k)
\leqslant k^n(2\pi)^{-n}\sum^q_{j=0}(-1)^{q-j}I^j(X,L)+o(k^n).
\end{equation}
If condition $Y(j)$ holds, for all $j=q,q+1,\ldots,n-1$, then
\begin{equation*}
\sum^{n-1}_{j=q}(-1)^{q-j}{\rm dim\,}H^j_b(X, L^k)
\leqslant k^n(2\pi)^{-n}\sum^{n-1}_{j=q}(-1)^{q-j}I^j(X,L)+o(k^n).
\end{equation*}
\end{thm}
\begin{proof}[Proof of Theorem \ref{th-app}]
The hypothesis of Theorem \ref{th-app} imply that $\mathbb R_{\phi(p),0}$ is nonempty and $\mathbb R_{\phi(p),1}=\emptyset$ for all $p\in X$. Thus the strong Morse inequalities \eqref{s7-e5} for $q=1$ imply the conclusion.
\end{proof}
\noindent
\begin{proof}[Proof of Theorem \ref{th-strip}]
Note that $X$ and $L$ satisfy the conditions of Theorem \ref{th-app}.
We have thus $\dim H^0_b(X,L^k)=O(k^n)$,
$k\to\infty$. Moreover, every CR function on $X$ extends
locally to a holomorphic function in a small open set of $M$. For $b<c$
denote by $M_b^c=\{b<\rho<c\}$. Thus, there exist
$b<a<c$ such that the restriction morphism $H^0(M_b^c,E)\to H^0(X,E)$ is
an isomorphism for any holomorphic line bundle $E\to M$. Moreover, we
know by \cite{AG:62} that the restriction $H^0(M,E)\to H^0(M_b^c,E)$ is
an isomorphism. Therefore
\begin{equation} \label{gm3.60}
\dim H^0(M,L^k)=O(k^n)\,,\quad k\to\infty.
\end{equation}
Now, $M$ is a $(n-2)$-concave manifold in the sense
of \cite{AG:62} in particular Andreotti-pseudo-concave (see
\cite[Def.\,3.4.3]{MM07}). By \cite[Th.\,3.4.5]{MM07}
there exists $C>0$ such that
\begin{equation} \label{gm3.6}
\dim H^0(M,L^k)\leqslant Ck^{\,\varrho_{k}},\,\quad \text{ for $k\geqslant 1$},
\end{equation}
where $\varrho_{k}=\max_{M\setminus B_k}\operatorname{rank}\Phi_k$ is the maximum rank of the Kodaira map
\begin{equation}
\Phi_k:M\setminus B_k\to \mathbb{P}(H^0(M,L^k)^*)\,,\quad
\Phi_k(p)=\{s\in H^0(M,L^k): s(p)=0\}\,,
\end{equation}
and $B_k$ is the base locus of $H^0(M,L^k)$.
Moreover, the field of meromorphic functions $\mathcal{K}_M$
is an algebraic field of transcendence degree $a(M)\leqslant \dim M$ and $\kappa(L):=\max_k\varrho_k\leqslant a(M)$.
By \eqref{gm3.60} and \eqref{gm3.6} we obtain that $\varrho_k=n$ for large $k$ and the desired conclusions follow.
\end{proof}
A gobal version of Theorem \ref{th-strip} goes like follows.
\begin{cor}\label{gr-riem}
Let $M$ be a projective manifold, $n=\dim_\mathbb C M$, and let $X=\rho^{-1}(0)\subset M$ be a compact hypersurface, where $\rho\in C^\infty(M,\mathbb R)$ satisfies $d\rho|_X\neq0$. We assume that the Levi form of $X$ is non-degenerate and has at least two negative and two positive eigenvalues.
Let $L\to M$ be a holomorphic line bundle whose curvature form
$R^L$ is positive on $X$ and the Levi form
$\partial\overline\partial\rho|_X$ satisfies the assumptions of Theorem~\ref{th-app}.
Then there exists a divisor $H\subset M\setminus X$ such that
\[
\dim H^0(M\setminus H,L^k)=\dim H^0(M,L^k\otimes[-H])=O(k^n)\,,\quad k\to\infty\,.
\]
\end{cor}
\begin{proof}
Let us first observe that under the given hypotheses, there exist $b<0<c$ sucht that $M'=\{b<\rho<c\}$
is a $(n-2)$-convex--concave strip which fulfills the assumptions of Theorem \ref{th-strip}. By \eqref{gm3.60}, $\dim H^0(M',L^k)=O(k^n)$, $k\to\infty$.
Since $M'$ is Andreotti-pseudoconcave, a theorem of Dingoyan \cite{Ding99} (which generalizes classical results of Barth, Chow and Rossi) shows that
there exists a divisor $H\subset M\setminus X$ such that the restriction morphisms $H^0(M\setminus H,L^k)\to H^0(M',L^k)$ and $H^0(M,L^k\otimes[-H])\to H^0(M',L^k)$ are isomorphisms.
\end{proof}
\subsection{Holomorphic line bundles over complex torus}
Let
\[T_n:=\mathbb C^n/(\sqrt{2\pi}\mathbb Z^n+i\sqrt{2\pi}\mathbb Z^n)\]
be the flat torus and let $L_\lambda$ be the holomorphic
line bundle over $T_n$
with curvature the $(1,1)$-form
$\Theta_\lambda=\sum^n_{j=1}\lambda_jdz_j\wedge d\overline z_j$,
where $\lambda_j$, $j=1,\ldots,n$, are given non-zero integers. More precisely, $L_\lambda:=\mathbb C^n\times\mathbb C/\sim$\,, where
$(z, \theta)\sim(\widetilde z, \widetilde\theta)$ if
\[
\widetilde z-z=(\alpha_1,\ldots,\alpha_n)\in \sqrt{2\pi}\mathbb Z^n+i\sqrt{2\pi}\mathbb Z^n\,,\quad
\widetilde\theta=\textstyle\exp\big(\sum^n_{j=1}\lambda_j(z_j\overline\alpha_j+\tfrac{1}{2}\abs{\alpha_j}^2\,)\big)\theta\,.
\]
We can check that $\sim$ is an equivalence relation and $L_\lambda$ is a holomorphic line bundle over $T_n$.
For $[(z, \theta)]\in L_\lambda$
we define the Hermitian metric by
\[
\big\vert[(z, \theta)]\big\vert^2:=\abs{\theta}^2\textstyle\exp(-\sum^n_{j=1}\lambda_j\abs{z_j}^2)
\]
and it is easy to see that this definition is independent of the choice of a representative $(z, \theta)$ of $[(z, \theta)]$. We write $\phi_\lambda(z)$ to denote this Hermitian fiber metric. Note that $\partial\overline\partial\phi_\lambda=\Theta_\lambda$.
From now on, we assume that $\lambda_j<0$, for $j=1,\ldots, n_-$ and $\lambda_j>0$, for $j=n_-+1,\ldots, n$
Let $L^*_\lambda$ be the
dual bundle of $L_\lambda$ and let $\norm{.}_{L^*_\lambda}$ be the norm of $L^*_\lambda$ induced by the Hermitian fiber metric on $L_\lambda$. Consider the compact CR manifold of dimension $2n+1$ $X=\{v\in L^*_\lambda;\, \norm{v}_{L^*_\lambda}=1\}$; this is the boundary of Grauert tube of $L^*_\lambda$.
Let $\pi:L^*_\lambda\rightarrow T_n$
be the natural projection from $L^*_\lambda$ onto $T_n$. Let $L_\mu$ be another holomorphic
line bundle over $T_n$ determined by the constant curvature form
$\Theta_\mu=\sum^n_{j=1}\mu_jdz_j\wedge d\overline z_j$,
where $\mu_j$, $j=1,\ldots,n$, are given non-zero integers. The pullback line bundle $\pi^*L_\mu$ is a holomorphic line bundle over $L^*_\lambda$. The Hermitian fiber metric $\phi_\mu$ on $L_\mu$ induces a Hermitian fiber metric on $\pi^*L_\mu$ that we
shall denote by $\psi$. If we restict $\pi^*L_\mu$ on $X$, then $\pi^*L_\mu$ is a CR line bundle over the CR manifold $X$.
The part of $X$ that lies over a fundamental domain of $T_n$ can be represented in local holomorphic coordinates
$(z, \xi)$, where $\xi$ is the fiber coordinates, as the set of all $(z, \xi)$ such that
$r(x, \xi):=\abs{\xi}^2\exp(\sum^n_{j=1}\lambda_j\abs{z_j}^2)-1=0$
and the fiber metric $\psi$ may be written as $\psi(z, \xi)=\sum^n_{j=1}\mu_j\abs{z_j}^2$.
We can identify $\mathcal{L}_p$ with $\frac{1}{\norm{dr(p)}}\sum^n_{j=1}\lambda_jdz_j\wedge d\overline z_j$.
It is easy to see that
$\partial\overline\partial\psi(p)|_{\Lambda^{1,0}T(X)}=\sum^n_{j=1}\mu_jdz_j\wedge d\overline z_j$.
We get for all $p\in X$, $s\in\mathbb R$,
\[
\partial\overline\partial\psi(p)|_{\Lambda^{1,0}T(X)}+s\mathcal{L}_p=\sum^n_{j=1}\Big(\mu_j+\frac{s}{\norm{dr(p)}}\lambda_j\Big)dz_j\wedge d\overline z_j\,.
\]
Thus, if $\mu_j=\lambda_j$, $j=1,\ldots,n$, and $q\neq n_-, n-n_-$, then $\mathbb R_{\phi(p),q}=\emptyset$, for all $p\in X$. From this and Theorem~\ref{s7-t1}, we obtain
\begin{thm} \label{s7-t3}
If $\mu_j=\lambda_j$, $j=1,\ldots,n$, and $q\neq n_-, n-n_-$, then
\[\dim H^q_b(X, (\pi^*L_\mu)^k)=o(k^{n+1})\,,\quad \text{as $k\to\infty$}\,.\]
\end{thm}
If $\mu_j=\abs{\lambda_j}$, $j=1,\ldots,n$, we can check that $\abs{\mathbb R_{\phi(p),0}}>0$, for all $p\in X$, where $\abs{\mathbb R_{\phi(p),0}}$ denotes the Lebesque measure of $\mathbb R_{\phi(p),0}$. Moreover, if $q>0$ and $q\neq n_-, n-n_-$, then $\mathbb R_{\phi(p),q}=\emptyset$, for all $p\in X$. From this observation
and strong Morse inequalities (Theorem~\ref{s7-t1}), we obtain
\begin{thm} \label{s7-t4}
If $\mu_j=\abs{\lambda_j}$, $j=1,\ldots,n$, and $Y(0)$, $Y(1)$ hold, then
\[\dim H^0_b(X, (\pi^*L_\mu)^k)=O(k^{n+1})\,,\quad \text{as $k\to\infty$}\,.\]
\end{thm}
\subsection{Compact Heisenberg groups: non-embedded cases} Let $\lambda_1,\ldots,\lambda_{n-1}$ be given non-zero integers.
Let $\mathscr CH_n=(\mathbb C^{n-1}\times\mathbb R)/_\sim$\,, where
$(z, \theta)\sim(\widetilde z, \widetilde\theta)$ if
\[
\textstyle
\widetilde z-z=\alpha\in\sqrt{2\pi}\mathbb Z^{n-1}+i\sqrt{2\pi}\mathbb Z^{n-1}\,,\quad
\widetilde\theta-\theta+i\sum^{n-1}_{j=1}\lambda_j(z_j\overline\alpha_j-\overline z_j\alpha_j)\in\pi\mathbb Z\,.
\]
We can check that $\sim$ is an equivalence relation
and $\mathscr CH_n$ is a compact manifold of dimension $2n-1$. The equivalence class of $(z, \theta)\in\mathbb C^{n-1}\times\mathbb R$ is denoted by
$[(z, \theta)]$.
For a given point $p=[(z, \theta)]$, we define
$\Lambda^{1,0}T_p(\mathscr CH_n)$ to be the space spanned by
\[
\textstyle
\big\{\frac{\partial}{\partial z_j}-i\lambda_j\overline z_j\frac{\partial}{\partial\theta},\ \ j=1,\ldots,n-1\big\}.
\]
It is easy to see that the definition above is independent of the choice of a representative $(z, \theta)$ for $[(z, \theta)]$.
Moreover, we can check that $\Lambda^{1,0}T(\mathscr CH_n)$ is a CR structure. Thus, $(\mathscr CH_n, \Lambda^{1,0}T(\mathscr CH_n))$ is a compact CR manifold of dimension $2n-1$. We take a Hermitian metric $(\ |\ )$ on the complexified tangent bundle $\mathbb C T(\mathscr CH_n)$ such that
\[
\Big\lbrace
\tfrac{\partial}{\partial z_j}-i\lambda_j\overline z_j\tfrac{\partial}{\partial\theta}\,, \tfrac{\partial}{\partial\overline z_j}+i\lambda_jz_j\tfrac{\partial}{\partial\theta}\,, \tfrac{\partial}{\partial\theta}\,;\, j=1,\ldots,n-1\Big\rbrace
\]
is an orthonormal basis. The dual basis of the complexified cotangent bundleis
\[
\Big\lbrace
dz_j\,,\, d\overline z_j\,,\, \omega_0:=d\theta+\textstyle\sum^{n-1}_{j=1}(i\lambda_j\overline z_jdz_j-i\lambda_jz_jd\overline z_j); j=1,\ldots,n-1
\Big\rbrace\,.
\]
The Levi form $\mathcal{L}_p$ of $\mathscr CH_n$ at $p\in\mathscr CH_n$ is given by
$\mathcal{L}_p=\sum^{n-1}_{j=1}\lambda_jdz_j\wedge d\overline z_j$.
From now on, we assume that $\lambda_1<0,\ldots,\lambda_{n_-}<0, \lambda_{n_-+1}>0,\ldots,\lambda_{n-1}>0$. Thus, the Levi form
has constant signature $(n_-, n-1-n_-)$.
Now, we construct a CR line bundle over $\mathscr CH_n$. Let $L=(\mathbb C^{n-1}\times\mathbb R\times\mathbb C)/_\equiv$ where $(z,\theta,\eta)\equiv(\widetilde z, \widetilde\theta, \widetilde\eta)$ if
\[
(z,\theta)\sim(\widetilde z, \widetilde\theta)\,,\quad
\widetilde\eta=\eta \exp(\textstyle\sum^{n-1}_{j=1}\mu_j(z_j\overline\alpha_j+\frac{1}{2}\abs{\alpha_j}^2))\,,\quad\text{for $\alpha=\widetilde z-z$}\,.
\]
where $\mu_1,\ldots,\mu_{n-1}$, are given non-zero integers. We can check that $\equiv$ is an equivalence relation and
$L$ is a CR line bundle over $\mathscr CH_n$. For $(z, \theta, \eta)\in\mathbb C^{n-1}\times\mathbb R\times\mathbb R$ we denote
$[(z, \theta, \eta)]$ its equivalence class.
It is easy to see that the pointwise norm
\[
\big\lvert[(z, \theta, \eta)]\big\rvert^2:=\abs{\eta}^2\exp\big(-\textstyle\sum^{n-1}_{j=1}\mu_j\abs{z_j}^2\big)
\]
is well-defined. In local coordinates $(z, \theta, \eta)$, the weight function of this metric is
$\phi=\sum^{n-1}_{j=1}\mu_j\abs{z_j}^2$. Note that
\[
\textstyle\overline\partial_b=\sum^{n-1}_{j=1}d\overline z_j\wedge(\frac{\partial}{\partial\overline z_j}+i\lambda_jz_j\frac{\partial}{\partial\theta})\,,\quad
\partial_b=\sum^{n-1}_{j=1}dz_j\wedge(\frac{\partial}{\partial z_j}-i\lambda_j\overline z_j\frac{\partial}{\partial\theta}).
\]
Thus
$d(\overline\partial_b\phi-\partial_b\phi)=2\sum^{n-1}_{j=1}\mu_jdz_j\wedge d\overline z_j$ and $M^\phi_p=\sum^{n-1}_{j=1}\mu_j dz_j\wedge d\overline z_j$.
Hence
\[
\textstyle
M^\phi_p+s\mathcal{L}_p=\sum^n_{j=1}(\mu_j+s\lambda_j)dz_j\wedge d\overline z_j\,,\quad\text{for all $p\in\mathscr CH_n$, $s\in\mathbb R$}.
\]
Thus, if $\mu_j=\lambda_j$, for all $j$, and $q\neq n_-, n-1-n_-$, then $\mathbb R_{\phi(p),q}=\emptyset$, for all $p\in X$. From this and Theorem~\ref{t-main2}, we obtain
\begin{thm} \label{s7-t5}
If $\mu_j=\lambda_j$, $j=1,\ldots,n-1$, and $q\neq n_-, n-1-n_-$, then
\[\dim H^q_b(\mathscr CH_n, L^k)=o(k^n)\,,\quad \text{as $k\to\infty$}\,.\]
\end{thm}
If $\mu_j=\abs{\lambda_j}$ for all $j$, we can check that $\abs{\mathbb R_{\phi(p),0}}>0$, for all $p\in X$, where $\abs{\mathbb R_{\phi(p),0}}$ denotes the Lebesque measure of $\mathbb R_{\phi(p),0}$. Moreover, if $q>0$ and $q\neq n_-, n-1-n_-$, then $\mathbb R_{\phi(p),q}=\emptyset$, for all $p\in X$. From this observation and the strong Morse inequalities (Theorem~\ref{t-main5}), we obtain
\begin{thm} \label{s7-t6}
If $\mu_j=\abs{\lambda_j}$, $j=1,\ldots,n-1$, and $Y(0)$, $Y(1)$ hold, then
\[
\dim H^0_b(\mathscr CH_n, L^k)=O(k^n)\,,\quad \text{as $k\to\infty$}\,.
\]
\end{thm}
|
1,314,259,996,726 | arxiv |
\section{Introduction}
Question Answering (QA) is an important and challenging research area in Natural Language Processing (NLP). QA systems enable efficient
\begin{figure}
\centering
\includegraphics[width=4.8 cm]{images/main_q.pdf} \caption{\footnotesize Samples from the MedMCQA dataset, along with the answer’s explanation. ({\cmark} : the correct answer)}
\label{fig:questions}
\end{figure}
access to the vast amount of information available that exists in text format.
In recent times, a significant amount of work has been done on constructing a question-answer dataset \citep{Rajpurkar2016,rajpurkar2018know,Reddy2019CoQAAC,Kwiatkowski2019,Yang2015} reading comprehension datasets \citep{Yang2018,Lai2017,Zellers2018,Yagcioglu2018,Dua2019,Bajaj2018,Huang2019}, extractive question answering \citep{Hermann2015,Trischler2017}, healthcare domain QA \citep{Jin2019,Clicr,CovidQA} and the organization of workshops \& competitions such as the Question Answering in the medical domain \& BioASQ Challenge \citep{MEDIQA, Nentidis_2020}
However, despite these successful efforts, automatic questions answering for real medical examination is still a challenge that is less explored. This type of real-world examination dataset on complex medical subjects like pharmacology, medicine, surgery, etc., is scarce. Apart from their scarcity, the requirement of a comprehensive understanding of the domain, matching human experts, makes them appealing for research pursuits.
Before this attempt, very few works have been done to construct biomedical MCQA datasets \citep{Vilares2019}, and they are (1) mostly small, containing up to few thousand questions, and (2) cover a limited number of Medical topics and Subjects.
Thus, a large-scale, diverse medical QA dataset is needed to accelerate research and facilitate more consistent and effective open-domain QA models in Medical-QA. This paper addresses the aforementioned limitations by introducing MedMCQA, a new large-scale, Multiple-Choice Question Answering (MCQA) dataset designed to address real-world medical entrance exam questions. The dataset consists of 194k high-quality medical domain MCQs covering 2.4k healthcare topics and 21 medical subjects to provide a reliable and diverse benchmark. Apart from the question, the correct answer(s), and other options., it also consists of various ancillary data, the primary being a detailed explanation of the solution.
Questions are taken from AIIMS \& NEET PG entrance exam MCQs, where graduate medical students are evaluated on their professional knowledge. Questions in these exams are challenging and generally require deeper domain and language understanding as it tests the 10+ reasoning abilities across a wide range of medical subjects \& topics. Hence a model must be trained to find relevant information from the open domain knowledge base, reason over them, and choose the correct answer.
Fig.\ref{fig:questions} shows two example questions, their corresponding explanation, and answers from the study dataset.
An in-depth analysis \& a thorough evaluation of the dataset are conducted. The baseline experiments on this dataset with the current state-of-the-art methods can only answer 47\% of the question correctly, which is far behind the performance of human candidates (merit candidates of these exams score an average of 90\% marks). Error analysis and results indicate possibilities for improvement in the current methods' reasoning and medical domain question answering. It is believed that this dataset would be an appropriate testbed for future research in this direction.
In brief, the contributions of this study are as follows.
\begin{itemize}
\item \textbf{Diversity and difficulty} This dataset offers several advantages over existing datasets: (i) Covers ~2.4k healthcare topics and 21 medical subjects with an average token length of 12.77, the diversity of questions in MedMCQA demonstrate challenges unique to the dataset. (ii) It is larger than pre-existing Medical QA datasets, (iii) As these questions are from real-world and mock examinations, all the questions and candidate options are created by human experts. These questions are a comprehensive evaluation of a medical practitioner's professional skills, (iv) The questions are difficult \& challenging. They test the 10+ reasoning abilities of a model across a wide range of medical subjects \& topics.
\item \textbf{Quality} Detailed statistics, analysis of the data, and fine-grained evaluation per medical subject are provided, yielding a more precise comparison between models. Each sample contains a question, correct answer(s), other options, and a detailed explanation of the solution.
\item \textbf{Evaluation of quality} Extensive experiments are conducted using high-performance pre-trained medical domain models. Error analysis is also provided to illustrate the major challenges of this task. The baseline experiments on this dataset with the most current state-of-the-art methods answer only 47\% of the question correctly, which is far behind the human performance of 90\%, indicating possibilities for improvement in models' reasoning ability \& constitutes a challenging benchmark for future research.
\item \textbf{Reproducible exam-based split} The dataset is split based on the exams instead of a question-based split (explained in section \ref{apd:gll}). This ensures that the evaluation is closer to the real-world examinations, model generalizability, and reusability. Individual Examinations tend to have similar questions or pattern of questions repeated periodically. Exam based split avoid this leakage of similar questions into test set, hence helping in generalizability of the dataset. The dataset code to reproduce the experiments \& the leaderboard to track the progress of MedMCQA are available at \url{medmcqa.github.io}
\end{itemize}
\begin{table*}[!ht]
\small
\centering
\begin{tabular}{lccccccc}
\toprule
{\bf Dataset} & {\bf \# Question} & {\bf \# Subject} & {\bf Publicly Available} & {\bf Explanation} & {\bf Split Type} & {\bf Open Domain} \\
\midrule
MedQA & 270,000 & - & {\xmark} & {\xmark} & random & {\cmark} \\
HEAD-QA & 13,530 & 6 & {\cmark} & {\xmark} & yearwise & {\cmark} \\
\textbf{MedMCQA} & 193,155 & 21 & {\cmark} & {\cmark} & exam-based & {\cmark} \\
\bottomrule
\end{tabular}
\caption{Comparison of MedMCQA with several existing MCQA datasets(MedQA\citep{zhang2018medical}, HEAD-QA\citep{Vilares2019}) in the medical domain. {\cmark} represents the dataset that has the feature and {\xmark} represents it does not}
\label{tab:comparison}
\vspace{-2ex}
\end{table*}
\section{The MedMCQA Dataset}
In this section, properties of the MedMCQA dataset are presented. Data collection, preparation, preprocessing, and train/test/development splits are discussed.
\subsection{Task Definition}
The MedMCQA task can be formulated as $\mathbf{X = \{Q, O\}}$ where $\mathbf{Q}$ represents the questions in the text, $\mathbf{O}$ represents the candidate options, multiple candidate answers are given for each question $\mathbf{O = \{O_{1}, O_{2}, ..., O_{n} \}}$. The goal is to select the single or multiple answers from the option set.The ground truth label of a data point is $y \in \mathbb{R}^{n}$ where ${y}^{i}$ $=$ $\mathbf{\{0,1\}}$ and ${n}$ is the number of options, the objective is to learn a prediction function ${f : X \rightarrow {y}}$
\subsection{Dataset collection}
All India Institute of Medical Sciences (AIIMS PG) \& National Eligibility cum Entrance Test (NEET PG) are the two medical entrance exams conducted by All India Institute for Medical Sciences (AIIMS) \& National Board of Examinations (NBE), respectively, for providing admission to the postgraduate medical courses. The applicants must have obtained an Bachelor of Medicine and Bachelor of Surgery (MBBS) from a recognized institute to appear for the exams. The exams are used to evaluate the candidates in a structured format, namely, Diagnostic Reasoning and Treatment, Pharmacology, Psychology, Biology, Physical Examination, General Management Strategies, Medical Knowledge, and many other aspects of health and general attitude demeanor of the patient and the examiners. These exams are a comprehensive evaluation of the professional skills of a medical practitioner.
In this paper, the raw data is collected from open websites and books that put together several mock tests and online test series created by medical professionals. In addition to the collected data, AIIMS \& NEET PG examination questions (1991- present) from the official websites are also used to create the MedMCQA.
The dataset contains MCQs with fine-grained human-labeled classes on various graduation level medical subjects. Each sample contains ID, question, correct answer, and options. Besides, an explanation of the solution is also provided.
\subsection{Preprocessing \& Quality Checks}
To ensure that all the questions are answerable using textual input only,
the following steps were taken to clean the raw data, considering questions from several data sources,
\begin{itemize}
\item Questions with an inconsistent format were excluded, e.g., a question where the number of options was not four(excluding punctuation marks).
\item Questions with no best answer and missing or null candidates were also omitted.
\item Questions whose validity relied on external information were filtered, i.e., the articles and questions containing images or tables.
\item Questions containing the keywords ``equation", ``India", ``graph", ``map" etc., were removed using a manually curated list of words.
\item Further, heuristic rules were also used. For example, in some cases, the question contained HTML tags, special symbols, URLs, extra whitespaces, and missing options. Different tools were used, e.g., a spell checker, an HTML parser, to identify and correct these cases.
\item A proofreading tool, `Grammarly' was used for all the questions, options, and explanations in the dataset to fix the grammar, punctuation, and spelling mistakes. Appropriate suggestions from the tool were applied to the content with human supervision to improve the dataset's quality. As a result, many errors could be corrected
\item Lastly, all duplicated questions were removed.
\end{itemize}
Additional data cleansing steps were carried out to ensure that the question has provided information that matches the data quality goals. The final dataset contains 193,155 questions.
\subsection{Split Criteria}
\label{apd:gll}
The goal of MedMCQA is to emulate the rigor of real word medical exams. To enable that, a predefined split of the dataset is provided. The split is by exams instead of the given questions. This also ensures the reusability and generalization ability of the models.
The training set of MedMCQA consists of all the collected mock \& online test series, whereas the test set consists of all AIIMS PG exam MCQs (years 1991-present). The development set consists of NEET PG exam MCQs (years 2001-present) to approximate real exam evaluation.
In the dataset, leakages of similar questions from the training data to test and dev could artificially inflate the models' performance. This is avoided by building the development and test set to include sufficiently different training data questions.
The Levenshtein distance between each pair of questions was computed in the entire dataset. If the similarity between the two documents was larger than 0.9, the question was excluded from the development and test set. The final dataset contains ~183K train examples, 6K in the development set, and 4K in the test set.
\begin{figure}
\centering
\includegraphics[width=6 cm]{images/question_types2.pdf}
\caption{ \footnotesize Relative sizes of Question Types in MedMCQA}
\label{fig:question_types}
\end{figure}
\begin{figure}[ht]
\centering
\begin{minipage}[b]{0.50\linewidth}
\includegraphics[width=0.90\linewidth]{images/PubMed_Context12.pdf}
\label{fig:stasdet0}
\vspace{2ex}
\end{minipage
\begin{minipage}[b]{0.50\linewidth}
\includegraphics[width=.90\linewidth]{images/Question.pdf}
\label{fig:stasdet1}
\vspace{2ex}
\end{minipage}
\begin{minipage}[b]{0.50\linewidth}
\includegraphics[width=.90\linewidth]{images/Answer.pdf}
\label{fig:stasdet2}
\vspace{2ex}
\end{minipage
\begin{minipage}[b]{0.50\linewidth}
\includegraphics[width=.90\linewidth]{images/Explanation.pdf}
\label{fig:stasdet3}
\vspace{2ex}
\end{minipage}
\caption{ \footnotesize (a) distribution of Pubmed context length (b) Distribution of question length (c) Distribution of answer length (d) Distribution of explanation length }
\label{fig:stasdet}
\end{figure}
\section{Data statistics}
This dataset covers many medical subjects based on the AIIMS \& NEET PG entrance exams. The train, development, and test set consist of 182,822 , 4,183 \& 6,150 questions with an average token length of 12.35, 13.91 \& 9.68, respectively.
The general statistics of preprocessed data are summarized in table \ref{tab:data_split}
An additional informative statistic is the count of unique tokens in the dataset plotted in Fig. \ref{fig:token_dist}. Vocabulary size is a good measure of linguistic and domain complexity associated with a text corpus and influences the models' performance. It is observed that the length of questions and the vocabulary size in the AIIMS PG exams (test set) are larger than that of the NEET PG exams (dev. set). Hence, it can be inferred that questions from AIIMS are more complex than NEET.
\begin{table}[!ht]
\footnotesize
\centering
\begin{tabular}{lcccc}
\toprule
& {\bf Train} & {\bf Test} & {\bf Dev} & {\bf Total} \\
\midrule
Question \# & 182,822 & 6,150 & 4,183 & 193,155 \\
Vocab & 94,231& 11,218 & 10,800 & 97,694 \\
Max Q tokens & 220 & 135 & 88 & 220 \\
Max A tokens & 38 & 21 & 25& 38 \\
Max E tokens & 3,155 & 651& 695& 3,155 \\
Avg Q tokens & 12.77 & 9.93 & 14.09& 12.71 \\
Avg A tokens & 2.69& 2.58& 3.19& 2.70 \\
Avg E tokens & 67.52 & 46.54 & 38.44& 66.22 \\
\bottomrule
\end{tabular}
\caption{MedMCQA dataset statistics, where Q, A, E represents the Question, Answer, and Explanation, respectively}
\label{tab:data_split}
\vspace{-2ex}
\end{table}
\section{Data Analysis}
An analysis of the dataset is presented in the subsequent sections. The difficulty and diversity of questions and the answers were analyzed to understand the MedMCQA dataset's properties. The complexity of MedMCQA is demonstrated by considering the question and reasoning types covered in the dataset.
\begin{figure*}[!ht]
\includegraphics[width=16cm]{images/vocab_diff3.pdf}
\caption{ \footnotesize Distribution of unique tokens \& Cumulative Frequency Graph in the union of Train, Test, and Development split in MedMCQA dataset. The vocabulary size in the AIIMS PG exams (Test Set) is larger than that of the NEET exams (Dev. Set). Thus indicating the correlation between vocabulary size and difficulty level of the exam.
}
\label{fig:token_dist}
\end{figure*}
\subsection{Difficulty and Diversity of Questions}
In clinical medicine, a diverse number of questions are possible as it is spread over a range of topics. For example, given the description of a patient's condition, the question might be asked for the most probable diagnosis/the most appropriate treatment or examination required/mechanism of a certain condition, etc.
The majority of the dataset questions are non-factoid and open-ended in nature and seek detailed
information about the health condition. Questions in MedMCQA are fairly long, with a mean length of 12.77 words, indicating the compositional nature of questions and different levels of complexity and details covered.
To understand the types of questions in MedMCQA, 25\% of questions were sampled, and their properties were analyzed manually. It was observed that 68\% of the questions started with an interrogative word, which generally tends to be open-ended. The dataset also contained many dichotomous questions, which often require explanations. The diversity of questions in the MedMCQA makes it a challenging dataset containing many aspects of medical knowledge. Another distinguishing factor of this dataset is that it has questions that were created for and by human domain experts.
\subsection{Answer types}
In the dataset, each question contains four options with an average length of 2.69 tokens. Out of which, 25\% examples were sampled from the development set, and the answer types are presented in Fig. \ref{fig:answer_types}. As shown, MedMCQA covers a broad range of answer types, which matches the analysis on questions' contribution.
The answers were manually categorized, and it was observed that answers regarding drug/medicine's name accounted for 22.49\%. Medical procedure/Treatment type aiming to determine, measure, or diagnose a condition or parameter accounted for 18.74\% of answers. In comparison, 11.24\% of answers were related to the quantity of dose(in unit). It was observed that side effects, causes \& affected body parts accounted for 12.74\%, 10.49\% \& 9.75\% of the dataset. The rest of the answer groups contained fewer instances of the time period, adverse events \& other types.
\subsection{Subject \& Topic Analysis}
Fig. 8(\ref{apd:first}) in the Appendix presents the distribution of medical topics per subject for the datasets. Almost 95\% of the subjects contain above 50 topics, while 70\% of subjects exceed 100 topics exhibiting a plethora of medical content. Topics range from Medicine (Endocrinology, Infection, Haematology, Respiratory, etc.), Surgery (General Surgery, Endocrinology, breast, and Vascular surgery, etc.) to Radiology \& Biochemistry. This wide range of topics increases the dataset's difficulty.
\subsection{Reasoning Types}
\label{apd:re_type}
To provide a detailed \& better understanding of the dataset\'s reasoning types, 25\% of questions from MedMCQA were sampled randomly. The reasoning types required to answer were manually analyzed. The procedure was followed, and the annotation types presented in \citep{Clark2018} were re-used to categorize them into the following reasoning types:
\begin{figure}
\includegraphics[width=7.5cm]{images/answer_types.pdf}
\caption{ \footnotesize Relative sizes of Answer Types in MedMCQA}
\label{fig:answer_types}
\end{figure}
\begin{itemize}
\item \textbf{Question logic} In this, the reasoning is tested by excluding the distractor.
\item \textbf{Factual} These are the questions that have facts as answers.
\item \textbf{Explanation/definition} The questions that require selection of definition or explanation or a term/phenomenon.
\item \textbf{MultiHop Reasoning} To answer these questions, the reasoning is required from multiple passages.
\item \textbf{Analogy} In these types of questions, the responder must select the most similar/analogous answer.
\item \textbf{Teleology/purpose} Requires understanding of the purpose of a phenomenon/a thing.
\item \textbf{Comparison} Questions that require reasoning by comparing multiple options.
\item \textbf{Fill in the blanks} The responder selects the most appropriate answer suitable to fill the blanks.
\item \textbf{Natural language inference} Determining whether a hypothesis is true, false (contradiction), or neutral given an assumption.
\item \textbf{Mathematical} Questions that require mathematical critical thinking and logical reasoning.
\item \textbf{Treatment} Questions that require selection of a correct treatment method for a given ailment / condition.
\item \textbf{Diagnosis} Questions that require selection of a correct cause of a given ailment / condition.
\end{itemize}
Fig. \ref{fig:reasoning_types} shows statistics \& examples of major reasoning types in the dataset.
\begin{figure}
\centering
\includegraphics[width=7 cm]{images/reasoning_types.pdf}
\caption{\footnotesize Relative sizes of Reasoning Types in MedMCQA}
\label{fig:reasoning_types}
\end{figure}
\section{Baseline Models}
\label{apd:bm}
The primary motivation of the baseline experiments is to understand the adequacy of the current models in answering multiple-choice questions meant for human domain experts (post-graduate medical students) and to understand the level of domain specificity required in the models. Therefore, models and knowledge sources with varying levels of specificity are selected. We consider four existing models in our baseline experiments.
They are based on different pre-trained language models using Transformers architecture \citep{NIPS2017_3f5ee243} , including BERT \citep{Devlin2019} , SciBERT \citep{Beltagy2019}, BioBERT \citep{c99d46c12d234e77957c3d847b64f5cf} and PubmedBERT\citep{Gu2020}. We fine-tuned these models on our training dataset in a multiclass classification fashion. We consider models of base size. BERT is evaluated for its out-domain pretraining, SciBERT and BioBERT for their mixed domain and in-domain continual training, and PubmedBERT for its in-domain pretraining. These models are explained in detail in the following section,
\subsection{SciBERT}
SciBERT \citep{Beltagy2019} is a pretrained language model based on BERT. The model has been pre-trained from scratch on 1.14M papers on the semantic scholar. Even though SciBERT has been pre-trained from scratch, it has a mix of computer science (18\%) and biomedical domain (82\%), making it a mix-domain pretrained model. The uncased version of the model that uses a vocabulary called scivocab is used, which is a domain-specific vocabulary of size 30K
\subsection{BioBERT}
BioBERT \citep{c99d46c12d234e77957c3d847b64f5cf} is the first biomedical domain-specific pretrained language model based on BERT. The model is initialized with standard BERT weights (pretrained from Wikipedia and BookCorpus), and continual pretraining is performed with PubMed abstracts and full texts. The model uses the same vocabulary as the standard BERT model. The base variant of the 1.1 version of the model is used in the experiments.
\subsection{PubMedBERT}
PubMedBERT \citep{Gu2020} is a recent domain-specific pre-trained language model that is first to pretrain only on in-domain texts (PubMed abstracts
\begin{figure*}
\centering
\includegraphics[width=10cm]{images/diagrammonk_main.pdf}
\caption{ \footnotesize The Retriever+Reader Pipeline for Open-Domain Question Answering system used in our experiments.Dense passage retrieval \citep{karpukhin2020dense} and PubMedBERT \citep{Gu2020} are used to evaluate Wikipedia and PubMed as knowledge bases respectively, while different transformer models (explained in section \ref{apd:bm}) as reader models.}
\label{fig:open_domain}
\end{figure*}
\noindent and full texts). The base version of the model trained with both abstracts and full texts is used in the experiments. This model is used to evaluate the performance of a fully in-domain pre-trained model on the dataset.
\subsection{Retriever models}
With the recent success of neural retrievers, dense passage retrieval \citep{karpukhin2020dense}, and PubMedBERT\citep{Gu2020} were utilized to evaluate Wikipedia and PubMed as knowledge bases, respectively. Dense passage retriever follows a siamese/bi-encoder architecture; One encoder encodes the documents and another to encode the query, originally trained with Maximum inner product search objective. The pretrained DPR model and Wikipedia index from Transformer's library \citep{Wolf2020} were used in the experiments.
\section{Experiments}
\label{apd:exp}
To complement the motivation stated in section \ref{apd:bm}, The reader models were chosen with varying domain specificity levels. The contribution of external knowledge sources (Wikipedia and PubMed) was evaluated by providing these sources as contexts. Furthermore, an ablation study was also performed on context by training and evaluating all the models without context. This was done to understand the contribution of external context and the usefulness of the internal knowledge stored in these domain-specific models. The baseline experiments are broadly classified as follows,
\begin{itemize}
\item \textbf{Out-Domain}:
Pre-trained models trained on out-domain corpora like Wikipedia and Book corpus were used in this experiment type.
\item \textbf{Mix domain (continual)}:
Pre-trained models trained on out-domain initially and later adapted to in-domain or trained from scratch on both out-domain and in-domain corpora were used in this experiment.
\item \textbf{In-Domain}:
Pre-trained models trained from scratch on in-domain corpora like PubMed abstracts and full texts were used in this experiment type.
\end{itemize}
All these experiments were repeated with and without external knowledge context.
\subsection{Pubmed Data Preprocessing}
Before encoding the passages, the passages were truncated to 250 token lengths to fit the memory.
\subsection{Retriever}
\label{apd:rt}
For the experiments that involve context, a retriever+reader pipeline approach was opted (as introduced in \citep{Chen2017}). The out-of-the-box retriever models were used (explained in the section \ref{apd:rt}) from Huggingface's Transformers library \citep{Wolf2020} to encode the passages and questions. The passage with the highest cosine similarity was retrieved and used as a context for training the reader models.
\subsection{Reader finetuning}
The finetuning approach was followed as in \citep{Devlin2019} to finetune the reader models. The highest scoring contexts for each question are retrieved from the retriever. These contexts are combined by \colorbox{mygreen}{ \footnotesize [SEP] } token with the concatenation of question and answer pair. This creates four input sequences per question.
\colorbox{mygreen}{ \footnotesize [CLS] }Context\colorbox{mygreen} { \footnotesize [SEP] }Question\colorbox{mygreen}{ \footnotesize [SEP] } Option\colorbox{mygreen}{ \footnotesize [SEP] }
A linear layer with softmax is applied over the output of the \colorbox{mygreen}{ \footnotesize [CLS]} token of the encoder. This is to select the most appropriate option for a question and context pair.
For the experiments that do not use context, question and answer pair concatenation is encoded, and a linear layer with softmax is applied over the output of the \colorbox{mygreen}{ \footnotesize [CLS] } token of the encoder to select the most appropriate option for a question.
\colorbox{mygreen}{ \footnotesize [CLS] } Question \colorbox{mygreen}{ \footnotesize [SEP] } Option \colorbox{mygreen}{ \footnotesize [SEP] }
The models were finetuned on two Tesla T4 GPUs for 5 epochs with a learning rate of 2e-4 and a batch size of 16. The model checkpoint with the highest validation score in the 5 epochs was selected and used to evaluate the Test Set.
\section{Error Analysis}\label{apd:error_anaysis}
The error analysis details on a sample set of mispredictions by the best baseline model (PubMedBERT) is given in this section. The analysis was done manually for about 100 mispredictions that were sampled.This could be used for further research to improve the models/methods on the dataset.
\begin{itemize}
\item \textbf{Multi-hop reasoning}: It was observed that the model often mispredicted the questions related to the cause of an event (diagnosis) and the right course of action (treatment) in a given medical situation. Such questions typically require information on multiple symptoms, ailments, and treatments to select the most appropriate choice. This multiplicity of information is not likely to be present in one passage, possibly the reason for the mispredictions.
\item \textbf{Incorrect context passages}: It is observed that inadequate contexts from the retriever are also major contributors to the mispredictions.
\item It is found that the models mispredicted the questions requiring arithmetic reasoning. This is in line with the observations in \citep{Dua2019} on BERT-based models.
\end{itemize}
\section{Result \& Discussion}
In this section, the results from the evaluation of the methods discussed in section \ref{apd:exp} are presented.
\begin{itemize}
\item It is observed that PubMedBERT performs better than other models in all the categories. This aligns with the results from \citep{Gu2020} where PubMedBERT surpasses all other biomedical models in the majority of BLURB tasks. Examples of correct and incorrect predictions of the model is presented in Table \ref{tab:predictions}
\item PubMedBERT is followed by SciBERT (mix domain pretraining) and BioBERT (continual pretraining) in accuracy. From this result, it can be inferred that the model's performance decreases with a decrease in domain specificity of the models and external knowledge sources.
\item It is observed that there is an insignificant improvement in the model's performance when Wikipedia is used as context compared to without context results, and the model variants trained on PubMed, which have a 4-7\% improvement in the performance. This can be attributed to the domain specificity of the external knowledge source required by the dataset. The majority of the reasoning types (Diagnosis, treatment, etc.) mentioned in \ref{apd:re_type} require domain expertise as these questions are intended for post-graduate medical students.
\item The subject wise accuracies of the top PubMedBERT model is presented in Table \ref{tab:subject_wise}
\end{itemize}
\begin{table}[!ht]
\small
\centering
\begin{tabular}{l|ccccc}
\toprule
{\bf Subject Name} & {\bf Test} & {\bf Dev} \\
\midrule
Anaesthesia & 0.47 & 0.26 \\
Anatomy & 0.40 & 0.39 \\
Biochemistry & 0.48 & 0.49 \\
Dental & 0.43 & 0.36 \\
ENT & 0.47 & 0.52 \\
FM & 0.48 & 0.35 \\
O\&G & 0.54 & 0.39 \\
Medicine & 0.49 & 0.47 \\
Microbiology & 0.50 & 0.44 \\
Ophthalmology& 0.60 & 0.51 \\
Orthopaedics & - & 0.33 \\
Pathology & 0.53 & 0.46 \\
Pediatrics & 0.39 & 0.45 \\
Pharmacology & 0.46 & 0.46 \\
Physiology & 0.47 & 0.47 \\
\rowcolor{rowgray}
\textbf{Psychiatry} & \textbf{0.67} & \textbf{0.56} \\
Radiology & 0.42 & 0.31 \\
Skin & 0.50 & 0.29 \\
PSM & 0.44 & 0.35 \\
Surgery & 0.50 & 0.43 \\
Unknown & 0.44 & 1.0 \\
\bottomrule
\end{tabular}
\caption{Fine-grained evaluation per medical subject in test and dev set}
\label{tab:subject_wise}
\vspace{-2ex}
\end{table}
\begin{table}[t!]
\small
\centering
\resizebox{0.45\textwidth}{!}{%
\begin{tabular}{@{}l|r@{ }r@{ }|r@{ }r@{ }|rrr@{ }}
\toprule
{} & \multicolumn{2}{c|}{\bf w/o
Context} & \multicolumn{2}{c|}{\bf Wiki} & \multicolumn{2}{c}{\bf PubMed}\\
\midrule
\bf Model & \multicolumn{1}{c}{\bf Test} & \multicolumn{1}{r|}{\bf Dev} & \multicolumn{1}{c}{\bf Test} & \multicolumn{1}{r|}{\bf Dev} & \multicolumn{1}{c}{\bf Test} & \multicolumn{1}{c}{\bf Dev}\\
\midrule
Bert$_{Base}$ & 0.33 & 0.35 & 0.33 & 0.35 & 0.37 & 0.35\\
BioBert & 0.37 & 0.38 & 0.39 & 0.37 & 0.42 & 0.39\\
SciBert & 0.39 & 0.39 & 0.38 & 0.39 & 0.43 & 0.41\\
\rowcolor{rowgray}
PubMedBERT & \textbf{0.41} & \textbf{0.40} & \textbf{0.42} & \textbf{0.41} & \textbf{0.47} & \textbf{0.43}\\
\bottomrule
\end{tabular}
}
\caption{Performance of all baseline models in accuracy (\%) on MedMCQA test-dev set
}
\label{tab:acc}
\end{table}
\section{Conclusion}
In this work, MedMCQA, a new large-scale, Multi-Choice Question Answering (MCQA) dataset, is presented, which requires a deeper domain and language understanding as it tests the 10+ reasoning abilities of a model across a wide range of medical subjects \& topics. It is demonstrated that the dataset is challenging for the current state-of-the-art methods and domain-specific models, with the best baseline achieving only 47\% accuracy. It is expected that this dataset would facilitate future research in this direction.
\section*{Institutional Review Board (IRB)}
This research does not require IRB approval.
|
1,314,259,996,727 | arxiv | \section{}
\begin{abstract}
For a cubic algebraic extension $K$ of $\mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_{K}(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for the sum
$$
\sum\limits_{n_{1}^2+n_{2}^2\leq x}a_{K}(n_{1}^2+n_{2}^2).
$$
\textbf{Keywords}: Non-normal extension; Ideal counting function; Rankin-Selberg convolution.
\end{abstract}
\section{Introduction}
Let $K$ be an algebraic extension of $\mathbb{Q}$ with degree $d$. The associated Dedekind zeta function is defined by
$$
\zeta_K(s)=\sum\limits_{\mathfrak{a}}\mathfrak{N}(\mathfrak{a})^{-s},\quad \Re s>1,
$$
where the sum runs over all integral ideals in $\mathcal{O}_K$, and $\mathfrak{N}(\mathfrak{a})$ is the norm of the integral ideal $\mathfrak{a}$.
Since the norm of an integral ideal is a positive rational integer, the Dedekind zeta function can be rewritten as an ordinary Dirichlet series
$$
\zeta_K(s)=\sum\limits_{n=1}^{\infty}a_K(n)n^{-s},\quad \Re s>1,
$$
where $a_K(n)$ counts the number of integral ideals $\mathfrak{a}$ with norm $n$ in $K$, we call it the ideal counting function. A great number of deep arithmetic properties of a number field are hidden within its Dedekind zeta function.
It is known that the ideal counting function $a_K(n)$ is a multiplicative function, and it has the upper bound
$$
a_K(n)\ll \tau^d(n),
$$
where $\tau(n)$ is the divisor function, see~\cite{Chandrasekharan K}.
Landau~\cite{Landau} in 1927 gave the average behavior of the ideal counting function
$$
\sum\limits_{n\leq x}a_K(n)=cx+O(x^{\frac{n-1}{n+1}+\varepsilon})
$$
for arbitrary algebraic number field of degree $d\geq2$.
Nowak~\cite{Nowak} then established the best estimation hitherto in any algebraic number field of degree $d\geq3$. By using the decomposition of prime number $p$ in $\mathcal{O}_{K}$ and the analytic properties of $L$-functions, in paper~\cite{Lv}, L\"{u} considered the average behavior of moments of the ideal function
$$
\sum\limits_{n\leq x}a_K^l(n),\quad l=1,2,\cdots
$$
and gave a sharper estimates for $l=1$ in the Galois extension over $\mathbb{Q}$, while later L\"{u} and the author~\cite{Yang} gave a bound for the sum
$$
\sum\limits_{n\leq x}a_K^l(n^2),\quad l=1,2,\cdots
$$
in the Galois extension over $\mathbb{Q}$.
For a non-normal extension $K$ of $\mathbb{Q}$, the decomposition of $p$ in $\mathcal{O}_{K}$ is complex, so we can not use the same method as the normal extension.
In 2008, by applying the so-called strong Artin conjecture, Fomenko~\cite{Fo} get
the results
$$
\sum\limits_{n\leq x}a_K^l(n),\quad l=2,\,3,
$$
when $K$ is a non-normal cubic field extension. Later, L\"{u}~\cite{Lv1} revised the error term.
In this paper,
the author will
be interested in the estimation of the following sum
\begin{equation}
\label{the aim}
\sum_{n_1^2+n_2^2\leq x}a_K(n_1^2+n_2^2).
\end{equation}
where $K$ is the cubic algebraic extension of $\mathbb{Q}$.
For the purpose, we first consider the arithmetic function $r(n)$ which is the number of representation of an integer $n$ as sums of two square integers. $i.e.$
$$
r(n)=\#\{n\in\mathbb{Z}| n=n_1^2+n_2^2\}.
$$
Then, we can rewrite the formula~\eqref{the aim} as
\begin{equation}
\sum_{n_1^2+n_2^2\leq x}a_K(n_1^2+n_2^2)=\sum\limits_{n\leq x}a_K(n)\sum_{n=n_1^2+n_2^2}1=\sum\limits_{n\leq x}a_K(n)r(n).
\end{equation}
It is known that $r(n)$ is the ideal counting function of the Gaussian number field $\mathbb{Q}(\sqrt{-1})$ and we have
$$
r(n)=\sum\limits_{d|n}\chi^{\prime}(d),
$$
where $\chi^{\prime}$ is a real primitive Dirichlet character modulo $4$.
For general quadratic number field $L$ with discriminant $D^{\prime}$, the ideal counting function of the field $L$ is
$$
a_{L}(n)=\sum\limits_{d|n}\chi^{\prime}(d),
$$
where $\chi^{\prime}$ is a real primitive Dirichlet character modulo $\left|D^{\prime}\right|$. It is an interesting question to consider the sum
$$
\sum\limits_{n\leq x}a_K(n)a_{L}(n).
$$
Fomenko~\cite{Fo1} consider this convolution sum when both $K$ and $L$ are quadratic field. However, we shall discuss a more general case. Assume that $q\geq1$ is an integer, $\chi$ is a primitive character modulo $q$, define the function
$$
f_\chi(n)=\sum\limits_{k|n}\chi(k),
$$
then we have the following results
\begin{theorem}
\label{theorem1}
Let $K$ be a cubic normal extension of $\mathbb{Q}$ and $q\geq1$ is an integer, $\chi$ a primitive Dirichlet character modulo $q$, then we have
\begin{equation}
\sum\limits_{n\leq x}a_K(n)f_\chi(n)=xP_{5}(\log x)+O(x^{5/8+\varepsilon}),
\end{equation}
where $P_{5}(t)$ is a polynomial in $t$ with degree $5$, and $\varepsilon>0 $ is an arbitrarily small constant.
\end{theorem}
\begin{theorem}
\label{thmnonnormal}
Let $K$ be a cubic non normal extension of $\mathbb{Q}$ and $q\geq1$ is an integer, $\chi$ a primitive Dirichlet character modulo $q$, then we have
\begin{equation}
\sum\limits_{n\leq x}a_K(n)f_\chi(n)=xP_{3}(\log x)+O(x^{5/8+\varepsilon}),
\end{equation}
where $P_{3}(t)$ is a polynomial in $t$ with degree $3$, and $\varepsilon>0$ is an arbitrarily small constant.
\end{theorem}
According to the theorems above, we obtain
\begin{corollary}
Let $K$ be a cubic normal extension of $\mathbb{Q}$, and $r(n)$ the number of representation of an integer $n$ as sums of two square integers. Then we have
$$
\sum\limits_{n\leq x}a_K(n)r(n)=xP_{5}(\log x)+O(x^{5/8+\varepsilon}),
$$
where $P_{5}(t)$ is a polynomial in $t$ with degree $5$.
\end{corollary}
\begin{corollary}
Let $K$ be a cubic non normal extension of $\mathbb{Q}$, and $r(n)$ the number of representation of an integer $n$ as sums of two square integers. Then we have
$$
\sum\limits_{n\leq x}a_K(n)r(n)=xP_{3}(\log x)+O(x^{5/8+\varepsilon}),
$$
where $P_{3}(t)$ is a polynomial in $t$ with degree $3$.
\end{corollary}
\bigskip
Assume that $K$ and $L$ are Galois extensions of $\mathbb{Q}$ with degree $d_{1}$, $d_{2}$, respectively. According to the theory of Artin $L$-functions, the ideal counting functions $a_{K}(n)$ and $a_{L}(n)$ can be represented by the sum of characters of the representations of $Gal(K/\mathbb{Q})$ and $Gal(L/\mathbb{Q})$, respectively.
\section{Preliminaries}
Let $K$ be a cubic algebraic extension of $\mathbb{Q}$, and $D=df^{2}$($d$ squarefree) its discriminant; the Dedekind zeta function of $K$ is
$$
\zeta_{K}(s)=\sum_{n=1}^{\infty}a_{K}(n)n^{-s},\quad \text{for} ~\Re s>1.
$$
It has the Euler product
$$
\zeta_{K}(s)=\prod_{p}\left(1+\frac{a_{K}(p)}{p^{s}}+\frac{a_{K}(p^{2})}{p^{2s}}+\cdots \right).
$$
We will give some results about Dedekind zeta function of cubic field $K$ in the following.
\begin{lemma}
\label{lemmazetaKnormal}
$K$ is a normal extension if and only if $D=f^{2}$. In this case
$$
\zeta_{K}(s)=\zeta(s)L(s,\varphi)L(s,\overline{\varphi}),
$$
where $\zeta(s)$ is the Riemann zeta function and $L(s,\varphi)$ is an ordinary
Dirichlet series (over $\mathbb{Q}$) corresponding to a primitive character $\varphi
$ modulo $f$.
\end{lemma}
\begin{proof}
See the lemma in ~\cite{Wolfgang}.
\end{proof}
By using lemma~\ref{lemmazetaKnormal}, the Euler product of Riemann zeta function $\zeta(s)$ and the Dirichlet $L$-functions, we have
\begin{lemma}
\label{aKnormal}
Assume that $a_{K}(n)$ is the ideal counting function of the cubic normal extension $K$ over $\mathbb{Q}$, we get
$$
a_{K}(n)=\sum\limits_{xy|n}\varphi(x)\overline{\varphi}(y),
$$
Here $x$ and $y$ are integers. In particular, when $n=p$ is a prime, we get
\begin{equation}
\label{aKp}
a_{K}(p)=1+\varphi(p)+\overline{\varphi}(p),
\end{equation}
where $\varphi$ is a primitive character modulo $f$.
\end{lemma}
\bigskip
Assume that $K$ is a non-normal cubic extension over $\mathbb{Q}$, which is given by an irreducible polynomial $f(x)=x^{3}+ax^{2}+bx+c$. Let $E$ denote the normal closure of $K$
that is normal over $\mathbb{Q}$ with degree $6$,
and denoted the Galois group $\text{Gal}(E/\mathbb{Q})=S_{3}$. Firstly, we will introduce some properties about $S_{3}$(See~\cite{Frohlich}, pp. 226-227 for detailed arguments).
The elements of $S_{3}$ fall into three conjugacy classes
$$
C_{1}:~(1);\quad C_{2}:~(1,2,3),~(3,2,1);\quad C_{3}:~(1,2),~(2,3),~(1,3).
$$
with the following three simple characters: the one dimensional characters $\psi_{1}$(the principal character) and $\psi_{2}$(the other character determined by the subgroup $C_{1}\cup
C_{2}$), and the two dimensional character $\psi_{3}$.
Let $D$ be the discriminant of $f(x)=x^{3}+ax^{2}+bx+c$ and $K_{2}=\mathbb{Q}(\sqrt{D})$. The fields $K_{2}$ and $K$ are the intermediate extensions fixed under the subgroups $A_{3}$ and $\{(1),(1,2)\}$, respectively. The extensions $K_{2}/\mathbb{Q}$, $E/K_{2}$ and $E/K$ are abelian. The Dedekind zeta function satisfy the relations
$$
\begin{array}{llll}
\zeta_{E}(s)&=&L_{\psi_{1}}L_{\psi_{2}}L_{\psi_{3}}^{2},\\
\zeta_{K_{2}}(s)&=&L_{\psi_{1}}L_{\psi_{2}},\\
\zeta_{K}(s)&=&L_{\psi_{1}}L_{\psi_{3}},\\
\zeta(s)&=&L_{\psi_{1}},
\end{array}
$$
where
$$
L_{\psi_{2}}=L(s,\psi_{2},E/\mathbb{Q})\quad\text{and}\quad L_{\psi_{3}}=L(s,\psi_{3},E/\mathbb{Q}),
$$
and $L_{\psi_{2}}=L(s,\psi_{2},E/\mathbb{Q})$ and $L_{\psi_{3}}=L(s,\psi_{3},E/\mathbb{Q})$ are Artin $L$-functions.
Kim in \cite{Kim} proved that the the strong Artin conjecture holds true for the group $S_{3}$. By using the strong Artin conjecture, the function $L_{\psi_{3}}$ also can be interpreted in
another way~\cite{Deligne}. Let $\rho: S_{3}\rightarrow GL_{2}(\mathbb{C})$ be the irreducible two-dimensional representation. Then $\rho$ gives rise to a cuspidal representation $\pi$
of $\text{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$. Let
$$
L(s,\pi)=\sum_{n=1}^{\infty}M(n)n^{-s}.
$$
Below we assume that $\rho$ is odd, $i.e.$ $D<0$, then $L(s,\pi)=L(s,f)$, where $f$ is a holomorphic cusp form of weight 1 with respect to the congruence group $\Gamma_{0}(|D|)$,
$$
f(z)=\sum_{n=1}^{\infty}M(n)e^{2\pi inz}.
$$
Here as usual, $L(s,\pi)$ denotes the $L$-function of the representation $\pi$, and $L(s,f)$ denotes the Hecke $L$-function of cusp form $f$. Thus $L_{\psi_{3}}=L(s,f)$ and
\begin{equation}
\label{zetaKL}
\zeta_{K}(s)=\zeta(s)L(s,f).
\end{equation}
The formula \eqref{zetaKL} implies that
\begin{lemma}
\label{nonaKn}
The symbols defined as above. we have
$$
a_{K}(n)=\sum_{d|n}M(d).
$$
In particular,
$$
a_{K}(p)=1+M(p),
$$
where $p$ is a prime integer.
\end{lemma}
\bigskip
To prove the theorem, we also need some well-known estimates of the relative $L$-functions in the following.
For subconvexity bounds, we have the following well-known estimates.
\begin{lemma}
For any $\varepsilon>0$, we have
\begin{equation}
\zeta(\sigma+it)\ll_{\varepsilon}(1+|t|)^{(1/3)(1-\sigma)+\varepsilon}
\end{equation}
uniformly for $1/2\leq\sigma\leq1$, and $|t|\geq 1$.
\end{lemma}
\begin{proof}
See theorem II 3.6 in the book~\cite{Tenenbaum}.
\end{proof}
For the Dirichlet $L$-series, By using the Phragmen-Lindel\"of principle for a strip~\cite{Iwaniec} and the estimates given by Heath-Brown~\cite{HeathB}, we have the similar results
\begin{equation}
\label{Lschi}
L(\sigma+it, \chi)\ll_{\varepsilon}(1+|t|)^{(1/3)(1-\sigma)+\varepsilon}
\end{equation}
uniformly for $1/2\leq\sigma\leq1$, and $|t|\geq 1$, where $\chi$ is a Dirichlet character modulo $q$, and $q$ is an integer.
\bigskip
For the mean values of the relative $L$-functions, we have
\begin{lemma}
\label{zetaonehalf}
For any $\varepsilon>0$, we have
\begin{equation}
\int\limits_{1}^{T}\left|\zeta(\frac{1}{2}+it)\right|^{A}\ll_{\varepsilon}T^{1+\varepsilon}
\end{equation}
uniformly for $T\geq 1$, where $A=2, 4$.
\end{lemma}
\begin{lemma}
\label{Lonehalf}
For any $\varepsilon>0$, and $q$ is an integer, $\chi$ is a character modulo $q$. We have
\begin{equation}
\int\limits_{1}^{T}\left|L(\frac{1}{2}+it,\ \chi)\right|^{A}\ll_{\varepsilon}T^{1+\varepsilon}
\end{equation}
uniformly for $T\geq 1$, where $A=2, 4$.
\end{lemma}
For Hecke $L$-functions defined in the formula \eqref{zetaKL}, we have
\begin{lemma}
For any $\varepsilon>0$, we have
\begin{equation}
\begin{array}{cccc}
\int\limits_{1}^{T}\left|L\left(\frac{1}{2}+it,\ f\right)\right|^{2}dt&\sim& CT\log T\\
\int\limits_{1}^{T}\left|L\left(\frac{1}{2}+it,\ f\right)\right|^{6}dt&\ll & T^{2+\varepsilon}
\end{array}
\end{equation}
uniformly for $T\geq 1$, and the subconvexity bound
$$
L(\sigma+it,\ f)\ll_{t,\ \varepsilon}(1+|t|)^{\max \{(2/3)(1-\sigma),\ 0\}+\varepsilon}
$$
uniformly for $1/2\leq \sigma\leq 2$ and $|t|\geq 1$.
\end{lemma}
\begin{proof}
The first and third results due to Good~\cite{Good}, and the second result was proved by Jutila~\cite{Jutila}.
\end{proof}
We also have the convexity bounds for the relative $L$-functions.
\begin{lemma}
Let $L(s,\ g)$ be a Dirichlet series with Euler product of degree $m\geq 2$, which means
$$
L(s,\ g)=\sum\limits_{n=1}^{\infty}a_{g}(n)n^{-s}=\prod\limits_{p<\infty}\prod\limits_{j=1}^{m}\left(1-\frac{\alpha_{g}(p,\ j)}{p^{s}}\right),
$$
where $\alpha_{g}(p,\ j),\ j=1,\ 2,\ \cdots,\ m$ are the local parameters of $L(s,\ g)$ at prime $p$ and $a_{g}(n)\ll n^{\varepsilon}$. Assume that this series and its Euler product are absolutely convergent for $\Re (s)>1$. Also, assume that it is entire except possibly for simple poles at $s=0,\ 1$, and satisfies a functional equation of Riemann type. Then for $0\leq \sigma\leq 1$ and any $\varepsilon>0$, we have
\begin{equation}
L(\sigma+it,\ g)\ll_{g,\ \varepsilon}(1+|t|)^{(m/2)(1-\sigma)+\varepsilon},
\end{equation}
and for $T\geq 1$, we have
\begin{equation}
\label{Lg}
\int\limits_{T}^{2T}\left|L(1/2+\varepsilon+it, g)\right|^{2}dt\ll_{g,\varepsilon} T^{m/2+\varepsilon}.
\end{equation}
\end{lemma}
\begin{proof}
The first result is form the lemma 2.2 in \cite{Guangshi Lv}, and the second result is from the lemma 2.1 in \cite{Guangshi Lv}.
\end{proof}
\section{Proof of Theorems}
Assume that $K$ is a cubic extension of $\mathbb{Q}$. The Dedekind zeta function of $K$ is
$$
\zeta_{K}(s)=\sum_{n=1}^{\infty}a_K(n)n^{-s}, \quad \Re s>1.
$$
Its Euler product is
$$
\zeta_{K}(s)=\prod_{p}\left(1+\frac{a_{K}(p)}{p^{s}}+\frac{a_{K}(p^{2})}{p^{2s}}+\cdots \right),\quad \Re s>1.
$$
Let $q$ be an integer, and $\chi$ a primitive Dirichlet character modulo $q$. Define the function
\begin{equation}
\label{fchin}
f_\chi(n)=\sum\limits_{k|n}\chi(k).
\end{equation}
It is easy to check that $f_{\chi}(m n)=f_{\chi}(m)f_{\chi}(n)$, when $(m,\,n)=1$.
Since $a_{K}(n)\ll n^{\varepsilon}$, so does $a_{K}(n)f_{\chi}(n)$. We can define an $L$-function associated to the function $a_{K}(n)f_{\chi}(n)$ in the half-plane $\Re s>1$,
\begin{equation}
L_{K,\,f_{\chi}}(s)=\sum_{n=1}^{\infty}a_{K}(n)f_{\chi}(n)n^{-s},
\end{equation}
which is absolutely convergent in this region. Both $a_{K}(n)$ and $f_{\chi}(n)$ are multiplicative function, then for $\Re s>1$, the function $L_{K,\,f_{\chi}}(s)$ can be expressed by the Euler product
$$
L_{K,\,f_{\chi}}(s)=\prod_{p}\left(1+\frac{a_{K}(p)f_{\chi}(p)}{p^{s}}+\frac{a_{K}(p^{2})f_{\chi}(p^{2})}{p^{2s}}+\cdots\right),
$$
where the product runs over all primes.
\subsection*{Proof of Theorem~\ref{theorem1}}
When $K$ is cubic normal extension, according to the formula~\eqref{aKp} and \eqref{fchin}, we get the formula
\begin{equation}
\label{aKpfchip}
a_{K}(p)f_{\chi}(p)=1+\varphi(p)+\overline{\varphi}(p)+\chi(p)+\varphi(p)\chi(p)+\overline{\varphi}(p)\chi(p),
\end{equation}
where $p$ is a nature prime number.
For $\Re s>1$, we can write
$$
M_{K,\,f_{\chi}}(s):=\zeta(s)L(s,\,\varphi)L(s,\,\overline{\varphi})L(s,\,\chi)L(s,\,\varphi\times \chi)L(s,\,\overline{\varphi}\times \chi)
$$
as an Euler product of the form
$$
\prod_{p}\left(1+\frac{A(p)}{p^{s}}+\cdots\right),
$$
where $A(p)=1+\varphi(p)+\overline{\varphi}(p)+\chi(p)+\varphi(p)\chi(p)+\overline{\varphi}(p)\chi(p)$, and the function $L(s,\,\varphi\times \chi)$ and $L(s,\,\overline{\varphi}\times \chi)$ are the Rankin-Selberg convolution $L$-function of the Dirichlet $L$-functions $L(s,\,\varphi)$, $L(s,\,\overline{\varphi})$ with the Dirichlet $L$-functions $L(s,\,\chi)$ respectively.
By comparing it with the Euler product of $L_{K,\,f_{\chi}}(s)$, and using the formula~\eqref{aKpfchip}, we obtain
\begin{equation}
L_{K,\,f_{\chi}}(s)=M_{K,\,f_{\chi}}(s)\cdot U_{1}(s),
\end{equation}
where $U_{1}(s)$ denotes a Dirichlet series, which is absolutely convergent for $\Re s>1/2$, and uniformly convergent for $\Re s>1/2+\varepsilon$. Therefore, the function $L_{K,\,f_{\chi}}(s)$ admits an analytic continuation into the half-plane $\sigma>1/2$, having as its only singularity a pole of order $6$ at $s=1$, because $\zeta(s)$ and each of the Dirichlet $L$-functions has a simple pole at $s=1$.
\smallskip
By using the well-known inversion formula for Dirichlet series, we obtain
$$
\sum_{n\leq x}a_{K}(n)f_{\chi}(n)=\frac{1}{2\pi i}\int_{b-iT}^{b+iT}L_{K,\,f_{\chi}}(s)\frac{x^{s}}{s}ds+O(\frac{x^{1+\varepsilon}}{T}).
$$
Where $b=1+\varepsilon$ and $1\leq T\leq x$ is a parameter to be chosen later.
Shifting the path of integration to the line $\sigma= 1/2+\varepsilon$. By using Cauchy's residue theorem, we have
\begin{eqnarray}
\label{1main}
\sum_{n\leq x}a_{K}(n)f_{\chi}(n)&=&\frac{1}{2\pi i}\left\{\int_{\frac{1}{2}+\varepsilon-iT}^{\frac{1}{2}+\varepsilon+iT}+\int_{\frac{1}{2}+\varepsilon+iT}^{b+iT}+\int_{b-iT}^{\frac{1}{2}+\varepsilon-iT}\right\}L_{K,\,f_{\chi}}(s)\frac{x^{s}}{s}ds\nonumber\\
& &+\text{Res}_{s=1}L_{K,\,f_{\chi}}(s)\frac{x^{s}}{s}+O(\frac{x^{1+\varepsilon}}{T})\nonumber\\
&=&xP_{5}(\log x)+J_{1}+J_{2}+J_{3}+O(\frac{x^{1+\varepsilon}}{T})
\end{eqnarray}
Where $P_{5}(t)$ is a polynomial in $t$ with degree $5$.
Using the lemmas in section 2 about the bound for the Dirichlet series, we will estimate the $J_{i}, i=1, 2, 3$ in the following.
For $J_{1}$, we have
\begin{eqnarray}
J_{1}\ll x^{1/2+\varepsilon}+x^{1/2+\varepsilon}\int_{1}^{T}\left|M_{K,\,f_{\chi}}(1/2+\varepsilon+it)\right|t^{-1}dt
\end{eqnarray}
where we have used that $U_{1}(s)$ is absolutely convergent in the region $\Re s\geq1/2+\varepsilon$ and behaves as $O(1)$ there.
By H\"older's inequality, we have
\begin{eqnarray}
\int_{1}^{T}\left|M_{K,\,f_{\chi}}(1/2+\varepsilon+it)\right|t^{-1}dt&\ll&\log T\sup_{1\leq T_{1}\leq T}T_{1}^{-1}\cdot T_{1}^{1/6+\varepsilon}\cdot T_{1}^{1/6+\varepsilon}\times\nonumber\\
& &I_{\zeta}(T_{1})^{1/4}I_{\varphi}(T_{1})^{1/4}I_{\overline{\varphi}}(T_{1})^{1/4}I_{\chi}(T_{1})^{1/4},\nonumber
\end{eqnarray}
where we have used the formula~\eqref{Lschi}, and
\begin{eqnarray}
I_{\zeta}(T_{1}) &:=&\int\limits_{T_{1}}^{2T_{1}}\left|\zeta(1/2+\varepsilon+it)\right|^{4}dt,\nonumber\\
I_{\varphi}(T_{1}) &:=&\int\limits_{T_{1}}^{2T_{1}}\left|L(1/2+\varepsilon+it,\varphi)\right|^{4}dt,\nonumber
\end{eqnarray}
\begin{eqnarray}
I_{\overline{\varphi}}(T_{1})&:=&\int\limits_{T_{1}}^{2T_{1}}\left|L(1/2+\varepsilon+it,\overline{\varphi})\right|^{4}dt,\nonumber\\
I_{\chi}(T_{1}) &:=&\int\limits_{T_{1}}^{2T_{1}}\left|L(1/2+\varepsilon+it,\chi)\right|^{4}dt.\nonumber
\end{eqnarray}
Now, by using lemma~\eqref{zetaonehalf} and lemma~\eqref{Lonehalf}, we have the estimation
$$
\int_{1}^{T}\left|M_{K,\,f_{\chi}}(1/2+\varepsilon+it)\right|t^{-1}dt\ll T^{1/3+\varepsilon}.
$$
So we can deduce that
\begin{eqnarray}
\label{1J1}
J_{1}\ll x^{1/2+\varepsilon}+x^{1/2+\varepsilon}T^{1/3+\varepsilon}.
\end{eqnarray}
For $J_{2}$ and $J_{3}$, we have
\begin{eqnarray}
\label{1J23}
J_{2}+J_{3}&\ll&\sup_{1/2+\varepsilon\leq \sigma\leq 1+\varepsilon}x^{\sigma}T^{-1}\left|M_{K,\,f_{\chi}}(\sigma+iT)\right|\nonumber\\
&\ll&\sup_{1/2+\varepsilon\leq \sigma\leq 1+\varepsilon}x^{\sigma}T^{-1}T^{(1/3+1/3+1/3+1/3+1/3+1/3)(1-\sigma)+\varepsilon}\nonumber\\
&\ll&\frac{x^{1+\varepsilon}}{T}+x^{1/2+\varepsilon}T^{\varepsilon}.
\end{eqnarray}
Form formula~\eqref{1main}, \eqref{1J1} and \eqref{1J23}, we have
\begin{equation}
\label{1last}
\sum_{n\leq x}a_{K}(n)f_{\chi}(n)=xP_{5}(\log x)+O(x^{1/2+\varepsilon}T^{1/3+\varepsilon})+O(\frac{x^{1+\varepsilon}}{T}).
\end{equation}
Taking $T=x^{3/8+\varepsilon}$ in \eqref{1last}, we have
$$
\sum_{n\leq x}a_{K}(n)f_{\chi}(n)=xP_{5}(\log x)+O(x^{5/8+\varepsilon}).
$$
We complete the proof of Theorem~\ref{theorem1}.
\subsection*{Proof of Theorem~\ref{thmnonnormal}}
Now, assume that $K$ is a cubic non-normal extension over $\mathbb{Q}$. According to the lemma~\ref{nonaKn} and the formula~\eqref{fchin}, we have
\begin{equation}
\label{nonaKpfchip}
a_{K}(p)f_{\chi}(p)=1+\chi(p)+M(p)+\chi(p)M(p),
\end{equation}
where $p$ is a nature prime number.
By virtue of \eqref{nonaKpfchip}, we have the relation
$$
L_{K,\,f_{\chi}}(s)=\zeta(s)L(s, \chi)L(s, f)L(s, f\times \chi)\cdot U_{2}(s),
$$
where $L(s, f\times \chi)$ is the Rankin-Selberg convolution $L$-function of $L(s,\, f)$ and $L(s,\,\chi)$, and $U_{2}(s)$ denotes a Dirichlet series, which is absolutely convergent for $\sigma>1/2$. Therefore, the function $L_{K,\,f_{\chi}}(s)$ admits an analytic continuation into the half-plane $\sigma>1/2$, having as its only singularity a pole of order $4$ at $s=1$, because $\zeta(s)$ and each of the relative $L$-functions has a simple pole at $s=1$.
The degree of $L(s,\ f\times \chi)$ is $2$, according to the formula~\eqref{Lg}, we have
$$
\int\limits_{T}^{2T}\left|L(1/2+\varepsilon+it, f\times\chi)\right|^{2}dt\ll_{g,\varepsilon} T^{2/2+\varepsilon}.
$$
Similarly as the proof of Theorem \ref{theorem1}, using the inversion formula for Dirichlet series and the estimates above,
we have the main term of the sum is
$$
\text{Res}_{s=1}{L_{K,\,f_{\chi}}(s)x^{s}s^{-1}}=xP_{3}(\log x),
$$
and the error term is $O(x^{5/8+\varepsilon})$.
The proof is over.
|
1,314,259,996,728 | arxiv | \section*{Acknowledgements}
\balance
\section{Case Study}
\label{sect:case-study}
In this section, we discuss our application of \textit{Gradeer}~in an end of year introductory Java programming assignment with 171 students' solutions.
\subsection{The Assignment}
The assignment required students to parse a series of structured input files into a provided data structure, then implement a set of methods that query this data.
The assignment also required students to plot graphs using this data in a GUI using Java's Swing library.
A primary goal of the assignment was to provide students with experience in working on a multi-faceted project with codependent systems, which are more akin to real software than the simpler introductory programs used earlier in the course.
As an end of year assessment, the assignment had a fairly wide span of learning outcomes.
Such learning outcomes included the use of polymorphism, bespoke data structures, the choice and use of various Java Collections, text manipulation, GUI programming, algorithm design, and the use of good quality code and programming style.
We first determined the overall assignment specification, then focused on creating a model solution that captured this specification.
We then created a set of grading unit tests, ensuring that they were valid and that the model solution passed on each of them.
Following this, we duplicated the model solution to create a skeleton project, from which we removed the classes and methods that students were to implement.
\subsection{Release}
We distributed the skeleton project to students.
We also provided the students with a set of input data files that were to be read by their implemented parsers. These data files were a subset of those that we later used when grading the assignment.
Around a week after we released the assignment, we also provided students with a set of public tests.
We designed these tests to ensure that students' code included the basic functionality of the assignment.
This provided students with a degree of feedback as they worked on the assignment, and dissuaded students from submitting solutions which are not compatible with our grading environment, such as including incorrect class names.
\subsection{Check Configuration}
We configured \textit{Gradeer}~to use 45 checks:
\begin{itemize}
\item 26 test suite checks (each check executed one unit test),
\item six PMD checks,
\item six Checkstyle checks, and
\item seven manual checks (for GUI functionality and subjective aspects of code review, such as variable names).
\end{itemize}
By using these checks together, we were able to use \textit{Gradeer}~to assess all of our learning outcomes.
The manual checks were important in this regard, since the design of the GUI and some aspects of code quality cannot be fully graded automatically.
\subsection{Assessment}
While \textit{Gradeer}~supports the use of all types of checks in a single execution, we split the checks across two separate execution configurations; one for automated checks and one for manual checks.
This was necessary since we anticipated that some solutions would be problematic, containing issues that would prevent compilation or execution.
As such, running manual checks on some of these solutions would have been a waste of effort if the solutions could not be executed properly.
By splitting the checks we were able to first compile the students' solutions and run the automated checks to identify any problematic solutions, and to assess the working solutions.
We identified 48 problematic solutions.
We repaired these solutions where possible so that they could still be graded with \textit{Gradeer}, but added a penalty for doing so when post-processing the grades.
We repeated the automated grading for these repaired solutions.
However, 11 of the solutions could not be repaired due to severe issues.
We wrote individual feedback for each of these solutions to explain the nature of these problems.
Finally, we re-executed \textit{Gradeer}~with only the manual checks on every working and repaired solution.
Table~\ref{tbl:times} shows the average amount of time that various aspects of running the assessment with \textit{Gradeer}~took for each applicable solution, alongside the time taken to manage problematic solutions.
\begin{table}[]
\centering
\caption{Average time to perform each assessment task on each applicable solution.}
\label{tbl:times}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{Assessment Task} & \textbf{Average Time Per Solution} \\ \midrule
\textit{Compilable Solutions} & \\
Compilation & $\sim$1.6 seconds \\
38 Automated Checks & $\sim$28.2 seconds \\
7 Manual Checks & $\sim$2 minutes \\ \midrule
\textit{Problematic Solutions} & \\
Problem Identification & $\sim$11.3 minutes \\%~675 seconds
Solution Repair & $\sim$11.4 minutes \\%~681 seconds
Individual Feedback & $\sim$10 minutes \\ \bottomrule
\end{tabular}
\end{table}
Once we completed grading the assignments, we
performed some post-processing on the results.
In particular, we added some more specific feedback and adjusted the weights of some of the checks.
Providing the additional feedback revealed the possible benefit of being able to add specific feedback when running \textit{Gradeer}, leading us to later implement the ability to add user entered feedback for manual checks.
We also provided more detailed and general feedback to the entire student cohort using the distribution of solutions' base scores for individual checks.
In addition, we used this check performance data to adjust the checks' weights.
For example, we found that the scores of some checks would vary considerably between solutions, such as a PMD check for cyclomatic complexity, for which approximately half of the solutions achieved $<0.5$.
In such cases, we increased the check's weight, as it better differentiated students' solutions.
However, we attempted to maintain similar total weights between the broader groups of learning outcomes, such as overall correctness and code quality, to assess students in a well-rounded manner.
\subsection{Benefits of \textit{Gradeer}}
We found that \textit{Gradeer's}~hybrid grading approach provided several benefits when assessing this programming assignment:
\subsubsection{Fast Manual Assessment}
\textit{Gradeer}~provides a particular benefit in allowing for quick manual assessment.
This is mostly due to \textit{Gradeer's}~solution inspector, which automatically executes students' solutions, and displays their source files in a text editor.
Without this feature, a tutor must manually open the correct directory, enter a command to run the solution, and open the source files, before beginning the manual assessment.
By removing the need to follow these steps for every solution, \textit{Gradeer}~removes a significant bottleneck in manual grading.
\subsubsection{Automated Grading}
By using automated grading wherever possible, we were able to reduce the number of manual checks.
For example, we used some static analysis checks to evaluate the style of students' solution programs, such as ensuring that they used camel case formatting in variable names.
By using these checks, the tutor did not have to look for these issues when performing the manual code inspection.
Similarly, the use of unit tests to assess correctness of some elements of the program removed the need for the tutor to identify faults in these elements manually.
The additional benefit of automated grading is that the checks are applied consistently across solutions.
Any two students' solutions which have the same faults will be assessed the exact same way.
\subsubsection{High Quality Feedback}
We found that \textit{Gradeer}~was capable of providing useful feedback to students.
While automated checks only provide simple feedback, the large number of these checks gave students a very wide range of feedback; they could gain a good understanding of where they succeeded and where they can improve.
This is supported by Falkner et al.'s findings that students' performance improves as more pieces of automated feedback are provided~\cite{Falkner2014}.
This feedback is further augmented by \textit{Gradeer's}~support for manual checks, the scores of which we used to determine which of several pieces of feedback to give to a student.
The ability to provide manual feedback at runtime in the current version of \textit{Gradeer}~supports this even further.
\subsubsection{Reusable}
In the past, we typically used unique autograding scripts for each assessment.
Developing these scripts is a time consuming process, and may involve repeated effort of implementing similar functionality across multiple assessments.
Conversely, \textit{Gradeer}~can be reused in different assessments, only requiring modifications to simple configuration files.
\subsection{Challenges}
When assessing the assignment, we found that uncompilable solutions introduced the greatest time cost.
Around 48 of the 171 solutions initially could not be compiled or executed, due to missing files, syntax errors, or modifying files that should be unmodified.
It is possible that such problems could be mitigated by preventing students from uploading broken solutions, such as by integrating \textit{Gradeer}~with the solution upload system, and reporting to students if an issue is detected.
Running the automated checks did take a considerable amount of time, at $\sim$28.2 seconds per solution using an AMD Ryzen 1700 CPU.
The main source of this time cost is setting up the test execution environment.
We plan to investigate a possible workaround for this issue in the future.
In addition, the version of \textit{Gradeer}~that we used for this assessment did not support multithreading.
After implementing multithreading, we observed an execution time of $\sim$10.9 seconds per solution.
We found that some static analysis rules can present a unique challenge in being used in an automated grader.
In particular, Checkstyle's indentation rules can only be used with one tutor defined indentation width, while indentation widths may vary between solutions.
This is an issue since several different indentation widths are commonly used in software, any of which may be acceptable provided that they are used consistently.
It may be possible to use multiple similar checks and only use the highest base score as a workaround.
While using software such as \textit{Gradeer}~requires less effort than writing a unique grading script, some tutors may be dissuaded by not understanding its internal functionality.
Providing tests may increase tutors' confidence in such tools.
\section{The \textit{Gradeer}~Grading Tool}
\label{sect:gradeer}
\textit{Gradeer}~is an assessment tool which provides tutors with the benefits of both automated and manual assessment in a single package.
The tool achieves this using a modular design, allowing a user to choose how to assess a programming task using simple configuration files, or even define their own modules for specific purposes.
To allow for manual assessment, \textit{Gradeer}~is designed to be used by tutors on personal computers, where the user can interact with the program via a CLI.
It is however possible for \textit{Gradeer}~to be integrated with a GUI or web interface.
\textit{Gradeer}~is implemented in Java, and allows for the assessment of Java programs.
Wider language support is planned for future versions of the tool.
This section describes our design of \textit{Gradeer}, alongside some of its benefits.
\subsection{Checks}
\label{sect:gradeer-checks}
We designed \textit{Gradeer}~with a focus on modular grading components, called \textit{checks}, each of which represents a single grading criterion.
Different types of checks are currently implemented, defining how a criterion's base score (a decimal value between zero and one) can be determined for a given process and student's solution.
Various checks of different types can be used together in a single run of \textit{Gradeer}, constructing a markscheme to assess several learning outcomes.
For example, users can configure \textit{Gradeer}~to use multiple checks to run various test suites, perform static analysis, and manually assess several aspects of a solution.
Users configure their checks in JSON files.
Users can also implement new checks to add the functionality of unique and domain-specific grading tools.
One currently implemented type of check is the \texttt{TestSuiteCheck}, which executes a given JUnit test class on a student's solution via Apache Ant~\cite{TheApacheSoftwareFoundation}, then calculates a score as the proportion of tests that pass.
Tutors can assess individual learning outcomes by grouping tests that evaluate the same outcome into one class.
We also implemented check types for two static analysis tools, Checkstyle and PMD\cite{Checkstyle, PMD}, in order to automatically assess the code quality of students' solutions.
Such checks search the output of their respective tool for a user defined rule violation.
The number of violations in each source file of a solution is recorded and used to compute a base score.
Users can also define a minimum and maximum number of violations, which yield base scores of one and zero, respectively.
To support manual assessment, we have implemented a \texttt{ManualCheck} type, which displays a user-defined prompt and score limit to the user when executed.
This check then parses numeric input from the user and normalises it to a score in the range of zero and one.
For example, the following response would produce a base score of 0.6:
\begin{lstlisting}[linewidth=\columnwidth, breaklines=true]
How informative are the variable names?
(0 = very poor, 10 = excellent)
# 6
\end{lstlisting}
Each check has an associated weight; a score multiplication factor to allow a test to have a greater or smaller impact on each solution's overall grade, as discussed in Section~\ref{sect:gradeer-output}.
This weight can be defined by the user.
In addition, each check has associated feedback to provide to a student for their solution.
For most checks, this feedback is determined by mapping a base score to one of several feedback values that have been pre-defined by the user.
For example, the above manual check may provide students with feedback for the base scores, \textit{bs}:
\begin{itemize}
\item $0.9\leq$~\textit{bs} $\leq1.0$: ``Your variable names are informative.''
\item $0.5\leq$~\textit{bs} $<0.9$: ``Some of your variable names could be more informative.''
\item $0.0\leq$~\textit{bs} $<0.5$: ``Most of your variable names could be more informative.''
\end{itemize}
Manual checks can also read text input from the user, allowing for additional feedback to be provided on an individual basis.
For example, a tutor may enter ``\texttt{a} is not an informative variable name, \texttt{leftMotor} would be better.''
\subsection{Execution}
\label{sect:gradeer-execution}
Figure~\ref{fig:gradeer-overview} shows an overview of \textit{Gradeer's}~execution process.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{input/gradeer-overview.pdf}
\caption{Overview of \textit{Gradeer's}~flow of execution. The dotted areas indicate different phases of the execution. Waved boxes are files, parallelograms are internal data, and regular boxes are processes.}
\label{fig:gradeer-overview}
\end{figure*}
\subsubsection{Compilation \& Check Loading}
First, \textit{Gradeer}~compiles every students' solution and every model solution (Section \ref{sect:gradeer-execution:model-solution}).
At this stage, any solutions which do not compile are flagged as such.
These solutions are reported to the tutor for review, and are excluded from further execution.
Next, \textit{Gradeer}~loads every check defined in the JSON files.
The tool also compiles the test classes that are provided by the user.
If enabled, \textit{Gradeer}~automatically generates a test suite check for each test class which does not have a matching check already defined by the user.
\subsubsection{Model Solution Execution}
\label{sect:gradeer-execution:model-solution}
The user can supply a set of one or more model solutions; entirely correct solutions to the programming task being assessed.
Users can choose to use multiple model solutions to define different correct implementations of the programming task.
In order to identify and remove invalid checks, \textit{Gradeer}~executes every check on each provided model solution.
Checks which attain a base score of less than one on any of the model solutions are considered to be invalid, and are removed; they falsely claim that a model solution is partly or completely incorrect.
This prevents invalid checks from being used in the assessment of students' solutions, preventing them from unfairly losing or gaining grades, or being given inaccurate feedback.
For example, uncompilable test classes will not pass on any solutions, so their checks are removed.
The names of invalid checks are stored in a file for the tutor to review and correct.
\subsubsection{Solution Grading (for each Student's Solution)}
\paragraph{Pre-checks}
In order for some checks to function properly, a series of pre-checks are executed on each solution.
For example, checks for Checkstyle and PMD require pre-checks which execute their corresponding static analysis tool on the solution under test and store its output in memory.
\paragraph{Solution Inspection}
To support effective manual grading, \textit{Gradeer}~includes a \textit{solution inspector} which can perform two processes, as configured by the user.
The first executes a student's solution in a separate thread before running any manual checks.
This allows the user to be able to interact with the solution, and to observe its user interface, which may be relevant to the rubric of manual checks.
The solution execution thread is closed following the completion of every manual check on a given solution.
The second opens each of the solution's source files in an external user defined text editor, such as Atom.
This allows for the user to perform manual code inspection, for example to determine the quality of variable names or comments.
The solution inspector removes the need for the user to manually run a student's solution to interact with it, or open its source files to inspect it, saving time.
\paragraph{Checks}
The final step of a solution's grading process is to run every check on it.
In order to improve execution time, \textit{Gradeer}~runs automated checks in parallel by default.
Manual checks are only executed in the main thread, however, as they require user input, and henceforth could result in the occurrence of race conditions otherwise.
In order to prevent some JUnit checks from taking too long to execute, \textit{Gradeer}~has a user configurable global test timeout, where any tests that take longer than this time are treated as failing.
This is particularly important, since some students' solutions may contain bugs that prevent them from halting, such as incorrect loop conditions.
\subsubsection{Output}
\label{sect:gradeer-output}
After executing every check on every solution, \textit{Gradeer}~stores the appropriate grades and feedback for each solution in various CSV files.
The final grade of each solution is stored in one CSV file.
This grade is calculated by:
\begin{align*}
\text{\textit{Grade} (s)} &= \frac{\sum_{c\in C} w(c)\cdot\text{\textit{Base Score}} (c, s)}{\sum_{c\in C}w(c)}\text{,}\\
\text{where }s &= \text{Student's solution,}\\
C &= \text{Set of enabled checks,}\\
w(c) &= \text{Weight of check }c
\end{align*}
Similarly, the combined feedback of each solution across all checks is also stored in a CSV file.
\textit{Gradeer}~also stores a CSV file with the individual base scores and feedback of every check for each solution.
This file also includes the weight of each check.
This allows for final changes to be made in spreadsheet software if absolutely necessary.
For example, the user can post-process the students' grades by adjusting the checks' weights, and recalculating the final grades in the same manner as \textit{Gradeer}.
Users can also gather valuable information on students' performance for the grading criteria, facilitating the provision of group feedback to the entire student cohort.
\subsection{State Restoration}
\label{sect:gradeer-state-restoration}
Following the completion of checks on a solution, \textit{Gradeer}~stores the results and feedback for every check in a JSON file.
When \textit{Gradeer}~is executed with such files present, it uses them to restore these check results for every applicable solution, and skips the corresponding checks when processing these solutions again.
This has numerous advantages:
\begin{itemize}
\item A tutor can effectively pause the grading process and come back to it at a later time.
This is particularly advantageous when using manual checks, as programming tasks with many students' solutions can take hours to manually assess.
State restoration allows this arduous process to be split into more manageable marking sessions.
\item Assessment tasks can be allocated to multiple users, such as TAs.
Tutors can adjust users' \textit{Gradeer}~configurations to use different solutions or checks.
By allocating different manual checks to different users, grading can be completed more quickly without reducing consistency.
By merging the users' JSON files and re-running \textit{Gradeer}, the final grades and feedback can be generated.
\item If \textit{Gradeer}~halts unexpectedly, perhaps due to a wider system error, the user's grading progress is not lost.
\item Tutors can either directly modify the result files to adjust the results of individual checks, or delete them outright to re-assess a solution.
Running \textit{Gradeer}~again will update the final output files (as described in Section~\ref{sect:gradeer-output}).
Tutors can also choose to add new checks after initial executions of the tool to capture additional assessment requirements.
\end{itemize}
\section{Introduction}
The demand for Computer Science and Software Engineering education has continued to increase over recent years, with educational institutions seeing larger cohorts of students enrolled in such courses\cite{BCSPressOffice-IncreaseStudents}.
As technology further advances, future generations of students will drive this demand further, with universities and schools facing several challenges in teaching a growing number of students.
One of these challenges is the assessment of a large number of students' solutions to programming tasks.
Assessment is particularly important, since it both has the ability to further students' development through the provision of detailed feedback, and serves to measure a student's understanding of a topic.
Automated grading and feedback techniques offer several benefits in assessing large numbers of students.
Their automated nature allows users to perform other tasks while grading is executed.
It is also often much quicker to run a series of automated processes than to manually assess individual students' solution programs.
This is especially important for courses with large numbers of students, where manual assessment would consume too much time, and manual feedback could be provided too late to be of relevance to students' learning.
In addition, automated feedback allows for a large amount of feedback to be generated, and providing more pieces of automated feedback has been shown to improve students' performance~\cite{Falkner2014}.
Automated grading is also more consistent than manual grading, especially if students' solutions are assessed manually by multiple people~\cite{Albluwi2018}, which would likely be necessary to improve assessment times.
There are, however, some issues with the use of automated assessment alone.
There is a significant initial time cost of using automated assessment, with the need to either develop or configure a tool before assessment can be performed.
Additionally, with the exception of test-based systems, tutors may find it difficult to adapt an automated assessment system to meet their requirements~\cite{Keuning2016}.
Similarly, there are a wide range of unique automated assessment approaches~\cite{Liu2019, Insa2018, Singh2013, Parihar2017, Wunsche2018, Sridhara2016}, some of which may be suited to certain tasks, but would require a significant degree of effort to combine into one grading tool.
Automated assessment also lacks some of the benefits of manual approaches.
Manual assessment has the ability to capture aspects of grading that are hard to automate, such as the usefulness of variable names, or the appearance of a GUI.
There is also evidence that manually provided feedback is of greater benefit to students' performance than automatically generated feedback~\cite{Leite2020}.
In this paper, we introduce \textit{Gradeer}, a hybrid modular grading system, with the goal of providing the benefits of both approaches, while mitigating their challenges (Section~\ref{sect:gradeer}).
We used \textit{Gradeer}~to assess an end of year assignment for an introductory programming course (Section~\ref{sect:case-study}).
We found that the tool's hybrid approach allowed for the use of a large number of consistent automated assessment criteria, and aided in the provision of detailed manual feedback to students.
\textit{Gradeer}~also provides a degree of automation to assist tutors in manual assessment, such as automatically launching students' programs and code inspectors.
We found that these features saved us a considerable amount of time when manually assessing students' solutions.
The modular nature of grading components allows a variety of automated grading techniques to be used in conjunction with one another, while minimising the effort required to combine their results.
\textit{Gradeer}~is available on GitHub under the GPLv3 license, which allows users to write their own extensions and integrations for the tool\cite{GradeerRepository}.
\section{Conclusions and Future Work}
In this paper we have presented \textit{Gradeer}, a modular grading tool to support both the automated and manual assessment of students' programs.
We have also discussed our experiences in using the tool to assess an end of year assignment for an introductory programming course.
We find that \textit{Gradeer}~can effectively support tutors in providing quality feedback to students, while maintaining a low time cost of assessment.
\textit{Gradeer}~also provides tutors with detailed data on students' performance, which can be used to inform and improve teaching quality, future assessment design, and feedback.
\textit{Gradeer}~is available at \url{https://github.com/ben-clegg/gradeer}~\cite{GradeerRepository}.
In our future work, we will extend our evaluation of \textit{Gradeer}, by comparing the time saved using our solution inspector versus manually running each solution, and by surveying more end users.
We plan to improve \textit{Gradeer}, such as enhancing its modularity, by further separating check modules from the rest of the system, and modularising other components (such as pre-checks and language-specific functionality) as well.
We also intend to add web integration to the tool, to inform students when they have submitted solutions with significant problems.
\section{Related Work}
Some existing automated grading tools also feature modular assessment elements~\cite{Zschaler2018}.
For example, Nexus's assessment components implemented as Docker micro-services~\cite{Zschaler2017}.
Web-CAT uses modular plug-ins~\cite{Edwards2008, Edwards-WhatIsWebCAT}.
JACK and ArTEMiS both use multiple software components that can be split across multiple servers, and interchanged to support different grading functionalities~\cite{Goedicke-2008, Krusche2018b}.
These tools are designed to be used as scalable web services, which can be beneficial for large courses and MOOCs.
Such approaches do have considerable advantages, and may allow tutors to view students' source code, but tutors cannot run and interact with students' solutions directly, which limits their ability to perform manual assessment.
By contrast, \textit{Gradeer}~specifically accommodates manual assessment.
It is not uncommon for assessment tools to take a ``semi-automatic'' approach, with support for user intervention and manual assessments alongside automated processes~\cite{Souza2016}.
Web-CAT allows tutors to manually inspect students' source code, and provide feedback or additional grades~\cite{Edwards-WhatIsWebCAT}.
Praktomat grants TAs the ability to provide manual feedback by adding comments to students' code~\cite{Brietner-2017}.
It also allows TAs to add manual scores for learning outcomes.
JACK enables tutors to provide manual corrections for generated grades, and manual feedback~\cite{Goedicke-2008}.
Jackson's grading tool displays the contents of a solution's files before reading the user's input to determine the scores of a series of manual assessment elements~\cite{Jackson2000}.
While these tools have provisions for manual assessment, none of them automate the process of launching students' programs for tutors to interact with them.
This may be problematic, as the bottleneck of manually running each solution is still present when evaluating user interaction.
\textit{Gradeer's}~solution inspector removes this bottleneck entirely.
\textit{Gradeer}~also combines the results of automated and manual checks into a single grade, without additional user intervention.
|
1,314,259,996,729 | arxiv |
\section{Introduction}\label{sec:refactoring}
Maintaining and adapting software takes up a substantial part of the entire
programming effort, both in time and money. Erlikh~\shortcite{Erlikh}
and Moad~\shortcite{Moad} both report on the proportion of maintenance
costs exceeding 90\% of the budget. About 75\% of these costs are spent on
providing enhancements (in the form of adaptive or perfective maintenance)
\cite{Nosek:Palvia,vanVliet}.
Before providing enhancements, it is recommended to improve the design
of the software in a preliminary step. This methodology, called
{\em refactoring}, emerged from a number of pioneer results in the
OO-community~\cite{Fowler:et:al,Opdyke:PhD,Roberts:Brant:Johnson} and
recently came to prominence for functional~\cite{Li:Reinke:Thompson} and
procedural~\cite{Garrido:Johnson} languages.
Refactoring is a disciplined technique for restructuring an existing body of
code, altering its internal structure without changing its external behavior.
Its heart is a series of small source-to-source program transformations,
called {\em refactorings}, that change program structure and organization,
but not program functionality. The major aim of refactoring is to improve
readability, maintainability and extensibility of the existing software.
While performance improvement is not considered as a crucial issue for
refactoring, it can be noted that well-structured software is more amenable to
performance tuning. We also observe that certain techniques that were developed
in the context of program optimization, such as dead-code elimination and
redundant argument filtering, can improve program organization and, hence,
can be used as refactoring techniques.
In this paper we study refactoring techniques for Prolog. Our goals are
threefold. Firstly, we want to show that refactoring is a viable technique
for Prolog and many of the existing techniques developed for refactoring
in general are applicable. Secondly, Prolog-specific refactorings are
possible and the application of some general techniques may be highly
specialized towards Prolog.
Finally, it should
be clear that refactoring is not only viable for Prolog but also very useful for
the maintenance of Prolog programs.
In order to achieve our goals we present a catalogue of refactoring
techniques for Prolog. The listed refactorings are a mix of general and
Prolog-specific ones.
Most of the refactorings proposed have been implemented in a prototype
refactoring browser {\tt ViPReSS}. {\tt ViPReSS} has been successfully applied for
refactoring a 50,000 lines-long legacy system.
As completeness of the catalogue is clearly not possible, we aimed to show a
wide range of possibilities for future work on combining the formal techniques
of program analysis and transformation with software engineering.
The formal elaboration of a particular topic may be a
substantial study on its own, as shows the work on detecting duplicate code
by Vanhoof~\shortcite{Vanhoof} that was inspired by a preliminary version
of our work.
\paragraph{Outline of the Paper}
First, Section \ref{sec:process} provides a brief overview of the refactoring process.
Next,
the use of several refactoring techniques is illustrated on a small example in
Section \ref{sec:example}. Then a catalogue of Prolog
refactorings is given in Section \ref{sec:catalogue}.
In Section \ref{sec:vipress} we introduce {\tt ViPReSS}, and discuss
its application in a case study. Finally,
in Section \ref{sec:conclusions} we conclude.
\section{The Refactoring Process}\label{sec:process}
The refactoring process consists of applying a number of refactorings, with both localized and global impact, to a
software system.
The individual significance of
a refactoring may be apparent, but
often a refactoring seems trivial on
its own and only in conjunction with other refactorings or intended changes
does the usefulness become clear. That is the reason why it is not feasible
to fully automate refactorings. They must be carefully considered in view
of the programmer's intentions.
For this reason the process of applying a single refactoring is to be split
into a number of distinct activities \cite{Mens:Tourwe}. These activities
involve decisions to be made by the programmer.
The first decision is {\em where} the software should
be refactored. Making this decision automatically can be a difficult
task on its own. Several ways to resolve this may be considered. For
instance, one can aim at identifying so called {\em bad smells}, i.e.,
``structures of the code that suggest (sometimes scream for) the possibility of
refactoring''~\cite{Fowler:et:al}. To this end program analysis can
be used. For example, it is common practice while ordering predicate arguments
to start with the input arguments and end with the output arguments. Mode
information can be used to detect when this rule is violated.
Next, one should determine {\em which} refactorings should be applied.
Sometimes, the correspondence between bad smells and refactorings
is clear. For instance, if the predicate arguments are not ordered according
to the ``input first output last'' rule, one can suggest to the user to reorder
the arguments. This refactoring is further discussed in Section~\ref{section:reorder:arguments}.
In more complex situations the relation becomes less obvious:
a number of different refactorings are applicable and the user has to choose between them.
For example, let module \texttt{A} contain a predicate that is mutually recursive
with predicate \texttt{p} from module \texttt{B}, and module
\texttt{C} contain a predicate that is mutually recursive
with predicate \texttt{q} from module \texttt{B}. This situation can be identified as problematic
since no clear hierarchy can be defined between these modules.
One possible solution would be to merge the three modules (Section~\ref{section:merge:modules}).
Alternatively, one may try to
first split \texttt{B} into \texttt{B1}, containing \texttt{p}, and \texttt{B2} containing \texttt{q}
such that there are no circular dependencies between \texttt{B1} and \texttt{B2} (Section~\ref{section:split:module}).
If this split is possible,
\texttt{A} could be merged with \texttt{B1}, and \texttt{C} with \texttt{B2}
(Section~\ref{section:merge:modules}).
Automatic refactoring tools, so called {\em
refactoring browsers}, can be expected to make suggestions on where refactoring
transformations should be applied. These suggestions can then be either
confirmed or rejected by the programmer.
By definition, refactorings should preserve the software's functionality.
Hence, the next step consists of {\em ensuring} that the behavior is indeed
preserved. This step, of course, depends on the definition of behavior.
In the case of logic programming, behavior comprises computed answers
semantics,
termination, and side effects such as input/output.
It should be observed that particular application domains might
require extending the notion of behavior to include such concepts as
efficiency or memory use. Moreover, in order for some refactorings to be
applicable certain preconditions should hold, like absence of user-defined
meta-predicates for dead-code elimination discussed in Section~\ref{section:remove:dead:code}.
Sometimes verification of the
preconditions cannot be done automatically, but must be delegated to the user.
Subsequently, {\em the chosen transformation is applied}. This step
might also require
user input. Consider for example a refactoring that renames a predicate:
while automatic tools can hardly be expected to guess the new predicate name,
they should be able to detect all program points affected by the change. This
refactoring is further studied in Section~\ref{section:rename:predicate}.
Finally, the {\em consistency} between the refactored program code and
other related artifacts should be maintained. By artifacts we understand
among others software documentation, specifications and test descriptions.
The ability to perform this task automatically strongly depends on the
formalisms used to express the corresponding artifacts.
For instance, documentation generators such as {\em lpdoc}~\cite{Hermenegildo}
make it possible to keep the documentation consistent automatically, whereas
ad hoc unstructured comments are much harder to update automatically. Ensuring
consistency is considered as future work.
\section{Detailed Prolog Refactoring Example}\label{sec:example}
\label{section:example}
We illustrate some of the techniques proposed by a detailed refactoring example.
Consider the following code fragment from O'Keefe's ``The Craft of
Prolog'' \shortcite{OKeefe}, p. 195. It describes three operations on a {\em reader}
data structure used to sequentially read terms from a file. The three
operations are \texttt{make\_reader/3}, which initializes the data structure,
\texttt{reader\_done/1}, which checks whether no more terms can be read, and
\texttt{reader\_next/3}, which gets the next term and advances the reader.
\begin{Verbatim}[commandchars=\\\{\},frame=single,fontsize=\small,framesep=1mm,label={[\listingcaption{O'Keefe's original version}]}]
\textbf{make_reader}(File,Stream,State) :-
open(File,read,Stream),
read(Stream,Term),
reader_code(Term,Stream,State).
\textbf{reader_code}(end_of_file,_,end_of_file) :- ! .
\textbf{reader_code}(Term,Stream,read(Term,Stream,Position)) :-
stream_position(Stream,Position).
\textbf{reader_done}(end_of_file).
\textbf{reader_next}(Term,read(Term,Stream,Pos),State)) :-
stream_position(Stream,_,Pos),
read(Stream,Next),
reader_code(Next,Stream,State).
\end{Verbatim}
We will now apply several refactorings to the above program in order to improve
its readability.
Firstly, we use if-then-else introduction (Section~\ref{section:replace:cut:by:if-then-else})
to get rid of the red cut\footnote{As defined in e.g. \cite{OKeefe}: a cut that alters the meaning.} in the \texttt{reader\_code/3} predicate
(modified code is underlined):
\begin{Verbatim}[commandchars=\\\{\},frame=single,fontsize=\small,framesep=1mm,label={[\listingcaption{Replace cut by if-then-else}]}]
\textbf{reader_code}(Term,Stream,State) :-
\underline{( Term = end_of_file,}
\underline{State = end_of_file ->}
true
\underline{;}
State = read(Term,Stream,Position),
stream_position(Stream,Position)
\underline{)}.
\end{Verbatim}
The result of this automatic transformation reveals two malpractices: the first is
producing output before the commit, something O'Keefe himself disapproves of
in \shortcite{OKeefe}. This malpractice and the ways to resolve it are further investigated
in~\ref{section:produce:output:after:commit}.
The problem is fixed to:
\begin{Verbatim}[commandchars=\\\{\},frame=single,fontsize=\small,framesep=1mm,label={[\listingcaption{Output after commit}]}]
\textbf{reader_code}(Term,Stream,State) :-
( Term = end_of_file ->
\underline{State = end_of_file}
;
State = read(Term,Stream,Position),
stream_position(Stream,Position)
).
\end{Verbatim}
The second malpractice is a unification in the condition
of the if-then-else where an equality test is meant.
Consider the case that the \texttt{Term} argument is a variable. Then
the binding of \texttt{Term} to the atom \texttt{end\_of\_file} is certainly unwanted behavior.
The transformation in question is discussed in Section~\ref{section:replace:unification:by:inequality:test}.
The following code does not exhibit the problematic behavior:
\begin{Verbatim}[commandchars=\\\{\},frame=single,fontsize=\small,framesep=1mm,label={[\listingcaption{Equality test}]}]
\textbf{reader_code}(Term,Stream,State) :-
( \underline{Term == end_of_file} ->
State = end_of_file
;
State = read(Term,Stream,Position),
stream_position(Stream,Position)
).
\end{Verbatim}
Next, we notice that the conjunction \texttt{read/2, reader\_code/3}
occurs twice. By applying predicate extraction (Section~\ref{section:extract:predicate:locally})
of this common sequence, we get:
\begin{Verbatim}[commandchars=\\\{\},frame=single,fontsize=\small,framesep=1mm,label={[\listingcaption{Predicate extraction}]}]
\textbf{make_reader}(File,Stream,State) :-
open(File,read,Stream),
\underline{read_next_state(Stream,State)}.
\textbf{reader_next}(Term,read(Term,Stream,Pos),State)) :-
stream_position(Stream,_,Pos),
\underline{read_next_state(Stream,State)}.
\underline{\textbf{read_next_state}(Stream,State) :-}
\underline{read(Stream,Term),}
\underline{reader_code(Term,Stream,State).}
\end{Verbatim}
Next we put the input argument first
and the output arguments last (Section~\ref{section:reorder:arguments} below),
a principle also advocated in \cite{OKeefe}:
\begin{Verbatim}[commandchars=\\\{\},frame=single,fontsize=\small,framesep=1mm,label={[\listingcaption{\label{lst:reorder_arguments}Argument reordering}]}]
\textbf{reader_next}\underline{(read(Term,Stream,Pos),Term,State)} :-
stream_position(Stream,_,Pos),
read_next_code(Stream,State).
\end{Verbatim}
Finally, note that the naming of the two builtins
\texttt{stream\_position/[2,3]} may be confusing to the user. It is easier to
distinguish between their functionality based on predicate name than based on
arity. We introduce the less confusing names \texttt{get\_stream\_position/2}
and \texttt{set\_stream\_position/3} respectively.
In addition, we provide a more consistent naming for \texttt{make\_reader},
more in line with the other two predicates in the interface. The importance of consistent naming conventions
is also stressed in \cite{OKeefe}.
Note that direct renaming of built-ins such as \texttt{stream\_position}
is not possible, but a similar effect can be achieved by
extracting the built-in into a new predicate with the desired name.
Extracting a predicate and renaming predicates
are considered in Sections~\ref{section:extract:predicate:locally} and
~\ref{section:rename:predicate}, respectively.
In order to avoid confusion between a built-in predicate
\texttt{read} and a functor \texttt{read} we rename the latter functor to
\texttt{reader}.
\begin{Verbatim}[commandchars=\\\{\},frame=single,fontsize=\small,framesep=1mm,label={[\listingcaption{\label{lst:rename_functor}Renaming}]}]
\underline{\textbf{reader_init}}(File,Stream,State) :-
open(File,read,Stream),
reader_next_state(Stream,State).
\textbf{reader_next}(\underline{reader}(Term,Stream,Pos),Term,State)) :-
\underline{set_stream_position}(Stream,Pos),
reader_next_state(Stream,State).
\textbf{reader_done}(end_of_file).
\textbf{reader_next_state}(Stream,State) :-
read(Stream,Term),
build_reader_state(Term,Stream,State).
\textbf{build_reader_state}(Term,Stream,State) :-
( Term == end_of_file ->
State = end_of_file
;
State = \underline{reader}(Term,Stream,Position),
\underline{get_stream_position}(Stream,Position)
).
\underline{\textbf{set_stream_position}(Stream,Position) :-}
\underline{ stream_position(Stream,_,Position).}
\underline{\textbf{get_stream_position}(Stream,Position) :-}
\underline{ stream_position(Stream,Position).}
\end{Verbatim}
This example demonstrates how the code readability can be ameliorated by performing
a series of relatively simple transformation steps. We have seen that some of these steps
required user's input. Clearly the changes can be performed manually.
However, refactoring browsers should be able to guarantee consistency, correctness
and furthermore can automatically single out opportunities for refactoring.
Techniques applied above are well-suited for local code improvement, i.e.,
the objects modified are predicates and clauses. In the next section
we also consider techniques for global code restructuring such as
duplicate predicates removal (Section~\ref{section:remove:duplicate:predicates}).
\section{A Catalogue of Prolog refactorings}\label{sec:catalogue}
In this section we present the refactorings that we have found to be useful
for Prolog programs.
The considered Prolog programs are
not limited to pure logic programs, but may contain various built-ins such as
those defined in the ISO standard \shortcite{ISO13211-1}. The only exception are
higher-order constructs that are not dealt with automatically, but manually.
This is done due to the fact that higher order constructs such as {\em call}
make it impossible to decide at the compile-time which predicate is going
to be called at the corresponding program point during execution.
Automating the detection and handling of higher-order predicates is an
important part of future work.
The refactorings in this catalogue are grouped by their scope. The scope expresses
the user-selected target of a particular refactoring. Hence, refactoring starts
by choosing an object in the specified scope. For instance, {\em split module}
(Section~\ref{section:split:module}) starts with selecting a module.
Then
the object is transformed.
For us, this means that the module is split. Finally, the changes propagate to
the affected code outside the selected scope. The latter might happen when
there is a dependency outside the scope. This corresponds
to updating import declarations in other modules of the system.
For Prolog programs we distinguish the following four scopes, based on the
code units of Prolog:
{\em system} scope (Section \ref{sub:system}),
{\em module} scope (Section \ref{sub:module}),
{\em predicate} scope (Section \ref{sub:predicate}) and
{\em clause} scope (Section \ref{sub:clause}).
As a starting point for this catalogue we used Fowler's~\shortcite{Fowler:catalogue}
for object-oriented languages. We selected those with clear Prolog counterparts, extended
the list with Prolog-specific transformations and some well-known program transformations, such as
dead code elimination.
In the current technical note we only include a short summary of the refactorings here
and refer to the companion technical report \cite{techrep}.
This report contains
the full catalogue with detailed description of the refactorings, examples,
preconditions and automatization techniques.
\subsection{System Scope Refactorings}\label{sub:system}
The system scope encompasses the entire code base. The user wants to consider
the system as a whole.
\begin{refactoring}{Eliminate explicit module qualification}
\label{section:eliminate:explicit:module:qualification}
In many Prolog systems, such as Quintus~\cite{Quintus:Manual}%
, the module system is non-strict, i.e. the normal visibility
rules can be overridden by a special construct, called {\em explicit module qualification}
and written as \texttt{m:q}
,
where \texttt{m} is a module that contains definition of the
predicate \texttt{q/0}. The refactoring
proposed adds import and export declarations to get rid of these special syntax constructions.
By forcing the code to conform to a strict module system a number of quality characteristics are improved.
First of all,
a strict module system better expresses the idea of information hiding,
which is important for software maintainability and readability~\cite{ParnasCriteria}. Moreover,
since not all Prolog systems support the above construct, code portability is improved.
\end{refactoring}
\begin{refactoring}{Extract common code into predicates}
\label{section:extract:common:code:into:predicates}
This refactoring looks for
common functionality across the system
and extracts it into new predicates. The common functionality consists of
identical subsequences of goals that are called in different predicate bodies, and extracts them
into new predicates.
The overall
readability of the program improves as the affected predicate bodies get
shorter,
and the calls to the new predicates can be more meaningful than what
they replace.
Moreover the increased sharing simplifies maintenance
as now only one copy
needs to be modified.
The problem of identifying identical subsequences of
of goals is related to determining longest repeated
subsequences~\cite{Crow:Smith,Pitkow:Pirolli}.
\end{refactoring}
\begin{refactoring}{Hide predicates}
\label{section:hide:predicates}
This refactoring removes export declarations for predicates that are not
imported in any other module. It simplifies the program by reducing the number of entry
points into modules and hence the intermodule dependencies.
\end{refactoring}
\begin{refactoring}{Remove dead code}
\label{section:remove:dead:code}
Dead code is code that can never be executed and therefore can
be safely eliminated without affecting correctness of the execution.
Dead code elimination is sometimes performed in compilers for efficiency
reasons, but it is also useful for developers: dead code clutters the program.
We consider a predicate definition as the unit of dead code.
\end{refactoring}
\begin{refactoring}{Remove duplicate predicates}
\label{section:remove:duplicate:predicates}
Predicate duplication or cloning is a well-known problem,
prominently caused by ``copy \& paste'' and
unawareness of available libraries and exported predicates in
other modules. The main problem with duplication is its bad
maintainability.
It is up to the user to decide whether to throw away some of the duplicates
and to use one of the remaining definitions instead
or to replace all the duplicate predicates by a new version in a new module.
\end{refactoring}
\begin{refactoring}{Rename functor}
\label{section:rename:functor}
This refactoring renames a term functor across the system. If the functor has
several different meanings and only one should be renamed, it is up to the user to
identify what occurrence corresponds with what meaning.
\end{refactoring}
\subsection{Module Scope Refactorings}\label{sub:module}
The module scope considers a particular module. Usually a module is
implementing a well-defined functionality and is typically contained in
one file.
\begin{refactoring}{Merge modules}
\label{section:merge:modules}
Merging several modules into one can be advantageous in case of strong
interdependency of the modules involved. Moreover, merging existing modules
and splitting the resulting module can lead to an improved module
structure.
\end{refactoring}
\begin{refactoring}{Remove dead intra-module code}
\label{section:remove:dead:code:intra-module}
Similar to {\em dead code removal} for an entire system (see Section
\ref{section:remove:dead:code}),
this refactoring works at the level of a single module.
It is useful for incomplete systems or library modules with an unknown number
of uses. Recall that determining the liveness of the code requires
knowledge of top-level predicates. In the case of intra-module dead
code elimination, the set of top level predicates is extended with,
or replaced by, the exported predicates of the module.
\end{refactoring}
\begin{refactoring}{Rename module}
\label{section:rename:module}
This refactoring applies when the name of the module no longer corresponds
to the functionality it implements e.g. due to other refactorings.
\end{refactoring}
\begin{refactoring}{Split module}
\label{section:split:module}
The refactoring is useful to split unrelated
parts of a module or make a large module more manageable.
Moores~\cite{Moores} has shown that the number of user-defined
predicates correlates with the number of errors
detected. Based on an empirical study he suggested a threshold of around
$35\pm 5$ predicates per program. While this is hardly reasonable as a
requirement for an entire Prolog system, trespassing the threshold should be
used as a guideline when the Split Module refactoring can be applied.
\end{refactoring}
\subsection{Predicate Scope Refactorings}\label{sub:predicate}
The predicate scope targets a single predicate. The code that
depends on the predicate may need updating as well. But this is considered
an implication of the refactoring of which either the user is alerted or
the necessary transformations are performed automatically.
\begin{refactoring}{Add argument}
\label{section:add:argument}
This refactoring should be applied when a callee needs more information
from its (direct or indirect) caller, which is very common in Prolog program
development.
Given a variable in the body of the caller and the name
of the callee, the refactoring browser should propagate
this variable along all possible computation paths from the caller to the
callee. This refactoring is an important preliminary step preceding
additional functionality integration or efficiency improvement.
\end{refactoring}
\begin{refactoring}{Move predicate}
\label{section:move:predicate}
This refactoring moves a predicate definition from one module
to another.
It can improve the overall structure of the program by bringing together
interdependent or related predicates,
hence improving both cohesion of each one
of the modules involved, and coupling of the pair.
{\em Move predicate} appears often
after predicate extraction, i.e., {\em extract common code} or {\em extract predicate locally},
discussed in Sections~\ref{section:extract:common:code:into:predicates}
and \ref{section:extract:predicate:locally}, respectively.
\end{refactoring}
\begin{refactoring}{Rename predicate}
\label{section:rename:predicate}
This refactoring can improve
readability and should be applied when the name of a predicate does not reveal
its purpose.
\end{refactoring}
\begin{refactoring}{Reorder arguments}
\label{section:reorder:arguments}
Our experience suggests that while writing predicate definitions Prolog
programmers tend to begin with the input arguments and to end with the output
arguments. This habit has been identified as a good practice and even
further refined by O'Keefe \shortcite{OKeefe} to more elaborate rules.
Unfortunately, this practice is difficult to maintain when additional arguments
are added later. We observed that failure to confirm to this ``input first
output last'' expectation pattern is experienced as very confusing.
\end{refactoring}
\begin{refactoring}{Specialize predicate}
\label{section:split:predicate}
By specializing a predicate we mean producing a (number of) more
specific version(s) of a given predicate provided some knowledge on the intended
uses of the predicate. Specialisation can simplify code as well as make a meaningful
distinction between different uses of a predicate.
\end{refactoring}
\begin{refactoring}{Remove redundant arguments}
\label{section:remove:redundant:arguments}
The basic intuition here is that parameters that are no longer used
by a predicate should be dropped.
It improves readability.
Leuschel and S{\o}rensen~\shortcite{Leuschel:Sorensen} established that the
redundancy property is undecidable and suggested two techniques to find
safe and effective approximations: top-down goal-oriented RAF (Redundant
Argument Filtering) and bottom-up goal-independent FAR (RAF ``upside-down'').
In the context of refactoring FAR is the more useful
technique, since only FAR deals correctly with exported predicates used
in unknown goals.
\end{refactoring}
\subsection{Clause Scope Refactorings}\label{sub:clause}
The clause scope affects a single clause in a predicate. Usually, this does
not affect any code outside the clause directly.
\begin{refactoring}{Extract predicate locally}
\label{section:extract:predicate:locally}
This refactoring is similar to the system-scope refactoring with the same name.
However, it does not aim to automatically
discover useful candidates for replacement.
The user is responsible for selecting the subgoal that
should be extracted, in order to improve the readability.
\end{refactoring}
\begin{refactoring}{Invert if-then-else}
\label{section:invert:if-then-else}
The order of ``then'' and ``else'' branches can be important for
code readability.
To enhance readability it might be worthwhile putting the shorter branch as
``then'' and the longer one as ``else''. Alternatively, the negation of
the condition may be more readable because, for example, a double negation can be
eliminated.
\end{refactoring}
\begin{refactoring}{Replace cut by if-then-else}
\label{section:replace:cut:by:if-then-else}
This technique aims at improving program readability by replacing
cuts (!) by the more declarative if-then-else ({\tt -> ; }). More detailed
discussion on replacing cut by if-then-else is deferred to {\em Related
work and extensions}.
\end{refactoring}
\begin{refactoring}{Replace unification by (in)equality test}
\label{section:replace:unification:by:inequality:test}
Often full unifications are used instead of equality or other tests.
O'Keefe in \shortcite{OKeefe} advocates the importance of steadfast code. Recall, that
steadfast code produces the right answers for all possible modes and inputs. A more moderate
approach is to write code that works for the intended mode only.
Unification succeeds in several modes and so does not convey a particular
intended mode. Equality ({\tt ==}, {\tt =:=}) and inequality ({\tt \verb+\==+},
{\tt \verb+=\=+}) checks usually only succeed for one particular mode and
fail or raise an error for other modes. Hence their presence makes it
easier in the code and at runtime to see the intended mode. Moreover, if
only a comparison was intended, then full unification may lead to unwanted
behaviour in unforeseen cases.
\end{refactoring}
\begin{refactoring}{Produce output after commit}
\label{section:produce:output:after:commit}
This refactoring addresses a similar issue as the previous one. Producing
output before the commit (cut) does not properly convey the intended mode
of a predicate. Moreover it may lead to unexpected results when used
in the wrong mode.
\end{refactoring}
\section{The {\tt ViPReSS} refactoring browser}\label{sec:vipress}
The refactoring techniques presented in Section \ref{sec:catalogue} have been implemented in
the prototype refactoring browser \mbox{{\tt ViPReSS}}\footnote{Vi(m) P(rolog)
Re(factoring) (by) S(chrijvers) (and) S(erebrenik)}.
It has been implemented
on the basis of VIM%
, a popular clone of the well-known VI editor. The text editing facilities of VIM
make it easy to implement techniques like {\em move
predicate} (Section~\ref{section:move:predicate}).
Most of the refactoring tasks have been implemented as SICStus Prolog
\cite{SICStus:Manual} programs inspecting source files and/or call
graphs. Updates to files have been implemented either directly in the
scripting language of VIM or, when many files need updating
at once, through \texttt{ed} scripts. VIM functions were written to
initiate the refactorings and to get user input.
{\tt ViPReSS} has been successfully applied to a large (more than 53 KLOC)
legacy system used at the Computer Science department of the Katholieke
Universiteit Leuven to manage the educational activities. The system, called
{\sf BTW},
has been developed and extended since the
early eighties by more than ten programmers, many of whom are no
longer employed by the department. The implementation has been done in
MasterProLog~\cite{MasterProLog}, which
is no longer supported.
Therefore, preparing the code for migration to a more
modern Prolog dialect and general structure improvement were essential for
further evolution of the system.
By using the refactoring techniques we succeeded in obtaining a better
understanding of this real-world system, in improving its structure and
maintainability, and in preparing it for intended changes:
porting it to a state-of-the-art Prolog system and adapting it to
new educational tasks the department is facing as a part of the unified
Bachelor-Master system in Europe.
A preliminary study revealed that many modules were unused. We brought in
an expert to help us identify the bulk of these unused modules, including
out-of-fashion user interfaces and outdated versions of program files. This
reduced the system size to a mere 20,000 lines.
Next, the actual refactoring process was started. As the first phase
we applied system-scope refactorings.
{\tt ViPReSS} was used to clean up after the bulk dead code removal:
299 predicates in the remaining modules were identified as dead.
This reduced the size by another 1,500 lines. Moreover
{\tt ViPReSS} discovered 79 pairwise identical predicates. In most of the cases,
identical predicates were moved to new modules used by the original ones. The
previous steps allowed us to improve the overall structure of the program
by reducing the number of files from 294 to 116 with a total of 18,000
lines. Very little time was spent to bring the system into this state. The
experts were sufficiently familiar with the system to identify
obsolete parts. The system-scope refactorings took only a few minutes each.
During this phase most of the work has been done by {\tt ViPReSS}, while the user's
involvement was limited to choosing a way to deal with duplicate predicates.
The second phase of refactoring consisted of a thorough code inspection aimed
at local improvement. Many malpractices were identified: excessive
use of cut (Section~\ref{section:replace:cut:by:if-then-else})
combined with output construction before commit (Section~\ref{section:produce:output:after:commit})
being the most notable one. Additional ``bad smells'' discovered include bad
predicate names such as {\tt q}, unused arguments and unifications
instead of identity checks or numerical equalities (Sections~\ref{section:rename:predicate}
and \ref{section:replace:unification:by:inequality:test},
respectively). Some of these were
located by {\tt ViPReSS} , others were recognised by the users, while {\tt ViPReSS}
performed the corresponding transformations. This step is more
demanding of the user. She has to consider all potential candidates for
refactoring separately and decide on what transformations apply. Hence,
the lion's share of the refactoring time is spent on these local changes.
In summary, from the case study we learned that automatic support for
refactoring techniques is essential and that {\tt ViPReSS} is well-suited
for this task. As the result of applying refactoring to {\sf BTW} we obtained
better-structured lumber-free code. Now it is not only more readable and
understandable but it also simplifies implementing the intended changes. From
our experience with refactoring this large legacy system and the relative
time investments of the global and the local refactorings, we recommend
starting out with the global ones and then selectively apply local refactorings
as the need occurs.
The current version of {\tt ViPReSS}
can be downloaded from \\
\texttt{http://www.cs.kuleuven.ac.be/\~{}toms/vipress}.
\section{Conclusions}\label{sec:conclusions}
In this paper we have studied refactoring techniques for Prolog. Firstly, we
have shown that refactoring is a viable technique for Prolog and that many of
the existing techniques developed for refactoring in general are applicable.
Our refactoring catalogue contains many such refactorings.
Secondly, Prolog-specific refactorings are possible and the application of some
general techniques may be highly specialized towards Prolog. In this context,
the companion technical report \cite{techrep} shows how refactoring fits
in with existing work on program analysis and transformation in the context
of Prolog and how many of these existing techniques may be adapted for the
purpose of partially automating the refactoring process.
Also, {\tt ViPReSS},
our refactoring browser integrates several automatable parts of the presented
refactorings in the VIM editor.
Finally, it should be clear that refactoring Prolog programs is not just viable
but very useful for the maintenance of Prolog programs. Refactoring helps
bridge the gap between prototypes and real-world applications. Indeed,
extending a prototype to provide additional functionality often leads to
cumbersome code. Refactoring allows software developers both to clean up code
after changes and to prepare code for future changes. These are important
benefits that also apply to logic programming.
As completeness of the catalogue is clearly not possible, we aimed to show a
wide range of possibilities for future work on combining the formal techniques
of program analysis and transformation with software engineering. Throughout
the catalogue many specific issues for future work have been mentioned. Below
we list related work and more general challenges for the future.
\subsection{Related and Future Work}
Logic programming has often been used to implement refactorings for other languages,
e.g. a meta-logic very similar to Prolog is used to detect, for instance,
obsolete parameters in \cite{Tourwe:Mens}.
Seipel {\em et al.}~\shortcite{Seipel:Hopfner:Heumesser}
include refactoring among the analysis and visualization techniques that
can be easily implemented by means of {\sc FnQuery}, a Prolog-inspired query
language for XML. However, the discussion stays at the level of an example.
The M.Sc. thesis of Steinke~\shortcite{Steinke} was dedicated
to refactoring of logic programs. A Catalogue of refactorings has been composed
and a prototype system has been implemented. However, only predicate-scope refactorings
have been considered and only the transformation step has been implemented.
In the logic programming community questions related to refactoring have been
intensively studied in the context of program transformation and specialisation.
There are two important differences with this line of work. Firstly,
refactoring improves readability, maintainability and extensibility rather
than performance. Secondly,
for refactoring user input is essential while in the mentioned
literature strictly automatic approaches were considered.
However, some of the transformations developed for
program optimization, e.g. {\em dead code elimination}, can be considered
as refactorings and have an important function in refactoring browsers.
To further increase the level of automation of particular refactorings
additional information such as types and modes can be used.
Future refactoring tools can also benefit from integration with Prolog development
environments.
Modern Prolog systems are often equipped with features
extending the ISO Standard such as constraint solving over different
domains and Constraint Handling Rules, coroutining, interfaces to foreign
languages, GUI-development systems and databases. In most of the cases,
the refactoring techniques described above can still be applied to
improve the code.
Certain refactorings may be specially designed for particular
extensions. For instance, our experience suggests that simplifying primitive
constraints may be useful in the case of CLP.
|
1,314,259,996,730 | arxiv | \section{Introduction}\label{sec:introduction}}
\section{Taxonomy of different methods}
Simultaneous Localization and Mapping (SLAM) have made the real time dense reconstruction possible increasing the prospects of robot navigation, tracking, and augmented reality problems . Some breakthroughs have been achieved in this regard during past few decades and more remarkable works are still going on. This paper presents an overview of SLAM approaches that have been developed till now. Kinect Fusion algorithm, its variants and further developed approaches are discussed in detailed. The algorithms and approaches are compared for their effectiveness in tracking and mapping based on Root Mean Square error over online available datasets.
\subsection{RGB-Depth Mapping}
As far as an optimal perception of phenomenal consciousness is concerned, theories based on representation of the mind are based on models of the information processing paradigm \cite{turan2018unsupervised}. These are as much in correspondence to the neurobiological or functional theories, at this point we are confronted with several arguments on the basis of inversion or absent qualia \cite{altuntacs20163}.
Such considerations exhibit a preceding pattern based on the assumption of holding complete knowledge of the neural and functional states that are in subservience to the occurrence of the consciousness that is phenomenal. This can still be conceived as the neural states which are also defined as the states with similar casual responsibilities or with similar representational function \cite{turan2017deep1}, \cite{turan2017endosensorfusion}.
\subsection{Spatially Extended and Moving Volume Kinetic Fusion}
These occur with no phenomenal content in any way or such states being accompanied by contents that are phenomenal with broad variation from the usual ones. In definition, visual information processing entails the visual cognitive skills that permit us the processing and interpretation of meaning from visualized information that we attain through eye sight.
Therefore, visual perception plays are vital role in aspects of cognitive and intelligence skills such as spelling, math and reading (\cite{angonese2016integration}). On the other hand, visual perceptual deficits can lead to challenges in learning, recognition and remembrance of letters, wording, and confusion of likeness as well as minor variations in addition to differentiating the main ideal from the details of insignificance.
\subsection{Scalable Real-Time Volumetric Reconstruction}
Visual perceptual processing can be sub segmented into the categories that comprise of visual discrimination, figure grounding, closure, memory, sequential memorization, constancy, spatial relations as well as visual motor integration.
Note should be taken of perception as active procedures of location and information extraction form the setting while learning entails the procedures of acquisition of information through experiences of information storage. In which case, thought is the manipulative stance upon information for solving challenges (\cite{ataer2016object}).
Such that it is eased to extract information (perception) which creates an ease in thought procedures becoming. In overall it is accepted that human vision takes the form of extreme powerful processing of information towards facilitation of the interaction of the world that surrounds us.
However, even in the face of extended and extensive efforts of research encompassing multiple fields of exploration, the fundamentals that underlay as well as operational principles of visual information procedures remain largely unknown.
\subsection{Segmentation-Based RGB-D Mapping}
We are still not able to ascertain the origin and distance along the route from eyes to the sensory input area known as the cortex. It is in this area that the conversion into object meaningful representation is undertaken under conscious manipulation of the brain (\cite{ataer2013tracking}).
Nearly half of the human brain in the cerebral cortex region is charged with the processes of visual information although even with extended and extensive research efforts that are encompassed a conundrum still persists. Present theories on visual information processing are held in the consideration of human visual information processing being interplay of the two inversely directed procedural streams.
\subsection{B-D Visual Odometry}
This is taking the form of a non-supervised, top down directed procedures that convey the regulations and guiding knowledge as a guide to linkage and binding of disjointed pieces of information to meaningful perceptual object images (\cite{altuntacs20163}). Most important in the idea of such a proposition not completely being new as in past research, there have been presentations in the form of depictions of the "faculty of appreciation" as a synthesized relation of "two constituents" which include the raw sensory data with the other being the cognitive "faculty of reason."
\subsection{Elastic Fusion}
Past research has presented a demonstration of distance and physical enviroment being among the aspects that impairs processing of information, although it remains unknown whether such impairment is on all the levels of information processing or in the onset states instead of the later stages.
Those faced with the condition of mapping algorithms suffer from deficiencies of attention that are impairment to the capability of selective procedures of visual information that is incoming. The early levels of information processing are held in the description of being those that entail the detection as well as response of simplified stimuli.
An assignment on the assessment of such function is the inspection time that has previously been demonstrated to entail sensitivity to pharmacological agents. This is as well as being the most reliable and validated within the cultural fairness of information processing measures of cognitive ability (\cite{angonese2016integration}).
Past assessment findings have also presented the impact of nicotine on information procedures as being held in the overall regard in the form of a measure of speed within the early levels of information processing. These include the speed of visual encoding that comprises of the ability of making observations or inspections on sensory input on which the discrimination of relative magnitude rests.
This is in contrast to assignments such as reaction time which is summarization entails the involvement of increased response oriented measures of complete decision making time that comprise of total information processing.
Although, there is no research of examination of the impacts administration of 3D scene construction in a similar response, there are limited studies based on the examination of the impacts of 3D scene construction in the early stages of information processing with utilization of other assignments (\cite{ataer2016object}). With the application of visual tracking assignments, it was ascertained that the speed of detection experienced impairment from 3D scene construction that that these impacts where greater in dual task settings with comparison to single task settings.
Such outcomes have been held in the description of being the deleterious impacts of 3D scene construction on the centralized processing capacity and on information processing availability on the capacity of information processing with time. Further investigations of early information processing are based on the examination of the mismatched negative component of auditory event relation potential as well as reports of reduced dosage of 3D scene construction attenuation of the event relation potential signal.
In this case, the mismatched negative component suppression was solid within stimuli deviation as reduced which the indication of relatively reduced blood 3D scene construction concentration is. The detection of minimal deviations for instance that needed in the course of the inspection time assignment more so in case of hampering in which case similar outcomes have been discovered in simplified reaction time assignments with double level of intensified stimuli.
These studies produced outcomes of an increase in response time as well as the impairment of stimuli detection which is a suggestion of the influence on sensory perceptual procedures and the measure of attentiveness (\cite{ataer2013tracking}).
\subsection{Bundle Fusion}
Current discoveries in the arena of visual information processing are based on the reflection of the elementary principles of vision as well as the utilization of visual information based on cognitive attributes.
This is based on the notion of such work leading to the verge of development based on the grounds of optimism within the several computational theories of sophistication that incorporate data that is neurobiological and behavioral.
These theories entail the flourishing of the skillful exploitation of the neural-imaging and computation of simulative technologies, these permits answering of questions that are subtle regarding the component subsystems within vision.
\section{Most relevant methods and description of their novelties and contributions and why are they published}
\subsection{GRAPH SLAM}
This algorithm applies information matrices sparsely production by the generation of graphs using observed interdependencies in case the observations are connected and if they contain information about the similar landmark.
Graph SLAM allows for the capability of constructing a map from an environment while simultaneously creating associated localization with the map for navigation in unknown settings when external referencing systems such as GPS are absent. This intuitive approach utilizes a graph with nodes in correspondence to the robot poses at varied points within time and whose edges are representative of the constraint in between the poses.
The latter is gained from environment observations of from movement actions as performed by the robot. Upon construction of a graph, the map could be computed by searching the nodes spatial configuration that is notably consistent with modeled measurements by the edges.
\begin{figure*}
\centering
\includegraphics{fig1}
\caption{Graph-pose SLAM estimation procedure}
\end{figure*}
From the image above, we note that particular nodes within the graph are in correspondence to the pose of the robot. Proximal poses are linked by the edges with model spatial constraints between the robot poses that are derived from measurements among the consecutive poses of model odometry measurements. This is whereas the other edges are representative of the spatial constraints based from several observations of the similar section of the environment.
The graph-based SLAM method develops a simple estimation challenge by abstraction of raw sensor readings. These readings as substituted by the graph edges which are viewed as "virtual measurements". Increased detail within an edge between the two nodes holds the label of a probability distribution over locations that are relative to the two poses with conditioning to mutual measurements.
\subsection{RGB-D Camera-Based Parallel Tracking and Meshing}
Visual real-time tracking in regard to established and unknown scenes is critical as well as an incontrovertible aspect in vision-based AR applications. Multiple algorithm contributions over the years. It is at this point that we introduce RGB-D Camera-Based Parallel Tracking and Meshing as an adaptation and updating of the algorithms utilized in estimating the motion of the camera as well as AR in accordance to the availability of the end user in computational abilities in permitting to gain impressive tracking outcomes in limited AR workspaces.
The fact is that estimation of camera motion using environment tracking as well as parallel constructing feature based sparse mapping that creates a possibility in part to the generalization of multi-core processors found in desktop and laptop computers. Of recent is has been revealed that increased computation power within a singular standard of a hand-held video camera is connected to a powerful computer using computational power gained from the Graphics Processing Unit (GPU).
The possibility to attain a dense representation of a desktop setting as well as increased texturing scenery whereas as undertaking tracking with the use RGB-D Camera-Based Parallel Tracking and Meshing. The online created map density can be increased with the use stereo-dense matching in addition to GPU founded implementations as shown by GPU to be utilized for effective replacement of the global bundle adjustment aspects of SLAM optimized based systems for instance RGB-D Camera-Based Parallel Tracking and Meshing as well as inherent parallelization refinement with step founded Monte Carlo simulations therefore freeing tools on the CPU for other assignments.
\subsection{LSD-SLAM (Large Scale Direct Monocular SLAM)}
The following algorithm is a novel direct monocular SLAM method that operates with direct image intensities rather than the use of key points for tracking and mapping. Camera tracking utilizes direct image alignment whereas geometry is estimated in the format of semi-dense depth maps gained through filtration of several pixel wise stereo comparisons. Thereafter, a Sim (3) pose-graph of key frames is created to permit for the development of scale-drift correction with large scale maps comprising of loop-closures. It should be noted that LSD-SLAM could be operated in real time on a CPU as well as smartphones.
This algorithm comprises of three core components namely tracking, map optimization and depth map estimation. The tracking feature persistently tracks new camera images by estimating the rigid body pose in regard to the present key frame and the uses of the pose in the past frame as a point of initialization. On the other hand, the depth map estimation feature applies tracked frames for either refinement or replacement of the present key frame.
Refinement of depth is achieved by filtration over several per-pixel, limited based line stereo comparisons as well as interleaved spatial regularization as the default proposition. Should the camera extend to far, initialization of a new key frame is implemented by projection of points from existent and proximal key frames.
Furthermore, upon replacement of a key frame as a reference tracking, the depth map will not be additionally refined but rather integrated into the global mapping with use of the map optimization feature. In this case, for detection of loop closures as well as scale drifting, the same transform to proximal key frames inclusive of the direct predecessor is estimated with use of the scale-aware and direct image alignment.
\subsection{S-PTAM (Stereo Parallel Tracking and Mapping)}
The algorithm holds the capability of computation of camera trajectory in real time with heavily exploitation of the parallel format of the SLAM challenge, separation of time constraints in pose estimation from less pressing issues for instance building of maps and refinement of assignments. In addition, the stereo setting permits for reconstruction of a metric 3D map for particular stereo frame d images, improvement of mapping procedure accuracy in respect to monocular SLAM and limiting the common bootstrapping challenge. Furthermore, the actual scale of the environment is a critical aspect for robots when in it comes to interaction with the surrounding workspace.
In order to permit for robotic mobile navigation and achieve autonomous assignments, it must be understood for its pose (position and orientation) as well as hold an environment (map) representation. In settings where robots do not have a past map and external information availed of the pose, it is necessitated to undertake both assignments simultaneously. The challenge of the robot and constraint the map of the environment in a simultaneous action is known as SLAM. However, in order to tackle the challenge of stereo vision, we introduced the S-PTAM (Stereo Parallel Tracking and Mapping) algorithm as an approach whose intention is to operate real-time of an extended duration of lengthy trajectories to permit for estimation of the pose with accuracy as it is built upon sparse mapped environments with a global coordinate system \cite{turan2017deep},\cite{turan2017endo}.
By using optimal performance, this algorithm is able to decouple localization and mapping assignments for the SLAM challenge with two independent threads which permits us to take the benefit of multi core processors. In addition to localization as well as mapping modules, the loop closure function is able to recognize locations from historically visited points. These detected loops are then applied for refinement of the map and trajectory estimation to effectively lower the accumulated error of the method. It is on this basis that S-PTAM operates on the visual features from extraction of images availed by the stereo camera.
\section{3D scene reconstruction and mapping}
3D scene reconstruction and mapping has been a crucial and important assignment within the arena of moveable robotics since it is critical need for various techniques specifically including path planning, semantic mapping, localization, navigation, and telepresence. Two major approaches towards 3D reconstruction are: offline multi-view stereo (MVS) based reconstruction and live incremental dense scene reconstruction. Many compelling results have been produced since past few years by exploring multi view stereo (MVS) and format derived on the basis of motion (SfM) techniques. Multi perspective stereo has been used extensively in photogrammetry for dense surface reconstruction (\cite{seitz2006comparison}) while the problem of accurate camera tracking has been cattered by SfM algorithms along with sparse reconstruction from large datasets of unordered images (\cite{agarwal2010reconstructing}). Although some groundbreaking results have been achieved but most of both SfM and MVS approaches have not been driven by live implementations.
Simultaneous Localization and Mapping (SLAM), unlike SfM and SVM, provides live motion tracking and re-structuring while applying input from a single commodity sensor but for a sequential ordered set of images. Various 3D mapping techniques offer different functionalities but all of them work almost on the same pipeline; of spatial aligning consecutive data frames at first, detecting the loop closures, and aligning the complete data sequence in a globally consistent manner. Although the developed systems provided satisfactory accuracy through point clouds and colored cameras but most of them are computationally exhaustive and inaccurate for dense depth reconstruction especially in dark environments or scenes with sparsely textured features \cite{turan2018magnetic}.
Based on the sensors used, 3D reconstruction can be achieved via three routes: using Multiview stereo, Laser scanning, or depth cameras. Multiview stereo is the traditional technique of photogrammetry where overlapping multi views of an object are captured for relative camera pose estimation and scene reconstruction is done via selected control points to get 3D coordinates of the object’s points through space intersection. Laser scanners work on the principle of time of flight where scene tracking is achieved via transmitted laser pulses that are received back by the scanner with high accuracy. The most recent and popular approaches are of constructing 3D scenes using RGB depth cameras that, working on the principle of time of flight, measure the pixel depth along with color information of the pixels.
Some early work on SLAM in 3D reconstruction over past few decades includes a range of approaches and their extensions. 3D reconstruction has been explored extensively with some point cloud models with real-time tracking like MonoSLAM (\cite{davison2007monoslam}) being the first successful effort on real-time tracking and active 3D mapping with only one camera. This had motivated many other works for online, though sparse, but fine and accurate reconstruction with freely moving hand-held cameras based on probabilistic models (\cite{weise2009hand}). Some later research focused on performing tracking and mapping in parallel instead of adopting probabilistic models. Parallel Tracking and Mapping system (PTAM) worked on the hierarchy of live tracking via feature optimization over spatially-distributed key frames for n-point pose estimation and expanding the maps obtained via bundle adjustment and global pose optimization (\cite{klein2007parallel}).
Although the mono SLAM approaches set the benchmarks in real-time 3D mapping and developed robust camera tracking systems, but the AR (Augmented Reality), and other live robust mapping and robot navigation applications cannot rely on sparse point clouds generated as a result of these systems. This triggered the work towards generating live dense maps using depth information of the scene via Multiview stereo approaches combined with PTAM for live camera tracking and robust pose estimation (\cite{newcombe2011kinectfusion}). But the availability of depth camera has made the task further easier and current approaches have set their focus on large scale 3D mapping using depth commodity sensors.
Considering the importance of SLAM approaches and their applications in field of robotics, this paper reveals a general understanding of the development of SLAM approaches for dense surface mapping and reconstruction in real-time using depth cameras as commodity sensors. An introduction of Kinect sensor is presented with its unique use in depth mapping and reconstruction for Augmented Reality (AR) applications. The focus is set on KinectFusion algorithms and marks achieved from them or their integration with other tracking and mapping algorithms. \cite{DBLP:journals/corr/abs-1804-01396}
\section{Depth Surface Mapping and Tracking Algorithms for 3D Reconstruction}
\subsection{RGB-Depth SLAM}
Depth cameras, with their ability to measure object’s depth from camera (based on time-of-flight or active stereo) in addition to RGB measurement, have paved a new wave of techniques in SLAM and Augmented Reality (AR). Incorporation of RGB-D cameras has allowed SLAM to benefit from range sensing along with visual data to handle the issues like data association and loop closures in visual Odometry along with visual SLAM (\cite{agarwal2010reconstructing}). Kinect sensor, among other RBG-D cameras, is the most notable depth device to be used in revolutionary approaches being developed for real-time tracking and surface mapping algorithms.
\subsection{Kinect Sensor}
Kinect sensor, a low-cost commodity platform mainly to detect human gestures in gaming and other entertainment applications, has shown its potential in simultaneous localization and mapping approaches to an unprecedented level. It applies an internal ASIC to generate 11-bit 640x480 depth map of a pixel at 30 Hz. Although map quality suffers from certain technical challenges (like motion blur at faster speeds), the information available is significant enough to be utilized by real-time 3D reconstruction algorithms. There have also been algorithms available to improve sensor accuracy (\cite{karan2015calibration}, \cite{butler2012shake}) depending upon sensor’s use or system’s requirements.
\subsection{Kinetic Fusion}
Developed by \cite{newcombe2011kinectfusion}, KinectFusion algorithm was the first attempt to real-time volumetric reconstruction of a scene in variable lightning conditions (\cite{newcombe2011kinectfusion}). Using information gained through Kinect sensor in form of input, while utilizing a coarse-to-fine iterative closest point (ICP) algorithm to simultaneously track camera pose and construct a medium sized 3D model in real-time by tracking a live depth frame relative to a global finished model. At a given time k, the transformation matrix given below was used to describe the 6 DOF, that mapped the camera coordinate frame to a global frame g, such as shown in \ref{eq1}.
\begin{equation}
\label{eq1}
T_{g,k} = \begin{bmatrix} R_{g,k} & t_{g,k} \\ 0 & 1 \end{bmatrix} \in \mathbb{SE}_3
\end{equation}
In equation \ref{eq1}, $\mathbb{SE}_3 := \{R,t|R \in \mathbb{SO}_3, t \in\mathbb{ R}^3\}$. This means, any point $P_k \in\mathbb{ R}^3$ in the camera frame is mapped to global coordinate frame via transformation $P_g=T_{g,k} P_k$. The algorithm was able to do real-time volumetric reconstruction in four steps – surface measurement, surface reconstruction update, surface prediction, and sensor pose estimation (Figure \ref{fig2}) – explained below:
\textit{Surface Measurement}: The first step is Pre-ICP registration and subsampling where raw depth measurements from Kinect sensor are used to build depth map pyramids (normal map and vertex map pyramid). A depth map $R_k (u) \in \mathbb{ R}$ obtained at every pixel $\bold{u} = (u,v)^T$ is calculated to obtain a metric point measurement $R_k (\bold{u}) K^{-1} \dot{\bold{u}}$ in the sensor frame at time k. A depth map $D_k$ is obtained after applying the bilateral filter (\cite{newcombe2011kinectfusion}), which after back projecting the value obtained into the sensor frame of reference gives the vertex map $V_k (\bold{u}) = D_k (\bold{u}) K^{-1} \dot{\bold{u}}$. Further, applying the rigid body transformation given in equation \ref{eq1}, the global frame vertex is obtained as: $V{_k^g} (\bold{u}) = T_{g,k} \dot{V}_k (\bold{u})$ while the equivalent mapping of normal vectors in the global frame is given as: $V{_k^g} (\bold{u}) = R_{g,k} N_k (\bold{u})$. \\
\textit{Surface Reconstruction Update}: a truncated signed distance function (TSDF) is computed from each input sensor frame. At each location, the global TSDF $S_k (\bold{p})$ at a given global point $\bold{p} \in \mathbb{ R}^3$ is a combination of local TSDF $F_k (\bold{p})$ and a weight $W_k (\bold{p})$ such that: $S_k (\bold{p}) \rightarrow [ F_k (\bold{p}) , W_k (\bold{p}) ]$. A volumetric integration fuses this TSDF into a discretized volumetric representation of the global coordinate space.\\
\textit{Surface Prediction from Ray-casting the TSDF}: It is a post-volumetric integration step in which the surface is visualized by ray-casting the signed distance function into an estimated frame and aligning this ray-casted view with live depth map. In pixel vise ray casting each ray $T_{g,k} K^{-1} \dot{\bold{u}}$ is started from a zero depth value till the surface crossing is found. \\
\begin{figure*}
\centering
\includegraphics[width=6in,height=1.5in]{fig2}
\caption{KineticFusion Workflow for real-time volumetric reconstruction}
\label{fig2}
\end{figure*}
\textit{Sensor Pose Estimation}: a multiscale ICP alignment is done between current sensor measurement and predicted surface to achieve sensor tracking. The algorithm is implemented on GPU to utilize all the data available at frame-rate.
The system shows notable accuracy for metrically consistent reconstruction from local loop closure trajectories without performing explicit global joint optimization (Figure \ref{fig3}). The system shows its effectiveness in reducing drift too by performing tracking relative to key frames instead of frame-to-frame matching. System can be easily scaled to available GPU memory with slightly lesser resolution.
\begin{figure*}
\centering
\includegraphics[width=6in,height=3in]{fig3}
\caption{Reconstruction results by Kinect fusion system through normal mapping (constant reconstruction resolution of 2563 voxels) with different sensor trajectories – Courtesy: Newcombe et al (2011)}
\label{fig3}
\end{figure*}
Tracking drifting occurs when sensor is faced with large planner scenes which accounts for system’s shortcomings, but Kinect fusion provides a powerful basis for large scale volumetric reconstruction and dense modeling with various approaches projected by \cite{newcombe2011kinectfusion}.
The Point Cloud Library developed by Rusu and Cousins \cite{rusu20113d} implements the Kinect fusion algorithm to develop Kinfu: an open source implementation hirearchy along with other methods for point clouds manipulation and 3D reconstruction.
Another extension of Kinfu is developed recently by Korn and Pauli with an alternative algorithm for ICP for increased voxel grid hence improving scene dynamics scaning \cite{korn2015kinfu}. The voxel grid data used by Kinfu is used to create vertex and normal maps that are registered with the maps obtained from sensor. But in doing so, unusual amount of information is lost. To cater this problem, Korn and Pauli have suggested direct matching of the maps obtained from sensor with voxel grid model. The ICP algorithm developed by them is also different from the original ICP algorithm adopted for Kinfu as they,ve removed the normal threshold and use the normals computed from the depth maps for point-to-plane error metric instead of using normals from voxel grid that has shown improved robustness in terms of pose estimations with moving objects.
\subsection{RGB-Depth Mapping}
As kinetic fusion algorithm provides consistent and accurate volumetric reconstruction of smaller indoor scenes, the problem of dense 3D mapping of large indoor environments is addressed by the RGB-Depth mapping algorithm by \cite{henry2012rgb}, a framework that uses RGB depth camera to generate dense 3D models of even darker and featureless planner indoor environments. A joint optimization computed over object shape as well as appearance matching (RGB features) is computed to develop alighnment between the frames followed bysparse features extraction and matching using RANSAC. Loop colosures are detected via matching data frames compared to a subset of earlier collected frames and finally an improved, globally consistent allignment is completed either via sparse bundle adjustment (SBA) or a more efficient pose grapgh optimization that is TORO in this case.
What lies at the core of RGB-D mapping is its novel ICP algorithm \cite{turan2017fully}, RGB-D ICP (Figure \ref{fig4}), that identifies the sparse feature points in each camera frame using the visual information. These identified point features then help in RANSAC optimization. An RGB-D frame $P_s$ is input to RGB-ICP algorithm along with target frame $P_t$. For an instant’s rotation R and translation t, the rigid transform is $T(p)=R_p+t$. RANSAC then finds the best optimized rigid transform $T^*$ in order to get best alignment as shown in Equation\ref{eq2}.
\begin{equation}
\label{eq2}
T^* = argminT\Bigg(\dfrac{1}{|A_f|} \Sigma_{i \in A_f} |Proj(T(f_s^i)) - Proj(f_t^i)|^2 \Bigg)
\end{equation}
In Equation\ref{eq2}, $A_f$ is the set containing correspondences between the features points of the target frame $f_t^i$ and source frame $f_s^i$ and the projection function finds the projection of feature points $(x,y,z) \in \mathbb{ R}^3$ in Euclidian space to feature points in camera space $(u,v,d) \in \mathbb{ R}^3$ where d is the depth of each pixel $(u,v)$. A joint optimization performed after that minimizes the re-projection error resulting in an even refine alignment. The joint optimization function is actually modeled to minimize the alignment error of both visual feature association and depth point association and is given as shown in Equation \ref{eq3}.
\begin{multline}
T^* = argminT\Bigg[\alpha\Bigg(\dfrac{1}{|A_f|} \Sigma_{i \in A_f} w_i |T(f_s^i) - f_t^i|^2 \Bigg) + \\
(1-\alpha)\Bigg(\dfrac{1}{|A_d|} \Sigma_{j \in A_d} w_i |\big(T(P_s^j) - P_t^j\big).n_t^j|^2 \Bigg)\Bigg]
\label{eq3}
\end{multline}
\begin{figure*}
\centering
\includegraphics[width=6in,height=2in]{fig4}
\caption{Workflow for RGB-D mapping for constructing 3D models of indoor scenes}
\label{fig4}
\end{figure*}
As shown in Figure \ref{fig4}, the system has a two-staged RGB-D ICP that combines the best properties of both sparse visual features and dense point clouds to get accurate and robust alignment in case of visual features or in complete darkness to determined accuracy. The features are detected using FAST feature detector combined with the Calonder feature descriptor to obtain even better results than using SIFT feature detector as used in the previous work of this study \cite{henry2010rgb}.
Surfel mapping (mapping via small colored surface patches) allows an efficient estimation of surface orientation, color extraction and visualization of the model \cite{krainin2011manipulator}. \cite{henry2012rgb} have proposed window-based sequential bundle adjustment to increase alignmnet accuracy of the system, while loop closure detection is sugested to improve by considering all the depth and color information instead of frame-to-frame visual matching.
\subsection{Spatially Extended and Moving Volume Kinetic Fusion}
As ICP based kinetic fusion suffers from the limitations of bounded reconstruction with poor tracking results for planner geometries. An alternative to ICP is Fast Odometry from Vision Systems (FOVIS) system by \cite{huang2017visual}, a framework based on standard stereo odometry pipeline that reduces mismatches and hence overall drift and increases robustness \cite{huang2017visual}. \cite{whelan2012robust} have updated the algorithm with a combination of ICP and FOVIS to cater for the two challenges stated above that generates a continuous fused map of a scene at an unprecedented extended scale through triangular mesh generation in real time. Unlike \cite{newcombe2011kinectfusion}, the focus of the kintinuous algorithm is to allow the area mapped by the TSDF to move over time to keep the origin of the TSDF at the center of TSDF volume. The movement threshold b is the distance in meters in all the directions from the current position of TSDF origin before re-centering of TSDF. Once the threshold is crossed (referred as boundary crossing in \cite{huang2017visual} given as $v_m = \dfrac{d}{v_s}$), the new camera pose is calculated as shown in Equation \ref{eq4}.
\begin{equation}
C_{i+1} = (R_{i+1}, t'_{i+1})
\label{eq4}
\end{equation}
In Equation \ref{eq4} $C_i$ and $t_i$ are the 6 DOF camera pose and translation vector at time i. And the new position of the global position is calculated by adding the number of number of whole voxel units crossed since the last boundary crossing $\bold{u}=[\dfrac{t_{i+1}}{v_m}]$ to the current global position of TSDF such that: $g_{i+1}=g_{i+\bold{u}}$.
In extension of their work in [16], Whelan et al present the complete workflow of GPU based kintinuous algorithms with a combination of ICP and FOVIS to generate high quality dense colored maps in real-time with robustness (Figure \ref{fig5}). They have adopted a novel switching strategy for which the system can switch between FOVIS and some other estimator depending on the error metric such that the system is accustomed to use FOVIS camera transform $T_F$ when $\bigg|\Big|\big|t_F\big|\Big|_2-\Big|\big|t_O\big|\Big|_2\bigg|> \mu$ otherwise, the system uses other estimator’s transform $T_O$ which is ICP for their system. $\mu$ Here is an experimentally set error metric which was kept 0.03m for system discussed in \cite{whelan2012kintinuous}. The chosen transform is then used to calculate next camera pose. The color and depth information is then combined in RGB-D and ICP integration step where two functions are weighted and combined in one cost function defined as: $E = E_{icp}+ \omega.E_{rgbd}$, where $\omega$ is the assigned weight on empirical calculation basis.
The voxels are streamed out of GPU based camera motion allowing space for new set of data. The streamed voxels are compressed to greedy mesh assuring a fine-quality reconstruction. Instead of restricting tracking and reconstruction to TSDF initialization points, Kintinuous has introduced a cyclic buffer type data structure that allows the area mapped by TSDF to move over time that results in an augments reconstruction of scene incrementally with time. A parallel processing system with multi-level threaded algorithms enables a continuous TSDF tracking and reconstruction of extended areas with unparalleled accuracy.
\begin{figure*}
\centering
\includegraphics[width=6in,height=1.8in]{fig5}
\caption{Extended Kintinuous pipeline for robust camera tracking and real-time colored dense mapping (i) vertex and normal maps are computed from Kinect fusion using the raw depth maps (ii) using a weighted running average (ray casting), the depth data is integrated into current TSDF resulting in a smooth surface reconstruction (iii) point cloud is extracted and triangulated to get mesh surface (iv) new region is registered to volume}
\label{fig5}
\end{figure*}
Rothe and Vona propose the same approach in \cite{roth2012moving} but for a moving volume, i.e, volume is also rotated while translated making the algorithms adaptable to free-roaming applications. Using a 6DOF visual odometry approach, at any time b the local camera pose ${_b^a}C$ is obtained relative to an earlier time a. Based on a threshold value it is determined if a new volume frame is needed or not. In case of a new volume frame, the new frame is remapped from the old by swapping the TSDF buffer and the one in the GPU memory such that new volume transform $P_{n+1}=C_t C_{t+1}^{t-1}$ is set. The idea is to translate and rotate the TSDF volumes that results in 6DOF visual odometry with camera poses constant relative to scene via volume-to-volume transforms hence generating an always-available 3D map of the scene. The linear $l_d$ and angular $a_d$ camera offsets are calculated using equation \ref{eq5} to obtain a local camera pose $C_t$ that determines whether there is a need of a new volume frame or not.
\begin{multline}
D = \begin{bmatrix} R_d & t_d \\ 0 & 1 \end{bmatrix} = C_s^{-1} C_t,
C_t = \begin{bmatrix} R_t & t_t \\ 0 & 1 \end{bmatrix},\\
l_d = \Big|\big| t_d \big|\Big|,
a_d = ||rodrigues^{-1}(R_d)||
\label{eq4}
\end{multline}
Some other approaches have also focused on multi-resolution RGB-D Scene Mapping. One such approach is Omnikinect (\cite{kainz2012omnikinect}) that offers a physical multiple-Kinect infrastructure along with tracking, filtering, and visual hull rendering software for real-time data acquisition and volumetric reconstruction at different resolutions. They have modified the Kinect fusion pipeline in a way that the new volumes are introduced as discrete histograms for the TSDF per voxel such that each voxel $v$ for a deviced, the TSDF is a function $f_{R_k} (v,d)$ for a distance $\eta$ to a measured depth is evaluated for the values $\Psi(\eta)$ in one histogram $\bold{\theta}(v)$ per voxel with an increment in the corresponding histogram bin with $\lambda$. After the TSDF for each value is evaluated the histograms are filtered separately giving filtered TSDF with reduced noise that improves the calibration accuracy of the pose estimation step of original Kinect fusion algorithm.
\subsection{Scalable Real-Time Volumetric Reconstruction}
Although algorithms like kintinuous and moving volume Kinect fusion allow more space for storing data by streaming voxels in real-time, the size of the active volume remains restricted which has triggered the research for saleable volumetric dense mapping and reconstruction. Two notable approaches for this are hierarchal scalable reconstruction by \cite{chen2013scalable} and direct volumetric reconstruction by \cite{niessner2013real}.
\begin{figure*}
\centering
\includegraphics[width=6in,height=2.3in]{fig6}
\caption{Volumetric reconstruction pipeline through Voxel Hashing}
\label{fig6}
\end{figure*}
Both works have developed an upgraded same volumetric method developed by \cite{curless1996volumetric} to generate scalable volumetric reconstruction in real-time. Chen et al adopts the standard Kinect fusion algorithm (Figure \ref{fig6}-a) while dynamically fusing and updating live depth maps and streaming the generated maps between the GPU structure and host all in a hierarchal manner. The basic idea of dense reconstruction through voxel hashing is to fuse an implicit SDF in the dense data structure avoiding a hierarchal data structure and incorporate a hashing scheme instead to efficiently store, access and update the scene reconstruction implicitly. Chen et al have adopted a GPU implementation of the system using CUDA where each hierarchical grid is saved as a sparse point structure in the GPU memory. The root level grid is a dense 3D array of GridDesc records which are initialized to null. Parallel reduction is performed to keep grid descriptors to nearSurface while minWeight field is used as a heuristic for garbage collection. With previous and current SDF $d_p$ and $d_c$ obtained, it is tested if zero crossing is obtained or not. If the surface is obtained then $t_z = t_p + \dfrac{d_p}{d_p - d_c}$; otherwise $d_c$ is set equal to $d_p$ and iteration is performed again. The gradient of the SDF at zero crossing gives the surface normal.
Figure \ref{fig6}-b illustrates the general pipeline of the voxel hashing system; provided a new depth map, the system allocates occupied voxels into hashed blocks (in surface integration) making an implicit surface that is then ray casted for camera pose estimation. \cite{niessner2013real} have proposed a hash table to allocate and retrieve the voxel blocks where the following hashing function maps the world coordinates $(x,y,z)$ to hash values $H(x,y,z)$ such that as shown in Equation \ref{eq5}.
\begin{equation}
H(x,y,z) = (x.p_1 \oplus y.p_2 \oplus z.p_3) mod n
\label{eq5}
\end{equation}
In Equation \ref{eq5} $p_1, p_2, p_3$ are very large prime numbers and n is the size of the hash table. The estimation is done frame-to-model by point-plane ICP hence alleviating any drift. In the final step the algorithm performs lossless bilateral streaming (by revisiting previously scanned areas) between the GPU and user. Streaming alleviates the restriction of range and scan revisiting unlike hierarchal approaches of \cite{whelan2012kintinuous} and \cite{chen2013scalable}.
\subsection{Segmentation-Based RGB-D Mapping }
Another rout to fine quality and globally consistent volumetric reconstruction is through segmentation based RGB-D mapping. 3D mapping with points of interest (POI) \cite{zhou2013dense} and with patch volumes \cite{henry2013patch} are two distinguished approaches developed in towards this stream. \cite{zhou2013dense} present the pipeline to construct scene geometry by detecting points of interest for a globally consistent mapping (Figure \ref{fig7}). For each planar set of localized points $\bar{P} = \{\bar{P}_i^k\} = \{\mathbb{P}_g (\tilde{T}_k p_i^k)\}$ the locations of points of interests POI are found by finding the modes in the density function induced by $\bar{P}$ such that as shown in Equation \ref{eq6}.
\begin{equation}
\rho(x) = \Sigma_{i} w_i^k K (\big|\Big|x-\bar{P}_i^k\Big|\big|/h)
\label{eq6}
\end{equation}
In Equation \ref{eq6}, $w_i^k = \tau exp \bigg(-\dfrac{d_i^{k^2}}{2\sigma^2}\bigg)$ is the weight assigned to a point $p_i^k$ and K is Epanechnikov kernel.
\begin{figure*}
\centering
\includegraphics{fig7}
\caption{Point of interest detection pipeline: the strong modes in the density function represent points of interest}
\label{fig7}
\end{figure*}
An offline optimization framework is incorporated with frame-to-model registration to detect loop closures. After detecting points of interest the framework performs trajectory segmentation followed by a two-pass registration in which for each frame k, a projective SDF $f_k (x)$ is calculated with the assigned weight function $w_k (x)$ for a local range of images $T_k D_k$. Camera pose estimation and global optimization done in this manner generates high quality dense 3D maps.
\cite{henry2013patch} propose an algorithm focusing on regions of interest – dense patches of space - instead of points of interest as a modification to general Kinect fusion algorithm. Previously developed volumetric fusion approaches suffer from a lack of scalability especially for larger scenes because of larger memory requirement for dense maps. Henry et al have addressed this problem by dividing the scene geometry into patches of volumes that can be replaced in the GPU memory. A relative corrected pose estimate is obtained by the Equation \ref{eq7}.
\begin{equation}
T^* = argminT\Sigma \Big|\big|\varepsilon\big|\Big|^2
\label{eq7}
\end{equation}
In Equation \ref{eq7}, $\varepsilon$ is the overall residual error of a weighted combination of geometric error $\varepsilon_g$ and color error $\varepsilon_c$. The error is minimized using the Jacobean equation: $J^TJ \Delta x = -J \epsilon $. The overall Jacobean to find a converged solution is modeled as shown in Equation \ref{eq8}.
\begin{equation}
J_g = \lambda \omega_g (- n_r^T J_{3D} + (p_f - p_r )^T J_{rot})
\label{eq8}
\end{equation}
In Equation \ref{eq8}, $n_r=( x_n, y_n, z_n )^T$ and $p_r=( x_p, y_p, z_p )^T$. And geometric rows are computed using the following equations \ref{eq9} and \ref{eq10}.
\begin{equation}
J_{3D} = \begin{bmatrix} 1 & 0 & 0 & 0 & 2z_n & -2y_n \\ 0 & 1 & 0 & -2z_n & 0 & 2x_n \\ 0 & 0 & 1 & 2y_n & -2x_n & 0 \end{bmatrix}
\label{eq9}
\end{equation}
\begin{equation}
J_{rot} = \begin{bmatrix} 0 & 0 & 0 & 0 & 2z_n & -2y_n \\ 0 & 0 & 0 & -2z_n & 0 & 2x_n \\ 0 & 0 & 0 & 2y_n & -2x_n & 0 \end{bmatrix}
\label{eq10}
\end{equation}
Further, to address the problem of drift because of loop closures, Henry et al have proposed a novel alignment procedure for global consistency in which they have divided the patch volumes into two sets: active patch volumes $S_{current}$ and inactive patch volumes $S_{old}$ where only path volumes of set $S_{current}$ take part into alignment and fusion. Only active patch volumes are used for computing loop closures along with re-rendering the inactive patch volumes.
\subsection{RGB-D Visual Odometry}
Iterative closest point algorithm, though a popular algorithm for image registration, is prone to local minima as image registration is based on suboptimal initial point correspondences. The constraint is alleviated by the steps following registration like RANSAC and bundle adjustment but this increases computational cost while reducing sparse key points reduces computational cost at the expense of data loss. F. Steinbr$\ddot{u}$cker et al have proposed an energy-based framework for computing visual odometry from input data directly hence reducing computational cost \cite{steinbrucker2011real}. They have suggested a computation of rigid body motion that minimizez the least-squares error and maximizes photoconsistency a by finding a twist $\xi$ that minimizes the least-squares error as shown in Equation \ref{eq11}.
\begin{equation}
E(\xi) = \int\limits_\Omega [I(\omega_\xi (w,t_1), t_1) - I(x, t_0), t_0]^2 dx
\label{eq11}
\end{equation}
They have proposed the neregy equation that minimizez the $\xi$ which is given as shown in Equation \ref{eq12}.
\begin{equation}
E_l (\xi) = \int\limits_\Omega \bigg( \dfrac{\delta I}{\delta t}+\Big( \nabla I.\dfrac{d\pi}{dG}.\dfrac{dG}{dg}.M_g\big|_{x,t_0}.\xi \Big) dx \bigg)
\label{eq12}
\end{equation}
An evaluation of the system over benchmark dataset by \cite{sturm2011towards} shows better accuracy compared to a general ICP algorithm that allows a better implementation of visual odometery computing algorithms on GPU.
These sollutions have also complimented the development of a surface mapping system for indoor environemnts in conjunction with visual odometery by Silva and Goncalves (\cite{silva2014visual}) that does not require a GPU for pose estimation. Unlike other feature detection through descriptor algorithms, Silva and Goncalves have adopted a short baseline optical flow tracker to track features for pose estimation. Features are tracked across consective image pairs $(I_{t-1}, I_t)$ via sparse optical flow (Figure \ref{fig8}) after initial Shi-Tomasi corner extraction. After the invalid points are removed, the resulting tracked points are added to robustly estimate the current camera pose $[R_t |t_t]$ using RANSAC. In the last setp the obtained RGB-D frame $I_t$ is registered with the previously estimated frame $I_{t-1}$ to obtain the 3D transform relative to the origin frame.
\begin{figure*}
\centering
\includegraphics[width=6in,height=1.3in]{fig8}
\caption{Visual Odometry approach for camera pose estimation and dense mapping}
\label{fig8}
\end{figure*}
The system has been evaluated for its localization accuracy based on Relative Position Error (RPE) compared with the RGB-SLAM system of Sturm et al using its publically available dataset \cite{sturm2011towards}. The evaluation shows competitive results of the approach proposed for volumetric reconstruction without using GPUs or any other expensive memory hardware.
\subsection{Elastic Fusion}
All the volumetric approaches described above are mainly variants of kinetic fusion algorithm with modifications of different steps. While sparse methods focus on pose graph optimization for reconstruction Whelan et al have recently presented a map-centric approach \cite{whelan2015elasticfusion} for dense SLAM. Based on the offline dense surface reconstruction framework provided by \cite{zhou2013dense}, elastic fusion approach focuses on early and repeatedly loop closure optimizations. This allows for a non-rigid space formation of surface map (via model-to-model loop closures) through an embedded sparse deformation graph instead of a pose graph or any post-processing steps in online incremental manner. While local loop closure optimization keeps the data closer to map distribution global loop closures are computed to recover from arbitrary drift that ensure global consistency. \cite{turan2017non}
Figure \ref{fig9} shows the output of the elastic fusion algorithm. The algorithm is developed based on the point-based fusion algorithm developed by \cite{keller2013real} with a different approach for pose estimation. In camera tracking, the pose estimation is followed by joint optimization minimizing the joint cost function of geometric $E_{icp}$ and photometric $E_{rgb}$ pose estimation such that $E_{track} = E_{icp} + \omega_{rgb} E_{rgb}$. At each iteration Gauss-Newton non-linear least-square resulting in improved camera transformation as shown in Equation \ref{eq13}.
\begin{equation}
T' = exp(\hat{\xi})T
\label{eq13}
\end{equation}
In Equation \ref{eq13}, $\hat{\xi} = \begin{bmatrix} [\omega]_x & x \\ 0 0 0 & 0 \end{bmatrix} $. For local loop closures, instead of performing geometric frame-to-modal tracking via splatted rendering, elastic fusion performs photometric frame-to-modal tracking via full colored splatted rendering. Furthermore, surfels are marked active or inactive based on a set time window threshold $\delta_t$ and only active surfels are utilized for estimating camera poses and in depth map fusion.
\begin{figure*}
\centering
\includegraphics[width=7in,height=2in]{fig9}
\caption{Main steps outlined for Elastic Fusion algorithm with a final global loop closure database containing sampled camera poses and underlying deformation graph}
\label{fig9}
\end{figure*}
For global loop closures, \cite{whelan2016elasticfusion} have utilized the randomized fern encoding approach developed by \cite{glocker2015real} for adding predicted views and finding matches. In case of detected matches, the views are registered for computing globally consistent maps into a non-rigid deformation to obtain a global surface alignment \cite{turan2017non1}.
Compared with other state of the art systems including, Kinect fusion, DVO SLAM, and RGB-D SLAM, the results obtained from Elastic Fusion algorithm (Tables \ref{table1} and \ref{table2}) show robustness and better accuracy in terms of camera trajectory and surface estimation with accepted computational performance requirements without pose graph estimation or any other post-processing steps. The detailed results demonstrated in \cite{whelan2015elasticfusion} also show the efficacy of this approach in surface mapping at even larger scale than that of rooms or buildings (dense mapping for $t \rightarrow \infty$).
\subsection{Bundle Fusion}
To address the tracking gaps lying in systems discussed above, \cite{dai2017bundlefusion}, have recently proposed a novel robust pose estimation scheme in \cite{dai2017bundlefusion} that employs an efficient hierarchal approach for optimizing per frame to obtain a global set of camera poses. Taking into account all the RGB-D history, bundle fusion approach sets its core to a novel two-stage global pose optimization strategy that results in an online globally-consistent 3D mapping. First, a sparse-then-dense global pose optimization is performed to obtain global alignment (Figure \ref{fig10}). A live RGB-D stream from the sensor is input to the system followed by an optimal rigid camera transforms by 3D correspondences between the frames. Correspondence filtering incorporates both geometric and photometric optimization to minimize outliers through key point correspondence filter, and surface area filter.
\begin{figure*}
\centering
\includegraphics[width=7in,height=2in]{fig10}
\caption{Bundle fusion pipeline for globally consistent 3D reconstruction with global pose optimization and on-the-fly surface re-integration – Courtesy: \cite{dai2017bundlefusion}}
\label{fig10}
\end{figure*}
Dai et al have followed hierarchical optimization approach in which input scene sequences are divided into small chunks of consecutive frames and local alignments are achieved within the chunks followed by global alignment in the next step. After performing a hierarchal optimization and pose estimation, dynamic 3D reconstruction is performed while continuous monitoring an updating consecutive poses through integration and disintegration of frame which helps in fixing the problem of accumulated drift and dead reckoning in feature-less regions. Same as the approach of \cite{silva2014visual}, for a parameter vector $X=(R_o, t_o, \cdots , R_{|S|}, t_{|S|})^T$ for $|S|$ frames, the pose alignment based on an energy optimization approach is achieved as given in Equation \ref{eq14}.
\begin{equation}
E_{align}(X) = \omega_{sparse} E_{sparse}(X) = \omega_{dense}E_{dense}(X)
\label{eq14}
\end{equation}
In Equation \ref{eq14}, $ω_{sparse}$ and $ω_{dense}$ are weights assigned to sparse and dense matching terms and $E_{sparse} (X)$ and $E_{dense} (X)$ are the sparse and dense matching terms respectively, such that: $E_{sparse} (X) = \Sigma_{i=1}^|S| \Sigma_{j=1}^|S| \Sigma_{(k,l) \in C(i,j)} \Big|\big| T_iP_{i,k} - T_j P_{j,l} \big|\Big|^2$ with $T_i$ being rigid camera transformation, $P_{i,k}$ the $k^{th}$ detected feature point in $i^{th}$ frame, and $C(i,j)$ being the set of pairwise correspondences between the $i^{th}$ and $j^{th}$ frame. And $E_{dense} (T) = \omega_{photo} E_{photo} (T) + \omega_{geo} E_{geo} (T)$ with $\omega_{photo}$ and $\omega_{geo}$ being the weights of photometric and geometric term and $E_{photo} (T)$ and $E_{geo} (T)$ being terms for calculating photometric and geometric alignment respectively.
Pose optimization is performed at every frame and reconstruction is updated accordingly along free-camera paths instead of temporal coherence. To make global pose alignment real-time and tractable a hierarchal local-to-global pose optimization is performed ensuring increased scalability for larger scene reconstruction. Dense surface reconstruction is achieved via volumetric fusion reconstruction pipeline proposed by Nießner et al with an added novelity of symmetric on-the-fly RGB-D farmes registration.
\begin{figure*}
\centering
\includegraphics{fig11}
\caption{Results showing Bundle Fusion's effectiveness in detecting grey overlay and recovering from tracking failures}
\label{fig11}
\end{figure*}
Where all frame-to-model tracking algorithms (\cite{newcombe2011kinectfusion}, \cite{chen2013scalable}, \cite{niessner2013real}) using ICP for closure detection face the problem of accumulated drift, bundle fusion outperforms them in terms of recovery from tracking failure (Figure \ref{fig11}) and reducing geometeric drift through an implicit global pose otimization approach to handle loop closures.
\section{Milestones about Kinect and bundle fusion and its derivations as a survey and history summarization}
\subsection{Kinetic fusion}
Kinetic Fusion refers to the 3D object scanning and model development resource that utilizes a Kinetic for Windows based operating system sensor. The user is able to paint a scene using the Kinetic camera while simultaneously viewing and interacting with an extensive 3D model of the scenery. Kinetic fusion is able to be operated within interactive rates with GPU support and is also able to operate on non-interactive rating on varied hardware.
However, it should be noted that operating at non-interactive rates can permit substantial volume reconstructions. Kinetic Fusion is able to process data by using either DirectX 11 GPU compatibility with C++ AMP with the alternative being on the CPU which is attained by placement of the reconstruction processor format in the course of the reconstruction volume development. The CPU processor is well suited for applications of offline processing which is restricted to DirectX 11 GPUs that permit for real time and interactive frame rating in the process of reconstruction.
\begin{figure*}
\centering
\includegraphics[width=7in,height=3in]{fig12}
\caption{Graphical chart representation of Kinetic fusion processing pipeline entailing multiple stages from raw depth to attain 3D reconstruction}
\label{fig12}
\end{figure*}
Recommended hardware entails Desktop computers with GHz or better utilizing a multi-core processor as well as a graphics card an additional dedicated on-board memory. It has also been tested on high end platforms of NVidia GeForce GTX680 as well as AMD Radeon HD 70.
The capability also exists of Kinetic Fusion to be operated on a laptop grade of DirectX11 hardware even as it normally operates on a significantly reduced performance in comparison of desktop grade hardware. In overall, the objective seeks to process a similar frame rate at 30fps as the Kinetic sensor to allow for the extensive robustness of tracking camera poses.
\subsection{Bundle fusion}
Bundle Fusion on the other hand allows for real-time, high grade 3D scanning of extensive scale scenery as a major consideration to mixture of reality and robotic applications. However, scalability introduces the challenge of drifting with estimation of poses and the introduction of significant errors in the accumulation models. This translates into approaches normally requiring extended hours for offline processing of international correct model errors.
Current online approaches has provide compelling outcomes even as they are limited from several issues. The first issue is the requirement of minutes to undertake online correction in the prevention of actual time usage. The second issue is the brittle frame to frame model pose estimation as an outcome in several tracking shortcomings. The third and final issue is their support is restricted to unstructured point based representation that is able to limit the quality of scanning and application.
\begin{figure*}
\centering
\includegraphics[width=7in]{fig13}
\caption{Graphical representation of real time, 3D scanning and high quality large scaling is vital to attaining a mixed reality which can be applied to robotic applications}
\label{fig13}
\end{figure*}
This can be addressed systematically to concerns where a novel, actual time and end to end reconstruction frame working. Within its core is the estimation of robust possess based on strategies that optimize the per frame rating for a global collection of camera posing with consideration of the entire history of RGB-D input using efficient hierarchical methods. The heavy reliance on temporal tracking is eliminated and persistently localized to the global optimization of frames.
The contribution is a parallel optimization framework that is able to apply correspondence on the basis of spatial feature and heavy geometric and photometric matching. The approach used is an estimation of global optimization which is to say bundle adjustment that places real time and support for robust tracking using gross tracking failure recovery which is to say re-localization and re-estimation of the 3D model in actual time for ensuring the global consistency all in entire single framework. This approach outperforms that modern online systems using quality on par to offline approaches even with unprecedented speeding and scan completion.
\section{Algorithms Characterization and Analysis}
The performance of the RGB-D SLAM and Kinect Fusion algorithm discussed above has been summarized in table 1 below in terms of absolute trajectory room mean square metric (ATE RMSE) on two datasets: of \cite{handa2014benchmark} and of \cite{sturm2012benchmark}. Comparison is done based on scene tracking, offline or online registration performance, camera revisiting the previous map, computing loop closures, and surface reconstruction.
The algorithms are usually in par with or are better than others in some features and lack in other and are well suited for various 3D reconstruction applications based on requirements and level of complexity. But some important points to note here are the computing performance of these algorithms to be implemented on commercial scale and more importantly efficacy of these algorithms in performing a live, real-time, large scale, and drift-free dense mapping of the scenes.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Comparison of trajectory estimation in terms of ATE RMSE results of state-of-the-art algorithms on the evaluated synthetic ICL-NUIM datasets of \cite{handa2014benchmark} Source: (Manipulator and object tracking for in-hand 3D object modeling) \cite{krainin2011manipulator}}
\label{table1}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
System & Kt0 & Kt1 & Kt2 & Kt3\\
\hline
DVO SlAM & 0.104m & 0.029m & 0.191m & 0.152m\\
\hline
RGB-D SLAM & 0.026m & 0.008m & 0.018m & 0.433m\\
\hline
MRSMap & 0.204m & 0.228m & 0.189m & 1.090m\\
\hline
Voxel Hashing & 0.014m & 0.004m & 0.018m & 0.120m\\
\hline
Kintinuous & 0.072m & 0.005m & 0.010m & 0.355m\\
\hline
Frame-to-model & 0.497m & 0.009m & 0.020m & 0.243m\\
\hline
Elastic Fusion & 0.009m & 0.009m & 0.014m & 0.106m\\
\hline
Bundle Fusion & 0.006m & 0.004m & 0.006m & 0.011m\\
\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Comparison of trajectory estimation in terms of ATE RMSE results of state-of-the-art algorithms on the evaluated synthetic TUM RGB-D dataset. Source: (Manipulator and object tracking for in-hand 3D object modeling) \cite{krainin2011manipulator}}
\label{table2}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
System & fri/desk & fr2/xyz & fr3/office & fr3/nst\\
\hline
DVO SlAM & 0.021m & 0.018m & 0.035m & 0.018m\\
\hline
RGB-D SLAM & 0.02m & 0.008m & 0.032m & 0.017m\\
\hline
MRSMap & 0.043m & 0.020m & 0.042m & 2.018m\\
\hline
Voxel Hashing & 0.023m & 0.022m & 0.023m & 0.087m\\
\hline
Kintinuous & 0.037m & 0.029m & 0.030m & 0.031m\\
\hline
Frame-to-model & 0.022m & 0.014m & 0.025m & 0.027m\\
\hline
Elastic Fusion & 0.020m & 0.011m & 0.017m & 0.016m\\
\hline
Bundle Fusion & 0.016m & 0.011m & 0.022m & 0.012m\\
\hline
\end{tabular}
\end{table}
The systems developed either perform implicit registration of poses and suffer from the issues of scalability because of limitations of uniform grid as in the case of Kinect Fusion \cite{newcombe2011kinectfusion}; or in case of more efficient volumetric fusion strategies (\cite{chen2013scalable} \cite{niessner2013real} \cite{whelan2012robust} \cite{roth2012moving}) face increased drift because of pose errors in pose estimation. The approaches performing globally consistent mapping suffer from a lack of live real-time reconstruction because of offline processing and larger memory and processor requirements (\cite{zhou2013dense} \cite{whelan2012kintinuous}).
As pointed out by \cite{dai2017bundlefusion}, a comprehensive and state-of-the-art system is required to meet the requirements of high-quality surface modeling that is able to model continuous surfaces while meeting the requirements of robust camera tracking, real-time rates, on-the-fly modal updates, scalability, and global model consistency. Although bundle fusion seems to address these issues in future providing a complete robust tracking and mapping system but the system still suffers from misalignments and memory issues.
Thus, for the environments that are too challenging or the robot dynamics are complex, the SLAM algorithms developed for real-time tracking and mapping may fail and there is still need for research in this arena \cite{turan2018sparse}. The SLAM algorithms are considered to have been entered into the third era of their development which Cadena et al called the robust-perception age \cite{cadena2016past}. The requirements of the robust-perception age the SLAM algorithms are currently in are pointed out by \cite{cadena2016past} as: robust performance of the SLAM system that allows fail-safe performance of the system with low failure over an extended period of time with self-tuning characteristics unlike current algorithms that require manual tuning; high-level understanding of the environment by the system, resource awareness in which the system can adapt to available sensing and computational resources; and task-driven perception of the system in which system is able to discard irrelevant data by itself allowing adaptive representation of maps whose complexity may vary depending on the type of task being performed.
The approaches discussed here all use the standard front-end sensing and back-end processing approach in which the front-end senses and gathers the data while the back-end does the processing of that data input by the front-end \cite{cadena2016past}. Algorithms relying on this approach are always prone to loop closure problem as the input of wrong loop closures to the back-end processing is unavoidable which further degrades the quality of the maps generated. Although the systems like in \cite{henry2012rgb} \cite{curless1996volumetric} \cite{glocker2015real} have adopted novel strategies to avoid the problem of wrong loop closures but the problem still requires more focus in terms of improving the robustness of the systems. a new line of research has been initiated specifically to deal with this problem as identified in \cite{sunderhauf2012towards} and \cite{latif2013robust}. Another problem of such algorithms is that they are prone to outliers \cite{cadena2016past} and this is where the fail-safe or the automatic recovery of the system comes as the key requirement for a robust system. Bundle fusion \cite{glocker2015real} shows a much better metric in this case than other SLAM systems discussed here.
In terms of scalability, most of the systems discussed in the paper show an effective performance for small indoor environment but have made improved scalability their future task showing there is much work to be done for systems’ scalability to map larger outdoor environments over an extended period of time. The approaches mentioned use the camera re-visiting approach using iterative linear solvers to register more scenes. This approach requires much larger memory for the mapping of larger environment \cite{sturm2012benchmark} and hence put a constraint on the scalability of the systems due to larger memory and computing requirements. However, the systems like the one developed by \cite{huang2017visual} that has incorporated GPU-based processing for a scalable mapping and such shown in \cite{kainz2012omnikinect} and \cite{chen2013scalable} show some prolific prospects for better scalability of the SLAM systems.
The use of Kinect sensor for simultaneous localization and mapping with considerable outcomes predicts an opportunity of better SLAM systems with improved novel sensors that allow active sensing and real-time scalable mapping by using such systems. New tools and technologies are needed to be integrated to obtain complete robust, fail-safe, self-tuning SLAM systems that would be able to predict, update, remember, or discard, the information according the requirements of the tasks in hand by utilizing the resources that are available and by adapting to the environmental requirements.
Several applications of relevant in robotics as well as computer vision need the capability of rapid acquisition of 3D models of the environment and the estimation of the camera pose in respect to the model. A robot for instance, requires ascertaining its location for navigation between places. This challenge is classical and difficult since camera localization in needs 3D models that is turn call for the camera pose according to \cite{niessner2013real}. As such, the camera trajectory and 3D model require estimation at the similar timing. Introducing Kinetic fusion platform avails for colored imagery and dense depth mapping for complete video frame rating.
It also permits for the creation of a novel method to SLAM integrates the scaling information of 3D depth sensing with the benefits for visual features to creating dense 3D setting representations. The estimation of the trajectory is segmented into a frontal end and a back end while the frontal extraction spatial relation among the specific observations provides for a back end optimization that places such observations in a so defined pose graph respecting non-linear error functions as interpreted by \cite{agarwal2010reconstructing}.
The frontal end utilizes visual imagery of the RGB-D sensor for detection of the key points as well as extracts descriptors. These can be matched to past extraction descriptors as well as the relative transformation among the sensor pose based on computation with use of the RANSAC by \cite{glocker2015real}.
Our analysis is based on the novel RGB-D SLAM system of visual odometry and information filter extension that does not need any other sensors or odometry. This is different to the approaches of graph optimization which is increasingly suitable for online applicability. The visual dead reckoning algorithm founded on the visual residuals is formulated which is applied in estimation of motion control input as presented by \cite{niessner2013real}. This is augmented with the utilization of the novel descriptor known a binary robust appearance and normalized descriptors (BRAND) for the extraction of features from the RGB-D frame and utilizing them in the form of landmarks according to \cite{karan2015calibration}.
In addition, with consideration of the 3D positions and landmark BRAND descriptors, we shall use an observation model that limits explicit data relation between the observations and mapping with the use of marginalization observation possibility over entire likely relations.
Experimental validity is availed in comparison with the proposed RGB-D SLAM algorithm using mere RGB-D visual odometry as well as graphing as provided by the dataset. The analytical results of the dataset reveal the self localization is broadly held in recognition as one of the most elementary challenges for robotics autonomy in regard to navigation according to \cite{curless1996volumetric}. This assignment can be undertaken well at the point when the environment is defined as priori even the mapping is not availed beforehand. Therefore, robot localization becomes highly challenging.
This can be attributed to the inadequate setting information of the movement of the robot in or the excessive expense of manually constructing a map based on objective. In this instance, the robot needs to simultaneously formulate a map of the environment followed by self-localization in it. This challenge defined as simultaneous localization and map construction is extensively examined.
The SLAM solutions challenge presented by far differs majorly for the setting description adopted and the employed estimation technique. The two main estimation methods we used for analysis are filter and graph based SLAM. Filter based SLAM entails estimation of the posterior using the means of the Bayes' rule shown in Equation \ref{eq15}.
\begin{equation}
p(\xi_t, m|z_{1:t}, u_{1:t}))
\label{eq15}
\end{equation}
In Equation \ref{eq15}, $\xi_t$ is the robot pose at time t, m denotes the map, $z_{1:t}$ as the observation sequence as well as $u_{1:t}$ as the odometry information. Also known as online SLAM, it uses an incremental past measurement and control alternative which are neglected upon processing. In accordance to varied approaches of tacking the posterior probability, there are several filter based techniques such as the extended Kalman filter (EKF), the extended information filter (EIF) as well as the particle filter method (PF).
Rather than estimation of the single post $\xi_t$, within filter based SLAM, the graph based estimation of the full trajectory $\xi_{1:t}$ as well as the map denoted by m is for the entire information observed by \cite{zhou2013dense}. Even as this approach is held in consideration as being time consuming and unable to satisfy actual time needs, however by techniques of efficient solving, graph based SLAM avails the more attention.
The initial analysis on the SLAM challenge places emphasis on the two dimensional setting such that can be normally applied in mobile robotics. Of recent, \cite{krainin2011manipulator} propose varied 3D SLAM algorithms have provided supports for 6-DOF (degree of freedom) estimation of pose such that the employed SLAM technique in varied platforms for instance quad rotors among others. Earlier 3D SLAM studies, costly sensors such as 2D and 3D-laser range finders were major applied. \cite{2018arXiv180307608S}
However, of recent, with the introduction of low cost Kinetic style sensors known a RGB-D cameras, they provide color imagery and depth data in a concurrent manner that is known as RGB-D SLAM \cite{turan2018deep}. For the greater part of our robotics SLAM, this was undertaken with a sensor that avails 2D scenery with the major reasoning being that in order to attain high quality 3D data, the cost is high. It is at this point that the cheap Kinetic technology provides immense interest in the capture and reconstruction of 3D environments using a RGB-D sensor that is moveable. It avails dense, increasingly resolution depth information at a lower cost and scope on the basis of data by \cite{sturm2012benchmark}.
We formulate SLAM application with the use of Kinetic and bundle adjustment framework for integration of the iterative closest point using visualized feature matching. Our research, our graph optimization uses a g2o also known as the general optimization framework to attain global alignment according to \cite{zhou2013dense}. This means we adopt the iterative closest point for pairwise alignment among the sequential framing and recovering of the rigid transformation within point clouds. The accuracy alignment of the iterative closest point is significantly dependant on the content of the scene. We therefore used color feature descriptors for improvement of our depth data correspondence.
We therefore propose a RGB-D slam approach that manages low dynamic scenarios with the use of pose graph structures where grouping of nodes is based on covariant values. Any constraints that are falsified are pruned on the basis of error metrics associated to the node factions according to \cite{keller2013real}. Our study therefore examines highly efficient pose graph optimization for instance tree based network optimizer (TORO).
Improvement of this algorithm when combined with features from accelerated segment test (FAST) a well a Calonder descriptors is able to provide an estimation of the pose with use of the re-projection error random sample consensus (RE-RANSAC) methodology for frame to frame based alignment as well as incorporation of ICP constraints into sparse bundle adjustment (SBA) for global optimization.
The algorithm core known as RGB-D ICP entails a noble iterative closest point variant that takes use of the extensive information within RGB-D data. Our methodology is highly efficient in complex indoor environments as its trajectory algorithmic estimation is based on the integration into singular global procedures that are not reliant on intermediate level features which has also been observed in works by \cite{dai2017bundlefusion}.
More so, the use of accurate pose measurement with techniques of localization and a compact photometric environment model is attained. In the correct rigid body motion of a handheld RGB-D camera, the energy based approach is the estimation.
This when combined with the technology of odometry sensor RGB-D for automated flight experimental analysis creates the possibility of planning complex 3D paths within a cluttered setting. This presents us with a novel GPU implementation on the foundation of an RGB-D visual odometry algorithm using the 6-DOF camera odometry estimation methodology for tracking and integrating RGB color information into the reconstruction process of Kinetic Fusion to permit a high quality mapping.
The analysis results reveal there is no necessity for the application of key frames and the outcomes of real-time colored volumetric surface reconstructions shows the several RGB-D SLAM techniques are restricted to geometrically structured environments. At this point we can propose a switched based algorithm with heuristic selection between RGB-D bundle adjustments on the basis of localized map building. Such maps are developed by the application of sparse bundle adjustments on an inclusion of two-step re-projection of RANSAC and ICP approach.
With a heuristic switching algorithm, we deal with multiple failure modes related with the RGB-D-BA bundle adjustment. The map linkage mechanism greatly lowers the computational expenses such that this algorithm holds immense benefit for application in a large scale setting. For evaluation of the system, we shall utilize the RGB-D benchmark that avails a Kinetic sequence dataset using synchronized ground truth. In addition, the benchmark avails an evaluation resource based on computation of the root mean square error when provided with an estimated trajectory.
In evaluating, we select the Freiburg 1 dataset comprising of nine sequences in placement with a normal indoor environment. Two of these sequences hold very simplified motions such that the outcome of the sequences reveals the capabilities of this technique in the best case.
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{System evaluation on a large set of sequences from RGB-D SLAM dataset with a 9.7cm and 3.95 degrees accuracy and a required approximation of 0.35 seconds for time per image processing. Source: (RGB-D mapping: Using Kinect-style depth cameras for dense 3D modeling of indoor environments) \cite{henry2012rgb}}
\label{table3}
\centering
\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{2cm}|p{2cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|}
\hline
Sequence Name & Length & Duration & Avg. Angular Velocity & Avg. Transl. Velocity & Frames & Total Runtime & g20 Runtime & Transl. RMSE & Rot. RMSE\\
\hline
FR1 360 & 5.82m & 28.69s & 41.60deg/s & 0.21m/s & 745 & 145s & 0.66s & 0.103m & 3.41deg\\
\hline
FR1 desk2 & 10.16m & 24.86s & 29.31deg/s & 0.43m/s & 614 & 176s & 0.68s & 0.102m & 3.81deg\\
\hline
FR1 desk & 9.26m & 23.40s & 23.33deg/s & 0.41m/s & 575 & 199s & 1.31s & 0.049m & 2.43deg\\
\hline
FR1 floor & 12.57m & 49.87s & 15.07deg/s & 0.26m/s & 1214 & 488s & 3.93s & 0.055m & 2.35deg\\
\hline
FR1 plant & 14.80m & 41.53s & 27.89deg/s & 0.37m/s & 1112 & 424s & 1.28s & 0.142m & 6.34deg\\
\hline
FR1 room & 15.99m & 48.90s & 29.88deg/s & 0.33m/s & 1332 & 423s & 1.56s & 0.219m & 9.04deg\\
\hline
FR1 rpy & 1.66m & 27.67s & 50.15deg/s & 0.06m/s & 687 & 243s & 10.26s & 0.042m & 2.50deg\\
\hline
FR1 teddy & 15.71m & 50.82s & 21.32deg/s & 0.32m/s & 1395 & 556s & 1.72s & 0.138m & 4.75deg\\
\hline
FR1 xyz & 7.11m & 30.09s & 8.92deg/s & 0.24m/s & 788 & 365s & 40.09s & 0.021m & 0.90deg\\
\hline
\end{tabular}
\end{table*}
However, outcomes such as this can normally be attained with the careful movement of the sensor in an indoor environment. For example, in the course of manual mapping recording before deploying a robot. The other datasets are increasingly complex since they entail coverage of larger sections as well as unrestricted camera motions.
The results of the accuracy of the SLAM system and evaluation are dependent on the accuracy of the selected feature detector and sensor frame rating \cite{turan2017deep1}. The second step is the investigation of the influence of multiple parameter4s on the system runtime. As observed on the entire nine sequences of the system in the FR1 table, the average camera velocities fall in the range of 9 to 42 degrees as well as 0.06 to 0.43 meters per second.
\subsection{Feature detection and descriptor extraction}
The method for detecting frames as well as extracting descriptors is increasingly apparent when specific to incoming image frames. The table below is a representation of the time comparison required for distinct feature types founded on early descriptives. From the tabulated information, ORB holds a faster speed compared to SURF and SIFTGPU according to single order magnitudes. However, the results based on revelations of increased errors in sequences of two or nine re-aimed angle at almost of 3.5x speeds when compared to SURF implementation that is non-parallel.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Feature runtime analysis with respect to a selected interest point detector and feature descriptor. Source: (RGB-D mapping: Using Kinect-style depth cameras for dense 3D modeling of indoor environments) \cite{henry2012rgb}}
\label{table4}
\centering
\begin{tabular}{|p{1cm}|p{3cm}|p{3cm}|}
\hline
Type & Avg. Count - Std. Dev & Runtime Detection + Extraction Avg. - Std. Dev.\\
\hline
SURF & 1733 - 153 & 0.34s + 0.34s\\
\hline
ORB & 1117 - 558 & 0.018s + 0.0086s\\
\hline
SIFTGPU & 1918 - 599 & 0.19s\\
\hline
\end{tabular}
\end{table}
Matching of features and estimating motion needs computation of single instances per frames in case the present frame is limited to matched single predecessors such that the output camera trajectory is rapidly increased with accumulation of errors when measured against time. Multiple frames feature matching is costly to compute more so since no assumption is held of the availed odometry information being restricted to the likelihood of closures within loops. The system can also be accurate of trajectories that are extended with augmented information concerning pairwise relative transformation that makes estimating of the trajectory highly robust to errors when estimated according to pose. However, this method is highly linked and requires increased optimization time. Therefore, we can understand that match current factors to past 20 frames provide a satisfactory compromise.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Runtime analysis of pair wise frame registration. Source: (RGB-D mapping: Using Kinect-style depth cameras for dense 3D modeling of indoor environments) \cite{henry2012rgb}}
\label{table5}
\centering
\begin{tabular}{|p{1cm}|p{2cm}|}
\hline
Matcher & Runtime (Avg. - Std. Dev.)\\
\hline
FLANN & 0.203s - 0.078s\\
\hline
Brute Force & 0.386s - 0120s\\
\hline
\end{tabular}
\end{table}
This is reflected in the tabulated information above in form of the mean run time in estimating and matching motion which is also a revelation of FLANN reducing the time required for registering frames by a factor of two. Pose graph optimization. Optimization of minimal graphs is faster to implement in the real time as well as for entire frames with lengthy sequences and dense connections with increases in time optimization. However, in case estimating motion is reliable, the overall method can be implemented at all levels according to findings by \cite{whelan2015real}.
In full sequences, graph optimization ratio when compared to the general time is lower than 6 percent. The novel method to visualization of SLAM algorithms with the use of RGB-D sensors means the method will entail the methodologies that entail extracting visual key points from colored images and applying depth images to localize. It is at this point that we can use RANSAC for robust estimation of changes between RGB-D frames and pose graph optimization using non-linear techniques. At the end, volumetric 3D maps of the setting can be used to localize the navigation of the robot and plan its path.
\section{Conclusion}
All the reconstruction approaches developed till now contribute towards a specific modification in 3D reconstruction algorithms for efficient camera tracking and surface mapping. But the systems developed till now still lack one crucial element: deployment of a complete robust and accurate real-time tracking and mapping system to be used for robotic navigation, scanning and mapping. RGB-D sensors have opened a new field of opportunities to handle this challenge and paved a way for improved (if not completely solved) SLAM systems in future. Prototype systems developed do show promising efficacy towards a universal real-time 3D reconstruction system implementation in future.
\bibliographystyle{IEEEtran}
|
1,314,259,996,731 | arxiv | \part{}
\section*{Contents}
\input{main.toc}
\input{sections/notation.tex}
\input{sections/visgo.tex}
\input{sections/app_logs.tex}
\input{sections/app_logsa.tex}
\input{sections/setproperties.tex}
\input{sections/discovery.tex}
\input{sections/rtest.tex}
\input{sections/app_consolidation.tex}
\input{sections/app_evaluation.tex}
\input{sections/auxresults.tex}
\end{document}
\section{Analysis of \pref{alg:LOGSSD}}\label{app:logs}
In this section, we assume the state space is finite (i.e., $S = |{\mathcal{S}}| < \infty$).
\subsection{Properties of the sets built by \pref{alg:LOGSSD}}
\begin{lemma}
\label{lem:calK.easy}
Denote by ${\mathcal{K}}_r$ the set ${\mathcal{K}}$ at the end of each round $r$, by $g^\star_r$ the goal selected in such a round, and by $\pi_{g^\star_r,r}$ its corresponding policy (computed by VISGO in \pref{line:compute V.easy}). With probability at least $1-\delta$ over the randomness of \pref{alg:LOGSSD}, we have that, for any round $r$,
\begin{itemize}
\item ${\mathcal{K}}_{r} \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$;
\item if \pref{line:goal condition.easy} is False, then $\|V^{\pi_{g^{\star}_r,r}}_{g^{\star}_r}\|_{\infty} \leq 4L$ which implies $\|V^{\star}_{{\mathcal{K}}_{r-1},g^{\star}_r}\|_\infty \leq 4L$;
\item for all $g \in {\mathcal{K}}_r$, $\|V^{{\widetilde{\pi}}_{g}}_{g}\|_{\infty} \leq 4L$ and $V^{{\widetilde{\pi}}_g}_g(s_0)\leq L(1+\epsilon)$.
\end{itemize}
\end{lemma}
\begin{proof}
Clearly, ${\mathcal{K}}_1 = \{s_0\} \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$. Then, consider a round $r\geq 2$ and suppose ${\mathcal{K}}_{r-1}\subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ (inductive hypothesis). If, in this round, the algorithm selects a goal $g^{\star}_r\in {\mathcal{U}}\setminus\calS^{\rightarrow}_{L(1+\epsilon)}$, \pref{line:goal condition.easy} is False, and a skip round is not triggered, then \pref{line:failure.easy} is reached. We now prove that the ``failure test'' in that line triggers.
Note that every time ${\mathcal{K}}$ is updated, the sampling at \pref{line:fill N.easy} guarantees that for all $(s,a) \in {\mathcal{K}}_{r-1} \times {\mathcal{A}}$, $\N_{r-1}(s,a) \geq O(L^2|{\mathcal{K}}_{r-1}|\log(S/\delta))$. By~\pref{lem:bounded error}, since ${\mathcal{K}}_{r-1} \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ (inductive hypothesis), we have that
\begin{equation}\label{eq:inductive.error.bound.easy}
\mathbb{P}\left(\forall g \in {\mathcal{S}} \setminus {\mathcal{K}}_{r-1} : V^{\pi_g}_g(s) \leq 2 V_{{\mathcal{K}}_{r-1}, g}(s) \right) \geq 1-\frac{\delta}{4r^2}.
\end{equation}
where $(\_, V_{{\mathcal{K}}_{r-1}, g},\_) = \textsc{VISGO}\xspace({\mathcal{K}}_{r-1},g,\xi_r,\N_{r-1},\frac{\delta}{4r^2 S^2})$ and $\xi_r$ is the value of $\epsilon_{\mathrm{VI}}$ used in round $r$.
Note that \textsc{VISGO}\xspace returns a value function that is either $\infty$ or bounded by $2L$ for all states (see Alg.~\ref{alg:VISGO}). Since $g^{\star}_r$ passes the test of \pref{line:goal condition.easy}, then $V^{\pi_{g^{\star}_r,r}}_{g^{\star}_r}(s) \leq 2 V_{{\mathcal{K}}_{r-1},g^{\star}_r }(s)\leq 4L$, for all $s \in {\mathcal{S}}$. Combining this with \pref{lem:V pi mean} and definition of $\lambda = N_{\textsc{Dev}}(32L, \frac{\epsilon}{256}, \frac{\delta}{4 r^2})$, we have $\widehat{\tau}\geq V^{\pi_{g^{\star}}}_{g^{\star}}(s_0) - L\epsilon/2$ with probability at least $1-\frac{\delta}{4r^2}$. By assumption on $g^\star_r$ and since $\pi_{g^{\star}_r,r}$ is restricted on ${\mathcal{K}}_{r-1}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$, we have $V^{\pi_{g^{\star}_r,r}}_{g^{\star}_r}(s_0) \geq V^{\star}_{{\mathcal{K}}_{r-1},g^{\star}_r}(s_0) \geq V^{\star}_{\calS^{\rightarrow}_{L(1+\epsilon)},g^{\star}_r}(s_0) > L(1+\epsilon)$, which implies that $\widehat{\tau}\geq L(1+\epsilon/2) \geq V_{{\mathcal{K}}_{r-1}, g^{\star}_r}(s_0) + \epsilon L/2$ with the same probability, where the last inequality is from the goal-selection rule. Therefore, the failure test of \pref{line:failure.easy} triggers and $g^\star_r$ is not added to ${\mathcal{K}}_r'$ or ${\mathcal{K}}_r$. Therefore, by the inductive hypothesis ${\mathcal{K}}_{r}\subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$. A union bound over all $r \geq 1$ yields the first statement with probability at least $1-\delta$.
To prove the second statement, note that we already proved above that $V^{\pi_{g^{\star}_r}}_{g^{\star}_r,r}(s) \leq 4L$ at any round $r$ where \pref{line:goal condition.easy} is False (i.e., where $g^{\star}_r$ reaches the policy evaluation step). Since $\pi_{g^{\star}_r,r}$ is restricted on ${\mathcal{K}}_{r-1}$, we clearly have $V^{\star}_{{\mathcal{K}}_{r-1},g^{\star}_r}(s) \leq V^{\pi_{g^{\star}_r,r}}_{g^{\star}_r}(s) \leq 4L$. This proves the second statement for any round $r$, which holds with the same $1-\delta$ probability.
Finally, the third statement is a simple consequence of the fact that any goal $g\in{\mathcal{K}}_r$ must have reached the policy evaluation step in some round $r' < r$ and the round was successful, and thus $\|V^{{\widetilde{\pi}}_{g}}_{g}\|_{\infty} \leq 4L$ by the second statement. Moreover, by the definition of success round, value of $\lambda$ and \pref{lem:V pi mean}, we have that, for each $g\in{\mathcal{K}}_r$, there exists $r' < r$ such that $V^{{\widetilde{\pi}}_g}_g(s_0)=V^{\pi_{g^{\star}_{r'},r'}}_{g^{\star}_{r'}}(s_0)\leq \widehat{\tau} + \frac{L\epsilon}{2} \leq V_{{\mathcal{K}}_{r'-1},g^{\star}_{r'}}(s_0) + L\epsilon \leq L(1+\epsilon)$. This holds with the same $1-\delta$ probability as above since we have already union bounded across the application of \pref{lem:V pi mean} for all $g^\star_r$ at all $r\geq 1$.
\end{proof}
\begin{lemma}\label{lem:calU.easy}
With probability at least $1-2\delta$, for any round $r \geq 1$ in which ${\mathcal{K}}_r$ is updated (i.e., \pref{line:compute calU'.easy} is executed), ${\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \subseteq {\mathcal{U}}_r$.
\end{lemma}
\begin{proof}
For any round $r$, let ${\mathcal{F}}_{r-1}$ denote the sigma-algebra generated by the history up to the previous round. Let $H_k$ denote the event ``\pref{line:compute calU'.easy} is executed at round $k$''. Note that $H_k$ is ${\mathcal{F}}_{k-1}$-measurable since no random step happens before \pref{line:compute calU'.easy} in round $r$. Moreover, define the events $E_r := \{\forall g\in{\mathcal{K}}_r : \|V_g^{\tilde\pi_g}\|_\infty \leq 4L\}$ and $E := \{\forall r\geq 1 : E_r\}$. Note that $E$ holds with probability at least $1-\delta$ by \pref{lem:calK.easy}. We have
\begin{align*}
\mathbb{P}\left( \exists r \geq 1: H_r, {\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \not\subseteq {\mathcal{U}}_r \right)
&\leq \mathbb{P}\left( \exists r \geq 1: H_r, {\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \not\subseteq {\mathcal{U}}_r, E \right) + \mathbb{P}\left( \neg E \right) \tag{union bound}
\\ &\leq \mathbb{P}\left( \exists r \geq 1: H_r, {\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \not\subseteq {\mathcal{U}}_r, E_r \right) + \delta \tag{\pref{lem:calK.easy}}
\\ &\leq \sum_{r\geq 1}\mathbb{P}\left( {\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \not\subseteq {\mathcal{U}}_r, E_r, H_r \right) + \delta. \tag{union bound}
\\ &\leq \sum_{r\geq 1}\mathbb{P}\left( {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L}) \not\subseteq {\mathcal{U}}_r, E_r, H_r \right) + \delta. \tag{${\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \subseteq {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L})$}
\end{align*}
Now take any round $r\geq 1$. Recall that ${\mathcal{U}}_r$ is built by sampling from each $(s,a) \in {\mathcal{K}}_r \times {\mathcal{A}}$ exactly $\mu_r := 2L\log(4SALr^2/\delta)$ times. For each $(s,a) \in {\mathcal{K}}_r \times {\mathcal{A}}$, let $s_{i,s,a}$ be the $i$-th sample (i.e., $s_{i,s,a} \sim P_{s,a}$) for $i\in[\mu_r]$. In order to collect each sample $s_{i,s,a}$, we must play the policy $\tilde\pi_s$ from $s_0$ until reaching $s$. Note that, under event $E_r$, $\|V_s^{\tilde\pi_s}\|_\infty \leq 4L$ for all $s\in{\mathcal{K}}_r$, hence all the states in ${\mathcal{K}}_r$ are reached with probability one (so $s_{i,s,a}$ is well defined for all $s,a,i$). Then, for any fixed ${\mathcal{K}}_r$,
\begin{align*}
\mathbb{P}\left( {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L}) \not\subseteq {\mathcal{U}}_r, E_r, H_r \mid {\mathcal{K}}_r \right)
&\leq \mathbb{P}\left(\exists s' \in {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L}), \forall (s,a) \in {\mathcal{K}}_r \times {\mathcal{A}}, \forall i \in [\mu_r]: s_{i,s,a} \neq s' \mid {\mathcal{K}}_r \right) \\
&\leq \sum_{s' \in {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L})} \mathbb P\left(\forall (s,a) \in {\mathcal{K}}_r \times {\mathcal{A}}, \forall i \in [\mu]: s_{i,s,a} \neq s'\right) \tag{union bound}
\\ &\leq \sum_{s' \in {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L})} \max_{(s,a) \in {\mathcal{K}}_r \times {\mathcal{A}}} \mathbb P\left(\forall i \in [\mu]: s_{i,s,a} \neq s'\right) \tag{trivial}
\\ & \leq \sum_{s' \in {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L})} \max_{(s,a) \in {\mathcal{K}}_r \times {\mathcal{A}}} \prod_{i \in [\mu_r]} (1-P(s'|s,a)) \tag{all $s_{i,s,a}$ are i.i.d.}
\\ & \leq \sum_{s' \in {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L})} \left(1-\frac{1}{2L}\right)^{\mu_r} \tag{definition of ${\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L})$}
\\ & \leq \sum_{s' \in {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L})}\frac{\delta}{4LASr^2} \leq \frac{\delta}{2r^2}.
\end{align*}
Now let $\Omega_{r-1}$ denote the sample space under which ${\mathcal{F}}_{r-1}$ is generated, such that $\sum_{\omega\in\Omega_{r-1}}\mathbb{P}(\omega) = 1$. Noting that ${\mathcal{K}}_r$ is measurable w.r.t. ${\mathcal{F}}_{r-1}$, define ${\mathcal{K}}_r(\omega)$ as the set ${\mathcal{K}}_r$ obtained after history $\omega$. Then,
\begin{align*}
\mathbb{P}\left( {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L}) \not\subseteq {\mathcal{U}}_r, E_r, H_r \right)
& = \sum_{\omega\in\Omega_{r-1}} \mathbb{P}\left( {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L}) \not\subseteq {\mathcal{U}}_r, E_r, H_r \mid \omega \right) \mathbb{P}(\omega)
\\ & = \sum_{\omega\in\Omega_{r-1} : E_r,H_r} \mathbb{P}\left( {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L}) \not\subseteq {\mathcal{U}}_r \mid \omega \right) \mathbb{P}(\omega)
\\ &= \sum_{\omega\in\Omega_{r-1} : E_r, H_r} \mathbb{P}\left( {\mathcal{N}}({\mathcal{K}}_r, \frac{1}{2L}) \not\subseteq {\mathcal{U}}_r \mid {\mathcal{K}}_r(\omega), E_r, H_r \right) \mathbb{P}(\omega) \leq \frac{\delta}{2r^2}.
\end{align*}
Plugging this into our initial inequality, we get $\mathbb{P}\left( \exists r \geq 1: H_r, {\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \not\subseteq {\mathcal{U}}_r \right) \leq 2\delta$.
\end{proof}
\begin{lemma}[Restricted Optimism]
\label{lem:V calK.easy}
With probability at least $1-\delta$ over the randomness of \pref{alg:LOGSSD}, for any $j\in[S]$ and any round $r\geq 1$, after executing \pref{line:compute V.easy}, if $\calK^{\star}_{j}\subseteq{\mathcal{K}}_r$, then
$V_{{\mathcal{K}}_r,g}(s) \leq V^{\star}_{\calK^{\star}_{j},g}(s)$ for any $s\in{\mathcal{S}}$ and $g\in\calK^{\star}_{j+1}\setminus{\mathcal{K}}_r$, where ${\mathcal{K}}_r$ is the set ${\mathcal{K}}$ immediately after the execution of \pref{line:compute V.easy}.
\end{lemma}
\begin{proof}
Let $j \in [S]$ and $g\in \calK^{\star}_{j+1} \setminus \calK^{\star}_{j}$. Fix some round $r\geq 1$ s.t. $\calK^{\star}_{j}\subseteq{\mathcal{K}}_r$. Let $\delta_r = \frac{\delta}{4r^2S^2}$ and $(Q_{\xi}, V_{\xi},\_) = \textsc{VISGO}\xspace(\calK^{\star}_{j},g,\xi,\N,\delta_r)$. By \pref{lem:opt} \footnote{Note that, by definition, $\|V_{\calK^{\star}_{j},g}^\star\|_{\infty} \leq L + 1 \leq 2L$ for all $g \in \calK^{\star}_{j+1} \setminus \calK^{\star}_{j}$ (which is a prerequisite of Lemma~\ref{lem:opt}).},
\begin{equation}
\label{eq:union.bound.restricted.opt.easy}
\begin{aligned}
\mathbb{P}\Big(\forall \xi > 0, s\in{\mathcal{S}} :
V_{\xi}(s)\leqV^{\star}_{\calK^{\star}_{j},g}(s) \Big) \geq 1 - \delta_r.
\end{aligned}
\end{equation}
Then, from a union bound and $|\calK^{\star}_{j+1} \setminus \calK^{\star}_{j}| \leq S$, the event above holds simultaneously across all $j\in[S]$, and $g\in \calK^{\star}_{j+1} \setminus \calK^{\star}_{j}$ with probability at least $1-\frac{\delta}{4r^2}$. This implies that the same result holds for all $g\in\calK^{\star}_{j+1}\setminus{\mathcal{K}}_r$ since $\calK^{\star}_{j+1}\setminus{\mathcal{K}}_r \subseteq \calK^{\star}_{j+1}\setminus\calK^{\star}_j$. A union bound implies that this holds at all rounds simultaneously with probability at least $1 - \delta$.
Now consider the execution of \pref{line:compute V.easy} and let ${\mathcal{K}}_r,\delta_r,\xi_r,\N_r$ be the values of the parameters used by VISGO in such a round, such that $\calK^{\star}_{j}\subseteq{\mathcal{K}}_r$ for some $j\in[S]$. For any $g\in\calK^{\star}_{j+1}\setminus{\mathcal{K}}_r$, let $(\_, V_{{\mathcal{K}}_r,g},\_) = \textsc{VISGO}\xspace({\mathcal{K}}_r,g,\xi_r, \N_r,\delta_r)$ and $(\_, V_{\calK^{\star}_{j},g},\_) = \textsc{VISGO}\xspace(\calK^{\star}_{j},g,\xi_r, \N_r,\delta_r)$.
Then, Eq.~\ref{eq:union.bound.restricted.opt.easy} implies that, for any $s\in{\mathcal{S}}$, $V_{\calK^{\star}_{j},g}(s)\leqV^{\star}_{\calK^{\star}_{j},g}(s)$. If $\calK^{\star}_{j}\subseteq{\mathcal{K}}_r$, by the update rule of \pref{alg:VISGO} and \pref{lem:subset opt}, we also have $V_{{\mathcal{K}}_r,g}(s) \leq V_{\calK^{\star}_{j},g}(s) \leq V^{\star}_{\calK^{\star}_{j},g}(s)$.
\end{proof}
The following lemma shows that if a set ${\mathcal{K}}^\star_{j}\subseteq {\mathcal{K}}$ at some round, at the next update of ${\mathcal{K}}$ it must be that ${\mathcal{K}}^\star_{j+1}\subseteq {\mathcal{K}}$ (if the algorithm does not terminate) and ensures correctness, in the sense that the algorithm returns a set of states including $\calS^{\rightarrow}_L$ with high probability.
\begin{lemma}[Correctness]\label{lem:update calK.easy}
Denote by ${\mathcal{K}}_r$ (resp ${\mathcal{U}}_r$) the set ${\mathcal{K}}$ (resp. ${\mathcal{U}}$) at the end of each round $r$. With probability at least $1-3\delta$, for any $j \geq 1$ and round $r \geq 1$ in which ${\mathcal{K}}_r$ is updated or returned (i.e., \pref{line:compute calU'.easy} is executed) and ${\mathcal{K}}_{r-1} \supseteq \calK^{\star}_{j}$, we have ${\mathcal{K}}^\star_{j+1} \subseteq {\mathcal{K}}_r$. Moreover, under the same probability, we have that, for any $r\geq 1$, $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}_{r}$ if the algorithm terminates at round $r$.
\end{lemma}
\begin{proof}
Define the event $E := \{ \forall r\geq 1 \text{ in which ${\mathcal{K}}_r$ is updated}: {\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \subseteq {\mathcal{U}}_r\}$. By \pref{lem:calU.easy}, it holds with probability at least $1-2\delta$. Let us carry out the proof conditioned on $E$ holding.
Take some round $r$ such that \pref{line:compute calU'.easy} is executed and ${\mathcal{K}}_{r-1} \supseteq \calK^{\star}_{j}$. Let $r'$ be the last round where ${\mathcal{K}}_{r'}$ was updated (and thus ${\mathcal{U}}_{r'}$ was created). Note that ${\mathcal{K}}_{r'} = {\mathcal{K}}_{r-1} \supseteq \calK^{\star}_j$. Then, event $E$ and the definition of the sets $(\calK^{\star}_j)_j$ directly imply that $\calK^{\star}_{j+1} := {\mathcal{T}}_L(\calK^{\star}_j) \subseteq {\mathcal{T}}_L({\mathcal{K}}_{r'}) \subseteq {\mathcal{U}}_{r'} \cup {\mathcal{K}}_{r'}$. Since ${\mathcal{K}}_r$ can only be formed by adding states in ${\mathcal{U}}_{r'}$ to ${\mathcal{K}}_{r'}$, and the union of these sets contains $\calK^{\star}_{j+1}$, if $\calK^{\star}_{j+1} \not\subseteq {\mathcal{K}}_r$, it must be that there exists $g\in{\mathcal{U}}_{r-1} \cap \calK^{\star}_{j+1}$ s.t. $V_{{\mathcal{K}}_{r-1},g}(s_0) > L$. However, \pref{lem:V calK.easy}, which holds with probability $1-\delta$, implies that, at any round $r\geq 1$, if $\calK^{\star}_{j}\subseteq{\mathcal{K}}_{r-1}$, then
$V_{{\mathcal{K}}_{r-1},g}(s_0) \leq V^{\star}_{\calK^{\star}_{j},g}(s_0) \leq L$ for any $g\in\calK^{\star}_{j+1}\setminus{\mathcal{K}}_{r-1}$. This is a contradiction, which implies that ${\mathcal{U}}_{r-1} \cap \calK^{\star}_{j+1} = \emptyset$ and, thus, all states in $\calK^{\star}_{j+1}$ must have been added to ${\mathcal{K}}_r$. A union bound over the application of \pref{lem:calU.easy} and \pref{lem:V calK.easy} yields the statement.
To prove the second statement, let us use the same events as above. First note that, since ${\mathcal{K}}_1 = \calK^{\star}_1 = \{s_0\}$, it must be that, at any round $r$, ${\mathcal{K}}_r \supseteq \calK^{\star}_j$ for some $j \geq 1$. Now take any round $r$ in which the algorithm terminates and suppose ${\mathcal{K}}_{r-1} \not\supseteq \calS^{\rightarrow}_L$. Let $j^\star$ be the largest $j$ s.t. ${\mathcal{K}}_r \supseteq \calK^{\star}_j$. By \pref{lem:SL.operator}, it must be that $j < J$, hence $\calK^{\star}_{j^\star +1} \supset \calK^{\star}_{j^\star}$. Let $r'$ be the last round at which ${\mathcal{K}}_{r'}$ was updated. Since the algorithm terminates at round $r$ it must be that ${\mathcal{K}}_{r-1}' = \emptyset$, i.e., no state in ${\mathcal{U}}_{r-1} = {\mathcal{U}}_{r'}$ has been found to be added to ${\mathcal{K}}_r$. From the same argument as above, under $E$ it must be that $\calK^{\star}_{j^\star+1} \subseteq {\mathcal{U}}_{r'} \cup {\mathcal{K}}_{r'}$. Since ${\mathcal{K}}_{r-1} \not\supseteq \calS^{\rightarrow}_L$, and no addition to ${\mathcal{K}}_{r-1}$ is performed as the algorithm stops at $r$, it must be that there exists $g\in{\mathcal{U}}_{r-1} \cap \calK^{\star}_{j^\star+1}$ s.t. $V_{{\mathcal{K}}_{r-1},g}(s_0) > L$. However, in the first part of the proof, we already found a contradiction for this case under the event of \pref{lem:V calK.easy}. This implies that the algorithm cannot stop at $r$ since some state must be added. Hence, whenever the algorithm stops it must be that ${\mathcal{K}}_r \supseteq \calS^{\rightarrow}_L$. This completes the proof.
\end{proof}
\begin{lemma}[Correctness under \pref{assum:id}]
\label{lem:calK id.easy}
Denote by ${\mathcal{K}}_r$ the set ${\mathcal{K}}$ at the end of each round $r$.
With \pref{assum:id}, with probability at least $1-5\delta$ over the randomness of Algorithm \ref{alg:LOGSSD}, for any round $r \geq 1$, we have that ${\mathcal{K}}_r = \calK^{\star}_j$ for some $j \in [S^{\rightarrow}_L]$ and ${\mathcal{K}}_{r}= \calS^{\rightarrow}_L$ if the algorithm terminates at round $r$.
\end{lemma}
\begin{proof}
By \pref{lem:calK.easy} and \pref{lem:update calK.easy}, with probability at least $1-4\delta$, we have $\calS^{\rightarrow}_L \subseteq {\mathcal{K}}_r \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ if the algorithm terminates at round $r$.
By~\pref{rem:id}, ${\mathcal{K}} = \calS^{\rightarrow}_L$.
Thus, it suffices to show that, at any round $r$, ${\mathcal{K}}_r=\calK^{\star}_j$ for some $j \leq |\calS^{\rightarrow}_L|$.
The algorithm is such that ${\mathcal{K}}_1 = \calK^{\star}_1 = \{s_0\}$. Suppose at, in some round $r\geq 1$, we have that ${\mathcal{K}}_{r}=\calK^{\star}_j$ for some $j\geq 1$. By \pref{lem:update calK.easy}, with the same probability as above, if the condition of \pref{line:update.K.easy} becomes True for the first time in some round $r'>r$ (i.e., the set ${\mathcal{K}}$ is updated in such round), then we must have $\calK^{\star}_{j+1}\subseteq{\mathcal{K}}_{r'}$ at then end of round $r'$. We shall prove that we also have ${\mathcal{K}}_{r'}\subseteq\calK^{\star}_{j+1}$, which implies the statement.
Take any round $r$ such that ${\mathcal{K}}_{r-1}=\calK^{\star}_j$ and $g^{\star}_r\in {\mathcal{U}}\setminus\calK^{\star}_{j+1}$. Since, the last time ${\mathcal{K}}$ was updated \pref{line:fill N.easy} was called, we must have $\N_{r-1}(s,a) \geq O(L^2|\calK^{\star}_j|\log(S/\delta))$ for all $(s,a) \in \calK^{\star}_j \times {\mathcal{A}}$. Then, by~\pref{lem:bounded error}, with probability at least $1-\frac{\delta}{4r^2}$, for all $s\in{\mathcal{S}}$, $V^{\pi_{g^{\star}_r}}_{g^{\star}_r}(s) \leq 2 V_{{\mathcal{K}}_{r-1}, {g^{\star}_r}}(s) \leq 4L$ due to properties of \textsc{VISGO}\xspace if \pref{line:goal condition.easy} is False. If a skip round is not triggered,
combining this with \pref{lem:V pi mean} and definition of $\lambda$, we have $\widehat{\tau}\geq V^{\pi_{g^{\star}_r}}_{g^{\star}_r}(s_0) - L\epsilon/2$ with probability at least $1-\frac{\delta}{4r^2}$.
By \pref{assum:id}, assumption on $g^\star_r$, and since $\pi_{g^{\star}_r}$ is restricted on ${\mathcal{K}}_{r-1}=\calK^{\star}_j$, we have $V^{\pi_{g^{\star}_r}}_{g^{\star}_r}(s_0) \geq V^{\star}_{\calK^{\star}_j,g^{\star}_r}(s_0) > L(1+\epsilon)$, which implies that $\widehat{\tau}\geq L(1+\epsilon/2) \geq V_{{\mathcal{K}}_{r-1}, g^{\star}_r}(s_0) + \epsilon L/2$ with the same probability, where the last inequality is from the fact that \pref{line:goal condition.easy} is False. Therefore, the failure test triggers and $g^\star_r$ is not added to ${\mathcal{K}}_r'$ or ${\mathcal{K}}_{r}$ since a failure round is triggered. This holds with probability at least $1-\delta$ across all rounds by a union bound. Therefore, for any round $r$ in which ${\mathcal{K}}$ is updated and ${\mathcal{K}}_{r-1} = \calK^{\star}_j$, we must have ${\mathcal{K}}_{r}\subseteq \calK^{\star}_{j+1}$. This concludes the proof, and the statement holds with probability at least $1-5\delta$ by a union bound.
\end{proof}
\subsection{Analysis of Policy Evaluation}
We consider the regret over the trajectories generated in the policy evaluation phase. We concatenate all policy evaluation episodes in all rounds and index them with $k \geq 1$.
To make the notation consistent with \pref{alg:SD}, we treat the whole learning procedure as an artificial trial.
Let ${\mathcal{K}}_k$, $V_k$, and $Q_k$ be the ${\mathcal{K}}$, $V_{{\mathcal{K}},g^{\star}}$, and $Q_{{\mathcal{K}},g^{\star}}$ in episode $k$. Let $\pi_k$ and $g_k$ be the corresponding policy $\pi_{g^{\star}}$ and goal $g^{\star}$.
Denote by ${\mathcal{F}}_k$ the $\sigma$-algebra of events up to episode $k$.
Let $K$ be the total number of episodes throughout the execution of \pref{alg:LOGSSD}.
For any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ with $\boldsymbol{1}_k\in{\mathcal{F}}_{k-1}$, define $R_{K',{\mathcal{I}}}=\sum_{k=1}^{K'}(I_k - V_k(s_0))\boldsymbol{1}_k$ and $C_{K'}=\sum_{k=1}^{K'} I_k$ for $K'\in[K]$.
Define $P^k_i=P_{s^k_i, a^k_i}$.
In episode $k$, when $s^k_i\in{\mathcal{K}}$, denote by $\P^k_i$, $\widetilde{P}^k_i$, $\N^k_i$, $b^k_i$ the values of $\P_{s^k_i,a^k_i}$, $\widetilde{P}_{s^k_i, a^k_i}$, $n^+(s^k_i, a^k_i)$, and $b^{(l)}(s^k_i, a^k_i)$, where $\P$, $n^+$, $b^{(l)}$ are used in \pref{alg:VISGO} to compute $V_k$ and $l$ is the final value of $i$ in \pref{alg:VISGO};
when $s^k_i\notin {\mathcal{K}}$, define $\P^k_i=\field{I}_{s_0}$, $\N^k_i=\infty$, and $b^k_i=0$.
Also define $\epsilon_k,\delta_k$ as the value of $\epsilon_{\text{VI}\xspace},\delta$ used in \pref{alg:VISGO} to compute $V_k$.
Note that $I_k<\infty$ with probability $1$ by \pref{line:skip.easy}, and $s^k_{I_k+1}\neq g$ only when a skip round is triggered in episode $k$.
\subsubsection{Regret bound without \pref{assum:id}}
\begin{lemma}
\label{lem:regret.easy}
For any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ with $\boldsymbol{1}_k\in{\mathcal{F}}_{k-1}$, we have, with probability at least $1-6\delta$, for any $K'\in[K]$,
$$R_{K',{\mathcal{I}}} \lesssim L \log(SAL/\delta)^2 \log(K)\sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}AK'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3.$$
Moreover, $C_{K'} \lesssim LK' + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3$.
\end{lemma}
\begin{proof}
We start by decomposing the regret as
\begin{align*}
\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k &\leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{1 + V_k(s^k_{i+1}) - V_k(s^k_i)}\boldsymbol{1}_k \tag{$\pm\sum_{i=1}^{I_k} V_k(s_{i+1}^k)$}\\
&\leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(\field{I}_{s^k_{i+1}} - P^k_i)V_k + (P^k_i - \P^k_i)V_k + (\P^k_i - \widetilde{P}^k_i)V_k+ b^k_i + \epsilon_k}\boldsymbol{1}_k, \tag{definition of $V_k$}
\end{align*}
where the last inequality uses that $V_k^{(l)}(s) = 1 + \widetilde{P}_{s,a}^k V_{k}^{(l-1)} - b_{s,a}^k$ for any $s\in{\mathcal{K}}_k,a\in{\mathcal{A}}$, where $l$ is the index of the last iteration of VISGO when called with $(\_, V_k, \pi_g)=\textsc{VISGO}\xspace({\mathcal{K}}_k, g_k, \epsilon_k, \N_k, \delta_k)$, and $\|V_k^{(l)} - V_k^{(l-1)}\|_{\infty} \leq \epsilon_k$ by definition of its termination condition (recall that $V_k$ is bounded since \pref{line:goal condition.easy} was passed).
Note that, if $s_i^k \notin {\mathcal{K}}_k$, then the $i,k$ term in the sum of the second line is clearly an upper bound to the corresponding term in the first line. We bound the terms above separately.
\paragraph{First term}
By \pref{lem:anytime freedman} and $\norm{V_k}_{\infty}\leq 2L$ (by \textsc{VISGO}\xspace and since \pref{line:goal condition.easy} was passed), with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (\field{I}_{s^k_{i+1}} - P^k_i)V_k\boldsymbol{1}_k \leq \sqrt{\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \boldsymbol{1}_k \field{V}(P^k_i, V_k)\iota} + L \iota,
\end{align*}
where $\iota = 9\log(16L^2 C_{K'}^3/\delta)$.
\paragraph{Second term}
Note that, by the event of \pref{lem:calK.easy}, ${\mathcal{K}}_k \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ in all episodes $k$. Moreover, when $s_i^k\notin {\mathcal{K}}_k$, the $k,i$ term in the sum is zero by definition of $P_i^k$ and $\P_i^k$. Therefore, we have all the preconditions to apply \pref{lem:dPV} on terms $(P^k_i - \P^k_i)V_k$ for all $i,k$ s.t. $s_i^k\in {\mathcal{K}}_k$, which yields, with probability $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k}(P^k_i - \P^k_i)V_k\boldsymbol{1}_k &\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\left(\sqrt{\frac{\Gamma_{L(1+\epsilon)}\field{V}(P^k_i, V_k)\iota'}{\N^k_i}} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}\iota'}{\N^k_i}\right),
\end{align*}
where $\iota' = O(\log \frac{SALC_{K'}}{\delta})$. Note that \pref{lem:dPV} already union bounds across all possible counts, value functions and state-action pair, so we do not need an extra union bound over episodes and steps here.
Then, by \pref{lem:sum N} and Cauchy-Schwarz inequality, with the same probability,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k}(P^k_i - \P^k_i)V_k\boldsymbol{1}_k \lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'',
\end{align*}
where $\iota'' = O(\log(SALC_{K'}/\delta)\log(C_{K'}))$.
\paragraph{Third term}
By the expressions of $\widetilde{P}_i^k$ and $\P_i^k$ (cf. \pref{alg:VISGO}) and \pref{lem:sum N},
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k}(\P^k_i - \widetilde{P}^k_i)V_k\boldsymbol{1}_k \leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\boldsymbol{1}_k \frac{(\P^k_i+\field{I}_g)V_k}{\N_i^k+1} \lesssim LS^{\rightarrow}_{L(1+\epsilon)}A \log(C_{K'}). \tag{$\field{I}_g(s')\triangleq\field{I}\{s'=g\}$}
\end{align*}
\paragraph{Fourth and fifth term}
By \pref{lem:sum b} and \pref{lem:sum eps}, with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (b^k_i+\epsilon_k)\boldsymbol{1}_k \lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^{1.5}A\iota'.
\end{align*}
\paragraph{Combining all terms}
Note that all the derived bounds can be absorbed into the one of the second term. Plugging everything back to our initial expression of the regret,
\begin{align*}
\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k
&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota''
\\ &\lesssim \sqrt{LS^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}AC_{K'}\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota''. \tag{\pref{lem:sum var Vk}}
\end{align*}
Note that $\iota'' \lesssim \log(SAL/\delta) (\log C_{K'})^2$. Now assuming $\boldsymbol{1}_k=1$ for all $k$, we can solve an inequality to find $C_K$. First, using that $\log(x) \leq x^\alpha/\alpha$ for any $x,\alpha > 0$ together with the derived regret bound, we can find the crude bound on $C_K$,
\begin{align*}
C_{K'} \lesssim \left(\sum_{k=1}^K V_k(s_0) + L{\calS^{\rightarrow}_{L(1+\epsilon)}}^2A\log(SAL/\delta) \right)^4 \leq \left(K'L + L{\calS^{\rightarrow}_{L(1+\epsilon)}}^2A\log(SAL/\delta) \right)^4.
\end{align*}
This implies that $\iota'' \lesssim (\log K')^2 \log(SAL/\delta)^3$. Plugging this into the regret bound, we get a quadratic inequality in $C_{K'}$. Solving it yields
\begin{align*}
C_{K'} \lesssim \sum_{k=1}^{K'} V_k(s_0) + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3 \leq LK' + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3.
\end{align*}
Plugging this back into the regret bound gives the stated bound. Throughout the proof we used following events with the corresponding probabilities:
\begin{itemize}
\item \pref{lem:anytime freedman}: $1-\delta$
\item \pref{lem:calK.easy}: $1-\delta$
\item \pref{lem:dPV}: $1-\delta$
\item \pref{lem:sum b}: $1-\delta$
\item \pref{lem:sum var Vk}: $1-2\delta$
\end{itemize}
A union bound concludes the proof.
\end{proof}
\subsubsection{Regret bound under \pref{assum:id}}
\begin{lemma}
\label{lem:regret-improved.easy}
Under \pref{assum:id}, for any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ with $\boldsymbol{1}_k\in{\mathcal{F}}_{k-1}$, we have, with probability at least $1-14\delta$, for any $K'\in[K]$,
$$R_{K',{\mathcal{I}}} \lesssim L \log(SAL/\delta)^2 \log(K')\sqrt{S^{\rightarrow}_{L(1+\epsilon)}AK'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3.$$
Moreover, $C_{K'} \lesssim LK' + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3$.
\end{lemma}
\begin{proof}
Note that, under \pref{assum:id} and by \pref{lem:calK id.easy}, in any episode, ${\mathcal{K}}=\calK^{\star}_j$ for some $j\leq J \leq |\calS^{\rightarrow}_{L(1+\epsilon)}| \leq S$ (cf. \pref{lem:SL.operator}). Moreover, by \pref{lem:calK.easy}, for any round in which $g^\star$ reaches the policy evaluation step, $\|V^{\star}_{{\mathcal{K}},g^\star}\|_\infty \leq 4L$, which implies that $\|V^{\star}_{\calK^{\star}_j,g^\star}\|_\infty \leq 4L$ for some $j$ in that round. Let $\mathcal{G}_j := \{g\in{\mathcal{S}} : \|V^{\star}_{\calK^{\star}_j,g}\|_\infty \leq 4L\}$. Consider the event
\begin{align*}
E := \left\{ \forall s\in{\mathcal{S}},a\in{\mathcal{A}}, j\in[S], g\in\mathcal{G}_j, \forall n(s,a) \geq 1 : |(\P_{s,a}^n-P_{s,a})V^{\star}_{\calK^{\star}_j,g}| \leq \sqrt{\frac{\field{V}(P_{s, a}, V^{\star}_{\calK^{\star}_j,g})\iota_{s,a}'}{n(s,a)}} + \frac{L\iota'_{s,a}}{n(s,a)} \right\},
\end{align*}
where $\iota'_{s,a} = 8\log(2S^3An(s,a)/\delta)$. Clearly, by \pref{lem:anytime bernstein} and a union bound, $E$ holds with probability at least $1-\delta$. Then, assuming $E$ and the events of \pref{lem:calK id.easy} and \pref{lem:calK.easy} hold, we clearly have, for all episodes $k$ and steps $i$,
\begin{align}\label{eq:bernstein-Vkstar}
(P^k_i -\P^k_i)V^{\star}_k\lesssim \sqrt{\frac{\field{V}(P^k_i, V^{\star}_k)\iota'}{\N^k_i}} + \frac{L\iota'}{\N^k_i},
\end{align}
where $\iota' = O(\log(SALC_{K'}/\delta))$. Note that we inflated the $\iota'$ term with an extra $L$ since it will simplify the bounds later. Now we split the regret as
\begin{align*}
\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k &\leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{1 + V_k(s^k_{i+1}) - V_k(s^k_i)}\boldsymbol{1}_k \tag{$\pm\sum_{i=1}^{I_k} V_k(s_{i+1}^k)$}\\
&\leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(\field{I}_{s^k_{i+1}} - P^k_i)V_k + (P^k_i - \P^k_i)V_k + (\P^k_i - \widetilde{P}^k_i)V_k+ b^k_i + \epsilon_k}\boldsymbol{1}_k, \tag{definition of $V_k$}
\end{align*}
where the last inequality uses that $V_k^{(l)}(s) = 1 + \widetilde{P}_{s,a}^k V_{k}^{(l-1)} - b_{s,a}^k$ for any $s\in{\mathcal{K}}_k,a\in{\mathcal{A}}$, where $l$ is the index of the last iteration of VISGO when called with $(\_, V_k, \pi_g)=\textsc{VISGO}\xspace({\mathcal{K}}_k, g_k, \epsilon_k, \N_k, \delta_k)$, and $\|V_k^{(l)} - V_k^{(l-1)}\|_{\infty} \leq \epsilon_k$ by definition of its termination condition (recall that $V_k$ is bounded since \pref{line:goal condition.easy} was passed). Note that, if $s_i^k \notin {\mathcal{K}}_k$, then the $i,k$ term in the sum of the second line is clearly an upper bound to the corresponding term in the first line.
We bound the terms above separately.
\paragraph{First term}
By \pref{lem:anytime freedman} and $\norm{V_k}_{\infty}\leq 2L$ (by \textsc{VISGO}\xspace and since \pref{line:goal condition.easy} was passed), with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (\field{I}_{s^k_{i+1}} - P^k_i)V_k\boldsymbol{1}_k \leq \sqrt{\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \boldsymbol{1}_k \field{V}(P^k_i, V_k)\iota} + L \iota,
\end{align*}
where $\iota = 9\log(16L^2 C_{K'}^3/\delta)$.
\paragraph{Second term}
Note that, from \eqref{eq:bernstein-Vkstar},
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i - \P^k_i)V_k| \boldsymbol{1}_k
&\leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i - \P^k_i)V_k|
\\ &= \sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{|(P^k_i-\P^k_i)V^{\star}_k| + |(P^k_i-\P^k_i)(V_k-V^{\star}_k)|}
\\ &\leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{\sqrt{\frac{\field{V}(P^k_i, V^{\star}_k)\iota'}{\N^k_i}} + \frac{L\iota'}{\N^k_i} + |(P^k_i-\P^k_i)(V_k-V^{\star}_k)|}.
\end{align*}
Note that, by the event of \pref{lem:calK.easy}, ${\mathcal{K}}_k \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ in all episodes $k$. Moreover, for all $k,i$, either $(s_i^k,a_i^k)\in {\mathcal{K}}_k \times {\mathcal{A}}$ or the second term above is zero. Since $\|V_k - V_k^\star\|_\infty \leq 6L$, we have all the preconditions to apply \pref{lem:dPV} on the terms $|(P^k_i-\P^k_i)(V_k-V^{\star}_k)|$, which yields, with probability $1-\delta$, for all $i,k$,
\begin{align*}
|(P^k_i-\P^k_i)(V_k-V^{\star}_k)| &\lesssim \sqrt{\frac{S^{\rightarrow}_{L(1+\epsilon)}\field{V}(P^k_i, V_k-V^{\star}_k)\iota'}{\N^k_i}} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}\iota'}{\N^k_i},
\end{align*}
where $\iota'$ was defined above. Note that \pref{lem:dPV} already union bounds across all possible counts, value functions and state-action pair, so we do not need an extra union bound over episodes and steps here. By $\textsc{Var}[X+Y]\leq 2(\textsc{Var}[X]+\textsc{Var}[Y])$, we have that $\field{V}(P^k_i, V^{\star}_k) \leq 2\field{V}(P^k_i, V_k - V^{\star}_k) + 2\field{V}(P^k_i, V_k)$ and thus
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i - \P^k_i)V_k| \leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{\sqrt{\frac{\field{V}(P^k_i, V_k)\iota'}{\N^k_i}} + \sqrt{\frac{S^{\rightarrow}_{L(1+\epsilon)}\field{V}(P^k_i, V_k-V^{\star}_k)\iota'}{\N^k_i}} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}\iota'}{\N^k_i}}.
\end{align*}
Then, by Cauchy-Schwarz inequality, with the same probability and \pref{lem:sum N},
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i - \P^k_i)V_k|
\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota''} + \sqrt{{S^{\rightarrow}_{L(1+\epsilon)}}^2A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k-V^{\star}_k)\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'',
\end{align*}
where $\iota'' = O(\log(SALC_{K'}/\delta)\log(C_{K'}))$. Now by \pref{lem:sum dV.easy}, with probability at least $1-2\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i,V_k^\star - V_k) \lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i-\P^k_i)V_k| + L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota'} + L^2{S^{\rightarrow}_{L(1+\epsilon)}}^{2}A\iota',
\end{align*}
where $\iota'$ was defined above. Let $Z_K := \sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i - \P^k_i)V_k|$. Plugging this into the previous inequality, using $\sqrt{xy} \leq x + y$ and $\iota'\leq\iota''$, we get
\begin{align*}
Z_{K'}
\lesssim
\sqrt{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL \iota''Z_{K'}}
+
\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota''.
\end{align*}
Solving thi quadratic inequality for $Z_{K'}$, we conclude with
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i - \P^k_i)V_k|
\lesssim
\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota''.
\end{align*}
\paragraph{Third term}
By the expressions of $\widetilde{P}_i^k$ and $\P_i^k$ (cf. \pref{alg:VISGO}) and \pref{lem:sum N},
\begin{align}\label{eq:regret-term3}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k}(\P^k_i - \widetilde{P}^k_i)V_k\boldsymbol{1}_k \leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\boldsymbol{1}_k \frac{(\P_i+\field{I}_g)V_k}{\N_i^k+1} \lesssim LS^{\rightarrow}_{L(1+\epsilon)}A \log(C_{K'}).
\end{align}
\paragraph{Fourth and fifth term}
By \pref{lem:sum b} and \pref{lem:sum eps}, with probability at least $1-\delta$,
\begin{align}\label{eq:regret-term4}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (b^k_i+\epsilon_k)\boldsymbol{1}_k \lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^{1.5}A\iota'.
\end{align}
\paragraph{Combining all terms}
Note that all the derived bounds can be absorbed into the one of the second term. Plugging everything back to our initial expression of the regret,
\begin{align*}
\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k
&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota''
\\ &\lesssim \sqrt{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}\iota''} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota''. \tag{\pref{lem:sum var Vk}}
\end{align*}
Note that $\iota'' \lesssim \log(SAL/\delta) (\log C_{K'})^2$. Now assuming $\boldsymbol{1}_k=1$ for all $k$, we can solve an inequality to find $C_{K'}$. First, using that $\log(x) \leq x^\alpha/\alpha$ for any $x,\alpha > 0$ together with the derived regret bound, we can find the crude bound on $C_{K'}$,
\begin{align*}
C_{K'} \lesssim \left(\sum_{k=1}^{K'} V_k(s_0) + L{\calS^{\rightarrow}_{L(1+\epsilon)}}^2A\log(SAL/\delta) \right)^4 \leq \left(K'L + L{\calS^{\rightarrow}_{L(1+\epsilon)}}^2A\log(SAL/\delta) \right)^4.
\end{align*}
This implies that $\iota'' \lesssim (\log K')^2 \log(SAL/\delta)^3$. Plugging this into the regret bound, we get a quadratic inequality in $C_{K'}$. Solving it yields
\begin{align*}
C_{K'} \lesssim \sum_{k=1}^{K'} V_k(s_0) + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3 \leq LK' + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A(\log K')^2 \log(SAL/\delta)^3.
\end{align*}
Plugging this back into the regret bound gives the stated bound. Throughout the proof we used following events with the corresponding probabilities:
\begin{itemize}
\item \pref{lem:calK id.easy}: $1-5\delta$
\item \pref{lem:calK.easy}: $1-\delta$
\item Event $E$ in this proof: $1-\delta$
\item \pref{lem:anytime freedman}: $1-\delta$
\item \pref{lem:dPV}: $1-\delta$
\item \pref{lem:sum b}: $1-\delta$
\item \pref{lem:sum dV.easy}: $1-2\delta$
\item \pref{lem:sum var Vk}: $1-2\delta$
\end{itemize}
A union bound concludes the proof.
\end{proof}
\subsection{Auxiliary results for policy evaluation}
\begin{lemma}
\label{lem:sum dV.easy}
With probability at least $1-2\delta$, for any $K'\in[K]$, if 1) $\norm{V_k}_{\infty}=\bigo{L}$ for any $k\in[K']$, and 2) $V_k(s)\leqV^{\star}_k(s)$ for any $k\in[K']$ and $s\in{\mathcal{S}}$, then
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i,V_k^\star - V_k) \lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i-\P^k_i)V_k| + L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota'} + L^2{S^{\rightarrow}_{L(1+\epsilon)}}^{2}A\iota',
\end{align*}
where $\iota' = O(\log(SALC_{K'}/\delta))$.
\end{lemma}
\begin{proof}
First note that, by Condition 1) and 2), for any $s\in{\mathcal{S}}$, $V_k^\star(s) - V_k(s) \geq 0$ and $V_k^\star(s) - V_k(s) \leq O(L)$. Thus, by \pref{lem:sum var}, with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i,V_k^\star - V_k) \lesssim \underbrace{\sum_{k=1}^{K'} (V_k^\star(s^k_{I_k+1}) - V_k(s^k_{I_k+1}))^2}_{(a)} + \underbrace{\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{(V^{\star}_k(s^k_i) - V_k(s^k_i))^2 - (P^k_i(V^{\star}_k-V_k))^2}}_{(b)} + L^2\iota,
\end{align*}
where $\iota = O(\log(LC_{K'}/\delta))$.
\paragraph{Bounding (a)}
Note that, since $V^{\star}_k(g_k)=V_k(g_k)=0$, we must have $(a) \leq \sum_{k=1}^{K'}\field{I}\{s^k_{I_k+1}\neq g\}$. Since the event $\{s^k_{I_k+1}\neq g\}$ happens only in skip rounds, it must be that $(a) \lesssim S^{\rightarrow}_{L(1+\epsilon)}A$.
\paragraph{Bounding (b)}
Using that $V_k(s)\leqV^{\star}_k(s)$ for all $s\in{\mathcal{S}}$ (Condition 2), $(a+b)(a-b)_+$ for $a,b\geq 0$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{(V^{\star}_k(s^k_i) - V_k(s^k_i))^2 - (P^k_i(V^{\star}_k-V_k))^2}
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (V^{\star}_k(s^k_i) - V_k(s^k_i) - P^k_iV^{\star}_k + P^k_iV_k)_+
\\
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (1 + P^k_iV_k - V_k(s^k_i))_+,
\end{align*}
where in the second inequality we used $V^{\star}_k(s^k_i)\leq 1 + P^k_iV^{\star}_k$ by definition of $V^{\star}_k$. Since, for all $i,k$, $V_k(s^k_i) \geq 1 + \widetilde{P}_i^k V_k - b_i^k - \epsilon_k$ (cf. \pref{alg:VISGO}), we also have
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{(V^{\star}_k(s^k_i) - V_k(s^k_i))^2 - (P^k_i(V^{\star}_k-V_k))^2}
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} ((P^k_i - \widetilde{P}_i^k)V_k + b_i^k + \epsilon_k)_+
\\ & = L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} ((P^k_i-\P^k_i)V_k + (\P^k_i - \widetilde{P}_i^k)V_k + b_i^k + \epsilon_k)_+
\\ & \leq L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (|(P^k_i-\P^k_i)V_k| + |(\P^k_i - \widetilde{P}_i^k)V_k| + b_i^k + \epsilon_k)
\end{align*}
All terms but the first one are bounded in \eqref{eq:regret-term3} and \eqref{eq:regret-term4}, which gives the following bound on (b) holding with probability at least $1-2\delta$,
\begin{align*}
(b) \lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k} |(P^k_i-\P^k_i)V_k| + L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota'} + L^2{S^{\rightarrow}_{L(1+\epsilon)}}^{2}A\iota',
\end{align*}
where $\iota' = O(\log(SALC_{K'}/\delta))$. Combining the bounds on (a) and (b) concludes the proof.
\end{proof}
\begin{lemma}
\label{lem:bound r.easy}
Assume that for any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ such that $\boldsymbol{1}_k\in{\mathcal{F}}_{k-1}$, we have $R_{K',{\mathcal{I}}}\lesssim c_1\sqrt{K'}\log^{p}(K')+c_2\log^{p}(K')$ and $C_{K'} \lesssim c_3 K' + \log^{p}(K')c_4$ for any $K'\in[K]$, where $c_1\geq L$ and $c_4 \gtrsim S^{\rightarrow}_{L(1+\epsilon)}A/\epsilon$.
Then, the total number rounds $r_{\text{tot}}$ with at least one episode is of order
\begin{align*}
\frac{c_1^2}{L^2} \log^{2p}\left(\frac{c_1c_4}{\epsilon}\right) + \left(\frac{c_2\epsilon}{L} + S^{\rightarrow}_{L(1+\epsilon)}A + \frac{c_1}{L}\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A}\right)\log^{p}\left(\frac{c_1c_2c_4}{\epsilon}S^{\rightarrow}_{L(1+\epsilon)}A\right).
\end{align*}
Moreover, $C_K \lesssim \frac{c_3 r_{\text{tot}}}{\epsilon^2} + c_4\log^p(r_\text{tot}/\epsilon)$ with probability at least $1-4\delta$.
\end{lemma}
\begin{proof}
Denote by $\bar{V}_r$, $\bar{\pi}_r$ and $\bar{g}_r$ the values of $V_{{\mathcal{K}},g^{\star}}$, $\pi_{g^{\star}}$, and $g^{\star}$ used for policy evaluation in round $r$ respectively.
For any $R'\geq 1$, let $K'$ be the total number of episodes in the first $R'$ rounds.
Denote by $r'_{\text{tot}}$ the total number of rounds with at least one episode and $r_f$ the number of failure rounds within the first $K'$ episodes.
The number of success rounds is at most $S^{\rightarrow}_{L(1+\epsilon)}$ by \pref{lem:calK.easy} (which holds with probability $1-\delta$), and the number of skip rounds is at most $\bigo{S^{\rightarrow}_{L(1+\epsilon)}A \log(C_{K'})}$ since we have a skip round only when the total number of steps or the number of visits of some state-action pair in ${\mathcal{K}}\times{\mathcal{A}}$ is doubled.
Therefore, $r'_{\text{tot}}\lesssim r_f + S^{\rightarrow}_{L(1+\epsilon)}A \log(C_{K'}) \lesssim r_f + S^{\rightarrow}_{L(1+\epsilon)}A \log(K') + S^{\rightarrow}_{L(1+\epsilon)}A \log(c_4)$, where the last inequality is by assumption on $C_{K'}$.
Define $\mathcal{W}=\{r: V^{\bar{\pi}_r}_{\bar{g}_r}(s_0)> \bar{V}_r(s_0)\}$.
Note that $\mathcal{W}$ includes all failure rounds with probability at least $1-\delta$. This is because, for any round $r\geq 1$ in which $V^{\bar{\pi}_r}_{g_r}(s_0)\leq \bar{V}_r(s_0)$ and the skip round condition is not triggered, by \pref{lem:V pi mean} and the value of $\lambda$ in \pref{alg:LOGSSD} in round $r$, we have $\widehat{\tau}\leq \bar{V}_r(s_0) + \epsilon L/2$ with probability at least $1-\frac{\delta}{2r^2}$. This implies that a success round is triggered. A union bound over all rounds proves that all failure rounds are indeed included in $\mathcal{W}=\{r: V^{\bar{\pi}_r}_{\bar{g}_r}(s_0)> \bar{V}_r(s_0)\}$ with probability at least $1-\delta$.
Define ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ such that $\boldsymbol{1}_k=\field{I}\{r\in\mathcal{W}\}\in{\mathcal{F}}_{k-1}$ for any episode $k$ in round $r$, the regret within these rounds satisfies
\begin{align*}
R_{K,{\mathcal{I}}} &\lesssim \left(\frac{c_1}{\epsilon}\sqrt{r_f + S^{\rightarrow}_{L(1+\epsilon)}A \log(K') + S^{\rightarrow}_{L(1+\epsilon)}A \log(c_4)} + c_2 \right) \log^{p}(K')
\\ & \lesssim \left(\frac{c_1}{\epsilon}\sqrt{r_f + S^{\rightarrow}_{L(1+\epsilon)}A \log(r_f/\epsilon) + S^{\rightarrow}_{L(1+\epsilon)}A \log(c_4)} + c_2 \right)\left( \log(r_f/\epsilon) + \log(c_4) \right)^p
\end{align*}
by $K = r'_{\text{tot}} \lambda \lesssim \frac{r'_{\text{tot}}}{\epsilon^2}$ (since $\lambda \lesssim 1/\epsilon^2$) and $\log(K') \lesssim \log(r_f/\epsilon) + \log(S^{\rightarrow}_{L(1+\epsilon)}A/\epsilon) \lesssim \log(r_f/\epsilon) + \log(c_4)$ by assumption on $c_4$. This shows that if we bound $r'_{\text{tot}}$ we can also control $C_{K'}$.
Now we build a lower bound to $R_{K',{\mathcal{I}}}$.
For each failure round $r$, let $C$ be the total number of steps within this round and $m$ the number of episodes within this round.
By definition, the regret within this round satisfies $C-m\bar{V}_r(s_0) \geq C-\lambda \bar{V}_r(s_0)=\lambda(\widehat{\tau}-\bar{V}_r(s_0))>\frac{\lambda\epsilon L}{2}=\lowo{L/\epsilon}$ (since $C/\lambda = \widehat{\tau} >\bar{V}_r(s_0) + \epsilon L/2$ in a failure round).
%
For any round $r\geq 1$, let $m$ be its number of episodes and $C$ be the total number of steps. By \pref{lem:V pi dev}, $mV^{\bar{\pi}_r}_{\bar{g}_r}(s_0) \leq C + L\sqrt{m}\ln^2\frac{mLr}{\delta}$ with probability at least $1-\frac{\delta}{2r^2}$. By a union bound, this holds simultaneously across all rounds with probability at least $1-\delta$. Then, with such probability, for each success and skip round $r$ in $\mathcal{W}$,
\begin{align*}
\sum_{j=u_r}^{u'_r}\rbr{I_j - \bar{V}_r(s_0)} \geq \sum_{j=u_r}^{u'_r-1}I_j - m V^{\bar{\pi}_r}_{\bar{g}_r}(s_0) - L \gtrsim -L\sqrt{\lambda}\log^2(\frac{\lambda r L}{\delta}) \gtrsim -\frac{L}{\epsilon},
\end{align*}
where $\{u_r,\ldots,u'_r\}$ are the episodes in round $r$, and we lower bound the regret in the last episode by $\lowo{-L}$ since the last trajectory in a skipped round is truncated. Note that the first inequality holds since $r\in\mathcal{W}$.
Since there are at most $\bigo{S^{\rightarrow}_{L(1+\epsilon)}A \log(C_{K'})} = \bigo{S^{\rightarrow}_{L(1+\epsilon)}A (\log(r_f/\epsilon) + \log(c_4))}$ of these rounds, we have
\begin{align*}
\frac{Lr_f}{\epsilon} &- \frac{LS^{\rightarrow}_{L(1+\epsilon)}A (\log(r_f/\epsilon) + \log(c_4))}{\epsilon} \lesssim R_{K',{\mathcal{I}}}
\\ & \lesssim \left(\frac{c_1}{\epsilon}\sqrt{r_f + S^{\rightarrow}_{L(1+\epsilon)}A \log(r_f/\epsilon) + S^{\rightarrow}_{L(1+\epsilon)}A \log(c_4)} + c_2 \right)\left( \log(r_f/\epsilon) + \log(c_4) \right)^p.
\end{align*}
This implies,
\begin{align*}
r_f &\lesssim
\left(\frac{c_1}{L}\sqrt{r_f} + \frac{c_2\epsilon}{L} + S^{\rightarrow}_{L(1+\epsilon)}A + \frac{c_1}{L}\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A}\right) (\log(r_f/\epsilon) + \log(c_4))^{p}.
\\ &\lesssim \left(\underbrace{\frac{c_1}{L}}_{:= a}\sqrt{r_f} + \underbrace{\frac{c_2\epsilon}{L} + S^{\rightarrow}_{L(1+\epsilon)}A + \frac{c_1}{L}\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A}}_{:= b} \right) \log(r_f \underbrace{c_4 /\epsilon}_{:= c})^{p}.
\end{align*}
By Lemma 28 of \cite{chen2021improved}, $a,b,c$ as defined above,
\begin{align*}
r_f \lesssim \frac{c_1^2}{L^2} \log^{2p}\left(\frac{c_1c_4}{\epsilon}\right) + \left(\frac{c_2\epsilon}{L} + S^{\rightarrow}_{L(1+\epsilon)}A + \frac{c_1}{L}\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A}\right)\log^{p}\left(\frac{c_1c_2c_4}{\epsilon}S^{\rightarrow}_{L(1+\epsilon)}A\right).
\end{align*}
The proof is concluded by $r'_\text{tot} \lesssim r_f + S^{\rightarrow}_{L(1+\epsilon)}A \log(r_f/\epsilon) + S^{\rightarrow}_{L(1+\epsilon)}A \log(c_4)$ as showed above and setting $K'=K$ (that is, $r'_{\text{tot}}=r_{\text{tot}}$).
\end{proof}
\subsection{Proof of \pref{thm:sd} and \pref{thm:sd id}}
\label{app:sd and id}
We restate and prove the two theorems together.
\begin{theorem}[Unified statement of \pref{thm:sd} and \pref{thm:sd id}]
\label{thm:sd.easy}
With probability at least $1-23\delta$, after collecting $N_{\text{tot}}$ samples, \pref{alg:LOGSSD} outputs ${\mathcal{K}}$ and $\{{\widetilde{\pi}}_g\}_{g\in{\mathcal{K}}}$ such that $\calS^{\rightarrow}_L \subseteq {\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and $V^{{\widetilde{\pi}}_g}_g(s_0)\leq L(1+\epsilon)$ for all $g\in{\mathcal{K}}$, where
\begin{itemize}
\item $N_{\text{tot}} = \bigO{\frac{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}AL}{\epsilon^2}\iota + \frac{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL}{\epsilon}\iota + L^3 {S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota}$ in the general case;
\item $N_{\text{tot}} = \bigO{\frac{S^{\rightarrow}_{L(1+\epsilon)}AL}{\epsilon^2}\iota + \frac{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL}{\epsilon}\iota + L^3 {S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota}$ with \pref{assum:id}.
\end{itemize}
Here $\iota = \log^{8}\left(\frac{SAL}{\epsilon\delta}\right)$.
\end{theorem}
\begin{proof}
By \pref{lem:calK.easy} and \pref{lem:update calK.easy}, with probability $1-4\delta$, the output ${\mathcal{K}}$ and $\{{\widetilde{\pi}}_g\}_{g\in{\mathcal{K}}}$ clearly satisfy the first statement.
Let us bound the sample complexity.
Each round can be classified into one of the following cases: 1) expansion of the sets (\pref{line:goal condition.easy} is true), and 2) policy evaluation is performed (from \pref{line:PE.easy}, so \pref{line:goal condition.easy} is false). Note that the sample complexity of case 2 is given by $C_K$. We shall bound it later.
In case 1), the algorithm terminates or at least one state is added into ${\mathcal{K}}$.
Thus, the number of rounds satisfying case 1) in each trial is at most $1+S^{\rightarrow}_{L(1+\epsilon)}$ by \pref{lem:calK.easy}. In a round satisfying case 1), if the algorithm terminates, then no samples are collected.
Otherwise, \pref{line:compute calU'.easy} and \pref{line:fill N.easy} are executed. Take any round $r$ in which this happens and denote by ${\mathcal{K}}_r$ the set ${\mathcal{K}}$ at the end of round $r$. Note that \pref{line:fill N.easy} collects at most $O(L^2|{\mathcal{K}}_r|\ln(Sr/\delta))$ for each $s\in{\mathcal{K}}_r$ and $a\in{\mathcal{A}}$, while \pref{line:compute calU'.easy} collects $O(L\log(SALr/\delta))$ samples from each state $s\in{\mathcal{K}}_r$ and $a\in{\mathcal{A}}$, so the total number of samples collected from each $s\in{\mathcal{K}}_r$ and $a\in{\mathcal{A}}$ is at most $n_r = O(L^2|{\mathcal{K}}_r|\ln(SALr/\delta))$.
Since, by \pref{lem:calK.easy}, at any round $r$, $\|V^{\tilde\pi_g}_{g}\|_{\infty} \leq 4L$ for each $g\in{\mathcal{K}}_r$, by \pref{lem:hitting}, with probability $1-\delta'$ it
takes no more than $8L\log(2/\delta')$ steps to reach the goal state $g$ following $\tilde\pi_g$.
Therefore, by setting $\delta'=\frac{\delta}{2r^2|{\mathcal{K}}_r||{\mathcal{A}}|n_r}$, with probability $1-\frac{\delta}{2r^2}$, all trajectories in round $r$ reach the goal within $8L\log(2/\delta')$ steps. Then, by a union bound over all rounds, with probability at least $1-\delta$, the total sample complexity is $\tilo{L^3 |{\mathcal{K}}_r|^2|{\mathcal{A}}|\log^2(SALr/\delta)}$ at any round $r$.
Note that, among these samples, only $\tilo{L |{\mathcal{K}}_r||{\mathcal{A}}|\log^2(SALr/\delta)}$ cumulate over rounds. This is because the sampling of \pref{line:fill N.easy} is performed only if the current counters are below the sampling requirement. Since the number of rounds in case 1) is at most $1+S^{\rightarrow}_{L(1+\epsilon)}$ and the total number of rounds $R$ performed by the algorithm satisfies $R \leq r_\text{tot} + S^{\rightarrow}_{L(1+\epsilon)} + 1$ (by summing the rounds in both cases) and $|{\mathcal{K}}_r| \leq S^{\rightarrow}_{L(1+\epsilon)}$ by \pref{lem:calK.easy}, we have that \pref{line:fill N.easy} contributes to at most $\tilo{L {S^{\rightarrow}_{L(1+\epsilon)}}^2A\log^2(SALr_\text{tot}/\delta)}$ sample complexity and the total sample complexity of Case 1) is thus $\tilo{L^3 {S^{\rightarrow}_{L(1+\epsilon)}}^2A\log^2(SALr_\text{tot}/\delta)}$.
We now conclude the sample complexity proof depending on whether \pref{assum:id} is considered or not.
\paragraph{Without \pref{assum:id}}
Plugging the regret bound of \pref{lem:regret.easy} into \pref{lem:bound r.easy}, using $p=2$, $c_1 = L \log(SAL/\delta)^2 \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A}$, $c_2 = L{S^{\rightarrow}_{L(1+\epsilon)}}^2A \log(SAL/\delta)^3$, $c_3 = L$, $c_4 = L{S^{\rightarrow}_{L(1+\epsilon)}}^2A \log(SAL/\delta)^3 / \epsilon$,
\begin{align*}
r_\text{tot} &\lesssim \left( \log(SAL/\delta)^4 S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A + {S^{\rightarrow}_{L(1+\epsilon)}}^2A \log(SAL/\delta)^3 \epsilon + \log(SAL/\delta)^2 S^{\rightarrow}_{L(1+\epsilon)}\sqrt{\Gamma_{L(1+\epsilon)}}A \right) \log^{4}\left(\frac{SAL}{\epsilon}\right)
\\ &\lesssim \left(S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A + {S^{\rightarrow}_{L(1+\epsilon)}}^2A \epsilon \right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right)
\end{align*}
and
\begin{align*}
C_K &\lesssim \frac{L}{\epsilon^2}\left(S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A + {S^{\rightarrow}_{L(1+\epsilon)}}^2A \epsilon \right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right) + \frac{L{S^{\rightarrow}_{L(1+\epsilon)}}^2A}{\epsilon} \log^{5}\left(\frac{SAL}{\epsilon\delta}\right),
\\ &\lesssim \left(\frac{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}AL}{\epsilon^2} + \frac{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL}{\epsilon} \right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right).
\end{align*}
Thus, the total sample complexity of the algorithm (which is given by $C_K$ plus the sample complexity of case 1) is
\begin{align*}
\left(\frac{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}AL}{\epsilon^2} + \frac{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL}{\epsilon} + L^3 {S^{\rightarrow}_{L(1+\epsilon)}}^2|{\mathcal{A}}|\right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right).
\end{align*}
\paragraph{With \pref{assum:id}}
Plugging the regret bound of \pref{lem:regret-improved.easy} into \pref{lem:bound r.easy}, using $p=2$, $c_1 = L \log(SAL/\delta)^2 \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A}$, $c_2 = L{S^{\rightarrow}_{L(1+\epsilon)}}^2A \log(SAL/\delta)^3$, $c_3 = L$, $c_4 = L{S^{\rightarrow}_{L(1+\epsilon)}}^2A \log(SAL/\delta)^3 / \epsilon$,
\begin{align*}
r_\text{tot} &\lesssim \left( \log(SAL/\delta)^4 S^{\rightarrow}_{L(1+\epsilon)}A + {S^{\rightarrow}_{L(1+\epsilon)}}^2A \log(SAL/\delta)^3 \epsilon + \log(SAL/\delta)^2 S^{\rightarrow}_{L(1+\epsilon)}\sqrt{\Gamma_{L(1+\epsilon)}}A \right) \log^{4}\left(\frac{SAL}{\epsilon}\right)
\\ &\lesssim \left(S^{\rightarrow}_{L(1+\epsilon)}A + {S^{\rightarrow}_{L(1+\epsilon)}}^2A \epsilon \right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right)
\end{align*}
and
\begin{align*}
C_K &\lesssim \frac{L}{\epsilon^2}\left(S^{\rightarrow}_{L(1+\epsilon)}A + {S^{\rightarrow}_{L(1+\epsilon)}}^2A \epsilon \right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right) + \frac{L{S^{\rightarrow}_{L(1+\epsilon)}}^2A}{\epsilon} \log^{5}\left(\frac{SAL}{\epsilon\delta}\right),
\\ &\lesssim \left(\frac{S^{\rightarrow}_{L(1+\epsilon)}AL}{\epsilon^2} + \frac{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL}{\epsilon} \right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right).
\end{align*}
Thus, the total sample complexity of the algorithm (which is given by $C_K$ plus the sample complexity of case 1) is
\begin{align*}
\left(\frac{S^{\rightarrow}_{L(1+\epsilon)}AL}{\epsilon^2} + \frac{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL}{\epsilon} + L^3 {S^{\rightarrow}_{L(1+\epsilon)}}^2|{\mathcal{A}}|\right) \log^{8}\left(\frac{SAL}{\epsilon\delta}\right).
\end{align*}
A union bound over the events of adopted lemmas (\pref{lem:calK.easy}, \pref{lem:update calK.easy}, Lemma 6 of \cite{rosenberg2020adversarial}, \pref{lem:bound r.easy}, and \pref{lem:regret.easy} without \pref{assum:id} or \pref{lem:regret-improved.easy} with \pref{assum:id}) yields the result with probability at least $1-23\delta$.
\end{proof}
\section{Notation}\label{app:notation}
Let $(x)_+=\max\{0, x\}$ and $\field{I}_s(s')=\field{I}\{s'=s\}$.
We say that a value function $V$ is \textbf{restricted} on a subset ${\mathcal{X}} \subseteq {\mathcal{S}}$, if there exists $v>0$ such that $V(s)=v$ for any $s\notin{\mathcal{X}}$.
When value function $V$ takes the same value within a subset of states $y$, we define $V(y)=V(s)$ for any $s\in y$.
For any subset $y\subseteq{\mathcal{S}}$ and distribution $P\in\Delta_{{\mathcal{S}}}$, define $P(y)=\sum_{s'\in y}P(s')$.
\paragraph{Trial} In \pref{alg:SD}, a trial is indexed by $\tau$, and each trial corresponds to a value of $z$ estimating $S^{\rightarrow}_{L(1+\epsilon)}$ (\pref{line:trial}).
In \pref{alg:LOGSSD} and \pref{alg:PC}, we assume the whole learning procedure lies in an artificial trial.
\begin{table*}[h]
\centering
\caption{The notation adopted in this paper.}
\label{tab:notation}
\resizebox{\textwidth}{!}{
\begin{tabular}{ll}
\toprule
Symbol & Meaning \\
\toprule
${\mathcal{S}}$ & State Space\\
${\mathcal{A}}$ & Action Space (including the \textsc{RESET} action)\\
$P$ & Transition function\\
$\pi : {\mathcal{S}} \to {\mathcal{A}}$ & A policy\\
$\Pi({\mathcal{X}})$ & Policies restricted to ${\mathcal{X}}$, \textsc{RESET}{} is taken outside ${\mathcal{X}}$\\
$L$ & Exploration radius\\
$\calS^{\rightarrow}_{L}$ & Incrementally $L$-controllable states\\
${\mathcal{N}}^{s, a}_L=\{s'\in\calS^{\rightarrow}_L: P_{s, a}(s')>0\}$ & States in $\calS^{\rightarrow}_L$ reachable from $(s,a)$\\
$\Gamma^{s, a}_L=|{\mathcal{N}}^{s, a}_L|, \Gamma_L=\max_{s\in\calS^{\rightarrow}_L, a}\Gamma^{s, a}_L$ & Cardinality of ${\mathcal{N}}^{s, a}_L$ and maximum value\\
${\mathcal{T}}_L({\mathcal{X}})=\{g \in {\mathcal{S}}: V^{\star}_{{\mathcal{X}},g}(s_0)\leq L\}$ & Set of $L$ controllable states restricted on ${\mathcal{X}}\subseteq{\mathcal{S}}$\\
$\{\calK^{\star}_j\}_j : \calK^{\star}_1 = \{s_0\}, \calK^{\star}_j ={\mathcal{T}}_L(\calK^{\star}_{j-1})$ & Layers defining $\calS^{\rightarrow}_L$\\
$\calO^{\rightarrow}_L=(s_1,\ldots,s_n)$ & Ordering of states in $\calS^{\rightarrow}_L$ defining the layer $\{\calK^{\star}_j\}$\\
$\calK^{\star}_{z,j}$ & $\calK^{\star}_{z,j}=\calK^{\star}_j$ when $|\calK^{\star}_j|< z$, and $\calK^{\star}_{z,j}=\{s_1,\ldots,s_z\}$ when $|\calK^{\star}_j|\geq z$\\
$\calK^{\star}_{z,z}=(s_1,\ldots,s_{z})$ & The first $z$ elements of $\calO^{\rightarrow}_L$ or $\calS^{\rightarrow}_L$\\
$\calU^{\star}_z=\rS{\calK^{\star}_{z,z}}{2L}$ & States reachable in $2L$ steps from $\calK^{\star}_{z,z}$\\
${\mathcal{N}}({\mathcal{X}}, p)=\{s'\notin{\mathcal{X}}: P(s'|s,a)\geq p\text{ for some }(s, a)\in{\mathcal{X}}\times{\mathcal{A}} \}$ & States not in ${\mathcal{X}}$ reachable with high probability from ${\mathcal{X}}$\\
$\bar{\calU}=\{s'\in{\mathcal{S}}: \exists s\in \calS^{\rightarrow}_{L(1+\epsilon)}, a\in{\mathcal{A}}, P(s'|s, a)\geq \frac{1}{2L}\}$ & States that are reachable from $\calS^{\rightarrow}_{L(1+\epsilon)}$ with high probability\\
\toprule
\multicolumn{2}{c}{Learning Algorithm} \\
\cmidrule{1-2}
$r \in \mathbb{N}_+$ & Round\\
$\tau \in \mathbb{N}_+$ & Trial\\
$z$ & An estimate of $|S^{\rightarrow}_{L(1+\epsilon)}|$.
The value of $z$ is updated at the beginning of each trial.\\
$\epsilon$ & accuracy\\
${\mathcal{K}}$ & Set of ``known'' states, such that $\calK^{\star}_j \subseteq {\mathcal{K}}$ for some $j$\\
${\mathcal{U}}$ & Set of ``unknown'' states\\
${\mathcal{K}}'$ & Increment to ${\mathcal{K}}$ leading to include layer $j+1$\\
$\N(s,a,s')$ & Number of visits to $(s,a,s')$\\
$\lambda$ & Number of episodes for policy evaluation\\
$\widehat{\tau}$ & Average number of steps to reach the goal by policy $\pi_{g^{\star}}$\\
\bottomrule
\end{tabular}
}
\end{table*}
\section{Auxiliary Results}\label{app:auxiliary}
\begin{lemma}
\label{lem:example 2L}
For any $S \geq 1$, $A \geq 2$, $\frac{3}{2} \leq L \leq \frac{1}{2} + \frac{\ln(S/2)}{2\ln(A)}$, and $0 < \epsilon < \frac{L-1}{L}$, there exists an MDP with $S$ states and $A$ actions (including action $\textsc{RESET}$) such that $S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}=1$ while $S^{\rightarrow}_{2L}\geq A^{2(L-1)}$.
\end{lemma}
\begin{proof}
Consider an MDP with the following structure.
At $s_0$, taking any action transits to one of $\{s_1,\ldots,s_L\}$ with probability $\frac{1}{L}$.
At any state in $\{s_1,\ldots,s_L\}$, taking any action transits to state $s^{\star}$.
States reachable from $s^{\star}$ form a full $A$-ary tree with depth $2(L-1)$.
The rest of the states are ignored (note that $S\geq 2A^{2L-1}\geq 1 + L + \sum_{i=0}^{2(L-1)}A^i$).
It is not hard to see that it takes $2L-1$ steps to reach any $s_i$ for $i\in[L]$ by a policy restricted on $\{s_0\}$.
Therefore, all unignored states are $2L$ incrementally controllable and thus $S^{\rightarrow}_{2L} \geq A^{2(L-1)}$ states.
On the other hand, by $L(1+\epsilon)<2L-1$, $\calS^{\rightarrow}_{L(1+\epsilon)}=\{s_0\}$ and $\Gamma_{L(1+\epsilon)}=1$ (note that the agent can reach $s_0$ from $s_0$ by taking $\textsc{RESET}$).
\end{proof}
\begin{remark}
The construction in \pref{lem:example 2L} also have $S^{\rightarrow}_{2L} = \lowo{S}$ while $S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)} = \bigo{1}$.
\end{remark}
\begin{lemma}
\label{lem:init bound}
For any ${\mathcal{X}}\subseteq{\mathcal{S}}$ and $g\in{\mathcal{S}}$, we have $\norm{V^{\star}_{{\mathcal{X}},g}}_{\infty} \leq 1 + V^{\star}_{{\mathcal{X}},g}(s_0)$.
\end{lemma}
\begin{proof}
Clearly $V^{\star}_{{\mathcal{X}},g}(g)=0\leq 1 + V^{\star}_{{\mathcal{X}},g}(s_0)$ and $V^{\star}_{{\mathcal{X}},g}(s)=1+V^{\star}_{{\mathcal{X}},g}(s_0)$ for any $s\in{\mathcal{S}}\setminus({\mathcal{X}}\cup\{g\})$.
For any $s\in{\mathcal{X}}\setminus\{g\}$, by Bellman optimality and $\textsc{RESET}\in{\mathcal{A}}$ we have $V^{\star}_{{\mathcal{X}},g}(s) \leq 1 + V^{\star}_{{\mathcal{X}},g}(s_0)$.
\end{proof}
\begin{lemma}
\label{lem:barPV to PV}
Let $n$ be a counter incrementally collecting samples from transition function $P$, and define $\P^n_{s,a}(s'):=\frac{n(s, a, s')}{n^+(s, a)}$.
Let $\mathcal{G}$ be the goal set such that $\calS^{\rightarrow}_{L(1+\epsilon)}\subseteq\mathcal{G}\subseteq{\mathcal{S}}$.
With probability at least $1-\delta$, for any status of $n$, $(s, a)\in\calS^{\rightarrow}_{L(1+\epsilon)}\times{\mathcal{A}}$, ${\mathcal{X}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$, $g\in\mathcal{G}\setminus{\mathcal{X}}$, and value function $V$ restricted on ${\mathcal{X}}\cup\{g\}$ with $\norm{V}_{\infty}\leq B$ for some $B>0$, we have $\field{V}(\P^n_{s,a}, V) \lesssim \field{V}(P_{s,a}, V) + \frac{\Gamma_{L(1+\epsilon)}B^2\iota'_{s,a}}{n^+(s, a)}$, where $\iota'_{s,a}=\bigo{\ln \frac{|\mathcal{G}|An^+(s,a)}{\delta}}$.
\end{lemma}
\begin{proof}
Note that
\begin{align*}
\field{V}(\P_{s,a}, V) &\leq \P_{s,a}(V - P_{s,a}V)^2 \tag{$\frac{\sum_ip_ix_i}{\sum_ip_i}=\argmin_z\sum_ip_i(x_i-z)^2$}\\
&= \field{V}(P_{s,a}, V) + (\P_{s,a}-P_{s,a})(V - P_{s,a}V)^2\\
&\lesssim \field{V}(P_{s,a}, V) + B\sqrt{\frac{\Gamma_{L(1+\epsilon)}\field{V}(P_{s,a}, V)\iota'_{s,a}}{n^+(s, a)}} + \frac{\Gamma_{L(1+\epsilon)}B^2\iota'_{s,a}}{n^+(s, a)} \tag{\pref{lem:dPV} and \pref{lem:quad}}\\
&\lesssim \field{V}(P_{s,a}, V) + \frac{\Gamma_{L(1+\epsilon)}B^2\iota'_{s,a}}{n^+(s, a)}. \tag{AM-GM inequality}
\end{align*}
This completes the proof.
\end{proof}
\begin{lemma}
\label{lem:dPV}
Let $n$ be a counter incrementally collecting samples from transition function $P$, and define $\P^n_{s,a}(s'):=\frac{n(s, a, s')}{n^+(s, a)}$.
Let $\mathcal{G}$ be the goal set such that $\calS^{\rightarrow}_{L(1+\epsilon)}\subseteq\mathcal{G}\subseteq{\mathcal{S}}$.\footnote{In most cases, we apply this lemma with $\mathcal{G}\in\{\calS^{\rightarrow}_{L(1+\epsilon)}, {\mathcal{S}}\}$.}
With probability at least $1-\delta$, for any status of $n$, $(s, a)\in\calS^{\rightarrow}_{L(1+\epsilon)}\times{\mathcal{A}}$, ${\mathcal{X}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$, $g\in\mathcal{G}\setminus{\mathcal{X}}$, and value function $V$ restricted on ${\mathcal{X}}\cup\{g\}$ with $\norm{V}_{\infty}\leq B$ for some $B>0$, we have
$$|(P_{s, a}-\P_{s, a}^n)V| \lesssim \sqrt{\frac{\min\{|{\mathcal{X}}|,\Gamma^{s, a}_{L(1+\epsilon)}\}\field{V}(P_{s, a}, V)\iota_{s,a}'}{n^+(s, a)}} + \frac{B\min\{|{\mathcal{X}}|, \Gamma^{s, a}_{L(1+\epsilon)}\}\iota_{s,a}'}{n^+(s, a)},$$
where $\iota_{s,a}' = \bigo{\ln \frac{S^{\rightarrow}_{L(1+\epsilon)}A\Gamma^2_{L(1+\epsilon)}|\mathcal{G}|n^+(s,a)}{\delta}}$.
\end{lemma}
\begin{proof}
By \pref{lem:anytime bernstein} and a union bound, for any $\delta'\in(0,1)$, with probability at least $1-\frac{\delta'}{S^{\rightarrow}_{L(1+\epsilon)}A\Gamma_{L(1+\epsilon)}{\Gamma^{s, a}_{L(1+\epsilon)} \choose i}|\mathcal{G}|}$, for each status of $n$, $(s, a)\in \calS^{\rightarrow}_{L(1+\epsilon)}\times{\mathcal{A}}$, size $i\in[\Gamma^{s,a}_{L(1+\epsilon)}]$, subset $y'\subseteq {\mathcal{N}}^{s,a}_{L(1+\epsilon)}$ with $|y'|=i$, and $g\in\mathcal{G}\setminus y'$,
\begin{align*}
|P_{s, a}(y) - \P_{s,a}^n(y)| \leq 2\sqrt{2\frac{P_{s,a}(y)(1-P_{s,a}(y))\ln(2n^+(s,a)/\delta')}{n^+(s, a)}} + \frac{\ln(2n^+(s,a)/\delta')}{n^+(s, a)},
\end{align*}
where $y={\mathcal{S}}\setminus(y'\cup\{g\})$. Let $y'={\mathcal{X}}'\triangleq{\mathcal{X}}\cap{\mathcal{N}}^{s,a}_{L(1+\epsilon)}$ such that $y={\mathcal{S}}\setminus({\mathcal{X}}'\cup\{g\})$. By another application of \pref{lem:anytime bernstein} and a union bound, for any $\delta'\in(0,1)$, with probability at least $1-\frac{\delta'}{|\mathcal{G}|}$, for all $s'\in{\mathcal{X}}'\cup\{g\} \subseteq \mathcal{G}$,
\begin{align*}
|P_{s, a}(s') - \P_{s,a}^n(s')| \leq 2\sqrt{2\frac{P_{s,a}(s')(1-P_{s,a}(s'))\ln(2n^+(s,a)/\delta')}{n^+(s, a)}} + \frac{\ln(2n^+(s,a)/\delta')}{n^+(s, a)}.
\end{align*}
Thus, setting $\delta' = \delta / 2S^{\rightarrow}_{L(1+\epsilon)}A\Gamma_{L(1+\epsilon)}{\Gamma^{s, a}_{L(1+\epsilon)} \choose i}|\mathcal{G}|$ and using ${n \choose i}\leq n^{\min\{i, n - i\}}$, the two inequalities above simplify as
\begin{align}
|P_{s, a}(y) - \P_{s,a}^n(y)| &\lesssim \sqrt{\frac{i\cdot P_{s,a}(y)(1-P_{s,a}(y))\iota'_{s,a}}{n^+(s, a)}} + \frac{i\iota'_{s,a}}{n^+(s, a)}, \label{eq:ineq1}
\\ |P_{s, a}(s') - \P_{s,a}^n(s')| &\lesssim \sqrt{\frac{P_{s,a}(s')(1-P_{s,a}(s')) \iota'_{s,a}}{n^+(s, a)}} + \frac{\iota'_{s,a}}{n^+(s, a)}.\label{eq:ineq2}
\end{align}
These hold with probability at least $1-\delta$. Now define, for all $s'\in{\mathcal{S}}$,
\begin{align*}
V'(s')=\begin{cases}
V(s'),& s'\in{\mathcal{X}}'\cup\{g\}\\
V({\mathcal{S}}\setminus({\mathcal{X}}\cup\{g\})),& \text{otherwise}
\end{cases}
\end{align*}
and $V_{\dagger}(s')=V'(s')-P_{s,a}V'$ for all $s'$.
Clearly, $V'$ and $V_{\dagger}$ are restricted on ${\mathcal{X}}'\cup\{g\}$.
Moreover, $V(s')\neq V'(s')\implies s'\in{\mathcal{X}}\setminus y'\implies s'\in{\mathcal{X}}\setminus{\mathcal{N}}^{s,a}_{L(1+\epsilon)}\implies P_{s,a}(s')=0$ by ${\mathcal{X}}\subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$.
Thus, $P_{s,a}V=P_{s,a}V'$, and
\begin{align*}
&(P_{s, a} - \P_{s, a}^n)V = (P_{s, a} - \P_{s, a}^n)V' = (P_{s,a}-\P^n_{s,a})V_{\dagger}\\
&= \sum_{s'\in{\mathcal{X}}'}(P_{s, a}(s') - \P_{s, a}^n(s'))V_{\dagger}(s') + (P_{s, a}(g) - \P_{s, a}^n(g))V_{\dagger}(g) + (P_{s, a}(y) - \P_{ s, a}^n(y))V_{\dagger}(y)\\
&\lesssim \sum_{s'\in{\mathcal{X}}'\cup\{g\}}\sqrt{\frac{P_{s, a}(s')\iota'_{s,a} }{n^+(s, a)}}|V_{\dagger}(s')| + \sqrt{\frac{|{\mathcal{X}}'| P_{s, a}(y)\iota'_{s,a}}{n^+(s, a)}}|V_{\dagger}(y)| + \frac{B|{\mathcal{X}}'|\iota'_{s,a}}{n^+(s, a)} \tag{\pref{eq:ineq1} and \pref{eq:ineq2}}\\
&\lesssim \sqrt{\frac{|{\mathcal{X}}'|\field{V}(P_{s, a}, V)\iota'_{s,a}}{n^+(s, a)}} + \frac{B|{\mathcal{X}}'|\iota'_{s,a}}{n^+(s, a)}.
\end{align*}
where in the last step we apply Cauchy-Schwarz inequality and
\begin{align*}
\sum_{s'}P_{s,a}(s')V_{\dagger}(s')^2 &= \sum_{s'}P_{s,a}(s')(V'(s') - P_{s,a}V)^2 \tag{$P_{s,a}V=P_{s,a}V'$}\\
&= \sum_{s'}P_{s,a}(s')(V(s') - P_{s,a}V)^2 \tag{$P_{s,a}(s')=0$ when $V'(s')\neq V(s')$}\\
&= \field{V}(P_{s,a},V).
\end{align*}
This completes the proof.
\end{proof}
\begin{lemma}
\label{lem:quad log}
If $x\leq a\sqrt{x\ln^p(dx)} + b\ln^p(dx) + c$ for some $a, b, c \geq 0$, $d>0$ and some absolute constant $p\geq 1$, then $x=\bigo{(a^2+b)\ln^p((a+b+c)d) + c}$.
\end{lemma}
\begin{proof}
By AM-GM inequality and $\ln x < x$ for $x>0$, we have
\begin{align*}
x\leq a\sqrt{x\ln^p(dx)} + b\ln^p(dx) + c \leq \frac{x}{2} + (a^2/2+b)\ln^p(dx) + c \leq \frac{x}{2} + (a^2/2+b)(2p)^p\sqrt{dx} + c.
\end{align*}
Solving a quadratic inequality w.r.t.~$x$ gives $x=\bigo{(a^2+b)^2d+c}$.
Plugging this back to the original inequality gives $x\leq a\sqrt{x\iota} + b\iota + c$, where $\iota=\ln^p((a+b+c)d)$.
Further solving a quadratic inequality w.r.t~$x$ completes the proof.
\end{proof}
\begin{lemma}\citep[Lemma 40]{chen2022reaching}
\label{lem:quad}
For any random variable $X\in[-B,B]$, for some $B>0$, we have $\textsc{Var}[X^2]\leq 4B^2\textsc{Var}[X]$.
\end{lemma}
\begin{lemma}\citep[Lemma C.2]{cai2022near}
\label{lem:mvp}
For some $B>0$, let $\Upsilon=\{v\in\field{R}^{{\mathcal{S}}}_{\geq 0}: v(g)=0, \norm{v}_{\infty}\leq B\}$ and $f:\Delta_{{\mathcal{S}}}\times\Delta_{{\mathcal{S}}}\times\Upsilon\times\field{R}_+\times\field{R}_+\rightarrow\field{R}$ with $f(\widetilde{p},p,v,n,\iota)=\widetilde{p} v - \max\cbr{c_1\sqrt{\frac{\field{V}(p, v)\iota}{n}}, c_2\frac{B\iota}{n}}$ with some constants $c_1\geq 0$ and $c_2\geq 2c_1^2$.
Then $f$ ensures for all $v$, $n$, $\iota$, and $\widetilde{p}$, $p$ s.t.~$\widetilde{p}(s)-\frac{1}{2}p(s)\geq0$ for all $s\neq g$,
\begin{enumerate}
\item $f(\widetilde{p},p,v,n,\iota)$ is non-decreasing in $v(s)$, that is,
\begin{align*}
\forall v, v'\in\Upsilon, v\leq v' \implies f(\widetilde{p},p,v,n,\iota) \leq f(\widetilde{p},p,v',n,\iota);
\end{align*}
\item if $\widetilde{p}(g)>0$, then $f(\widetilde{p},p,v,n,\iota)$ is $\rho_{\widetilde{p}}$-contractive in $v(s)$, with $\rho_{\widetilde{p}}=1-\widetilde{p}(g)<1$, that is,
\begin{align*}
\forall v,v'\in\Upsilon, \abr{f(\widetilde{p},p,v,n,\iota) - f(\widetilde{p},p,v',n,\iota)} \leq \rho_{\widetilde{p}}\norm{v-v'}_{\infty}.
\end{align*}
\end{enumerate}
\end{lemma}
\begin{lemma}
\label{lem:V pi mean}
There exist a function $N_{\textsc{Dev}}(L_0, \epsilon, \delta) =\bigo{ \ln^4\frac{L_0}{\epsilon\delta}/\epsilon^2}$, such that for any $g\in{\mathcal{S}}$ and policy $\pi$ with $\norm{V^{\pi}_g}_{\infty}\leq L_0$ for some $L_0>0$, we have with probability at least $1-\delta$, for all $n\geq N_{\textsc{Dev}}(L_0, \epsilon, \delta)$ simultaneously, $|\widehat{\tau}_n - V^{\pi}_g(s_0)| \leq \norm{V^{\pi}_g}_{\infty}\epsilon$, where $\widehat{\tau}_n=\frac{1}{n}\sum_{i=1}^n C_i$ and each $C_i$ is a realization of the total cost incurred by following $\pi$ starting from $s_0$ with goal state $g$.
\end{lemma}
\begin{proof}
By \pref{lem:V pi dev}, with probability at least $1-\delta$, $\abr{\widehat{\tau}_n - V^{\pi}_g(s_0)}\leq \frac{8\norm{V^{\pi}_g}_{\infty}}{\sqrt{n}}\ln^2\frac{8n^2\norm{V^{\pi}_g}_{\infty}}{\delta}$ for all $n\geq 1$.
Solving the range of $n$ for the inequality $\frac{8\norm{V^{\pi}_g}_{\infty}}{\sqrt{n}}\ln^2\frac{8n^2L_0}{\delta}\leq \norm{V^{\pi}_g}_{\infty}\epsilon$ (\pref{lem:quad log}) completes the proof.
\end{proof}
\begin{lemma}
\label{lem:V pi dev}
For any $g\in{\mathcal{S}}$ and policy $\pi$ with $\norm{V^{\pi}_g}_{\infty}\leq L_0$ for some $L_0\geq 1$, we have with probability at least $1-\delta$, for all $n\geq 1$ simultaneously, $|\widehat{\tau}_n - V^{\pi}_g(s_0)| \leq \frac{8L_0}{\sqrt{n}}\ln^2\frac{8n^2L_0}{\delta}$, where $\widehat{\tau}_n=\frac{1}{n}\sum_{i=1}^n C_i$ and each $C_i$ is a realization of the total cost incurred by following $\pi$ starting from $s_0$ with goal state $g$.
\end{lemma}
\begin{proof}
By \pref{lem:hitting} and a union bound,
\begin{align*}
\mathbb{P}\left( \exists i \geq 1 : C_i > 4L_0\ln\frac{8i^2L_0}{\delta}\right) \leq \sum_{i\geq 1}\mathbb{P}\left( C_i > 4L_0\ln\frac{8i^2L_0}{\delta}\right) \leq \sum_{i \geq 1} \frac{\delta}{4i^2L_0} \leq \frac{\delta}{2}.
\end{align*}
Then, under the complement of the event above (which holds with probability at least $1-\frac{\delta}{2}$), we have $\bar{\tau}_n=\widehat{\tau}_n$ for all $n\geq 1$, where $\bar{\tau}_n=\frac{1}{n}\sum_{i=1}^n C_i\field{I}\{C_i\leq 4L_0\ln\frac{8n^2L_0}{\delta}\}$. Moreover, by \pref{lem:azuma} and a union bound,
\begin{align*}
\mathbb{P}\left( \exists n \geq 1 : |\bar{\tau}_n-\field{E}[\bar{\tau}_n]| > 4L_0\ln\frac{8n^2L_0}{\delta}\sqrt{\frac{2\ln\frac{8n^2}{\delta}}{n}}\right) \leq \sum_{n \geq 1} \frac{\delta}{4n^2} \leq \frac{\delta}{2}.
\end{align*}
A union bound on the complement of the two events above yields that, with probability at least $1-\delta$, for all $n\geq 1$ simultaneously,
\begin{align*}
\widehat{\tau}_n - V^{\pi}_g(s_0) = \bar{\tau}_n - V^{\pi}_g(s_0) \leq \bar{\tau}_n - \field{E}[\bar{\tau}_n] \leq 4L_0\ln\frac{8n^2L_0}{\delta}\sqrt{\frac{2\ln\frac{8n^2}{\delta}}{n}},
\end{align*}
and by \pref{lem:hitting},
\begin{align*}
V^{\pi}_g(s_0) - \widehat{\tau}_n \leq \field{E}[\bar{\tau}_n] - \bar{\tau}_n + L_0\cdot\frac{1}{2nL_0} \leq 4L_0\ln\frac{8n^2L_0}{\delta}\sqrt{\frac{2\ln\frac{8n^2}{\delta}}{n}} + \frac{1}{2n}.
\end{align*}
Combining these two cases gives $\abr{\widehat{\tau}_n - V^{\pi}_g(s_0)}\leq \frac{8L_0}{\sqrt{n}}\ln^2\frac{8n^2L_0}{\delta}$.
\end{proof}
\begin{lemma}{\citep[Lemma B.5]{cohen2020near}}
\label{lem:hitting}
For a given $g\in{\mathcal{S}}$, let $\pi$ be a policy such that $\norm{V^{\pi}_g}_{\infty}\leq\tau$.
Then, for any $n\in\mathbb{N}$, the probability that the cost of $\pi$ to reach the goal state starting from any state is more than $n$, is at most $2e^{-\frac{n}{4\tau}}$.
\end{lemma}
\begin{lemma}[Azuma's inequality]
\label{lem:azuma}
Let $\{X_t\}_{t=1}^n$ be a martingale difference sequence with $|X_t|\leq B$.
Then with probability at least $1-\delta$, $|\sum_{t=1}^nX_i|\leq B\sqrt{2n\ln\frac{2}{\delta}}$.
\end{lemma}
\begin{lemma}\citep[Lemma 34]{chen2021implicit}
\label{lem:anytime bernstein}
Let $\{X_t\}_t$ be a sequence of i.i.d random variables with mean $\mu$, variance $\sigma^2$, and $0\leq X_t \leq B$.
Then with probability at least $1-\delta$, the following holds for all $n\geq 1$ simultaneously:
\begin{align*}
\abr{\sum_{t=1}^n(X_t-\mu)} &\leq 2\sqrt{2\sigma^2 n\ln\frac{2n}{\delta}} + 2B\ln\frac{2n}{\delta}.\\
\abr{\sum_{t=1}^n(X_t-\mu)} &\leq 2\sqrt{2\hat{\sigma}^2_nn\ln\frac{2n}{\delta}} + 19B\ln\frac{2n}{\delta}.
\end{align*}
where $\hat{\sigma}_n^2=\frac{1}{n}\sum_{t=1}^nX_t^2 - (\frac{1}{n}\sum_{t=1}^nX_t)^2$.
\end{lemma}
\begin{lemma}\citep[Lemma 50]{chen2022policy}
\label{lem:anytime freedman}
Let $\{X_i\}_{i=1}^{\infty}$ be a martingale difference sequence adapted to the filtration $\{{\mathcal{F}}_i\}_{i=0}^{\infty}$ and $|X_i|\leq B$ for some $B>0$.
Then with probability at least $1-\delta$, for all $n\geq 1$ simultaneously,
\begin{align*}
\abr{\sum_{i=1}^nX_i}\leq 3\sqrt{\sum_{i=1}^n\field{E}[X_i^2|{\mathcal{F}}_{i-1}]\ln\frac{4B^2n^3}{\delta} } + 2B\ln\frac{4B^2n^3}{\delta}.
\end{align*}
\end{lemma}
\subsection{Properties of the sets built by \pref{alg:SD}}
\begin{lemma}[Restricted Optimism]
\label{lem:V calK}
With probability at least $1-\delta$ over the randomness of \pref{alg:SD}, at any trial and any round, after executing \pref{line:compute V.improved}, if $\calK^{\star}_{z,j}\subseteq{\mathcal{K}}$ for some $j\in[z]$, then
$V_{{\mathcal{K}},g}(s) \leq V^{\star}_{\calK^{\star}_{z,j},g}(s)$ for any $s\in{\mathcal{S}}$ and $g\in\calK^{\star}_{z,j+1}\setminus{\mathcal{K}}$.
\end{lemma}
\begin{proof}
For any $\tau'\geq 1$, $z'\geq 1$, $j\in[z']$, $g\in\calK^{\star}_{z',j+1}\setminus\calK^{\star}_{z',j}$, by \pref{lem:opt} and $\norm{V^{\star}_{\calK^{\star}_{z',j},g}}_{\infty}\leq L+1$ (\pref{lem:init bound}), with probability at least $1-\frac{\delta}{4(z')^4(\tau')^2}$, for any status of $\N$ and $\xi>0$, we have $V(s)\leq V^{\star}_{\calK^{\star}_{z',j},g}(s)$ for all $s\in{\mathcal{S}}$ where $(\_,V,\_)=\textsc{VISGO}\xspace(\calK^{\star}_{z',j}, g, \xi, \N, \frac{\delta}{4(\tau')^2(z')^4AL})$.
By a union bound, all events above hold simultaneously with probability at least $1-\delta$.
At any trial $\tau$ and round, after executing \pref{line:compute V.improved}, let $(\_,V_{\calK^{\star}_{z,j}, g}, \_)=\textsc{VISGO}\xspace(\calK^{\star}_{z,j}, g, \epsilon_{\text{VI}\xspace},\N,\delta')$ (no need to compute explicitly) for any $j\in[z]$, and $g\in\calK^{\star}_{z,j+1}\setminus\calK^{\star}_{z,j}$, where $\delta'=\frac{\delta}{4\tau^2z^4AL}$.
The union bound above implies that $V_{\calK^{\star}_{z,j}, g}(s) \leq V^{\star}_{\calK^{\star}_{z,j}, g}(s)$ for any $s\in{\mathcal{S}}$.
Then by \pref{lem:subset opt}, we also have $V_{{\mathcal{K}},g}(s) \leq V^{\star}_{\calK^{\star}_{z,j},g}(s)$ if $\calK^{\star}_{z,j}\subseteq{\mathcal{K}}$ ($V_{{\mathcal{K}},g}$ is computed in \pref{line:compute V.improved}).
\end{proof}
\begin{lemma}
\label{lem:update calK}
For a given trial $(\tau, z)$, denote by ${\mathcal{K}}_r$ the set ${\mathcal{K}}$ at the end of each round $r$. With probability at least $1-2\delta$, for any $j \geq 1$ and round $r \geq 1$ in any trial in which ${\mathcal{K}}_r$ is updated or returned (i.e., \pref{line:compute calU'.easy} is executed) and ${\mathcal{K}}_{r-1} \supseteq \calK^{\star}_{j}$, we have ${\mathcal{K}}^\star_{j+1} \subseteq {\mathcal{K}}_r$.
\end{lemma}
\begin{proof}
In this lemma we denote by ${\mathcal{U}}_r$ the value of ${\mathcal{U}}$ \textit{at the end of} round $r$.
Define the event $E := \{ \text{for any trial, }\forall r\geq 1 \text{ in which ${\mathcal{K}}_r$ is updated}: {\mathcal{T}}_L({\mathcal{K}}_r) \setminus {\mathcal{K}}_r \subseteq {\mathcal{U}}_r\}$. By \pref{lem:calU}, it holds with probability at least $1-\delta$. Let us carry out the proof conditioned on $E$ holding.
In any trial, take some round $r$ such that \pref{line:compute calU'.easy} is executed and ${\mathcal{K}}_{r-1} \supseteq \calK^{\star}_{j}$. Let $r'<r$ be the last round where ${\mathcal{K}}_{r'}$ was updated (and thus ${\mathcal{U}}_{r'}$ was created). Note that ${\mathcal{K}}_{r'} = {\mathcal{K}}_{r-1} \supseteq \calK^{\star}_j$. Then, event $E$ and the definition of the sets $(\calK^{\star}_j)_j$ directly imply that $\calK^{\star}_{j+1} := {\mathcal{T}}_L(\calK^{\star}_j) \subseteq {\mathcal{T}}_L({\mathcal{K}}_{r'}) \subseteq {\mathcal{U}}_{r'} \cup {\mathcal{K}}_{r'}$. Since ${\mathcal{K}}_r$ can only be formed by adding states in ${\mathcal{U}}_{r'}$ to ${\mathcal{K}}_{r'}$, and the union of these sets contains $\calK^{\star}_{j+1}$, if $\calK^{\star}_{z,j+1} \not\subseteq {\mathcal{K}}_r$, it must be that there exists $g\in{\mathcal{U}}_{r-1} \cap \calK^{\star}_{z,j+1}$ s.t. $V_{{\mathcal{K}}_{r-1},g}(s_0) > L$. However, \pref{lem:V calK}, which holds with probability $1-\delta$, implies that, at any round $r\geq 1$, if $\calK^{\star}_{j}\subseteq{\mathcal{K}}_{r-1}$ (which implies that $z>|\calK^{\star}_j|$ and $\calK^{\star}_j=\calK^{\star}_{z,j}$ by \pref{line:z.improved}), then
$V_{{\mathcal{K}}_{r-1},g}(s_0) \leq V^{\star}_{\calK^{\star}_{j},g}(s_0) \leq L$ for any $g\in\calK^{\star}_{z,j+1}\setminus{\mathcal{K}}_{r-1}$. This is a contradiction, which implies that ${\mathcal{U}}_{r-1} \cap \calK^{\star}_{z,j+1} = \emptyset$ and, thus, all states in $\calK^{\star}_{z,j+1}$ must have been added to ${\mathcal{K}}_r$.
Moreover, since a new trial is not triggered in round $r$, by \pref{line:z.improved}, we have $z>|\calK^{\star}_{z,j+1}|$ and $\calK^{\star}_{z,j+1}=\calK^{\star}_{j+1}$.
This completes the proof.
\end{proof}
\begin{lemma}
\label{lem:calK}
For a given trial $(\tau, z)$, denote by ${\mathcal{K}}_r$ the set ${\mathcal{K}}$ at the end of each round $r$ inside the trial.
With probability at least $1-4\delta$, at any trial $(\tau,z)$, we have ${\mathcal{K}}_r \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ for any round $r$, and $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}_r$ if the algorithm terminates at round $r$.
\end{lemma}
\begin{proof}
Fix any trial $(\tau,z)$.
Clearly, ${\mathcal{K}}_1\subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$.
To prove the first statement, consider a round $r\geq 1$ and suppose ${\mathcal{K}}_r\subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$. If, in this round, the algorithm selects a goal $g^{\star}\in {\mathcal{U}}\setminus\calS^{\rightarrow}_{L(1+\epsilon)}$, $\pi_{g^{\star}}$ passes the test of \pref{line:rtest.improved}, and a skip round is not triggered, then we show that the ``failure test'' in \pref{line:failure.improved} is triggered.
Since $\pi_{g^{\star}}$ passed the test of \pref{line:rtest.improved}, we have $\|V^{\pi_{g^{\star}}}_{g^{\star}}\|_\infty \leq 32L$ with probability at least $1-\delta$ by \pref{lem:rtest} and a union bound over all trials and rounds. Combining this with \pref{lem:V pi mean} and the value of $\lambda$ (\pref{line:PE.improved}) (again by a union bound over all trials and rounds), we have $\widehat{\tau}\geq V^{\pi_{g^{\star}}}_{g^{\star}}(s_0) - L\epsilon/2$ with probability at least $1-2\delta$. By assumption on $g^\star$ and since $\pi_{g^{\star}}$ is restricted on ${\mathcal{K}}_r\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$, we have $V^{\pi_{g^{\star}}}_{g^{\star}}(s_0) \geq V^{\star}_{{\mathcal{K}}_r,g^{\star}}(s_0) \geq V^{\star}_{\calS^{\rightarrow}_{L(1+\epsilon)},g^{\star}}(s_0) > L(1+\epsilon)$, which implies that $\widehat{\tau}\geq L(1+\epsilon/2) \geq V_{{\mathcal{K}}_r, g^{\star}}(s_0) + \epsilon L/2$, where the last inequality is from the goal-selection rule.
Therefore, the failure test triggers and $g^\star$ is not added to ${\mathcal{K}}'$.
Overall, any $g\notin\calS^{\rightarrow}_{L(1+\epsilon)}$ will never be added to ${\mathcal{K}}$ or ${\mathcal{K}}'$ throughout the execution of \pref{alg:SD}.
To prove the second statement, let us consider any trial $(\tau,z)$ where the algorithm stops.
Clearly, $\calK^{\star}_1\subseteq {\mathcal{K}}_1$ at the end of round $r=1$ in this last trial.
Then, if $r$ is the round where the algorithm terminates, and $\calK^{\star}_j\subseteq{\mathcal{K}}_{r-1}$ for some $j\geq 1$, we have $\calK^{\star}_{j+1}\subseteq{\mathcal{K}}_r$ with probability at least $1-2\delta$ by \pref{lem:update calK}.
Moreover, since ${\mathcal{K}}'=\varnothing$ in round $r$, we have $\calK^{\star}_{j+1}\subseteq{\mathcal{K}}_{r-1}={\mathcal{K}}_r$.
By a recursive application of \pref{lem:update calK}, we have $\calK^{\star}_j\subseteq{\mathcal{K}}_r$ for any $j\geq 1$ (note that ${\mathcal{K}}'=\varnothing$ at the beginning of round $r$).
\pref{lem:SL.operator} then implies the statement.
\end{proof}
\begin{lemma}
\label{lem:bcalU}
Conditioned on the events of \pref{lem:calU} and \pref{lem:calK}, ${\mathcal{U}}\subseteq\bar{\calU}$ at the beginning of any round in any trial.
\end{lemma}
\begin{proof}
This is clearly true at the beginning of the first round of any trial since ${\mathcal{U}}=\varnothing$.
Then by the events of \pref{lem:calU} and \pref{lem:calK}, ${\mathcal{U}} \subseteq \rS{{\mathcal{K}}}{2L}\setminus{\mathcal{K}}\subseteq\bar{\calU}$ every time after executing \pref{line:add}.
Moreover, we only remove elements from ${\mathcal{U}}$ except when executing \pref{line:add}.
This completes the proof.
\end{proof}
\begin{lemma}
\label{lem:calK id}
Denote by ${\mathcal{K}}_r$ the set ${\mathcal{K}}$ at the end of each round $r$.
With \pref{assum:id}, with probability at least $1-8\delta$ over the randomness of \pref{alg:SD}, we have that ${\mathcal{K}}_r = \calK^{\star}_j$ for some $j \in [S^{\rightarrow}_L]$ at any round $r$ and, ${\mathcal{K}}_{r}=\calS^{\rightarrow}_L$ if the algorithm terminates at round $r$.
\end{lemma}
\begin{proof}
By \pref{lem:calK}, with probability at least $1-4\delta$, we have $\calS^{\rightarrow}_L \subseteq {\mathcal{K}} \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ if the algorithm terminates.
By \pref{rem:id}, ${\mathcal{K}} = \calS^{\rightarrow}_L$.
Thus, it suffices to show that at any trial ${\mathcal{K}}=\calK^{\star}_j$ for some $j \leq S^{\rightarrow}_L$.
The algorithm is such that $\calK^{\star}_1 = {\mathcal{K}}_1 = \{s_0\}$.
Suppose at the end of a round $r$ we have that ${\mathcal{K}}_r=\calK^{\star}_j$ for some $j\geq 1$.
By \pref{lem:update calK}, with probability at least $1-2\delta$, if the condition of \pref{line:goal condition.improved} is verified the first time in some round $r'>r$, then we must have $\calK^{\star}_{j+1}\subseteq{\mathcal{K}}_{r'}$.
If we also have ${\mathcal{K}}_{r'}\subseteq\calK^{\star}_{j+1}$, then the statement is proved.
In any round $r$ such that ${\mathcal{K}}=\calK^{\star}_j$, $g^{\star}\in {\mathcal{U}}\setminus\calK^{\star}_{j+1}$, $\pi_{g^{\star}}$ passes the test of \pref{line:rtest.improved}, and a skip round is not triggered, by \pref{lem:V pi mean}, the value of $\lambda$, and \pref{lem:rtest} (applying a union bound over all trials and rounds), we have $\widehat{\tau}\geq V^{\pi_{g^{\star}}}_{g^{\star}}(s_0) - L\epsilon/2$ with probability at least $1-2\delta$.
By assumption on $g^\star$ and since $\pi_{g^{\star}}$ is restricted on ${\mathcal{K}}\subseteq\calK^{\star}_j$, we have $V^{\pi_{g^{\star}}}_{g^{\star}}(s_0) \geq V^{\star}_{{\mathcal{K}},g^{\star}}(s_0) \geq V^{\star}_{\calK^{\star}_j,g^{\star}}(s_0) > L(1+\epsilon)$, which implies that $\widehat{\tau}\geq L(1+\epsilon/2) \geq V_{{\mathcal{K}}, g^{\star}}(s_0) + \epsilon L/2$, where the last inequality is from the goal-selection rule. Therefore, the failure test triggers and $g^\star$ is not added to ${\mathcal{K}}'$ or ${\mathcal{K}}$.
This proves ${\mathcal{K}}\subseteq\calK^{\star}_{j+1}$ in round $r'$.
\end{proof}
\begin{lemma}
\label{lem:calU id}
With \pref{assum:id}, conditioned on the events of \pref{lem:calU} and \pref{lem:calK id}, in any trial, ${\mathcal{U}}\subseteq\calU^{\star}_z$ at the beginning of any round.
\end{lemma}
\begin{proof}
By \pref{lem:calK id}, in any trial, we have ${\mathcal{K}} = \calK^{\star}_j\subseteq\calK^{\star}_{z,z}$ for some $j\leq z$ at the end of any round.
Then by \pref{lem:calU}, we have ${\mathcal{U}}\subseteq \rS{{\mathcal{K}}}{2L}\setminus{\mathcal{K}}\subseteq\calU^{\star}_z$ every time \pref{line:add} is executed.
\end{proof}
\section{Analysis of Policy Consolidation}\label{app:consolidation}
In this section, we bound the sample complexity of \pref{alg:PC}.
\paragraph{Notation} We assume that all episodes lie in one (artificial) trial.
Let $g_k$, $\tset_k$, $V_k$ $V^{\star}_k$ be the values of $g^{\star}$, $\tset\setminus\{g^{\star}\}$, $\widehat{V}$, and $V^{\star}_{\tset,g^{\star}}$ in episode $k$ respectively.
Denote by $I_k$ the number of steps in episode $k$.
Note that $I_k<\infty$ with probability $1$ by \pref{line:skip PC}, and $s^k_{I_k+1}\neq g_k$ only when a skip round is triggered in episode $k$.
Denote by ${\mathcal{F}}_k$ the $\sigma$-algebra of events up to episode $k$.
Define $K$ as the total number of episodes throughout the execution of \pref{alg:PC}.
For any $K'\leq K$, define $R_{K'}=\sum_{k=1}^{K'}(I_k - V_k(s_0))$ and $C_{K'}=\sum_{k=1}^{K'} I_k$.
Define $P^k_i=P_{s^k_i,a^k_i}$.
In episode $k$, when $s^k_i\in{\mathcal{K}}$, denote by $\P^k_i$, $\widetilde{P}^k_i$, $\N^k_i$, $b^k_i$ the values of $\P_{s^k_i,a^k_i}$, $\widetilde{P}_{s^k_i, a^k_i}$, $n^+(s^k_i, a^k_i)$, and $b^{(l)}(s^k_i, a^k_i)$, where $\P$, $n^+$, $b^{(l)}$ are used in \pref{alg:VISGO} to compute $V_k$ and $l$ is the final value of $i$ in \pref{alg:VISGO};
when $s^k_i\notin {\mathcal{K}}$, define $\P^k_i=\field{I}_{s_0}$, $\N^k_i=\infty$, and $b^k_i=0$.
Also define $\epsilon_k$ as the value of $\epsilon_{\text{VI}\xspace}$ used in \pref{alg:VISGO} to compute $V_k$.
In this section, ${\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ is an input of \pref{alg:PC} and thus does not have randomness.
\begin{proof}[\pfref{thm:PC}]
By \pref{lem:output PC}, the output policies $\{{\widetilde{\pi}}_g\}_g$ clearly satisfies the statement.
Define $\iota=\ln\left(\frac{LS^{\rightarrow}_{L(1+\epsilon)}A}{\delta\epsilon}\right)$.
It suffices to bound the number of samples collected in \pref{line:nu} and policy evaluation.
With probability at least $1-\delta$, the number of samples collected in \pref{line:nu} is of order $\bigo{L^3{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota^2}$ by \pref{lem:sc fillc} and \pref{lem:bounded error fresh}.
With probability at least $1-16\delta$, by \pref{lem:bound C PC} and \pref{lem:reg PC} ($c_1=\sqrt{LS^{\rightarrow}_{L(1+\epsilon)}A}$, $c_2=L{S^{\rightarrow}_{L(1+\epsilon)}}^2A$, and $p=2$), the number of samples collected in policy evaluation is of order $\tilO{\frac{LS^{\rightarrow}_{L(1+\epsilon)}A\iota^{10}}{\epsilon^2}+\frac{L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota^{10}}{\epsilon}}$.
Combining all cases completes the proof.
\end{proof}
\begin{lemma}
\label{lem:bound C PC}
With probability at least $1-4\delta$, if $R_{K'}\lesssim c_1\sqrt{\sum_{k=1}^{K'} V_k(s_0)\ln^p(c_3K')}+c_2\ln^p(c_3K')$ for any $K'\geq 1$ with $c_1,c_2\geq 1$ and $c_3=\frac{LS^{\rightarrow}_{L(1+\epsilon)}A}{\delta}$, then $C_K \lesssim \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\iota^8}{\epsilon^2} + \frac{c_1^2\iota^{p+8}}{\epsilon^2} + \frac{c_2\iota^{p+4}}{\epsilon}$, where $\iota=\ln\frac{c_1c_2c_3}{\epsilon\delta}$.
\end{lemma}
\begin{proof}
For any $R'\geq 1$, let $K'$ be the total number of episodes in the first $R'$ rounds.
Let $Z_{K'}=\sum_{k=1}^{K'} V_k(s_0)$.
First note that the regret gives $C_{K'}\lesssim Z_{K'} + c_1\sqrt{Z_{K'}\ln^p(c_3K')} + c_2\ln^p(c_3K')$ and thus $\ln(C_{K'})\lesssim \ln(c_1c_2c_3Z_{K'})$.
By $K'\lesssim C_{K'}$ and solving a ``quadratic'' inequality (\pref{lem:quad log}), we have $C_{K'}\lesssim Z_{K'} + (c_1^2+c_2)\ln^p(c_1c_2c_3Z_{K'})$.
Denote by $\bar{g}_r$, $\bar{V}_{r}$, $\bar{\pi}_r$ the value of $g^{\star}$, $\widehat{V}$, and $\widehat{\pi}$ in round $r$ respectively.
For each failure round $r$, let $C$ be the total cost within this round and $m$ the number of episodes within this round.
By definition, regret within this round satisfies $C-m\bar{V}_r(s_0) \geq C-\lambda \bar{V}_r(s_0)=\lambda(\widehat{\tau}-\bar{V}_r(s_0))>\frac{\lambda\epsilon \bar{V}_r(s_0)}{2}=\lowo{\bar{V}_r(s_0)/\epsilon}$.
For each success and skip round $r$, by \pref{lem:opt PC}, \pref{lem:nu}, \pref{lem:V pi dev}, and the value of $\lambda$, we have
\begin{align*}
\sum_{j=u_r}^{u'_r}\rbr{I_j - \bar{V}_r(s_0)} \gtrsim \sum_{j=u_r}^{u'_r-1}\rbr{I_j - V^{\bar{\pi}_r}_{\bar{g}_r}(s_0)} - L \gtrsim -L\sqrt{\lambda}\ln^2\frac{L\lambda}{\delta} \gtrsim -\frac{L}{\epsilon}\ln^4\frac{Lr}{\delta\epsilon} \gtrsim -\frac{L}{\epsilon}\ln^4\frac{LC_{K'}}{\delta\epsilon},
\end{align*}
where $\{u_r,\ldots,u'_r\}$ are the episodes in round $r$, and we lower bound the regret in the last episode by $\lowo{-L}$ since the last trajectory in a skipped round is truncated.
Denote by ${\mathcal{R}}_f$ the total number of failure rounds within the first $R'$ rounds.
By the assumption in \pref{alg:PC} that $\tset\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$, in the first $R'$ rounds, the number of success round is at most $S^{\rightarrow}_{L(1+\epsilon)}$ and the number of skip rounds is at most $\bigo{S^{\rightarrow}_{L(1+\epsilon)}A\ln(C_{K'})}$.
Since there are at most $\bigo{S^{\rightarrow}_{L(1+\epsilon)}A\ln(C_{K'})}$ these rounds, in each round there are at most $\tilo{\frac{\ln^4\frac{LC_{K'}}{\delta\epsilon}}{\epsilon^2}}$ episodes (\pref{line:PE PC}), and $\bar{V}_r(s_0)\leq 2L$ in any round $r$ by \pref{lem:opt PC}, we have
\begin{align*}
Z_{K'}
&\lesssim \frac{\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)\ln^4\frac{LC_{K'}}{\delta\epsilon}}{\epsilon^2} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\ln^5\frac{c_1c_2c_3Z_{K'}}{\delta\epsilon}}{\epsilon^2} \\
&\lesssim \frac{\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)\ln^4\frac{c_1c_2c_3Z_{K'}}{\delta\epsilon}}{\epsilon^2} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\ln^5\frac{c_1c_2c_3Z_{K'}}{\delta\epsilon}}{\epsilon^2}.
\end{align*}
By \pref{lem:quad log}, this gives
\begin{align*}
Z_{K'} \lesssim \frac{\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)\ln^4(c_4\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0))}{\epsilon^2} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\ln^5(c_4\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0))}{\epsilon},
\end{align*}
and $\ln(Z_{K'})\lesssim \ln(\frac{c_1c_2c_3\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)}{\delta\epsilon})\triangleq \ln(c_4\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0))$, where $c_4=\frac{c_1c_2c_3}{\delta\epsilon}$.
Therefore, the regret upper and lower bound and $\ln(K')\leq \ln(C_{K'})\lesssim \ln(c_1c_2c_3Z_{K'})\lesssim \ln(c_4\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0))$ give
\begin{align*}
&\frac{\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)}{\epsilon} - \frac{LS^{\rightarrow}_{L(1+\epsilon)}A}{\epsilon}\ln^4\frac{LC_{K'}}{\delta\epsilon} \lesssim c_1\sqrt{Z_{K'}\ln^p(c_3K')}+c_2\ln^p(c_3K')\\
&\lesssim \frac{c_1}{\epsilon}\sqrt{\rbr{\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0) + LS^{\rightarrow}_{L(1+\epsilon)}A\ln\rbr{c_4\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)}}\ln^{p+4}\rbr{c_4\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)} } + c_2\ln^p\rbr{c_4\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)}.
\end{align*}
Applying \pref{lem:quad log} gives $\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0) \lesssim LS^{\rightarrow}_{L(1+\epsilon)}A\ln^4(c_4) + c_1^2\ln^{p+4}(c_4) + c_2\epsilon\ln^p(c_4)$ and $\ln(\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0))\lesssim \ln(c_4)$.
Now by the regret bound and AM-GM inequality, we have
\begin{align*}
C_{K'} &\lesssim Z_{K'} + c_1\sqrt{Z_{K'}\ln^p(c_3K')} + c_2\ln^p(c_3K') \lesssim Z_{K'} + (c_1^2 + c_2)\ln^p(c_4)\\
&\lesssim \frac{\sum_{r\in{\mathcal{R}}_f}\bar{V}_r(s_0)\ln^4(c_4Z_{K'})}{\epsilon^2} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\ln^5(c_4Z_{K'})}{\epsilon^2} + (c_1^2 + c_2)\ln^p(c_4)\\
&\lesssim \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\ln^8(c_4)}{\epsilon^2} + \frac{c_1^2\ln^{p+8}(c_4)}{\epsilon^2} + \frac{c_2\ln^{p+4}(c_4)}{\epsilon}.
\end{align*}
Setting $R'$ to be the total number of rounds, we have $K'=K$ and the proof completes.
\end{proof}
\begin{lemma}
\label{lem:output PC}
With probability at least $1-4\delta$, we have $V^{{\widetilde{\pi}}_g}_g(s_0) \leq V^{\star}_{\tset,g}(s_0)(1 + \epsilon)$ for $g\in\tset$ throughout the execution of \pref{alg:PC}.
\end{lemma}
\begin{proof}
By \pref{lem:nu} and \pref{lem:init bound}, with probability at least $1-2\delta$, we have $V^{\widehat{\pi}}_{g^{\star}}(s) \leq 2V^{\star}_{\tset,g^{\star}}(s)\leq 4V^{\star}_{\tset,g^{\star}}(s_0)\leq \min\{8L, 4V^{\widehat{\pi}}_{g^{\star}}(s_0)\}$ for any $s\in{\mathcal{S}}$ throughout the execution.
For any $g\in\tset$, at the round that ${\widetilde{\pi}}_g$ is determined (where $g^{\star}=g$), by \pref{lem:V pi mean}, value of $\lambda$ and definition of success round, $V^{{\widetilde{\pi}}_g}_g(s_0) = V^{\widehat{\pi}}_g(s_0) \leq \widehat{\tau} + \frac{\epsilon}{256}\norm{V^{\widehat{\pi}}_g}_{\infty} \leq \widehat{\tau} + \frac{\epsilon}{4}V^{\widehat{\pi}}_g(s_0) \leq \widehat{V}(s_0)(1+\frac{\epsilon}{2}) + \frac{\epsilon}{4}V^{\widehat{\pi}}_g(s_0)$.
This gives $V^{{\widetilde{\pi}}_g}_g(s_0) \leq \frac{1 + \frac{\epsilon}{2}}{1-\frac{\epsilon}{4}}\widehat{V}(s_0) \leq (1+\epsilon)V^{\star}_{\tset,g}(s_0)$ by $\widehat{V}(s_0)\leqV^{\star}_{\tset,g}(s_0)$ (\pref{lem:opt PC}) and $\epsilon\in(0, 1]$.
\end{proof}
\begin{lemma}
\label{lem:reg PC}
With probability at least $1-12\delta$, for any $K'\leq K$, we have
$R_{K'}\lesssim \sqrt{LS^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'} V_k(s_0)\iota} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota$, where $\iota=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AK'}{\delta}$.
\end{lemma}
\begin{proof}
By \pref{lem:anytime bernstein} and a union bound on $\{V^{\star}_{\tset,g}\}_{g\in\tset}$ and $(s, a)\in\tset\times{\mathcal{A}}$, with probability at least $1-\delta$, $(P^k_i -\P^k_i)V^{\star}_k\lesssim \sqrt{\frac{\field{V}(P^k_i, V^{\star}_k)\iota'}{\N^k_i}} + \frac{L\iota'}{\N^k_i}$ for any $k\in[K']$ and $i\in[I_k]$ (note that this holds even if $s^k_i\notin{\mathcal{K}}$), where $\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$.
Moreover, with probability at least $1-\delta$,
\begin{align*}
&\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)} \leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{1 + V_k(s^k_{i+1}) - V_k(s^k_i)}\\
&\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(\field{I}_{s^k_{i+1}} - P^k_i)V_k + (P^k_i - \P^k_i)V_k + b^k_i + \epsilon_k} \tag{\pref{lem:def Vk}}\\
&\lesssim \sqrt{\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\ln\frac{LC_{K'}}{\delta}} + \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(P^k_i-\P^k_i)V^{\star}_k + (P^k_i-\P^k_i)(V_k-V^{\star}_k) + b^k_i} + L\ln\frac{LC_{K'}}{\delta}.
\end{align*}
where the last step is by \pref{lem:sum eps} and \pref{lem:anytime freedman}.
Now note that with probability at least $1-2\delta$,
\begin{align*}
&\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(P^k_i-\P^k_i)V^{\star}_k + (P^k_i-\P^k_i)(V_k-V^{\star}_k) + b^k_i}\\
&\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{\sqrt{\frac{\field{V}(P^k_i, V^{\star}_k)\iota'}{\N^k_i}} + \sqrt{\frac{\Gamma_{L(1+\epsilon)}\field{V}(P^k_i, V_k-V^{\star}_k)\iota'}{\N^k_i}} + \frac{\Gamma_{L(1+\epsilon)}L\iota'}{\N^k_i} + b^k_i} \tag{\pref{lem:dPV}, $\norm{V^{\star}_k}_{\infty}\leq 2L+1$, $\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$}\\
&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k-V^{\star}_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota',
\end{align*}
where in the last step $\iota'=\ln^2\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$ and we apply \pref{lem:sum N}, Cauchy-Schwarz inequality, \pref{lem:sum b}, and $\textsc{Var}[X+Y]\leq 2(\textsc{Var}[X]+\textsc{Var}[Y])$.
Thus, by \pref{lem:sum dV} with \pref{lem:opt PC} and AM-GM inquality, with probability at least $1-8\delta$, we continue with
\begin{align*}
C_{K'} - \sum_{k=1}^{K'} V_k(s_0)&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'\\
&\lesssim \sqrt{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota', \tag{\pref{lem:sum var Vk}}
\end{align*}
where $\iota'=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$.
Solving a ``quadratic'' inequality w.r.t $C_{K'}$ (\pref{lem:quad log}), we have $C_{K'}\lesssim \sum_{k=1}^{K'} V_k(s_0) + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AK'}{\delta}$.
Plugging this back to the last inequality above completes the proof.
\end{proof}
\begin{lemma}
\label{lem:nu}
With probability at least $1-2\delta$, throughout the execution of \pref{alg:PC}, $V_{g^{\star}}^{\widehat{\pi}}(s)\leq 2V^{\star}_{\tset,g^{\star}}(s)$ for any $s\in{\mathcal{S}}$.
\end{lemma}
\begin{proof}
By \pref{lem:opt PC}, value of $\nu$ (\pref{line:nu}), and applying \pref{lem:bounded error fresh} with ${\mathcal{X}}=\tset\setminus\{g\}$ for each $g\in\tset$, we have $V^{\widehat{\pi}}_{g^{\star}}(s)\leq 2\widehat{V}(s)\leq 2V^{\star}_{\tset,g^{\star}}(s)$ for all $s\in{\mathcal{S}}$.
\end{proof}
\begin{lemma}
\label{lem:opt PC}
With probability at least $1-\delta$, throughout the execution of \pref{alg:PC}, $\widehat{V}(s)\leqV^{\star}_{\tset,g^{\star}}(s)$ for any $s\in{\mathcal{S}}$.
\end{lemma}
\begin{proof}
This is simply by the value of $\widehat{V}$ in each round and applying \pref{lem:opt} on $\{V^{\star}_{\tset,g}\}_{g\in\tset}$.
\end{proof}
\section{Introduction}
A distinctive feature of intelligent beings is the ability to explore an unknown environment without any supervision or extrinsic reward while learning skills that solve tasks (e.g., reaching goal states) of increasing difficulty. \citet{lim2012autonomous} first proposed a formal framework of \textit{autonomous exploration} in reinforcement learning (RL) as the process of progressively discovering states within a certain distance from an initial state $s_0$ at the same time as learning near-optimal policies to reach them. \citet{lim2012autonomous} also devised the first sample efficient exploration algorithm (\textsc{UcbExplore}\xspace) for this setting, while its sample complexity and optimality guarantees were later improved by \textsc{DisCo}\xspace~\citep{tarbouriech2020improved} and \textsc{VALAE}\xspace~\citep{cai2022near}.
In this paper, we make several contributions to this problem:
\begin{itemize}[topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=10pt]
\item Given an initial state $s_0$, the autonomous exploration objective is built upon the concept of incrementally $L$-controllable states, i.e., states that can be reached within $L$ steps from $s_0$ by only traversing incrementally $L$-controllable states\footnote{We say that a state $s$ is $L$-controllable if there exists a policy that reaches $s$ from $s_0$ in less than $L$ steps on average. In general an $L$-controllable state may be reached by policies traversing states that are not $L$-controllable themselves.}. While the original definition of the set of incrementally $L$-controllable states ${\mathcal{S}}_L^{\rightarrow}$ involves considering all possible partial orders of states in the environment, we derive an equivalent constructive definition that reveals the \emph{layered} structure of ${\mathcal{S}}_L^{\rightarrow}$, where each layer can be obtained as the set of states that can be reached in $L$ steps by only traversing states in the previous layers (see~\pref{sec:sL.constructive}).
\item We then leverage the layered structure of ${\mathcal{S}}_L^{\rightarrow}$ to design Layered Autonomous Exploration (\textsc{LAE}\xspace), a novel algorithm that keeps exploring the environment to learn policies to reach newly discovered states until a new layer can be consolidated and a new step of discovery and learning is started. We prove that the sample complexity of \textsc{LAE}\xspace is bounded as $\tilo{LS^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A/\epsilon^2}$, where $L$ is the exploration radius, $S^{\rightarrow}_{L(1+\epsilon)}$ is the number of states that are incrementally controllable from the initial state within $L(1+\epsilon)$ steps, $\Gamma_{L(1+\epsilon)}$ is the branching factor of the transition function over such states, $A$ is the number actions, and $\epsilon$ is target accuracy. As illustrated in Table~\ref{tab:summary}, this improves the sample complexity of \textsc{DisCo}\xspace by a factor of $L^2$ and it avoids the scaling with $S^{\rightarrow}_{2L}$ of \textsc{VALAE}\xspace, which in some MDPs may be much larger than $S^{\rightarrow}_{L(1+\epsilon)}$, thus making the bound of \textsc{LAE}\xspace preferable. Indeed, in~\pref{lem:example 2L} in appendix we show that $S^{\rightarrow}_{2L}$ may be even exponentially larger than $S^{\rightarrow}_{L(1+\epsilon)}$.
\item Under a certain layer identifiability condition (see ~\pref{assum:id}), we further improve the sample complexity of \textsc{LAE}\xspace to $\tilo{LS^{\rightarrow}_{L}A/\epsilon^2}$, which improves w.r.t.\ \textsc{VALAE}\xspace and matches the lower bound in~\citep{cai2022near}.
\item Similar to existing algorithms, the sample complexity of \textsc{LAE}\xspace still depends on the logarithm of the total number of states $S$. Since in autonomous exploration the state space is unknown and possibly unbounded, such dependency is highly undesirable. We then design an alternative version of \textsc{LAE}\xspace, which preserves its original sample complexity but replaces the dependency on $\ln S$ with $\lnS^{\rightarrow}_{L(1+\epsilon)}$, without requiring any prior knowledge of $S^{\rightarrow}_{L(1+\epsilon)}$ (see~\pref{sec:log.adaptivity}).
\item \textsc{LAE}\xspace also leverages a novel procedure, \textsc{PolicyConsolidation}\xspace, that takes a set of states ${\mathcal{K}}$ as input and returns goal-conditioned policies reaching each state in ${\mathcal{K}}$ with \emph{multiplicative} $\epsilon$-optimality guarantees, which is stronger than previous algorithms and better suited to the autonomous exploration setting (see~\pref{sec:pc}).
\end{itemize}
\paragraph{Related Work}
In reinforcement learning (RL), several approaches to \emph{unsupervised exploration} have been proposed often grounded in concepts such as curiosity~\citep{schmidhuber1991possibility}, intrinsic motivation~\citep{singh2004intrinsically,oudeyer2009intrinsically,bellemare2016unifying,colas2020intrinsically} and with the objective of learning skills in an unsupervised fashion~\citep{gregor2016variational,eysenbach2018diversity,pong2019skew,bagaria2021skill,kamienny2021direct}. On the other hand, a rigorous formalization and theoretical understanding of unsupervised exploration has been rather sparse until recently. \citet{tarbouriech2020active} studied unsupervised exploration for model estimation, \citet{hazan2019provably} formalized the maximum entropy exploration objective, while reward-free RL~\citep[e.g.,][]{jin2020reward,kaufmann2021adaptive,menard2021fast,zhang2021near,tarbouriech2021provably,tarbouriech2022adaptive} studies how to efficiently explore an environment to solve any downstream task near-optimally. As autonomous exploration seeks to learn goal-conditioned policies, it also carries strong technical and algorithmic connections with exploration in the stochastic shortest path problem~\citep[e.g.][]{bertsekas2013stochastic,tarbouriech2020no,tarbouriech2021stochastic,chen2021finding,chen2022near}.
\section{Conclusion}
We introduced a layered decomposition of the set of incrementally $L$-controllable states. We built on this decomposition and showed that our algorithm \textsc{LAE}\xspace attains the strongest performance guarantee $\text{AX}^+$, does not need to know $S$ and thus can be used with a countably-infinite state space, and is minimax-optimal when the layers can be uniquely identified.
The natural future directions include 1) designing an algorithm with minimax sample complexity without~\pref{assum:id}; 2) extending the problem to continuous states and function approximation; 3) identifying benchmarks that can be used to evaluate practical progresses towards the $\text{AX}$ capability.
\section{Lemmas for Policy Evaluation}
\label{app:evaluation}
In this section, we present a set of lemmas related to regret analysis shared among \pref{alg:LOGSSD}, \pref{alg:SD}, and \pref{alg:PC}.
In \pref{alg:SD}, a trial is indexed by $\tau$, and each trial corresponds to a value of $z$ estimating $S^{\rightarrow}_{L(1+\epsilon)}$ (\pref{line:trial}).
In \pref{alg:LOGSSD} and \pref{alg:PC}, we assume the whole learning procedure lies in an artificial trial.
Note that when lemmas below are involved, we have $b^k_i=0$, $\N^k_i=\infty$, and $\P^k_i=\field{I}_{s_0}$ when $s^k_i\notin{\mathcal{K}}_k$.
\begin{lemma}
\label{lem:sum var Vk}
Let $\mathcal{G}$ be the goal set such that $\calS^{\rightarrow}_{L(1+\epsilon)}\subseteq\mathcal{G}\subseteq{\mathcal{S}}$.
In any trial, with probability at least $1-2\delta$, for any $K'\in[K]$, if ${\mathcal{K}}_k\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and $g_k\in\mathcal{G}\setminus{\mathcal{K}}_k$ for any $k\in[K']$, then $\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\lesssim LC_{K'} + L^2\Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}A\iota$, where $\iota = \bigo{\ln(|\mathcal{G}|ALC_{K'}/\delta)\ln(C_{K'})}$.
\end{lemma}
\begin{proof}
Note that $\norm{V_k}_{\infty}\leq 2L$ by the stopping condition (\pref{line:bound V}) of \pref{alg:VISGO}, and with probability at least $1-\delta$,
\begin{align*}
&\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{V_k(s^k_i)^2 - (P^k_iV_k)^2} \lesssim L\sum_{k=1}^K\sum_{i=1}^{I_k}(V_k(s^k_i) - P^k_iV_k)_+ \tag{$a^2-b^2\leq (a+b)(a-b)_+$ for $a,b\geq 0$} \\
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{1 + (\P^k_i - P^k_i)V_k + \frac{1}{\N^k_i} + \epsilon_k}_+ \tag{\pref{lem:def Vk}}\\
&\lesssim LC_{K'} + L\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{\sqrt{\frac{\Gamma_{L(1+\epsilon)}\field{V}(P^k_i, V_k)\iota'}{\N^k_i}} + \frac{L\Gamma_{L(1+\epsilon)}\iota'}{\N^k_i} + \epsilon_k} \tag{\pref{lem:dPV} and $\N^k_i=\infty$ when $s^k_i\notin{\mathcal{K}}_k$}\\
&\lesssim LC_{K'} + L\sqrt{\Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'\ln(C_{K'})} + L^2\Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}A\iota'\ln(C_{K'}),
\end{align*}
where $\iota'=\ln(|\mathcal{G}|AC_{K'}/\delta)$, and the last step is by Cauchy-Schwarz inequality, \pref{lem:sum N}, and \pref{lem:sum eps}.
Now let $Z_{K'}=\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)$.
Applying \pref{lem:sum var} and $\sum_{k=1}^{K'} V_k(s^k_{I_k+1})^2\lesssim L^2S^{\rightarrow}_{L(1+\epsilon)}A\iota'$ (this is because $V_k(s^k_{I_k+1})$ is non-zero only in skip rounds), we have with probability a least $1-\delta$,
\begin{align*}
Z_{K'} \lesssim LC_{K'} + L\sqrt{\Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}AZ_{K'}\iota} + L^2\Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}A\iota,
\end{align*}
where $\iota = \bigo{\ln(|\mathcal{G}|ALC_{K'}/\delta)\ln(C_{K'})}$.
Solving a quadratic inequality completes w.r.t.~$Z_{K'}$ the proof.
\end{proof}
\begin{lemma}
\label{lem:sum dV}
In any trial, with probability at least $1-5\delta$, for any $K'\in[K]$ if 1) $\{V^{\star}_k\}_{k\in[K']}\subseteq{\mathcal{V}}$ where ${\mathcal{V}}$ is determined at the beginning of the trial, $|{\mathcal{V}}|$ is upper bounded by polynomials of $S^{\rightarrow}_{L(1+\epsilon)}$, and $\norm{V}_{\infty}=\bigo{L}$ for any $V\in{\mathcal{V}}$, 2) $V_k(s)\leqV^{\star}_k(s)$ for any $k\in[K']$ and $s\in{\mathcal{S}}$, 3) ${\mathcal{K}}_k\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ for any $k\in[K']$, and 4) $g_k\in\bar{\calU}\setminus{\mathcal{K}}_k$ for any $k\in[K']$, then $\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V^{\star}_k - V_k)\lesssim L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota'} + L^2{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'$, where $\iota'=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$.
\end{lemma}
\begin{proof}
First note that
\begin{align*}
&\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(V^{\star}_k(s^k_i) - V_k(s^k_i))^2 - (P^k_i(V^{\star}_k-V_k))^2}\\
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k}(V^{\star}_k(s^k_i) - V_k(s^k_i) - P^k_iV^{\star}_k + P^k_iV_k)_+ \tag{$V_k(s)\leqV^{\star}_k(s)$ for all $s$ and $a^2-b^2\leq (a+b)(a-b)_+$ for $a,b\geq 0$}\\
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k}(1 + P^k_iV_k - V_k(s^k_i))_+ \tag{$V^{\star}_k(s^k_i)\leq 1 + P^k_iV^{\star}_k$}.
\end{align*}
Let $\P_{s,a}(s')=\frac{\N(s,a,s')}{\N^+(s,a)}$.
By \pref{lem:anytime bernstein}, with probability at least $1-\delta$, for any $(s, a)\in\calS^{\rightarrow}_{L(1+\epsilon)}\times{\mathcal{A}}$, $V\in{\mathcal{V}}$, and status of counter $\N$:
\begin{align}
(P_{s,a} - \P_{s,a})V \lesssim \sqrt{\frac{\field{V}(P_{s,a}, V)\iota'}{\N(s, a)}} + \frac{L\iota'}{\N(s, a)},\label{eq:calV}
\end{align}
where $\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$.
By \pref{lem:def Vk}, with probability at least $1-2\delta$, we continue with
\begin{align*}
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k}( (P^k_i-\P^k_i)V^{\star}_k + (P^k_i-\P^k_i)(V_k - V^{\star}_k) + b^k_i + \epsilon_k)_+\\
&\lesssim L\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{\sqrt{\frac{\field{V}(P^k_i, V^{\star}_k)\iota'}{\N^k_i}} + \sqrt{\frac{\Gamma_{L(1+\epsilon)}\field{V}(P^k_i, V_k-V^{\star}_k)\iota'}{\N^k_i}} + \frac{\Gamma_{L(1+\epsilon)}L\iota'}{\N^k_i} + b^k_i + \epsilon_k} \tag{\pref{eq:calV}, \pref{lem:dPV}, conditions 3) and 4), $\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$}\\
&\lesssim L\rbr{\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + \sqrt{{S^{\rightarrow}_{L(1+\epsilon)}}^2A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k - V^{\star}_k)\iota'}} + L^2{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota',
\end{align*}
where in the last step $\iota'=\ln^2\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$ and we apply $\textsc{Var}[X_1+X_2]\leq\textsc{Var}[X_1]+\textsc{Var}[X_2]$, Cauchy-Schwarz inequality, \pref{lem:sum N}, \pref{lem:sum eps}, and \pref{lem:sum b}.
Then applying \pref{lem:sum var} with $\norm{V^{\star}_k-V_k}_{\infty}\lesssim L$ and solving a quadratic inequality w.r.t.~$\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V^{\star}_k-V_k)$, we have with probability at least $1-\delta$,
\begin{align*}
&\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V^{\star}_k - V_k)\\
&\lesssim \sum_{k=1}^{K'} (V^{\star}_k(s^k_{I_k+1})-V_k(s^k_{I_k+1}))^2 + L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota'} + L^2{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'. \tag{$\iota'=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$}
\end{align*}
The proof is completed by noting that $V^{\star}_k(g)=V_k(g)=0$ and $\sum_{k=1}^{K'}\field{I}\{s^k_{I_k+1}\neq g\}\lesssim S^{\rightarrow}_{L(1+\epsilon)}A$.
\end{proof}
\begin{lemma}
\label{lem:sum var}
Let $K\in\mathbb{N}$ and $\{V_k\}_{k\in[K]}$ be a sequence of value functions with $V_k\in[0, B]^{{\mathcal{S}}}$ for $B>0$. With probability at least $1-\delta$, for any $K'\in [K]$, $$\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i,V_k)\lesssim \sum_{k=1}^{K'} V_k(s^k_{I_k+1})^2 + \sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{V_k(s^k_i)^2 - (P^k_iV_k)^2} + B^2\iota,$$
where $\iota = \ln(BC_{K'}/\delta)$.
\end{lemma}
\begin{proof}
We decompose the sum as follows:
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k) = \sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{P^k_i(V_k)^2 - V_k(s^k_{i+1})^2}
+ \sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{V_k(s^k_{i+1})^2 - V_k(s^k_i)^2} + \sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{V_k(s^k_i)^2 - (P^k_iV_k)^2}.
\end{align*}
For the first term, by \pref{lem:anytime freedman}, \pref{lem:quad}, and $I_k<\infty$ for any $k\in[K]$ by the skip-round condition, with probability at least $1-\delta$, for all $K'\in[K]$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \rbr{P^k_i(V_k)^2 - V_k(s^k_{i+1})^2} &\lesssim \sqrt{\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, (V_k)^2)\iota} + B^2\iota\\
&\lesssim B\sqrt{\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i, V_k)\iota} + B^2\iota,
\end{align*}
where $\iota = \bigo{\ln(BC_{K'}/\delta)}$.
The second term is clearly upper bounded by $\sum_{k=1}^{K'} V_k(s^k_{I_k+1})^2$.
Putting everything together and solving a quadratic inequality w.r.t.~$\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \field{V}(P^k_i,V_k)$ completes the proof.
\end{proof}
\begin{lemma}
\label{lem:sum b}
Let $\mathcal{G}$ be the goal set such that $\calS^{\rightarrow}_{L(1+\epsilon)}\subseteq\mathcal{G}\subseteq{\mathcal{S}}$.
In any trial, with probability at least $1-\delta$, for any $K'\in[K]$, if ${\mathcal{K}}_k\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and $g_k\in\mathcal{G}\setminus{\mathcal{K}}_k$ for any $k\in[K']$, then
$\sum_{k=1}^{K'}\sum_{i=1}^{I_k} b^k_i\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)} A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota} + L{S^{\rightarrow}_{L(1+\epsilon)}}^{1.5}A\iota$, where $\iota=\ln(|\mathcal{G}|AC_{K'}/\delta)$.
\end{lemma}
\begin{proof}
Note that with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} b^k_i &\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{\sqrt{\frac{\field{V}(\P^k_i, V_k)\iota}{\N^k_i}} + \frac{L\iota}{\N^k_i}} \tag{definition of $b^k_i$ and $\max\{a,b\}\leq a + b$}\\
&\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{ \sqrt{\frac{\field{V}(P^k_i, V_k)\iota}{\N^k_i}} + \frac{L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}}\iota}{\N^k_i} } \tag{\pref{lem:barPV to PV}}\\
&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota} + L{S^{\rightarrow}_{L(1+\epsilon)}}^{1.5}A\iota. \tag{Cauchy-Schwarz inequality and \pref{lem:sum N}}
\end{align*}
This completes the proof.
\end{proof}
\begin{lemma}
\label{lem:sum N}
In any trial, for any $K'\in[K]$, if ${\mathcal{K}}_k \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ for any $k\in[K']$, we have $\sum_{k=1}^{K'}\sum_{i=1}^{I_k} \frac{1}{\N^k_i}\lesssim S^{\rightarrow}_{L(1+\epsilon)}A\log_2(C_{K'})$.
\end{lemma}
\begin{proof}
Note that, for any $i,k$, if $s_i^k \notin \calS^{\rightarrow}_{L(1+\epsilon)}$ we must have $s_i^k \notin {\mathcal{K}}_k$, which implies that the corresponding count $N_i^k$ is $\infty$. Then,
\begin{align*}
\sum_{k=1}^K\sum_{i=1}^{I_k}\frac{1}{\N^k_i}
&\leq \sum_{s \in \calS^{\rightarrow}_{L(1+\epsilon)},a\in{\mathcal{A}}} ~\sum_{0\leq h \leq \log_2(C_K)} \sum_{k=1}^K\sum_{i=1}^{I_k} \field{I}\big[(s_i^k,a_i^k) = (s,a), \N_i^k(s,a)= 2^h\big] \frac{1}{2^h}\\
&\leq |\calS^{\rightarrow}_{L(1+\epsilon)}| A \log_2(C_k).
\end{align*}
\end{proof}
\begin{lemma}
\label{lem:sum eps}
In any trial, for any $K'\in[K]$, $\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\epsilon_k=\bigo{\ln C_{K'}}$.
\end{lemma}
\begin{lemma}
\label{lem:def Vk}
In any trial, $1 + \P^k_iV_k - 2b^k_i - \epsilon_k\leq V_k(s^k_i)\leq 1 + \P^k_iV_k + \epsilon_k$ for any $k\in[K], i\in[I_k]$.
\end{lemma}
\begin{proof}
When $s^k_i\notin{\mathcal{K}}_k$, we have $b^k_i=\frac{1}{\N^k_i}=0$ and $\P^k_iV_k=V_k(s_0)$.
Thus, the statement holds.
When $s^k_i\in{\mathcal{K}}_k$, by the definition of $V_k$ and the stopping rule of \pref{alg:VISGO}, we have
\begin{align*}
V_k(s^k_i) &\geq 1 + \widetilde{P}^k_iV_k - b^k_i - \epsilon_k \geq 1 + \P^k_iV_k - b^k_i - \epsilon_k - \frac{\P^k_iV_k}{\N^k_i} \tag{definition of $\widetilde{P}^k_i$}\\
&\geq 1 + \P^k_iV_k - 2b^k_i - \epsilon_k,
\end{align*}
where the last step is by $\frac{\P^k_iV_k}{\N^k_i}\leq\frac{2L}{\N^k_i}\leq b^k_i$.
Moreover, $V_k(s^k_i)\leq 1 + \widetilde{P}^k_iV_k + \epsilon_k \leq 1 + \P^k_iV_k + \epsilon_k$.
This completes the proof.
\end{proof}
\section{Analysis of \pref{alg:SD}}
\label{app:logsa}
\begin{algorithm2e*}[t]
\caption{Improved Layer-Aware State Discovery (\textsc{LASD}\xspace{}\textsuperscript{+}\xspace)}
\label{alg:SD}
\SetKwProg{proc}{Procedure}{}{}
\SetKwFunction{add}{ComputeU}
\DontPrintSemicolon
\LinesNumbered
\KwIn{$L\geq1$, $\epsilon\in(0, 1]$, and $\delta\in(0, 1)$.}
Let $\tau\leftarrow 1$, $\mathfrak{N}=\{2^j\}_{j\geq 0}$, $z\leftarrow 2$.\label{line:trial}\;
\While{True}{\label{line:size.improved}
Let ${\mathcal{K}}\leftarrow \varnothing, {\mathcal{U}} \leftarrow \varnothing$, ${\mathcal{K}}'\leftarrow \{s_0\}$, $\Pi_{{\mathcal{K}}} = \{{\widetilde{\pi}}_{s_0}\text{ a random policy}\}$, $\N(\cdot, \cdot)\leftarrow 0, \N(\cdot,\cdot,\cdot) \leftarrow 0$, $n_{\min}\leftarrow 1$, $k\leftarrow 0$.\;
\For{round $r=1,\ldots$}{\label{line:round.improved}
\lIf{$|{\mathcal{K}}\cup{\mathcal{K}}'| \geq z$}{$z\leftarrow 2|{\mathcal{K}}\cup{\mathcal{K}}'|$, $\tau\overset{+}{\leftarrow}1$, and return to \pref{line:size.improved}.}\label{line:z.improved}
$\epsilon_{\text{VI}\xspace}\leftarrow 1/\max\{16, \sum_{s,a}\N(s,a)\}$.\;
Let $g^{\star}=\argmin_{g\in{\mathcal{U}}}\big\{V_{{\mathcal{K}},g}(s_0)\big\}$ where $(Q_{{\mathcal{K}},g}, V_{{\mathcal{K}},g}, \pi_g)=\textsc{VISGO}\xspace({\mathcal{K}}, g, \epsilon_{\text{VI}\xspace}, \N, \frac{\delta}{4\tau^2z^4AL})$ (see \pref{alg:VISGO}).\label{line:compute V.improved}\;
\uIf{$g^{\star}$ does not exist or $V_{{\mathcal{K}},g^{\star}}(s_0)>L$}{\label{line:goal condition.improved}
\tcc{Expand or Terminate}
\lIf{${\mathcal{K}}'=\varnothing$}{\textbf{return} ${\mathcal{K}}$ and $\Pi_{{\mathcal{K}}}$.}\label{line:terminate.improved}
Set ${\mathcal{K}}\leftarrow{\mathcal{K}}\cup{\mathcal{K}}'$, ${\mathcal{K}}'=\varnothing, {\mathcal{U}}=\varnothing$.
\;
${\mathcal{U}} \leftarrow $\add{${\mathcal{K}}$, $\Pi_{{\mathcal{K}}}$, $\frac{\delta}{4\tau^2r^2}$}.\label{line:add}
}
\uElseIf{$\textsc{RTest}\xspace(\Pi_{{\mathcal{K}}}, \pi_{g^{\star}}, g^{\star}, \frac{\delta}{4(\tau r)^2})=$ \textbf{False} (see \pref{alg:rtest})}{\label{line:rtest.improved}
$n_{\min}\leftarrow 2n_{\min}$.\;
$(\N,\_) \leftarrow \textsc{Explore}\xspace({\mathcal{K}}, \Pi_{{\mathcal{K}}},\N,n_{\min})$ (see \pref{alg:fillc}).
}\Else{
\tcc{Policy evaluation}
Let $\widehat{\tau}\leftarrow 0$, $\lambda\leftarrow N_{\textsc{Dev}}(32L, \frac{\epsilon}{256}, \frac{\delta}{2 r^2})\lesssim \frac{1}{\epsilon^2}\ln^4(\frac{Lr}{\epsilon\delta})$ (defined in \pref{lem:V pi mean}).\label{line:PE.improved}
\For{$j=1,\ldots,\lambda$}{\label{line:episode.improved}
$k\overset{+}{\leftarrow}1$, $i\leftarrow 1$, and reset to $s^k_1\leftarrow s_0$ by taking action $\textsc{RESET}$.\;
\While{$s^k_i\neq g^{\star}$}{
Take $a^k_i=\pi_{g^{\star}}(s^k_i)$, and transits to $s^k_{i+1}$.
Increase $\N(s^k_i, a^k_i)$, $\N(s^k_i, a^k_i, s^k_{i+1})$, and $i$ by $1$.\;
\lIf{$\sum_{s,a}\N(s,a)\in\mathfrak{N}$ or ($s^k_i\in{\mathcal{K}}$ and $\N(s^k_i, a^k_i)\in\mathfrak{N}$)}{ return to \pref{line:round.improved} (skip round).\label{line:skip.improved}}
Set $\widehat{\tau}\overset{+}{\leftarrow} \frac{c(s^k_i, a^k_i)}{\lambda}$.
}
\lIf{$\widehat{\tau}>V_{{\mathcal{K}},g^{\star}}(s_0) + \epsilon L/2$}{ return to \pref{line:round.improved} (failure round). \label{line:failure.improved}}
}
${\mathcal{K}}'\leftarrow{\mathcal{K}}'\cup\{g^{\star}\}$, ${\mathcal{U}}\leftarrow{\mathcal{U}}\setminus\{g^{\star}\}$, $\Pi_{{\mathcal{K}}}=\Pi_{{\mathcal{K}}} \cup \{{\widetilde{\pi}}_{g^{\star}}:=\pi_{g^{\star}}\}$
(success round).
}
}
}
\proc{\add{${\mathcal{X}}$, $\Pi_{{\mathcal{X}}}$, $\delta$}}{
$(\_,{\mathcal{U}}')\leftarrow\textsc{Explore}\xspace({\mathcal{X}},\Pi_{{\mathcal{X}}}, 0, 2L\ln\frac{4LA|{\mathcal{X}}|}{\delta})$ (see~\pref{alg:fillc}). \label{line:compute calU'.improved}\;
$(\N', \_) \leftarrow \textsc{Explore}\xspace({\mathcal{X}},\Pi_{{\mathcal{X}}}, 0, N_1(|{\mathcal{X}}|, \frac{\delta}{4|{\mathcal{U}}'|}))$ where $N_1$ is defined in \pref{lem:bounded error fresh}.\label{line:gather N'.improved}\;
Let ${\mathcal{U}}=\{g\in{\mathcal{U}}': V'_{{\mathcal{X}},g}(s_0)\leq L\}$ where $(\_,V'_{{\mathcal{X}},g},\pi'_g)=\textsc{VISGO}\xspace({\mathcal{X}},g,\frac{1}{16},\N',\frac{\delta}{4|{\mathcal{U}}'|})$.\label{line:filter calU'.improved}\;
\Return{${\mathcal{U}}$}
}
\end{algorithm2e*}
\paragraph{Notation} Define ${\mathcal{N}}({\mathcal{K}}, p)=\{s'\notin{\mathcal{K}}: P(s'|s,a)\geq p\text{ for some }(s, a)\in{\mathcal{K}}\times{\mathcal{A}} \}$.
Fix any ordering $\calO^{\rightarrow}_L=(s_1,\ldots,s_n)$ of states in $\calS^{\rightarrow}_L$ such that it can be partitioned into $J$ (defined in \pref{lem:SL.operator}) segments with states in the $j$-th segment belonging to $\calK^{\star}_j\setminus\calK^{\star}_{j-1}$.
For an arbitrary $z\in\field{N}_+$, also define $\{\calK^{\star}_{z,j}\}_j$, such that $\calK^{\star}_{z,j}=\calK^{\star}_j$ when $|\calK^{\star}_j|< z$, and $\calK^{\star}_{z,j}=\{s_1,\ldots,s_z\}$ when $|\calK^{\star}_j|\geq z$.
Therefore, $\calK^{\star}_{z,z}=(s_1,\ldots,s_{z})$ (the first $z$ elements of $\calO^{\rightarrow}_L$) or $\calS^{\rightarrow}_L$ by definition.
Define $\calU^{\star}_z=\rS{\calK^{\star}_{z,z}}{2L}$.
Clearly, $\calU^{\star}_z\subseteq\{s'\in{\mathcal{S}}: \exists s\in \calK^{\star}_{z,z}, a\in{\mathcal{A}}, P(s'|s, a)\geq \frac{1}{2L}\}$, and thus $|\calU^{\star}_z|\leq 2zAL$.
\subsection{\pfref{thm:sd id improved}}
\label{app:sd id improved}
\begin{proof}
We condition on the events of \pref{lem:output}, \pref{lem:calU}, and \pref{lem:calK}, which happen with probability at least $1-7\delta$.
By the events of \pref{lem:calK} and \pref{lem:output}, the output ${\mathcal{K}}$ and $\Pi_{{\mathcal{K}}}=\{{\widetilde{\pi}}_g\}_{g\in{\mathcal{K}}}$ clearly satisfy the statement.
By \pref{lem:bound z}, there are at most $\bigo{\lnS^{\rightarrow}_{L(1+\epsilon)}}$ trials.
Thus, it suffices to bound the number of samples used in each trial.
Define $\iota=\ln\frac{LS^{\rightarrow}_{L(1+\epsilon)}A}{\delta\epsilon}$.
Each round in a trial can be classified into one of the following cases: 1) \pref{line:goal condition.improved} is verified, 2) \pref{line:rtest.improved} is verified, and 3) policy evaluation is performed (\pref{line:PE.improved}).
In case 1), the algorithm terminates or at least one state is added into ${\mathcal{K}}$ (\pref{line:terminate.improved}).
Thus, the number of rounds satisfying case 1) in each trial is at most $1+S^{\rightarrow}_{L(1+\epsilon)}$ by \pref{lem:calK}.
By \pref{lem:bound nmin} and the update rule of $n_{\min}$, the number of rounds satisfying case 2) is of order $\bigo{\ln(LS^{\rightarrow}_{L(1+\epsilon)})}$.
By \pref{lem:bound r} and \pref{lem:regret}, with probability at least $1-8\delta$, the total number of rounds satisfying case 3) is of order $\bigo{ S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A\iota^6 + {S^{\rightarrow}_{L(1+\epsilon)}}^2A\epsilon\iota^6}$.
So the total number of rounds in each trial is at most $\bigo{ S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A\iota^6 + {S^{\rightarrow}_{L(1+\epsilon)}}^2A\epsilon\iota^6 }$.
Now it suffices to bound the number of samples collected in a round satisfying each of the cases above in a trial.
In a round satisfying case 1), if the algorithm terminates, then no samples are collected.
Otherwise, \add is called, and $\bigo{L^3{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota^2}$ samples are collected with probability at least $1-\delta$ by \pref{lem:calU each} (\pref{line:add} and a union bound over all trials and rounds).
In a round satisfying case 2), with probability at least $1-4\delta$, $\bigo{LS^{\rightarrow}_{L(1+\epsilon)}\iota^2}$ samples are collected in performing \textsc{RTest}\xspace by \pref{lem:output} and \pref{lem:rtest} (\pref{line:rtest.improved} and a union bound over all trials and rounds), and $\bigo{L^3{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota^2}$ samples are collected in executing \textsc{Explore}\xspace by \pref{lem:bound nmin} and \pref{lem:sc fillc}.
In a round satisfying case 3), with probability at leat $1-\delta$, $\bigo{LS^{\rightarrow}_{L(1+\epsilon)}\iota^2}$ samples are collected in performing \textsc{RTest}\xspace similar to that of case 2), and $\bigo{L\iota^5/\epsilon^2}$ samples are collected by the value of $\lambda$ and the fact that $\pi_{g^{\star}}$ passes the test in \pref{line:rtest.improved} (\pref{lem:rtest} and a union bound over all trials and rounds).
Thus, the total sample complexity is
\begin{align*}
&\sum_{i=1}^3\text{[\#rounds satisfying case $i$}]\cdot[\text{\#samples in a round satisfying case $i$}]\cdot\iota\\
&\lesssim S^{\rightarrow}_{L(1+\epsilon)}\cdot L^3{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota^3 + L^3{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota^4 + (S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A + {S^{\rightarrow}_{L(1+\epsilon)}}^2A\epsilon)\cdot\rbr{\frac{L}{\epsilon^2}+LS^{\rightarrow}_{L(1+\epsilon)}}\iota^{12}\\
&\lesssim \rbr{\frac{LS^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A}{\epsilon^2} + \frac{L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\epsilon}{\epsilon} + L^3{S^{\rightarrow}_{L(1+\epsilon)}}^3A}\iota^{12}.
\end{align*}
This completes the proof.
To prove the second statement, we can simply follow the proof above except that we involve \pref{lem:regret-improved} instead of \pref{lem:regret} when applying \pref{lem:bound r} to bound the total number of rounds satisfying case 3), which holds with probability at least $1-20\delta$.
\end{proof}
\begin{lemma}
\label{lem:bound nmin}
With probability at least $1-2\delta$, if the events of \pref{lem:calK} and \pref{lem:bcalU} hold, then $n_{\min}\lesssim L^2S^{\rightarrow}_{L(1+\epsilon)}\lnS^{\rightarrow}_{L(1+\epsilon)}$ throughout the execution of \pref{alg:SD}.
\end{lemma}
\begin{proof}
In any trial $\tau$, when $n_{\min}\geq N^{\rightarrow}_0(\frac{\delta}{4\tau^2z^4AL})$ (defined in \pref{lem:bounded error}), we have with probability at least $1-\frac{\delta}{2\tau^2}$, $\norm{V^{\pi_{g^{\star}}}_{g^{\star}}}_{\infty}\leq 2\norm{V_{{\mathcal{K}},g^{\star}}}_{\infty}\leq 2(1 + V_{{\mathcal{K}},g^{\star}}(s_0)) \leq 4L$ in any round such that $g^{\star}$ exists and $V_{{\mathcal{K}},g^{\star}}(s_0)\leq L$.
This implies that with probability at least $1-\sum_{r=1}^{\infty}\frac{\delta}{4\tau^2r^2}\geq 1-\frac{\delta}{2\tau^2}$, the condition of \pref{line:rtest.improved} is always false by \pref{lem:rtest}, and the value of $n_{\min}$ will no longer change within this trial.
A union bound over all trials and noting the update rule of $n_{\min}$ completes the proof.
\end{proof}
\begin{lemma}
\label{lem:bound z}
Conditioned on the event of \pref{lem:calK}, we have $z\leq 2S^{\rightarrow}_{L(1+\epsilon)}+2$ and $\tau\leq 1 + \log_2(S^{\rightarrow}_{L(1+\epsilon)}+1)$ throughout the execution of \pref{alg:SD}.
\end{lemma}
\begin{proof}
The proof of \pref{lem:calK} shows that $s\notin\calS^{\rightarrow}_{L(1+\epsilon)}$ will never be added to ${\mathcal{K}}'$, which implies ${\mathcal{K}}\cup{\mathcal{K}}'\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ throughtout the execution of \pref{alg:SD}.
Thus, when $z\geq S^{\rightarrow}_{L(1+\epsilon)}+1$, $z$ will not be updated again.
Then, the statement is proved by the update rule of $z$ and $\tau$.
\end{proof}
\subsection{Lemmas for Policy Evaluation}
\paragraph{Notation} Let $g_k$, ${\mathcal{K}}_k$, $V_k$, $Q_k$, $V^{\star}_k$ be the values of $g^{\star}$, ${\mathcal{K}}$, $V_{{\mathcal{K}},g^{\star}}$, $Q_{{\mathcal{K}},g^{\star}}$, and $V^{\star}_{{\mathcal{K}},g^{\star}}$ in episode $k$ respectively.
Denote by $I_k$ the number of steps in episode $k$.
Note that $I_k<\infty$ with probability $1$ by \pref{line:skip.improved}, and $s^k_{I_k+1}\neq g_k$ only when a skip round is triggered in episode $k$.
Denote by ${\mathcal{F}}_k$ the $\sigma$-algebra of events up to episode $k$.
Define $K$ as the total number of episodes throughout the execution of \pref{alg:SD}.
For any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ and $K'\leq K$, define $R_{K',{\mathcal{I}}}=\sum_{k=1}^{K'}(I_k - V_k(s_0))\boldsymbol{1}_k$ and $C_{K'}=\sum_{k=1}^{K'} I_k$.
Define $P^k_i=P_{s^k_i,a^k_i}$.
In episode $k$, when $s^k_i\in{\mathcal{K}}$, denote by $\P^k_i$, $\widetilde{P}^k_i$, $\N^k_i$, $b^k_i$ the values of $\P_{s^k_i,a^k_i}$, $\widetilde{P}_{s^k_i, a^k_i}$, $n^+(s^k_i, a^k_i)$, and $b^{(l)}(s^k_i, a^k_i)$, where $\P$, $n^+$, $b^{(l)}$ are used in \pref{alg:VISGO} to compute $V_k$ and $l$ is the final value of $i$ in \pref{alg:VISGO};
when $s^k_i\notin {\mathcal{K}}$, define $\P^k_i=\field{I}_{s_0}$, $\N^k_i=\infty$, and $b^k_i=0$.
Also define $\epsilon_k$ as the value of $\epsilon_{\text{VI}\xspace}$ used in \pref{alg:VISGO} to compute $V_k$.
\begin{lemma}
\label{lem:regret}
With probability at least $1-5\delta$, if the events of \pref{lem:calK} and \pref{lem:bcalU} hold, then in any trial, for any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ with $\boldsymbol{1}_k\in{\mathcal{F}}_{k-1}$, we have $R_{K',{\mathcal{I}}} \lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}AL^2K'\iota} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota$ for any $K'\leq K$, where $\iota=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AK'}{\delta}$.
\end{lemma}
\begin{proof}
Note that by \pref{lem:def Vk},
\begin{align*}
\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k &\leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{1 + V_k(s^k_{i+1}) - V_k(s^k_i)}\boldsymbol{1}_k\\
&\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(\field{I}_{s^k_{i+1}} - P^k_i)V_k + (P^k_i - \P^k_i)V_k + b^k_i + \epsilon_k}\boldsymbol{1}_k.
\end{align*}
We bound the sums above separately.
By \pref{lem:anytime freedman} and $\norm{V_k}_{\infty}\leq 2L$, with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (\field{I}_{s^k_{i+1}} - P^k_i)V_k\boldsymbol{1}_k \lesssim \sqrt{\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\ln\frac{LC_{K'}}{\delta}} + L\ln\frac{LC_{K'}}{\delta}.
\end{align*}
By \pref{lem:dPV}, ${\mathcal{K}}_k\in\calS^{\rightarrow}_{L(1+\epsilon)}$ (\pref{lem:calK}), $g_k\in \bar{\calU}\setminus{\mathcal{K}}_k$ (\pref{lem:bcalU}), Cauchy-Schwarz inequality, and \pref{lem:sum N}, with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k}(P^k_i - \P^k_i)V_k\boldsymbol{1}_k &\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\boldsymbol{1}_k\sqrt{\frac{\Gamma_{L(1+\epsilon)}\field{V}(P^k_i, V_k)\iota'}{\N^k_i}} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}\iota'}{\N^k_i} \tag{$\N^k_i=\infty$ when $s^k_i\notin{\mathcal{K}}_k$ and $\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$}\\
&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'. \tag{$\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}\ln(C_{K'})$}
\end{align*}
Finally, by \pref{lem:sum b} and \pref{lem:sum eps}, with probability at least $1-\delta$,
\begin{align*}
\sum_{k=1}^{K'}\sum_{i=1}^{I_k} (b^k_i+\epsilon_k)\boldsymbol{1}_k \lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^{1.5}A\iota'. \tag{$\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$}
\end{align*}
Plugging these back, we have with probability at least $1-2\delta$,
\begin{align}
\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k &\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'\notag\\
&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}ALC_{K'}\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota', \label{eq:var}
\end{align}
where $\iota'=\ln\frac{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}\ln(C_{K'})$ and in the last step we apply \pref{lem:sum var Vk}.
Now assuming $\boldsymbol{1}_k=1$ for all $k$ and solving a ``quadratic'' inequality (\pref{lem:quad log}) w.r.t.~$C_{K'}$, we have
\begin{align*}
C_{K'}\lesssim \sum_{k=1}^{K'} V_k(s_0) + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota' \lesssim LK' + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'. \tag{$\iota'=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AK'}{\delta}$}
\end{align*}
Plugging this back to \pref{eq:var} completes the proof.
\end{proof}
\begin{lemma}
\label{lem:regret-improved}
With \pref{assum:id}, with probability at least $1-12\delta$, if the events of \pref{lem:calU}, \pref{lem:bound z}, \pref{lem:calK id}, and \pref{lem:calU id} hold, in any trial, for any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ with $\boldsymbol{1}_k\in{\mathcal{F}}_{k-1}$, we have $R_{K',{\mathcal{I}}} \lesssim L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}AK'\iota} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota$ for any $K'\leq K$, where $\iota=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AK'}{\delta}$.
\end{lemma}
\begin{proof}
Note that with \pref{assum:id} and by \pref{lem:calK id} and \pref{lem:calU id}, in any episode, ${\mathcal{K}}=\calK^{\star}_j$ for some $j\leq z$ and $g^{\star}\in\calU^{\star}_z$.
Thus by \pref{lem:anytime bernstein} and a union bound over $\{V^{\star}_{\calK^{\star}_{z,j},g}\}_{j\in[z], g\in\calU^{\star}_z}$ and $(s, a)\in\calS^{\rightarrow}_{L(1+\epsilon)}\times{\mathcal{A}}$, we have with probability at least $1-\delta$,
\begin{equation}
(P^k_i -\P^k_i)V^{\star}_k\lesssim \sqrt{\frac{\field{V}(P^k_i, V^{\star}_k)\iota'}{\N^k_i}} + \frac{L\iota'}{\N^k_i},\label{eq:dP optV}
\end{equation}
where $\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$.
Thus, with probability at least $1-\delta$,
\begin{align*}
&\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k \leq \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{1 + V_k(s^k_{i+1}) - V_k(s^k_i)}\boldsymbol{1}_k\\
&\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(\field{I}_{s^k_{i+1}} - P^k_i)V_k + (P^k_i - \P^k_i)V_k + b^k_i + \epsilon_k}\boldsymbol{1}_k \tag{\pref{lem:def Vk}}\\
&\lesssim \sqrt{\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\ln\frac{LC_{K'}}{\delta}} + L\ln\frac{LC_{K'}}{\delta} + \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(P^k_i-\P^k_i)V^{\star}_k\boldsymbol{1}_k + (P^k_i-\P^k_i)(V_k-V^{\star}_k)\boldsymbol{1}_k + b^k_i },
\end{align*}
where the last step is by \pref{lem:anytime freedman} and \pref{lem:sum eps}.
Note that by \pref{eq:dP optV}, \pref{lem:dPV}, and $\norm{V^{\star}_k}_{\infty}\leq 2L+1$ by \pref{lem:calU} and \pref{lem:init bound}, with probability at least $1-2\delta$,
\begin{align*}
&\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{(P^k_i-\P^k_i)V^{\star}_k\boldsymbol{1}_k + (P^k_i-\P^k_i)(V_k-V^{\star}_k)\boldsymbol{1}_k + b^k_i }\\
&\lesssim \sum_{k=1}^{K'}\sum_{i=1}^{I_k}\rbr{\sqrt{\frac{\field{V}(P^k_i, V^{\star}_k)\iota'}{\N^k_i}} + \sqrt{\frac{\Gamma_{L(1+\epsilon)}\field{V}(P^k_i, V_k-V^{\star}_k)\iota'}{\N^k_i}} + \frac{L\Gamma_{L(1+\epsilon)}\iota'}{\N^k_i} + b^k_i} \tag{$\iota'=\ln\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$}\\
&\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + \sqrt{{S^{\rightarrow}_{L(1+\epsilon)}}^2A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k-V^{\star}_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota' \tag{$\iota'=\ln^2\frac{S^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$}.
\end{align*}
where the last step is by \pref{lem:sum N}, Cauchy-Schwarz inequality, $\textsc{Var}[X+Y]\leq 2(\textsc{Var}[X]+\textsc{Var}[Y])$, and \pref{lem:sum b}.
Plugging this back, applying \pref{lem:sum dV} with \pref{lem:opt} on $\{V^{\star}_{\calK^{\star}_j,g}\}_{j\in[z], g\in\calU^{\star}_z\setminus\calK^{\star}_j}$ (where all $V^{\star}_k$ lies in), \pref{lem:calK id}, and \pref{lem:calU id}, and then applying AM-GM inequality, we have with probability at least $1-8\delta$,
\begin{align*}
\sum_{k=1}^{K'}\rbr{I_k - V_k(s_0)}\boldsymbol{1}_k &\lesssim \sqrt{S^{\rightarrow}_{L(1+\epsilon)}A\sum_{k=1}^{K'}\sum_{i=1}^{I_k}\field{V}(P^k_i, V_k)\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'\\
&\lesssim \sqrt{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}\iota'} + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota', \tag{\pref{lem:sum var Vk}}
\end{align*}
where $\iota'=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AC_{K'}}{\delta}$.
Now assuming $\boldsymbol{1}_k=1$ for all $k$ and solving a ``quadratic'' inequality (\pref{lem:quad log}), we have
\begin{align*}
C_{K'} \lesssim \sum_{k=1}^{K'} V_k(s_0) + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota' \leq LK' + L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota'. \tag{$\iota'=\ln^2\frac{LS^{\rightarrow}_{L(1+\epsilon)}AK'}{\delta}$}
\end{align*}
Plugging this back completes the proof.
\end{proof}
\begin{lemma}
\label{lem:bound r}
In any trial, with probability at least $1-8\delta$, if for any sequence of indicators ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ with $\boldsymbol{1}_k\in{\mathcal{F}}_{k-1}$, we have $R_{K',{\mathcal{I}}}\lesssim c_1\sqrt{K'\ln^p(c_3K')}+c_2\ln^p(c_3K')$ with $c_1,c_2\geq 1$, and $c_3=\frac{LS^{\rightarrow}_{L(1+\epsilon)}A}{\delta}$ for any $K'\leq K$, then the total number of rounds with at least one epsiode is of order $\bigo{S^{\rightarrow}_{L(1+\epsilon)}A\iota^4 + \frac{c_1^2}{L^2}\iota^{p+4} + c_2\epsilon\iota^p/L}$, where $\iota=\ln\frac{c_1c_2c_3}{\epsilon\delta}$.
\end{lemma}
\begin{proof}
For any $R'\geq 1$, let $K'$ be the total number of episodes in the first $R'$ rounds.
Denote by $r_{\text{tot}}$ the total number of rounds with at least one episode, and $r_f$ the number of failure rounds in the first $R'$ rounds.
First note that by $V_k(s_0)\leq L$ (\pref{line:goal condition.improved}) and setting $\boldsymbol{1}_k=1$, the regret guarantee in the assumption gives $C_{K'}\lesssim LK' + c_1\sqrt{K'\ln^p(c_3K')} + c_2\ln^p(c_3K')$, which gives $\ln(C_{K'})\lesssim \ln(c_1c_2c_3K')$.
Moreover, $K'\lesssim \frac{r_{\text{tot}}}{\epsilon^2}\ln^4\frac{Lr_{\text{tot}}}{\epsilon\delta}$ by the value of $\lambda$ in each round (\pref{line:PE.improved}).
Thus, $\ln(C_{K'})\lesssim \ln\frac{c_1c_2c_3r_{\text{tot}}}{\epsilon\delta}$ and $\ln(c_3K')\lesssim \ln\frac{c_1c_2c_3r_{\text{tot}}}{\epsilon\delta}$.
Fixed a trial, denote by $\bar{V}_r$, $\bar{\pi}_r$ and $\bar{g}_r$ the values of $V_{{\mathcal{K}},g^{\star}}$, $\pi_{g^{\star}}$, and $g^{\star}$ used for policy evaluation in round $r$ respectively.
It is clear that in the first $R'$ rounds, the number of success round is at most $S^{\rightarrow}_{L(1+\epsilon)}$ by \pref{lem:calK}, and the number of skip rounds is at most $\bigo{S^{\rightarrow}_{L(1+\epsilon)}A\ln(C_{K'})}$ since we have a skip round only when the total number of steps or the number of visits of some state-action pair in ${\mathcal{K}}\times{\mathcal{A}}$ is doubled.
Therefore, $r_{\text{tot}}\lesssim r_f + S^{\rightarrow}_{L(1+\epsilon)}A\ln(C_{K'}) \lesssim r_f + S^{\rightarrow}_{L(1+\epsilon)}A\ln\frac{c_1c_2c_3r_{\text{tot}}}{\epsilon\delta}$.
By \pref{lem:quad log}, we have $r_{\text{tot}} \lesssim r_f + S^{\rightarrow}_{L(1+\epsilon)}A\ln\frac{c_1c_2c_3r_f}{\epsilon\delta}$.
Now define $\iota(r_f)=\ln\frac{c_1c_2c_3r_f}{\epsilon\delta}$.
It remains to bound $r_f$.
Define $\mathcal{W}=\{r: V^{\bar{\pi}_r}_{\bar{g}_r}(s_0)>\bar{V}_r(s_0)\}$.
Note that $\mathcal{W}$ includes all failure rounds with probability at least $1-\delta$, since when $V^{\bar{\pi}_r}_{\bar{g}_r}(s_0)\leq \bar{V}_r(s_0)$ and $r$ is not a skip round, by \pref{lem:V pi mean} and the value of $\lambda$ in round $r$ we have $\widehat{\tau}\leq \bar{V}_r(s_0) + \epsilon L/2$ in round $r$.
Define ${\mathcal{I}}=\{\boldsymbol{1}_k\}_k$ such that $\boldsymbol{1}_k=\field{I}\{r\in\mathcal{W}\}\in{\mathcal{F}}_{k-1}$ for any episode $k$ in round $r$, the regret within these rounds satisfies $R_{K',{\mathcal{I}}}\lesssim \frac{c_1}{\epsilon}\sqrt{r_f+S^{\rightarrow}_{L(1+\epsilon)}A} + c_2$.
\begin{align*}
R_{K',{\mathcal{I}}} &\lesssim c_1\sqrt{K'\ln^p(c_3K')}+c_2\ln^p(c_3K') \lesssim \frac{c_1}{\epsilon}\sqrt{(r_f + S^{\rightarrow}_{L(1+\epsilon)}A\iota(r_f))\iota(r_f)^{p+4}} + c_2\iota(r_f)^p\\
&\lesssim \frac{c_1}{\epsilon}\sqrt{r_f\iota(r_f)^{p+4}} + \frac{c_1^2\iota(r_f)^{p+4}}{L\epsilon} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\iota(r_f)}{\epsilon}+c_2\iota(r_f)^p. \tag{AM-GM inequality}
\end{align*}
For each failure round $r$, let $C$ be the total cost within this round and $m$ the number of episodes within this round.
By definition, regret within this round satisfies $C-mV_{{\mathcal{K}},g^{\star}}(s_0) \geq C-\lambda V_{{\mathcal{K}},g^{\star}}(s_0)=\lambda(\widehat{\tau}-V_{{\mathcal{K}},g^{\star}}(s_0))>\frac{\lambda\epsilon L}{2}=\lowo{L/\epsilon}$.
By \pref{lem:V pi dev}, with probability at least $1-\delta$, for each success and skip round $r$ in $\mathcal{W}$ ($V^{\bar{\pi}_r}_{g_r}(s_0)>\bar{V}_r(s_0)$),
\begin{align*}
\sum_{j=u_r}^{u'_r}\rbr{I_j - \bar{V}_r(s_0)} \gtrsim \sum_{j=u_r}^{u'_r-1}\rbr{I_j - V^{\bar{\pi}_r}_{\bar{g}_r}(s_0)} - L \gtrsim -L\sqrt{\lambda}\ln^2\frac{L\lambda}{\delta} = -\frac{L}{\epsilon}\ln^4\frac{Lr}{\delta\epsilon},
\end{align*}
where $\{u_r,\ldots,u'_r\}$ are the episodes in round $r$, and we lower bound the regret in the last episode by $\lowo{-L}$ since the last trajectory in a skipped round is truncated.
Since there are at most $\tilo{S^{\rightarrow}_{L(1+\epsilon)}A}$ these rounds, we have
\begin{align*}
\frac{Lr_f}{\epsilon} - \frac{LS^{\rightarrow}_{L(1+\epsilon)}A}{\epsilon}\ln^4\frac{Lr_f}{\epsilon\delta} \lesssim \frac{c_1}{\epsilon}\sqrt{r_f\iota(r_f)^{p+4}} + \frac{c_1^2\iota(r_f)^{p+4}}{L\epsilon} + \frac{LS^{\rightarrow}_{L(1+\epsilon)}A\iota(r_f)}{\epsilon}+c_2\iota(r_f)^p.
\end{align*}
This gives $r_f \lesssim S^{\rightarrow}_{L(1+\epsilon)}A\iota^4 + \frac{c_1^2}{L^2}\iota^{p+4} + c_2\epsilon\iota^p/L$, where $\iota=\ln\frac{c_1c_2c_3}{\epsilon\delta}$.
Setting $R'$ to be the total number rounds completes the proof.
\end{proof}
\begin{lemma}
\label{lem:output}
With probability at least $1-2\delta$, throughout the execution of \pref{alg:SD}, for each $g\in{\mathcal{K}}$ we have $V^{{\widetilde{\pi}}_g}_g(s_0)\leq L(1+\epsilon)$ and $\norm{V^{{\widetilde{\pi}}_g}_g}_{\infty}\leq 32L$.
\end{lemma}
\begin{proof}
By \pref{lem:rtest} and a union bound over all trials and rounds, with probability at least $1-\delta$, we have $\norm{V^{{\widetilde{\pi}}_g}_g}_{\infty}\leq 32L$ for each $g\in{\mathcal{K}}$, since ${\widetilde{\pi}}_g$ passes the test in \pref{line:rtest.improved}.
Moreover, by the definition of success round, value of $\lambda$, and \pref{lem:V pi mean}, with probability at least $1-\delta$, for each $g\in{\mathcal{K}}$, in the round that $g$ is added to ${\mathcal{K}}$, we have $V^{{\widetilde{\pi}}_g}_g(s_0)=V^{\pi_g}_g(s_0)\leq \widehat{\tau} + \frac{L\epsilon}{2} \leq V_{{\mathcal{K}},g}(s_0) + L\epsilon \leq L(1+\epsilon)$.
\end{proof}
\subsection{\textsc{RTest}\xspace and \textsc{Explore}\xspace}
Here we show auxiliary algorithms and related lemmas used in \pref{alg:SD}.
\begin{algorithm2e}[t]
\DontPrintSemicolon
\caption{\textsc{Explore}\xspace}
\label{alg:fillc}
\KwIn{States ${\mathcal{X}}$, policies $\Pi = \{\pi_x\}_{x\in{\mathcal{X}}}$ such that $\norm{V^{\pi_x}_x}_{\infty} = \bigo{L}$, counters $n$, target value $\bar{n}$.}
$\calS_{\text{next}}\leftarrow \varnothing$.\;
\For{$(x,a)\in{\mathcal{X}}\times{\mathcal{A}}$}{
\While{$n(x, a) < \bar{n}$}{
Reset to $s_0$ and execute $\pi_x$ until reaching $x$.\;
Execute action $a$, observe $x'\sim P_{x,a}$, and update $n(x,a,x') \overset{+}{\leftarrow} 1$.\;
\lIf{$x'\notin {\mathcal{X}}$}{$\calS_{\text{next}}\leftarrow\calS_{\text{next}}\cup\{x'\}$.}
}
}
\Return{$n$ and $\calS_{\text{next}}$.}
\end{algorithm2e}
\begin{algorithm2e}[t]
\DontPrintSemicolon
\caption{\textsc{RTest}\xspace}
\label{alg:rtest}
\KwIn{reaching policy $\{\pi_s\}_{s\in{\mathcal{X}}}$, test policy $\pi\in\Pi({\mathcal{X}})$, goal state $g$, and failure probability $\delta$.}
Let $n=2^{10}\ln\frac{2|{\mathcal{X}}|}{\delta}$.
\For{$s\in{\mathcal{X}}$}{
$i_s\leftarrow 0$.
\For{$j=1,\ldots,n$}{
Reset to $s_0$ and execute $\pi_s$ until $s$ is reached.
Execute $\pi$ until $g$ is reached or $8L$ steps is taken.
\lIf{$g$ is reached}{$i_s\overset{+}{\leftarrow}1$}
}
\lIf{$i_s/n < \frac{7}{16}$}{\Return{\textsc{False}\xspace.} }
}
\Return{\textsc{True}\xspace.}
\end{algorithm2e}
\begin{lemma}
\label{lem:rtest}
For any ${\mathcal{X}}\subseteq{\mathcal{S}}$, $\{\pi_g\}_{g\in{\mathcal{X}}}$, policy $\overline{\pi}\in\Pi({\mathcal{X}})$, goal state $g\in{\mathcal{S}}$, and $\delta\in(0,1)$, we have
\begin{align*}
\mathbb{P}\rbr{\left. \textsc{RTest}\xspace({\mathcal{X}}, \{\pi_g\}_{g\in{\mathcal{X}}}, \pi, g, \delta) = \textsc{True}\xspace \right| \norm{V^{\overline{\pi}}_g}_{\infty} \leq 4L} &\geq 1- \delta,\\
\mathbb{P}\rbr{ \textsc{RTest}\xspace({\mathcal{X}}, \{\pi_g\}_{g\in{\mathcal{X}}}, \pi, g, \delta) = \textsc{True}\xspace \implies \norm{V^{\overline{\pi}}_g}_{\infty}\leq 32L } &\geq 1- \delta.
\end{align*}
Moreover, if $\norm{V^{\pi_g}_g}_{\infty}=\bigo{L}$ for any $g\in{\mathcal{X}}$, then with probability at least $1-\delta$, the sample complexity is $\tilo{L|{\mathcal{X}}|\ln^2\frac{|{\mathcal{X}}|}{\delta}}$.
\end{lemma}
\begin{proof}
Let $\{\eta_i\}_{i\in [n]}$ be rollouts of length at most $\bar{l}$ generated running $\overline{\pi}$ from state $s$, and denote by $p_{\bar{l},g}^{\overline{\pi}}(s)$ the probability of reaching the goal $g$ in at most $\bar{l}$ steps by following policy $\overline{\pi}$ starting from $s$.
Let $\boldsymbol{1}(\eta) = 1$ if the goal has been reached in rollout $\eta$, zero otherwise.
$X_i = \boldsymbol{1}_g(\eta_i) - p_g^\pi(s)$ is a martingale difference sequence ($|X_i| \leq 1$) and by Azuma's inequality (see \pref{lem:azuma}), setting $n = 2^{10}\ln(\frac{2|{\mathcal{X}}|}{\delta})$, we have
\begin{equation}
\mathbb{P} \left( \forall s \in {\mathcal{X}}, \frac{1}{n}\abr{\sum_{i=1}^n X_i} \leq \frac{1}{16} \right) \geq 1 - \delta.
\label{eq:azuma.is}
\end{equation}
1) If $\norm{V^{\overline{\pi}}_g}_{\infty} \leq 4L$, by Markov's inequality, $p_{\bar{l},g}^{\overline{\pi}}(s) \geq 1/2$ when $\bar{l}= 8L$.
This gives $\frac{i_s}{n} = \sum_i \frac{\boldsymbol{1}_g(\eta_i)}{n} \geq p_g^{\overline{\pi}}(s) - \frac{1}{16}\geq\frac{7}{16}$ for any $s\in{\mathcal{X}}$, and thus the algorithm returns \textsc{True}\xspace on termination.
2) If the output is \textsc{True}\xspace, then $\frac{i_s}{n} \geq \frac{7}{16}$ for all $s \in{\mathcal{X}}$. By~\eqref{eq:azuma.is}, we have that $p_g^{\overline{\pi}}(s) \geq \frac{i_s}{n} - \frac{1}{16} \geq \frac{3}{8}$.
Thus for any $s\in{\mathcal{X}}$, $V^{\pi}_g(s)\leq 8L + \frac{5}{8}\norm{V^{\pi}_g}_{\infty}$, which gives $\norm{V^{\pi}_g}_{\infty}\leq 1+8L+\frac{5}{8}\norm{V^{\pi}_g}_{\infty}$ by $\pi\in\Pi({\mathcal{X}})$.
This implies $\norm{V^{\pi}_g}_{\infty}\leq 32L$.
\paragraph{Sample complexity.}
If $\norm{V^{\pi_s}_{s}}_{\infty} = \bigo{L}$ for any $s \in {\mathcal{X}}$, by \pref{lem:hitting}, with probability $1-\delta$, all trajectories generated by $\pi_s$ for some $s\in{\mathcal{X}}$ reaches state $s$ in $\bigo{L\ln(2n|{\mathcal{X}}|/\delta)}$ steps.
Noting that we generate $n$ trajectories for each $s\in{\mathcal{X}}$ completes the proof.
\end{proof}
\begin{lemma}
\label{lem:sc fillc}
For any ${\mathcal{X}}\subseteq{\mathcal{S}}$, $\Pi=\{\pi_x\}_{x\in{\mathcal{X}}}$, counter $n$, threshold $\bar{n}\geq 1$, and $\delta\in(0, 1)$, with probability at least $1-\delta$, the sample complexity of $\textsc{Explore}\xspace({\mathcal{X}},\Pi,n,\bar{n})$ is $\bigo{L|{\mathcal{X}}|A\bar{n}\ln\frac{|{\mathcal{X}}|A\bar{n}}{\delta}}$.
\end{lemma}
\begin{proof}
For any $x\in{\mathcal{X}}$, since $\|V^{\pi_x}_{x}\|_{\infty} = \bigo{L}$, by \pref{lem:hitting}, with probability $1-\delta'$ it takes $\bigo{L\ln(1/\delta')}$ steps to reach the goal state following $\pi_x$ from any $s \in {\mathcal{X}}$.
Therefore, by setting $\delta'=\frac{\delta}{|{\mathcal{X}}|A\bar{n}}$, with probability $1-\delta$, all trajectories reach the desired goal state within $\bigo{L\ln(1/\delta')}$ steps.
Given that there are at most $|{\mathcal{X}}|A\bar{n}$ trajectories, with probability at least $1-\delta$, the total sample complexity is
$\bigo{L|{\mathcal{X}}|A\bar{n}\ln\frac{|{\mathcal{X}}|A\bar{n}}{\delta}}$.
\end{proof}
\section{$\text{AX}_L$ through Layer Discovery}
\begin{algorithm2e*}[t]
\caption{Layer-Aware State Discovery (\textsc{LASD}\xspace)}
\label{alg:LOGSSD}
\DontPrintSemicolon
\LinesNumbered
\small
\KwIn{$L\geq1$, $\epsilon\in(0, 1]$, $\delta\in(0, 1)$.}
Let $\mathfrak{N}=\{2^j\}_{j\geq 0}$, ${\mathcal{K}}\leftarrow \varnothing, {\mathcal{U}} \leftarrow \varnothing$, ${\mathcal{K}}'\leftarrow \{s_0\}, \Pi_{{\mathcal{K}}} = \{{\widetilde{\pi}}_{s_0}\text{ a random policy}\}$, $\N(\cdot, \cdot)\leftarrow 0, \N(\cdot,\cdot,\cdot) \leftarrow 0$.\;
\For{round $r=1,\ldots$}{\label{line:round.easy}
$\epsilon_{\text{VI}\xspace}\leftarrow 1/\max\{16, \sum_{s,a}\N(s,a)\}$.\;
\tcc{Policy optimisation and goal selection}
Let $g^{\star}=\argmin_{g\in{\mathcal{U}}}\big\{V_{{\mathcal{K}},g}(s_0)\big\}$ where $(Q_{{\mathcal{K}},g}, V_{{\mathcal{K}},g}, \pi_g)=\textsc{VISGO}\xspace({\mathcal{K}}, g, \epsilon_{\text{VI}\xspace}, \N, \frac{\delta}{4r^2S^2})$ (see \pref{alg:VISGO}).\label{line:compute V.easy}\;
\eIf{$g^{\star}$ does not exist or $V_{{\mathcal{K}},g^{\star}}(s_0)>L$}{\label{line:goal condition.easy}
\tcc{Expand or Terminate}
\lIf{${\mathcal{K}}'=\varnothing$}{\textbf{return} ${\mathcal{K}}$ and $\Pi_{{\mathcal{K}}}$.}\label{line:terminate.easy}
Set ${\mathcal{K}}\leftarrow{\mathcal{K}}\cup{\mathcal{K}}'$, ${\mathcal{K}}'=\varnothing, {\mathcal{U}}=\varnothing$. \label{line:update.K.easy}\;
$(\_,{\mathcal{U}})\leftarrow\textsc{Explore}\xspace({\mathcal{K}},\Pi_{{\mathcal{K}}}, 0, 2L\log(4SALr^2/\delta))$ (see~\pref{alg:fillc}). \label{line:compute calU'.easy}\;
Set $n_{\min} \leftarrow N_0({\mathcal{K}},\frac{\delta}{4r^2S^2}) \lesssim L^2|{\mathcal{K}}|\ln(Sr/\delta)$ (defined in~\pref{lem:bounded error}).\;
$(\N,\_) \leftarrow \textsc{Explore}\xspace({\mathcal{K}},\Pi_{{\mathcal{K}}},\N,n_{\min})$. \label{line:fill N.easy}
}{
\tcc{Policy evaluation}
Let $\widehat{\tau}\leftarrow 0$, $\lambda\leftarrow N_{\textsc{Dev}}(32L, \frac{\epsilon}{256}, \frac{\delta}{4 r^2}) \lesssim \frac{1}{\epsilon^2} \ln^4\Big(\frac{Lr}{\epsilon \delta}\Big)$ (defined in \pref{lem:V pi mean}).\label{line:PE.easy}\;
\For{$j=1,\ldots,\lambda$}{\label{line:episode.easy}
$k\overset{+}{\leftarrow}1$, $i\leftarrow 1$, and reset to $s^k_1\leftarrow s_0$ by taking action $\textsc{RESET}$.\;
\While{$s^k_i\neq g^{\star}$}{
Take $a^k_i=\pi_{g^{\star}}(s^k_i)$, and transits to $s^k_{i+1}$.
Increase $\N(s^k_i, a^k_i)$, $\N(s^k_i, a^k_i, s^k_{i+1})$, and $i$ by $1$.\;
\lIf{ $\sum_{s,a}\N(s,a)\in\mathfrak{N}$ or ($s^k_i\in{\mathcal{K}}$ and $\N(s^k_i, a^k_i)\in\mathfrak{N}$)}{ return to \pref{line:round.easy} (skip round).\label{line:skip.easy}}
Set $\widehat{\tau}\overset{+}{\leftarrow} \frac{c(s^k_i, a^k_i)}{\lambda}$.
}
\lIf{$\widehat{\tau}>V_{{\mathcal{K}},g^{\star}}(s_0) + \epsilon L/2$}{ return to \pref{line:round.easy} (failure round). \label{line:failure.easy}}
}
${\mathcal{K}}'\leftarrow{\mathcal{K}}'\cup\{g^{\star}\}$, ${\mathcal{U}}\leftarrow{\mathcal{U}}\setminus\{g^{\star}\}$, $\Pi_{{\mathcal{K}}}=\Pi_{{\mathcal{K}}} \cup \{{\widetilde{\pi}}_{g^{\star}}:=\pi_{g^{\star}}\}$
(success round).
}
}
\end{algorithm2e*}
\pref{alg:LOGSSD} illustrates Layer-Aware State Discovery (\textsc{LASD}\xspace), a novel algorithm for $\text{AX}_L$ based on the iterative construction of $\calS^{\rightarrow}_L$ introduced in \pref{lem:SL.operator}.
In \pref{sec:pc}, we then introduce a policy consolidation procedure that achieves $\text{AX}^+$ when combined with \textsc{LASD}\xspace, leading to the $\textsc{LAE}\xspace$ algorithm.
\textsc{LASD}\xspace maintains a set ${\mathcal{K}}$ of ``known'' states, i.e., states for which a policy ${\widetilde{\pi}}_s \in \Pi({\mathcal{K}})$ with $V^{{\widetilde{\pi}}_s}_s(s_0) \leq L(1+\epsilon)$ has been learned. These policies are stored in $\Pi_{{\mathcal{K}}}$. The set ${\mathcal{K}}$ is updated only when the algorithm is confident enough to have identified a new layer. To this purpose, ${\mathcal{K}}'$ is used as a buffer for the new layer, i.e., for states that have been found to be $L$-controllable by policies restricted on ${\mathcal{K}}$ and that are waiting to be merged with ${\mathcal{K}}$. Finally, any other state discovered over time (and potential candidate to be in $\calS^{\rightarrow}_L$) is stored in ${\mathcal{U}}$.
At each round, \textsc{LASD}\xspace first uses the samples collected so far to compute an optimistic policy for each state in ${\mathcal{U}}$ through \textsc{VISGO}\xspace (\pref{alg:VISGO}), a slight variant of the state-of-the-art algorithm for exploration-exploitation in stochastic shortest paths~\citep{tarbouriech2021stochastic}, and it selects the state that is optimistically closer to $s_0$ as candidate goal $g^\star$.
If the optimistic distance of $g^\star$ from $s_0$ is larger than $L$, then no additional state can be confidently added to the current layer ${\mathcal{K}}'$ and a \emph{set expansion} round is triggered. \textsc{LASD}\xspace updates the set of known states by adding the new layer ${\mathcal{K}}'$ (${\mathcal{K}} = {\mathcal{K}} \cup {\mathcal{K}}'$) and starts a discovery process where policies in $\Pi_{{\mathcal{K}}}$ are used to reach all states in ${\mathcal{K}}$, then it executes all possible actions in these states, and it adds newly observed states to ${\mathcal{U}}$. Notice that the samples obtained during this process are not included in the policy improvement of \textsc{VISGO}\xspace to avoid statistical dependencies.
The sequence of expansion rounds is designed to approximate the sequence $\{\calK^{\star}_j\}_j$. With high probability, every update of ${\mathcal{K}}$ is not smaller than the application of ${\mathcal{T}}_L$, i.e., if, for some $j$, $\calK^{\star}_j\subseteq {\mathcal{K}} \not\supseteq \calK^{\star}_{j+1}$ before an update (this holds for $\calK^{\star}_1 = \{s_0\}$ at the first round), then $\calK^{\star}_{j+1} = {\mathcal{T}}_{L}(\calK^{\star}_{j})\subseteq {\mathcal{K}}$ after the update. Thus, ${\mathcal{K}}'$ is the increment to ${\mathcal{K}}$ to include the next layer. At the end of the expansion round \textsc{LASD}\xspace executes an additional exploration step to ensure that a minimum number of samples is available for each $(s,a) \in {\mathcal{K}} \times {\mathcal{A}}$ (see~\pref{line:fill N.easy}).
On the other hand, if the optimistic distance of $g^\star$ is smaller than $L$, \textsc{LASD}\xspace performs a \emph{policy evaluation} round by running $\pi_{g^{\star}}$ to estimate whether the current policy is indeed able to reach $g^\star$ in less than $L$ steps. If the number of visits to some state-action pair is doubled within the current round, then the current round is classified as a \emph{skip round}. If the test on the policy performance fails, then the current round is classified as a \emph{failure round}. In both cases, a new round is started.
Otherwise, the current round is classified as a success round and $g^{\star}$ is added to the new layer ${\mathcal{K}}'$. The samples collected in policy evaluation rounds are stored and used in all estimation and planning steps of the algorithm.
\textsc{LASD}\xspace terminates whenever the candidate goal $g^\star$ has an optimistic distance larger than $L$ and the new layer is empty, indicating that previous policy evaluation rounds could not identify any good policy and, thus, all states in $\calS^{\rightarrow}_L$ have been identified with high probability.
We prove that \textsc{LASD}\xspace achieves the following guarantee, the proof can be found in~\pref{app:sd and id}.
\begin{theorem}
\label{thm:sd}
Suppose ${\mathcal{S}}$ is finite. For any $L \geq 1$, $\epsilon \in (0,1]$ and $\delta \in (0,1)$, with probability at least $1-\delta$, \textsc{LASD}\xspace (\pref{alg:LOGSSD}) outputs a set ${\mathcal{K}}$ such that $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$
and $\Pi_{{\mathcal{K}}}$ such that $V^{\pi_g}_g(s_0)\leq L(1+\epsilon)$ for any $\pi_g \in \Pi_{{\mathcal{K}}}$, with sample complexity bounded by
\[
\resizebox{\columnwidth}{!}
$
\bigO{\frac{S^{\rightarrow}_{L(1+\epsilon)}\Gamma_{L(1+\epsilon)}AL}{\epsilon^2}\iota + \frac{{S^{\rightarrow}_{L(1+\epsilon)}}^2AL}{\epsilon}\iota + L^3 {S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota}
$}
\]
where $\iota =\ln^8\left( \frac{SAL}{\epsilon\delta} \right)$.
\end{theorem}
Compared to the lower bound (see~\pref{tab:summary}), \textsc{LASD}\xspace still suffers from an extra $\Gamma_{L(1+\epsilon)}$ dependence.
This is because in the analysis we use a Bernstein-like concentration inequality to control the deviation $(P- \P)V$, where $\P$ are the estimated transitions, for any value function $V$ restricted on ${\mathcal{K}}$ (i.e., $V$ is constant on all states outside ${\mathcal{K}}$). Unfortunately, we cannot leverage refined concentration inequalities since ${\mathcal{K}}$ is random and can take an exponentially large amount of values throughout the execution of \textsc{LASD}\xspace.
However, by inspecting the proof of \cite{cai2022near}, we note that the construction of the lower bound leverages a certain separation condition defined as follows.
\begin{assumption}[identifiability of $\{\calK^{\star}_j\}_j$]
\label{assum:id}
We say $\{\calK^{\star}_j\}_j$ is $\epsilon$-identifiable, if for any $j \geq 2, g\notin\calK^{\star}_j$, we have $V^{\star}_{\calK^{\star}_{j-1}, g}(s_0)>L(1+\epsilon)$.
\end{assumption}
This means that each layer $\calK^{\star}_j$ can be identified exactly by an algorithm run with accuracy $\epsilon$
since states that do not belong to the immediate next layer are clearly separated, i.e., they are more than $L(1+\epsilon)$-steps away. This leads to following remark.
\begin{remark}
\label{rem:id}
\pref{assum:id} implies that $\calS^{\rightarrow}_L=\calS^{\rightarrow}_{L(1+\epsilon)}$.
\end{remark}
The fact that states $g \notin \calK^{\star}_j$ are not reachable in $L(1+\epsilon)$ steps from $\calK^{\star}_{j-1}$ allows \textsc{LASD}\xspace to uniquely identify the layers. Indeed, under~\pref{assum:id}, \textsc{LASD}\xspace behaves as the operator ${\mathcal{T}}_L$ and, after each expansion, we have that ${\mathcal{K}} = \calK^{\star}_j$ for some $j \in [\calS^{\rightarrow}_L]$.
Thanks to this property, we can show that \textsc{LASD}\xspace is minimax optimal.\footnote{Minimax optimality holds for $\epsilon \leq \min\{1/S^{\rightarrow}_L, 1/L\}$, which makes the first term in \pref{thm:sd id} dominant \citep{cai2022near}.}
\begin{theorem}
\label{thm:sd id}
Suppose that ${\mathcal{S}}$ is finite.
For any $L \geq 1$, $\epsilon \in (0,1]$ and $\delta \in (0,1)$, if~\pref{assum:id} holds, with probability at least $1-\delta$, \textsc{LASD}\xspace (\pref{alg:LOGSSD}) outputs ${\mathcal{K}}=\calS^{\rightarrow}_{L(1+\epsilon)} = \calS^{\rightarrow}_{L}$ and $\Pi_{{\mathcal{K}}}$ such that $V^{\pi_g}_g(s_0)\leq L(1+\epsilon)$ for any $\pi_g \in \Pi_{{\mathcal{K}}}$, with sample complexity bounded by
%
\[
\bigO{\frac{S^{\rightarrow}_{L}AL}{\epsilon^2}\iota + \frac{{S^{\rightarrow}_{L}}^2AL}{\epsilon}\iota + L^3 {S^{\rightarrow}_{L}}^2A\iota},
\]
%
where $\iota =\ln^8\left( \frac{SAL}{\epsilon\delta} \right)$.
\end{theorem}
The trick to remove the $\Gamma_{L(1+\epsilon)}$ from \pref{thm:sd} is that, since layers are uniquely identified by the algorithm, we only need to concentrate the term $(P- \P)V$ for any value function in the set $\{V^\star_{\calK^{\star}_j}\}_{j \in [S^{\rightarrow}_{L}]}$.
\subsection{Proof Sketch}
Here we report a sketch of the proof, while the detailed one can be found in~\pref{app:logs}. All the statements we report here are to be considered to hold with high probability.
The first step of the proof (see~\pref{lem:calK.easy}) is to show by induction that, at each round, ${\mathcal{K}} \subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$. Thanks to the fact that $\tilo{L^2|{\mathcal{K}}|}$ samples are always available for each $(s,a) \in {\mathcal{K}} \times {\mathcal{A}}$ (\pref{line:fill N.easy}) and the properties of \textsc{VISGO}\xspace, it is possible to show that, for the goal $g^{\star}$ selected at the current round, $\|V^{\pi_{g^{\star}}}_{g^{\star}}\| \leq 2\|V^{\pi_{g^{\star}}}_{{\mathcal{K}},g^{\star}}\|\leq 4L$ if \pref{line:goal condition.easy} is passed.
Combining this with the properties of policy evaluation and the inductive hypothesis, we have that $\widehat{\tau}\geq L(1+\epsilon/2) \geq V^{\pi_{g^{\star}}}_{{\mathcal{K}},g^{\star}}(s_0) - L\epsilon/2$ if $g^{\star} \in {\mathcal{U}} \setminus \calS^{\rightarrow}_{L(1+\epsilon)}$. Thus a failure test is triggered and $g^{\star}$ is never added to ${\mathcal{K}}$. This shows that states outside $\calS^{\rightarrow}_{L(1+\epsilon)}$ are not added to ${\mathcal{K}}$.
By the same reasoning, we can show that if a goal $g^{\star}$ is added to ${\mathcal{K}}'$, the corresponding policy has bounded value function (important prerequisite for policy consolidation) and satisfies $\text{AX}_L$.
Furthermore, by properly selecting the number of rollouts in the expansion phase (\pref{line:compute calU'.easy}), we can show that ${\mathcal{U}}$ always contains at least those states that are reachable in $L$ steps from ${\mathcal{K}}$ (see~\pref{lem:calU.easy}), i.e., ${\mathcal{T}}_L({\mathcal{K}}) \setminus {\mathcal{K}} \subseteq {\mathcal{U}}$.
Combining these results with optimism restricted on $\calK^{\star}_j$ (see~\pref{lem:V calK.easy}), we are able to show (see~\pref{lem:update calK.easy}) that ${\mathcal{K}}$ always expands by at least one layer at each update. Formally, if $\calK^{\star}_j \subseteq {\mathcal{K}}$ at a certain update, then ${\mathcal{K}} \cup {\mathcal{K}}' \supseteq \calK^{\star}_{j+1}$ at the next update in~\pref{line:update.K.easy} (i.e., $\calK^{\star}_{j+1} = {\mathcal{T}}_L(\calK^{\star}_j) \subseteq {\mathcal{K}}$), see~\pref{lem:calK}.
If \pref{assum:id} holds, thanks to the identifiability of the layers, we show that ${\mathcal{K}} = {\mathcal{T}}_L(\calK^{\star}_{j}) = \calK^{\star}_{j+1}$, i.e., the algorithm replicates the ${\mathcal{T}}_L$ operator (see~\pref{lem:calK id}). In this case, ${\mathcal{K}}'$ is exactly the set of states needed to move from $\calK^{\star}_j$ to $\calK^{\star}_{j+1}$. By induction, we conclude that $\calS^{\rightarrow}_L \subseteq {\mathcal{K}}$ when the algorithm stops, ${\mathcal{K}} = \calS^{\rightarrow}_L$ with~\pref{assum:id}.
These results provide $\text{AX}_L$ guarantees when the algorithm stops. For computing the sample complexity we use a reduction to a regret analysis of a stochastic shortest path problem (SSP). We define the SSP regret as $R=\sum_{k=1}^K(I_k - V_k(s_0))$ where $K$ is the total number of episodes done in policy evaluation, $I_k$ is the length of episode $k$, and $V_k$ is the optimistic value function of the goal selected at episode $k$.
Then, $C_K = \sum_{k=1}^K I_k$ is the sample complexity of policy evaluation. Through the SSP regret analysis we can show that $R \lesssim c_1\sqrt{K} +c_2$ and $C_K \lesssim LK$, where $c_1 = L\sqrt{\Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}A}$ (resp. $c_1 = L\sqrt{S^{\rightarrow}_{L(1+\epsilon)}A}$ under~\pref{assum:id}) and $c_2 = L {S^{\rightarrow}_{L(1+\epsilon)}}^2A$, see~\pref{lem:regret.easy} and~\pref{lem:regret-improved.easy}. To conclude the analysis of the sample complexity we need to bound $K$.
We note that $K = r_{\text{tot}}\lambda\lesssim r_{\text{tot}}/\epsilon^2$ where $r_{\text{tot}}$ is the total number of rounds and $\lambda$ is the maximum number of episodes per round.
Moreover, $r_{\text{tot}} \lesssim \frac{c_1^2}{L^2} + \frac{c_2\epsilon}{L}$ can be controlled since the regret is sublinear (see~\pref{lem:bound r.easy}).
In the expansion phases we execute policies that reach any state $s \in {\mathcal{K}}$ almost surely since, as mentioned above, $\|V^{\pi_{s}}_{s}\| \leq 4L$. By~\citep[][Lemma 6]{rosenberg2020adversarial} we can bound the number of steps required to reach the goal by $8L$. Then, considering the number of samples that needs to be collected and that there are $\bigo{S^{\rightarrow}_{L(1+\epsilon)}}$ of such phases, the total sample complexity of the expansion phases is $\tilo{L^3 {S^{\rightarrow}_{L(1+\epsilon)}}^2A}$. Summing everything together concludes the proof (see~\pref{thm:sd.easy}).
\section{Policy Consolidation}
\label{sec:pc}
Algorithms introduced in previous sections discover a set ${\mathcal{K}}$ such that $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and a set of goal-conditioned policies satisfying $\text{AX}_L$.
In this section, we introduce an algorithm that takes a set ${\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ as input, and learns a set of goal-condition policies satisfying the $\text{AX}^+$ condition.
Our algorithm has a similar structure as \valve: the whole learning procedure is divided into rounds.
In each round, we randomly select an ``unknown'' goal state from ${\mathcal{L}}$ and compute a policy to reach it (\pref{line:}).
We then evaluate the performance of this policy by $\tilo{\frac{1}{\epsilon^2}}$ rollouts, and based on the evaluation result, the current round is classified into success, skip, or failure round similar to that in \pref{alg:LOGSSD}.
The crucial difference compared to \valve is the condition of success round (\pref{line:}), which has a form similar to $\text{AX}^+$.
Thus, one can treat \pref{alg:PC} as an improved version of \valve.
The guaranetee of \pref{alg:PC} is as follows.
\begin{theorem}
\label{thm:PC}
Given a target state space $\tset\subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ for some $\epsilon\in(0, 1)$ and a set of initial policy $\Pi'=\{\pi'_g\}_{g\in{\mathcal{K}}}$ such that $\norm{V^{\pi'_g}_g}_{\infty}\lesssim L$ for each $\pi'\in\Pi'$, with probability at least $1-\bigo{\delta}$, after collecting $\tilO{\frac{LS^{\rightarrow}_{L(1+\epsilon)}A\ln^p(S^{\rightarrow}_{L(1+\epsilon)})}{\epsilon^2}}$ (ignoring lower order terms) samples from the environment for some absolute constant $p\geq 1$, \pref{alg:PC} outputs a set of policies $\{{\widetilde{\pi}}_g\}_{g\in\tset}$ such that $V^{{\widetilde{\pi}}_g}_g(s_0)\leq V^{\star}_{\tset,g}(s_0)(1+\epsilon)$ for all $g\in\tset$.
\end{theorem}
We develop an improved regret-based analysis to bound the total sample complexity of \pref{alg:PC}.
The key technical contribution here is that instead of bounding the total number of rounds as in \valve, we directly bound the total number of steps in all rounds, which takes varying length of trajectories in different rounds into consideration.
This enables \pref{alg:PC} to achieve a better sample complexity compared to \valve.
Note that \pref{alg:PC} can be combined with any set discovery algorithm (such as \textsc{UcbExplore}\xspace and \textsc{DisCo}\xspace) and give an algorithm satisfying $\text{AX}^+$.
\subsection{Properties of ${\mathcal{U}}$}
Given ${\mathcal{X}}$, $\Pi_{{\mathcal{X}}}=\{\pi_g\}_{g\in{\mathcal{X}}}$ and $\delta$ as input of \add, let ${\mathcal{D}}_0$ and ${\mathcal{D}}_1$ be the random samples collected respectively in \pref{line:compute calU'.improved} and \pref{line:gather N'.improved}.
Define
\begin{align*}
{\mathcal{E}}_0({\mathcal{D}}_0) &= \left\{{\mathcal{N}}({\mathcal{X}}, \frac{1}{2L})\not\subseteq{\mathcal{U}}' \right\},\\
{\mathcal{E}}_1({\mathcal{D}}_0, {\mathcal{D}}_1) &= \left\{ \exists g \in {\mathcal{U}}', V'_{{\mathcal{X}},g}(s_0) > V^\star_{{\mathcal{X}}, g}(s_0)\right\},\\
{\mathcal{E}}_2({\mathcal{D}}_0, {\mathcal{D}}_1) &= \left\{ \exists g \in {\mathcal{U}}', V^{\pi_g}_g(s) > 2 V'_{{\mathcal{X}},g}(s) \right\}.
\end{align*}
In this section we use $\mathbb{E}$ and $\mathbb{P}$ to denote expectation and probability w.r.t.\ these two random generation processes.
\begin{lemma}
\label{lem:calU each}
With any ${\mathcal{X}}$, $\{\pi_g \in \Pi({\mathcal{X}})\}_{g \in {\mathcal{X}}}$ such that $\norm{V^{\pi_g}_g}_{\infty} = \bigo{L}$, and $\delta \in (0,1)$ as input, \add ensures
\[
\mathbb{P}\rbr{\rS{{\mathcal{X}}}{L}\setminus {\mathcal{X}} \subseteq {\mathcal{U}} \subseteq \rS{{\mathcal{X}}}{2L}\setminus{\mathcal{X}} } \geq 1-\delta.
\]
With the same probability, the sample complexity of \add is bounded by $\bigo{L^3|{\mathcal{X}}|^2A\ln^2\frac{L|{\mathcal{X}}|A}{\delta}}$.
\end{lemma}
\begin{proof}
Denote by $\{s_{i,s,a}\}_{i,s,a}$ the set of next state samples collected in \pref{line:compute calU'.improved} for each $(s, a)$.
Let $\mu = 2L\ln(4LA|{\mathcal{X}}|/\delta)$, then
\begin{align*}
\mathbb{P}\left( {\mathcal{E}}_0({\mathcal{D}}_0)\right) &= P\left(\exists s' \in {\mathcal{N}}({\mathcal{X}}, \frac{1}{2L}), \forall (s,a) \in {\mathcal{X}} \times {\mathcal{A}}, \forall i \in [\mu]: s_{i,s,a} \neq s'\right) \\
&\leq \sum_{s' \in {\mathcal{N}}({\mathcal{X}}, \frac{1}{2L})} P\left(\forall (s,a) \in {\mathcal{X}} \times {\mathcal{A}}, \forall i \in [\mu]: s_{i,s,a} \neq s'\right)\\
& \leq \sum_{s' \in {\mathcal{N}}({\mathcal{X}}, \frac{1}{2L})} \prod_{(s,a) \in {\mathcal{X}} \times {\mathcal{A}}} \prod_{i \in [\mu]} (1-P(s'|s,a)) \leq \sum_{s' \in {\mathcal{N}}({\mathcal{X}}, \frac{1}{2L})} \left(1-P(s'|\bar{s},\bar{a})\right)^{\mu} \tag{$\bar s, \bar a$ such that $P(s'|\bar s, \bar a)\geq\frac{1}{2L}$}\\
& \leq \sum_{s' \in {\mathcal{N}}({\mathcal{X}}, \frac{1}{2L})} \left(1-\frac{1}{2L}\right)^{\mu} \leq \sum_{s' \in {\mathcal{N}}({\mathcal{X}}, \frac{1}{2L})}\frac{\delta}{4LA|{\mathcal{X}}|} \leq \delta/2. \tag{$|{\mathcal{N}}({\mathcal{X}},\frac{1}{2L})|\leq 2LA|{\mathcal{X}}|$}
\end{align*}
Let $N_1$ be defined as in \pref{lem:bounded error fresh}. Then, from \pref{lem:opt} and \pref{lem:bounded error fresh}, by using $\delta/(4|{\mathcal{U}}'|)$, we have that $\mathbb{P}\left( {\mathcal{E}}_1({\mathcal{D}}_0,{\mathcal{D}}_1) | {\mathcal{D}}_0\right) \leq \delta/4$ and $\mathbb{P}\left( {\mathcal{E}}_2({\mathcal{D}}_0,{\mathcal{D}}_1) | {\mathcal{D}}_0\right) \leq \delta/4$.
Then, we can write that
\begin{align*}
\mathbb{P}({\mathcal{E}}_0({\mathcal{D}}_0) \cup {\mathcal{E}}_1({\mathcal{D}}_0,{\mathcal{D}}_1)\cup {\mathcal{E}}_2({\mathcal{D}}_0,{\mathcal{D}}_1))
&\leq
\mathbb{P}({\mathcal{E}}_0({\mathcal{D}}_0)) + \mathbb{P}({\mathcal{E}}_1({\mathcal{D}}_0,{\mathcal{D}}_1) \cup {\mathcal{E}}_2({\mathcal{D}}_0,{\mathcal{D}}_1))\\
&\leq \delta/2 + \sum_{{\mathcal{D}}_0} \mathbb{P}({\mathcal{D}}_0) \underbrace{\mathbb{P}({\mathcal{E}}_1({\mathcal{D}}_0,{\mathcal{D}}_1)\cup {\mathcal{E}}_2({\mathcal{D}}_0,{\mathcal{D}}_1) | {\mathcal{D}}_0)}_{\leq \delta/2, \forall {\mathcal{D}}_0} = \delta
\end{align*}
We then carry out the proof under event $E = \neg ({\mathcal{E}}_1({\mathcal{D}}_0)\cup {\mathcal{E}}_1({\mathcal{D}}_0,{\mathcal{D}}_1)\cup {\mathcal{E}}_2({\mathcal{D}}_0,{\mathcal{D}}_1))$ which hold with probability $1-\delta$.
Since $\pi'_g$ is restricted on ${\mathcal{X}}$, we have that $V^\star_{{\mathcal{X}},g}(s_0) \leq V^{\pi'_g}_g(s_0)$ by the definition of optimal policy.
We have that, for any $g\in{\mathcal{U}}$, $V^{\star}_{{\mathcal{X}},g}(s_0)\leq V^{\pi'_g}_g(s_0)\leq 2V'_{{\mathcal{X}},g}(s_0)\leq 2L$ by the definition of ${\mathcal{U}}$. This implies that ${\mathcal{U}} \subseteq \rS{{\mathcal{X}}}{2L}\cap{\mathcal{U}}'\subseteq \rS{{\mathcal{X}}}{2L}\setminus{\mathcal{X}}$ since ${\mathcal{U}}' \cap {\mathcal{X}} = \emptyset$ by definition.
Finally, note that, by the definition of $\rS{{\mathcal{X}}}{L}$ and the event $\neg {\mathcal{E}}_0$, $\rS{{\mathcal{X}}}{L}\setminus {\mathcal{X}} \subseteq {\mathcal{N}}({\mathcal{X}}, \frac{1}{2L}) \subseteq {\mathcal{U}}'$ w.h.p. Furthermore, under the event $\neg {\mathcal{E}}_1({\mathcal{D}}_0,{\mathcal{D}}_1)$, we have that for any $g \in {\mathcal{U}}'$, if $V^\star_{{\mathcal{X}},g}(s_0) \leq L$, then $V'_{{\mathcal{X}},g}(s_0) \leq V^\star_{{\mathcal{X}},g}(s_0) \leq L$. Thus, $\rS{{\mathcal{X}}}{L}\setminus {\mathcal{X}} \subseteq {\mathcal{U}}$.
\paragraph{Sample complexity.} Since $\|V^{\pi_g}_{g}\|_{\infty} = \bigo{L}$, by \pref{lem:sc fillc} with $\bar{n}=\mu$ and $N_1(|{\mathcal{X}}|, \frac{\delta}{4|{\mathcal{U}}'|})$, with probability at least $1-\delta$, the sample complexity is $\bigo{L|{\mathcal{X}}|An'\ln\frac{|{\mathcal{X}}|An'}{\delta}}$,
where $n'=\mu+N_1(|{\mathcal{X}}|, \delta/(4|{\mathcal{U}}'|)$.
Given that $N_1(|{\mathcal{X}}|, \frac{\delta}{4|{\mathcal{U}}'|}) = \bigo{L^2 |{\mathcal{X}}| \ln(|{\mathcal{U}}'||{\mathcal{X}}|/\delta)}$ (see \pref{lem:bounded error fresh}), we have $n'=\bigo{L^2 |{\mathcal{X}}| \ln(L|{\mathcal{X}}|A/\delta)}$.
Plugging this back, the sample complexity is $\bigo{L^3|{\mathcal{X}}|^2A\ln^2\frac{L|{\mathcal{X}}|A}{\delta}}$.
\end{proof}
\begin{lemma}
\label{lem:calU}
With probability at least $1-\delta$ over the randomness of \pref{alg:SD}, at any trial and round, $\rS{{\mathcal{K}}}{L}\setminus {\mathcal{K}} \subseteq {\mathcal{U}} \subseteq \rS{{\mathcal{K}}}{2L}\setminus{\mathcal{K}}$ after executing \pref{line:add} (if it is executed).
\end{lemma}
\begin{proof}
This is simply by \pref{lem:calU each} and the choice of confidence level in \pref{line:add} in each trial and round.
\end{proof}
\section{Improved Algorithms}
In this section, we present two improvements to \textsc{LASD}\xspace that allow to \emph{i)} replace the $\ln(S)$ dependence with a much milder $\ln(\calS^{\rightarrow}_{L(1+\epsilon)})$; \emph{ii)} move from $\text{AX}_L$ to $\text{AX}^+$.
\subsection{Log-Adaptivity to $\calS^{\rightarrow}_{L(1+\epsilon)}$}\label{sec:log.adaptivity}
Inspired by intrinsically motivated learning agents, \citet{lim2012autonomous} originally focused on a learning scenario where the environment is possibly infinite or at least no prior knowledge about it is available. Unfortunately, all the existing algorithms fail in dealing with this scenario since they require prior knowledge of the cardinality of the state space ${\mathcal{S}}$. While the sample complexity only depends logarithmically on $S$, this shows that inability of the algorithms to exclusively focus on the portion of environment discovered and consolidated over time and it thus prevents from dealing with arbitrarily large or infinite environments.
In this section, we carefully identify all the aspects of the algorithm causing this problem in \textsc{LASD}\xspace, and propose an improved algorithm \textsc{LASD}\xspace{}\textsuperscript{+}\xspace (\pref{alg:SD} in \pref{app:logsa}) that replaces the $\ln(S)$ dependency by $\ln(S^{\rightarrow}_{L(1+\epsilon)})$. This is a much favorable dependency since $S^{\rightarrow}_{L(1+\epsilon)}$ is finite even when ${\mathcal{S}}$ is countably infinite~\citep[][Prop. 6]{lim2012autonomous}.
Below we list each source of $\ln(S)$ dependency and the corresponding modification to fix it.
\paragraph{A) Limiting the set of candidate goals.} In the expansion phase, \textsc{LASD}\xspace uses all the newly discovered states to build the set ${\mathcal{U}}$ of candidates states for $\calS^{\rightarrow}_L$. This phase could potentially discover any state $s\in{\mathcal{S}}$ as long as the transition probability to $s$ from ${\mathcal{K}}$ is non-zero. This means that any $s \in {\mathcal{S}}$ can be considered in the goal selection step (\pref{line:compute V.easy}), requiring a union bound over ${\mathcal{S}}$ when analyzing the concentration of the estimated value functions. To overcome this issue, \textsc{LASD}\xspace{}\textsuperscript{+}\xspace performs a step of state filtering in the construction of ${\mathcal{U}}$ (\pref{alg:SD}-\pref{line:filter calU'.improved}).\footnote{A similar filter is used in \textsc{DisCo}\xspace to reduce computational complexity, but as it does not use fresh samples, it still requires a union bound over ${\mathcal{S}}$ to deal with statistical dependencies.} The idea is to include in ${\mathcal{U}}$ only goal states with estimated hitting time upper bounded by $L$. To break statistical dependencies we estimate the hitting time of each candidate goal state using fresh samples (i.e., samples that are discarded after this step). It can be showed (see \pref{lem:bcalU}) that using this filtering scheme, ${\mathcal{U}}$ only includes states that are $\bigo{L}$-controllable by policies restricted on ${\mathcal{K}}$, which is a much smaller candidate set of order $S^{\rightarrow}_{L(1+\epsilon)}$.
\paragraph{B) Scaling the confidence bounds.}
While the state filtering step allows to consider only states in $\calS^{\rightarrow}_{L(1+\epsilon)}$ rather than ${\mathcal{S}}$, the knowledge of $S^{\rightarrow}_{L(1+\epsilon)}$ is required to properly set the confidence level when computing the estimated value functions (\pref{alg:SD}-\pref{line:compute V.improved}). We thus maintain an estimate $z$ of $S^{\rightarrow}_{L(1+\epsilon)}$. Each attempt on a specific value of $z$ is a trial indexed by $\tau$ (\pref{alg:SD}-\pref{line:size.improved}) that ends when the total number of ``known'' states ($|{\mathcal{K}} \cup {\mathcal{K}}'|$) exceeds the estimated dimension $z$ (\pref{alg:SD}-\pref{line:z.improved}). In this case, we double the value of $z$. We can show (see~\pref{lem:bound z}) that the total number of trials is bounded $\tau \lesssim \log_2(S^{\rightarrow}_{L(1+\epsilon)})$ and $z\lesssim S^{\rightarrow}_{L(1+\epsilon)}$.
\paragraph{C) Controlling the policy quality.}
An important step in \textsc{LASD}\xspace is to gather a minimum number of samples for each ``known'' state (\pref{line:fill N.easy}) to ensure a reasonable performance of the policy being evaluated. The right number of samples also depends on $S^{\rightarrow}_{L(1+\epsilon)}$. Unfortunately, we cannot leverage $z$ to compute this threshold since $z$ is likely to be smaller than $S^{\rightarrow}_{L(1+\epsilon)}$ throughout the execution of the algorithm.
Using $z$ will invalidate the properties of policy evaluation that may lead to halt prematurely, without satisfying the $\text{AX}$ properties (e.g., $\calS^{\rightarrow}_L \subseteq {\mathcal{K}}$). This failure mode is not captured by the condition used in~\pref{alg:SD}-\pref{line:z.improved} to increase $z$.
We thus introduce a Monte-Carlo reachability test (\pref{alg:SD}-\pref{line:rtest.improved}) before policy evaluation. Intuitively, if the test fails \textsc{LASD}\xspace{}\textsuperscript{+}\xspace gathers new samples to improve the estimate of the MDP, otherwise the test guarantees that $\|V^{\pi_{g^{\star}}}_{g^{\star}}\|_{\infty} \lesssim L$ (see~\pref{lem:rtest}).
Combining these three changes, we are able to obtain the following sample complexity guarantee (see~\pref{app:sd id improved}), which is $S$-independent.
\begin{theorem}
\label{thm:sd id improved}
For any $L \geq 1$, $\epsilon \in (0,1]$ and $\delta \in (0,1)$, with probability at least $1-\delta$, \textsc{LASD}\xspace{}\textsuperscript{+}\xspace (\pref{alg:SD}) outputs $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and $\Pi_{{\mathcal{K}}}$ such that $V^{\pi_g}_g(s_0)\leq L(1+\epsilon)$ for any $\pi_g \in \Pi_{{\mathcal{K}}}$, with sample complexity bounded by
\[
\bigO{\frac{LMA\iota}{\epsilon^2} +\frac{LS^{\rightarrow}_{L(1+\epsilon)}A\iota}{\epsilon}+ L^3{S^{\rightarrow}_{L(1+\epsilon)}}^3A\iota},
\]
where $\iota =\ln^{12}(\frac{S^{\rightarrow}_{L(1+\epsilon)}AL}{\epsilon\delta})$ and $M = \Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}$. If~\pref{assum:id} holds, then $M = S^{\rightarrow}_{L}$ and $S^{\rightarrow}_{L(1+\epsilon)} = S^{\rightarrow}_L$.
\end{theorem}
\begin{algorithm2e}[t]
\caption{Policy Consolidation (\textsc{PC}\xspace)}
\label{alg:PC}
\DontPrintSemicolon
\LinesNumbered
\small
\KwIn{$L\geq 1$, $\epsilon\in(0,1]$, $\delta\in(0, 1)$, target state space $\tset\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$, and initial policies $\Pi'=\{\pi'_g\}_{g\in\tset}$.}
Set $k\leftarrow 1$, $\mathfrak{N}=\{2^j\}_{j\geq 0}$, ${\mathcal{L}}=\tset$, $\Pi^+_{{\mathcal{K}}}= \{{\widetilde{\pi}}_{s_0} \text{ a random policy}\}$, $\N(\cdot, \cdot), \N(\cdot,\cdot,\cdot) \leftarrow 0$.\;
$(\N,\_)\leftarrow\textsc{Explore}\xspace({\mathcal{K}}, \Pi', \N, N_1(|\tset|-1, \frac{\delta}{|\tset|}))$ (see \pref{alg:fillc}; $N_1 \lesssim L^2|{\mathcal{K}}|\ln(\frac{|{\mathcal{K}}|}{\delta})$ is defined in \pref{lem:bounded error fresh}).\label{line:nu}\;
\For{$r=1,\ldots$}{\label{line:round mge}
\lIf{${\mathcal{L}}=\varnothing$}{
\textbf{return} $\Pi^+_{{\mathcal{K}}}$.
}
$\epsilon_{\text{VI}\xspace}\leftarrow 1/\max\{16, \sum_{s,a}\N(s,a)\}$.\;
Pick $g^{\star}\in{\mathcal{L}}$ arbitrarily and compute $(\widehat{Q}, \widehat{V}, \widehat{\pi})=\textsc{VISGO}\xspace(\tset\setminus\{g\}, g, \epsilon_{\text{VI}\xspace}, \N, \frac{\delta}{|\tset|})$.
\label{line:pc.goal.selection}\;
Let $\lambda\leftarrow N_{\textsc{Dev}}(32L, \frac{\epsilon}{256}, \frac{\delta}{2r^2}) \lesssim \frac{1}{\epsilon^2} \ln^4\left(\frac{Lr}{\epsilon\delta}\right)$ (defined in \pref{lem:V pi mean}) and $\widehat{\tau}\leftarrow 0$.\label{line:PE PC}\;
\For{$j=1,\ldots,\lambda$}{\label{line:episode PC}
$k\overset{+}{\leftarrow}1$, $i\leftarrow 1$, and reset to $s^k_1\leftarrow s_0$ by taking action $\textsc{RESET}$.\;
\While{$s^k_i\neq g^{\star}$}{
Take $a^k_i=\widehat{\pi}(s^k_i)$, and transits to $s^k_{i+1}$.\;
Increase $\N(s^k_i, a^k_i)$, $\N(s^k_i, a^k_i, s^k_{i+1})$, and $i$ by $1$.\;
\lIf{$\sum_{s,a}\N(s,a)\in\mathfrak{N}$ or ($s^k_i\in{\mathcal{K}}$ and $\N(s^k_i, a^k_i)\in\mathfrak{N}$)}{ return to \pref{line:round mge} (skip round).}\label{line:skip PC}
Set $\widehat{\tau}\overset{+}{\leftarrow} \frac{c(s^k_i, a^k_i)}{\lambda}$.
}
\lIf{$\widehat{\tau}>\widehat{V}(s_0)(1+\epsilon/2)$}{ return to \pref{line:round mge} (failure round).}
}
${\mathcal{L}}\leftarrow{\mathcal{L}}\setminus\{g^{\star}\}$, $\Pi^+_{{\mathcal{K}}} \leftarrow \Pi^+_{{\mathcal{K}}} \cup \{{\widetilde{\pi}}_{g^{\star}}= \widehat{\pi}\}$ (success round).
}
\end{algorithm2e}
\subsection{Policy Consolidation}\label{sec:pc}
Both \textsc{LASD}\xspace and \textsc{LASD}\xspace{}\textsuperscript{+}\xspace discover a set ${\mathcal{K}}$ such that $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and a set of goal-conditioned policies satisfying $\text{AX}_L$. We now introduce a procedure that, given a set ${\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and associated goal-reaching policies $\Pi_{{\mathcal{K}}}$ with bounded value function, learns a set of goal-condition policies satisfying the $\text{AX}^+$ condition.
\textsc{PolicyConsolidation}\xspace (\pref{alg:PC}) is an algorithm for Multi-Goal Exploration (MGE)~\citep[e.g.,][]{tarbouriech2022adaptive} over ${\mathcal{K}}$.
In each round, \textsc{PolicyConsolidation}\xspace randomly selects an ``unknown'' goal state from ${\mathcal{L}}$ and computes a policy to reach it (\pref{line:pc.goal.selection}).
It then evaluates the performance of this policy by $\tilo{\frac{1}{\epsilon^2}}$ rollouts, and based on the evaluation result, the current round is classified into success, skip, or failure round similar to that in \pref{alg:LOGSSD}.
While it shares a similar structure with \textsc{VALAE}\xspace, the crucial difference is the condition of success round (\pref{line:round mge}), which has a form similar to $\text{AX}^+$.
Thus, one can consider \pref{alg:PC} as an improved version of \textsc{VALAE}\xspace.
Its simplicity and high sample efficiency, allow \textsc{PolicyConsolidation}\xspace to be integrated with any existing algorithm for $\text{AX}_L$ or $\text{AX}^\star$ at no cost. As showed in the following lemma, the sample complexity of policy consolidation matches the lower-bound for $\text{AX}$, thus providing a ``minor'' contribution to the overall sample complexity.
Details are deferred to \pref{app:consolidation}.
\begin{theorem}
\label{thm:PC}
Given a target state space $\tset\subseteq \calS^{\rightarrow}_{L(1+\epsilon)}$ for some $\epsilon\in(0, 1)$ and a set of initial policies $\Pi'=\{\pi'_g\}_{g\in{\mathcal{K}}}$ such that $\norm{V^{\pi'_g}_g}_{\infty}\lesssim L$, with probability at least $1-\delta$,
\textsc{PolicyConsolidation} (\pref{alg:PC}) outputs a set of policies $\{{\widetilde{\pi}}_g\}_{g\in\tset}$ such that $V^{{\widetilde{\pi}}_g}_g(s_0)\leq V^{\star}_{\tset,g}(s_0)(1+\epsilon)$ for all $g\in\tset$, with sample complexity bounded by
\[
\tilO{\frac{LS^{\rightarrow}_{L(1+\epsilon)}A\iota}{\epsilon^2} + \frac{L{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota}{\epsilon} + L^3{S^{\rightarrow}_{L(1+\epsilon)}}^2A\iota},
\]
where $\iota=\ln^{10}(\frac{S^{\rightarrow}_{L(1+\epsilon)}AL}{\epsilon\delta})$.
\end{theorem}
To achieve this result we developed an improved regret-based analysis. Instead of bounding the total number of rounds as in \textsc{VALAE}\xspace, we directly bound the total number of steps in all rounds, which takes varying length of trajectories in different rounds into consideration.
This enables \textsc{PolicyConsolidation} to achieve a better guarantee on the performance of the learned policies compared to \textsc{VALAE}\xspace, preserving the same sample complexity.
\subsection{$AX^+$ through Layer Discovery and Consolidation}
We combine all these improvement into Layered Autonomous Exploration (\textsc{LAE}\xspace) whose pseudo code is reported in~\pref{alg:ax.plus}. Combining the previous results, we can state the following guarantee for $\text{AX}^+$.
\begin{corollary}
\label{cor:sd id}
For any $L \geq 1$, $\epsilon \in (0,1]$ and $\delta \in (0,1)$, with probability at least $1-\delta$, \textsc{LAE}\xspace (\pref{alg:ax.plus}) outputs $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ and $\Pi_{{\mathcal{K}}}$ such that $V^{\pi_g}_g(s_0)\leq V_{{\mathcal{K}},g}^\star(s_0)(1+\epsilon)$, for any $\pi_g \in \Pi_{{\mathcal{K}}}$, with sample complexity
\[
\bigO{\frac{LMA\iota}{\epsilon^2} +\frac{LS^{\rightarrow}_{L(1+\epsilon)}A\iota}{\epsilon}+ L^3{S^{\rightarrow}_{L(1+\epsilon)}}^3A\iota}
\]
where $\iota=\ln^{12}\rbr{\frac{S^{\rightarrow}_{L(1+\epsilon)}AL}{\epsilon\delta}}$ and $M = \Gamma_{L(1+\epsilon)}S^{\rightarrow}_{L(1+\epsilon)}$. If~\pref{assum:id} holds, then $M = S^{\rightarrow}_{L}$ and $S^{\rightarrow}_{L(1+\epsilon)} = S^{\rightarrow}_{L}$.
\end{corollary}
This shows that \textsc{LAE}\xspace is the first algorithm able to i) achieve the strongest performance $\text{AX}^+ \Rightarrow \text{AX}^\star \Rightarrow \text{AX}_L$, ii) match the lower-bound under certain settings, and iii) completely remove the dependence on $S$. In particular, the latter was an open problem since the initial work by~\citet{lim2012autonomous}.\footnote{\textsc{UcbExplore}\xspace originally considered a countable, possibly infinite state space; however this leads to a
technical issue in the analysis~\citep[][Footnote 2]{tarbouriech2020improved}.}
\textbf{Comparisons.}
\textsc{LASD}\xspace/\textsc{LASD}\xspace{}\textsuperscript{+}\xspace shares similarities with both \textsc{UcbExplore}\xspace and \textsc{VALAE}\xspace. While we leverage the same condition as in \textsc{VALAE}\xspace for the failure test of policy evaluation, the policy evaluation in \textsc{VALAE}\xspace is only for learning goal-conditioned policies and not for consolidating states. In fact, they first run \textsc{DisCo}\xspace for state discovery, and then learn goal-conditioned policies on a potentially much larger set subsuming $\calS^{\rightarrow}_{2L}$. However, $\calS^{\rightarrow}_{2L}$ can be exponentially larger than $S^{\rightarrow}_{L(1+\epsilon)}$ (see \pref{lem:example 2L}) in general and thus the sample complexity of \textsc{VALAE}\xspace is incomparable to other algorithms.
Therefore, \textsc{VALAE}\xspace only improves the sample complexity of policy learning but not that of state discovery.
Similarly to \textsc{UcbExplore}\xspace, we perform state and policy identification simultaneously. Our evaluation phase is much more sample efficient compared to \textsc{UcbExplore}\xspace, which saves a $L^2/\epsilon$ factor in the leading-order term.
Compared to \textsc{DisCo}\xspace, our algorithm saves a $L^2$ factor by i) adaptively collecting samples to estimate state values instead of prescribing a fixed number of samples to guarantee a uniformly-accurate transition estimate over ${\mathcal{K}}$,
and ii) leveraging variance information.
The tool enabling all these improvements is a new Bernstein-type concentration inequality for restricted value functions (see \pref{lem:dPV}). The key difficulty in our analysis is that the set on which value functions are restricted is random since we learn ${\mathcal{K}}$ and $\Pi_{{\mathcal{K}}}$ simultaneously. In comparison, in \textsc{VALAE}\xspace the set ${\mathcal{K}}$ is fixed after the initial phase of state discovery, which makes the analysis much simpler.
Specifically, leveraging the fact that the learned goal-conditioned policies are all restricted on $\calS^{\rightarrow}_{L(1+\epsilon)}$, we are able to make use of the variance information without incurring a polynomial dependency on $S$.
\begin{algorithm2e}[t]
\caption{Layered Autonomous Exploration (\textsc{LAE}\xspace)}
\label{alg:ax.plus}
\DontPrintSemicolon
\LinesNumbered
\small
\KwIn{$L\geq 1$, $\epsilon\in(0,1]$, and $\delta\in(0, 1)$.}
$({\mathcal{K}}, \Pi_{{\mathcal{K}}}^L) = \textsc{LASD}\xspace{}\textsuperscript{+}\xspace\big(L, \epsilon, \delta \big)$ see \pref{alg:SD} in appendix (or \textsc{LASD}\xspace for $\log S$).\tcp*{$\text{AX}_L$}
$\Pi^+_{{\mathcal{K}}} = \textsc{PC}\xspace\big(L, \epsilon, \delta, {\mathcal{K}}, \Pi_{{\mathcal{K}}}^L\big)$.\tcp*{$\text{AX}^+$}
\Return{${\mathcal{K}}$ and $\Pi^+_{{\mathcal{K}}}$.}
\end{algorithm2e}
\section{Preliminaries}
We consider a reward-free Markov Decision Process ${\mathcal{M}}=({\mathcal{S}}, {\mathcal{A}}, s_0, P)$, where ${\mathcal{S}}$ is a countable state space,
${\mathcal{A}}$ is a finite action space, $s_0$ is the initial state, and $P=\{P_{s, a}\}_{(s, a)\in{\mathcal{S}}\times{\mathcal{A}}}$ with $P_{s,a}\in\Delta_{{\mathcal{S}}}$ is the transition function, where $\Delta_{{\mathcal{S}}}$ is the simplex over ${\mathcal{S}}$.
In a general MDP, the learner may get stuck in undesirable states and be unable to return to $s_0$.
To avoid this issue, we make the following assumption.
\begin{assumption}
The action space contains a $\textsc{RESET}$ action such that $P_{s, \textsc{RESET}}(s_0)=1$ for all $s\in{\mathcal{S}}$.
\end{assumption}
A deterministic stationary policy $\pi\in{\mathcal{A}}^{{\mathcal{S}}}$ is a mapping that assigns an action $\pi(s)$ to each state $s$, and we define $\Pi={\mathcal{A}}^{{\mathcal{S}}}$ as the set of all policies.
To explicitly characterize the behavior of a policy, we say a policy $\pi$ is \emph{restricted} on ${\mathcal{X}}\subseteq{\mathcal{S}}$ if $\pi(s)=\textsc{RESET}$ for any $s\notin{\mathcal{X}}$, and we denote by $\Pi({\mathcal{X}})$ the set of policies restricted on ${\mathcal{X}}$.
We measure the performance of a policy in navigating the MDP as follows.
For any policy $\pi \in \Pi$ and a pair of states $(s, g) \in \mathcal{S}^2$, let $V^{\pi}_g(s) \in [0, + \infty]$ be the expected number of steps it takes to reach $g$ (that is, the \emph{hitting time} of $g$) starting from $s$ when executing policy $\pi$, that is,
\begin{align*}
V^{\pi}_g(s) &\triangleq \mathbb{E}^\pi\sbr{\left. \omega_g \right|s_1=s}, \\
\omega_g &\triangleq \inf \cbr{ i \geq 0: s_{i+1} = g }.
\end{align*}
Note that $V^{\pi}_g(s) = +\infty$ if $g$ is unreachable by playing $\pi$ starting from $s$.
For any subset ${\mathcal{X}} \subseteq \mathcal{S}$
and any goal state $g$, define $V^{\star}_{{\mathcal{X}}, g}(s) = \min_{\pi \in \Pi({\mathcal{X}})} V^{\pi}_g(s)$ as the minimum hitting time of $g$ following a policy restricted on ${\mathcal{X}}$. Note that, if ${\mathcal{X}}\subseteq{\mathcal{X}}'$, then $V^{\star}_{{\mathcal{X}}',g}(s)\leq V^{\star}_{{\mathcal{X}},g}(s)$ for any $s,g\in{\mathcal{S}}$.
The objective of the learner is to efficiently navigate in the vicinity of $s_0$.
A state $s$ is \textit{$L$-controllable} if there exists a policy $\pi$ such that $V^{\pi}_s(s_0)\leq L$.
While discovering all $L$-controllable states may be a reasonable objective for exploring the vicinity of $s_0$~\citep{tarbouriech2022adaptive}, \citet{lim2012autonomous} showed that this may still require the learner to explore the whole state space, since reaching a $L$-controllable state may require navigating through non-$L$-controllable states.
To this end, \citet{lim2012autonomous} propose to only focus on navigating among \textit{incrementally $L$-controllable states}: states that are $L$-controllable by policies restricted on other incrementally controllable states.
\begin{definition}[Incrementally $L$-controllable states $\mathcal{S}_L^{\rightarrow}$]\label{def:sl.original}
Given a partial order $\prec$ on ${\mathcal{S}}$, we define ${\mathcal{S}}_L^{\prec}$ recursively as 1) $s_0\in\mathcal{S}_L^{\prec}$ and 2) if there exists a policy $\pi \in \Pi\big(\{ s' \in \mathcal{S}_L^{\prec}: s' \prec s \}\big)$ with $V^{\pi}_s(s_0) \leq L$, then $s \in \mathcal{S}_L^{\prec}$. The set $\calS^{\rightarrow}_L$ of incrementally $L$-controllable states is defined as $\calS^{\rightarrow}_L \triangleq \cup_{\prec} \mathcal{S}_L^{\prec}$, where the union is over all partial orders.
\end{definition}
Instead of exploring the potentially infinite state space, the objective of the learner is to discover the \emph{finite} set $\calS^{\rightarrow}_L$~\citep[][Prop. 6]{lim2012autonomous} and learn a corresponding set of policies that reliably reach each state in $\calS^{\rightarrow}_L$. We introduce three different formulations of the objective.
\begin{definition}[AX sample complexity]\label{def:ax}
For any given length $L\geq 1$, error threshold $\epsilon>0$, and confidence level $\delta\in(0,1)$, the sample complexities ${\mathcal{C}}(\mathfrak{A}, L,\epsilon,\delta)$, ${\mathcal{C}}^{\star}(\mathfrak{A}, L,\epsilon,\delta)$, and ${\mathcal{C}}^{+}(\mathfrak{A}, L,\epsilon,\delta)$ are defined as the number of steps required by a learning algorithm $\mathfrak{A}$ to identify a set of states ${\mathcal{K}}$ and a set of policies $\{\pi_s\}_{s\in{\mathcal{K}}}$ such that, with probability at least $1-\delta$, we have $\calS^{\rightarrow}_L\subseteq{\mathcal{K}}$ and
\quad ($\text{AX}_L$)\quad $\forall s\in\calS^{\rightarrow}_L$, $V^{\pi_s}_s(s_0)\leq L(1+\epsilon)$,
\quad ($\text{AX}^{\star}$)\quad $\forall s\in\calS^{\rightarrow}_L$, $V^{\pi_s}_s(s_0)\leq V^{\star}_{\calS^{\rightarrow}_L,s}(s_0) + L\epsilon$,
\quad ($\text{AX}^+$)\quad $\forall s\in\calS^{\rightarrow}_L$, $V^{\pi_s}_s(s_0)\leq V^{\star}_{\calS^{\rightarrow}_L,s}(s_0)(1 + \epsilon)$.
\end{definition}
Note that the three formulations above are increasingly more demanding.
$\text{AX}_L$ only requires to reach each state in $\calS^{\rightarrow}_L$ within $L(1+\epsilon)$ steps, which could correspond to a quite poor performance for a state $s$ with $V^{\star}_{\calS^{\rightarrow}_L,s}(s_0) \ll L$.
$\text{AX}^{\star}$ requires to learn a near-optimal policy for reaching each state in $\calS^{\rightarrow}_L$.
However, the allowed error threshold (i.e., $L\epsilon$) is uniform across all goal states, which again could correspond to a bad performance for a state $s$ with $V^{\star}_{\calS^{\rightarrow}_L,s}(s_0) \ll L$. $\text{AX}^+$ solves this issue by requiring a \emph{multiplicative} threshold. This implies that the allowed error for reaching state $s$ (i.e., $V^{\star}_{\calS^{\rightarrow}_L,s}(s_0)\epsilon$) scales with the optimal value $V^{\star}_{\calS^{\rightarrow}_L,s}(s_0)$ itself, hence making this formulation adaptive to the hardness of reaching each goal state.
No existing algorithm is able to achieve $\text{AX}^+$ guarantees, see \pref{tab:summary}.
Note that these conditions cannot be checked at algorithmic time since $\calS^{\rightarrow}_L$ is unknown to the algorithm. Existing algorithms verify these conditions directly on the computed set ${\mathcal{K}}$.
Since they guarantee that $\calS^{\rightarrow}_L \subseteq {\mathcal{K}}$, $ V^{\star}_{{\mathcal{K}},g}(s_0) \leq V^{\star}_{\calS^{\rightarrow}_L,g}(s_0)$ for any $g \in \calS^{\rightarrow}_L$ and thus they satisfy the performance in~\pref{def:ax}.
\textbf{Other notation} Let $S=|{\mathcal{S}}|$ and $A=|{\mathcal{A}}|$.
For any $L\geq 1$, define $S^{\rightarrow}_L=|\calS^{\rightarrow}_L|$, ${\mathcal{N}}^{s, a}_L=\{s'\in\calS^{\rightarrow}_L: P_{s, a}(s')>0\}$, $\Gamma^{s, a}_L=|{\mathcal{N}}^{s, a}_L|$ and $\Gamma_L=\max_{s\in\calS^{\rightarrow}_L, a}\Gamma^{s, a}_L$.
For simplicity, we often write $a=\bigo{b}$ as $a\lesssim b$.
For $n\in\field{N}_+$, define $[n]=\{1,\ldots,n\}$.
\subsection{A Constructive Definition of $\calS^{\rightarrow}_L$}\label{sec:sL.constructive}
While~\citet[][Proposition 6]{lim2012autonomous} showed that there exists a partial order $\prec$ such that $\calS^{\rightarrow}_L = {\mathcal{S}}_L^{\prec}$, no explicit characterization of such partial order is provided. In the following, we develop an alternative definition of $\calS^{\rightarrow}_L$ that leads to an explicit constructive procedure to build the set.
This alternative definition is the main inspiration for the design of our algorithms.
We introduce an operator ${\mathcal{T}}_L$ which, given a set ${\mathcal{X}}\subseteq {\mathcal{S}}$, selects all the states that are reachable in $L$ steps by a policy restricted on ${\mathcal{X}}$ and show its connection with $\calS^{\rightarrow}_L$.
\begin{lemma}\label{lem:SL.operator}
Let $\mathsf{P}(\mathcal{S})$ be the set of all subsets of $\mathcal{S}$.
For any $L\geq 1$, define the operator $\mathcal{T}_L : \mathsf{P}(\mathcal{S}) \rightarrow \mathsf{P}(\mathcal{S})$ as follows: for any ${\mathcal{X}} \subseteq {\mathcal{S}}$, $\mathcal{T}_L({\mathcal{X}}) = \{ s \in {\mathcal{S}} : V_{{\mathcal{X}},s}^\star(s_0) \leq L \}$. Then,
\begin{enumerate}
\item $\calS^{\rightarrow}_L$ is the fixed-point of ${\mathcal{T}}_L$ of smallest cardinality, i.e., $\calS^{\rightarrow}_L\subseteq{\mathcal{X}}$ if ${\mathcal{X}}={\mathcal{T}}_L({\mathcal{X}})$.
\end{enumerate}
Let us denote by $\{\calK^{\star}_j\}_{j\in\mathbb{N}}$ the unique sequence such that $\calK^{\star}_1=\{s_0\}$, $\calK^{\star}_j = {\mathcal{T}}_L(\calK^{\star}_{j-1})$. Then,
\begin{enumerate}
\setcounter{enumi}{1}
\item For any $j\geq 1$, $\calK^{\star}_j\subseteq\calK^{\star}_{j+1} \subseteq \calS^{\rightarrow}_L$;
\item There exists $J\leq S^{\rightarrow}_L$ such that $\calK^{\star}_j=\calS^{\rightarrow}_L$ for all $j\geq J$ (i.e., ${\mathcal{T}}^{J}_L(\calK^{\star}_{1}) = \lim_{j\rightarrow\infty} {\mathcal{T}}^j_L(\calK^{\star}_{1}) = \calS^{\rightarrow}_L$).
\end{enumerate}
\end{lemma}
\begin{proof}
Note that there exists a partial ordering $\prec^\star$ such that $\calS^{\rightarrow}_L={\mathcal{S}}^{\prec^\star}_L$~\citep[][Proposition 6]{lim2012autonomous}.
Let ${\mathcal{X}}$ be s.t.\ $\calS^{\rightarrow}_L \not\subseteq {\mathcal{X}}$. If $\calS^{\rightarrow}_L \cap {\mathcal{X}} = \emptyset$, then $s_0 \notin {\mathcal{X}}$, which implies that ${\mathcal{T}}_L({\mathcal{X}}) = \{s_0\}$ since $V^\star_{{\mathcal{X}},s_0}(s_0) = 0 \leq L$ and $V^\star_{{\mathcal{X}},g}(s_0) = \infty$ for all $g\neq s_0$. Thus, ${\mathcal{X}}$ cannot be a fixed point of ${\mathcal{T}}_L$. Then, assume that $\calS^{\rightarrow}_L \cap {\mathcal{X}} \neq \emptyset$. Order the states in ${\mathcal{X}} \cap \calS^{\rightarrow}_L$ according to the ordering $\prec^\star$. Let $s_i \in {\mathcal{S}}_L^{\prec^\star}$ be the first state s.t.\ $s\notin{\mathcal{X}}$ (it exists since $\calS^{\rightarrow}_L \not\subseteq {\mathcal{X}}$). By definition of $\prec^\star$ and $\calS^{\rightarrow}_L$, $V_{\{s_0,\ldots, s_{i-1}\},s_i}^\star(s_0) \leq L$, which implies that $s_i \in {\mathcal{T}}_L({\mathcal{X}})$. As a consequence, ${\mathcal{X}} \neq {\mathcal{T}}_L({\mathcal{X}})$. Thus, if ${\mathcal{X}} = {\mathcal{T}}_L({\mathcal{X}})$, we must have $\calS^{\rightarrow}_L \subseteq {\mathcal{X}}$. This proves the first point.
Let us prove that $\calK^{\star}_j\subseteq\calK^{\star}_{j+1}$ for all $j\geq 1$. Clearly, $\calK^{\star}_2 = {\mathcal{T}}_L(\calK^{\star}_1) = \{s\in{\mathcal{S}} : V_{\{s_0\},s}(s_0)\leq L\} \supseteq \{s_0\} = \calK^{\star}_1$. Then, suppose that $\calK^{\star}_{j-1}\subseteq\calK^{\star}_{j}$ for some $j \geq 2$. By definition, for all $ s \in \calK^{\star}_j$, $V^{\star}_{\calK^{\star}_{j-1},s}(s_0) \leq L$, which implies that $V^{\star}_{\calK^{\star}_{j},s}(s_0) \leq L$ by the inductive hypothesis. Then, $\calK^{\star}_{j+1} = {\mathcal{T}}_L(\calK^{\star}_j) = \{s\in{\mathcal{S}} : V_{\calK^{\star}_j,s}(s_0)\leq L\} \supseteq \calK^{\star}_j$.
Now let us prove that $\calK^{\star}_j \subseteq \calS^{\rightarrow}_L$ for all $j\geq 1$. Clearly, $ \calK^{\star}_1 \subseteq \calS^{\rightarrow}_L$. Suppose that $\calK^{\star}_j \subseteq \calS^{\rightarrow}_L$ for some $j\geq 1$. Then, if $s\in\calK^{\star}_{j+1}$ for some $s \notin \calS^{\rightarrow}_L$, it must be that $V_{\calK^{\star}_j,s}(s_0)\leq L$. By the inductive hypothesis, this implies that we found an ordering of the states in which $s$ is reachable in $L$ steps by a policy restricted on states of $\calS^{\rightarrow}_L$. Hence, $s\in\calS^{\rightarrow}_L$, which is a contradiction. This proves point 2.
Let us enumerate over $\calS^{\rightarrow}_L=\{s_0,\ldots,s_{S^{\rightarrow}_L-1}\}$ in a way that obeys $\prec^\star$. We prove by induction that $s_{j}\in\calK^{\star}_{j+1}$ for any $0 \leq j < S^{\rightarrow}_L$. Given point 2, this implies point $3$.
Clearly, $s_0\in\calK^{\star}_1$.
Now suppose that $\{s_0,\ldots,s_j\}\in\calK^{\star}_{j+1}$ for $0 \leq j \leq S^{\rightarrow}_L - 2$.
Then, we clearly have $s_{j+1}\in\calK^{\star}_{j+2}$ by the definition of $\calK^{\star}_{j+2}$ and the fact that $s_{j+1}$ is $L$-controllable by a policy restricted on $\{s_0,\ldots,s_j\}$.
\end{proof}
This lemma shows that $\calS^{\rightarrow}_L$ is a fixed-point solution of ${\mathcal{T}}_L$. Most importantly, it provides an iterative procedure to construct $\calS^{\rightarrow}_L$.
Starting from $\{s_0\}$ or $\emptyset$, ${\mathcal{T}}_L$ acts as an expansive operator over sets (i.e., $T^{j}(\{s_0\}) \subset T^{j+1}(\{s_0\})$) until the set $\calS^{\rightarrow}_L$ is built. From this point, ${\mathcal{T}}_L$ acts as an identity map since $\calS^{\rightarrow}_L$ is a fixed point. In other words, this procedure builds $\calS^{\rightarrow}_L$ iteratevely starting from $\calK^{\star}_1$, expanding it to $\calK^{\star}_2 = {\mathcal{T}}_L(\calK^{\star}_1)$, and so on until reaching $\calS^{\rightarrow}_L$. For this reason, we shall refer to the sets $(\calK^{\star}_j)_j$ as \emph{layers}.
This process is learnable since it evolves only through subsets of $\calS^{\rightarrow}_L$ and it is at the core of the design of our algorithm.
It is worth noticing that not all the fixed-point solutions of ${\mathcal{T}}_L$ are learnable. In fact, Proposition 4 of~\citet{lim2012autonomous} implies that there exist MDPs with fixed points ${\mathcal{X}} = {\mathcal{T}}_L({\mathcal{X}}) \neq \calS^{\rightarrow}_L$ which may require an exponential number of samples to be learned. For example, there exist MDPs where the whole set of states ${\mathcal{S}}$ is itself a fixed point of ${\mathcal{T}}_L$ (that is, all states are $L$-controllable) but ${\mathcal{S}}$ is exponentially larger than $\calS^{\rightarrow}_L$. This reveals an interesting connection between the existence of a \emph{unique} iterative process to reach the fixed-point corresponding to $\calS^{\rightarrow}_L$ and its learnability.
\section{Analysis of VISGO}\label{app:visgo}
\begin{algorithm2e}[t]
\DontPrintSemicolon
\caption{\textsc{VISGO}\xspace}
\label{alg:VISGO}
\KwIn{state subset ${\mathcal{X}}$, goal state $g\notin{\mathcal{X}}$, precision $\epsilon_{\text{VI}\xspace}$, counter $n$, and failure probability $\delta$.}
\textbf{Require:} $\norm{V^{\star}_{{\mathcal{X}},g}}_{\infty}\leq 8L$.
Let $c_1=3$, $c_2=512$, and $\iota_{s,a}=\ln\left(\frac{2|{\mathcal{X}}|An(s,a)}{\delta}\right)$ for all $(s, a)$.
Let $\P_{s,a}(s')=\frac{n(s,a,s')}{n^+(s,a)}$ and $\widetilde{P}_{s,a}(s')=\frac{n(s,a)}{n(s,a)+1}\P_{s,a}(s') + \frac{\field{I}\{s'=g\}}{n(s,a)+1}$ for all $(s,a,s')$.
\textbf{Initialize:} $V^{(0)}(\cdot)\leftarrow 0$, $i\leftarrow 0$.
\While{$i=0$ or $\norm{V^{(i)}-V^{(i-1)}}_{\infty}>\epsilon_{\text{VI}\xspace}$}{
\nl \lIf{$\norm{V^{(i)}}_{\infty} > 2L$}{\textbf{return} $(\infty, \infty, \pi)$ with $\pi$ being a random policy.}\label{line:bound V}
$i\leftarrow i + 1$.
\For{$s\in{\mathcal{X}}$}{
$b^{(i)}(s,a)\leftarrow \max\cbr{c_1\sqrt{\frac{\field{V}(\P_{s, a}, V^{(i-1)})\iota_{s,a}}{n^+(s, a)}}, \frac{c_2L\iota_{s,a}}{n^+(s, a)} }$.
$Q^{(i)}(s, a) \leftarrow \max\cbr{0, 1 + \widetilde{P}_{s, a}V^{(i-1)} - b^{(i)}(s,a) }$ for $a\in{\mathcal{A}}$.
$V^{(i)}(s) \leftarrow \min_aQ^{(i)}(s, a)$
}
$V^{(i)}(s)\leftarrow (1 + V^{(i-1)}(s_0))\field{I}\{s\neq g\}$ for $s\notin{\mathcal{X}}$.
}
\Return{$(Q^{(i)}, V^{(i)}, \pi)$ with $\pi(s)=\argmin_aQ^{(i)}(s, a)$ for $s\in{\mathcal{X}}$ and $\pi_g(s)=\textsc{RESET}$ for $s\notin{\mathcal{X}}$.}
\end{algorithm2e}
The convergence of \textsc{VISGO}\xspace has been proved in \citep[Lemma C.4]{cai2022near}.
We further introduce some properties of the algorithm.
\begin{lemma}[Optimism]
\label{lem:opt}
Let ${\mathcal{X}}\subseteq{\mathcal{S}}$, $g\in{\mathcal{S}}\setminus{\mathcal{X}}$, $n$ be a counter incrementally collecting samples from transition function $P$, and $\delta\in(0,1)$ be such that $\|V_{{\mathcal{X}},g}^\star\|_{\infty} \leq 8L$. For any precision $\xi > 0$, define $(Q_{\xi}, V_{\xi},\_) = \textsc{VISGO}\xspace({\mathcal{X}},g,\xi,n,\delta)$ as the output of \pref{alg:VISGO}. Let $\mathbb{P}$ be the probability operator on the process generating the counter $n$ and assume that ${\mathcal{X}}$ and $g$ are independent of $n$. Then,
\begin{align*}
\mathbb{P}\Big( \forall \xi > 0, s\in{\mathcal{S}},a\in{\mathcal{A}} : Q_{\xi}(s, a)\leqQ^{\star}_{{\mathcal{X}},g}(s, a), V_{\xi}(s)\leqV^{\star}_{{\mathcal{X}},g}(s) \Big) \geq 1 - \delta.
\end{align*}
\end{lemma}
\begin{proof}
First, by \pref{lem:anytime bernstein} and a union bound over $(s,a)\in{\mathcal{X}}\times{\mathcal{A}}$, we have with probability at least $1-\delta$, for any $(s,a)\in{\mathcal{X}}\times{\mathcal{A}}$,
\begin{align}
\abr{(\P_{s,a}-P_{s,a})V^{\star}_{{\mathcal{X}},g}} &\leq 2\sqrt{\frac{2\field{V}(\P_{s,a},V^{\star}_{{\mathcal{K}},g})\ln\frac{2|{\mathcal{X}}|An(s,a)}{\delta}}{n^+(s,a)}} + \frac{19\cdot 8L\ln\frac{2|{\mathcal{X}}|An(s,a)}{\delta}}{n^+(s, a)}\notag\\
&\leq \frac{c_1}{2}\sqrt{\frac{\field{V}(\P_{s, a}, V^{\star}_{{\mathcal{X}},g})\iota_{s,a}}{n^+(s, a)}} + \frac{c_2L\iota_{s,a}}{2n^+(s, a)},\label{eq:optV}
\end{align}
with $\iota_{s,a}$, $c_1$, and $c_2$ are defined in \pref{alg:VISGO}. We then carry out the proof assuming that such event holds.
Fix a configuration $({\mathcal{X}},g,\xi,n,\delta)$ of the inputs of VISGO and let $(Q^{(i)}, V^{(i)})_{i\geq 0}$ be the iterates of the algorithm. It suffices to show that for any $i\geq 0$, $Q^{(i)}(s, a) \leq Q^{\star}_{{\mathcal{X}},g}(s,a)$ for all $(s,a)\in{\mathcal{X}}\times{\mathcal{A}}$ and $V^{(i)}(s)\leqV^{\star}_{{\mathcal{X}},g}(s)$ for all $s\in{\mathcal{S}}$. We prove it by induction.
Note that $Q^{(0)}(\cdot) = V^{(0)}(\cdot) = 0$, thus the statement clearly holds for the base case $i=0$. Suppose it holds at some iteration $i-1 \geq 0$. Under event of \pref{eq:optV}, for any $i>0$ and $(s,a)\in{\mathcal{X}}\times{\mathcal{A}}$,
\begin{align*}
&1 + \widetilde{P}_{s, a}V^{(i-1)} - \max\cbr{c_1\sqrt{\frac{\field{V}(\P_{s, a}, V^{(i-1)})\iota_{s,a}}{n^+(s, a)}}, \frac{c_2L\iota_{s,a}}{n^+(s, a)} }\\
&\leq 1 + \widetilde{P}_{s, a}V^{\star}_{{\mathcal{X}},g} - \max\cbr{c_1\sqrt{\frac{\field{V}(\P_{s, a}, V^{\star}_{{\mathcal{X}},g})\iota_{s,a}}{n^+(s, a)}}, \frac{c_2L\iota_{s,a}}{n^+(s, a)} } \tag{induction step and \pref{lem:mvp}}\\
&\leq 1 + \P_{s,a}V^{\star}_{{\mathcal{X}},g}- \max\cbr{c_1\sqrt{\frac{\field{V}(\P_{s, a}, V^{\star}_{{\mathcal{X}},g})\iota_{s,a}}{n^+(s, a)}}, \frac{c_2L\iota_{s,a}}{n^+(s, a)} } \tag{definition of $\widetilde{P}_{s,a}$}\\
&\leq 1 + P_{s,a}V^{\star}_{{\mathcal{X}},g} + (\P_{s,a}-P_{s,a})V^{\star}_{{\mathcal{X}},g} - \frac{c_1}{2}\sqrt{\frac{\field{V}(\P_{s, a}, V^{\star}_{{\mathcal{X}},g})\iota_{s,a}}{n^+(s, a)}} - \frac{c_2L\iota_{s,a}}{2n^+(s, a)} \tag{$\max\{a,b\}\geq\frac{a+b}{2}$}\\
&\leq Q^{\star}_{{\mathcal{X}},g}(s, a). \tag{\pref{eq:optV}}
\end{align*}
This also proves that $V^{(i)}(s)\leq V^{\star}_{{\mathcal{X}},g}(s)$ for all $s\in{\mathcal{X}}$. Moreover, for $s\notin{\mathcal{X}}, s\neq g$, $V^{(i)}(s) = 1 + V^{(i-1)}(s_0) \leq 1 + V^{\star}_{{\mathcal{X}},g}(s_0) = V^{\star}_{{\mathcal{X}},g}(s)$. Finally, $ V^{(i)}(g) = V^{\star}_{{\mathcal{X}},g}(g) = 0$. This proves that $V^{(i)}(s)\leq V^{\star}_{{\mathcal{K}},g}(s)$ for all $s\in{\mathcal{S}}$, thus concluding the proof.
\end{proof}
\begin{lemma}[Bounded Error]
\label{lem:bounded error}
There exists a function $N_0(z_0,z'_0,\delta_0,\delta)\lesssim L^2z_0\ln\frac{z'_0}{\delta_0\delta}$ such that, for goal set $\mathcal{G}$ with $\calS^{\rightarrow}_{L(1+\epsilon)}\subseteq\mathcal{G}\subseteq{\mathcal{S}}$ and $\delta_0\in(0,1)$, with probability at least $1-\delta$ over the randomness of a counter $n$ incrementally collecting samples from transition function $P$, for any ${\mathcal{X}}\subseteq\calS^{\rightarrow}_{L(1+\epsilon)}$ with $|{\mathcal{X}}|\leq z_0$, $g\in\mathcal{G}\setminus{\mathcal{X}}$, precision $\xi\in(0, \frac{1}{8})$, and $\delta'\in[\delta_0,1)$, if $z'_0\geq|\mathcal{G}|$ and $n(s, a)\geq N_0(z_0,z'_0,\delta_0,\delta)$ for all $(s, a)\in{\mathcal{X}}\times{\mathcal{A}}$, then $V^{\pi_g}_g(s)\leq 2V(s)$ for all $s\in{\mathcal{S}}$, where $(\_, V,\pi_g)=\textsc{VISGO}\xspace({\mathcal{X}},g,\xi,n,\delta')$ is the output of \pref{alg:VISGO}.
Also define $N_0(z_0,\delta)=N_0(z_0,S,\delta,\delta)$ and $N^{\rightarrow}_0(\delta)=N_0(S^{\rightarrow}_{L(1+\epsilon)},|\bar{\calU}|,\delta,\delta)$ (recall that $|\bar{\calU}|\leq 2LAS^{\rightarrow}_{L(1+\epsilon)}$).
\end{lemma}
\begin{proof}
Note that the statement clearly holds if VISGO returns a value function $V=\infty$. Otherwise, $\norm{V^{(i)}}_{\infty}\leq 2L$ for any $i\leq l$, where $l$ is the index of the last iteration in \pref{alg:VISGO}. By \pref{lem:dPV}, with probability at least $1-\delta$\footnote{this holds under the same good event of \pref{lem:dPV}, which does not depend on the chosen ${\mathcal{X}},g,\delta',\xi$}, for any status of $n$, $(s, a)\in{\mathcal{X}}\times{\mathcal{A}}$, and $V$ s.t. $\|V\|_\infty \leq 2L$,
\begin{align*}
\abr{(P_{s,a} - \widetilde{P}_{s,a})V} &\leq \abr{(P_{s,a} - \P_{s,a})V} + \abr{(\P_{s,a}-\widetilde{P}_{s,a})V}\\
&\lesssim L\sqrt{\frac{z_0\iota'}{n(s, a)}} + \frac{Lz_0\iota'}{n(s, a)} + \frac{(\P_{s,a}+\field{I}_g)V}{n(s,a)+1},
\end{align*}
where $\widetilde{P}_{s,a}$ and $\P_{s,a}$ are as defined in \pref{alg:VISGO} with counter $n$ and $\iota'=\tilo{\ln\frac{z'_0}{\delta}}$ by $|\mathcal{G}|\leq z'_0$. Clearly, there exists $n_1=\tilo{L^2z_0\ln(|\mathcal{G}|/\delta)}$, such that when $n(s, a)\geq n_1$, we have $|(P_{s,a}-\widetilde{P}_{s,a})V|\leq \frac{1}{8}$.
Moreover, we have
\begin{align*}
b^{(l)}(s,a) \lesssim \max\cbr{\sqrt{\frac{\field{V}(\P_{s, a}, V^{(l-1)})}{n(s, a)}}, \frac{L}{n(s, a)} } \lesssim \frac{L}{\sqrt{n(s, a)}}.
\end{align*}
Then there exist $n_2=\tilo{L^2\ln(1/\delta_0)}$ such that when $n(s,a)\geq n_2$, $b^{(l)}(s,a)\leq\frac{1}{8}$.
Thus when $n(s,a)\geq\max\{n_1,n_2\}$ for all $s\in{\mathcal{X}},a\in{\mathcal{A}}$, we can apply the same conclusion as in the proof of \pref{lem:bounded error fresh} as get the desired result.
\end{proof}
\begin{lemma}[Bounded Error with Fresh Samples]
\label{lem:bounded error fresh}
There exists a function $N_1(x,\delta_0,\delta)\lesssim L^2x\ln \frac{x}{\delta_0\delta}$ (also define $N_1(x,\delta)=N_1(x,\delta,\delta)$) such that for ${\mathcal{X}}\subseteq{\mathcal{S}}$, $g\in{\mathcal{S}}\setminus{\mathcal{X}}$, $\delta_0\in(0, 1)$, $\delta\in(0,1)$, $n$ a counter incrementally collecting samples from transition function $P$, and assume that ${\mathcal{X}},g,\delta_0$ are independent of $n$,
with probability at least $1-\delta$, for any precision $\xi\in(0, \frac{1}{8})$ and $\delta'\in[\delta_0,1)$, if $n(s, a)\geq N_1(|{\mathcal{X}}|,\delta_0,\delta)$ for all $(s, a)\in{\mathcal{X}}\times{\mathcal{A}}$, then $V^{\pi_g}_g(s)\leq 2V(s)$ for all $s\in{\mathcal{S}}$, where $(\_, V,\pi_g)=\textsc{VISGO}\xspace({\mathcal{X}},g,\xi,n,\delta')$ is the output of \pref{alg:VISGO}.
\end{lemma}
\begin{proof}
Let $y={\mathcal{S}}\setminus({\mathcal{X}}\cup\{g\})$ and $\iota^n_{s,a}=\ln\frac{4|{\mathcal{X}}|^2 A n(s,a)}{\delta}$.
Consider the following events:
\begin{align*}
E_1 &:= \left\{ \forall s\in{\mathcal{X}}, a\in{\mathcal{A}}, s' \in {\mathcal{X}}, n(s,a) \geq 1 : |P_{s,a}(s') - \P_{s,a}(s')| \leq 2\sqrt{\frac{2P_{s,a}(s') \iota^n_{s,a}}{n(s,a)}} + \frac{2\iota^n_{s,a}}{n(s,a)} \right\},
\\ E_2 &:= \left\{\forall s\in{\mathcal{X}}, a\in{\mathcal{A}}, n(s,a) \geq 1 : |P_{s,a}(y) - \P_{s,a}(y)| \leq 2\sqrt{\frac{2 P_{s,a}(y)\iota^n_{s,a}}{n(s,a)}} + \frac{2\iota^n_{s,a}}{n(s,a)} \right\}.
\end{align*}
By \pref{lem:anytime bernstein} and a union bound, they hold simultaneously with probability at least $1-\delta$. We carry out the proof conditioned on these events holding.
For any ${\mathcal{X}},g,\xi,n,\delta'$, the statement clearly holds if $V=\infty$.
Otherwise, $\norm{V^{(i)}}_{\infty}\leq 2L$ for any $i\leq l$, where $l$ is the index of the last iteration in \pref{alg:VISGO}. Take any status of counter $n$, precision $\xi \in(0, \frac{1}{8})$, $\delta'\in[\delta_0,1)$. Let $V$ and $\pi_g$ be the output of \pref{alg:VISGO} with these parameters such that $\|V\|_\infty \leq 2L$. Since $V$ is restricted on ${\mathcal{X}}\cup\{g\}$, we have $V(s') = 1 + V^{(l-1)}(s_0)$ for any $s' \notin {\mathcal{X}}\cup\{g\}$. Then, for any $(s, a)\in{\mathcal{X}}\times{\mathcal{A}}$,
\begin{align*}
&\abr{(P_{s,a} - \widetilde{P}_{s,a})V} \leq \abr{(P_{s,a} - \P_{s,a})V} + \abr{(\P_{s,a}-\widetilde{P}_{s,a})V}
\\ &\leq \abr{\sum_{s'\in{\mathcal{X}}} (P_{s,a}(s') - \P_{s,a}(s'))V(s')} + \abr{(P_{s,a}(y) - \P_{s,a}(y))(1 + V^{(l-1)}(s_0))} + \abr{(\P_{s,a}-\widetilde{P}_{s,a})V}
\\ &\leq 2L\sum_{s'\in{\mathcal{X}}}\abr{ P_{s,a}(s') - \P_{s,a}(s')} + 2L\abr{P_{s,a}(y) - \P_{s,a}(y)} + \abr{(\P_{s,a}-\widetilde{P}_{s,a})V}
\\ &\lesssim\frac{L\sqrt{|{\mathcal{X}}| \ln(|{\mathcal{X}}|)}}{\sqrt{n(s, a)}} + \frac{L|{\mathcal{X}}| \ln(|{\mathcal{X}}|)}{n(s, a)} + \frac{(\P_{s,a}+\field{I}_g)V}{n(s,a)+1},
\end{align*}
where in the last step we applied Cauchy-Schwarz inequality, the good events, the definition of $\widetilde{P}_{s,a}$, and removed logarithmic terms and constants.
Clearly, there exists $n_1=\tilo{L^2|{\mathcal{X}}|\ln (|{\mathcal{X}}|/\delta)}$, such that when $n(s, a)\geq n_1$, we have $|(P_{s,a}-\widetilde{P}_{s,a})V|\leq \frac{1}{8}$.
Moreover, we have
\begin{align*}
b^{(l)}(s,a) \lesssim \max\cbr{\sqrt{\frac{\field{V}(\P_{s, a}, V^{(l-1)})}{n(s, a)}}, \frac{L}{n(s, a)} } \lesssim \frac{L}{\sqrt{n(s, a)}}.
\end{align*}
Then there exist $n_2=\tilo{L^2\ln(1/\delta_0)}$ such that when $n(s,a)\geq n_2$, $b^{(l)}(s,a)\leq\frac{1}{8}$.
Thus when $n(s,a)\geq\max\{n_1,n_2\}$ for all $s\in{\mathcal{X}},a\in{\mathcal{A}}$, for any $s\in{\mathcal{X}}$,
\begin{align*}
&V(s) = V^{(l)}(s) \geq 1 + \widetilde{P}_{s,\pi_g(s)}V^{(l-1)}(s) - b^{(l)}(s, \pi_g(s))
\\ &\geq 1 - \xi + \widetilde{P}_{s,\pi_g(s)}V^{(l)} - b^{(l)}(s,\pi_g(s))
\\ &\geq 1 - \xi + P_{s,\pi_g(s)}V - \abr{(P_{s,\pi_g(s)} - \widetilde{P}_{s,\pi_g(s)})V} - b^{(l)}(s,\pi_g(s))
\geq \frac{1}{2} + P_{s,\pi_g(s)}V(s),
\end{align*}
where we used the definition of $V^{(l)}$, the stopping condition of VISGO, and the previously derived bounds.
For $s\notin{\mathcal{X}}$, we have $V(s)=(1 + V^{(l-1)}(s_0))\field{I}\{s\neq g\}\geq (\frac{1}{2}+V(s_0))\field{I}\{s\neq g\}$.
Applying this recursively gives $V(s)\geq \frac{1}{2}V^{\pi_g}_g(s)$.
This completes the proof.
\end{proof}
\begin{lemma}
\label{lem:subset opt}
For any subsets ${\mathcal{X}}$ and ${\mathcal{X}}'$ such that ${\mathcal{X}}\subseteq{\mathcal{X}}'\subseteq{\mathcal{S}}$, any $g\in{\mathcal{S}}\setminus{\mathcal{X}}'$, $\xi > 0$, counter $n$, and $\delta\in(0, 1)$, we have $V_{{\mathcal{X}}'}(s)\leq V_{{\mathcal{X}}}(s)$ for any $s\in{\mathcal{S}}$, where we define $V_{{\mathcal{X}}''}=\textsc{VISGO}\xspace({\mathcal{X}}'',g,\xi,n,\delta)$ (see \pref{alg:VISGO}) for any ${\mathcal{X}}''\subseteq{\mathcal{S}}$.
\end{lemma}
\begin{proof}
For any ${\mathcal{X}}''\subseteq{\mathcal{S}}$,
denote by $Q^{(i)}_{{\mathcal{X}}''}$ and $V^{(i)}_{{\mathcal{X}}''}$ the values of $Q^{(i)}$ and $V^{(i)}$ in \pref{alg:VISGO} respectively when computing $V_{{\mathcal{X}}''}$.
It suffices to prove that $V^{(i)}_{{\mathcal{X}}'}(s)\leq V^{(i)}_{{\mathcal{X}}}(s)$ for any $s\in{\mathcal{S}}$ and $i\geq 0$ by induction.
The base case $i=0$ is clearly true by initialization.
When $i>0$, we consider three disjoint cases:
1) if $s\in{\mathcal{X}}$, by the induction step and \pref{lem:mvp}, for any $a\in{\mathcal{A}}$,
\begin{align*}
&1 + \widetilde{P}_{s, a}V^{(i-1)}_{{\mathcal{X}}'} - \max\cbr{c_1\sqrt{\frac{\field{V}(\P_{s, a}, V^{(i-1)}_{{\mathcal{X}}'})\iota_{s,a}}{n^+(s, a)}}, \frac{c_2L\iota_{s,a}}{n^+(s, a)} }\\
&\leq 1 + \widetilde{P}_{s, a}V^{(i-1)}_{{\mathcal{X}}} - \max\cbr{c_1\sqrt{\frac{\field{V}(\P_{s, a}, V^{(i-1)}_{{\mathcal{X}}})\iota_{s,a}}{n^+(s, a)}}, \frac{c_2L\iota_{s,a}}{n^+(s, a)} }.
\end{align*}
This implies that $V^{(i)}_{{\mathcal{X}}'}(s)\leq V^{(i)}_{{\mathcal{X}}}(s)$ for $s\in{\mathcal{X}}$.
2) if $s\in{\mathcal{X}}'\setminus{\mathcal{X}}$, we have:
$V^{(i)}_{{\mathcal{X}}'}(s) \leq Q^{(i)}_{{\mathcal{X}}'}(s, \textsc{RESET}) \leq 1 + \widetilde{P}_{s,\textsc{RESET}}V^{(i-1)}_{{\mathcal{X}}'} \overset{\text{(i)}}{\leq} 1 + V^{(i-1)}_{{\mathcal{X}}'}(s_0) \overset{\text{(ii)}}{\leq} 1 + V^{(i-1)}_{{\mathcal{X}}}(s_0) = V^{(i)}_{{\mathcal{X}}}(s)$,
where step (i) is by $P_{s,\textsc{RESET}}(s_0)=1$ and step (ii) is by the induction step.
3) if $s\in{\mathcal{S}}\setminus{\mathcal{X}}'$, by the induction step we have $V^{(i)}_{{\mathcal{X}}'}(s)=(1+V^{(i-1)}_{{\mathcal{X}}'}(s_0))\field{I}\{s\neq g\}\leq (1+V^{(i-1)}_{{\mathcal{X}}}(s_0))\field{I}\{s\neq g\}=V^{(i)}_{{\mathcal{X}}}(s)$.
Combining these three cases completes the proof.
\end{proof} |
1,314,259,996,732 | arxiv | \section{introduction}
By analyzing the precise cross sections for $e^+ e^- \to \omega \chi_{c0} $\cite{Ablikim:2014qwy}, $e^+ e^- \to \pi^+ \pi^- J/\psi $ \cite{Ablikim:2016qzw}, $e^+ e^- \to \pi^+ \pi^- h_c$ \cite{BESIII:2016adj} and $e^+ e^- \to \pi^+ D^0 D^{\ast-} $ \cite{open-bes}, the BESIII Collaboration reported a series of vector charmonium-like states, which are $Y(4220)$, $Y(4320)$ and $Y(4390)$. The charmonium-like state $Y(4220)$ have been reported in $\chi_{c0} \omega$, $\pi^+ D^0 D^{\ast -}$, $\pi^+ \pi^- h_c$ and $\pi^+\pi^- J/\psi$ channels at present. Its width were reported to be around 40 MeV by analyzing the cross sections for $e^+ e^- \to \chi_{c0} \omega$ and $e^+ e^- \to \pi^+ \pi^- J/\psi$, while it were measured to be about 70 MeV in the cross sections for $e^+ e^- \to \pi^+ \pi^- h_c$ process. As for $Y(4320)$, it was a broad charmonium-like state and only reported in $\pi^+ \pi^- J/\psi$ process. The charmonium-like state $Y(4390)$ is also a broad state and was observed in the spin flipped $\pi^+ \pi^- h_c$ channel.
These newly observed charmonium-like states make resonances with $J^{PC}=1^{--}$ between $4.0\sim 4.5 $ GeV overcrowed and the nature of these charmonium-like states becomes an intriguing question. As for $Y(4220)$, it has been observed in various channels. In the $\pi^+ \pi^- J/\psi$ channel, a structure, $Y(4260)$ was firstly reported by BaBar Collaboration \cite{Aubert:2005rm} and then confirmed by Belle Collaboration \cite{Yuan:2007sj}. Recent precise analysis from BESIII Collaboration indicates the structure $Y(4260)$ should contain two charmonium-like state, $Y(4220)$ and $Y(4320)$ \cite{Ablikim:2016qzw}. The former one is consistent with the one observed in the channels of $\chi_{c0} \omega$, $\pi^+ \pi^- h_c$ and $\pi^+ D^0 D^{\ast -}$. Since $Y(4260)/Y(4220)$ is close to $D_1(2420) \bar{D}$ threshold, it could be considered as a molecular state composed of $D_1(2420) \bar{D}$ \footnote{The charge conjugate states are implied throughout this work}\cite{Ding:2008gr, Cleven:2016qbn, Chen:2016byt, Xue:2017xpu, Cleven:2013mka, Dong:2013kta}. While, the QCD sum rule estimations indicate that $Y(4260)$ could be a mixed charmonium-tetraquark state \cite{Dias:2012ek, Wang:2016mmg}.
Before the observations of $Y(4220)$, we predicted a narrow $\psi(4S)$ around 4.2 GeV in Ref. \cite{He:2014xna}, while $\psi(4415)$ was considered as $\psi(5S)$. After the observation of $Y(4220)$ in the $\chi_{c0} \omega$ channel, The possibility of $Y(4220)$ as $\psi(4S)$ was further evaluated \cite{Chen:2014sra, Chen:2015bma}. As for $Y(4390)$, it is only observed in the $\pi^+ \pi^- h_c$ channel. In Refs. \cite{Chen:2017abq, He:2017mbh}, the possibility of $Y(4390)$ as a $D^\ast D_1(2420)$ molecular state were investigated. While in Ref. \cite{Chen:2017uof}, the lineshapes of the cross sections for $e^+ e^- \to \pi^+ \pi^- J/\psi,\ \pi^+ \pi^- h_c,\ \pi^+ D^- D^{\ast -}$ could be well reproduced by interferences of the well established charmonia $\psi(4160)$ and $\psi(4415)$ as well as $Y(4220)$.
As for $Y(4320)$, it was also observed in the $\pi^+ \pi^- J/\psi$ channel firstly. Actually, in the $\pi^+ \pi^- \psi(2S)$ channel, there exists a charmonium-like state $Y(4360)$ near the newly observed $Y(4320)$ \cite{Aubert:2007zz, Wang:2007ea}. The mass of $Y(4360)$ was fitted to be $4324 \pm 24$ MeV by BaBar Collaboration \cite{Aubert:2007zz}, which is consistent with the mass of $Y(4320)$. In addition, with recent precise data, the analysis in Ref. \cite{Zhang:2017eta} also indicates that the charmonium-like states $Y(4360)$ in the $\pi^+ \pi^- \psi(2S)$ channel and $Y(4320)$ in the $\pi^+ \pi^- J/\psi$ channel should be the same state.
\begin{table*}
\caption{Mass spectra and $R$ values of $D$-wave charmonia. SP, GI and MGI refer to the screen potential model \cite{Li:2009zu}, Godfrey-Isgur relativistic quark model \cite{Godfrey:1985xj} and modified Godfrey-Isgur relativistic quark model \cite{Wang:2019mhs}. The values in the bracket are the effective $R$ values of the corresponding states in unit of $\rm GeV^{-1}$. \label{Tab:Dwave}}
\begin{tabular}{p{2.5cm}<{\centering} p{3.5cm}<{\centering} p{3.cm}<{\centering} p{3cm}<{\centering} p{3cm}<{\centering}}
\toprule[1pt]
States & Experiment & SP Model \cite{Li:2009zu} & GI Model \cite{Godfrey:1985xj} & MGI Model \cite{Wang:2019mhs}\\
$\eta_{c2}(1D)$ &---&3796 & 3837 & 3848\\
$\psi_1(1D)$ & $3773.13 \pm 0.15$ \cite{Tanabashi:2018oca} & 3783 (2.59) & 3821 (1.84) & 3830 (1.88) \\
$\psi_2(1D)$ & $3822.2 \pm 1.2$ \cite{Tanabashi:2018oca} & 3798 & 3838 & 3848 \\
$\psi_3(1D)$ & $3842.71 \pm 1.6 \pm0.12 $ \cite{LHCbnew} & 3799 & 3846 & 3858 \\
\midrule[1pt]
$\eta_{c2}(2D)$ &--- & 4099 & 4207 & 4137\\
$\psi_1(2D)$ & $4191\pm 5$ \cite{Tanabashi:2018oca} & 4089 (3.12) & 4197 (2.09) & 4125 (2.38) \\
$\psi_2(2D)$ &---& 4100 & 4209 & 4137 \\
$\psi_3(2D)$ &---& 4103 & 4215 & 4144\\
\midrule[1pt]
$\eta_{c2}(3D)$ &---&4326 & 4531 & 4343\\
$\psi_1(3D)$ & --- &4317 (3.59) & 4522 (2.24) & 4334 (2.85)\\
$\psi_2(3D)$ &---& 4327 & 4532 & 4343\\
$\psi_3(3D)$ &---& 4331 & 4536 & 4348\\
\bottomrule[1pt]
\end{tabular}
\end{table*}
In our previous work, we have categorized $Y(4220)$ as $\psi(4S)$ and $\psi(4415)$ as $\psi(5S)$ \cite{He:2014xna, Chen:2014sra, Chen:2015bma}. In such a scenario, there are no additional room left for $Y(4320)$ in the $S$-wave vector charmonium and in the vicinity of $Y(4320)$, there is no charmed mesons pair threshold. However, if one further checks the charmonium spectroscopy, one can find that in the $D$-wave charmonium sector, $\psi(3770)$ and $\psi(4160)$ are well established as $\psi(1^3D_1)$ and $\psi(2^3D_1)$ states, respectively. The higher $D$-wave vector charmonia have not been observed experimentally. On the theoretical side, the masses of $D$ wave charmonia have been predicted in the quark model as shown in Table \ref{Tab:Dwave}. One can find the mass of $\psi(3^3D_1)$ was predicted to be $4519$ MeV by the relativistic quark model \cite{Godfrey:1985xj}. However, for the higher charmonia, the couple channel effects will shift their mass to the open-charm threshold \cite{Li:2009ad, Kalashnikova:2005ui, Ortega:2009hj}, thus the predicted mass of $\psi(3^3D_1)$ in Ref. \cite{Godfrey:1985xj} should be too large since the coupled-channel effects are not included. In Refs. \cite{Li:2009zu, Wang:2019mhs}, the screened potential model were employed to depict the couple channel effect in the charmonium, the predicted mass of $\psi(3^3D_1)$ is $4317$ MeV and $4334$ MeV, respectively, which is well consistent with the one of $Y(4320)$. Thus, $Y(4320)$ could be a good candidate of $\psi(3^3D_1)$ state.
Moreover, very recently, the LHCb collaboration reported their measurements of the near threshold $D\bar{D}$ spectroscopy \cite{LHCbnew}. In the $D\bar{D}$ mass spectrum, the $D$ wave charmonium $\psi(3770)$ was observed in the hadronproduction process for the first time \cite{LHCbnew}. In the same spectroscopy, a new narrow state (named $\psi(3842)$ hereafter) was reported.
As shown in Table. \ref{Tab:Dwave}, the mass of this newly observed state is consistent with one of $\psi(1^3D_3)$ state predicted by quark model \cite{Godfrey:1985xj, Li:2009zu, Wang:2019mhs} and the narrow width could result from the higher partial wave suppression since $\psi(1^3D_3)$ decays into $D\bar{D}$ via a $F$ wave with $L=3$. Moreover, another $D$-wave charmonia candidate, $\psi(3823)$, was firstly observed by Belle Collaboration \cite{Bhardwaj:2013rmw} and then confirmed by BES III Collaboration \cite{Ablikim:2015dlj}. Considering $\psi(3823)$ as $\psi_2(1D)$ state, together with the newly observed $\psi(3842)$ as $\psi_3(1D)$ state, the $D$ wave ground spin triplets have been well established. As for $2D$ charmonia, one can find only $\psi_1(2D)$ state has been observed experimentally. Thus, searching the missing highly excited $D$ wave charmonia experimentally will be intriguing. Unlike to the electron-positron annihilation process, the states produced in the hadronproduction process have more possibility of $J^{PC}$ quantum numbers, while states involved in the electron-positron annihilation process have fixed $J^{PC}$ quantum numbers, which are $1^{--}$. Thus, the hadronproduction process in the LHCb Collaboration provide us a powerful platform of searching for charmonium states with various $J^{PC} $ quantum numbers, which includes the missing highly excited $D$ wave charmonia.
On the theoretical side, it will be intriguing to comb the $D$ wave charmonium states. In the present work, we take $\psi(3770)$, $\psi(4160)$ and $Y(4320)$ as the $\psi(1^3D_1)$, $\psi(2^3D_1)$ and $\psi(3^3D_1)$ charmonia, and take these states as scales to investigate the open charm decays of other $D$ wave charmonium states, which could, to some extend, cancel the uncertainties of quark model.
This work is organized as follows. After introduction, a short review of quark pair creation model and the formula of open-charm decays of $D$ wave charmonium states are presented in Section II. Our numerical results and discussions are given in Section III. Section IV is devoted to summary.
\section{Quark pair Creation model and open charm decays of $\psi(^3D_1)$ charmonium}
\subsection{Review of quark pair creation model}
Here, we adopt the quark pair creation (QPC) model (also named $^3P_0$ model since the $J^{PC}$ quantum numbers of the quark pair created from the vacuum are $0^{++}$) to estimate the open charm decays of charmonia. The QPC model was first proposed by Micu \cite{Micu:1968mk, LeYaouanc:1972vsx, LeYaouanc:1973ldf, LeYaouanc:1977fsz} and then widely used to estimate the OZI allowed strong decay processes \cite{Liu:2009fe, Close:2005se, Song:2014mha, Godfrey:2015dia, Chen:2016iua, Barnes:2003vb, Wang:2019mhs, Song:2015fha, Song:2015nia}. In the QPC model, the related $S-$ matrix of $A\to BC$ process reads,
\begin{eqnarray}
\langle BC \left| S \right| A \rangle = I- i 2\pi \delta(E_f-E_i) \langle BC \left|\mathcal{T}\right| A\rangle, \label{Eq:SMat}
\end{eqnarray}
where the transition operator $\mathcal{T}$ is,
\begin{eqnarray}
\mathcal{T} =&&-3 \gamma \sum_m \langle 1m;1-m|00\rangle \int d \mathbf{k}_3 \mathbf{k}_4 \delta^3 (\mathbf{k}_3 +\mathbf{k}_4) \nonumber \\
&&\times \mathcal{Y}_{1m}\left( \frac{\mathbf{k}_3-\mathbf{k}_4}{2} \right) \chi_{1,-m}^{34} \varphi_0^{34} \omega_{0}^{34} d_{3i}^{\dagger} (\mathbf{k}_3) b_{4j}^{\dagger} (\mathbf{k}_4), \label{Eq:Tran}
\end{eqnarray}
where $\mathcal{Y}_{1m}(\mathbf{k} )=|\mathbf{k}| Y_{1m}(\theta ,\phi)$, $\chi_{1,-m}^{34}$, $\varphi_0^{34}= (u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$ and $\omega_0^{34} =\delta_{\alpha_3 \alpha_4}$ are the space, spin, flavor and color parts of the wave functions, respectively. $\alpha_3$ and $\alpha_4$ are the color indexes of the created quark pair. In the QPC model, the parameter $\gamma $ is introduced to represent the strength of the quark-antiquark pair creation from the vacuum and it could be fixed by fitting the decay data. In the present work, we take $\gamma=6.3$ for the up/down quark pair and $\gamma_s =\gamma/\sqrt{3}$ for strange quark pair creation \cite{Chen:2016iua, Liu:2009fe}.
In the initial rest frame, the matrix element of the transition operator is
\begin{eqnarray}
&&\langle BC \left| \mathcal{T} \right| A\rangle = \sqrt{8 E_A E_B E_C} \gamma \sum_{\substack{M_{L_A}, M_{L_B}, M_{L_C}, \\M_{S_A}, M_{S_B}, M_{S_C}}} \langle 1m, 1-m|00\rangle \nonumber\\
&&\hspace{5mm}\times \langle L_A, M_{L_A}, S_A M_{S_A}|J_A, M_A\rangle
\langle L_B, M_{L_B}, S_B M_{S_B}|J_B, M_B\rangle \nonumber\\
&&\hspace{5mm}\times \langle L_C, M_{L_C}, S_C M_{S_C} |J_C, M_C\rangle \langle \varphi_{B}^{13} \varphi_{C}^{24} | \varphi_{A}^{12} \varphi_0^{34} \rangle \nonumber\\
&&\hspace{5mm}\times \langle \chi_{S_B M_{S_B}}^{13} \chi_{S_C M_{S_C}}^{24} | \chi_{S_A M_{S_A}}^{12} \chi^{34}_{1-m} \rangle I_{M_{L_B} M_{L_C}}^{M_{L_A} m} (\mathbf{K}),
\end{eqnarray}
where $\langle \varphi_{B}^{13} \varphi_{C}^{24} | \varphi_{A}^{12} \varphi_0^{34} \rangle$ and $\langle \chi_{S_B M_{S_B}}^{13} \chi_{S_C M_{S_C}}^{24} | \chi_{S_A M_{S_A}}^{12} \chi_{1-m}^{34}\rangle$ are the flavor matrix element and spin matrix element, respectively. While the color matrix element $\langle \omega_{B}^{13} \omega_{C}^{24} | \omega_{A}^{12} \omega_0^{34} \rangle =1/3$ cancels out the factor $3$ in the transition operator defined in Eq. (\ref{Eq:Tran}). The matrix element of the spatial part reads
\begin{eqnarray}
&& I^{M_{L_A},m}_{M_{L_B},M_{L_C}}(\textbf{K}) = \int\!\rm
d\mathbf{k}_1\rm d\mathbf{k}_2\rm d\mathbf{k}_3\rm
d\mathbf{k}_4\,\delta^3(\mathbf{k}_1+\mathbf{k}_2)\delta^3(\mathbf{k}_3+\mathbf{k}_4) \nonumber\\ &&
\hspace{5mm} \times \delta^3
(\textbf{K}_B-\mathbf{k}_1-\mathbf{k}_3)\delta^3(\textbf{K}_C-\mathbf{k}_2-\mathbf{k}_4) \Psi^*_{n_B L_B
M_{L_B}}(\mathbf{k}_1,\mathbf{k}_3)\nonumber\\
&&\hspace{5mm} \times\Psi^*_{n_C L_C
M_{L_C}}(\mathbf{k}_2,\mathbf{k}_4)
\Psi_{n_A L_A M_{L_A}}(\mathbf{k}_1,\mathbf{k}_2)
\mathcal{Y}_{1m}\Big(\frac{\mathbf{k}_3-\mathbf{k}_4}{2}\Big),
\label{Eq:I}
\end{eqnarray}
which reflects the overlap of the spatial wave functions of the initial state and final states. The amplitude of the decay process is
\begin{eqnarray}
\langle BC \left| \mathcal{T} \right| A\rangle =\delta^3(\mathbf{K}_B+\mathbf{K}_C-\mathbf{K}_A) \mathcal{M}^{M_{J_A} M_{J_B} M_{J_C}}.
\end{eqnarray}
By the Jacobi-Wick rotation, the amplitude can be transformed into partial wave amplitude, which is,
\begin{eqnarray}
\mathcal{M}^{JL}(A\to BC) &=&\frac{\sqrt{2L+1}}{2J_A +1} \sum_{M_{J_B},M_{J_C}} \langle L0 JM_{J_A} | J_A M_{J_A}\rangle \nonumber\\ &&\times \langle J_B M_{J_B } J_C M_{J_C} | J_A M_{J_A} \rangle \mathcal{M}^{M_{J_A} M_{J_B} M_{J_C}}. \label{Eq:PWA}\nonumber\\
\end{eqnarray}
In terms of the partial wave amplitude, the partial width is
\begin{eqnarray}
\Gamma = \pi^2 \frac{\left|\mathbf{K} \right|}{m_{A}^2} \sum_{JL} \left| \mathcal{M}^{JL}\right| ,
\end{eqnarray}
where $\left| \mathbf{K}\right| = \lambda^{1/2} (m_A^2, m_B^2, m_C^2)$ with the K$\mathrm{\ddot{a}}$llen function $\lambda(x,y,z)=x^2 +y^2 +z^2 -2xy -2yz -2xz$.
\subsection{Open charm decays of $D$-wave charmonia}
\begin{table}[htb]
\centering
\caption{The masses and $R$ values of the involved mesons. Here $(\pm)$ and $(0)$ indicate the charge of the mesons. \label{Tab:mass}}
\begin{tabular}{p{1.5cm}p{3.5cm}<{\centering}p{2.5cm}<{\centering}}
\toprule[1pt]
Meson & Mass (MeV) & $R\ (\mathrm{GeV}^{-1})$ \cite{Godfrey:1986wj} \\
\midrule[1pt]
$D$ & $1864.83 (0), 1869.58(\pm)$ & $1.52$\\
$D^\ast$ & $2006.85(0), 2010.26(\pm)$ & $1.85$\\
$D_0(2400)$ & $2318(0) 2351(\pm)$ & $1.85$\\
$D_1(2420)$ & $2420.8(0), 2423.2 (\pm)$ &$2.00$\\
$D_1^\prime(2430)$ & $2427 (0),\ 2427(\pm) $ & $2.00$ \\
$D_2(2460)$ & $2460.7 (0),\ 2465.4(\pm) $ & $2.00$ \\
$D_s$ & $1968.28 (\pm)$ & $1.41$ \\
$D_s^\ast$ &$2112.1 (\pm)$& $1.69$\\
\midrule[1pt]
$\psi(3770)$ &$3773.13$ &---\\
$\psi_3(3842)$& $3842.71$&---\\
$\eta_{c2}(2D)$ & 4201& ---\\
$\psi(4160)$ &$4191$ &---\\
$\psi_2(2D)$ & $4203$ &---\\
$\psi_3(2D)$ & $4209$ &---\\
$\eta_{c2}(3D)$ & $4330$ & ---\\
$Y(4320)$ &$4320.0$&---\\
$\psi_2(3D)$ & $4330$ &---\\
$\psi_3(3D)$ & $4335$ &---\\
\bottomrule[1pt]
\end{tabular}
\end{table}
\begin{table*}[t]
\caption{The open charm decay modes of $D$ wave charmonia. \label{Tab:mode} }
\begin{tabular}{ccccccccccccc}
\toprule[1pt]
Channel &$\eta_ {c2}(1D)$ & $\psi_1(1D)$ & $\psi_2(1D)$ & $\psi_3(1D)$ & $\eta_{c2 }(2D)$ & $\psi_1(2D)$ & $\psi_2(2D)$ & $\psi_3(2D)$ & $\eta_ {c2}(3D)$ & $\psi_1(3D)$ & $\psi_2(3D)$ & $\psi_3(3D)$ \\
\midrule[1pt]
$ D \bar{D} $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ \\
$ D \bar{D}^{\ast} $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ \\
$ D^{\ast} \bar{D}^{\ast} $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ \\
$D_s^+ D_s^- $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ \\
$D_s^+ D_s^{\ast-} $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ \\
$D_s^{\ast +}D_s^{\ast-} $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ \\
$ D\bar{D}_0 $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ \\
$ D\bar{D}_1(2420) $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ \\
$ D\bar{D}_1(2430) $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\checkmark $ & $\checkmark $ & $\checkmark $ \\
$ D^{\ast}\bar{D}_0(2400) $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\checkmark $ \\
$D_s^+D_{s0}^-(2317) $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\dots $ & $\checkmark $ & $\dots $ & $\checkmark $ & $\dots $ \\
\bottomrule[1pt]
\end{tabular}
\end{table*}
In the present work, we perform a system estimation of the open charm decays of $D$ wave charmonia. To evasion the uncertainties of quark model, we take $\psi(3770)$, $\psi(4160)$ and $Y(4320)$ as the scale of $1D$, $2D$ and $3D$ charmonium states. The masses of the involved charmonium states and charmed mesons are listed in Table \ref{Tab:mass}. As for the charmed mesons and already established charmonia, i.e., $\psi(3770)$ and $\psi(4160)$, we adopt the center values of the PDG average \cite{Tanabashi:2018oca}. As for $Y(4320)$ and $\psi_3(3842)$, we take the measurement one in Ref. \cite{Ablikim:2016qzw, LHCbnew}. It's interesting to notice the mass splitting between the same spin multiplets are predicted to be very similar for different quark model \cite{Li:2009zu, Wang:2019mhs, Godfrey:1985xj}.
For example, the mass splitting $\Delta m_{a} = m_{\eta_{c2}(2D)}- m_{\psi_1(2D)} $ are predicted to 10, 12 and 10 MeV for GI model, MGI model and SP model, respectively. By using the mass splitting estimated in SP model and taking $\psi (4160)$ as the scale of $2D$ states, the masses of the missing $2D$ states can be estimated, for example, $m_{\eta_{c2}(2D)} =m_{\psi(4160)}+\Delta m_{a} =4201 \ \rm{MeV}$. In the same way, the masses of the missing $3D$ states can be evaluated by taking $Y(4320)$ as the scale.
Considering the $J^{PC}$ conservation and kinetics limit, we list all the possible open charm decay modes of $D$ wave charmonia in Table \ref{Tab:mode}. As for ground states, $\psi_1(1D)$ and $\psi_3(1D)$ can decay into $D\bar{D}$ via $P$ wave and $F$ wave, respectively, while $\eta_{c2}(1D)$ and $\psi_2(1D)$ have no open charm decay mode, although they are above the threshold of $D\bar{D}$. With Eqs. (\ref{Eq:SMat})-(\ref{Eq:PWA}), one can get the partial wave amplitudes of the involved process as shown in Table \ref{Tab:mode}. The estimated particular expressions of these partial wave amplitudes are listed in Table \ref{Tab:amp1}-\ref{Tab:amp4} in Appendix \ref{Sec:App}.
For $3D$ charmonia, their masses are above the threshold of $D_1^\prime(2430) \bar{D}$ and $D_1(2420) \bar{D}$. The charmed meson $D_1^\prime(2430)$ and $D_1(2420)$ are the mixture of the $1^3P_1$ and $1^1P_1$ states and the mixing scheme is,
\begin{eqnarray}
\left(
\begin{array}{c}
|D_1^\prime(2430) \rangle \\
|D_1(2420) \rangle
\end{array}
\right)=
\left(
\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}
\right)
\left(
\begin{array}{c}
|1^1P_1 \rangle \\
|1^3P_1 \rangle
\end{array}
\right),
\end{eqnarray}
where the mixing angle $\theta =-54.7^\circ$, which is determined by the heavy quark limit \cite{Godfrey:1986wj, Barnes:2002mu, Matsuki:2010zy}.
\section{Numerical Results and discussions}
With above preparations, we could investigate the open charm decays of the $D$ wave charmonia. In Eq. (\ref{Eq:I}), the spatial wave functions of the mesons are involved. In principle, these wave functions could be estimated by the constitute quark model. However, as we discussed in the introduction, there exist some uncertainties in the quark models. Thus, in the present work, we employ the simple harmonic oscillator wave function to simulate the spatial distribution of the quark-antiquark in meson. the detailed form of the spatial wave function in the momentum representation is
\begin{eqnarray}
\Psi_{n\ell m_\ell} (R, \mathbf{k})&=& \frac{(-1)^n (-i)^\ell R^{3/2}}{\sqrt[4]{\pi}}\sqrt{\frac{2^{\ell -n+2} (2\ell +2n +1)!!}{n! (2\ell +1)!!^2}} (k R)^\ell \nonumber\\ &&\times F\left(-n,\ell+\frac{3}{2}, R^2 k^2\right) e^{-R^2 k^2 /2} Y_{\ell ,m_\ell}(\hat{\mathbf{k}})
\end{eqnarray}
where $n,\ \ell$ and $m_\ell$ are the radial, angular momentum and magnetic quantum numbers, respectively. $F(-n, \nu, x)$ and $Y_{\ell m_{\ell}}$ indicate the hypergeometric function and spherical harmonic function, respectively.
In the spatial wave function, a parameter $R$ is introduced. As for the lowest charmed mesons, the predictions of the relativistic quark model are well consistent with the experimental measurements. Thus, in the present work, the values of parameter $R$ for the charmed and charmed-strange mesons are fixed such that it reproduces the root mean square radius estimated by the relativistic quark model \cite{Godfrey:1986wj}. In Ref. \cite{Close:2005se, Song:2014mha, Godfrey:2015dia, Barnes:2003vb, Chen:2016iua}, the simple harmonic oscillator wave function with a parameter $R$ has been used to investigate the decay behavior of mesons and the estimated results could well reproduce the corresponding experimental data, which proves such an approach is reliable to investigate the strong decays of the hadrons. As for the charmonia, the $R$ values are quite different in different quark model as shown in Table \ref{Tab:Dwave}. For example, the $R$ value of $1D$ states are estimated to be $2.59$, $1.84$ and $1.88 \ \rm{GeV}^{-1}$ by SP, GI and MGI model, respectively. It should be noticed that the mass of charm quark is taken as $1.4045 $ GeV and $1.65$ GeV in the SP and MGI models, respectively. The mass spectra and the wave function depend on both the quark mass and the potential between quark and anti-quark. Thus, the large discrepancy of $R$ values in SP and MGI model could be understood. In the $^3P_0$ model, the constituent quark masses for the charm, up/down, strange quarks are adopted to be 1.60, 0.22 and 0.419 GeV, respectively \cite{Chen:2016iua, Liu:2009fe}. The mass of the charm quark in $^3P_0$ model is very close to the one in MGI model. Thus in the present work, we vary the $R$ values of the charmonia around the one of MGI model to check the $R$ dependence of the decay widths. In addition, similar to the case of determining the masses of the missing $D$ wave states, the $R$ values of $1D$, $2D$ and $3D$ states could be determined such that it could reproduce the widths of $\psi(3770)$, $\psi(4160)$ and $Y(4320)$, respectively, which could also reduce the uncertainties of quark model.
The masses and $R$ values of the involved mesons are presented in Table \ref{Tab:mass}.
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{1.0}{\includegraphics{psi1d.eps}}
\caption{(Color online). The partial width of $\psi(1D) \to D\bar{D}$ (left panel) and $\psi_3(1D) \to D\bar{D}$ (right panel). The cyan band indicates the PDG average of the corresponding partial width. The $R$ value estimated by MGI model are marked by red arrow. \label{Fig:psi1D}}
\end{figure}
\subsection{Open Charm Decays of $1D$ States}
As for $1D$ charmonia, their masses are all above the $D\bar{D}$ threshold, however, $\eta_{c2}$ and $\psi_2(1D)$ can not decay into $D\bar{D}$ due to $J^P$ quantum numbers violation. As for the $\psi(3770)$, the only open charm decay mode is $D\bar{D}$ due to kinematics limit. The $R$ dependence of the partial width of $\psi(3770)\to D\bar{D}$ is presented in the left panel of Fig. \ref{Fig:psi1D}. The $R$ value estimated by MGI model is marked by the red arrow and with this $R$ value, the partial width of $\psi(3770) \to D\bar{D}$ is evaluated to be 29.1 MeV. The PDG average of the the branching ratio for $\psi(3770) \to D\bar{D} $ is $(93^{+8}_{-9}) \%$ and the width of the $\psi(3770)$ is $27.2\pm 1$ MeV \cite{Patrignani:2016xqp}. Thus, the measured partial width of $\psi(3770) \to D\bar{D}$ is $22.7 \sim 27.3 $ MeV, which indicates the partial width with $R$ value in MGI model is approximately consistent with the experimental measurement. Moreover, we vary $R$ value from $1.6\ \mathrm{GeV}^{-1}$ to $2.0\ \mathrm{GeV}^{-1}$ and find that the estimated partial width of $\psi(3770) \to D \bar{D}$ with $R=1.6 \sim 1.76 \ \mathrm{GeV}^{-1}$ could well reproduce the experimental measurement. Taking $\psi(3770)$ as a scale of $1D$ charmonia, the partial width of $\psi_3(1D) \to D\bar{D}$ is $0.34 \sim 0.46$ MeV, which is consistent with the theoretical estimations in Ref. \cite{Barnes:2005pb, Deng:2016stx} and safely under the measured width from the LHCb Collaboration \cite{LHCbnew}
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{psi12d.eps}}
\caption{(Color online). Partial and total widths of $\psi(4160)$. The cyan horizontal band is the PDG average of the total width and the light vertical grey band is the $R$ range determined by the overlap of theoretical estimation and experimental data. The $R$ value estimated by MGI model are marked by red arrow. \label{Fig:psi12d}}
\end{figure}
\subsection{Open Charm Decays of $2D$ States}
The $R$ dependent partial and total widths of $\psi(4160)$ are presented in Fig. \ref{Fig:psi12d}. By taking the $R$ value determined in MGI model, the width of $\psi(4160)$ is estimated to be 147.4 MeV, which is about two time larger than the PDG average one, i.e., $70 \pm 10$ MeV. It should noticed that the mass of $\psi(2D)$ is estimated to be 4.125 GeV in the MGI model, which is much smaller than the measured one, thus we discuss the the decay behavior of $2D$ states in the $R$ range determined by comparing the estimated width with the experimental data \cite{Patrignani:2016xqp}, which is $R= (1.82 \sim 1.97) \ \mathrm{GeV}^{-1}$. Moreover, the determined $R$ value of $\psi(4160)$ is a bit larger than the one of $\psi(3770)$, which is consistent with the expectation. In this $R$ range, our results indicates the $\psi(4160)$ dominantly decays into $D\bar{D}$, $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$, while the partial widths of $D_s^+ D_s^-$ and $D_s^+ D_s^{\ast -}$ are less than 1 MeV. In this $R$ range, the ratios of the partial widths of open charmed processes are estimated to be
\begin{eqnarray}
\Gamma(\psi(4160)\to D\bar{D})\over \Gamma(\psi(4160) \to D^\ast \bar{D}^\ast) &=&0.71 \sim 1.72 \nonumber\\
\Gamma(\psi(4160)\to D^\ast \bar{D})\over\Gamma(\psi(4160) \to D^\ast \bar{D}^\ast)&=&0.01 \sim 0.31 \nonumber
\end{eqnarray}
These ratios are evaluated to be 0.46/0.01 and 0.2/0.05 by the QPC model with relativistic quark model and linear potential model, respectively\cite{Barnes:2005pb, Gui:2018rvv}. In Ref. \cite{Eichten:2005ga} , by using the Connell coupled- channel mode, the ratios are determined to be 0.08 and 0.16. On the experimental side, the BaBar collaboration performed a measurement of the exclusive production of $D\bar{D}$, $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$, the ratios were measured to be $0.02 \pm 0.03 \pm 0.02$ and $0.34 \pm 0.14 \pm 0.05$ \cite{Aubert:2009aq}, respectively, which is different from the QPC model estimations in the present work. It should be noticed that in Ref. \cite{Aubert:2009aq}, the data are fitted with three charmonia with fixed mass and width, which are $\psi(4040)$, $\psi(4160)$ and $\psi(4415)$. From the current situation, there should exist more vector states in this energy range and thus the fitted results will be changed if more states are included. Moreover, in the analysis, the mass and width of $\psi(4160)$ are fixed to be $4153$ MeV and $103$ MeV, respectively \cite{Amsler:2008zzb}. The values of the resonance parameters used in Ref. \cite{Aubert:2009aq} are much different from latest PDG average, which are $4191$ MeV and $70$ MeV, respectively \cite{Patrignani:2016xqp}. We expect the new precise measurement and analysis of the open charm decays of $\psi(4160)$ at BESIII, BelleII and LHCb could determine these ratios and test the results in the present work.
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{etac22d.eps}}
\caption{(Color online). Partial and total widths of $\eta_{c2}(2D)$. The light grey band is the $R$ range determined by the comparison of $\psi(4160)$ total width with the experimental data. The $R$ value estimated by MGI model are marked by the arrow. \label{Fig:etac22d}}
\end{figure}
Taking the $R$ range determined by the width of $\psi(4160)$, we can investigate the open charm decays of other $2D$ charmonium states. As for $\eta_{c2}(2D)$, the partial and total widths depending on $R$ value are presented in Fig. \ref{Fig:etac22d}. The total width of $\eta_{c2}(2D)$ is estimated to be $48\sim 64 $ MeV. The dominant decay modes are $D^\ast \bar{D}^\ast$ and $D^\ast \bar{D}$ and the ratio of these two decay channels is estimated to be
\begin{eqnarray}
\frac{\Gamma(\eta_{c2}(2D) \to D^\ast \bar{D})}{\Gamma(\eta_{c2}(2D) \to D^\ast \bar{D}^\ast)} = 0.7 \sim 1.5,
\end{eqnarray}
which indicates the partial width of $\Gamma(\eta_{c2}(2D) \to D^\ast \bar{D})$ and $\Gamma(\eta_{c2}(2D) \to D^\ast \bar{D}^\ast)$ are very similar. As for the $D_s^{\ast+} D_s^{-}$ and $D \bar{D}_0$ modes, their partial widths are less than 1 MeV.
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{psi22d.eps}}
\caption{(Color online). The same as Fig. \ref{Fig:etac22d} but for $\psi_2(2D)$ charmonium. \label{Fig:psi22d}}
\end{figure}
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{psi32d.eps}}
\caption{(Color online). The same as Fig. \ref{Fig:etac22d} but for $\psi_3(2D)$ charmonium. \label{Fig:psi32d}}
\end{figure}
As for $\psi_2(2D)$, the total width is very weakly dependent on the $R$ values in the determined $R$ range, and it is estimated to be $50 \sim 52$ MeV. Similar to the case of $\eta_{c2}(2D)$, the dominant decay modes of $\psi_2(2D)$ are also $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$, and the ratio of the partial widths for these two channels are estimated to be,
\begin{eqnarray}
\frac{\Gamma(\psi_2(2D) \to D^\ast \bar{D})}{\Gamma(\psi_{2}(2D) \to D^\ast \bar{D}^\ast)} = 0.6 \sim 1.4.
\end{eqnarray}
As for $\psi_3(2D)$, the total width are estimated to be $52 \sim 76$ MeV in the determined $R$ range. Such a large width mainly comes from the $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$ modes since $\psi_3(2D)$ decays into $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$ are also via $P$ wave. The partial widths ratio of these two channel is predicted to be,
\begin{eqnarray}
\frac{\Gamma(\psi_{3}(2D) \to D^\ast \bar{D})}{\Gamma(\psi_{3}(2D) \to D^\ast \bar{D}^\ast)} = 0.5 \sim 0.8.
\end{eqnarray}
Compared with the above two channels, the partial width of $D\bar{D}$ mode is much smaller due to the high partial wave suppression.
\subsection{Open Charm Decays of $3D$ States}
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{psi13d.eps}}
\caption{(Color online). The same as Fig. \ref{Fig:psi12d} but for $\psi_1(3D)$\label{Fig:psi13d}}
\end{figure}
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{etac23d.eps}}
\caption{(Color online). Partial and total widths of $\eta_{c2}(3D)$. The light grey band is the $R$ range determined by the comparison of $Y(4320)$ total width with the experimental data, where $Y(4320)$ is assigned as $\psi_1(3D)$ charmonium. \label{Fig:etac23d}}
\end{figure}
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{psi23d.eps}}
\caption{(Color online). The same as Fig. \ref{Fig:etac23d} but for $\psi_2(3D)$ charmonium. \label{Fig:psi23d}}
\end{figure}
\begin{figure}[htb]
\vspace{0.25cm}
\scalebox{0.8}{\includegraphics{psi33d.eps}}
\caption{(Color online). The same as Fig. \ref{Fig:etac23d} but for $\psi_3(3D)$ charmonium. \label{Fig:psi33d}}
\end{figure}
The $R$ dependent total and partial widths of the open charm decays of $Y(4320)$ are presented in Fig. \ref{Fig:psi13d}, where $Y(4320)$ is assigned as $\psi(3^3D_1)$ charmonium. In the MGI model, the $R$ value of $3D$ charmonia is $2.85 \ \mathrm{GeV}^{-1}$ and with this $R$ value, the width of $\psi(3^3D_1)$ is $121.7$ MeV, which is consistent with one of $Y(4320)$, i.e., $(101.4^{+25.3}_{-19.7} \pm 10.2) \ \mathrm{MeV}$ \cite{Ablikim:2016qzw}. To further check the $R$ dependence of the total width and partial width of $\psi(3^3D_1)$, we vary $R$ value from $2.5 \ \mathrm{GeV}^{-1}$ to $3.0 \ \mathrm{GeV}^{-1}$. Our estimations indicate that when $R=2.50 \sim 2.92 \ \mathrm{GeV}^{-1}$ the evaluated total width are consistent with the measured one from the BESIII Collaboration \cite{Ablikim:2016qzw}. This $R$ value for $\psi(3^3D_1)$ is larger than the one of $\psi(4160)$ and $\psi(3770)$, which is consistent with our expectation. In this $R$ range, $\psi(3^3D_1)$ dominantly decays into $D\bar{D}$, $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$. The partial width of $\psi(3^3D_1) \to D\bar{D}$ weakly depend on the paramter $R$, and in the determined $R$ range, we find $\Gamma(\psi(3^3D_1)\to D\bar{D})= 45.2 \sim 48.0\ \mathrm{MeV}$. And in this $R$ range, the partial widths of $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$ are estimated to be $24.3 \sim 46.0 $ and $4.9 \sim 11.9 \ \mathrm{MeV}$, respectively. The ratios of the partial widths of these dominant decay channels are predicted to be
\begin{eqnarray}
\Gamma(Y(4320)\to D \bar{D})\over \Gamma(Y(4320) \to D^\ast \bar{D}^\ast) &=&4.0 \sim 9.2 \nonumber\\
\Gamma(Y(4320)\to D^\ast \bar{D})\over\Gamma(Y(4320) \to D^\ast \bar{D}^\ast)&=&3.8 \sim 4.5, \nonumber
\end{eqnarray}
respectively.
As for $\eta_{c2}(3D)$, the total and partial widths depending on the model parameter $R$ are presented in Fig. \ref{Fig:etac23d}. In the $R$ range determined by $Y(4320)$, we find the total width of $\eta_{c2}(3D)$ is strongly dependent on the model parameter. In particular, the total width is estimated to be $47 \sim 114$ MeV in this $R$ range. Moreover, our estimation indicates that the $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$ should be the dominant decay modes of $\eta_{c2}(3D)$ and the partial widths ratio of these two modes is estimated to be,
\begin{eqnarray}
\frac{\Gamma(\eta_{c2} \to D^\ast \bar{D})}{ \Gamma(\eta_{c2} \to D^\ast \bar{D}^\ast)} =1.6 \sim 3.8.
\end{eqnarray}
The total and partial widths of $\psi_2(3D)$ are presented in Fig. \ref{Fig:psi23d}. The total width is estimated to be $54 \sim 113$ MeV and in the determined $R$ range, $D^\ast \bar{D}$ is the dominant decay modes, the branching ratio of this decay mode is $(76 \sim 86) \%$. As shown in Fig. \ref{Fig:psi33d}, the total width of $\psi_3(3D)$ is also strong dependent on the model parameter and predicted to be $17\sim 89$ MeV. In the determined $R$ range, the dominant decay mode is $D^\ast \bar{D}^\ast$, and its branching ratio is estimated to be $(86 \sim 93) \%$.
It should be noticed the measured width of $Y(4320)$ has a relative large uncertainty, thus in a large $R$ range our estimation could overlap with the measured data. Then the predicted total and partial widths for $\eta_{c2}(3D),\ \psi_{2}(3D)$ and $\psi_3(3D)$ vary in a relative large range. However, the dominant decay modes and the partial widths ratios are weakly dependent on the model parameter, which are helpful for searching these missing $3D$ states.
\section{Summary}
The observations of the vector charmonium-like states in the $e^+ e^-$ annihilation processes make the states between $4.0$ and $ 4.5$ GeV overcrowed. Besides the higher excited $J/\psi$ state, these charmonium-like states could also be higher $\psi(^3D_1)$ states. Moreover, a $\psi_3(1D)$ candidate was observed in the $D\bar{D}$ invariant mass spectroscopy very recently by the LHCb Collaboration. These experimental measurements stimulate us to comb $D$ wave charmonium states. In the present work, by investigating the open charm decay behaviors, we evaluate the possibility of $Y(4320)$ as $\psi(3^3D_1)$ charmonium and then take $\psi(3770)$, $\psi(4160)$ and $Y(4320)$ as the scales to evaluate the open charm decays of other $1D$, $2D$ and $3D$ charmonium states.
Our estimations indicate that the total widths of $\psi(3770)$ and $\psi(4160)$ can be reproduced in a proper $R $ range, which are $R=1.60 \sim 1.76\ \mathrm{GeV}^{-1}$ and $1.82\sim 1.97)\ \mathrm{GeV}^{-1}$ for $\psi(3770)$ and $\psi(4160)$, respectively. As for $Y(4320)$, the estimated total width can overlap with the measured one when we take $R=2.50 \sim 2.92 \ \mathrm{GeV}^{-1}$, which indicates $Y(4320)$ can be a $\psi_1(3D)$ candidate. It should be notice that the $R$ range for $1D$ is very close to the one of estimated in MGI model, and for $3D$ charmonium, the $R$ value estimated in MGI model is consistent with the $R$ range determined by the width of $\psi(3^3D_1)$ state. However, as for $2D$ state, the $R$ value estimated in MGI model is much larger than the range determined by the width of $\psi(4160)$.
Taking $\psi(3770)$, $\psi(4160)$ and $\psi(4320)$ as the scale of $1D$, $2D$ and $3D$ charmonia, respectively, we can estimated the open charm decays of other $D$ wave charmonium states. From our estimations, we find,
\begin{itemize}
\item $\psi(4382)$ could be assigned as $\psi_3(1D)$ state. The narrow width resulted from the high partial wave suppression and our estimated partial width $\psi_3(1D) \to D\bar{D}$ is safely under the measured width of $\psi(3842)$ and consistent with the theoretical estimations of other group.
\item As for $2D$ states, we predict the total widths of $\eta_{c2}(2D)$, $\psi_2(2D)$ and $\psi_3(2D)$ to be $48 \sim 64 $, $50\sim 52$ and $52 \sim 76$ MeV, respectively. We also find that the dominant decay modes of these three $D$ wave charmonia are $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$, respectively. Furthermore, the partial widths ratios of these dominant channels are also predicted.
\item As for $3D$ states, the predicted total widths of $\eta_{c2}(3D)$, $\psi_2(3D)$ and $\psi_3(3D)$ are in a relative large range due to the large uncertainty of $R$ determined by the width of $Y(4320)$. However, we find the predominant decay modes are $D^\ast\bar{D}$, $D^\ast\bar{D}$ and $D^\ast\bar{D}^\ast$ for $\eta_{c2}(3D)$, $\psi_2(3D)$ and $\psi_3(3D)$, respectively. Moreover, some partial widths ratios are predicted, which are nonsensitive to the model parameter $R$.
\end{itemize}
The open charm decay channels are the important observation channels of higher charmonia since they are the dominant decay channels of these higher charmonia. The charmonia produced in hadronproduction process have more possible $J^{PC}$ quantum numbers. Thus, hadronproduction is one of most promising process of searching for the higher $D$ wave charmonia in the open charm mass spectroscopy. All the estimations in the present work could be helpful for searching for the missing $2D$ and $3D$ charmonia in the open charm decay channels in the further experimental measurements at LHCb.
\section*{Acknowledgement}
The authors would like to thank Jun-Zhang Wang for useful discussion. This project is supported by the National Natural Science Foundation of China under Grant No. 11775050, No. 11375240 and No. 11675228, Nature Science Foundation Projects of Qinghai Office of Science and Technology, No. 2017-ZJ-748, the Chunhui Plan of China’s Ministry of Education, No. Z2017054, the Natural Science Foundation of Jiangsu Province of China under contract No. BK20171349.
|
1,314,259,996,733 | arxiv | \section{Supplemental Material}
\subsection{S.1. Multifractal analysis}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=.5 \textwidth] {SFig.eps}
\end{center}
\caption{(Color online) $\mathrm{MIPR}$ as a function of $1/L^2$ with (a) $\lambda=0.8$, (b) $\lambda=2.0$, and (c) $\lambda=3.5$, respectively. The solid line is a fitting line. Here, $\alpha_{x}=(\sqrt{5}-1)/2$, $\alpha_y=\sqrt{3}$, and under OBCs.}\label{SFig}
\end{figure}
To further strength the localization of the 2D non-Hermitian quasicrystal with $\alpha_x \ne \alpha_y$, we study the scaling behavior of eigenstats by performing a multifracal analysis. We define the mean inverse participation ratio (MIPR): $\mathrm{MIPR}=\sum_{j} \mathrm{IPR}_j /L^2$, where $\mathrm{IPR}_j$ represents the IPR of the $j$-th normalized eigenstate. Figure \ref{SFig} shows $\mathrm{MIPR}$ as a function of $1/L^2$ for different $\lambda$ with $\alpha_{x}=(\sqrt{5}-1)/2$, $\alpha_y=\sqrt{3}$ under OBCs. In the extended regime shown in Fig. \ref{SFig}(a) with $\lambda=0.8$, $\mathrm{MIPR} \propto L^{-2}$, and in the large size limit, the MIPR approaches to $0$. When $\lambda$ is localized in the intermediate regime, $\mathrm{MIPR}$ decays as $L^{-2}$ with the increase of $L$, and finial tends to a finite value about $0.1178$ [see Fig. \ref{SFig}(b) with $\lambda=2.0$]. As shown in Fig. \ref{SFig}(c) with $\lambda=3.5$ in the localized regime, $\mathrm{MIPR}$ keeps a finite finite value about $0.6838$ with the increase of the size of the system.
\subsection{S.2. Winding number}
To calculate the topological number of the 2D non-Hermitian quasicrystal described by Eq. (3) in the main text, we take a Fourier transformation
\begin{equation}\label{seq1}
f_{l_x,l_y}=\frac{1}{L}\sum_{m,n} e^{i 2\pi (\alpha_x l_x m+\alpha_y l_y n)} \psi_{mn}.
\end{equation}
In the thermodynamics limit, the duality equation can be written as
\begin{equation}\label{seq2}
\lambda f_{l_x+1,l_y+1}-2[\cos{2\pi\alpha_x l_x}+\cos{2\pi\alpha_y l_y}]f_{l_x,l_y}=\omega f_{l_x,l_y}.
\end{equation}
It is easy to find that the 2D duality equation can be reduced to 1D chains along the diagonal direction, and these 1D chains are decoupled. Hence, for a infinite 2D system, it can be equivalent to a serial of 1D chains labeled by different $l=l_x-l_y$. For a given $l$, the eigenvalue equation of the 1D chain is given as follows:
\begin{equation}\label{seq3}
\lambda f_{l_x+1}-2[\cos{(2\pi\alpha_x l_x)}+\cos{(2\pi\alpha_y l_x-\theta)}]f_{l_x}=\omega f_{l_x},
\end{equation}
with $\theta=2\pi \alpha_y l$. The winding number of the 1D non-Hermitian chain is defined as
\begin{equation}\label{seq4}
w=\frac{1}{2\pi i}\int_{0}^{2\pi} d\phi \partial_{\phi} \ln{\det{[\mathcal{H}(\phi)-\omega_B]}},
\end{equation}
where $\mathcal{H}$ is the Hamiltonian of the 1D non-Hermitian chain under the periodic boundary condition, the phase $\phi$ is considered as a magnetic flux through a non-Hermitian ring, and the base frequency $\omega_B=0$ is chosen. The matrix form of the 1D non-Hermitian chain with a magnetic flux is written as
\begin{equation}\label{seq5}
\mathcal{H}(\phi)=\begin{pmatrix} -V_1 & \lambda & 0 & 0 & \cdots & 0 & 0 \\
0 & -V_2 & \lambda & 0 & \cdots & 0 & 0\\
0 & 0 & -V_3 & \lambda & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\
0 & 0 & 0 & 0 & \cdots & -V_{L-1} & \lambda\\
\lambda e^{i\phi} & 0 & 0 & 0 & \cdots & 0 & -V_L \\
\end{pmatrix},
\end{equation}
with $V_{l_x}=2[\cos{(2\pi\alpha_x l_x)}+\cos{(2\pi\alpha_y l_x-\theta)}]$. Hence, the determination of $\mathcal{H}(\phi)$
\begin{equation}\label{seq6}
D(\phi)=\det{[\mathcal{H}(\phi)]}=(-1)^L\left(\prod_{l_x=1}^{L}V_{l_x}-\lambda^L e^{i\phi}\right).
\end{equation}
Since the winding number of the non-Hermitian case describes the complex spectral trajectory encircles the base point $\omega_B=0$ when the flux varies from $0$ to $2\pi$, we can rewrite the winding number as follows:
\begin{equation}\label{seq7}
w=\frac{1}{2}\sum_{j} \mathrm{sgn}\left\{\mathrm{Re}[D(\phi_j)]\right\}\cdot \mathrm{sgn} \left\{\frac{d \mathrm{Im}[D(\phi_j)]}{d\phi}\right\},
\end{equation}
where $\mathrm{Re}[D(\phi)]=(-1)^L\left(\prod_{l_x=1}^{L}V_{l_x}-\lambda^L\cos{\phi}\right)$, $\mathrm{Im}[D(\phi)]=(-1)^{L+1}\lambda^L\sin{\phi}$, and $\mathrm{sgn}(\cdots)$ denotes the sign operator. $\phi_j$ is the $j$th solution of $\mathrm{Im}[D(\phi)]=0$ with $\phi_1=0$ and $\phi_2=\pi$, respectively. We have
\begin{align}\label{seq8}
w &=\frac{1}{2}\left[\mathrm{sgn}\left( \prod_{l_x=1}^{L}V_{l_x}+\lambda^L\right)- \mathrm{sgn}\left( \prod_{l_x=1}^{L}V_{l_x}-\lambda^L\right) \right] \notag \\
&= S\left(\lambda^L-\left|\prod_{l_x=1}^{L}V_{l_x}\right|\right),
\end{align}
with the step function
\begin{equation}\label{seq9}
S(x) = \begin{cases} 0, \quad x<0 \\
1/2,\quad x=0 \\
1,\quad x>0.
\end{cases}
\end{equation}
The topological phase transition point is determined by
\begin{equation}\label{seq10}
\lambda^L=\left|\prod_{l_x=1}^{L} V_{l_x}\right|.
\end{equation}
In the large $L$ limit, we have
\begin{align}\label{seq11}
\ln{\lambda} = &\lim_{L\to\infty} \frac{1}{L} \sum_{l_x=1}^{L}\ln{|V_{l_x}|} \notag \\
= & \int_{0}^{1} \ln{\left|2[\cos{(2\pi\alpha_x Lx)}+\cos{(2\pi\alpha_y Lx-\theta)}]\right|} dx \notag \\
= & 2\ln{2}+\int_{0}^{1} \ln{\left|\cos{\frac{2\pi(\alpha_x+\alpha_y)Lx-\theta}{2}}\right|} dx \notag \\
& + \int_{0}^{1} \ln{\left|\cos{\frac{2\pi(\alpha_x-\alpha_y)Lx+\theta}{2}}\right|} dx \notag \\
= & 2\ln{2}+\frac{1}{\pi(\alpha_x+\alpha_y)L}\mathcal{L}\left[\pi(\alpha_x+\alpha_y)L\right] \notag \\
& +\frac{1}{\pi(\alpha_x -\alpha_y)L}\mathcal{L}\left[\pi(\alpha_x-\alpha_y)L\right] \notag \\
\approx & 0,
\end{align}
where
\begin{align}\label{seq12}
\mathcal{L}(x) &=\int_0^{x} \ln{\cos{(x^{\prime})}}dx^{\prime} \notag \\
&=-x\ln{2}+\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\sin(2kx)}{k^2}.
\end{align}
It means that in thermodynamic limit $L\to \infty$, the topological phase transition point at $\lambda_{c1}=e^{0}=1$.
\subsection{S.3. Energy spectrum of the 2D non-Hermitian quasicrystal with the finite size}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=.5 \textwidth] {SFig1.eps}
\end{center}
\caption{(Color online) Eigenvalues of the 2D non-Hermitian quasicrystal with (a) $\lambda=0.8$, (b) $\lambda=2.0$, and (c) $\lambda=3.5$, respectively. Here, $\alpha_{x}=(\sqrt{5}-1)/2$, $\alpha_y=\sqrt{3}$, $L\times L=80\times 80$, and under OBCs.}\label{SFig1}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=.5 \textwidth] {SFig2.eps}
\end{center}
\caption{(Color online) (a) $R_{\mathrm{Im}}$ as a function of $1/L^2$ with $\lambda=0.8$. The solid line is a fitting line. (b) $R_{\mathrm{Im}}$ vs $\lambda$ with $L\times L=80\times 80$. Here, $\alpha_{x}=(\sqrt{5}-1)/2$, $\alpha_y=\sqrt{3}$, and under OBCs.}\label{SFig2}
\end{figure}
In the main text, we study a real-complex transition of the energy spectrum for a 2D non-Hermitian quasicrystal in the large $L$ limit. For our model with an infinite size, the system in the duality space can be reduced to a serial of decoupled chains along the diagonal direction labeled by $l=l_x-l_y$. Our results imply that the real-complex transition point corresponds to the topological phase transition point $\lambda_{c1}$ in the thermodynamic limit. However, for an arbitrary finite system $L\times L$ in duality space, the length of the 1D chains shrinks along the anti-diagonal direction from the center to the corners, and finally, the 1D chain turns to a single lattice at the corners. It means the finite-size effect is remarkable in our system. Unlike in the infinite case that all the eigenvalues are real in $\lambda<\lambda_{c1}=1$, a finite number of complex energies is observed, which is shown in Fig. \ref{SFig1}(a) with $L\times L=80\times 80$ and $\lambda=0.8$ under OBCs. Due to the finite-size effect, the parts of eigenvalues deviate from that in the thermodynamic limit, as shown in Figs. \ref{SFig1}(b) and \ref{SFig1}(c). We define $R_{\mathrm{Im}}=N_{\mathrm{Im}}/L^2$, where $N_{\mathrm{Im}}$ is the number of eigenvalues with the modulus of the imaginary parts above the cutoff $10^{-6}$. Figures \ref{SFig2}(a) shows $R_{\mathrm{Im}}$ as a function of $1/L^2$ with $\lambda=0.8$ under OBCs. The results imply that in the current geometric configuration, the number of the complex eigenvalues of the finite 2D non-Hermitian quasicrystal keeps finite. We also calculate the behavior of $R_{\mathrm{Im}}$ versus $\lambda$. For $\lambda<\lambda_{c2}$, $R_{\mathrm{Im}}$ displays a monotonous increase, and when $\lambda\ge \lambda_{c2}$, $R_{\mathrm{Im}}=1$.
\end{document}
|
1,314,259,996,734 | arxiv | \section{Introduction}\label{sec:intro}
When $G$ is an algebraic group acting on a smooth algebraic variety $X$ over $\bb{C}$, it is a natural problem to describe the simple $G$-equivariant holonomic $\mathcal{D}$-modules on $X$. When $G$ acts with finitely many orbits, all such $\mathcal{D}$-modules have regular singularities, and they are classified via the Riemann--Hilbert correspondence by the $G$-equivariant simple local systems on the orbits of the group action. Describing these $\mathcal{D}$-modules explicitly is however a difficult problem (see Open Problem 3 in \cite[Section~6]{mac-vil}, and \cite{vilonen}). In this paper we consider the case when $X$ is a vector space of matrices (general, symmetric, or skew-symmetric), and $G$ is a natural rank preserving group of symmetries. In all these cases $G$ is a reductive group and the $\mathcal{D}$-modules are \defi{$G$-admissible} representations (they decompose into a direct sum of irreducible representations, each appearing with finite multiplicity). The purpose of this paper is to describe these representations (which we will refer to as the \defi{characters} of the equivariant $\mathcal{D}$-modules) and to realize these $\mathcal{D}$-modules explicitly. The motivation for this work is two-fold:
\begin{itemize}
\item {\bf Computing local cohomology.} In \cites{raicu-weyman,raicu-weyman-witt,raicu-weyman-loccoh} we describe the characters, and the $\mathcal{D}$-module composition factors of the local cohomology modules $\mc{H}_Y^{\bullet}(X,\mc{O}_X)$ in the case when $X$ is a space of matrices (general, symmetric, or skew-symmetric), and $Y$ is any orbit closure for the natural group action on $X$. We expect that the combination of $\mathcal{D}$-module and commutative algebra techniques that we employ to study local cohomology in the case of matrices will apply to other cases of interest \cite[Appendix]{levasseur}. We note that character calculations in the context of analyzing local cohomology modules appear also in \cites{kempf,VdB:loccoh}: in both cases, the representations are $T$-admissible for $T$ a maximal torus in $G$; the equivariant $\mathcal{D}$-modules that we study in this paper are $G$-admissible, but in general they are too large to be $T$-admissible.
\item {\bf Levasseur's conjecture.} For a class of multiplicity-free $G$-representations $X$, Levasseur conjectured \cite[Conjecture~5.17]{levasseur} an equivalence between the category $\mc{C}$ of equivariant holonomic $\mathcal{D}$-modules whose characteristic variety is a union of conormal varieties to the orbits of the group action, and a module category admitting a nice quiver description. His formulation is equivalent to the fact that any simple $\mathcal{D}$-module $\mc{M}$ in $\mc{C}$ contains sections which are invariant under the action of the derived subgroup $G'=[G,G]$. Our character description provides a direct proof of this conjecture for general and skew-symmetric matrices, and yields counterexamples for symmetric matrices.
\end{itemize}
Our work complements the existing literature that studies the categories of $\mathcal{D}$-modules on rank stratifications \cites{nang-nxnmat,nang-skew} (see also \cite{braden-grinberg} for the corresponding categories of perverse sheaves), in that we realize concretely the simple objects of these categories and discuss some applications, filling some gaps in the arguments and generally painting a more transparent picture. To give a flavor of the level of concreteness that we seek, we begin with the following ($\bb{Z}^n_{\operatorname{dom}}$ denotes the set of \defi{dominant weights} $\ll=(\ll_1\geq\cdots\geq\ll_n)\in\bb{Z}^n$, and $S_{\ll}$ denotes the \defi{Schur functor} associated to $\ll$; throughout the paper we use the convention $\ll_s=\infty$ for $s\leq 0$, $\ll_s=-\infty$ for $s>n$):
\begin{theorem}\label{thm:nxnbasicthm}
Let $X=\bb{C}^{n\times n}$ be the vector space of $n\times n$ matrices, and let $S=\bb{C}[x_{i,j}]$ be the coordinate ring of $X$. If we write $\det=\det(x_{i,j})$, and let $S_{\det}$ be the localization of $S$ at $\det$, then we have a filtration
\[0\subsetneq S\subsetneq\langle\det^{-1}\rangle_{\mathcal{D}}\subsetneq\cdots\subsetneq\langle\det^{-n}\rangle_{\mathcal{D}}=S_{\det},\]
where $F_s=\langle \det^{-s}\rangle_{\mathcal{D}}$ denotes the $\mathcal{D}$-submodule of $S_{\det}$ generated by $\det^{-s}$ for $s=0,\cdots,n$ (and $F_{-1}=0$). The successive quotients $A_s=F_s/F_{s-1}$, $s=0,\cdots,n$ are the simple $\operatorname{GL}_n(\bb{C})\times\operatorname{GL}_n(\bb{C})$-equivariant holonomic $\mathcal{D}$-modules on $X$ (for the natural action by row and column operations) and their characters are given by
\[A_s=\bigoplus_{\substack{\ll\in\bb{Z}^n_{\operatorname{dom}} \\ \ll_s\geq s\geq\ll_{s+1}}}S_{\ll}\bb{C}^n\otimes S_{\ll}\bb{C}^n.\]
\end{theorem}
In the case of symmetric matrices, the $\mathcal{D}$-modules obtained as in Theorem~\ref{thm:nxnbasicthm} cover roughly half of the simple equivariant $\mathcal{D}$-modules. The remaining half are more mysterious, and they provide counterexamples to \cite[Conjecture~5.17]{levasseur}. In the case of $m\times n$ matrices with $m>n$, as well as in the case of skew-symmetric matrices of odd size, the simple equivariant $\mathcal{D}$-modules arise as local cohomology modules, while in the case of skew-symmetric matrices of even size the simple equivariant $\mathcal{D}$-modules arise, just as in Theorem~\ref{thm:nxnbasicthm}, from the pole order filtration associated with the Pfaffian of the generic skew-symmetric matrix. Most of our simple $\mathcal{D}$-modules have irreducible characteristic variety, but for roughly half of the ones arising from symmetric matrices the characteristic variety has two connected components: this is deduced in Remark~\ref{rem:charvarieties} as a consequence of the character information.
As suggested by Theorem~\ref{thm:nxnbasicthm}, one motivation behind our investigation is that the simple $\mathcal{D}$-modules are the building blocks for many $\mathcal{D}$-modules of interest that one would like to understand. More precisely, every holonomic $\mathcal{D}$-module $\mc{M}$ has finite length, i.e. it has a finite filtration (\defi{composition series}) whose successive quotients (\defi{composition factors}) are simple holonomic $\mathcal{D}$-modules. When $G$ is connected and $\mc{M}$ is $G$-equivariant, the composition factors are also $G$-equivariant \cite[Prop.~3.1.2]{VdB:loccoh}. We are mainly interested in two types of $G$-equivariant holonomic $\mathcal{D}$-modules:
\begin{itemize}
\item {\bf Local cohomology modules.} If $Y\subset X$ is a $G$-invariant subset, then the local cohomology modules $\mc{H}_Y^{\bullet}(X,\mc{O}_X)$ are $G$-equivariant $\mathcal{D}$-modules. If $Y$ is smooth and irreducible, and if we write $c=\operatorname{codim}_X(Y)$ for the codimension of $Y$ inside $X$, then $\mc{H}_Y^c(X,\mc{O}_X)$ is the unique non-vanishing local cohomology module and it is simple. In general, for an irreducible subvariety $Y\subset X$ one can define an \defi{intersection homology} $\mathcal{D}$-module $\mc{L}(Y,X)$ which is simple (and it is $G$-equivariant when $Y$ is a $G$-subvariety), and we have an inclusion $\mc{L}(Y,X)\subset\mc{H}_Y^c(X,\mc{O}_X)$, whose cokernel is suported on a proper subset of $Y$. The case when $X=\bb{C}^{n\times n}$ and $Y$ is the subvariety of singular matrices is implicitly described in Theorem~\ref{thm:nxnbasicthm}: $c=1$, $\mc{L}(Y,X)=A_1$, $\mc{H}^1_Y(X,\mc{O}_X)=S_{\det}/S$, and the cokernel $\mc{H}^1_Y(X,\mc{O}_X)/\mc{L}(Y,X)$ has composition factors $A_2,\cdots,A_n$. In general, the local cohomology modules $\mc{H}_Y^i(X,\mc{O}_X)$ for $i\neq c$ may be non-zero, but they are all supported on proper subsets of $Y$: it is an interesting problem to decide their (non)vanishing, or at a more refined level to understand their $\mathcal{D}$-module composition factors.
\item {\bf The $\mathcal{D}$-module (generated by) $f^{\a}$.} For a non-zero polynomial $f\in S=\bb{C}[x_1,\cdots,x_N]$ and a complex number $\a$, we can define $\langle f^{\a}\rangle_{\mathcal{D}}$ -- the (holonomic) $\mathcal{D}$-module generated by $f^{\a}$ (see \cite{walther-fs} for a recent survey). A strict inclusion $\langle f^{\a+1}\rangle_{\mathcal{D}}\subsetneq\langle f^{\a}\rangle_{\mathcal{D}}$ implies that $\a$ is a root of the Bernstein-Sato polynomial of $f$ (this can happen only when $\a$ is rational and negative \cite{kashiwara-bs}). It is an interesting question to decide whether each root $\a$ gives rise to such a strict inclusion \cite[Question~2.1]{walther-fs}, \cite[Question~1,~Section~4]{saito}. More generally, one may be interested in the composition factors of $\langle f^{\a}\rangle_{\mathcal{D}}$. For $\a\in\bb{Z}$ and $f=\det$ this is completely answered by Theorem~\ref{thm:nxnbasicthm}. When $\a\notin\bb{Z}$, $\langle\det^{\a}\rangle_{\mathcal{D}}$ is a simple $\mathcal{D}$-module (see the proof of Theorem~\ref{thm:nonequivLa}). Similar conclusions are obtained when $f$ is the symmetric determinant, or the Pfaffian of a skew-symmetric matrix of even size.
\end{itemize}
Before stating our results in more detail, we give a simple example to illustrate how character calculations alone can allow one to determine the $\mathcal{D}$-module composition factors.
\begin{example}\label{ex:TactsCN}
Let $X=\bb{C}^N$ be the $N$-dimensional affine space, and let $G=(\bb{C}^*)^N$ be the $N$-dimensional torus. The orbits $X_I$ of the $G$-action are indexed by subsets $I\subset[N]=\{1,\cdots,N\}$, where
\[X_I=\{x\in\bb{C}^N:x_i\neq 0\rm{ if and only if }i\in I\}.\]
The stabilizer of each $X_I$ is connected, so there is a one-to-one correspondence between orbits and simple $G$-equivariant holonomic $\mathcal{D}$-modules $D_I$ (Theorem~\ref{thm:equivRH}, Remark~\ref{rem:Ltrivial}), given by $D_I=\mc{L}(Y_I,X)$, where $Y_I=\ol{X_I}$ is the corresponding orbit closure. Since $Y_I$ is an affine space of codimension $N-|I|$, it is in particular smooth, and therefore the $\mathcal{D}$-module $D_I$ is just a local cohomology module $D_I=H_{Y_I}^{N-|I|}(X,\mc{O}_X)$. If we write $S=\bb{C}[x_1,\cdots,x_N]$ for the coordinate ring of $X$, then each $Y_I$ is defined by the ideal generated by the variables $x_j$, $j\notin I$. Using the \v Cech complex description of local cohomology we get
\[D_I=\bigoplus_{\substack{\ll\in\bb{Z}^N \\ \ll_i\geq 0\rm{ if and only if }i\in I}} \bb{C}\cdot x_1^{\ll_1}\cdots x_N^{\ll_N}\]
which is a decomposition into irreducible $G$-representations. If we take $f=x_1\cdots x_N$ then we get
\[S_f = \bigoplus_{\ll\in\bb{Z}^N}\bb{C}\cdot x_1^{\ll_1}\cdots x_N^{\ll_N}.\]
The torus weights appearing in the $D_I$'s form a partition of those appearing in $S_f$, so each $D_I$ appears as a $\mathcal{D}$-module composition factor of $S_f$ with multiplicity one. Using a similar argument for $X=\bb{C}^{n\times n}$ we obtain a proof of Theorem~\ref{thm:nxnbasicthm} (see Section~\ref{sec:mxnmatrices}).
\end{example}
\subsection*{Symmetric matrices}
Our results run in parallel for the three spaces of matrices (general, symmetric, and skew-symmetric). We have made the effort to apply a uniform strategy to all three cases, but we weren't able to treat the combinatorial details uniformly. For the sake of brevity, we have chosen to treat only the case of symmetric matrices in full detail, and only indicate the changes that are required in the other cases. Two features that make the case of symmetric matrices more interesting are: (a) the presence of non-trivial equivariant local systems; (b) the existence of counterexamples to Levasseur's conjecture.
For each positive integer $n$ and for $s=0,\cdots,n$, we consider the collections of dominant weights
\begin{equation}\label{eq:defmcC^isn}
\begin{aligned}
\mc{C}^1(s,n)&=\{\ll\in\bb{Z}^{n}_{\operatorname{dom}}:\ll_i\overset{(\operatorname{mod}\ 2)}{\equiv} s+1\rm{ for }i=1,\cdots,n,\ll_{s}\geq s+1\geq\ll_{s+2}\},\\
\mc{C}^2(s,n)&=\left\{\ll\in\bb{Z}^{n}_{\operatorname{dom}}:\ll_i\overset{(\operatorname{mod}\ 2)}{\equiv}
\begin{cases}
s+1 & \rm{for }i=1,\cdots,s\\
s & \rm{for }i=s+1,\cdots,n
\end{cases},
\ll_{s}\geq s+1,\ll_{s+1}\leq s
\right\}.
\end{aligned}
\end{equation}
Note that $\mc{C}^1(n,n)=\mc{C}^2(n,n)$. For a positive integer $n$, we identify $\operatorname{Sym}^2\bb{C}^n$ with the vector space $M^{\operatorname{symm}}$ of $n\times n$ symmetric matrices, where the squares $w^2$, $w\in\bb{C}^n$, correspond to matrices of rank at most one. We write $M^{\operatorname{symm}}_i$ for the subvariety of matrices of rank at most $i$. For $s=0,\cdots,n$, and $j=1,2$, we define
\[\mf{C}^j_s=\bigoplus_{\ll\in\mc{C}^j(s,n)} S_{\ll}\bb{C}^n.\]
\begin{charsym*}[Section~\ref{sec:symm}]
There exist $(2n+1)$ simple $\operatorname{GL}_n(\bb{C})$-equivariant holonomic $\mathcal{D}$-modules on $M^{\operatorname{symm}}$, whose characters are $\mf{C}^j_s$, $s=0,\cdots,n$, $j=1,2$. More precisely, if we denote by $C^j_s$ the $\mathcal{D}$-module with character $\mf{C}^j_s$ then $C^1_n=C^2_n=\mc{L}(\{0\},M^{\operatorname{symm}})$ is the simple holonomic $\mathcal{D}$-module supported at the origin, and for $s<n$
\[
C^j_s=\begin{cases}
\mc{L}(M^{\operatorname{symm}}_{n-s},M^{\operatorname{symm}}) & \rm{if }j\equiv s\ (\operatorname{mod}\ 2), \\
\mc{L}(M^{\operatorname{symm}}_{n-s},M^{\operatorname{symm}};1/2) & \rm{if }j\equiv s+1\ (\operatorname{mod}\ 2). \\
\end{cases}
\]
Here $\mc{L}(M^{\operatorname{symm}}_{n-s},M^{\operatorname{symm}})$ is the usual intersection homology $\mathcal{D}$-module, while $\mc{L}(M^{\operatorname{symm}}_{n-s},M^{\operatorname{symm}};1/2)$ is the intersection homology $\mathcal{D}$-module associated to the non-trivial irreducible $\operatorname{GL}_n(\bb{C})$-equivariant local system on the orbit of rank $(n-s)$ matrices.
We let $S=\bb{C}[x_{i,j}]$ be the coordinate ring of $M^{\operatorname{symm}}$, where $x_{i,j}=x_{j,i}$. We write $\operatorname{sdet}=\det(x_{i,j})$ for the determinant of the generic symmetric matrix, and let $S_{\operatorname{sdet}}$ be the localization of $S$ at $\operatorname{sdet}$. We consider $F_s=\langle \operatorname{sdet}^{-s/2}\rangle_{\mathcal{D}}$, the $\mathcal{D}$-submodule of $S_{\operatorname{sdet}}$ (or of $S_{\operatorname{sdet}}\cdot\operatorname{sdet}^{1/2}$) generated by $\operatorname{sdet}^{-s/2}$ for $s=0,\cdots,n+1$ (and $F_{-1}=0$). We have that $C^2_0=F_0=S$, and $C^1_s=F_{s+1}/F_{s-1}$ for $s=0,\cdots,n$.
\end{charsym*}
\begin{remark}\label{rem:failureLev}
The $\mathcal{D}$-modules $C^2_s$ for $s=1,\cdots,n-1$ contain no $\operatorname{SL}_n(\bb{C})$-invariant sections, so they provide counterexamples to \cite[Conjecture~5.17]{levasseur}. It may be interesting to note that when $n\geq 3$, among these counterexamples there are the intersection homology $\mathcal{D}$-modules $\mc{L}(M^{\operatorname{symm}}_{n-s},M^{\operatorname{symm}})$ with $s$ even, so the failure of Levasseur's conjecture can't be solely explained by the presence of non-trivial local systems!
\end{remark}
\begin{remark}\label{rem:bsato}
We can now give a quick derivation for the \defi{Bernstein-Sato polynomial} of $\operatorname{sdet}$ \cite[Appendix]{kimura}:
\begin{equation}\label{eq:bsatosdet}
b_{\operatorname{sdet}}(s)=\prod_{i=1}^n \left(s+\frac{1+i}{2}\right).
\end{equation}
It follows from Cayley's identity that $b_{\operatorname{sdet}}(s)$ divides $\prod_{i=1}^n \left(s+\frac{1+i}{2}\right)$, while for each $i=1,\cdots,n$ the strict inclusion $F_{i-1}\subsetneq F_{i+1}$ shows that $-\frac{1+i}{2}$ is a root of $b_{\operatorname{sdet}}(s)$. This is enough to conclude the equality (\ref{eq:bsatosdet}).
\end{remark}
\begin{remark}\label{rem:charvarieties}
It is interesting to note that the character calculation allows us to determine the characteristic varieties for the $\mathcal{D}$-modules $C_s^j$. The Fourier transform $\mc{F}$ (see Section~\ref{subsec:Fourier}) permutes the $\mathcal{D}$-modules $C_s^j$, and ``rotates'' their characteristic varieties by $90^{\circ}$ (note that ``rotating'' the conormal variety to the orbit of rank $s$ matrices yields the conormal variety to rank $(n-s)$ matrices). The formula (\ref{eq:defGFourierU}) where $U=\bigwedge^2\bb{C}^n$, together with (\ref{eq:defmcC^isn}), shows that $\mc{F}(C_s^1)=C_{n-s-1}^1$ for $s=0,\cdots,n-1$, and $\mc{F}(C_s^2)=C_{n-s}^2$ for $s=0,\cdots,n$. Since $C_s^1$ has support $M^{\operatorname{symm}}_{n-s}$ and $\mc{F}(C_s^1)$ has support $M^{\operatorname{symm}}_{s+1}$, it follows that the characteristic variety of $C_s^1$ has two components, namely the conormal varieties to the orbits of rank $(n-s)$ and rank $(n-s-1)$ matrices. Since $C_s^2$ has support $M^{\operatorname{symm}}_{n-s}$ and $\mc{F}(C_s^2)$ has support $M^{\operatorname{symm}}_s$, it follows that the characteristic variety of $C_s^2$ is irreducible, namely it is the conormal variety to the orbit of rank $(n-s)$ matrices. Similar considerations show that for general and skew-symmetric matrices, the characteristic varieties of the simple equivariant $\mathcal{D}$-modules are irreducible. The calculation of characteristic varieties can also be deduced from~\cite{braden-grinberg}.
\end{remark}
\subsection*{Strategy for computing the characters of equivariant $\mathcal{D}$-modules}\label{subsec:computingcharacters}
Our approach to computing characters of equivariant $\mathcal{D}$-modules is based on performing Euler characteristic calculations using the $\mathcal{D}$-module functoriality together with some combinatorial and geometric methods. More precisely, for the inclusion of an orbit $\iota:O\hookrightarrow X$, the $\mathcal{D}$-module direct image $\int_{\iota}\mc{O}_O$ is an object in the derived category of $G$-equivariant $\mathcal{D}_X$-modules, whose cohomology groups $\int^j_{\iota}\mc{O}_O$ are (in the cases that we study) $G$-admissible representations. Analyzing the inclusion $\iota$ directly is complicated, so we make use of a resolution of singularities $Z$ of the orbit closure $\ol{O}$. The variety $Z$ is a vector bundle over a Grassmannian $\bb{G}$ (or a product of Grassmannians), and the inclusion $j:O\hookrightarrow Z$ is an affine open immersion. The map $\pi:Z\to X$ factors as $p\circ s$
\[
\xymatrix{
O \ar@{^{(}->}[r]^{j} \ar@{=}[d] & Z \ar@{^{(}->}[r]^{s} \ar[d] \ar[dr]_{\pi} & X\times\bb{G} \ar[d]_{p} \\
O \ar@{^{(}->}[r] & \ol{O} \ar@{^{(}->}[r] & X \\
}
\]
where $s$ is a regular embedding and $p$ is the projection onto the first factor. We compute the Euler characteristic of $\int_{\iota}\mc{O}_O$ as a virtual admissible $G$-representation, by using the factorization $\iota=p\circ s\circ j$. If we pretend that there is a one-to-one correspondence between simple equivariant $\mathcal{D}$-modules and orbits (which is true for general and skew-symmetric matrices), and write $X_s$ for the $\mathcal{D}$-module corresponding to matrices of rank $s$, then the Euler characteristic calculations together with general considerations regarding the structure of $\mathcal{D}$-module direct images, allow us to write down an upper-triangular matrix with ones on the diagonal, that represents the change of coordinates in the Grothendieck group of admissible representations, from $(X_s)_s$ to appropriately defined linearly independent characters $(\mf{X}_s)_s$. The Fourier transform on one hand preserves this matrix, and on the other hand it makes it lower-triangular, which allows us to conclude that the matrix is in fact the identity and therefore $X_s=\mf{X}_s$ for all $s$ (see Section~\ref{subsec:linalg}).
In the process of computing Euler characteristics, we are led to the following combinatorial problem. Let $X=\bb{G}(k,\bb{C}^n)$ be the Grassmannian of $k$-dimensional quotients of $\bb{C}^n$, with $\mc{O}_X(1)$ denoting the Pl\"ucker line bundle, and $\Omega^i_X$ denoting the sheaf of differential $i$-forms on $X$, and define the virtual $\operatorname{GL}_n(\bb{C})$-representation
\[p_{k,r}=\sum_{i=0}^{k\cdot(n-k)}(-1)^i\cdot\chi(X,\Omega_X^i(r)).\]
The problem is to compute $p_{k,r}\otimes S_{\ll}\bb{C}^n$ for $\ll\in\bb{Z}^n_{\operatorname{dom}}$. When $k=1$, $p_{1,r}$ corresponds to the $r$-th power sum symmetric function, and the answer is given in \cite[Exercise~I.3.11(1)]{macdonald}. The relevance of this formula for computing Euler characteristics is as follows: if we write $E=\mc{L}(\{0\},X)$ for the simple holonomic $\mathcal{D}$-module supported at the origin, $O_k$ for the orbit of rank $k$ matrices, and $\iota_k$ for the inclusion of $O_k$ into the ambient space, then (up to minor adjustments, depending on which space of matrices we analyze)
\[\chi\left(\int_{\iota_k}\mc{O}_{O_k}\right)=\sum_{j\in\bb{Z}}(-1)^j\int_{\iota_k}^j\mc{O}_{O_k}=\lim_{r\to\infty}p_{k,r}\otimes E,\]
where the limit is taken in the Grothendieck group of admissible representations (see Section~\ref{subsubsec:admissible} for a precise formulation, and Section~\ref{sec:lims} for the calculations).
\subsection*{Organization}
In Section~\ref{sec:prelim} we establish the notation and basic results concerning the representation theory of general linear groups and $\mathcal{D}$-modules that will be used throughout the rest of the paper. In Section~\ref{sec:lims} we compute the relevant Euler characteristics as limits in the Grothendieck group of $\operatorname{GL}$-admissible representations. In Sections~\ref{sec:symm},~\ref{sec:mxnmatrices}, and~\ref{sec:skew} we prove the main results on characters of equivariant $\mathcal{D}$-modules. Finally, in Section~\ref{sec:simplesrankstrat} we discuss the simple $\mathcal{D}$-modules that arise from non-equivariant local systems on the orbits, and prove Levasseur's conjecture for skew-symmetric and general matrices.
\section{Preliminaries}\label{sec:prelim}
\subsection{Representation Theory {\cite[Ch.~2]{weyman}}}\label{subsec:repthy}
Let $W$ be a complex vector space of dimension $\dim(W)=n$, and denote by $\operatorname{GL}(W)$ the group of invertible linear transformations of $W$. The irreducible finite dimensional $\operatorname{GL}(W)$-representations, denoted $S_{\ll}W$, are indexed by \defi{dominant weights} $\ll=(\ll_1\geq\cdots\geq\ll_n)\in\bb{Z}^n$. A dominant weight $\ll$ is said to be a \defi{partition} if all its \defi{parts} $\ll_1,\cdots,\ll_n$ are nonnegative. The \defi{size} of $\ll$ is $|\ll|=\ll_1+\cdots+\ll_n$. The \defi{conjugate partition} $\ll'$ is defined by transposing the associated Young diagram: $\ll'_i$ is the number of $j$'s for which $\ll_j\geq i$; for example $(5,2,1)'=(3,2,1,1,1)$. Write $[n]$ for the set $\{1,\cdots,n\}$, and for a given a subset $I\subset[n]$ and an integer $u$, let $(u^I)$ be the sequence $\mu\in\bb{Z}^n$ having $\mu_i=u$ when $i\in I$, and $\mu_i=0$ when $i\notin I$. When $I=[k]$ for $k\leq n$, we simply write $(u^k)$ instead of $(u^I)$. We have that $S_{(1^k)}W=\bigwedge^k W$ is the $k$-th exterior power of $W$, and we let $\det(W)$ denote the top exterior power $\bigwedge^n W$.
\subsubsection{Admissible representations}\label{subsubsec:admissible}
Given a reductive algebraic group $G$, we write $\Lambda$ for the set of (isomorphism classes of) finite dimensional irreducible $G$-representations. We will be mainly interested in the case when $G=\operatorname{GL}(W)$ is a general linear group: we write $\Gamma(G)=\Gamma(W)$, and $\Lambda=\{S_{\ll}W:\ll\in\bb{Z}^n_{\operatorname{dom}}\}$. We also consider $G=\operatorname{GL}(W_1)\times\operatorname{GL}(W_2)$, $\dim(W_1)=m$, $\dim(W_2)=n$, and write $\Gamma(G)=\Gamma(W_1,W_2)$ and $\Lambda=\{S_{\d}W_1\otimes S_{\ll}W_2:\d\in\bb{Z}^m_{\operatorname{dom}},\ll\in\bb{Z}^n_{\operatorname{dom}}\}$. An \defi{admissible $G$-representation} decomposes as
\[M=\bigoplus_{L\in\Lambda} L^{\oplus a_L},\]
where each $a_L\in\bb{Z}_{\geq 0}$. We say that $M$ is \defi{finite} if only finitely many of the $a_L$'s are non-zero. We define the \defi{Grothendieck group $\Gamma(G)$ of admissible representations} to be $\bb{Z}^{\Lambda}$, the direct product of copies of $\bb{Z}$, indexed by the set $\Lambda$. We call the elements of $\Gamma(G)$ \defi{virtual representations}. We write a typical element $U\in\Gamma(G)$ as
\[U=\sum_{L\in\Lambda} a_{L}\cdot L,\]
where $a_L\in\bb{Z}$ and define $\scpr{U}{L}=a_L$ to be the \defi{multiplicity} of $L$ inside $U$. A sequence $(U_r)_r$ of virtual representations is said to be \defi{convergent} (in $\Gamma(G)$) if for every $L\in\Lambda$, the sequence of integers $\scpr{U_r}{L}$ is eventually constant. If $(U_r)_r$ is convergent, we write $a_L=\lim_{r\to\infty}\scpr{U_r}{L}$ for each $L\in\Lambda$. We define $U=\sum_{L\in\Lambda}a_L\cdot L$ to be \defi{the limit} of $(U_r)_r$, and write
\[\lim_{r\to\infty}U_r=U.\]
\subsubsection{Combinatorics of weights}\label{subsubsec:combinatoricsweights}
It will be convenient to make sense of $S_{\ll}W$ even when $\ll\in\bb{Z}^n$ is not dominant. In order to do so we let $\d=(n-1,n-2,\cdots,1,0)$ and consider $\ll+\d=(\ll_1+n-1,\ll_2+n-2,\cdots,\ll_{n-1}+1,\ll_n)$. We write $\rm{sort}(\ll+\d)$ for the sequence obtained by rearranging the entries of $\ll+\d$ in non-increasing order. If $\ll+\d$ has non-repeated entries, we let $\operatorname{sgn}(\ll)$ denote the sign of the unique permutation realizing the sorting of the sequence $\ll+\d$. We define
\[\tl{\ll}=\rm{sort}(\ll+\d)-\d,\]
and let $S_{\ll}W$ be the element of $\Gamma(W)$ defined by
\begin{equation}\label{eq:defSll}
S_{\ll}W = \begin{cases}
\operatorname{sgn}(\ll)\cdot S_{\tl{\ll}}W & \rm{if }\tl{\ll}\rm{ is dominant (i.e. if }\ll+\d\rm{ has non-repeated entries);} \\
0 & \rm{otherwise}.
\end{cases}
\end{equation}
For example, we have $S_{(2,1,4,3)}W=0$ and $S_{(1,1,0,7)}W=-S_{(4,2,2,1)}W$. Note that in particular
\begin{equation}\label{eq:vanishSll}
S_{\ll}W=0\rm{ if }\ll_{i+1}=\ll_i+1\rm{ for some }i=1,\cdots,n-1.
\end{equation}
We denote by ${[n]\choose k}$ the collection of subsets $I\subset[n]$ of size $|I|=k$, and write $P(k,n-k)$ for the set of partitions $\mu=(\mu_1\geq\cdots\geq\mu_k)$ with $\mu_1\leq n-k$. There is a one-to-one correspondence between sets $I\in{[n]\choose k}$ and partitions $\mu\in P(k,n-k)$ given by
\begin{equation}\label{eq:Itomu}
I = \{\mu_k+1,\mu_{k-1}+2,\cdots,\mu_2+(k-1),\mu_1+k\}.
\end{equation}
If we write $\mu'\in P(n-k,k)$ for the conjugate partition of $\mu$ then the complement of $I$ in $[n]$ is given by
\begin{equation}\label{eq:Icfrommu'}
I^c=[n]\setminus I=\{k+1-\mu'_1,k+2-\mu'_2,\cdots,n-\mu'_{n-k}\}.
\end{equation}
For every $\ll\in\bb{Z}^n$, $I\in{[n]\choose k}$ and $r\in\bb{Z}$, we define $\ll(r,I)\in\bb{Z}^n$ as follows: we write the elements of $I$ and $I^c$ in increasing order
\begin{equation}\label{eq:listIIc}
I=\{i_1<\cdots<i_k\},\ I^c=\{i_{k+1}<\cdots<i_{n}\},
\end{equation}
and let
\begin{equation}\label{eq:defllrI}
\ll(r,I)_t=\begin{cases}
r+t+\ll_{i_t}-i_t & \rm{for }t=1,\cdots,k; \\
t+\ll_{i_t}-i_t & \rm{for }t=k+1,\cdots,n.
\end{cases}
\end{equation}
We define $\ll^1(I)\in\bb{Z}^k$ and $\ll^2(I)\in\bb{Z}^{n-k}$ via
\begin{equation}\label{eq:defll12}
\begin{aligned}
\ll^1(I)_t &= t+\ll_{i_t}-i_t,\rm{ for }t=1,\cdots,k, \\
\ll^2(I)_{t-k} &= t+\ll_{i_t}-i_t,\rm{ for }t=k+1,\cdots,n,
\end{aligned}
\end{equation}
so that $\ll(r,I)$ is the concatenation of $\ll^1(I)+(r^k)$ and $\ll^2(I)$. In particular
\[\ll^1([k])=(\ll_1,\cdots,\ll_k)\rm{ and }\ll^2([k])=(\ll_{k+1},\cdots,\ll_n).\]
We define the permutation $\sigma(I)$ of $[n]$ via
\begin{equation}\label{eq:defsI}
\sigma(I)_t=i_t\rm{ for }t=1,\cdots,n.
\end{equation}
With this notation we obtain
\begin{equation}\label{eq:Sll+rI=Sll(rI)}
S_{\ll+(r^I)}W=\operatorname{sgn}(\sigma(I))\cdot S_{\ll(r,I)}W=(-1)^{|\mu|}\cdot S_{\ll(r,I)}W,
\end{equation}
and note that if $\ll$ is dominant and $r$ is sufficiently large then $\ll(r,I)$ is also dominant.
We define for $h,j\in\bb{Z}/2\bb{Z}$ the sets of partitions
\begin{equation}\label{eq:defPhjab}
P^{h,j}(a,b)=\{\mu\in P(a,b):\mu_i\equiv h\ (\operatorname{mod}\ 2)\rm{ for }i=1,\cdots,a,\mu'_i\equiv j\ (\operatorname{mod}\ 2)\rm{ for }i=1,\cdots,b\}.
\end{equation}
A quick counting argument yields the following:
\begin{lemma}\label{lem:countpartitions}
The cardinality of $P^{h,j}(a,b)$ is computed by:
\[
|P^{0,0}(a,b)|={\lfloor \frac{a}{2}\rfloor + \lfloor \frac{b}{2}\rfloor \choose \lfloor \frac{b}{2}\rfloor},\quad
|P^{0,1}(a,b)|=\begin{cases}
\displaystyle{\lfloor \frac{a-1}{2}\rfloor + \frac{b}{2} \choose \frac{b}{2}} & b\rm{ even} \\
0 & b\rm{ odd}
\end{cases},\]
\[
|P^{1,0}(a,b)|=\begin{cases}
\displaystyle{\frac{a}{2} + \lfloor \frac{b-1}{2}\rfloor \choose \lfloor \frac{b-1}{2}\rfloor} & a\rm{ even} \\
0 & a\rm{ odd}
\end{cases},\quad
|P^{1,1}(a,b)|=\begin{cases}
\displaystyle{\frac{a-1}{2} + \frac{b-1}{2} \choose \frac{b-1}{2}} & a,b\rm{ odd} \\
0 & \rm{otherwise}
\end{cases}.
\]
\end{lemma}
\subsubsection{A generalized Pieri rule}\label{subsubsec:genlPieri}
The Grothendieck group $\Gamma(W)$ is a module over the \defi{representation ring $R(W)$} of finite dimensional $\operatorname{GL}(W)$-representations. As a ring, $R(W)$ is generated by the exterior powers $\bigwedge^k W$, $k\leq n$, and by the inverse $\det(W)^{-1}=\bigwedge^n W^*=S_{(-1^n)}W$ of $\det(W)$. We have $S_{\ll}W\otimes\det(W)=S_{\ll+(1^n)}W$. The following lemma generalizes this by describing the multiplicative action of the exterior powers $\bigwedge^k W$ on $\Gamma(W)$ (since the multiplication is continuous, i.e. it commutes with limits, it suffices to determine its action on the indecomposables $S_{\ll}W$):
\begin{lemma}[Pieri's rule]\label{lem:Pieri}
For every $\ll\in\bb{Z}^n$ we have the following equality in $\Gamma(W)$:
\begin{equation}\label{eq:Pieri}
\left(\bigwedge^k W\right)\otimes S_{\ll}W=\sum_{I\in{[n]\choose k}}S_{\ll+(1^I)}W.
\end{equation}
\end{lemma}
\begin{proof}
We may assume without loss of generality that $\ll$ is dominant. If $\ll_{i+1}=\ll_i$ and $I$ is such that $i\notin I$ and $(i+1)\in I$ then it follows from (\ref{eq:vanishSll}) that $S_{\ll+(1^I)}W=0$. For all the other terms appearing on the right hand side of (\ref{eq:Pieri}) we have that $\mu=\ll+(1^I)$ is dominant and $\mu/\ll$ is a \defi{vertical strip} (i.e. $\mu_i-\ll_i\in\{0,1\}$ for all $i$) of size $k$. (\ref{eq:Pieri}) then follows from the usual Pieri formula \cite[Corollary~2.3.5]{weyman}.
\end{proof}
We define elements $p_{k,r}(W)\in R(W)$ for every $r\in\bb{Z}$ and $0\leq k\leq n$, by
\begin{equation}\label{eq:defpkr}
p_{k,r}(W)=\sum_{I\in{[n]\choose k}} S_{(r^I)}W,
\end{equation}
and note that $p_{k,1}(W)=\bigwedge^k W$. We have the following generalization of Pieri's rule:
\begin{lemma}\label{lem:genlPieri}
For every $\ll\in\bb{Z}^n$ we have the following equality in $\Gamma(W)$:
\begin{equation}\label{eq:genlPieri}
p_{k,r}(W)\otimes S_{\ll}W=\sum_{I\in{[n]\choose k}}S_{\ll+(r^I)}W.
\end{equation}
\end{lemma}
\begin{proof} When $k=0$, $p_{0,r}(W)=\bb{C}$ is the identity element of $R(W)$, so the conclusion is trivial. We may thus assume that $k>0$. As before, we also assume that $\ll$ is dominant. Multiplication by $\det(W)$ is an invertible operation, so proving (\ref{eq:genlPieri}) for $\ll$ is equivalent to proving it for $\ll+(1^{n})$. In particular, we may assume that $\ll$ is a partition and that moreover $\ll_n = 0$.
We consider the ordering of the partitions $\ll$ with at most $n$ parts induced by the graded reverse lexicographic order on their conjugates: more precisely, we say that $\ll\succ\mu$ if $|\ll|>|\mu|$, or if $|\ll|=|\mu|$ and for the largest index $i$ for which $\ll'_i\neq\mu'_i$ one has $\ll'_i>\mu'_i$. We prove (\ref{eq:genlPieri}) for all partitions $\ll$, by induction with respect to the said ordering. When $\ll$ is the empty partition, (\ref{eq:genlPieri}) coincides with (\ref{eq:defpkr}).
Assume now that $\ll_1>0$ and consider the parititon $\mu$ obtained from $\ll$ by removing the last column of its Young diagram: the conjugate $\mu'$ is given by $\mu'_i=\ll'_i$ for $i<\ll_1$ and $\mu'_i=0$ for $i\geq\ll_1$. We let $l=\ll'_{\ll_1}$ denote the size of the column removed from $\ll$. Using the induction hypothesis for $\mu\prec\ll$ and Lemma~\ref{lem:Pieri} we get
\begin{equation}\label{eq:pkrSmubw^l}
(p_{k,r}(W)\otimes S_{\mu}W)\otimes\left(\bigwedge^l W\right)=\sum_{J\in{[n]\choose l}}\left(\sum_{I\in{[n]\choose k}} S_{\mu+(r^I)+(1^J)}W\right).
\end{equation}
Consider the collection of partitions $\mc{P}=\{\a:\a/\mu\rm{ is a vertical strip of size }l\}$, so that
\[S_{\mu}W\otimes\left(\bigwedge^l W\right)=\sum_{\a\in\mc{P}} S_{\a}W,\]
and note that $\ll\in\mc{P}$ and that $\a\prec\ll$ for every $\ll\neq\a\in\mc{P}$. We can then rewrite the left hand side of (\ref{eq:pkrSmubw^l}) as
\[\sum_{\a\in\mc{P}}p_{k,r}(W)\otimes S_{\a}W,\]
so in order to prove (\ref{eq:genlPieri}) for $\ll$ it is sufficient to show that the right hand side of (\ref{eq:pkrSmubw^l}) is equal to
\[\sum_{\a\in\mc{P}}\left(\sum_{I\in{[n]\choose k}} S_{\a+(r^I)}W\right).\]
Since $\mc{P}=\{\mu+(1^J):J\in{[n]\choose l},\mu+(1^J)\rm{ dominant}\}$, we only have to check that when $\a=\mu+(1^J)$ is not dominant then
\begin{equation}\label{eq:sumSar^I=0}
\sum_{I\in{[n]\choose k}} S_{\a+(r^I)}W=0.
\end{equation}
Note that the only way in which $\a=\mu+(1^J)$ can fail to be dominant is if for some index $j$, $\mu_j=\mu_{j+1}$ and $j\notin J$, $(j+1)\in J$. Fix such an index $j$, and note that $\a_{j+1}=\a_j+1$. It follows from (\ref{eq:vanishSll}) that when $I\subset[n]$ is such that both $j,j+1\in I$, or both $j,j+1\notin I$, then $S_{\a+(r^I)}W=0$. To show (\ref{eq:sumSar^I=0}) it is then enough to prove that
\begin{equation}\label{eq:sumjxorj+1=0}
\sum_{\substack{I\in{[n]\choose k} \\ j\in I,(j+1)\notin I}} S_{\a+(r^I)}W + \sum_{\substack{I'\in{[n]\choose k} \\ j\notin I',(j+1)\in I'}} S_{\a+(r^{I'})}W = 0.
\end{equation}
There is a one-to-one correspondence between the collection of subsets $I$ with $j\in I$, $(j+1)\notin I$, and subsets $I'$ with $j\notin I'$, $(j+1)\in I'$, given by $I'=(I\cup\{j+1\})\setminus\{j\}$. Moreover, for such a pair $I,I'$ it follows from (\ref{eq:defSll}) that $S_{\a+(r^I)}W=-S_{\a+(r^{I'})}W$ (because $\a+(r^I)+\d$ is obtained from $\a+(r^{I'})+\d$ by switching the $j$-th part with the $(j+1)$-st part), which proves (\ref{eq:sumjxorj+1=0}) and concludes the proof of the lemma.
\end{proof}
\subsection{Bott's theorem for Grassmannians {\cite[Ch.~4]{weyman}}}\label{subsec:bott}
We consider $X=\bb{G}(k,V)$, the Grassmannian of $k$-dimensional quotients of $V$ (or $k$-dimensional subspaces of $W=V^*$), with the tautological sequence
\begin{equation}\label{eq:tautGr}
0\longrightarrow\mc{R}\longrightarrow V\otimes\mc{O}_X\longrightarrow\mc{Q}\longrightarrow 0,
\end{equation}
where $\mc{Q}$ is the tautological rank $k$ quotient bundle, and $\mc{R}$ is the tautological rank $(n-k)$ sub-bundle. Bott's Theorem for Grassmannians \cite[Corollary~4.1.9]{weyman} computes the cohomology of a large class of $\operatorname{GL}$-equivariant bundles on $X$. We only need a weaker version that computes Euler characteristics.
Suppose that $\mc{M}$ is a quasi-coherent $\operatorname{GL}(W)$-equivariant sheaf on $X$. We say that $\mc{M}$ has \defi{admissible (resp. finite) cohomology} if its cohomology groups $H^j(X,\mc{M})$ are admissible (resp. finite) for $j=0,\cdots,\dim(X)$. We can therefore make sense of the Euler characteristic of $\mc{M}$ as an element of $\Gamma(W)$ (resp. $R(W)$). We define the \defi{Euler characteristic} of $\mc{M}$ to be the virtual representation
\begin{equation}\label{eq:defEulerChar}
\chi(X,\mc{M}) = \sum_{j=0}^{k\cdot(n-k)}(-1)^j H^j(X,\mc{M}).
\end{equation}
\begin{theorem}[Bott]\label{thm:bott}
Let $\a\in\bb{Z}^k_{\operatorname{dom}}$ and $\b\in\bb{Z}^{n-k}_{\operatorname{dom}}$ be dominant weights, and let $\ll=(\a,\b)\in\bb{Z}^n$ be their concatenation. The Euler characteristic of $S_{\a}\mc{Q}\otimes S_{\b}\mc{R}$ is given (with the convention (\ref{eq:defSll})) by
\[\chi(X,S_{\a}\mc{Q}\otimes S_{\b}\mc{R})=S_{\ll}V.\]
\end{theorem}
We can now give an alternative interpretation of the elements $p_{k,r}$ introduced in (\ref{eq:defpkr}):
\begin{lemma}\label{lem:pkrispfwd}
If we let $\Omega_X^i=\bigwedge^i(\mc{R}\otimes\mc{Q}^*)$ denote the sheaf of $i$-differential forms on $X$, and write $\mc{O}_X(1)=\det(\mc{Q})$ for the Pl\"ucker line bundle on $X$, then
\[p_{k,r}(V)=\sum_{i=0}^{k\cdot(n-k)}(-1)^i\cdot\chi(X,\Omega_X^i(r)).\]
\end{lemma}
\begin{proof}
Cauchy's formula \cite[Cor.~2.3.3]{weyman} yields
\[\bigwedge^i(\mc{R}\otimes\mc{Q}^*)=\bigoplus_{\mu\in P(k,n-k),\ |\mu|=i} S_{\mu}\mc{Q}^*\otimes S_{\mu'}\mc{R}.\]
Twisting by $\mc{O}_X(r)=\det(\mc{Q})^{\otimes r}=S_{(r^k)}\mc{Q}$, and taking Euler characteristics, we get using Theorem~\ref{thm:bott}
\[\chi(X,\Omega^i_X(r))=\sum_{\mu\in P(k,n-k),\ |\mu|=i}S_{(r-\mu_k,r-\mu_{k-1},\cdots,r-\mu_1,\mu'_1,\cdots,\mu'_{n-k})}V.\]
Using (\ref{eq:Sll+rI=Sll(rI)}) with $\ll=0$, we get $S_{(r-\mu_k,r-\mu_{k-1},\cdots,r-\mu_1,\mu'_1,\cdots,\mu'_{n-k})}V=(-1)^{|\mu|}\cdot S_{(r^I)}V$, so
\[\sum_{i=0}^{k\cdot(n-k)}(-1)^i\cdot\chi(X,\Omega_X^i(r))=\sum_{I\in{[n]\choose k}}S_{(r^I)}V=p_{k,r}(V).\qedhere\]
\end{proof}
\subsection{$\mathcal{D}$-modules {\cite{borel}, \cite{hottaetal}}}\label{subsec:Dmods}
For a smooth algebraic variety $X$ over $\bb{C}$, we let $\mathcal{D}_X$ denote the sheaf of \defi{differential operators} on $X$ \cite[Section~1.1]{hottaetal}. A \defi{$\mathcal{D}$-module} $\mc{M}$ on $X$ (or a $\mathcal{D}_X$-module) is a quasi-coherent sheaf $\mc{M}$ on $X$, with a left module action of $\mathcal{D}_X$.
\begin{definition}\label{def:GequivariantDmod}
Let $G$ be an algebraic group acting on $X$, and let $\mc{M}$ be a $\mathcal{D}_X$-module. Differentiating the action of $G$ on $X$ yields a map $d:\operatorname{Lie}(G)\to\operatorname{Der}_X$ from the Lie algebra of $G$ to the vector fields on $X$. The $\mathcal{D}_X$-module operation
\begin{equation}\label{eq:Dmodaction}
\mathcal{D}_X\otimes\mc{M}\to\mc{M},
\end{equation}
composed with $d$ yields an action of $\operatorname{Lie}(G)$ on $\mc{M}$. The $\mathcal{D}_X$-module $\mc{M}$ is \defi{$G$-equivariant} if
\begin{enumerate}
\item[(a)] $\mc{M}$ admits an action of $G$ compatible with (\ref{eq:Dmodaction}) (see \cite[Def.~11.5.2]{hottaetal} for a precise meaning of compatibility).
\item[(b)] The action of $\operatorname{Lie}(G)$ on $\mc{M}$ obtained by differentiating the action of $G$ on $\mc{M}$ coincides with the one induced from $d:\operatorname{Lie}(G)\to\operatorname{Der}_X$ and (\ref{eq:Dmodaction}).
\end{enumerate}
\end{definition}
As discussed in the Introduction, examples of $G$-equivariant holonomic $\mathcal{D}_X$-modules are $\mc{O}_X$, and for a $G$-invariant subset $Y\subset X$, the local cohomology modules $\mc{H}_Y^{\bullet}(X,\mc{O}_X)$, as well as the intersection homology $\mathcal{D}$-modules $\mc{L}(Y,X)$. When $X=U$ is a vector space, and $Y=\{0\}$ is the origin, we let
\begin{equation}\label{eq:defE}
E = \mc{L}(\{0\},U) = \mc{H}_{\{0\}}^{\dim(U)}(U,\mc{O}_U),
\end{equation}
be the unique simple $\mathcal{D}_U$-module supported at the origin. As a vector space (and a $G$-representation)
\begin{equation}\label{eq:E=detSym}
E=\det(U)\otimes\operatorname{Sym}(U).
\end{equation}
The following theorem gives a classification of the simple equivariant holonomic $\mathcal{D}$-modules, for a group action with finitely many orbits (see \cite[Section~11.6]{hottaetal}):
\begin{theorem}\label{thm:equivRH}
Let $G$ be an algebraic group acting with finitely many orbits on a smooth algebraic variety~$X$. There is a one-to-one correspondence between:
\begin{enumerate}
\item[(a)] Simple $G$-equivariant holonomic $\mathcal{D}_X$-modules.
\item[(b)] Pairs $(O,L)$ where $O$ is a $G$-orbit, and $L$ is an irreducible $G$-equivariant local system on $O$.
\item[(b')] Pairs $(O,L)$ where $O$ is a $G$-orbit, and $L$ is an irreducible representation of the component group of the isotropy group of $O$.
\end{enumerate}
\end{theorem}
\noindent Here by the \defi{isotropy group} of $O$ we mean the stabilizer of any element in $O$ (they are all isomorphic). For an algebraic group $H$, we denote by $H_0$ the connected component of the identity, which is a normal subgroup of $H$. The quotient $H/H_0$ is called the \defi{component group} of $H$.
\begin{remark}\label{rem:Ltrivial}
When the representation $L$ in Theorem~\ref{thm:equivRH}(b') is trivial, the corresponding $\mathcal{D}_X$-module in part (a) is $\mc{L}(\ol{O},X)$, where $\ol{O}$ is the closure of $O$. It follows that in the case when the isotropy groups for the $G$-action on $X$ are connected, there is a one-to-one correspondence between simple $G$-equivariant $\mathcal{D}_X$-modules and orbits of the group action.
\end{remark}
Let $m\geq n$ be positive integers and consider the complex vector spaces $M$ of general $m\times n$ matrices, $M^{\operatorname{symm}}$ of $n\times n$ symmetric, and $M^{\skew}$ of $n\times n$ skew-symmetric matrices respectively. These spaces admit a natural action of a group $\operatorname{GL}$ via row and column operations: $\operatorname{GL}_m(\bb{C})\times\operatorname{GL}_n(\bb{C})$ acts on $M$, and $\operatorname{GL}_n(\bb{C})$ acts $M^{\operatorname{symm}}$ and $M^{\skew}$. We write $M_s$ (resp. $M^{\operatorname{symm}}_s$) for the subvariety of $M$ (resp. $M^{\operatorname{symm}}$) consisting of matrices of rank at most $s$, for $s=0,\cdots,n$, and $M^{\skew}_s$ for the subvariety of $M^{\skew}$ consisting of skew-symmetric matrices of rank at most $2s$, for $s=0,\cdots,\lfloor n/2\rfloor$. We have the following:
\begin{theorem}[Classification of simple $\operatorname{GL}$-equivariant holonomic $\mathcal{D}$-modules on spaces of matrices]\label{thm:classificationDmods}\
\begin{itemize}
\item (General matrices). There are $(n+1)$ simple $\operatorname{GL}$-equivariant $\mathcal{D}$-modules on the vector space $M$ of $m\times n$ matrices, namely the intersection homology $\mathcal{D}$-modules $\mc{L}(M_{s},M)$, $s=0,\cdots,n$.
\item (Symmetric matrices). There are $(2n+1)$ simple $\operatorname{GL}$-equivariant $\mathcal{D}$-modules on the vector space $M^{\operatorname{symm}}$ of $n\times n$ symmetric matrices, $(n+1)$ of which are the intersection homology $\mathcal{D}$-modules $\mc{L}(M^{\operatorname{symm}}_{s},M^{\operatorname{symm}})$, $s=0,\cdots,n$, while the remaining ones are the intersection homology $\mathcal{D}$-modules $\mc{L}(M^{\operatorname{symm}}_{s},M^{\operatorname{symm}};1/2)$, $s=1,\cdots,n$, corresponding to the non-trivial irreducible equivariant local systems on the orbits.
\item (Skew-symmetric matrices). There are $(\lfloor n/2\rfloor+1)$ simple $\operatorname{GL}$-equivariant $\mathcal{D}$-modules on the vector space $M^{\skew}$ of $n\times n$ skew-symmetric matrices, namely $\mc{L}(M^{\skew}_{s},M^{\skew})$, $s=0,\cdots,\lfloor n/2\rfloor$.
\end{itemize}
\end{theorem}
\begin{proof}
The theorem follows from Theorem~\ref{thm:equivRH} and Remark~\ref{rem:Ltrivial}, since the isotropy groups for general and skew-symmetric matrices are connected, while for symmetric matrices the isotropy groups of the non-zero orbits have two connected components.
\end{proof}
\subsection{Computing Euler characteristics}\label{subsec:pfwdDmods}
Let $X$ be a smooth complex projective algebraic variety and denote its dimension by $d_X$. Consider a finite dimensional vector space $U$, and a short exact sequence
\begin{equation}\label{eq:basicses}
0\longrightarrow\xi\longrightarrow U\otimes\mc{O}_X\longrightarrow\eta\longrightarrow 0,
\end{equation}
where $\xi,\eta$ are locally free sheaves on $X$. We think of $U^*$ as an affine space, and of~$U$ as linear forms on~$U^*$. We let $Y=\rm{Tot}_X(\eta^*)$ denote the total space of the bundle $\eta^*$, and define a morphism $\pi:Y\to U^*$ via the commutative diagram
\begin{equation}\label{eq:diagrGeneric}
\xymatrix{
Y=\rm{Tot}_X(\eta^*) \ar@{^{(}->}[r] \ar[dr]_{\pi} & U^*\times X \ar[d] \\
& U^* \\
}
\end{equation}
where the top map is the inclusion of $\eta^*$ into the trivial bundle~$U^*$, and the vertical map is the projection onto the $U^*$ factor. We will be interested in understanding the (Euler characteristic of the) $\mathcal{D}$-module pushforward $\int_{\pi}\mc{M}$ along the map $\pi$ for certain $\mathcal{D}_Y$-modules $\mc{M}$. For affine morphisms $X'\to X$, we will identify freely quasi-coherent sheaves on $X'$ with quasi-coherent $\mc{O}_{X'}$-modules on $X$ as in \cite[Exercise~II.5.17(e)]{hartshorne}.
We let $\mc{S}=\operatorname{Sym}_{\mc{O}_X}(\eta)$, so that $Y=\ul{\operatorname{Spec}}_X(\mc{S})$, and consider a locally free sheaf $\mc{L}$ of rank one with $\mc{L}\subset\operatorname{Sym}^i(\eta)$ for some $i>0$. We pull-back $\mc{L}$ to $Y$, define $L=\rm{Tot}_Y(\mc{L}^*)$ to be the total space of the line bundle $\mc{L}^*$, and write $p:L\to Y$ for the natural map. The inclusion $\mc{L}\subset\operatorname{Sym}^i(\eta)$ defines a section $z:Y\to L$ of $p$ \cite[Exercise~II.5.18(c)]{hartshorne}, and we define $Z$ to be the zero-locus of $z$. If $X=\rm{Spec}(\bb{C})$ then $Y$ is an affine space, $\mc{S}$ is the ring of polynomial functions on $Y$, $\mc{L}$ corresponds to (the vector space spanned by) a polynomial $f\in\mc{S}$ of degree $i$, and $Z$ is the vanishing locus of $f$.
We consider the complement $Y^0=Y\setminus Z$ and let $j:Y^0\to Y$ denote the inclusion. Since $j$ is an affine open immersion, $\int_j\mc{O}_{Y^0}=\mc{O}_{Y^0}$ can be thought of as a quasi-coherent sheaf of algebras on $Y$ (or on $X$):
\[\mc{O}_{Y^0}=\varinjlim_r \mc{L}^{-r}\otimes\mc{O}_Y=\varinjlim_r \mc{L}^{-r}\otimes\mc{S}.\]
In the case when $X=\rm{Spec}(\bb{C})$, we have $\mc{O}_{Y^0}=\mc{S}_f$ is the localization of $\mc{S}$ at $f$, which is a $\mathcal{D}$-module on the affine space $Y$. We define the quasi-coherent sheaf $\mc{S}^{\vee}$ on $X$ (the \defi{graded dual of $\mc{S}$}) by
\[\mc{S}^{\vee} = \det(\eta^*)\otimes\operatorname{Sym}_{\mc{O}_X}(\eta^*).\]
\begin{proposition}\label{prop:EuleraslimPr}
With the notation above, we assume that $X$ admits an action of a reductive group $G$, that $U$ is a finite dimensional $G$-representation, and that $\xi,\eta,\mc{L}$ are $G$-equivariant locally free sheaves. Assume further that we have an isomorphism of $G$-equivariant quasi-coherent sheaves on $X$
\begin{equation}\label{eq:twolims}
\mc{O}_{Y^0}\simeq\varinjlim_r\mc{L}^{r}\otimes\mc{S}^{\vee}.
\end{equation}
Let $\mc{M}$ be a $\mathcal{D}_Y$-module which is isomorphic, as a quasi-coherent $G$-equivariant sheaf on $X$, to $\mc{O}_{Y^0}\otimes_{\mc{O}_X}\mc{L}'$, with $\mc{L}'$ a line bundle on $X$. We denote by $\Omega^i_X$ the sheaf of $i$-differential forms on $X$, and assume that for every $i=0,\cdots,d_X$ the sheaves $\Omega^i_X\otimes\mc{M}\otimes\det(\xi^*)\otimes\operatorname{Sym}_{\mc{O}_X}(\xi^*)$ have $G$-admissible cohomology.
If we define the sequence $P_r(X,\mc{L};\mc{L}')\in\Gamma(G)$ via
\[P_r(X,\mc{L};\mc{L}')=\sum_{i=0}^{d_X} (-1)^{d_X-i}\cdot\chi(X,\mc{L}^r\otimes\mc{L}'\otimes\Omega^i_X),\]
then
\begin{equation}\label{eq:chiMlimitPr}
\chi\left(\int_{\pi}\mc{M}\right)=\lim_{r\to\infty} P_r(X,\mc{L};\mc{L}')\otimes\det(U^*)\otimes\operatorname{Sym}_{\bb{C}}(U^*).
\end{equation}
\end{proposition}
\begin{remark}\label{rem:Euleraslim}
We will apply Proposition~\ref{prop:EuleraslimPr} in the case when $X=\bb{G}(k,V)$ is a Grassmann variety, and $\mc{L}=\mc{O}_X(1)$ is the Pl\"ucker line bundle (or its square). It follows from Lemma~\ref{lem:pkrispfwd} that
\[P_r(X,\mc{O}_X(1);\mc{O}_X)=(-1)^{k(n-k)}\cdot p_{k,r}(V).\]
It follows that if $X=\bb{G}(k,V_1)\times\bb{G}(k,V_2)$, where $\dim(V_1)=m$, $\dim(V_2)=n$, and if $\mc{L}=\mc{O}_X(1,1)$ then $P_r(X,\mc{O}_X(1);\mc{O}_X)=(-1)^{k\cdot(m-n)}\cdot p_{k,r}(V_1)\otimes p_{k,r}(V_2)$.
\end{remark}
\begin{proof}[Proof of Proposition~\ref{prop:EuleraslimPr}]
Since the sheaves $\Omega^i_X\otimes\mc{M}\otimes\det(\xi^*)\otimes\operatorname{Sym}_{\mc{O}_X}(\xi^*)$ have admissible cohomology, it follows from \cite[Corollary~2.10]{raicu-VeroDmods} that
\begin{equation}\label{eq:intM}
\chi\left(\int_{\pi}\mc{M}\right) = \sum_{i=0}^{d_X} (-1)^{d_X-i}\cdot\chi(X,\Omega^i_X\otimes\mc{M}\otimes\det(\xi^*)\otimes\operatorname{Sym}_{\mc{O}_X}(\xi^*)).
\end{equation}
Computing Euler characteristics commutes with colimits and associated graded constructions. By (\ref{eq:basicses}) we get a filtration of $U^*\otimes\mc{O}_X$ with $\operatorname{gr}(U^*\otimes\mc{O}_X)=\xi^*\oplus\eta^*$, which yields a filtration of $\operatorname{Sym}_{\mc{O}_X}(U^*\otimes\mc{O}_X)$ with $\operatorname{gr}(\operatorname{Sym}_{\mc{O}_X}(U^*\otimes\mc{O}_X))=\operatorname{Sym}_{\mc{O}_X}(\xi^*)\otimes\operatorname{Sym}_{\mc{O}_X}(\eta^*)$. We also get that $\det(U^*\otimes\mc{O}_X)=\det(\xi^*)\otimes\det(\eta^*)$, and therefore
\[
\begin{aligned}
\chi(X,\Omega_X^i\otimes\mc{L}'\otimes\mc{L}^r\otimes\mc{S}^{\vee}\otimes\det(\xi^*)&\otimes\operatorname{Sym}_{\mc{O}_X}(\xi^*)) = \chi(X,\Omega_X^i\otimes\mc{L}'\otimes\mc{L}^r\otimes\det(U^*)\otimes\operatorname{Sym}_{\mc{O}_X}(U^*\otimes\mc{O}_X)) \\
&\overset{U^*\rm{ is a trivial bundle}}{=} \chi(X,\Omega_X^i\otimes\mc{L}'\otimes\mc{L}^r)\otimes\det(U^*)\otimes\operatorname{Sym}_{\bb{C}}(U^*).
\end{aligned}
\]
Multiplying this equality by $(-1)^{d_X-i}$, summing over $i=0,\cdots,d_X$, taking the limit as $r\to\infty$, and using the identification (\ref{eq:twolims}) tensored with $\mc{L}'$, we get (\ref{eq:chiMlimitPr}).
\end{proof}
\subsection{The Weyl algebra and the Fourier transform}\label{subsec:Fourier}
For a positive integer $N$, the \defi{Weyl algebra}
\begin{equation}\label{eq:defWeyl}
\bb{C}[x_1,\cdots,x_N,\partial_1,\cdots,\partial_N],\ \partial_i=\frac{\partial}{\partial x_i}
\end{equation}
is the ring of differential operators on $\bb{C}^N$. In this section we give a coordinate independent description of the Weyl algebra, and use it to describe the Fourier transform.
Given a finite dimensional $\bb{C}$--vector space $U$ of dimension $N$, we write $\scpr{}{}$ for the natural pairing $U\times U^*\to\bb{C}$. We let $\tl{U}=U\oplus U^*$ and define a non--degenerate skew--symmetric form $\omega:\tl{U}\otimes\tl{U}\to\bb{C}$ by
\[\omega(u,u')=
\begin{cases}
\scpr{u}{u'} & \textrm{if }u\in U,\ u'\in U^*, \\
-\scpr{u'}{u} & \textrm{if }u'\in U,\ u\in U^*, \\
0 & \textrm{otherwise}. \\
\end{cases}
\]
We write $T_n(\tl{U})$ for the tensor product $\tl{U}^{\otimes n}$, and let $T(\tl{U})=\bigoplus_{n\geq 0}T_n(\tl{U})$ denote the \defi{tensor algebra} on $\tl{U}$. We have a natural inclusion $\bigwedge^2\tl{U}\subset T_2(\tl{U})$, and define the \defi{Weyl algebra} $\mc{D}_{U^*}$ as the quotient
\begin{equation}\label{eq:WeylAlgebra}
\mc{D}_{U^*}=T(\tl{U})/\langle x-\omega(x):x\in\bigwedge^2\tl{U}\rangle
\end{equation}
of the tensor algebra by the bilateral ideal generated by differences $x-\omega(x)$, with $x\in\bigwedge^2\tl{U}$. Note that $\mathcal{D}_{U^*}$ is the ring of differential operators on the vector space $U^*$. If we choose a basis $x_1,\cdots,x_N$ of $U$, and the dual basis $\partial_1,\cdots,\partial_N$ of $U^*$, then $\mathcal{D}_{U^*}$ coincides with (\ref{eq:defWeyl}).
\begin{lemma}[Fourier transform]\label{lem:Fourier}
If $M$ is a (left) $\mc{D}_{U}$-module, then $\det(U^*)\otimes M$ has the structure of a (left) $\mc{D}_{U^*}$-module.
\end{lemma}
\begin{example}
The most basic example is when $M=\operatorname{Sym}(U^*)$ is the coordinate ring of $U$. In that case $\det(U^*)\otimes\operatorname{Sym}(U^*)$ is equal to $E$, the simple holonomic $\mathcal{D}_{U^*}$-module supported at the origin (see (\ref{eq:defE})).
\end{example}
\begin{proof}[Proof of Lemma~\ref{lem:Fourier}]
Using the identification of $\tl{U}^*$ with $\tl{U}$ coming from the natural isomorphism $U^*\oplus U\simeq U\oplus U^*$, it is easy to see that $\mc{D}_{U^*}\simeq\mc{D}_{U}^{\operatorname{op}}$, where \textsuperscript{op} denotes the opposite ring. Since $M$ is a left $\mc{D}_U$-module, it is also a right $\mc{D}_U^{\operatorname{op}}$-module, i.e. it can be identified with a right $\mc{D}_{U^*}$-module. The canonical sheaf $\omega_{U^*}$ on the vector space $U^*$ is a free rank one module generated by $\det(U)$. By \cite[Prop.~1.2.12]{hottaetal}, the association $M\mapsto\omega_{U^*}^{-1}\otimes M=\det(U^*)\otimes M$ gives an equivalence between the categories of right $\mc{D}_{U^*}$-modules and left $\mc{D}_{U^*}$-modules.
\end{proof}
Motivated by Lemma~\ref{lem:Fourier}, we define a \defi{Fourier transform relative to $U$}, denoted $\mc{F}_U$, on the Grothendieck group $\Gamma(G)$ of admissible $G$-representations as follows:
\begin{equation}\label{eq:defGFourierU}
\mc{F}_U\left(\sum a_i\cdot M_i\right)=\sum a_i\cdot(\det(U^*)\otimes M_i^*).
\end{equation}
The context in which we apply the Fourier transform is as follows: we will have constructions which are functorial in $U$ for certain $\mc{D}_U$-modules $M_U$ which are admissible representations for some group $G$, in such a way that
\[M_U=\bigoplus_i M_i^{\oplus a_i}\rm{ if and only if }M_{U^*}=\bigoplus_i (M^*_i)^{\oplus a_i}.\]
By Lemma~\ref{lem:Fourier}, the Fourier transform of the $\mc{D}_{U^*}$-module $M_{U^*}$ has character equal to $\mc{F}_U(\sum_i a_i\cdot M_i)$. We will slightly imprecisely refer to this as the character of the Fourier transform of $M_U$.
\subsection{A little linear algebra}\label{subsec:linalg}
Consider a finite partially ordered set $\mc{P}$, and let $\mc{A}$ denote the free abelian group with basis $\{\mf{v}_p:p\in\mc{P}\}$. We write $p\succ q$ to indicate that $p$ is strictly larger than $q$ with respect to the partial order, and $p\succeq q$ when we allow equality. Assume that $\mc{F}:\mc{P}\longrightarrow\mc{P}$ is an order reversing bijection, i.e. $p\succeq q$ if and only if $\mc{F}(q)\succeq\mc{F}(p)$. By abuse of notation, we also write $\mc{F}:\mc{A}\longrightarrow\mc{A}$ for the induced automorphism of $\mc{A}$, given by $\mc{F}(\mf{v}_p)=\mf{v}_{\mc{F}(p)}$. We have the following:
\begin{lemma}\label{lem:uppertriangularFourier}
Suppose that we have a collection of elements $v_p\in\mc{A}$ for $p\in\mc{P}$, for which there exist relations
\begin{equation}\label{eq:vtomfv}
v_p=\mf{v}_p+\sum_{q\succ p} a^p_q\cdot\mf{v}_q,\rm{ for some integers }a^p_q.
\end{equation}
If the automorphism $\mc{F}$ of $\mc{A}$ permutes the elements $v_p$ then $v_p=\mf{v}_p$ for all $p\in\mc{P}$ (and hence all $a^p_q=0$).
\end{lemma}
\begin{proof}
Write $\mc{F}(v_p)=v_{\sigma(p)}$ for some permutation $\sigma:\mc{P}\longrightarrow\mc{P}$. Applying $\mc{F}$ to (\ref{eq:vtomfv}) we get
\[v_{\sigma(p)}=\mf{v}_{\mc{F}(p)}+\sum_{q\succ p} a^p_q\cdot\mf{v}_{\mc{F}(q)},\]
which is necessarily a permutation of the relations (\ref{eq:vtomfv}). Since $\mc{F}$ is order-reversing, it follows that $\sigma(p)=\mc{F}(q)$ for some $q\succeq p$, and if $\sigma(p)=\mc{F}(p)$ then one also has $v_{\sigma(p)}=\mf{v}_{\mc{F}(p)}$, i.e. $a^p_q=0$ for all $q\succ p$. We get that $\mc{F}(p)\succeq\sigma(p)$ for all $p\in\mc{P}$, and the equality $\mc{F}(p)=\sigma(p)$ implies $v_{\sigma(p)}=\mf{v}_{\mc{F}(p)}$. An easy induction on the \defi{height} of $\mc{F}(p)$, defined by $\operatorname{ht}(\mc{F}(p))=\#\{q:\mc{F}(p)\succ q\}$, shows that $\mc{F}(p)=\sigma(p)$ for all $p$, which concludes the proof of the lemma.
\end{proof}
\section{Some limit calculations in the Grothendieck group of admissible representations}\label{sec:lims}
Recall the terminology from Sections~\ref{subsubsec:admissible}--\ref{subsubsec:genlPieri} which we will be using freely throughout this section. In particular recall the notation $\Gamma(G)$ for the Grothendieck group of admissible $G$-representations for some group $G$, and the definition of $p_{k,r}(V)$ from (\ref{eq:defpkr}) (also Lemma~\ref{lem:pkrispfwd}). When $W$ is a vector space, we write $V=W^*$ for its dual. In this section we compute in three cases limits in $\Gamma(G)$ of the type
\begin{equation}\label{eq:limspkrE}
\lim_{r\to\infty} p_{k,r}\otimes E,
\end{equation}
where $(p_{k,r})_{r}$ is a sequence of finite virtual $G$-representations, $E=\det(U)\otimes\operatorname{Sym}(U)$ is the (character of the) simple $\mathcal{D}_U$-module supported at the origin (\ref{eq:defE}), where $U$ is a finite dimensional $G$-representation:
\begin{itemize}
\item $U=\operatorname{Sym}^2 W$, $G=\operatorname{GL}(W)$ (so that $\Gamma(G)=\Gamma(W)$), $p_{k,r}=p_{k,r}(V)$. The limit (\ref{eq:limspkrE}) does not exist if $r$ is arbitrary, but instead we have to consider the cases when $r$ is even resp. odd separately.
\item $U=W_1\otimes W_2$, $G=\operatorname{GL}(W_1)\times\operatorname{GL}(W_2)$ (so that $\Gamma(G)=\Gamma(W_1,W_2)$), $p_{k,r}=p_{k,r}(V_1)\otimes p_{k,r}(V_2)$.
\item $U=\bigwedge^2 W$, $G=\operatorname{GL}(W)$, $p_{k,r}=p_{k,r}(V)$ with $k$ even.
\end{itemize}
As mentioned in the Introduction and explained in Section~\ref{subsec:pfwdDmods}, the limits (\ref{eq:limspkrE}) correspond to Euler characteristic calculations for certain $\mathcal{D}$-module direct images. They are essential to the character calculations in Sections~\ref{sec:symm}--\ref{sec:skew} below. The reader who is not interested in the details of the limit calculations may wish to record the results of Propositions~\ref{prop:limsym},~\ref{prop:limsgeneral}, and~\ref{prop:limskew} below, and skip to Section~\ref{sec:symm}.
\subsection{Symmetric matrices}\label{subsec:limsym}
We let $W$ be a vector space of dimension $n$. For $s=0,\cdots,n$ and $j=1,2,$ we define the elements $\mf{C}^j_s\in\Gamma(W)$ via
\begin{equation}\label{eq:defcjs}
\mf{C}^j_s=\bigoplus_{\ll\in\mc{C}^j(s,n)} S_{\ll}W.
\end{equation}
where $\mc{C}^j(s,n)$ is defined in (\ref{eq:defmcC^isn}).
\begin{proposition}\label{prop:limsym}
If $E=\det\left(\operatorname{Sym}^2 W\right)\otimes\operatorname{Sym}\left(\operatorname{Sym}^2 W\right)$ then for $k=0,\cdots,n$,
\[
(-1)^{k(n-k)}\cdot\left(\lim_{\substack{r\to\infty \\ r\equiv k+1\ (\operatorname{mod}\ 2)}} p_{k,r}(V)\otimes E\right)=\begin{cases}
\displaystyle\sum_{\substack{s=n-k \\ s\rm{ even}}}^n {\frac{s-2}{2}\choose\frac{n-k-2}{2}}\cdot \mf{C}_s^2 + \sum_{\substack{s=n-k+1 \\ s\rm{ odd}}}^n {\frac{s-1}{2}\choose\frac{n-k}{2}}\cdot \mf{C}_s^1 & \rm{if }n-k\rm{ even,} \\
& \\
\displaystyle\sum_{\substack{s=n-k \\ s\rm{ odd}}}^n {\frac{s-1}{2}\choose\frac{n-k-1}{2}}\cdot \mf{C}_s^1 - \sum_{\substack{s=n-k+1 \\ s\rm{ even}}}^n {\frac{s-2}{2}\choose\frac{n-k-1}{2}}\cdot \mf{C}_s^2 & \rm{if }n-k\rm{ odd.}
\end{cases}
\]
\[
(-1)^{k(n-k)}\cdot\left(\lim_{\substack{r\to\infty \\ r\equiv k\ (\operatorname{mod}\ 2)}} p_{k,r}(V)\otimes E\right)=\begin{cases}
\displaystyle\sum_{\substack{s=n-k \\ s\rm{ even}}}^n {\frac{s}{2}\choose\frac{n-k}{2}}\cdot \mf{C}_s^1 & \rm{if }n-k\rm{ even,} \\
& \\
\displaystyle\sum_{\substack{s=n-k \\ s\rm{ odd}}}^n {\frac{s-1}{2}\choose\frac{n-k-1}{2}}\cdot \mf{C}_s^2 & \rm{if }n-k\rm{ odd.}
\end{cases}
\]
\end{proposition}
When $k=0$ the above equalities are easy to verify: $p_{k,r}(V)=\bb{C}$ is the trivial representation, so the left hand side reduces to $E$, regardless of the parity of $r$; the right hand side is either $\mf{C}_n^1$ or $\mf{C}_n^2$, but $E=\mf{C}_n^1=\mf{C}_n^2$. We therefore fix $1\leq k\leq n$ for the rest of this section. We begin with some notation and preliminary results before proving the proposition. For $j\in\bb{Z}/2\bb{Z}$ we let
\begin{equation}\label{eq:defmcCjk}
\begin{aligned}
\mc{C}^j &=\{\ll\in\bb{Z}^{k}_{\operatorname{dom}}:\ll_{i}\equiv n+1+j\ (\operatorname{mod}\ 2)\rm{ for }i=1,\cdots,k\}, \\
\mc{C}^j_{\geq n+1} &=\{\ll\in\bb{Z}^{n-k}_{\operatorname{dom}}:\ll_{i}\equiv n+1+j\ (\operatorname{mod}\ 2)\rm{ for }i=1,\cdots,n-k,\rm{ and }\ll_{n-k}\geq n+1\}.
\end{aligned}
\end{equation}
With the convention $\ll_0=\infty$, $\ll_{n+1}=-\infty$, we define for $s=0,\cdots,n$,
\begin{equation}\label{eq:ZsnC}
\mc{Z}(s)=\{\ll\in\bb{Z}^n_{\operatorname{dom}}:\ll_s\geq s+1\geq\ll_{s+1}\},
\end{equation}
and note that the sets $\mc{Z}(s)$, $s=0,\cdots,n$ form a partition of $\bb{Z}^n_{\operatorname{dom}}$. For $h,j\in\bb{Z}/2\bb{Z}$ we let
\begin{equation}\label{eq:defChj}
\mc{C}^{h,j}(s)=\left\{\ll\in\mc{Z}(s):\ll_i\overset{(\operatorname{mod}\ 2)}{\equiv}\begin{cases}
h, & \rm{for }i=1,\cdots,s, \\
j, & \rm{for }i=s+1,\cdots,n.
\end{cases}
\right\},\quad
\mf{C}^{h,j}_s=\sum_{\ll\in\mc{C}^{h,j}(s)}S_{\ll}W.
\end{equation}
Comparing with (\ref{eq:defmcC^isn}) we get that $\mc{C}^1(s,n)=\mc{C}^{s+1,s+1}(s)\cup\mc{C}^{s+1,s+1}(s+1)$ and $\mc{C}^2(s,n)=\mc{C}^{s+1,s}(s)$ so
\begin{equation}\label{eq:ChjtoCj}
\mf{C}^1_s=\mf{C}^{s+1,s+1}_s+\mf{C}^{s+1,s+1}_{s+1},\quad\rm{and}\quad\mf{C}^2_s=\mf{C}^{s+1,s}_s.
\end{equation}
\begin{lemma}\label{lem:ll^iinmcC}
If $I\in{[n]\choose k}$, $\ll^1(I)\in\mc{C}^{k+1+j}$ and $\ll^2(I)\in\mc{C}^0_{\geq n+1}$ then
\begin{itemize}
\item $\ll\in\mc{Z}(s)$ for some $s=n-k,\cdots,n$.
\item $\{s+1,\cdots,n\}\subset I$.
\item $\ll_{s+1}\equiv\cdots\equiv\ll_n\equiv j\ (\operatorname{mod}\ 2)$.
\end{itemize}
\end{lemma}
\begin{proof} Consider the unique $s$ for which $\ll\in\mc{Z}(s)$. Let $s'$ be the maximal element of $I^c$ and assume that $s'>s$. We have (using (\ref{eq:listIIc})) that $i_n=s'$ and therefore $\ll_{s'}\leq\ll_{s+1}\leq s+1$ and
\[\ll^2(I)_{n-k}\overset{(\ref{eq:defll12})}{=}n+\ll_{i_n}-i_n=n+\ll_{s'}-s'\leq n+s+1-s'<n+1,\]
which contradicts $\ll^2(I)\in\mc{C}^0_{\geq n+1}$. It follows that $s'\leq s$ and hence $\{s+1,\cdots,n\}\subset I$, which implies $n-s\leq k$, or $s\geq n-k$. From (\ref{eq:listIIc}) we get
\[i_t=t+n-k \rm{ for }t=k-n+s+1,\cdots,k,\]
which using the fact that $\ll^1(I)\in\mc{C}^{k+1+j}$ yields for $t=k-n+s+1,\cdots,k$
\[n+1+k+1+j\overset{(\operatorname{mod}\ 2)}{\equiv}\ll^1(I)_t=t+\ll_{i_t}-i_t=t+\ll_{t+n-k}-(t+n-k)=\ll_{t+n-k}+k-n\]
so $\ll_{t+n-k}\equiv j\ (\operatorname{mod}\ 2)$, concluding the proof of the lemma.
\end{proof}
\begin{lemma}\label{lem:lemPllkj}
Assume that $\ll\in\mc{Z}(s)$ and that there exists an index $1\leq i<s$ such that $\ll_i\not\equiv\ll_{i+1}\ (\operatorname{mod}\ 2)$. For any $j\in\bb{Z}/2\bb{Z}$, consider the collection
\begin{equation}\label{eq:defmcPllkj}
\mc{P}_{\ll}(j)=\left\{I\in{[n]\choose k}:\ll^1(I)\in\mc{C}^{k+1+j},\ll^2(I)\in\mc{C}^0_{\geq n+1}\right\}.
\end{equation}
We have (using notation (\ref{eq:defsI}))
\begin{equation}\label{eq:sumsgnsI=0}
\sum_{I\in\mc{P}_{\ll}(j)}\operatorname{sgn}(\sigma(I))=0.
\end{equation}
\end{lemma}
\begin{proof}
We show that if $I\in\mc{P}_{\ll}(j)$ then exactly one of $i,i+1$ is contained in $I$. Moreover, we show that the assignment $I'=I\setminus\{i\}\cup\{i+1\}$ establishes a bijection between
\begin{equation}\label{eq:corrII'}
\{I\in\mc{P}_{\ll}(j):i\in I\}\quad\rm{and}\quad\{I'\in\mc{P}_{\ll}(j):i+1\in I'\}.
\end{equation}
Since $\operatorname{sgn}(\sigma(I'))=-\operatorname{sgn}(\sigma(I))$, the conclusion (\ref{eq:sumsgnsI=0}) follows.
Assume that $I$ is such that $i,i+1$ are both in $I$, or both in $I^c$. We can then find $t<k$ or $t>k$ such that $i_t=i$ and $i_{t+1}=i+1$. If $t<k$ then $\ll^1(I)_t\not\equiv\ll^1(I)_{t+1}\ (\operatorname{mod}\ 2)$, contradicting $\ll^1(I)\in\mc{C}^{k+1+j}$. If $t>k$ then $\ll^2(I)_{t-k}\not\equiv\ll^2(I)_{t-k+1}\ (\operatorname{mod}\ 2)$, contradicting $\ll^2(I)\in\mc{C}^0_{\geq n+1}$.
Choose now a set $I$ with $i\in I$, $i+1\in I^c$, and choose $t_0\leq k$, $t_1\geq k+1$, such that $i_{t_0}=i$, $i_{t_1}=i+1$. If we let $I'=I\setminus\{i\}\cup\{i+1\}$ then $\ll^1(I)_t=\ll^1(I')_t$ for $t\neq t_0$, and $\ll^2(I)_t=\ll^2(I')_t$ for $t\neq t_1-k$. We have
\[\ll^1(I)_{t_0}=t_0+\ll_i-i,\quad \ll^1(I')_{t_0}=t_0+\ll_{i+1}-(i+1),\]
\[\ll^2(I)_{t_1-k}=t_1+\ll_{i+1}-(i+1),\quad \ll^2(I')_{t_1-k}=t_1+\ll_i-i,\]
and since $\ll_i\not\equiv\ll_{i+1}\ (\operatorname{mod}\ 2)$, we get $\ll^1(I)_t\equiv\ll^1(I')_t\ (\operatorname{mod}\ 2)$ and $\ll^2(I)_t\equiv\ll^2(I')_t\ (\operatorname{mod}\ 2)$ for all $t$. Since $\ll^2(I)_{t_1-k}\leq\ll^2(I')_{t_1-k}$, the only way in which the correspondence $I\leftrightarrow I'$ could fail to induce a bijection (\ref{eq:corrII'}) is if for some $I,I'$ we get $t_1=n$ and $\ll^2(I)_{n-k}\leq n<n+1\leq\ll^2(I')_{n-k}$, in which case $\ll^2(I)\not\in\mc{C}^0_{\geq n+1}$, but $\ll^2(I')\in\mc{C}^0_{\geq n+1}$. However, the inequality $\ll^2(I)_{n-k}\leq n$ would imply
\[\ll^2(I)_{t_1-k}=t_1+\ll_{i_{t_1}}-i_{t_1}=n+\ll_{i+1}-(i+1)\leq n\rm{ or equivalently }\ll_{i+1}\leq i+1.\]
Since $i<s$ by hypothesis, we get $\ll_s\leq\ll_{i+1}\leq(i+1)\leq s$, contradicting the fact that $\ll\in\mc{Z}(s)$.
\end{proof}
\begin{lemma}\label{lem:Plljtoparts}
If $\ll\in\mc{C}^{h,j}(s)$, $s\geq n-k$, then there is a one-to-one correspondence between elements $\mc{P}_{\ll}(j)$ and the set $P^{n-k+j-h,s+1-h}(k-n+s,n-k)$ (defined in (\ref{eq:defPhjab})).
Moreover, for every $I\in\mc{P}_{\ll}(j)$ we have
\[\operatorname{sgn}(\sigma(I))=(-1)^{(n-k)\cdot(k+h)},\]
and $\mc{P}_{\ll}(j)$ is empty if $h\equiv s\equiv j+1\ (\operatorname{mod}\ 2)$.
\end{lemma}
\begin{proof} The correspondence between sets $I\in{[n]\choose k}$ (resp. their complements $I^c$) and partitions $\mu\in P(k,n-k)$ (resp. their conjugates $\mu'$) is given in (\ref{eq:Itomu}) (resp. (\ref{eq:Icfrommu'})). If $I\in\mc{P}_{\ll}(j)$ then it follows from Lemma~\ref{lem:ll^iinmcC} that $s+1,\cdots,n$ are the largest elements of $I$, namely $i_{k-n+s+1},\cdots,i_k$, so $\mu_1=\cdots=\mu_{n-s}=n-k$. The set $I$ is then determined by $\ol{\mu}=(\mu_{n-s+1},\cdots,\mu_k)\in P(k-n+s,n-k)$. Since $\ll\in\mc{C}^{h,j}(s)$, the condition $\ll^1(I)\in\mc{C}^{k+1+j}$ is equivalent to $\ol{\mu}_i\equiv n-k+j-h\ (\operatorname{mod}\ 2)$. The condition $\ll^2(I)\in\mc{C}^0_{\geq n+1}$ is equivalent to $\mu'_i\equiv n+1-h\ (\operatorname{mod}\ 2)$, which in turn is equivalent to $\ol{\mu}'_i\equiv s+1-h\ (\operatorname{mod}\ 2)$. It follows that $I\in\mc{P}_{\ll}(j)$ if and only if $\ol{\mu}\in P^{n-k+j-h,s+1-h}(k-n+s,n-k)$, which establishes the desired bijection. Moreover
\[\operatorname{sgn}(\sigma(I))=(-1)^{|\mu|}=(-1)^{|\mu'|}=(-1)^{(n-k)\cdot(n+1-h)}=(-1)^{(n-k)\cdot(k+h)},\]
where the last equality follows from the fact that $(n-k)\cdot(n+1-k)$ is even. If $h\equiv s\equiv j+1\ (\operatorname{mod}\ 2)$ then $|\mc{P}_{\ll}(j)|=|P^{n-k+1,1}(k-n+s,n-k)|=0$ by Lemma~\ref{lem:countpartitions}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:limsym}]
We have
\[\scpr{S_{\ll}W}{p_{k,r}(V)\otimes E}=\scpr{S_{\ll}W\otimes p_{k,r}(W)}{E}\overset{(\ref{eq:genlPieri}),(\ref{eq:Sll+rI=Sll(rI)})}{=}\sum_{I\in{[n]\choose k}}\operatorname{sgn}(\sigma(I))\cdot\scpr{S_{\ll(r,I)}W}{E}.\]
Since $\det(\operatorname{Sym}^2 W)=\det(W)^{\otimes(n+1)}=S_{(n+1)^n}W$, we get using Cauchy's formula \cite[Prop.~2.3.8]{weyman} that
\[E=\det\left(\operatorname{Sym}^2 W\right)\otimes\operatorname{Sym}\left(\operatorname{Sym}^2 W\right)=\bigoplus_{\substack{\ll\in\bb{Z}^n_{\operatorname{dom}},\ll_n\geq n+1 \\ \ll_i\equiv n+1\ (\operatorname{mod}\ 2)}}S_{\ll}W.\]
Using notation (\ref{eq:defllrI}--\ref{eq:defll12}) and (\ref{eq:defmcCjk}) we obtain for $r\gg 0$
\[
\scpr{S_{\ll(r,I)}W}{E} = \begin{cases}
1, & \rm{if }\ll^1(I)\in\mc{C}^r\rm{ and }\ll^2(I)\in\mc{C}^0_{\geq n+1}, \\
0, & \rm{otherwise}.
\end{cases}
\]
It follows (using notation (\ref{eq:defmcPllkj})) that for $j\in\bb{Z}/2\bb{Z}$
\begin{equation}\label{eq:limpkrvsym1}
\lim_{\substack{r\to\infty \\ r\equiv k+1+j\ (\operatorname{mod}\ 2)}} p_{k,r}(V)\otimes E=\sum_{\ll\in\bb{Z}^n_{\operatorname{dom}}}\left(\sum_{I\in\mc{P}_{\ll}(j)}\operatorname{sgn}(\sigma(I))\right)\cdot S_{\ll}W,
\end{equation}
and by Lemmas~\ref{lem:ll^iinmcC} and~\ref{lem:lemPllkj} we only need to consider $\ll\in\mc{Z}(s)$ for $s\geq n-k$ such that (for some $h\in\bb{Z}/2\bb{Z}$)
\[\ll_1\equiv\cdots\equiv\ll_s\equiv h\ (\operatorname{mod}\ 2)\quad\rm{ and }\quad\ll_{s+1}\equiv\cdots\equiv\ll_n\equiv j\ (\operatorname{mod}\ 2),\]
i.e. $\ll\in\mc{C}^{h,j}(s)$. Multiplying both sides of (\ref{eq:limpkrvsym1}) by $(-1)^{k\cdot(n-k)}$ and using Lemma~\ref{lem:Plljtoparts} we get
\[(-1)^{k\cdot(n-k)}\cdot\left(\lim_{\substack{r\to\infty \\ r\overset{(\operatorname{mod}\ 2)}{\equiv} k+1+j}} p_{k,r}(V)\otimes E\right)=\sum_{\substack{n-k\leq s\leq n \\ h=j,j+1}}(-1)^{(n-k)\cdot h}\cdot|P^{n-k+j-h,s+1-h}(k-n+s,n-k)|\cdot\mf{C}^{h,j}_s.\]
We separate the contributions of the right hand side according to two cases:
\ul{Terms with $h=j+1$:} By Lemma~\ref{lem:Plljtoparts} we can consider only the terms with $s\equiv j\ (\operatorname{mod}\ 2)$, in which case we get from (\ref{eq:ChjtoCj}) that $\mf{C}^{h,j}_s=\mf{C}^{s+1,s}_s=\mf{C}^2_s$. We have
\[
|P^{n-k+j-h,s+1-h}(k-n+s,n-k)|=|P^{n-k+1,0}(k-n+s,n-k)|\overset{\rm{Lemma }\ref{lem:countpartitions}}{=}\begin{cases}
\displaystyle{\lfloor\frac{s-1}{2}\rfloor \choose \frac{n-k-1}{2}} & n-k\rm{ odd}, \\
\\
\displaystyle{\frac{s-2}{2} \choose \frac{n-k-2}{2}} & n-k\rm{ and }s\rm{ even}, \\
\\
0 & \rm{otherwise.}
\end{cases}
\]
Comparing the coefficient of $\mf{C}^2_s$ in Proposition~\ref{prop:limsym} with $(-1)^{(n-k)\cdot h}\cdot|P^{n-k+j-h,s+1-h}(k-n+s,n-k)|$ in each of the cases $j=0,1$, and $(n-k)$ even and odd, we see that they agree.
\ul{Terms with $h=j$:} The terms with $s\equiv j+1\ (\operatorname{mod}\ 2)$ contribute $\mf{C}^{h,j}_s=\mf{C}^{s+1,s+1}_s$ with coefficient $(-1)^{(n-k)\cdot h}\cdot|P^{n-k,0}(k-n+s,n-k)|$. The terms with $s\equiv j\ (\operatorname{mod}\ 2)$ contribute $\mf{C}^{h,j}_s=\mf{C}^{s,s}_s$ with coefficient $(-1)^{(n-k)\cdot h}\cdot|P^{n-k,1}(k-n+s,n-k)|$. For $s=n-k$ we get $|P^{n-k,1}(k-n+s,n-k)|=0$ so $\mf{C}^{s,s}_s$ only appears for $s>n-k$. Observing that $|P^{n-k,0}(k-n+s,n-k)|=|P^{n-k,1}(k-n+s+1,n-k)|$ for $s\geq n-k$, and using $\mf{C}^1_s=\mf{C}^{s+1,s+1}_s+\mf{C}^{s+1,s+1}_{s+1}$ in (\ref{eq:ChjtoCj}), we conclude that the terms with $h=j$ contribute \[\sum_{s\equiv j+1\ (\operatorname{mod}\ 2)}(-1)^{(n-k)\cdot h}\cdot|P^{n-k,0}(k-n+s,n-k)|\cdot\mf{C}^1_s,\]
where
\[|P^{n-k,0}(k-n+s,n-k)|\overset{\rm{Lemma }\ref{lem:countpartitions}}{=}\begin{cases}
\displaystyle{\lfloor\frac{s}{2}\rfloor \choose \frac{n-k}{2}} & n-k\rm{ even}, \\
\\
\displaystyle{\frac{s-1}{2} \choose \frac{n-k-1}{2}} & n-k\rm{ and }s\rm{ odd}, \\
\\
0 & \rm{otherwise.}
\end{cases}\]
Comparing with the coefficient of $\mf{C}^1_s$ in Proposition~\ref{prop:limsym} we conclude the proof of the proposition.
\end{proof}
\subsection{General matrices}\label{subsec:limgeneral}
For positive integers $m\geq n$ and for $s=0,\cdots,n$, we let
\begin{equation}\label{eq:defAsn}
\mc{A}(s;m,n)=\{\ll\in\bb{Z}^n_{\operatorname{dom}}:\ll_s\geq s+m-n,\ll_{s+1}\leq s\}.
\end{equation}
If $\ll\in\mc{A}(s;m,n)$ then we define a dominant weight $\ll(s)\in\bb{Z}^m_{\operatorname{dom}}$ by
\begin{equation}\label{eq:deflls}
\ll(s) = (\ll_1-(m-n),\cdots,\ll_s-(m-n),\underbrace{s,\cdots,s}_{m-n},\ll_{s+1},\cdots,\ll_n)
\end{equation}
For vector spaces $W_1$, $W_2$, with $\dim(W_1)=m$, $\dim(W_2)=n$, and $s=0,\cdots,n$ we define $\mf{A}_s\in\Gamma(W_1,W_2)$~by
\begin{equation}\label{eq:defas}
\mf{A}_s=\bigoplus_{\ll\in\mc{A}(s;m,n)} S_{\ll(s)}W_1\otimes S_{\ll}W_2.
\end{equation}
\begin{proposition}\label{prop:limsgeneral}
We write $W=W_1\otimes W_2$. If $E=\det(W)\otimes\operatorname{Sym}(W)$ then for $k=0,\cdots,n,$
\[(-1)^{k\cdot(m-n)}\cdot\left(\lim_{r\to\infty} p_{k,r}(V_1)\otimes p_{k,r}(V_2)\otimes E\right)=\sum_{s=n-k}^n (-1)^{(m-n)\cdot(n-k-s)}\cdot {s\choose s-n+k}\cdot \mf{A}_s.\]
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:limsgeneral}]
Consider dominant weights $\tl{\d}\in\bb{Z}^m_{\operatorname{dom}}$ and $\tl{\ll}\in\bb{Z}^n_{\operatorname{dom}}$, and let
\begin{equation}\label{eq:tldtlll}
\d=\tl{\d}-(n^m),\quad\ll=\tl{\ll}-(m^n).
\end{equation}
We obtain using (\ref{eq:Sll+rI=Sll(rI)}), (\ref{eq:genlPieri}), and easy manipulations that $\scpr{S_{\tl{\d}}W_1\otimes S_{\tl{\ll}}W_2}{p_{k,r}(V_1)\otimes p_{k,r}(V_2)\otimes E}$ equals
\[\sum_{I\in{[m]\choose k},\ J\in{[n]\choose k}}\operatorname{sgn}(\sigma(I))\cdot\operatorname{sgn}(\sigma(J))\cdot\scpr{S_{\d(r,I)}W_1\otimes S_{\ll(r,J)}W_2}{\operatorname{Sym}(W)}.\]
Using (\ref{eq:defll12}) and writing $\mu | (0^{m-n})$ for the sequence obtained by appending $m-n$ zeros to $\mu$, we get for~$r\gg 0$
\[
\scpr{S_{\d(r,I)}W_1\otimes S_{\ll(r,J)}W_2}{\operatorname{Sym}(W)} = \begin{cases}
1, & \rm{if }\d^1(I)=\ll^1(J),\ \d^2(I)=\ll^2(J) | (0^{m-n}),\rm{ and }\d^2(I)\in\bb{Z}^{m-k}_{\geq 0}, \\
0, & \rm{otherwise}.
\end{cases}
\]
Let $u\in\{0,\cdots,m\}$ be the unique index such that $\d_u\geq u-m\geq\d_{u+1}$. The condition $\d^2(I)\in\bb{Z}^{m-k}_{\geq 0}$ is equivalent to the inclusion $\{u+1,\cdots,m\}\subset I$, which implies $u\geq m-k$. When $m>n$, the last $m-n$ entries of $\d^2(I)$ being $0$ forces $\d_u=\d_{u-1}=\cdots=\d_{u-m+n+1}=u-m$, and all the elements $u,u-1,\cdots,u-m+n+1$ to be contained in $I^c=[m]\setminus I$. We modify $\d$ and $I$ as follows: we consider $\ol{\d}\in\bb{Z}^n_{\operatorname{dom}}$ and $\ol{I}\in{[n]\choose k}$ defined by
\[\ol{\d}=(\d_1,\cdots,\d_{u-m+n},\d_{u+1}-(m-n),\cdots,\d_m-(m-n)),\]
\[\quad\ol{I}=\{i_1,\cdots,i_{k-m+u},u+1-(m-n),u+2-(m-n),\cdots,n\},\]
so that $\ol{I}^c=[n]\setminus\ol{I}=I^c\setminus\{u,u-1,\cdots,u-m+n+1\}$. The conditions $\d^1(I)=\ll^1(J)$ and $\d^2(I)=\ll^2(J) | (0^{m-n})$ are then equivalent to $\ol{\d}^1(\ol{I})=\ll^1(J)$ and $\ol{\d}^2(\ol{I})=\ll^2(J)$. Since both $\ll,\ol{\d}$ are dominant weights, these equalities can only hold for $\ol{\d}=\ll$ and $\ol{I}=J$. Note that the freedom in choosing $I$ (or $\ol{I}=J$) is in the choice of an increasing sequence $i_1<\cdots<i_{k-m+u}$ inside $\{1,\cdots,u\}$, i.e. there are ${u-m+n\choose k-m+u}$ choices for $I$ once we fix $\d$. Writing $s=u-m+n$ we get
\[s\geq(m-k)-m+n=n-k,\]
\[\ll_s=\d_s=\d_{u-m+n}\geq\d_u\geq u-m=s-n,\]
\[\ll_{s+1}=\d_{u+1}-(m-n)\leq(u-m)-(m-n)=(s-n)-(m-n)=s-m,\]
and moreover
\[\d=(\ll_1,\cdots,\ll_s,\underbrace{s-n,\cdots,s-n}_{m-n},\ll_{s+1}+(m-n),\cdots,\ll_n+(m-n)).\]
It follows (using (\ref{eq:deflls}), (\ref{eq:tldtlll})) that $\tl{\d}=\tl{\ll}(s)$. Since $\operatorname{sgn}(\sigma(I))=(-1)^{(m-n)\cdot(m-u)}\cdot\operatorname{sgn}(\sigma(\ol{I}))$, it follows that if $\ol{I}=J$ and $m-u=n-s$ then
\[\operatorname{sgn}(\sigma(I))\cdot\operatorname{sgn}(\sigma(J))=(-1)^{(m-n)\cdot(n-s)}.\]
Putting everything together, and using ${u-m+n\choose k-m+u}={s\choose s-n+k}$, we obtain for $r\gg 0$
\[\scpr{S_{\tl{\d}}W_1\otimes S_{\tl{\ll}}W_2}{p_{k,r}(V_1)\otimes p_{k,r}(V_2)\otimes E}=\begin{cases}
\displaystyle(-1)^{(m-n)\cdot(n-s)}\cdot{s\choose s-n+k} & \rm{if }\tl{\ll}\in\mc{A}(s;m,n)\rm{ and }\tl{\d}=\tl{\ll}(s)\\
& \rm{for some }s\geq n-k,\\
0 & \rm{otherwise.}
\end{cases}
\]
Multiplying by $(-1)^{k\cdot(m-n)}$ and taking the limit $r\to\infty$ yields the desired conclusion.
\end{proof}
\subsection{Skew-symmetric matrices}\label{subsec:limskew}
For a positive integer $m$ and for $s=0,\cdots,m$, we let
\begin{equation}\label{eq:defmcBsn}
\begin{aligned}
\mc{B}(s,2m) &= \{\ll\in\bb{Z}^{2m}_{\operatorname{dom}}:\ll_{2s}\geq(2s-1),\ll_{2s+1}\leq 2s,\ll_{2i-1}=\ll_{2i}\rm{ for all }i\}, \\
\mc{B}(s,2m+1) &= \{\ll\in\bb{Z}^{2m+1}_{\operatorname{dom}}:\ll_{2s+1}=2s,\ll_{2i-1}=\ll_{2i}\rm{ for }i\leq s,\ll_{2i}=\ll_{2i+1}\rm{ for }i>s\}.
\end{aligned}
\end{equation}
For a vector space $W$ with $\dim(W)=n$, and for $s=0,\cdots,m=\lfloor n/2\rfloor$ we define $\mf{B}_s\in\Gamma(W)$ via
\begin{equation}\label{eq:defbs}
\mf{B}_s=\bigoplus_{\ll\in\mc{B}(s,n)} S_{\ll}W.
\end{equation}
\begin{proposition}\label{prop:limskew}
If $E=\det\left(\bigwedge^2 W\right)\otimes\operatorname{Sym}\left(\bigwedge^2 W\right)$ and $k=0,\cdots,m=\lfloor n/2\rfloor$, then
\[\lim_{r\to\infty} p_{2k,r}(V)\otimes E=\sum_{s=m-k}^m {s\choose m-k}\cdot \mf{B}_s.\]
\end{proposition}
With $m=\lfloor n/2\rfloor$ and $1\leq k\leq m$, we define the following collections of dominant weights
\begin{equation}\label{eq:defmcBs}
\begin{aligned}
\mc{B} &=\{\ll\in\bb{Z}^{2k}_{\operatorname{dom}}:\ll_{2i-1}=\ll_{2i}\rm{ for }i=1,\cdots,k\}, \\
\mc{B}_{\geq n-1} &=\{\ll\in\bb{Z}^{n-2k}_{\operatorname{dom}}:\ll_{2i-1}=\ll_{2i}\geq n-1\rm{ for }i=1,\cdots,m-k,\ll_{n-2k}=n-1\rm{ if }n\rm{ is odd}\}.
\end{aligned}
\end{equation}
We partition $\bb{Z}^n_{\operatorname{dom}}$ into the following collections of dominat weights $\mc{Y}(u)$, $u=0,\cdots,n$, defined by
\begin{equation}\label{eq:Yu}
\mc{Y}(u)=\{\ll\in\bb{Z}^n_{\operatorname{dom}}:\ll_u\geq u-1\geq\ll_{u+1}\},
\end{equation}
In analogy with Lemma~\ref{lem:ll^iinmcC} one can prove:
\begin{lemma}\label{lem:ll^iinmcB}
If $I\in{[n]\choose 2k}$, then the conditions $\ll^1(I)\in\mc{B}$ and $\ll^2(I)\in\mc{B}_{\geq n-1}$ are equivalent to
\begin{itemize}
\item $\ll\in\mc{Y}(u)$ for some $u=n-2k,\cdots,n$.
\item $\{u+1,\cdots,n\}\subset I$.
\item $\ll_{i_{2t-1}}=\ll_{i_{2t}}$ and $i_{2t}=i_{2t-1}+1$ for all $t=1,\cdots,m$. If $n$ is odd then $u\in I^c$ is odd and $\ll_u=u-1$.
\end{itemize}
\end{lemma}
\begin{lemma}\label{lem:permi2t-1i2t}
Assume that $\ll,I$ satisfy the equivalent conditions in Lemma~\ref{lem:ll^iinmcB}. If $n=2m$ is even then
\begin{equation}\label{eq:permi2teven}
\{(i_1,i_2),(i_3,i_4),\cdots,(i_{2m-1},i_{2m})\} = \{(1,2),(3,4),\cdots,(2m-1,2m)\}.
\end{equation}
If $n=2m+1$ is odd and if we write $u=2s+1$, then we have
\begin{equation}\label{eq:permi2todd}
\{(i_1,i_2),(i_3,i_4),\cdots,(i_{2m-1},i_{2m})\} = \{(1,2),\cdots,(2s-1,2s),(2s+2,2s+3),\cdots,(2m,2m+1)\}.
\end{equation}
Moreover, we have that $\ll\in\mc{B}(s,n)$ for some $s=m-k,\cdots,m$.
\end{lemma}
\begin{proof}
The conclusions (\ref{eq:permi2teven}--\ref{eq:permi2todd}) follow from the fact that $i_1,\cdots,i_n$ give a permutation of $[n]$ with $i_{2t}=i_{2t-1}+1$, and $i_n=u$ is odd when $n$ is odd. If $n$ is odd and $u=2s+1$, it follows from $u\geq n-2k$ that $s\geq m-k$. Moreover, we have $\ll_{2s+1}=\ll_u=u-1=2s$, and it follows from (\ref{eq:permi2todd}) that $\ll\in\mc{B}(s,n)$.
Assume now that $n=2m$ is even. It follows from (\ref{eq:permi2teven}) that $i_n$ is even, so we can write $i_n=2s'$. Since $i_n+1,\cdots,n\in I$, we get $n-i_n\leq 2k$, i.e. $s'\geq m-k$. Since $i_n\leq u$, we have $\ll_{2s'}\geq\ll_u\geq u-1\geq 2s'-1$. We have by (\ref{eq:permi2teven}) that $\ll_{2i-1}=\ll_{2i}$ for $i=1,\cdots,m$, so taking $s$ to be the maximal index for which $\ll_{2s}\geq 2s-1$ we find that $s\geq s'$ and $\ll_{2s+1}=\ll_{2s+2}<2s+1$, i.e. $\ll\in\mc{B}(s,n)$.
\end{proof}
\begin{lemma}\label{lem:sizeBsn/2-sj}
Let $m=\lfloor n/2\rfloor$, and for $m-k\leq s\leq m$ define the collection of partitions
\begin{equation}\label{eq:defBsn/2-sj}
\begin{aligned}
\mc{B}(k,n/2-k,s)=\{\mu\in P(2k,n-2k):&\mu'_i\rm{ even for }i=1,\cdots,n-2k,\mu'_{n-2k}=2m-2s,\\
&\mu_i\rm{ even for }i=2m-2s+1,\cdots,2k\}.
\end{aligned}
\end{equation}
Every partition $\mu\in\mc{B}(k,n/2-k,s)$ has even size, and the cardinality of the set $\mc{B}(k,n/2-k,s)$ is given by
\[
|\mc{B}(k,n/2-k,s)| = \begin{cases}
{s\choose m-k} & \rm{ if }n=2m+1\rm{ is odd}; \\
{s-1\choose m-1-k} & \rm{ if }n=2m\rm{ is even}. \\
\end{cases}
\]
\end{lemma}
\begin{proof}
Since each $\mu_i'$ is even, $|\mu|=|\mu'|$ is even. To compute the size of $\mc{B}(k,n/2-k,s)$ we first note that the condition $\mu'_{n-2k}=2m-2s$ implies that $\mu_1=\cdots=\mu_{2m-2s}=n-2k$, so any $\mu\in\mc{B}(k,n/2-k,s)$ is determined by $\ol{\mu}=(\mu_{2m-2s+1},\cdots,\mu_{2k})\in P(2(k+s-m),n-2k)$. Since $\mu'_{n-2k}=2m-2s$, we must have $\ol{\mu}_1<n-2k$. The condition $\mu\in\mc{B}(k,n/2-k,s)$ is then equivalent (using (\ref{eq:defPhjab})) to
\[\ol{\mu}\in\begin{cases}
P^{0,0}(2(k+s-m),2(m-k)) & \rm{if }n\rm{ is odd}, \\
P^{0,0}(2(k+s-m),2(m-1-k)) & \rm{if }n\rm{ is even}. \\
\end{cases}
\]
By Lemma~\ref{lem:countpartitions}, the number of choices for $\ol{\mu}$ is ${s\choose m-k}$ if $n$ is odd, respectively ${s-1\choose m-1-k}$ if $n$ is even.
\end{proof}
\begin{lemma}\label{lem:corrImcBkn/2s}
Assume that $\ll\in\mc{B}(s,n)$ for some $s\geq m-k$. The collection of subsets $I\in{[n]\choose 2k}$ for which $\ll^1(I)\in\mc{B}$ and $\ll^2(I)\in\mc{B}_{\geq n-1}$ corresponds via (\ref{eq:Itomu}) to $\mc{B}'(k,n/2-k,s)$, where
\[
\mc{B}'(k,n/2-k,s)=\begin{cases}
\mc{B}(k,n/2-k,s) & \rm{if }n\rm{ is odd}, \\
\displaystyle\bigcup_{s'=m-k}^s\mc{B}(k,n/2-k,s') & \rm{if }n\rm{ is even}.
\end{cases}
\]
\end{lemma}
\begin{proof} Consider $\ll\in\mc{B}(s,n)$ for $s\geq m-k$, and $I\in{[n]\choose 2k}$ satisfying the conditions of Lemma~\ref{lem:ll^iinmcB}. If $n=2m+1$ is odd then $i_n=2s+1$ and $I$ contains $2s+2,\cdots,n$, i.e. the corresponding $\mu\in P(2k,n-2k)$ has
\[\mu_1=\cdots=\mu_{2m-2s}=n-2k,\quad \mu_{2m-2s+1}<n-2k,\]
so $\mu'_{n-2k}=2m-2s$. For $2k<t<n$ we have that $i_t\leq 2s$, so $\mu'_{t-2k}=t-i_t$ is even by (\ref{eq:permi2todd}). The set of $\mu_i$ with $2m-2s<i\leq 2k$ coincides with that of differences $i_t-t$ for $1\leq t\leq 2(k-m+s)$, which are all even again by (\ref{eq:permi2todd}) and the fact that $i_t\leq 2s$ for $t\leq 2(k-m+s)$.
Assume next that $n=2m$ is even, and use (\ref{eq:permi2teven}) to write $i_n=2s'$. As in the previous paragraph, this implies $\mu'_{n-2k}=2m-2s'$. By (\ref{eq:permi2teven}) all the differences $t-i_t$ are even, so all $\mu_i,\mu'_i$ are even. This shows that $I\in\mc{B}(k,n/2-k,s')$. Since $i_n+1,\cdots,n\in I$ we get as before $s'\geq m-k$. If $s'>s$ then $\ll^2(I)_{n-2k}=\ll_{i_n}+n-i_n=\ll_{2s'}+n-2s'\leq\ll_{2s+1}+n-2s'\leq 2s+n-2s'\leq n-2$, a contradiction.
The verification that $\mu\in\mc{B}'(k,n/2-k,s)$ yields a subset $I$ with $\ll^1(I)\in\mc{B}$ and $\ll^2(I)\in\mc{B}_{\geq n-1}$ follows easily by tracing back the arguments.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:limskew}]
We have
\[\scpr{S_{\ll}W}{p_{2k,r}(V)\otimes E}=\scpr{S_{\ll}W\otimes p_{2k,r}(W)}{E}\overset{(\ref{eq:genlPieri}),(\ref{eq:Sll+rI=Sll(rI)})}{=}\sum_{I\in{[n]\choose 2k}}\operatorname{sgn}(\sigma(I))\scpr{S_{\ll(r,I)}W}{E}.\]
Using notation (\ref{eq:defll12}) and (\ref{eq:defmcBs}) we get that for $r\gg 0$
\[
\scpr{S_{\ll(r,I)}W}{E} = \begin{cases}
1, & \rm{if }\ll^1(I)\in\mc{B}\rm{ and }\ll^2(I)\in\mc{B}_{\geq n-1}, \\
0, & \rm{otherwise}.
\end{cases}
\]
It follows that
\[\lim_{r\to\infty} p_{2k,r}(V)\otimes E = \sum_{\substack{\ll\in\bb{Z}^n_{\operatorname{dom}},\ I\in{[n]\choose 2k}\\ \ll^1(I)\in\mc{B},\ \ll^2(I)\in\mc{B}_{\geq n-1}}}\operatorname{sgn}(\sigma(I))\cdot S_{\ll}W\] \[\overset{Lemmas~\ref{lem:permi2t-1i2t}-\ref{lem:corrImcBkn/2s}}{=}\sum_{s=m-k}^m\left(\sum_{\substack{\ll\in\mc{B}(s,n) \\ \mu\in\mc{B}'(k,n/2-k,s)}} (-1)^{|\mu|} S_{\ll}W\right)\overset{|\mu|\rm{ even}}{=} \sum_{s=m-k}^m\left(\sum_{\ll\in\mc{B}(s,n)} |\mc{B}'(k,n/2-k,s)|\cdot S_{\ll}W\right)\]
If $n=2m+1$ is odd, then $|\mc{B}'(k,n/2-k,s)|=|\mc{B}(k,n/2-k,s)|={s\choose m-k}$, and the desired equality follows. Similarly, when $n=2m$ is even, we get
\[|\mc{B}'(k,n/2-k,s)|=\sum_{s'=m-k}^s |\mc{B}(k,n/2-k,s')|=\sum_{s'=m-k}^s {s'-1\choose m-1-k}={s\choose m-k}.\qedhere\]
\end{proof}
\section{Equivariant $\mathcal{D}$-modules on symmetric matrices}\label{sec:symm}
In this section we compute the characters of the $\operatorname{GL}$-equivariant $\mathcal{D}$-modules on the vector space $M^{\operatorname{symm}}$ of symmetric $n\times n$ matrices. We let $W$ denote a complex vector space of dimension $n$, $V=W^*$, and we identify $\operatorname{Sym}^2 W$ with $M^{\operatorname{symm}}$, where squares $w^2$ correspond to matrices of rank one. If we write $\operatorname{GL}=\operatorname{GL}(W)$, and let $M^{\operatorname{symm}}_s$ denote the subvariety of matrices of rank at most $s$ then the main result of this section is:
\begin{theorem}\label{thm:symmetric}
There exist $(2n+1)$ simple $\operatorname{GL}$-equivariant holonomic $\mathcal{D}$-modules on $M^{\operatorname{symm}}$, namely
\[
C^j_s=\begin{cases}
\mc{L}(M^{\operatorname{symm}}_{n-s},M^{\operatorname{symm}}) & \rm{if }j\equiv s\ (\operatorname{mod}\ 2), \\
\mc{L}(M^{\operatorname{symm}}_{n-s},M^{\operatorname{symm}};1/2) & \rm{if }j\equiv s+1\ (\operatorname{mod}\ 2), \\
\end{cases}
\rm{ for }s=0,\cdots,n-1,\ j=1,2,
\]
and $C_n^1=C_n^2=\mc{L}(\{0\},M^{\operatorname{symm}})$. For all $s,j$, the character of $C_s^j$ is $\mf{C}_s^j$ (as defined in (\ref{eq:defcjs})).
\end{theorem}
The remaining assertion of the Theorem on Equivariant $\mathcal{D}$-modules on Symmetric Matrices described in the Introduction is the identification $C_s^1=F_{s+1}/F_{s-1}$ for $s=0,\cdots,n$: its proof follows closely the proof of Theorem~\ref{thm:nxnbasicthm} in the next section, so we leave the details to the interested reader. The classification of $\operatorname{GL}$-equivariant holonomic simple $\mathcal{D}$-modules is explained in Theorem~\ref{thm:classificationDmods}, so we only need to check that $\mf{C}_s^j$ is the character of $C_s^j$. For $k=1,\cdots,n$, we consider the situation of Section~\ref{subsec:pfwdDmods}, with $X=X_k=\bb{G}(k,V)$ and $\mc{R},\mc{Q}$ as in (\ref{eq:tautGr}). We let $U=\operatorname{Sym}^2 V$, $\eta=\operatorname{Sym}^2\mc{Q}$. If we write $Y=Y_k$, $\pi=\pi_k$, then (\ref{eq:diagrGeneric}) becomes
\begin{equation}\label{eq:diagrsym^2W}
\xymatrix{
Y_k=\rm{Tot}_{X_k}(\operatorname{Sym}^2\mc{Q}^*) \ar@{^{(}->}[r] \ar[dr]_{\pi_k} & \operatorname{Sym}^2 W\times\bb{G}(k,V) \ar[d] \\
& \operatorname{Sym}^2 W \\
}
\end{equation}
Locally on $X_k$, $\mc{Q}^*$ trivializes to a vector space of dimension $k$, and $Y_k$ gets identified with the space of $k\times k$ symmetric matrices. We take $\mc{L}=(\det\mc{Q})^{\otimes 2}$, consider its $\operatorname{GL}$-equivariant inclusion $\mc{L}\subset\operatorname{Sym}^k\eta$ and note that $\mc{L}$ is locally generated by the symmetric determinant. If we let $Y_k^0\subset Y_k$ be the open set defined locally by the non-vanishing of the determinant, $\mc{M}_k^0=\mc{O}_{Y_k^0}$ is a $\mc{D}_{Y_k}$-module. Note that $Y_k^0$ maps isomorphically via $\pi_k$ to the orbit of symmetric matrices of rank $k$. As a $\operatorname{GL}$-equivariant quasi-coherent sheaf on $X_k$
\begin{equation}\label{eq:defM0symm}
\mc{M}_k^0=\bigoplus_{\ll\in\bb{Z}^k_{\operatorname{dom}},\ll_i\rm{ even}}S_{\ll}\mc{Q}=\varinjlim_{r\equiv k+1\ (\operatorname{mod}\ 2)} (\det\mc{Q})^{\otimes r}\otimes\det\left(\operatorname{Sym}^2\mc{Q}^*\right)\otimes\operatorname{Sym}\left(\operatorname{Sym}^2\mc{Q}^*\right),
\end{equation}
so condition (\ref{eq:twolims}) is satisfied in our context. The Euler characteristic of the $\mathcal{D}$-module pushforward $\int_{\pi_k}\mc{M}_k^0$ is now easily computed as a consequence of Proposition~\ref{prop:EuleraslimPr} and of Remark~\ref{rem:Euleraslim}:
\begin{equation}\label{eq:Dmodpfwdsym0}
\chi\left(\int_{\pi_k}\mc{M}_k^0\right) = (-1)^{k\cdot(n-k)}\cdot\left(\lim_{\substack{r\to\infty \\ r\equiv k+1\ (\operatorname{mod}\ 2)}} p_{k,r}(V)\otimes\det\left(\operatorname{Sym}^2 W\right)\otimes\operatorname{Sym}\left(\operatorname{Sym}^2 W\right)\right),
\end{equation}
which is evaluated explicitly in Proposition~\ref{prop:limsym}.
We next explain why $\mc{M}_k^0\otimes\det(\mc{Q})$ also has the structure of a $\mathcal{D}_{Y_k}$-module. Consider the \'etale double cover $Y_k^{1/2}$ of $Y_k^0$ defined locally by the square-root of the symmetric determinant. The structure sheaf $\mc{O}_{Y_k^{1/2}}$ is naturally a $\mathcal{D}_{Y_k^0}$-module \cite{coutinho-levcovitz} and hence also a $\mathcal{D}_{Y_k}$-module. It contains $\mc{M}_k^0$ so we can define $\mc{M}_k^1$ as the cokernel of the inclusion $\mc{M}_k^0\subset\mc{O}_{Y_k^{1/2}}$. As a $\operatorname{GL}$-equivariant quasi-coherent sheaf on $X_k$, $\mc{M}_k^1$ is given~by
\begin{equation}\label{eq:defM1symm}
\mc{M}_k^1=\bigoplus_{\ll\in\bb{Z}^k_{\operatorname{dom}},\ll_i\rm{ odd}}S_{\ll}\mc{Q}=\mc{O}_{Y_k^0}\otimes\det(\mc{Q}).
\end{equation}
It follows that $\mc{M}_k^1$ satisfies the setting of Proposition~\ref{prop:EuleraslimPr} with $\mc{L}'=\det(\mc{Q})$ so we can compute the Euler characteristic of its direct image via $\pi_k$ as
\begin{equation}\label{eq:Dmodpfwdsym1}
\chi\left(\int_{\pi_k}\mc{M}_k^1\right) = (-1)^{k\cdot(n-k)}\cdot\left(\lim_{\substack{r\to\infty \\ r\equiv k\ (\operatorname{mod}\ 2)}} p_{k,r}(V)\otimes\det\left(\operatorname{Sym}^2 W\right)\otimes\operatorname{Sym}\left(\operatorname{Sym}^2 W\right)\right),
\end{equation}
which is evaluated in Proposition~\ref{prop:limsym}. We are now ready to prove the main result of this section:
\begin{proof}[Proof of Theorem~\ref{thm:symmetric}] The classification of simple $\mathcal{D}$-modules follows from Theorem~\ref{thm:classificationDmods}, so it remains to check that in $\Gamma(W)$ we have the equalities $C_s^j=\mf{C}_s^j$ for $s=0,\cdots,n$ and $j=1,2$. The equalities (\ref{eq:Dmodpfwdsym0}--\ref{eq:Dmodpfwdsym1}) together with Proposition~\ref{prop:limsym} yield for $s=1,\cdots,n$ and $j=1,2$,
\[C_{n-s}^j=\mf{C}_{n-s}^j+\sum_{i=n-s+1}^n (a^s_i\cdot \mf{C}^1_i+b^s_i\cdot\mf{C}^2_i),\rm{ for some integers }a^s_i,b^s_i.\]
Since $C_n^j=\det(\operatorname{Sym}^2 W)\otimes\operatorname{Sym}(\operatorname{Sym}^2 W)$ has character $\mf{C}_n^j$ (by Cauchy's formula \cite[Prop.~2.3.8]{weyman}), the equation above is also satisfied for $s=0$. The Fourier transform $\mc{F}$ permutes the modules $C_s^j$, and it takes
\[
\mc{F}(\mf{C}_s^1)=\mf{C}_{n-s}^1\rm{ for }s=0,\cdots,n\rm{ and }\quad\mc{F}(\mf{C}_s^2)=\mf{C}_{n-s-1}^2\rm{ for }s=0,\cdots,n-1.
\]
We can then apply Lemma~\ref{lem:uppertriangularFourier} to the poset $\mc{P}=\{(s,j):s=0,\cdots,n-1,j=1,2\}\cup\{(n,1)\}$ with the lexicographic ordering given by $(s,j)<(s',j')$ if and only if $s<s'$, or $s=s'$ and $j<j'$. We let $\mf{v}_{(s,j)}=\mf{C}_s^j$ and $v_{(s,j)}=C_s^j$, and conclude using Lemma~\ref{lem:uppertriangularFourier} that $C_s^j=\mf{C}_s^j$ for all $s=0,\cdots,n$ and $j=1,2$.
\end{proof}
\section{Equivariant $\mathcal{D}$-modules on $m\times n$ matrices}\label{sec:mxnmatrices}
In this section we compute the characters of the $\operatorname{GL}$-equivariant $\mathcal{D}$-modules on the vector space $M$ of $m\times n$ matrices, for $m\geq n$. We consider $W_1,W_2$ vector spaces of dimension $\dim(W_1)=m$, $\dim(W_2)=n$, let $V_i=W_i^*$, and identify $W=W_1\otimes W_2$ with $M$, where tensor products $w_1\otimes w_2$ correspond to matrices of rank one. If we write $\operatorname{GL}=\operatorname{GL}(W_1)\times\operatorname{GL}(W_2)$, let $M_s$ denote the subvariety of matrices of rank at most $s$, and recall the notation (\ref{eq:defas}) for the characters $\mf{A}_s$, then the main result of this section is:
\begin{chargenl*}
The simple $\operatorname{GL}$-equivariant holonomic $\mathcal{D}$-modules on $M$ are $A_s=\mc{L}(M_{n-s},M)$, $s=0,\cdots,n$, and for each $s$ the character of $A_s$ is $\mf{A}_s$. When $m=n$, $A_s$ is as described in Theorem~\ref{thm:nxnbasicthm}, while for $m>n$ it can be expressed in terms of local cohomology:
\begin{equation}\label{eq:Asloccoh}
A_s=\mc{H}^{1+s\cdot(m-n)}_{M_{n-1}}(M,\mc{O}_M)=\mc{H}^{\operatorname{codim}(M_{n-s})}_{M_{n-s}}(M,\mc{O}_M).
\end{equation}
\end{chargenl*}
We only need to show that $\mf{A}_s$ is the character of $A_s$, and to prove Theorem~\ref{thm:nxnbasicthm}. The classification of $\operatorname{GL}$-equivariant holonomic simple $\mathcal{D}$-modules is explained in Theorem~\ref{thm:classificationDmods}, while (\ref{eq:Asloccoh}) follows by comparing $\mf{A}_s$ with the characters of local cohomology modules from \cite[Thm.~4.5]{raicu-weyman-witt} and \cite[Thm.~6.1]{raicu-weyman}.
\begin{proof}[Proof of Theorem~\ref{thm:nxnbasicthm}] Let's assume for now that $\mf{A}_s$ is the character of $A_s$, and write $W_1=W_2=\bb{C}^n$. Using Cauchy's formula \cite[Cor.~2.3.3]{weyman}, we get an equality of $\operatorname{GL}$-representations
\[S_{\det}=\bigoplus_{\ll\in\bb{Z}^n_{\operatorname{dom}}}S_{\ll}W_1\otimes S_{\ll}W_2=\bigoplus_{i=0}^n\mf{A}_i.\]
As in Example~\ref{ex:TactsCN}, this shows that $A_0,\cdots,A_n$ are the $\mathcal{D}$-module composition factors of $S_{\det}$, each appearing with multiplicity one. It remains to check that $A_s=F_s/F_{s-1}$ where $F_s=\langle\det^{-s}\rangle_{\mathcal{D}}$.
We prove by induction on $s$ that the $\mathcal{D}$-module composition factors of $F_s$ are $A_0,\cdots,A_s$, which is clearly true for $s=0$. Assume that $s>0$ and that the induction hypothesis is valid for $F_{s-1}$, so that $S_{\det}/F_{s-1}=\bigoplus_{i=s}^n\mf{A}_i$ as $\operatorname{GL}$-representations. We must then have for some $i\geq s$ an inclusion of $\mathcal{D}$-modules $A_i\subset S_{\det}/F_{s-1}$. Using the character description, $A_i$ must contain the class of $\det^{-i}$ inside the quotient $S_{\det}/F_{s-1}$, and therefore it must also contain the classes of $\det^{-i+1},\det^{-i+2},\cdots$. If $i>s$ this contradicts the formula for the character of $A_i$. We conclude that $i=s$ and that we have an inclusion $A_s\subset S_{\det}/F_{s-1}$. Since $A_s$ is simple, it is generated by the class of $\det^{-s}$, so the image of $A_s$ is $F_s/F_{s-1}$.
\end{proof}
We note that, just as in Remark~\ref{rem:bsato}, the strict inclusions $F_{i-1}\subsetneq F_i$, $i=1,\cdots,n$, in Theorem~\ref{thm:nxnbasicthm} combined with Cayley's identity show that the $b$-function of the generic determinant is $b_{\det}(s)=\prod_{i=1}^n(s+i)$.
We conclude by showing that $\mf{A}_s$ is the character of $A_s$. For $k=1,\cdots,n$, we consider the situation of Section~\ref{subsec:pfwdDmods}, with $X=X_k=\bb{G}(k,V_1)\times\bb{G}(k,V_2)$ and $\mc{R}_1,\mc{Q}_1,\mc{R}_2,\mc{Q}_2$ as in (\ref{eq:tautGr}). We let $U=V_1\otimes V_2$, $\eta=\mc{Q}_1\otimes\mc{Q}_2$, and write $Y=Y_k$, $\pi=\pi_k$ in (\ref{eq:diagrGeneric}). We note that locally on $X_k$, $\mc{Q}_1^*,\mc{Q}_2^*$ trivialize to vector spaces of dimension $k$, and $Y_k$ gets identified with the space of $k\times k$ matrices. We take the line bundle $\mc{L}=\det\mc{Q}_1\otimes\det\mc{Q}_2$, consider its $\operatorname{GL}$-equivariant inclusion $\mc{L}\subset\operatorname{Sym}^k\eta$, and note that $\mc{L}$ is locally generated by the function that assigns to a matrix its determinant. If we let $Y_k^0\subset Y_k$ be the open set defined locally by the non-vanishing of the determinant, then as a $\operatorname{GL}$-equivariant quasi-coherent sheaf on $X_k$, $\mc{O}_{Y_k^0}$ is given~by
\[\mc{O}_{Y_k^0}=\bigoplus_{\ll\in\bb{Z}^k_{\operatorname{dom}}}S_{\ll}\mc{Q}_1\otimes S_{\ll}\mc{Q}_2=\varinjlim_r \mc{L}^{\otimes r}\otimes\det\left(\mc{Q}_1^*\otimes\mc{Q}_2^*\right)\otimes\operatorname{Sym}\left(\mc{Q}_1^*\otimes\mc{Q}_2^*\right),\]
so condition (\ref{eq:twolims}) is satisfied in our context. The Euler characteristic of the $\mathcal{D}$-module pushforward $\int_{\pi_k}\mc{O}_{Y_k^0}$ is now easily computed as a consequence of Propositions~\ref{prop:EuleraslimPr} and \ref{prop:limsgeneral}, and of Remark~\ref{rem:Euleraslim}:
\[
\begin{aligned}
\chi\left(\int_{\pi_k}\mc{O}_{Y_k^0}\right) &= (-1)^{k\cdot(m-n)}\cdot\lim_{r\to\infty} p_{k,r}(V_1)\otimes p_{k,r}(V_2)\otimes\det(W)\otimes\operatorname{Sym}(W) \\
&=\sum_{s=n-k}^n (-1)^{(m-n)\cdot(n-k-s)}\cdot{s\choose s-n+k}\cdot \mf{A}_s.
\end{aligned}
\]
Since $\mc{O}_{Y_k^0}$ maps isomorphically via $\pi_k$ to the orbit of rank $k$ matrices in $M$, the conclusion that $\mf{A}_s$ is the character of $A_s$ follows as in the proof of Theorem~\ref{thm:symmetric} by the linear algebra trick in Section~\ref{subsec:linalg}.
\section{Equivariant $\mathcal{D}$-modules on skew-symmetric matrices}\label{sec:skew}
In this section we compute the characters of the $\operatorname{GL}$-equivariant $\mathcal{D}$-modules on the vector space of skew-symmetric $n\times n$ matrices. We let $W$ denote a complex vector space of dimension $n$, $V=W^*$, and we identify $\bigwedge^2 W$ with the vector space $M^{\skew}$ of $n\times n$ skew-symmetric matrices, where exterior products $w_1\wedge w_2$ correspond to matrices of rank two. If we write $\operatorname{GL}=\operatorname{GL}(W)$, $m=\lfloor n/2\rfloor$, let $M^{\skew}_s$ denote the subvariety of matrices of rank at most $2s$, and recall the notation (\ref{eq:defbs}) for the characters $\mf{B}_s$ then we have:
\begin{charskewsym*}
The simple $\operatorname{GL}$-equivariant holonomic $\mathcal{D}$-modules on $M^{\skew}$ are $B_s=\mc{L}(M_{m-s}^{\skew},M^{\skew})$, $s=0,\cdots,m$, and for each $s$ the character of $B_s$ is $\mf{B}_s$. If $n=2m+1$ is odd then for $s=1,\cdots,m$, $B_s$ can be described in terms of local cohomology:
\begin{equation}\label{eq:Bsloccoh}
B_s=\mc{H}^{2s+1}_{M^{\skew}_{m-1}}(M^{\skew},\mc{O}_{M^{\skew}})=\mc{H}^{\operatorname{codim}(M_{m-s}^{\skew})}_{M_{m-s}^{\skew}}(M^{\skew},\mc{O}_{M^{\skew}}).
\end{equation}
If $n=2m$ is even, we let $\operatorname{Pf}$ be an equation defining the hypersurface $M^{\skew}_{m-1}$. We let $S$ denote the coordinate ring of $M^{\skew}$, and consider $F_s=\langle \operatorname{Pf}^{-2s}\rangle_{\mathcal{D}}$, the $\mathcal{D}$-submodule of the localization $S_{\operatorname{Pf}}$ generated by $\operatorname{Pf}^{-2s}$ for $s=0,\cdots,m$ (and $F_{-1}=0$). We have that $B_s=F_s/F_{s-1}$ for $s=0,\cdots,m$.
\end{charskewsym*}
The classification of $\operatorname{GL}$-equivariant holonomic simple $\mathcal{D}$-modules is explained in Theorem~\ref{thm:classificationDmods}, while the equality (\ref{eq:Bsloccoh}) follows from \cite[Theorem~5.5]{raicu-weyman-witt} and \cite[(1.4)]{raicu-weyman-loccoh}. When $n=2m$, we get that $B_s=F_s/F_{s-1}$ just as in the proof of Theorem~\ref{thm:nxnbasicthm}. Note that Cayley's identity shows that $b_{\operatorname{Pf}}(s)$ divides $\prod_{i=1}^m(s+2\cdot i-1)$, which in turn implies that $\langle \operatorname{Pf}^{-2i}\rangle_{\mathcal{D}}=\langle \operatorname{Pf}^{-2i+1}\rangle_{\mathcal{D}}$. The strict inclusions $F_{i-1}\subsetneq F_i$ then force $2\cdot i-1$ to be a root of $b_{\operatorname{Pf}}(s)$ for $i=1,\cdots,m$, so in fact $b_{\operatorname{Pf}}(s)=\prod_{i=1}^m(s+2\cdot i-1)$.
To prove the theorem, it remains to check that $\mf{B}_s$ is the character of~$B_s$. For $k=1,\cdots,m$, we consider the situation of Section~\ref{subsec:pfwdDmods}, with $X=X_k=\bb{G}(2k,V)$ and $\mc{R},\mc{Q}$ as in (\ref{eq:tautGr}). We let $U=\bigwedge^2 V$, $\eta=\bigwedge^2\mc{Q}$, and write $Y=Y_k$, $\pi=\pi_k$ in (\ref{eq:diagrGeneric}). Locally on $X_k$, $\mc{Q}^*$ trivializes to a vector space of dimension $2k$, and $Y_k$ gets identified with the space of $2k\times 2k$ skew-symmetric matrices. We take the line bundle $\mc{L}=\det\mc{Q}$ to be the Pl\"ucker line bundle on $X$, consider its $\operatorname{GL}$-equivariant inclusion $\mc{L}\subset\operatorname{Sym}^k\eta$, and note that $\mc{L}$ is locally generated by the function that assigns to a skew-symmetric matrix its Pfaffian. If we let $Y_k^0\subset Y_k$ be the open set defined locally by the non-vanishing of the Pfaffian, then we get using Cauchy's formula \cite[Prop.~2.3.8]{weyman} that condition (\ref{eq:twolims}) is satisfied. As a consequence of Propositions~\ref{prop:EuleraslimPr} and \ref{prop:limskew}, and of Remark~\ref{rem:Euleraslim} we obtain
\begin{equation}\label{eq:Dmodpfwdskew}
\chi\left(\int_{\pi_k}\mc{O}_{Y_k^0}\right) = \lim_{r\to\infty} p_{2k,r}(V)\otimes\det\left(\bigwedge^2 W\right)\otimes\operatorname{Sym}\left(\bigwedge^2 W\right)=\sum_{s=m-k}^m {s\choose m-k}\cdot \mf{B}_s.
\end{equation}
Since $\mc{O}_{Y_k^0}$ maps isomorphically via $\pi_k$ to the orbit of rank $2k$ matrices in $M^{\skew}$, we conclude as in the proof of Theorem~\ref{thm:symmetric} that $\mf{B}_s$ is the character of $B_s$ for all $s$.
\section{The simple regular holonomic $\mathcal{D}$-modules on rank stratifications}\label{sec:simplesrankstrat}
We let $X$ denote any of the vector spaces of general, symmetric, or skew-symmetric matrices, with the natural group action by row and column operations of the corresponding group $G$ as considered in the previous sections. We denote by $\Lambda$ the union of conormal varieties to the orbits of $G$, and consider the category $\mc{C}=\rm{mod}_{\Lambda}^{rh}(\mathcal{D}_X)$ of regular holonomic $\mathcal{D}_X$-modules whose characteristic variety is contained in $\Lambda$. The goal of this section is to describe explicitly the simple objects in $\mc{C}$ and obtain as a corollary a direct proof of Levasseur's conjecture \cite[Conj.~5.17]{levasseur} in the case of general and skew-symmetric matrices.
Via the Riemann--Hilbert correspondence, the simple objects in $\mc{C}$ are classified by irreducible local systems on the $G$-orbits. When the local systems are $G$-equivariant, the corresponding $\mathcal{D}_X$-modules have been described in the previous sections. The only orbits with irreducible non-equivariant local systems are the orbits $O\subset X$ of rank $n$ matrices, when $X$ is the vector space of $n\times n$ general or symmetric matrices, or when $X$ is the vector space of $2n\times 2n$ skew-symmetric matrices. In each of these cases, the complement of $O$ in $X$ is defined by a single polynomial $f$ which is the determinant of the generic (symmetric) $n\times n$ matrix in the first two cases, and it is the Pfaffian of the generic $2n\times 2n$ skew-symmetric matrix in the last case. The fundamental group of $O$ is equal to $\bb{Z}$, so the monodromy of the corresponding local system is given by a non-zero complex number $\ll=e^{2\pi i\a}$ with $\a\in\bb{C}/\bb{Z}$. We let $S$ denote the coordinate ring of $X$ and for $\a\in\bb{C}$ we consider the $\mathcal{D}_X$-module $F_{\a}=S_f\cdot f^{\a}$ (which only depends on the class of $\a$ in $\bb{C}/\bb{Z}$).
\begin{theorem}\label{thm:nonequivLa}
With notation as above, consider the irreducible local system $L_{\a}$ on $O$ whose monodromy is given by $\ll=e^{2\pi i\a}$. If $L_{\a}$ is not $G$-equivariant then the corresponding simple object in $\operatorname{mod}_{\Lambda}^{rh}(\mathcal{D}_X)$ is $F_{\a}$.
\end{theorem}
\begin{proof} The restriction of $F_{\a}$ to $O$ is a rank one integrable connection whose corresponding local system has monodromy given by $\ll=e^{2\pi i\a}$. It follows that in order to prove the theorem we need to check that $F_{\a}$ is a simple $\mathcal{D}_X$-module. The condition that $L_{\a}$ is not $G$-equivariant is equivalent to (see Theorems~\ref{thm:equivRH} and~\ref{thm:classificationDmods})
\begin{itemize}
\item $\a\notin\bb{Z}$ if $X$ is the space of general or skew-symmetric matrices.
\item $\a\notin\frac{1}{2}\bb{Z}$ if $X$ is the space of symmetric matrices.
\end{itemize}
From now on we assume that $L_{\a}$ is not $G$-equivariant. It follows from the Cayley's identity (and its symmetric and skew-symmetric versions) that $F_{\a}$ is generated as a $\mathcal{D}_X$-module by $f^{\a}$ (or by $f^{r+\a}$ for any $r\in\bb{Z}$). In order to prove that $F_{\a}$ is simple, it is then sufficient to show that any non-zero $\mathcal{D}_X$-submodule $F\subset F_{\a}$ contains $f^{r+\a}$ for $r\gg 0$. Fix any such $F$.
We write $\mf{g}$ for the Lie algebra of $G$, and note that any $\mathcal{D}_X$-module is a $\mf{g}$-representation. In particular this is true about $F\subset F_{\a}$. Since $F_{\a}$ has a multiplicity free decomposition into irreducible $\mf{g}$-representations of the form $M\cdot f^{\a}$, where $M\subset S_f$ is an irreducible integral $\mf{g}$-representation, we may assume that $F$ contains one such $M\cdot f^{\a}$. Replacing $\a$ by $\a-r$ and $M$ by $M\cdot f^r$ for $r\in\bb{Z}$, we may assume that $M\subset S$. Since $M$ generates a non-zero ideal which is invariant under the action of $G$, it defines set-theoretically a proper closed $G$-invariant subset of $X$, which is necessarily contained in the zero locus of $f$ (the complement of $f$ is a dense orbit for the $G$-action). We obtain that the ideal in $S$ generated by $M$ contains all large enough powers of $f$, and therefore that $F$ contains $f^{r+\a}$ for $r\gg 0$, which concludes the proof of the theorem.
\end{proof}
We end by remarking that Theorem~\ref{thm:nonequivLa} yields a proof of Levasseur's conjecture in the case of general and skew-symmetric matrices. We have already seen that the irreducible $G$-equivariant local systems on the orbits of the group action give rise to simple $\mathcal{D}_X$-modules containing (and hence generated by) non-zero sections invariant under the action of the derived subgroup $G'$. By Theorem~\ref{thm:nonequivLa}, the remaining simple objects of $\mc{C}$ are all of the form $F_{\a}=S_f\cdot f^{\a}$. Since $f$ is a $G'$-invariant, the same is true about $f^{\a}$, so $F_{\a}$ contains non-zero $G'$-invariant sections.
\section*{Acknowledgments}
I am grateful to Nero Budur, David Eisenbud, Sam Evens, Mircea Musta\c t\u a, Uli Walther and Jerzy Weyman for interesting conversations and helpful advice, as well as to the anonymous referee for suggesting many improvements to the presentation. Experiments with the computer algebra software Macaulay2 \cite{M2} have provided numerous valuable insights. This work was supported by the National Science Foundation Grant No.~1458715.
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1,314,259,996,735 | arxiv |
\section{Introduction}\label{Sec:intro}
Quantum chromodynamics (QCD) is the theory of strong interactions of quarks and gluons collectively called partons, the basic constituents of all nuclear matter. Its non-abelian character manifests in nature in the form of two remarkable properties: color confinement and asymptotic freedom \cite{Gross:1973id,Politzer:1973fx, Hanada:2019czd}. Confinement forbids the existence of quarks and gluons in free form or separated by long distances, allowing only colorless bound states such as mesons and baryons. Asymptotic freedom on the other hand states that at short distances, quarks and gluons interact weakly due to smallness of the coupling constant $\alpha_{s}$ at asymptotically high energies. The latter property formed the theoretical basis behind the parton model \cite{Bjorken:1968dy,Bjorken:1969ja,Feynman:1969ej} allowing the use of perturbation theory while leading to the successful description of a plethora of experimental results from fixed target experiments to high energy colliders. A multi-decade endeavor to quantify the structure of hadrons was launched with the introduction of the universal parton distribution functions (PDFs) which obey the DGLAP (Dokshitzer-GribovLipatov-Altarelli-Parisi) renormalization group equation~\cite{Gribov:1972ri,Lipatov:1974qm,Altarelli:1977zs,Dokshitzer:1977sg}. These PDF's describe the parton densities as functions of the longitudinal momentum fraction $x$. Beyond this one-dimensional picture, transverse momentum dependent (TMD) PDFs have been introduced to characterize the three-momenta of partons inside hadrons \cite{Collins:1981uw,Mulders:2000sh,Meissner:2007rx}. More recently, the complementary generalized parton distributions (GPDs) have been defined by furnishing PDFs with the two dimensional transverse spatial distribution of partons resulting in a tomographic picture of hadrons and nuclei \cite{Ji:1996ek,Radyushkin:1997ki,Mueller:1998fv}.
At high energies (or small $x$), gluon densities grow quickly resulting in a large parton occupation number. This growth is expected to be controlled by non-linear QCD effects at sufficiently small-$x$ \cite{Gribov:1984tu,Mueller:1985wy}. An appropriate description of the fundamental degrees of freedom of hadrons and nuclei in this regime is in terms of classical strong gluon fields, replacing the usual partonic description. A description of this high density regime is given by the Color Glass Condensate (CGC), a semi-classical effective field theory (EFT) for small-$x$ gluons~\cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd,Ayala:1995kg,Ayala:1995hx}.
High energy nuclear and particle physics experiments have spent the past decades quantifying the structure of protons and nuclei in terms of their fundamental constituents confirming extraordinary behaviour of matter at extreme density and pressure conditions. In the process they have also measured seemingly unexpected phenomena which will need a new generation of theoretical efforts as well as pertinent collider experiments. These last decades have also resulted on number of gluon saturation theoretical reviews which have paved the foundation for the current document~\cite{Iancu:2003xm,Gelis:2010nm,Kovchegov:2012mbw,Albacete:2014fwa,Blaizot:2016qgz}.
This paper is organized as follows, having given a brief introduction we then proceed to Sec.~\ref{Sec:CGC} which describes the principal underlying theoretical tools for the description of a gluon saturated state. Sec.~\ref{Sec:experiment} gives a selected overview of experimental signatures to date while Sec.~\ref{Sec:EIC} discusses future facilities pertinent for the discovery and quantification of gluon saturation. Sec~\ref{Sec:conclusions} gives an outlook for our field followed by acknowledgements (Sec.~\ref{Sec:Ack}) to the people and institutions that made this manuscript possible.
\section{Color Glass Condensate effective field theory}\label{Sec:CGC}
This section will serve as a description to the underlying principles behind the Color Glass Condensate (CGC) as the effective field theory for saturated gluons. We will discuss the separation of degrees of freedom into sources and fields and the computation of observables in the CGC in Sec.\,\ref{sec:sources_vs_fields}. Light-like Wilson lines and their correlators in the context of high energy scattering will follow in Sec.\,\ref{SSec:wilson} which will include a discussion on the dipole correlator and the saturation scale. Sec.\,\ref{sec:DIS_to_pA} will provide some examples of high energy processes in the CGC and discuss basic features of saturation. We briefly discuss the elements of small-$x$ quantum evolution in Sec.\,\ref{sec:quantum_evolution_CGC}, where we introduce the Balitsky-Kovchegov equation and the JIMWLK evolution for the evolution of the sources. We note that a detailed exposition of the subject can be found in the reviews~\cite{Iancu:2003xm,Gelis:2010nm} as well as the text book~\cite{Kovchegov:2012mbw}.
\subsection{Separation of degrees of freedom: Sources and fields}
\label{sec:sources_vs_fields}
The CGC is an effective field theory for high energy QCD \cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd,Ayala:1995kg,Ayala:1995hx}. For a hadron moving in the plus light-cone direction with large momentum $P^+$ probed at the scale $x_0 P^+$, with $x_0 \ll 1$, the CGC separates the partonic content of hadrons according to their longitudinal momentum $k^+ =x P^+$ where
$x$ refers to the longitudinal momentum fraction of the parton probed in the nucleus/nucleon.
Partons carrying large longitudinal momentum fraction $x \gtrsim x_0$ (large-$x$ partons) are treated as static and localized color sources $\rho$. Heisenberg's uncertainty principle justifies this view: the degree of localization of partons $\Delta z^-$ is much smaller than the longitudinal resolution $1/(x_0 P^+)$ of the probe:
\begin{align}
\Delta z^- \sim \frac{1}{k^+} = \frac{1}{x P^+} \ll \frac{1}{x_0 P^+} \,.
\end{align}
Similarly, the time scale $\Delta z^+$ for the evolution of these large-$x$ partons is much larger than the time scale of the probe $\tau \sim \frac{2 x_0 P^+}{k_\perp^2}$, where $\vect{k}$ is the transverse momentum of the produced quark:
\begin{align}
\Delta z^+ \sim \frac{1}{k^-} = \frac{2k^+}{k_\perp^2} = \frac{2 x P^+}{k_\perp^2} \gg \frac{2 x_0 P^+}{k_\perp^2} \,.
\end{align}
From the point of view of a probe, large-$x$ partons are localized in the longitudinal direction $z^-$ and frozen in light-cone time $z^+$. Their color charge distribution is non-perturbative and will be dictated by a gauge invariant weight functional $W_{x_0}[\rho]$ as will be discussed hereafter.
For a hadron/nucleus moving close to the light-cone in the plus direction, these sources generate a current independent of the light-cone time $z^+$:
\begin{align}
J^{\mu,a}(z) = \delta^{\mu+} \rho^a(z^-,\vect{z}) \,,
\label{eq:current_static sources}
\end{align}
where the support of $\rho$ along the minus light-cone direction is small.
The partons possessing a small momentum fraction $x \lesssim x_0 $ are treated as a delocalized dynamical field $A^{\mu,a}(z)$ (small-$x$ partons). This classical treatment of $A^{\mu,a}_{\rm cl}(z)$ is justified by noting that the occupation number of small-$x$ partons is large $\langle A_{\rm cl} A_{\rm cl} \rangle \sim 1/\alpha_s$.
Sources and fields are related by the Yang-Mills equations $[D_\mu,F^{\mu\nu}] = J^{\nu}$, where $D_\mu = \partial_\mu - ig A_{\mu}$. The independence of $z^+$ of the current in Eq.\,\eqref{eq:current_static sources} is consistent with the conservation equation $\left[ D_\mu, J^\mu \right] = 0$ when working in an appropriate gauge ($A^{-}=0$). For this choice of gauge condition, the classical gauge field adopts a simple solution:
\begin{align}
A^{\mu,a}_{\rm cl}(z) = \delta^{
\mu+} \alpha^a(z^-,\vect{z}) \,, \label{eq:A_classical_sol}
\end{align}
where $\alpha^a(z^-,\vect{z})$ satisfies the two-dimensional Poisson equation $\nabla_\perp^2 \alpha^a = - \rho^a$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.33]{source_fields.pdf}
\caption{In the CGC EFT partons are organized as color sources or fields according to their longitudinal momentum fraction $x$ relative to the characteristic momentum fraction of the probe $x_0$. Sources are stochastic and their distribution is characterized by a gauge invariant weight functional $W_{x_0}[\rho]$ (represented in blue). The gauge field is a solution to Yang-Mills equations in the presence of the sources (represented in red).}
\label{fig:CGC_seperation_dofs}
\end{figure}
As a consequence of the separation of degrees of freedom described above (see Fig.\,\ref{fig:CGC_seperation_dofs}), the expectation value of any observable $\mathcal{O}$ is computed in the CGC in a two step process:
\begin{enumerate}
\item Compute the quantum expectation value/path integral $\mathcal{O}[\rho] = \langle \mathcal{O} \rangle_{\rho}$ in the presence of sources $\rho$ drawn from $W_{x_0}[\rho]$.
\item Average over all possible configurations given by an appropriate gauge invariant weight functional $W_{x_0}[\rho]$.
\end{enumerate}
This procedure is summarized in the following expression:
\begin{align}
\left \langle \mathcal{O} \right \rangle_{x_0} = \int \left[\mathcal{D} \rho \right] W_{x_0}[\rho] \mathcal{O} [\rho] \,.
\label{eq:double_average_CGC}
\end{align}
For observables that involve longitudinal momentum fraction $x$ close to $x_0$, the path integral $\langle \mathcal{O} \rangle_{\rho}$ is dominated by the classical solution in Eq.\,\eqref{eq:A_classical_sol}. When the observable is probed at significantly smaller values of $x \ll x_0$ one must account for quantum evolution. We will return to this point in Sec.\,\ref{sec:quantum_evolution_CGC}.
The most widely used choice for the weight function is the McLerran-Venugopalan (MV) model \cite{McLerran:1993ni,McLerran:1993ka}. For a sufficiently large nucleus, the MV model invokes the central limit theorem, thus constructing a distribution following Gaussian statistics (for a detailed exposition see \cite{Jeon:2004rk}):
\begin{align}
W_{x_0}[\rho] = \mathcal{N} \exp \left\{-\frac{1}{2} \int \mathrm{d} x^- \mathrm{d}^2 \vect{x} \frac{\rho^a(x^-,\vect{x}) \rho^a(x^-,\vect{x})}{\lambda_{x_0}(x^-)} \right\}\,.
\end{align}
The function $\lambda_{x_0}(x^-)$ is related to the transverse color charge density distribution inside the nucleus. An energetic probe will interact coherently with the partons encountered along its longitudinal trajectory. Considering the contribution from the valence quarks only one finds the quantity
\begin{align}
\mu^2 = \int \mathrm{d} x^- \lambda_{x_0}(x^-) = \frac{2\pi g^2 A}{R_A^2} \sim A^{1/3} \label{eq:mu2_MVmodel} \,,
\end{align}
where $A$ is the nuclear mass number, $R_A \sim A^{1/3}$ is the nuclear radius and $g$ is the strong coupling constant. This new quantity $\mu^2$ is closely related to the saturation scale $Q_s^2$ as we will see in the next section where we introduce the high energy correlators.
\subsection{High energy scattering: light-like Wilson lines and correlators}\label{SSec:wilson}
The interaction of a highly energetic color charged particle with large $k^-$ momentum, and small $k^+ = k_\perp^2/(2k^-)$, with the classical field $A_{\rm cl}$ created by a nucleus is more easily described in mixed space $(k^-,\vect{x})$, where $\vect{x}$ is conjugate to $\vect{k}$. In the eikonal approximation the scattering rotates the color state of the particle while keeping the longitudinal momentum $k^-$, transverse coordinates $\vect{x}$ and any additional quantum numbers (e.g. polarization or helicities) unchanged. The effect of the rotation is encoded in the light-like Wilson lines which for quark and gluon read
\begin{align}
V_{ij}(\vect{x}) = \mathcal{P}\left( ig \int_{-\infty}^\infty A_{\rm cl}^{+,c}(z^- ,\vect{x}) t^c_{ij} \ dz^- \right)\,, \\
U_{ab}(\vect{x}) = \mathcal{P}\left( ig \int_{-\infty}^\infty A_{\rm cl}^{+,c}(z^- ,\vect{x}) T^c_{ab} \ dz^- \right) \,,
\end{align}
respectively, where $t^c$ and $T^c$ are generators of $SU(3)$ in the fundamental and adjoint representations respectively. Here $\mathcal{P}$ denotes path ordered exponential, $(i,j)$ and $(a,b)$ are color indices, and $\vect{x}$ is the transverse location at which the color charged particle interacts with the background-field $A_{\rm cl}$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{quark_propagator.pdf}
\caption{The interaction of a quark with the background field of the nucleus is encoded in a light-like Wilson line which resums multiple eikonal scatterings.}
\end{figure}
Light-like Wilson lines resum multiple interactions $(gA_{\rm cl}^{+})^n$ with the background field. These are the fundamental degrees of freedom in high energy QCD scattering. Production cross-sections are expressed as convolutions of Wilson line correlators with perturbatively calculable and process-dependent impact factors, as we will see in Sec.\,\ref{sec:DIS_to_pA}.
The simplest and most important of such correlators is the two-point correlator or dipole:
\begin{align}
S^{(2)}_{x_0}(\vect{x},\vect{y}) = \frac{1}{N_c} \left \langle \mathrm{Tr} \left[ V(\vect{x}) V^\dagger(\vect{y})\right] \right \rangle_{x_0} \,,
\end{align}
where $N_c=3$ is the number of colors. This correlator represents the scattering amplitude of a quark anti-quark dipole interacting with the background field of a nucleus at transverse locations $\vect{x}$ and $\vect{y}$. It is the building block of many processes in high energy QCD such as the total Deep Inelastic Scattering (DIS) cross-section at small-$x$.
In the MV model the dipole correlator only depends on the dipole separation $r_\perp = |\vect{x}-\vect{y}|$ and takes the form:
\begin{align}
S^{(2)}_{x_0}(r_\perp) = \exp \left\{ -\frac{1}{4} \alpha_s C_F \mu^2 r_\perp^2 \log\left(\frac{1}{\Lambda r_\perp }+e\right) \right\} \,, \label{eq:dipole_MVmodel}
\end{align}
where $\mu^2$ was introduced in Eq.\eqref{eq:mu2_MVmodel}. $\alpha_s$ is the fine structure constant, $C_F = (N_c^2-1)/(2N_c)$ is the Casimir in the fundamental representation, and $\Lambda$ is an infrared cut-off.
For small separations $r_\perp $, the dipole correlator behaves as a color neutral object, and thus the scattering amplitude is close to unity (i.e. the scattering matrix $\mathcal{S} \approx \mathbbm{1}$, no scattering), this is known as color transparency. Mathematically, this follows from the unitarity of the Wilson lines. On the other hand, at large distances $r_\perp$ the dipole correlator vanishes as the Wilson lines decorrelate as expected from the black-disk limit.
The transition between these two regimes is delineated by defining the saturation scale $Q_s^2$ as
\begin{align}
Q_s^2 = \frac{2}{r_{\perp,s}^2}\,, \quad \mathrm{where} \quad S^{(2)}_{x_0}(r_{\perp,s}) = \exp(-c) \,, \label{eq:sat_scale}
\end{align}
where the constant $c$ is typically chosen to be $1/2$.
The inverse of the saturation scale provides a measure of the correlation length of the Wilson line pair.
By examining Eq.\,\eqref{eq:dipole_MVmodel}, it follows that the saturation scale is proportional to the color charge density $Q_s^2 \sim \mu^2 \sim A^{1/3}$; hence growing with larger nuclei, this is also refered to as the nuclear \emph{oomph} factor. In Sec.\,\ref{sec:quantum_evolution_CGC} we will argue that the saturation scale also grows with decreasing values of $x$ (or equivalently with increasing energies). This results in the relation
\begin{align}
Q_s^2 \sim \frac{A^{1/3}}{x^{\lambda}}\,,
\end{align}
where $\lambda \approx 0.3$ and its arises from estimates of the energy dependence of the saturation momentum from DIS and nucleus-nucleus (A-A) scattering experiments~\cite{Gotsman:2015gba}.
In Fig.\,\ref{fig:dipole_amplitude} we plot the dipole amplitude $D(r_\perp) = 1- S(r_\perp)$ with two different values of the saturation scale, which can be interpreted as examining different nuclei species, or a nucleus at two different energies~\footnote{This is an oversimplified view point, as the small-$x$ evolution will not only change the value of $Q_s$ but also the functional form of the dipole. In the most general case, the saturation scale will also depend on the impact parameter $b_\perp$ as more color charge densities are expected in the center of the nucleus than in its periphery, modulo fluctuations.}. As expected the larger saturation scale leads to a more rapid transition to the strong scattering regime, where eventually the dipole amplitude approaches unity.
\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{dipole_amplitude_diagram.pdf}
\caption{Dipole amplitude $D(r_\perp) = 1- S^{(2)}(r_\perp)$ in the MV model (see Eq.\,\eqref{eq:dipole_MVmodel}) with two different values of the saturation scale $Q_s^2$ defined in Eq.\,\eqref{eq:sat_scale} displaying the transition between weak and strong scattering regimes.}
\label{fig:dipole_amplitude}
\end{figure}
More complex correlators of light-like Wilson lines appear in less inclusive processes and in the small-$x$ evolution equations. Two notable examples are the double dipole correlator:
\begin{align}
S^{(2,2)}_{x_0}(\vect{x},\vect{y};\vect{y}',\vect{x}') = \frac{1}{N_c^2} \left \langle \mathrm{Tr} \left[V(\vect{x})V^\dagger(\vect{y}) \right]\mathrm{Tr} \left[V(\vect{y}')V^\dagger(\vect{x}')\right] \right \rangle_{x_0}\,,
\end{align}
and the quadrupole correlator
\begin{align}
S^{(4)}_{x_0}(\vect{x},\vect{y};\vect{y}',\vect{x}') = \frac{1}{N_c} \left \langle \mathrm{Tr} \left[V(\vect{x})V^\dagger(\vect{y})V(\vect{y}')V^\dagger(\vect{x}')\right] \right \rangle_{x_0} \,.
\end{align}
As in the dipole case, these correlators implicitly contain the saturation scale $Q_s^2$. It is noted that this could be explicitly realized in the MV model, where the Gaussian approximation allows expressing any $n-$point Wilson line correlator as a non-linear function of the dipole. Other correlators involving Wilson lines in the adjoint representation $U(\vect{z})$ appear in the scattering/production of gluons.
In the next section we will see the manifestations of these correlators of light-like Wilson lines in concrete high energy processes in QCD.
\subsection{From DIS to proton-nucleus (pA) collisions}
\label{sec:DIS_to_pA}
As discussed in the previous section, high energy scattering processes in the CGC are expressed in terms of correlators of light-like Wilson lines with impact factors that can be systematically computed in perturbation theory. In this section we provide a few examples.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{DIS_diagram.pdf}
\caption{Feynman diagram for the forward scattering amplitude $\mathcal{M}^{\gamma^* A \to \gamma^*A}$ of virtual photon-nucleus collision. The amplitude contains two light-like Wilson lines, which appear from the interaction of the quark anti-quark pair with the nucleus. This amplitude is related to the total DIS cross-section by virtue of the optical theorem $\sigma^{\gamma^*A} = 2 \mathrm{Im} (\mathcal{M}^{\gamma^* A \to \gamma^*A})$. In the high energy limit, the forward amplitude is purely imaginary.}
\label{fig:forward_scat_amplitude_DIS}
\end{figure}
The total DIS cross-section, at small-$x$, for a virtual photon scattering off a nucleus (see Fig.\,\ref{fig:forward_scat_amplitude_DIS}) can be expressed with the help of the optical theorem as \cite{Gelis:2002nn}:
\begin{align}
\sigma^{\gamma^*A}_{\lambda}(x,Q^2) = 2 \int \mathrm{d}^2 \vect{r} \mathrm{d}^2 \vect{b} \int_0^1 \mathrm{d} z \ \left | \Psi^{\gamma^*}_{\lambda}(\vect{r},Q^2,z) \right |^2 \left[1 - S^{(2)}_{x}\left(\vect{b}+\frac{\vect{r}}{2}, \vect{b}-\frac{\vect{r}}{2}\right) \right] \,,
\end{align}
where $Q^2=-q^2$ and $\lambda$ are the virtuality and polarization of the photon respectively. Here $\Psi^{\gamma^*}_{\lambda}(\vect{r},Q^2,z)$ is the light-cone wave-function of the splitting of the virtual photon into a quark anti-quark pair which only depends on the dipole separation $\vect{r}=\vect{x}-\vect{y}$ and it can be calculated from perturbation theory. The longitudinal momentum fraction of the quark relative to that of the photon is denoted as $z$ and that of the anti-quark is $1-z$ by momentum conservation. The dipole correlator arises from the interaction of the quark and anti-quark with the background field of the nucleus. In addition to the dependence on the dipole vector $\vect{r}$, the dipole correlator can generally depend on the impact parameter vector defined as $\vect{b} = \frac{1}{2}(\vect{x} +\vect{y})$.
The longitudinal momentum fraction $x$ is given by Bjorken $x=Q^2/W^2$, where $W$ is the center of mass energy per nucleon of the virtual photon-nucleus system.
In order to access the saturated regime one has to probe dipole sizes $r_\perp \sim 1/Q_s$ (see Fig.\,\ref{fig:dipole_amplitude}). The light-cone wave-functions $\Psi^{\gamma^*}_\lambda$ rapidly suppress dipoles with sizes $r^2_\perp \gtrsim 1/Q^2$ (more precisely $r^2_\perp \gtrsim 1/ \left[z(1-z)Q^2 \right]$). These two observations imply that saturation effects in DIS at small-$x$ are more visible at lower values of photon virtuality $\Lambda_{QCD}^2 \ll Q^2 \lesssim Q_s^2$. At high virtualities one probes the weak scattering regime where gluon saturation has not yet set in.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{quark_pA_diagram.pdf}
\caption{Feynman diagram for the amplitude $\mathcal{M}^{pA \to q + X}$ for quark production in proton-nucleus collisions . A light-like Wilson line appears in the amplitude; thus the cross-section will feature a dipole correlator.}
\label{fig:amplitude_pA}\end{figure}
Another process which features the dipole correlator is the forward quark production in proton-nucleus scattering (Fig.~\ref{fig:amplitude_pA}), which can be studied via forward jet or hadron production. Within the \emph{hybrid factorization} approach, the differential cross-section reads \cite{Gelis:2002nn}
\begin{align}
\frac{\mathrm{d} \sigma^{pA \to q X}}{\mathrm{d} \eta \mathrm{d}^2 \vect{k}} = \frac{1}{(2\pi)^2} x_p q(x_p) C_{x_A}(\vect{k}) \,,
\end{align}
where $\vect{k}$ and $\eta$ are the transverse momentum and rapidity of the produced quark, $x_p q(x_p)$ is
the quark distribution in the proton for a collinear quark with longitudinal momentum
fraction $x_p$. In this case the dipole correlator appears from a light-like Wilson line $V(\vect{x})$ in the amplitude and another one $V^\dagger(\vect{y})$ in the complex conjugate amplitude. Here $x_A$ refers to the longitudinal momentum fraction of the gluon probed in the dense nucleus.
The function $C_{x_A}(\vect{k})$ is the Fourier transform of the dipole amplitude:
\begin{align}
C_{x_A}(\vect{k}) = \int \mathrm{d}^2 \vect{x} \mathrm{d}^2 \vect{y} e^{-i \vect{k} \cdot (\vect{x}-\vect{y})} S^{(2)}_{x_A}(\vect{x},\vect{y}) \,
\end{align}
This function determines the transverse momentum kick acquired by a collinear quark as it multiple-scatters from the nucleus. In Fig.\,\ref{fig:dipole_FT_MV} we plot the function $C(k_\perp)$ corresponding to the dipoles shown in Fig.\,\ref{fig:dipole_amplitude}, where we normalized by an overall factor of the transverse area as the MV model is translationally invariant. We observe a clear difference in the behavior of $C(k_\perp)$ between the small and large $k_\perp$ regions. In the perturbative limit it behaves as a power law
\begin{align}
C_x(\vect{k}) \sim \frac{Q_s^2(x)}{k_\perp^4}, \quad k_\perp \gtrsim Q_s \,
\end{align}
whereas in the saturated regime it approaches a constant
\begin{align}
C_x(\vect{k}) \sim \frac{1}{Q_s^2(x)}, \quad k_\perp \lesssim Q_s \,
\end{align}
In Fig.\,\ref{fig:dipole_amplitude} we see that as the saturation scale is increased, the distribution $C_x(\vect{k})$ is pushed to larger values of $k_\perp$. This is one of the consequences of saturation: as the energy of the collision is increased, the saturation scale $Q_s$ grows and radiated gluons at small-$x$ are pushed to larger values of $k_\perp$ since the phase space $k_\perp \lesssim Q_s$ is overoccupied.
The function $k_\perp C_x(k_\perp)$, where the additional factor of $k_\perp$ arises from the phase space, determines the transverse momentum acquired by the quark as it multiply scatters from the nucleus. It can be verified that this function peaks around $k_\perp \sim Q_s$. While it is possible to parametrize $C_x(k_\perp)$ as a function of both $k_\perp$ and $x$; however, in the CGC usually its Fourier conjugate, the dipole correlator is constrained by HERA data.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{Two_dipole_TMDs.pdf}
\caption{Fourier transform of the dipole correlator as a function of $k_\perp$ for two different values of the saturation scale. A transition between saturation and perturbative regime is observed near $k_\perp \sim Q_s$. The $x$ dependence of the distribution is effectively accounted by the saturation scale. In a more careful treatment, the functional shape of the distribution also depends on $x$. } \label{fig:dipole_FT_MV}
\end{figure}
Appearing both in DIS and proton-nucleus collisions, the dipole correlator is a universal building block in high energy collisions. Evidently, its manifestation is different for each process, thus one can constrain different features of this object from independent measurements either by studying the $(x,Q^2)$ dependence in DIS or the $(\eta,\vect{k})$ distribution of quark jet production in proton-nucleus collisions.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{dijet_DIS_diagram.pdf}
\caption{Feynman diagram for the amplitude $\mathcal{M}^{\gamma^*A \to q\bar{q} + X}$ for the quark anti-quark dijet production in virtual photon-nucleus collisions. The amplitude contains the production of two light-like Wilson lines; thus the amplitude will feature a quadrupole (and dipole) correlator. }
\label{fig:dijet_DIS_CGC_LO}
\end{figure}
We provide one more example of a high energy process, the production of a dijet or a dihadron pair in DIS as shown in Fig.\,\ref{fig:dijet_DIS_CGC_LO}. In the CGC for general small-$x$ kinematics, this process depends on the quadrupole correlator in a non-trivial way. However, in the limit where the dijets or dihadrons are back-to-back in transverse space, it is possible to establish a TMD factorization \cite{Dominguez:2011wm}. The corresponding Wilson line correlator is given by the small-$x$ Weizsäcker-Williams (WW) gluon TMD $x G_{WW}(x,\vect{k})$, where the transverse momentum $\vect{k}$ refers to the imbalance of the dijet/dihadron system. The differential cross-section reads \cite{Dominguez:2011wm}:
\begin{align}
\frac{\mathrm{d} \sigma_{\lambda}^{\gamma^*A \to q\bar{q}+X}}{\mathrm{d} z_1 \mathrm{d} z_2 \mathrm{d}^2 \vect{k} \mathrm{d}^2 \vect{P} } = \delta(1-z_1-z_2) H_{\gamma^*g \to q\bar{q}}^{ij,\lambda}(Q^2,\vect{P},z) x G^{ij}_{WW}(x,\vect{k}) \,,
\end{align}
with $H_{\gamma^*g \to q\bar{q}}^{ij,\lambda}$ a perturbatively calculable factor, $z_{1,2}$ are the longitudinal momentum fraction of the jets/hadrons relative to the virtual photon and $\vect{P}$ denotes the mean transverse momenta of the jets/hadrons.
The WW gluon TMD is computed from a correlator of light-like Wilson lines and its derivatives:
\begin{align}
x G^{ij}_{WW}(x,\vect{k}) = \frac{4}{(2\pi)^3} \int \mathrm{d}^2 \vect{b} \mathrm{d}^2 \vect{b}' e^{-i \vect{k} \cdot (\vect{b}- \vect{b}')}\left\langle \Tr\left[A^i(\vect{b}) A^{j}(\vect{b}') \right] \right \rangle_x \,,
\end{align}
where $A^i(\vect{b}) = \frac{i}{g} V(\vect{b}) \partial^i V^\dagger(\vect{b})$ is the transverse gauge field in the light-cone gauge $A^+=0$.
Unlike the Fourier transform of the dipole correlator, this distribution has a probability density interpretation. In Fig.\,\ref{fig:WW_TMD_MVmodel} we plot the WW gluon TMD for two different values of the saturation scale. As expected we observe a transition in the behavior of this function near $k_\perp \sim Q_s$. In the perturbative limit, this distribution has the following power law tail
\begin{align}
xG^{ii}(x,k_\perp) \sim \frac{Q_s^2(x)}{k_\perp^2} , \quad k_\perp \gtrsim Q_s \,,
\end{align}
and a slow logarithmic growth in the saturated regime:
\begin{align}
xG^{ii}(x,k_\perp) \sim \log\left(\frac{Q_s^2(x)}{k_\perp^2} \right), \quad k_\perp \gtrsim Q_s \,.
\end{align}
The transverse momentum imbalance of produced dihadrons/dijets which originated from the virtual photon (with zero transverse momentum, i.e. in the Breit frame) is dictated by the WW gluon TMD distribution. The comparison of the azimuthal angle distribution of dijets/dihadrons near the back-to-back configuration is one of the promising observables for the search of saturation as we will review in the later sections.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{Two_gluon_TMDs.pdf}
\caption{WW gluon TMD distribution as a function of $k_\perp$ for two different values of the saturation scale. A transition between saturation and perturbative regime is observed near $k_\perp \sim Q_s$. The $x$ dependence of the distribution is effectively accounted by the saturation scale. In a more careful treatment, the functional shape of the distribution also depends on $x$. }
\label{fig:WW_TMD_MVmodel}
\end{figure}
It is worth pointing out that it might be possible to directly parametrize the WW gluon distribution $xG^{ii}(x,k_\perp)$ as a function of $x$ and $k_\perp$. In the CGC and within the Gaussian approximation the WW gluon distribution object is typically constructed from the dipole correlator which is constrained by HERA data.
\subsection{Quantum evolution}
\label{sec:quantum_evolution_CGC}
We close this section on the CGC EFT by very briefly reviewing the crucial aspect of quantum evolution and the renormalization group equations at small-$x$. Thus far we have focused on observables and light-like Wilson line correlators computed using the classical solutions to the Yang-Mills equations for color sources drawn from the MV model. This procedure is appropriate when the observables of interest are probed at a longitudinal momentum fraction $x$ close to $x_0$ at which the weight functional is constructed (for MV $x_0 \approx 0.01$). However, quantum fluctuations around the classical solution are enhanced by terms proportional to $\alpha_s \log(x_0/x)$. These terms can be of order $1$ for sufficiently small $x$, and thus require resummation. Physically, these contributions arise from gluon emissions in the interval $[x,x_0]$.
At large $N_c$, the resummation of these terms results in the Balitsky-Kovchegov (BK) equation for the small-$x$ evolution of the dipole correlator \cite{Balitsky:1995ub,Kovchegov:1999yj} (diagrams shown in Fig.\,\ref{fig:dipole_evolution_diagrams}):
\begin{align}
\frac{\mathrm{d} S_x^{(2)}(r_\perp)}{d \log(1/x)} = \int \mathrm{d}^2 \vect{r}' \frac{\vect{r}^2}{\vect{r}'^2 (\vect{r}-\vect{r}')^2}\left[S_x^{(2)}(r_\perp') S_x^{(2)}(|\vect{r}-\vect{r}'|) - S_x^{(2)}(r_\perp) \right] \,.
\end{align}
The terms quadratic in $S^{(2)}$ arise from the real emission diagrams in which the gluon crosses the shock-wave, while those linear in $S^{(2)}$ appear from virtual contributions. The BK equation reduces to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation \cite{Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic} in the weak scattering regime $D_x(r_\perp)=1-S^{(2)}(r_\perp) \ll 1$ .
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{evolution_diagrams.pdf}
\caption{A subset of Feynman diagrams for the quantum evolution of the dipole correlator. Upper diagrams correspond to real gluon emission, while lower diagrams correspond to virtual contributions.}
\label{fig:dipole_evolution_diagrams}
\end{figure}
Remarkably, an alternative way to resum large logarithmic contributions is by the evolution of the weight-functional from the scale $x_0$ to $x$ following the equation:
\begin{align}
\frac{\mathrm{d} W_x[\rho]}{\mathrm{d} \log(1/x)} = -\mathcal{H}_{\rm JIMWLK} W_x[\rho]
\label{eq:evolution_weightfunctional}
\end{align}
where $\mathcal{H}_{\rm JIMWLK}$ is the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner (JIMWLK) Hamiltonian \cite{JalilianMarian:1996xn,JalilianMarian:1997dw,Kovner:2000pt,Iancu:2000hn,Iancu:2001ad,Ferreiro:2001qy}. Physically, this procedure correspond to absorbing the quantum fluctuations in the interval $[x_0 - dx,x_0]$ into stochastic fluctuations of the color sources by redefinition of the sources $\rho$ (Fig.~\ref{fig:CGC_seperation_dofs_evolution}). Iterating this process through a self-similar Wilsonian renormalization group (RG) procedure results in Eq.\eqref{eq:evolution_weightfunctional}. This procedure is equivalent to an infinite hierarchy (known as the B-JIMWLK hierarchy) of non-linear coupled equations dictating the evolution of $n$-point Wilson line correlators \cite{Balitsky:1995ub,JalilianMarian:1996xn,JalilianMarian:1997dw,Kovner:2000pt,Iancu:2000hn,Iancu:2001ad,Ferreiro:2001qy,Weigert:2000gi}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.33]{source_fields_renormalized.pdf}
\caption{Schematic representation of the quantum evolution of the weight-functional. Quantum fluctuations in the interval $[x_1,x_0]$ (shown in yellow) are absorbed into stochastic fluctuations of the color sources by redefinition of the weight functional $W_{x_0}[\rho] \to W_{x_1}[\rho]$ (compare with Fig.\,\ref{fig:CGC_seperation_dofs}). The choice of the scale separating small-$x$ and large-$x$ partons is thus arbitrary and different choices are related by the JIMWLK renormalization group equations. The long right bracket represents how large-$x$ partons and fluctuations are considered as sources after properly evolving of weight functional.}
\label{fig:CGC_seperation_dofs_evolution}
\end{figure}
We end this section by illustrating the effect of small-$x$ JIMWLK evolution equations on the dipole amplitude. The quantum evolution effectively increases the color charge density as more partons are introduced as sources. This in turn implies an increase in the saturation scale which for fixed dipole size $r_\perp$ drives the dipole amplitude closer to one (strong scattering). The small-$x$ evolution of the dipole amplitude and the corresponding evolution of the saturation scale are illustrated in Fig.\,\ref{fig:dipoleQs2__evolution}. It is worth mentioning that the small-$x$ evolution also changes the functional shape of the dipole; thus, it is not sufficient to simply parametrize the saturation scale.
\begin{figure}[H]
\centering
\includegraphics[scale=0.18]{dipole_evolution.pdf}
\includegraphics[scale=0.33]{Qs2_evolution.pdf}
\caption{Left: Dipole amplitude $D_x(r_\perp)$ small-$x$ evolution drives a more rapid transition to the strong scattering regime. Right: small-$x$ evolution of the saturation scale $Q_s^2$ normalized by the saturation scale at $x_0$.}
\label{fig:dipoleQs2__evolution}
\end{figure}
\section{Experimental signatures to date}\label{Sec:experiment}
Having introduced the underlying theoretical principles and techniques that address a gluon saturated state, we now move on to an interpretation of experimental signatures at colliders from HERA to the LHC. While our list is not meant to be exhaustive it does provide a summary of the published results which are the pillars to Sections ~\ref{Sec:EIC} and ~\ref{Sec:conclusions}. The results described in this section have been tied to a number of phenomena including gluon saturation; throughout this document we will caution against competing mechanisms that can also explain the measurements without invoking saturation. As many of the competing mechanisms are process dependent, we will address them case by case depending on the observable under consideration.
This section is organized as follows, first we will introduce structure functions and their historic relevance followed by diffraction and semi-inclusive measurements. We apologize for the omission or superficial description of some the work of the scientists in our field for the sake of maintaining this manuscript of reasonable length.
\subsection{Structure functions}\label{subsec:structure functions}
One of the major achievements of the deep inelastic scattering experiments at HERA is the determination of the structure functions $F_2$ and $F_L$ of the proton ~\cite{ZEUS:1993ppj}, which can be determined from the total DIS cross-sections:
\begin{align}
F_2(x,Q^2) &= \frac{Q^2}{4\pi \alpha_{em}} \left[\sigma^{\gamma*A}_T(x,Q^2) + \sigma^{\gamma*A}_L(x,Q^2) \right] \,, \\
F_L(x,Q^2) &= \frac{Q^2}{4\pi \alpha_{em}} \sigma^{\gamma*A}_L(x,Q^2) \,.
\end{align}
From these objects it is possible to extract the PDFs. PDFs are universal parton densities containing long-distance structure of hadrons and are independent of the colliding system (the same in DIS and proton-proton (pp) ). In the collinear framework, PDFs known at an initial scale $Q_0$ are evolved according to the DGLAP renormalization group equations to a different scale $Q$. At high energies, or equivalently small $x$, DGLAP evolution must be supplemented with BFKL dynamics which re-sums $\alpha_s \log(1/x)$ \cite{Lipatov:1976zz,Kuraev:1977fs,Balitsky:1978ic}. Compelling evidence of BFKL dynamics has been suggested in a recent analysis of HERA data with small-$x$ re-summation \cite{Ball:2017otu, Hentschinski:2013id}. Yet at even higher energies (or smaller $x$), the rapid rise of gluon densities cannot grow unchecked as it would violate unitarity, in other words probability conservation. It is expected that at high gluon densities, the nonlinear dynamics of QCD can result in the competing effect of gluon recombination taming growth of gluon distributions.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{predictionsForShadowing_bCGC-0705-3037.pdf}
\caption{Predictions for shadowing compared to data from the New Muon Collaboration. Center: predictions for Right: Predictions for Q$^{2}= 5$ GeV 2 as a function of $x$. Figure from ~\cite{Kowalski:2007rw}}
\label{fig:NMC}
\end{figure}
Comparisons to New Muon Collaboration data (Fig.~\ref{fig:NMC}) and comprehensive analyses of the proton structure functions from HERA data have been performed in the saturation framework. The first comparisons to HERA data were performed over 20 years ago in \cite{Golec-Biernat:1998zce} using the Golec-Biernat Wusthoff (GBW) model, which assumed a simple parametrization of the dipole amplitude $D_x(r_\perp) = 1- \exp(-\frac{1}{4} r_\perp^2 Q_s^2(x))$. We point out that an additional parameter $\sigma_0/2$ that effectively accounts for the transverse area is required as this model does not capture impact parameter dependence. Even so the GBW model had a reasonable agreement with data \footnote{Unfortunately, the GBW model fails to describe other observables such as single hadron inclusive spectra in pA due to its exponential tail, rather than the expected power law behavior.} and led to the observation that the total DIS cross-section can be described by a single variable $\tau = Q^2/Q_s^2(x)$. This phenomena is known as geometric scaling \cite{Stasto:2000er}.
Models that do incorporate the impact parameter $b_\perp$ dependence have been introduced e.g the IPsat model \cite{Kowalski:2003hm} and the the bCGC model \cite{Watt:2007nr}. Roughly speaking, these models incorporate the $b_\perp$ dependence of the saturation scale $Q_s^2(b_\perp) \sim T_p(b_\perp)$ by introducing the thickness function $T_p(b_\perp)$ which parametrizes the gluon density inside the proton. The impact parameter dependence is typically constrained by exclusive processes such as vector meson production, comparisons of these models with HERA data can be found in \cite{Rezaeian:2012ji,Rezaeian:2013tka}. A drawback of these frameworks is the lack of a rigorous treatment of small-$x$ evolution since it would require to incorporate significant contributions from confining physics. Some attempts to tackle this problem can be found in \cite{Berger:2011ew,Bendova:2019psy}.
One of the most comprehensive studies of structure functions in the saturation framework was performed in \cite{Albacete:2010sy}. These studies used the running coupling BK equation supplemented with suitable initial conditions and accounted for contributions of heavy quarks. This is one of the first attempts at a rigorous description of HERA data using modern theoretical tools of the saturation framework. More recently studies have built upon this work incorporating collinear resummations in the BK equation \cite{Albacete:2015xza,Iancu:2015joa,Ducloue:2019jmy}.
We conclude the discussion by highlighting that the state of the art was achieved in \cite{Beuf:2020dxl}. In this work, the authors compared HERA data to the predictions of the CGC at next-to-leading order (NLO) including impact factor and small-$x$ evolution equations. Their comparisons for the reduced cross-section (Eq.~\ref{Eq:reduced_xs}) and the longitudinal structure function are shown in Fig.\,\ref{fig:HERA_CGCNLO}. As compared to the CGC leading order fits, the authors find that the evolution speed is naturally reduced by the NLO corrections without the need to introduce a large factor in the running coupling. It is worth mentioning that this study only included light-quark contribution as the computation for the impact factor for massive quark contributions is ongoing \cite{Beuf:2021qqa}. For completeness we include the reduced cross-section form which is given by
\begin{align}
\sigma_r(x,y,Q^2) = F_{2}(x,Q^2) - \frac{y^2}{1+(1-y)^2} F_{L}(x,Q^2) \,,
\label{Eq:reduced_xs}\end{align}
where $y$ is inelasticity of the collision.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{plt-sigmar-hera-x001-uksi-ursi-utsi-fan-xbj.pdf}
\includegraphics[scale=0.4]{plt-H1fl-hera-uksi-ursi-utsi-x001.pdf}
\caption{Comparison of the CGC at NLO compared to HERA data. Left: reduced cross-section at small-$x$. Right: $F_L$ structure function. Figure from \cite{Beuf:2020dxl}}
\label{fig:HERA_CGCNLO}
\end{figure}
\subsubsection{Competing mechanisms in structure functions}
While structure functions extracted from HERA and the NMC data have been influential, some aspects relevant to gluon saturation need to be confronted. A main feature to address is the impact of the non-linear phenomena of saturation for the description of the structure functions. This difficulty finds its roots in the large non-perturbative contributions to the determination of structure function at low to moderate $Q^2$. More specifically, structure functions $F_2$ and $F_L$ in the dipole picture can have a significant contribution from non-perturbatively large dipoles. This has been demonstrated in \cite{Mantysaari:2018nng} and \cite{Mantysaari:2018zdd} where the authors study the contribution to $F_2$ and $F_L$ from large dipoles as shown in Fig.\,\ref{fig:dipole_size_contribution_F2FL}. Large dipole contributions arise from the so called \emph{aligned jet} configuration where either the quark or anti-quark carries most of the longitudinal momentum $(z \to 0,1)$. This configuration is more important for $F_2$ than
$F_L$ due to the different structure of the light-cone wave-function between transversely and longitudinally polarized photons. Fig.\,\ref{fig:dipole_size_contribution_F2FL} shows that it is necessary to go to very large virtualities to suppress non-perturbatively large dipoles $(r_\perp \gtrsim \ 1.0\ \rm{GeV})$; however, at large $Q^2$ one expects less sensitivity to gluon saturation. This problem is ameliorated when studying charm structure functions as the mass of the quarks serve as an infrared cut-off.
\begin{figure}[H]
\centering
\includegraphics[scale=0.32]{IPsat_f2_maxr.pdf}
\includegraphics[scale=0.32]{IPsat_fl_maxr.pdf}
\caption{Contribution to the structure functions $F_2$ (left) and $F_L$ (right) from dipole sizes smaller than $r_{\rm max}$ at different photon virtualities. $F_L$ sensitivity to large dipoles is reduced compared to $F_2$ due to the different structure of light-cone wavefunctions between longitudinally and transversely polarized photons. Figure from \cite{Mantysaari:2018nng}.}
\label{fig:dipole_size_contribution_F2FL}
\end{figure}
Another difficulty in unambiguously determining the need of non-linear/gluon saturation effects in the description of HERA data structure functions is due to the parameter freedom allowed in the fits. To illustrate this flexibility, the authors in \cite{Mantysaari:2018nng} fitted structure functions using the IPsat model and its linearized version (IPnonsat) which is expected to exclude gluon saturation dynamics. When independently fitted, both models result in almost indistinguishable results across a large phase space in $(x,Q^2)$, (see Fig.\,\ref{fig:IPsat_vs_IPnonsat}). This suggests that non-linear effects are not visible in the proton structure functions at HERA alone. It might be possible to reduce the freedom of these models by applying them to other physical processes. In Section ~\ref{Sec:EIC}, we will briefly discuss the improved potential to discover gluon saturation in the study of nuclear structure functions at future DIS collider experiments.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{sigmar.pdf}
\caption{IPsat and IPnonsat (linearized IPsat) independent fits to inclusive reduced cross-section HERA data. Both fits result in almost indistinguishable results, hindering the extraction of a signal of gluon saturation at HERA. Figure from \cite{Mantysaari:2018nng}.}
\label{fig:IPsat_vs_IPnonsat}
\end{figure}
\subsection{Diffractive reactions}
\label{sec:diffractive_reactions}
Diffractive observables are characterized by a rapidity gap (the absence of particles produced in a given rapidity window) which originates from a color neutral exchange between the two colliding systems. Since gluons carry color, this exchange requires at least two gluons which must be in the color singlet state. As consequence, diffractive measurements are sensitive to the "square" of the the gluon distribution (at lowest order in perturbation theory). Compared to inclusive measurements, this enhanced sensitivity to the gluon distribution makes diffractive observables excellent candidates for gluon saturation searches at small-$x$. In this section, we will mostly focus on diffractive production in the collision of a photon (virtual or real) with a proton or nucleus. This can be realized in deep inelastic scattering (DIS) and ultra-peripheral collisions (UPCs).
Diffractive DIS observables have been extensively studied at HERA. The first hints of saturation were observed in the analysis of inclusive and diffractive cross-sections using the GBW model in \cite{Golec-Biernat:1998zce,Golec-Biernat:1999qor}. The authors found that in the saturation framework the ratio of diffractive to inclusive events was almost constant with only a mild logarithmic dependence on the virtuality $Q^2$ and Bjorken-$x$. The results from these saturation models compared well to data, especially after they were furnished with DGLAP evolution \cite{Bartels:2002cj}. More refined studies for the description of the diffractive structure functions \cite{Marquet:2007nf} using the IPsat and bCGC model were carried out in \cite{Kowalski:2008sa}.
The exclusive production of vector particles (photons and vector mesons) are also powerful tools to study the gluon content of protons and nuclei. In addition to the energy and virtuality dependence of their production cross-sections, the momentum transfer squared $t$ dependence, and the dependence on the mass $M_{V}$ of the produced vector particle give more detailed insight into the gluon structure of nuclei at high energies. One can distinguish two cases: (i) coherent events in which the target (proton/nucleus) remains intact, and (ii) incoherent events in which the target (proton/nucleus) breaks up.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{plot-jpsi-wf.eps}
\includegraphics[scale=0.5]{plot-jpsi-tf.eps}
\caption{$J/\psi$ exclusive electroproduction data from HERA compared to saturation models (bCGC and IPsat). Left: energy dependence. Right: $|t|$ spectra. Figure from \cite{Rezaeian:2013tka}.}
\label{fig:Jpsi_t_and_energydep}
\end{figure}
Coherent events are dominant at low $t \lesssim 1/R^2$, where $R$ is the size of the target, and they are sensitive to the average color density profile of the target. Both spectra and energy dependence for coherent vector meson production were compared to HERA data within a GBW model incorporating impact parameter dependence in \cite{Kowalski:2006hc} and later using the bCGC and IPsat models \cite{Watt:2007nr,Rezaeian:2012ji,Rezaeian:2013tka,Goncalves:2014wna} (see Fig.\ref{fig:Jpsi_t_and_energydep}). The impact parameter dependence of the dipole models is crucial as it is responsible for the cross-section not vanishing at non-zero momentum transfers. This dependence is typically modeled and the parameters are part of the fit. More complex studies using JIMWLK evolution have shown that the impact parameter dependence of the average color charge density evolves with energy: the gradients of color charge become smoother and the overall size of the profile grows \cite{Schlichting:2014ipa}.
Exclusive vector meson production can also occur in ultraperipheral nucleus-nucleus collisions, where either of the nuclei acts as a source of Weizsäcker-Williams real photons, which then interact with the other nucleus. These processes have been studied at RHIC and the LHC with the saturation models providing a good description of the data \cite{Armesto:2014sma,Goncalves:2014swa}. Recently, the energy evolution has been studied comparing models that incorporate either BFKL (linear) or BK (non-linear) evolution \cite{ArroyoGarcia:2019cfl}. The authors argue the onset of gluon saturation as they find the need for non-linear evolution to describe the vector meson photo-production data from HERA, RHIC and LHC.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{csont_ratio.eps}
\caption{Photo-production of $J/\psi$ at very low values of $t$ compared to Starlight data \cite{Klein:2016yzr}, leading twist approach (LTA) \cite{Guzey:2016qwo}, and b-BK saturation model \cite{Bendova:2020hbb}. Figure from \cite{ALICE:2021tyx}.}
\label{fig:very_lowt_Jpsi}
\end{figure}
One of the consequences of saturation is a steeper $t$-distribution compared to one obtained from the form factor or Fourier transform of the density profile~\cite{Klein:2016yzr}. This has recently received some attention \cite{Lappi:2021ieu} in light of the very precise ALICE measurements of $J/\psi$ photo-production at very low $t$ \cite{ALICE:2021tyx} as shown in Fig.\ref{fig:very_lowt_Jpsi}. Note that even after saturation is included as in the model in \cite{Bendova:2020hbb}, the spectrum is not steep enough to reproduce the lowest $t$ bin.
\begin{figure}[H]
\centering
\includegraphics[scale=0.42]{density_ipglasma.pdf}
\includegraphics[scale=0.42]{ipglasma_w_75.pdf}
\caption{Left: color charge density for four different events. Right: Coherent and incoherent H1 data compared to predictions from CGC incorporating subnucleonic (and $Q_s$ saturation fluctuations).}\label{Fig:densityPlasma}
\end{figure}
In contrast to coherent production, incoherent events dominate at large values of $t$ and the spectrum is sensitive to event-by-event fluctuations \cite{Frankfurt:1993qi,Frankfurt:2008vi,Dominguez:2008aa,Lappi:2010dd}. In \cite{Mantysaari:2016ykx,Mantysaari:2016jaz} the authors found that in order to reproduce the data from HERA, it is necessary to incorporate subnucleonic fluctuations in terms of hotspots of color charge density (Fig.~\ref{Fig:densityPlasma}. These studies have been extended to UPCs at RHIC in \cite{Mantysaari:2017dwh} where the effect of fluctuations also significantly increases the distribution at large $t$. Furthermore, they have also been explored for LHC energies in \cite{Cepila:2016uku,Cepila:2018zky} where a model for the energy dependence of the number of hotspots has been introduced. For a comprehensive review on the subject of proton and nuclear shape fluctuations see \cite{Mantysaari:2020axf}).
It might also be possible to single out saturation effects by studying azimuthal correlations in the diffractive production of dihadron or dijets. This subject has been investigated recently for DIS and UPCs in \cite{Altinoluk:2015dpi,Mantysaari:2019csc,Salazar:2019ncp,Shi:2020djm,Boer:2021upt,ATLAS:2021jhn}.
\subsubsection{Competing mechanisms and systematic uncertainties}
A significant source of theoretical uncertainty in the production of vector mesons arises from the description of their light-cone wave-function and the model for the dipole amplitude. Different parametrizations for these objects have been recently compared when studying rapidity distributions \cite{Goncalves:2017wgg} where the authors find large systematics uncertainties. Recent developments on relativistic corrections to vector meson light-cone wave-functions can be found in \cite{Lappi:2020ufv}.
As previously noted, the free parameters in the dipole models are typically chosen to reproduce HERA data, models are then used for predictions at RHIC and the LHC. However, saturation effects at HERA might be weak, as we argued in our discussion of the structure functions. The authors of \cite{Mantysaari:2018nng} find a similar description of HERA data when comparing fits with IPsat and IPnonsat, signaling that gluon saturation effects are weak; thus arguing for the need of nuclear DIS.
We close this section by mentioning that other compelling frameworks such as the leading twist approach \cite{Frankfurt:2011cs} based on QCD factorization theorems and nuclear shadowing can provide a good description of coherent vector meson photo-production \cite{Guzey:2013xba,Guzey:2013qza,Guzey:2016qwo,Guzey:2016piu} and diffractive dijet photo-production \cite{Guzey:2016tek}.
\subsection{Semi-inclusive reactions}\label{sec:semi_inclusive_reactions}
Semi-inclusive measurements defined where one or more particles are tagged, provide more detailed information about the dynamics of gluons than fully inclusive measurements such as structure functions. Experimental signatures of gluon saturation are expected to be imprinted in the transverse momentum and rapidity distributions of particles produced in hadronic collisions \cite{Kharzeev:2004bw}.
\subsubsection{ Single inclusive production}
Single inclusive particle production has been extensively studied in the saturation framework. The first ideas can be traced back to over 20 years ago, where inclusive forward gluon production \cite{Kovchegov:1998bi,Dumitru:2001jn,Kovner:2001vi} and inclusive forward quark production \cite{Dumitru:2002qt} in proton-nucleus where studied in the \emph{hybrid} formalism. In this framework incoming partons inside the proton are treated within the DGLAP collinear approximation and subsequently scatter eikonally off the strong field produced by the nucleus via correlators of light-like Wilson lines (see Sec.\,\ref{sec:DIS_to_pA}). These studies opened up the possibility to access the saturated gluon regime with semi-inclusive measurements in the collisions of a small dilute nucleus with a larger saturated nucleus. The conceptions soon capitalized in \cite{Kharzeev:2002pc,Kharzeev:2003wz}, where the authors argued that the high $p_\perp$ suppression observed in BRAHMS \cite{BRAHMS:2004xry} at forward rapidities in $d$-$Au$ collisions was a signature of the onset of gluon saturation.
To characterize the suppression observed in these experiments, one defines the nuclear modification factor:
\begin{align}
R_{A_1 A_2} = \frac{1}{N_{\mathrm{coll}}} \frac{\mathrm{d} \sigma^{A_1 A_2 \to h X} }{\mathrm{d}^2 \boldsymbol{p}_\perp \mathrm{d} \eta} \Big / \frac{\mathrm{d} \sigma^{pp \to h X} }{\mathrm{d}^2 \boldsymbol{p}_\perp \mathrm{d} \eta}\,,
\end{align}
where $N_{\mathrm{coll}}$ is the number of binary collisions. This ratio is expected to be unity if nuclear collisions were a simple incoherent superposition of collisions with individual nucleons, while deviations from unity indicate coherent effects at play.
In the MV model, the presence of saturated gluons with typical momentum $k_\perp \sim Q_s$ induce a broadening of the transverse momentum distribution of the produced particles. More specifically, the nuclear modification factor is suppressed for $k_\perp \lesssim Q_s$ and enhanced for $k_\perp \gtrsim Q_s$. This enhancement is also called the Cronin peak \cite{Cronin:1974zm}. Indeed, in \cite{Kharzeev:2003wz} the authors explicitly show that not integrated gluon distributions (at small-$x$ there are two kinds: the dipole and the Weizsäcker-Williams type) obtained from the MV model satisfy a sum rule \footnote{The factor of $A$ in Eq.\,\eqref{eq:UGD_sum_rule_MV} arises from an $A^{2/3}$ overall area, and $A^{1/3}$ from the scaling of the saturation momentum.}:
\begin{align}
\int \mathrm{d}^2 \vect{p} \phi^{A}_{\rm MV}(p_\perp) = A \int \mathrm{d}^2 \vect{p} \phi^{p}_{\rm MV}(p_\perp) \,.
\label{eq:UGD_sum_rule_MV}
\end{align}
where $\phi^{p}_{\rm MV}(p_\perp)$ and $\phi^{A}_{\rm MV}(p_\perp)$ are the unintegrated gluon distributions for the proton and nucleus respectively.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{rda0.eps}
\includegraphics[scale=0.5]{rda3.eps}
\caption{BRAHMS data on the nuclear modification factor $R_{dAu}$ of charged particles at midrapidity and forward rapidity (data from \cite{BRAHMS:2004xry}) compared with the saturation based results in \cite{Kharzeev:2004yx}. The midrapidity $p_\perp$ distribution is characterized by a Cronin peak, while at forward rapidities this peak is washed away.}
\label{fig:nuclear_mod_factor_hadron_DAu}
\end{figure}
The authors also find by analytic arguments, that after sufficient quantum small-$x$ evolution, the sum rule in Eq.\,\eqref{eq:UGD_sum_rule_MV} is turned into an inequality: the $p_\perp$ integrated distribution is suppressed for larger nuclei. These analytic arguments were confirmed in \cite{Albacete:2003iq} by evaluation of the corresponding unintegrated gluon distribution using numerical solutions of the BK equation. A quantitative comparison with RHIC data was carried out in \cite{Kharzeev:2004yx} (See Fig.\,\ref{fig:nuclear_mod_factor_hadron_DAu}). It is worth noting that in this work the authors shifted the nuclear saturation scale $Q_s^2 \to Q_s^2 + \kappa^2 A^{1/3}$ with $\kappa \sim 1.0 \ \rm{GeV}$ to describe the data, where they argued it was due to non-perturbative low energy rescattering which is absent in the saturation framework. Nuclear suppression due to gluon saturation has also been studied at the LHC in \cite{Rezaeian:2012ye,Albacete:2012xq,Lappi:2013zma,Ducloue:2016ywt,Mantysaari:2019nnt}, e.g. $D$-meson production measurements~\cite{Ducloue:2016ywt} at the ALICE experiment\cite{ALICE:2018vhm} which was well described by CGC predictions. Notwithstanding, further studies are necessary as these data can also be well described without saturation see e.g. \cite{Eskola:2019bgf}.
Another opportunity to study gluon saturation is enabled by prompt or direct photon production in proton-nucleus collisions~\cite{Gelis:2002ki,Jalilian-Marian:2005qbq,Helenius:2014qla,JalilianMarian:2012bd,Benic:2016uku,Ducloue:2017kkq,Golec-Biernat:2020cah,SampaiodosSantos:2020raq}. Direct photons are differentiated from virtual or fragmentation photons as real photons originating from the electromagnetic vertex. High energy direct photons are a particularly interesting probe as they do not suffer from non-perturbative fragmentation. Experimentally on the other hand they are challenging to identify. Photons are neutral final state particles which often rely on energy depositions in calorimeters or material conversions as part of the identification strategy. Their relative cross-sections are much smaller as compared to single hadron production which can be a competing background via photon decays. This difference in cross-sections is due to an additional factor of the electromagnetic structure constant $\alpha_{em}$. General and beam related a backgrounds as will be discussed in the last Sec~.\ref{Sec:conclusions} may also inundate and overlap in the detectors which can make a precise identification difficult in particular at moderate to high (or very low) $p_\perp$ depending on the resolution limits of the detectors used. LHC results from the ALICE experiment have been measured~\cite{Acharya:2018dqe} at large $x$ and mid-rapidity. While there is little indication that the ALICE $x$ range/rapidity probed is ideal for a clean gluon saturation signal, sensitivity is anticipated, mainly, a suppression in the nuclear modification factor in pA collisions is expected featuring a Cronin peak at midrapidity and its disappearance at forward rapidities. Since the photon is colorless it is expected that its distribution will be less influenced by final state interactions. There are number of active efforts at the LHC to measure similar photon observables at forward rapidity as well as with mid- and forward rapidity correlations to investigate gluon saturation effects~\cite{ALICE:2020mso, Boettcher:2019kxa} with dedicated calorimetry.
\label{Sec:quarkonia}
We wrap up the single inclusive discussion with quarkonia production. Quarkonium or hidden charm in proton and heavy ion collisions also provides valuable opportunities to study gluon saturation at high scattering energies.
One of the advantages of quarkonia in terms of pQCD calculations is that the charm quark mass is larger than the typical QCD scale of $\Lambda_{QCD}$, making pQCD calculations meaningful via factorization. On the other hand the evolution into a bound state is intrinsically non perturbative due to the size of the $q\bar{q}$ system and the inverse of the binding energy not being small enough.
A key consequence of the saturation scale defined as: $Q^{2}_{sat}\sim A^{1/3} x^{-0.3}$ with $A$ the mass number, is that the same scale for heavy nuclei in heavy ion colliders is comparable with heavy quark mass. We borrow from a relevant formalism for quarkonia given in ~\cite{Watanabe:2016ert, Kharzeev:2005zr} for the arguments hereafter. In the case of pA collisions and in the nucleus (A) rest frame, the interaction time when proton scatters off the nucleus is characterized $\tau_{int}\sim R_{A}$. In a heavy ion collision a heavy quark pair ($q\bar{q}$) is produced over the time $\tau_{P}\sim \frac{E_{g}}{(2m_{q})^{2}}$, where $m_{q}$ is the quark mass and $E_{g}$ is the energy of the parent gluon ($E_{g}=x_{p}E_{p}$). The gluon will ultimately split into the $q\bar{q}$ pair. Momentum conservation for partonic scattering dictates that $\tau_{P}\sim \frac{1}{2x_{A}M_{N}}$, with $x_{a}$ representing the longitudinal momentum fraction of target nucleus carried by the incident gluons and $M_{A}$ being the mass of the nucleon. This latter indicates that in a \emph{proton going direction} or forward rapidity the $q\bar{q}$ has a longer production time ($\tau_{P}$) than $\tau_{int}$ owing to the Lorentz time dilation and the coherent interaction of the proton with the nucleus. As the biding energy of quarkonium is smaller than its mass $m_{q}$~\cite{Kharzeev:2005zr, Kharzeev:1999bh}, quarkonium production is thus shorter than its formation time, $\tau_{F}\gg \tau_{P}\gg \tau_{int}$.
This tells us that the dynamics of the $q\bar{q}$ bound state formation can be thus decoupled with cold nuclear effects thanks to its formation occurring well outside the target nucleus.
A body of theoretical work exists which has compared
low $p_{T}$ quarkonia production at LHC at forward rapidity using Color Evaporation Models (CEM), Color Singlet Models (CSM) and Non Relativistic QCD coupled with gluon saturation (NRQCD$+$CGC).
In the NRQCD$+CGC$ approach both Color Singlet and Color Octet state of the $c\bar{c}$ pairs are considered. The relative contribution of the states is parametrized using a finite set of universal long distance matrix elements (LDME), fitted to a subset of the data (e.g. Tevatron). Non-relativistic QCD is used to factorize the short-distance scale (annihilation), set by the heavy quark mass $M$, from the longer-distance scales (production). The short distance scales are expressed in terms of non-perturbative matrix elements of 4-fermion operators in non-relativistic QCD, with coefficients that can be computed using perturbation theory in the coupling constant $\alpha_{s}$. The matrix elements are organized into a hierarchy according to their scaling with $v$, the typical velocity of the heavy quark~\cite{Bodwin:1994jh} making the heavy quark's velocity ($v$) and $\alpha_{s}$ the two key parameters which are employed to describe quarkonium production.
Fig~\ref{fig:Onia-ALICE} illustrates a comprehensive comparison of $J/\psi$ and $\psi^{'}$ production at all center of energies that the LHC has collided protons and at forward rapidity. These results are all well described by a number of independent CGC coupled with NRQCD calculations~\cite{Ma:2010jj, Ma:2014mri, Butenschoen:2010rq}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{Figure8ALICE-AstridHugo.png}
\caption{ $J/\psi$ and $\psi(2S)$ at five center of mass energies from the ALICE experiment compared to summed CGC-NRQCD and FONLL
calculations~\cite{Acharya:2017hjh}.
}
\label{fig:Onia-ALICE}
\end{figure}
Figs.~\ref{fig:StarHQ}, ~\ref{fig:Onia} complete the global picture with experimental data at the lower center of mass energies available at RHIC as well as more central rapidities both at the LHC and RHIC. It is noted that CGC$+$NRQCD and CEM and their respective charmonium factorization approaches have been subject of many comparisons over the last two decades, one such comparison is documented in reference~\cite{Bodwin:2005hm}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{Xsec_diedimu_fonll-STAR.png}
\caption{$J/\psi$ Production cross sections as a function of $p_\perp$ in pp collisions at $\sqrt{s}=$ 510 and 500 GeV measured through the $\mu^{+}\mu^{-}$ (blue stars) and $e^{+}e^{-}$ decay channels (red circles). From the STAR experiment~\cite{Adam:2019mrg}}
\label{fig:StarHQ}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{Xsec_dimu-ALLColliders.pdf}
\caption{The $\psi^{'}$ (top curve) and $J/\psi$(other four curves) differential cross section as a function of p$_T$. Figure from~\cite{Ma:2014mri}.
}
\label{fig:Onia}
\end{figure}
\subsubsection{Competing mechanisms in single inclusive production}
We begin by cautioning that the behavior of the nuclear modification factor shown in Fig.\,\ref{fig:nuclear_mod_factor_hadron_DAu} can also be described by other mechanisms: the data at mid-rapidity can be reproduced well within the leading twist approach using nuclear PDFs \cite{Eskola:2009uj} and within Glauber like multiple scattering \cite{Accardi:2003jh}. At forward rapidities, it has been argued in \cite{Kopeliovich:2005ym} that the disappearance of the Cronin peak follows from energy-momentum conservation \footnote{Note that the partons in the dilute projectile (in this case the deuteron) involved in the forward production carry large momentum fraction $x$ close to the kinematic limit $x\sim 1$.}. Accordingly, the searches of saturation at RHIC using single hadron production need to be supplemented by similar studies at the LHC, where the kinematic coverage in rapidity and transverse momentum is substantially larger and might allow for the distinction of these mechanisms and gluon saturation.
For the quarkonia measurements, as it is the case for all results presented here, caution must be exercised as number of other effects may be at play and can also account for the experimental observations, this include and not are limited to collective effects (CE) and Multiple Parton Interactions (MPI). For an interesting review on the subject of small systems and Cold Nuclear Matter effects such as MPI and CE, see ~\cite{2018EPJWC.17111001A}
\subsubsection{Double inclusive production}
\label{sec:double_inclusive}
We now move on to the study of inclusive two particle particle production in proton-nucleus or deuteron-nucleus collisions. For this process, it is very natural to study azimuthal angle correlations, which involve measuring the $\Delta\phi$ of particle pairs in an event. Typically, one observes two peaks in the distribution: a near side peak ($\Delta\phi \sim 0$) dominated by fragmentation of the leading jet, and an away side peak ($\Delta\phi \sim \pi$) produced by $2 \to 2$ back-to-back scatterings. It has been suggested that the emergence of gluon saturation might be studied in modifications to the away side peak by comparing pp to $p(d)$-$p(A)$ collisions as we will see shortly.
RHIC has measured a depletion of the back-to-back peak in the production of forward dihadrons in $d$-$Au$ collisions when compared to the same distributions from pp collisions. This effect was predicted in the CGC formalism \cite{Marquet:2007vb} as a consequence of multiple scattering on the dense nucleus and the quantum evolution. If the two particles originate from the same parton, the collinear framework dictates that their transverse momenta must be (almost) back-to-back following momentum conservation. On the other hand, in the saturation framework (within the hybrid factorization) the scattered partons acquire a momentum imbalance from the dense nucleus with characteristic momentum scale $Q_s$. Since the saturation scale $Q_s$ grows with nuclear species, one expects a systematic enhancement of the suppression when the collision involves larger nuclei, higher energies or when the particles are produced at more forward rapidities.
\begin{figure}[H]
\centering
\includegraphics[scale=0.35]{cp_phenix_pp_130912.pdf}
\includegraphics[scale=0.35]{cp_phenix_300712.pdf}
\caption{ Azimuthal correlation for $\pi^0$ production compared to PHENIX data \cite{PHENIX:2011puq}. Top: proton-proton collision. Bottom: central deuteron-gold collisions. Figures from \cite{Lappi:2012nh,Lappi:2012xe}. More modern version of this work both experimentally~\cite{Chu:2749297} and theoretically exist with forthcoming publications.}
\label{fig:dAu_supression}
\end{figure}
The first comparison of dihadron correlations at RHIC to the result of a CGC calculation was made in \cite{Albacete:2010pg}, while the important inelastic contribution (quadrupole) was included in \cite{Lappi:2012nh}. Both studies only considered the quark initiated channel (from deuteron), which is expected to be dominant at RHIC energies\footnote{At the LHC one has to include the gluon initiated channels as well.}. In addition, there is an angle independent contribution (pedestal) arising from double parton scattering \cite{Strikman:2010bg} that must be taken into account. Results show that gluon saturation qualitatively reproduces the systematics of suppression (see Fig.\,\ref{fig:dAu_supression}). We note that a more modern experimental work from RHIC using pAl and pAu collisions has been recently presented in ~\cite{Chu:2749297} where a forthcoming publication is expected.
An important theoretical advancement was made in \cite{Dominguez:2011wm} where the authors established the connection between the CGC formalism and the TMD framework. These findings significantly simplified the theoretical computations allowing to include other channels (e.g. gluon initiated) and to incorporate higher order contributions. In the back-to-back limit the most important NLO contribution is the Sudakov factor derived in \cite{Mueller:2013wwa} which leads to a suppression of the back-to-back peak. In recent years, theoretical comparisons have been made using the TMD approximation: including both quark and gluon channels and rcBK evolution \cite{Albacete:2018ruq} and including Sudakov resummation within the GBW model \cite{Stasto:2018rci}.
Similar studies have been carried out for dijet production at the LHC. Despite the fact that Sudakov resummation plays a dominant role (due to the higher $p_\perp$ required for insufficient jet reconstruction compared to hadron measurements), the results show that it is possible to distinguish this effect from gluon saturation. The first studies were performed without Sudakov in \cite{Kutak:2012rf,vanHameren:2014lna} and with Sudakov resummation in \cite{vanHameren:2014ala}. More recently, these results have been supplemented with kinematic power corrections within the so called Improved TMD (ITMD) framework \cite{Kotko:2015ura,vanHameren:2016ftb} extending the agreement with data to large non back-to-back configurations \cite{vanHameren:2019ysa} (see also for UPC studies \cite{Kotko:2017oxg}).
Finally, we point out that a similar depletion of the back-to-back peak was proposed in photon-hadron, photon-pion, and photon-jet correlations
at RHIC and the LHC \cite{Jalilian-Marian:2012wwi,Rezaeian:2012wa,Rezaeian:2016szi,Benic:2017znu,Goncalves:2020tvh}. It would be interesting to update these studies to include Sudakov resummation which is known to impact azimuthal correlations near the back-to-back peak \cite{Mueller:2013wwa}.
\subsubsection{Competing mechanisms in double inclusive production}
As discussed above a common challenge to uncover gluon saturation in these observables is to assess the impact of Sudakov resummation. Sudakov double logarithms can appear in these processes as a result of the incomplete cancellation between real and virtual contributions~\cite{Mueller:2012uf, Mueller:2013wwa} and are enhanced when the transverse momentum of the produced particles/jets is large. As a consequence, the searches for saturation are restricted to low to moderate transverse momentum phase space, where the Sudakov does not overwhelm the effects of gluon saturation but where jet reconstruction or effects of hadronization might obscure signals of saturation.
Other key physics mechanisms which have been used to explain the dihadron suppression observed at PHENIX and STAR are energy loss in the medium, final state radiation and coherent power corrections~\cite{Kang:2011bp, Xing:2012ii}.
\subsection{High multiplicity and small systems}
We conclude with a brief note of recent observables that have been highlighted in \emph{small systems} and high energy collisions (pA, pp). These observables entail the classification of some of the results discussed through this document, into \emph{high activity} or \emph{high multiplicity} environment classes. High activity has been recently used as a proxy to what is more commonly known and used in heavy ion collisions as \emph{centrality}. Centrality has been traditionally used for describing and classifying the heavy ion collision system size according to their impact parameter where the colliding nuclei are viewed as hard spheres with radius $R$. In pp collisions this description is not as clear since until recently, the primary models used to describe centrality assumed the proton a point-like particle. In pA collisions it has additionally raised biases due to detector geometry and triggering effects potentially present in the experiments~\cite{Armesto:2015kwa, ALICE:2017svf, Connors:2017ptx}.
Nevertheless a plethora of experimental results are currently available and a comprehensive paper that compared a number of cold nuclear matter effects to LHC data in small systems was published in~\cite{Albacete:2017qng}. In this report among other subjects, the multiplicity distributions of charged identified/unidentified hadrons in pp collisions at 7~TeV center of mas energies ~\cite{ALICE:2010mty,CMS:2012xvn} were compared to the IP-Glasma model~\cite{Schenke:2012wb, Schenke:2016lrs}. These comparisons showed an important milestone: the reproduction of overall mass ordering trends in data over the whole multiplicity range. Recent quarkonia \emph{high multiplicity} measurements from the ALICE experiment ~\cite{Khatun:2019slm, Abelev:2012rz, Thakur:2018dmp, Acharya:2020pit} have been also compared recently to a CGC approach by E. Levin, I. Schmidt and M. Siddikov~\cite{Siddikov:2019xvf} in which quarkonia data from p-p collisions and at forward rapidity is successfully described by the CGC framework as shown in Fig.~\ref{fig:13TevLevin}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{dndeta-ch-pp.png}
\includegraphics[scale=0.5]{IP-GlasmaCMS.pdf}
\caption{IP-Glasma predictions for 7~TeV center of mass energies pp collisions for the multiplicity dependence of (top) unidentified charged hadrons, (center and bottom) identified hadrons. Figures from~\cite{Albacete:2017qng} and \cite{Schenke:2016lrs}}
\label{fig:IPGlasma-Albacete}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{13TeVLevin-IPsat.png}
\caption{Multiplicity dependence of $J/\psi$ in $\sqrt{13}$ center of mass energies pp collisions at the ALICE experiment. Left figure corresponds to ALICE data in the forward region~\cite{Khatun:2019slm, Acharya:2020pit}. Theory comparisons from ~\cite{Siddikov:2019xvf}}
\label{fig:13TevLevin}
\end{figure}
\subsubsection{A final note on competing mechanisms }
Single inclusive probes poses a number of challenges for evidencing gluon saturation. A concern that is raised with the mid-rapidity ~$\sim 0$ region is the competing Glauber like mechanisms ~\cite{Accardi:2003jh} in addition to the energy scale which may be be less sensitive in the equivalent $x$ range. Energy conservation plays a non-negligible role which may be manifested in terms of energy loss mainly via radiation and hadronization effects. In addition as many of these results are produced at low $p_{T}$ where the bulk of particle production is \emph{soft}, in other words the mean free path of the particles in the medium is very small and the phenomena can be explained with hydrodynamic evolution.
\section{A new generation of high energy DIS colliders}\label{Sec:EIC}
In 2020 the Department of Energy (DOE) granted approval for the Electron-Ion Collider (EIC) to be built in the USA~\cite{CD0} marking the beginning of a new chapter for high-energy nuclear physics (HENP). The EIC is a key international facility that will collide electrons and protons/ions (e-p/A) at high energy with unprecedented luminosity. Amongst its rich physics program, the potential discovery of gluon saturation is one of the key missions of the EIC \cite{Boer:2011fh,Accardi:2012qut,Aschenauer:2017jsk,AbdulKhalek:2021gbh}. The prospect for the discovery for gluon saturation is facilitated thanks to its ability to collide electrons with large ions, where the saturation scale is enhanced $Q_s^2 \sim A^{1/3}$ as compared to the electron-proton collisions at HERA. Another future project of interest to our field is the Large Hadron electron Collider (LHeC). The LHeC is an ongoing accelerator study which would upgrade the existing LHC storage ring colliding an intense electron beam with a proton or ion beam from the High Luminosity–Large Hadron Collider (HL-LHC)~\cite{Agostini:2020fmq, Goncalves:2020ywm}. Among its rich and diverse physics capabilities, it would also present an opportunity to study gluon saturation in ep collisions while probing Bjorken-$x$ values as low as 10$^{-6}$.
The experimental discovery of gluon saturation will require comprehensive analyses at future colliders. By the same token quantifying its characteristics will need an in-depth energy, $Q^2$ and mass number dependence scan on a number of quantities. Critical measurements that can help us in the discovery of a gluon saturated state can be classified once again into three main groups: structure functions, exclusive reactions, and semi-inclusive reactions. Below we will discuss the expected manifestations of gluon saturation for each of these processes while keeping an open tab to other mechanisms that could shroud its attributes.
\subsection{Structure functions}
In Sec.\,\ref{subsec:structure functions} we discussed the searches of gluon saturation in proton structure functions at HERA. Major obstacles for its clean extraction are: (i) the contribution from non-perturbatively large dipoles at low to moderate virtualities $Q^2$, and (ii) the non-perturbatively small size of the momentum saturation scale $Q_s^2$ accessed in measurements at HERA.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{bartels3dCombo.pdf}
\caption{Relative difference between longitudinal structure function $F_{L}$ obtained from the saturation framework to that of the leading twist formalism for proton (left) and Gold (right) \cite{Bartels:2009tu}. Figure from \cite{Accardi:2012qut}.}
\label{fig:FL_sat_vs_LT}
\end{figure}
While the expected top center of mass energy of the future EIC is lower than that at HERA, the possibility to collide electrons with large nuclei results in accessing larger values of the saturation scale as compared to electron-proton collisions. A comparison of the saturation scales accessible at HERA and the EIC was done in~\cite{Aschenauer:2017jsk}.
The enhancement in the saturation scale is known as nuclear \emph{oomph} factor, and it is a result of the coherent interaction of the probe with partons along its path of propagation. Larger saturation scales at the EIC are expected to manifest as more pronounced differences between the saturation framework and the leading twist formalism when computing the structure functions. In Fig.\,\ref{fig:FL_sat_vs_LT} we see the effects of gluon saturation, embodied in higher twist corrections, on the structure function $F_L$. As expected, the largest difference is observed in the small $(x,Q^2)$ corner.
\begin{figure}[H]
\centering
\includegraphics[scale=0.32]{F2.pdf}
\caption{Comparison of the structure function $F_2$ obtained from the rcBK solutions to those from extrapolated nuclear PDFs. Figure from \cite{Marquet:2017bga}.}
\label{fig:sat_vs_nuclearPDF_EIC}
\end{figure}
To more explicitly analyze the impacts of gluon saturation on the structure functions at the EIC, the authors of \cite{Marquet:2017bga} generated pseudodata for electron-gold collisions, using the running-coupling
Balitsky-Kovchegov evolution equation, and found tension between the compatibility of
these saturated pseudodata with extrapolations of the existing nuclear PDFs (see Fig.\,\ref{fig:sat_vs_nuclearPDF_EIC}). While this tension might result in a signature of gluon saturation, it remains possible that refittings of nPDFs could accommodate for these differences as nPDFs are not well constrained in the low $x$ regime. We have confidence that the high statistics of the EIC combined with forthcoming precise theoretical computations will distinguish both scenarios.
\begin{figure}[H]
\centering
\includegraphics[scale=0.38]{pulls-5gev2.pdf}
\includegraphics[scale=0.38]{pulls-50gev2.pdf}
\caption{Pull (defined in Eq.\,\eqref{eq:pull_pseudodata_dglap}) between pseudodata for reduced cross-section and the fit based on DGLAP. The pseudodata has been generated either by DGLAP or by the GBW saturation model. Figure from \cite{Agostini:2020fmq}.}
\label{fig:F2_LHeC_DGLAP_GBW}
\end{figure}
Complementary to the EIC, the LHeC will reach very low values of $x$, providing potential to discriminate linear DGLAP evolution from the non-linear QCD evolution via BK/JIMWLK equations. In a recent preliminary study that can be found in \cite{Agostini:2020fmq}, the authors generate pseudo data for the reduced cross-section using two models: (i) a DGLAP based model, and (ii) a saturation (GBW) based model. Subsequently, they fit each pseudodata set using a DGLAP calculation. As expected, the fit is excellent for the DGLAP generated pseudodata, while significant tension is observed when the DGLAP fit is applied to the pseudodata based on the saturation model. This is quantified by studying the pull:
\begin{align}
P(x,Q^2) = \frac{\mathcal{F}_{\rm dat}(x,Q^2)-\mathcal{F}_{\rm fit}(x,Q^2)}{\delta_{\rm exp} \mathcal{F}(x,Q^2)} \,,
\label{eq:pull_pseudodata_dglap}
\end{align}
where $\mathcal{F}_{\rm fit}$ is the central value of the result of the fit for the observable $\mathcal{F}$, $\mathcal{F}_{\rm dat}$ corresponds to the pseudodata generated by model (i) or (ii), and $\delta_{\rm exp} \mathcal{F}$ represented the experimental uncertainty. Their results are shown in Fig.\,\ref{fig:F2_LHeC_DGLAP_GBW}, where the magnitude of the pull is close to $0$ for the DGLAP pseudodata, and significantly larger than $0$ for the GBW (saturation) pseudodata. This tension suggests that if gluon saturation is present at LHeC, a DGLAP fit will not be able to conceal it.
\subsection{Diffractive measurements}
Building upon the observables discussed in Sec.\,\ref{sec:diffractive_reactions}, we briefly discuss the potential for diffractive measurements at future colliders.
The ability to collide electrons with different nuclei opens up the possibility to study the nuclear modification factor for the production of diffractive events. Computations within the saturation framework result in the enhancement of nuclear diffractive structure functions when the invariant mass of the final state $M_{X}$ is small, which is dominated by the dipole Fock state. On the other hand, at large invariant masses the dominant state is that of a tripole ($q\bar{q}$+ gluon) which is absorbed more strongly in a denser target, resulting in a suppression of the nuclear structure function \cite{Kowalski:2008sa}. Another prediction of saturation models is that the number of diffractive events relative to all events is larger for nuclei than for protons, which can be quantified by a double ratio \cite{Kowalski:2008sa} as shown in Fig.\,\ref{fig:diffractive-to-inclusive}. The leading twist formalism on the other hand predicts a slight suppression of diffractive events.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{ratio-diffractive-inclusive-ep-eA.png}
\caption{ Ratio of diffractive to inclusive DIS cross-section in eA normalized to pA (double ratio). Comparison between saturation predictions and the leading twist approach. Figure from \cite{Accardi:2012qut}. }
\label{fig:diffractive-to-inclusive}
\end{figure}
Another possibility is the study of coherent vector meson electroproduction off nuclei. As in proton DIS, the spectrum of the exclusively produced particle in nuclear DIS provides a tomographic picture of the color charge density profile of the nuclear target. Predictions from the saturation framework are shown in Fig.\,\ref{fig:VM_production_eA} based on the calculations in \cite{Toll:2012mb,Toll:2013gda}. This figure shows that saturation results in spectra which deviate from the form factor (Fourier transform of the nuclear density profile). These deviations grow with energy and when the produced vector mesons are less massive\footnote{More massive vector mesons probe shorter distances where saturation effects are suppressed.}. To quantify saturation effects it is also necessary to compare these results to predictions obtained from competing mechanisms such as the leading twist nuclear shadowing framework, where deviations from the form factor are also expected due to multiple scattering \cite{Frankfurt:2011cs}. Furthermore, theoretical control over the uncertainties for the light-cone wave-functions is necessary to distinguish saturation from non-saturation models. On the experimental side, enough statistics are necessary to resolve the peaks and dips of the spectra; particularly in the region in which the cross-section might be overwhelmed by incoherent (break-up of target) events or general beam-induced backgrounds as it will be discussed in Sec.~\ref{Sec:conclusions}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{dsigma_dt_jpsi_phi.pdf}
\caption{Transverse momentum spectra for the diffractive coherent production of vector mesons in electron-gold ion collisions. Left: $J/\psi$ production. Right: $\rho$ production. The figures show the comparison between models with and without saturation. Figure from \,\cite{Accardi:2012qut}. }
\label{fig:VM_production_eA}
\end{figure}
Incoherent production is also interesting as one expects sensitivity to subnucleonic fluctuations of various kinds. At the EIC, in addition to incoherent events in heavy nucleus DIS, we will also be able to study DIS off light-nuclei and study the interplay of short range correlations and gluon saturation \cite{Miller:2015tjf,Mantysaari:2019jhh,Tu:2020ymk}.
Finally we note that performing these measurements require detection on the forward or backward region in a center of mass instrumentation design. Due to this a number of non-trivial experimental challenges are present that need to be considered for a statistically significant and minimally biased measurement. These challenges include: control of machine related backgrounds such as synchrotron radiation and beam-gas interactions, magnetic field strengths, Interaction Region (IR) designs, up-to-date adequate particle detection technologies including designs which includes beam-line detectors capable to discern final state hadrons essential for identifying exclusive DIS events.
\subsection{Semi-inclusive measurements}
In this section we revisit some of the observables discussed in Sec.\,\ref{sec:semi_inclusive_reactions} in the context of Semi Inclusive Deep Inelastic (SIDIS) measurements. SIDIS at colliders offer several advantages over proton-proton and proton-nucleus collisions:
(i) The kinematics of the electromagnetic probe (the virtual photon exchanged between the electron and the ion) can be fully reconstructed by measuring the scattered electron. In contrast to pp/pA collisions where the probes are quarks or gluons whose kinematics cannot be retrieved, but require convolutions with parton distribution functions.
(ii) The number of mechanisms is less in electron-nucleus collisions as compared to proton-nucleus collisions, since in the former the probe is a virtual photon where in the latter one can have both quarks and gluons.
(iii) The virtuality $Q^2$ of the exchanged photon can be used as a knob to scan between the non-linear saturated and linear QCD regimes.
We begin by discussing forward dihadron azimuthal correlations in ep, eA collisions. Motivated by the studies in pp and dAu collisions at RHIC and the LHC (c.f. Sec.\ref{sec:double_inclusive}), this process has received considerable attention in recent years and it is considered a promising channel for gluon saturation searches at the EIC (see also \cite{Kolbe:2020tlq} for photon-hadron azimuthal correlations).
The away side peak in the azimuthal angle distribution of dihadron production is expected to be suppressed in nuclear DIS compared to proton DIS due to the momentum imbalance imparted by the saturated gluon inside the nucleus. At small-$x$ and in the TMD approximation this process involves only the Weizsäcker-Williams (WW) gluon distribution \cite{Dominguez:2011wm} given that the dominant partonic channel is virtual photon-gluon fusion\footnote{In electron-nucleus collisions there are no initial state interactions in the gauge links, in the language of TMDs, since the exchange photon is colorless. This is in contrast to proton-nucleus collisions, where the collinear quark or gluon to the proton carry color and thus initial interactions in the gauge links are present \cite{Mulders:2000sh,Dominguez:2011wm,Petreska:2018cbf}}. An advantage over proton-nucleus collisions is the absence of the pedestal arising from double parton scattering.
The first feasibility study for this process at the EIC has been carried out in \cite{Zheng:2014vka} employing a GBW model to compute the WW gluon TMD and including the Sudakov factor \cite{Mueller:2012uf,Mueller:2013wwa}. Their results for the correlation function (dihadron production normalized by single hadron production) are shown in the right panel of Fig.\,\ref{fig:dihadron_suppresion_Adep} showing a clear depletion of the back-to-back peak, while the left panel shows the nuclear dependence of the suppression.
\begin{figure}[H]
\centering
\includegraphics[scale=0.35]{3_16-replace-dihadrons.pdf}
\caption{Left: Dihadron correlation function in electron collisions with different nuclei showing a depletion of the back-to-back peak. Right: Comparison of the correlation function with and without saturation. The gray band is a result of varying the saturation scale. Figure from\,\cite{Accardi:2012qut}. }
\label{fig:dihadron_suppresion_Adep}
\end{figure}
To better interpret future results it would be necessary to update the predictions in Fig.~\ref{fig:dihadron_suppresion_Adep}
to include a more realistic WW gluon distribution, e.g. obtained from the solution to the rcBK equation. Initial steps in this direction have been recently taken in~\cite{vanHameren:2021sqc} where the authors employ a model capturing rcBK evolution and also included kinematic power corrections~\cite{Altinoluk:2019fui,Altinoluk:2021ygv,Boussarie:2021lkb}. It is also necessary to further investigate competing mechanisms which may deplete the away side peak due the momentum broadening; these can include cold nuclear matter energy loss and coherent power corrections as proposed in~\cite{Xing:2012ii}.
As for other observables that can be measured in DIS, recently, new signatures of gluon saturation have been proposed by studying single inclusive particle production~\cite{Marquet:2009ca,Iancu:2020jch}. These measurements are analogous to those at RHIC and the LHC, where the nuclear modification factor R$_{eA}$ develops a Cronin-like peak at mid rapidity which is then suppressed by saturation. Preliminary studies in~\cite{Iancu:2020jch} show very characteristic features of the transverse momentum distribution when studied at different rapidities and different virtualities and in DIS. The authors propose that this observable can be studied at perturbative virtualities $Q^2 \gtrsim 1 \ \mathrm{GeV}^2$ and that sensitivity to the saturated regime is enhanced for hadrons that carry a large longitudinal momentum fraction $z$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.7]{RpA_total_cs.pdf}
\includegraphics[scale=0.7]{RpA_total_cs_rapidity.pdf}
\caption{Nuclear modification factor for single semi-inclusive hadron production displaying a Cronin peak and its disappearance. Left: dependence on the virtuality. Right: dependence on rapidity of the produced particle. In this figures $\bar{Q}^2 = z (1-z) Q^2$, where $z$ is the longitudinal momentum fraction of the hadron relative to the virtual photon, and is chosen to be close to unity. Figure from\,\cite{Iancu:2020jch}. }
\end{figure}
\section{Discussion and concluding remarks}\label{Sec:conclusions}
\subsection{Theoretical advances}
In this document we have presented various observables that may pave the road for the discovery of gluon saturation at existing and future collider experiments. However, there are still significant sources of theoretical uncertainties that could complicate a systematic extraction from current and ensuing measurements if left unchecked. The CGC framework has entered a new era of theoretical developments which aim to push the precision of the saturation framework to the standards of collinear pQCD. In this section we briefly review recent advances in the field which align to a discovery direction.
Most of the observables presented have been calculated in the CGC at leading order (LO) in the impact factor and with leading logarithmic (LL) small-$x$ evolution equations with running coupling corrections for the BK \cite{Kovchegov:2006vj,Balitsky:2006wa,Albacete:2007yr} and for the JIMWLK \cite{Lappi:2012vw}. Active efforts by the CGC theoretical community are being conducted to promote these observables to higher loop order accuracy. This requires the determination of the next-to-leading order (NLO) impact factors, and the numerical implementation of the next-to-leading logarithmic (NLL) small-$x$ evolution equations. The NLL small-$x$ evolution equations have been derived in \cite{Balitsky:2008zza} for BK and in \cite{Balitsky:2013fea,Kovner:2013ona,Kovner:2014lca,Lublinsky:2016meo} for JIMWLK. Only the former has been implemented numerically in \cite{Lappi:2015fma}, where it was found that evolution is unstable for initial conditions of phenomenological interest. This issue of instability was resolved in \cite{Beuf:2014uia,Iancu:2015vea,Ducloue:2019ezk} by resummation of (anti-)collinear logarithms, with a numerical implementation realized in \cite{Lappi:2016fmu}. While a numerical implementation of the full NLL JIMWLK equation is not yet available, numerical codes exist \cite{Korcyl:2020orf,Cali:2021tsh} based on the collinearly improved JIMWLK equation proposed in \cite{Hatta:2016ujq}.
To achieve precise computations of physical processes, one also needs high order computations of the corresponding impact factors. Significant progress has been made in this direction for a variety of processes at NLO in the CGC, which include: DIS structure functions \cite{Balitsky:2012bs,Beuf:2011xd,Beuf:2016wdz,Beuf:2017bpd,Lappi:2016oup,Hanninen:2017ddy,Ducloue:2017ftk,Beuf:2020dxl,Beuf:2021qqa}, exclusive dijet \cite{Boussarie:2016bkq,Boussarie:2016ogo,Boussarie:2019ero} and exclusive vector meson \cite{Mantysaari:2021ryb} in eA, inclusive di-jet+photon in in eA \cite{Roy:2019hwr}, single inclusive particle production in pA \cite{Chirilli:2011km,Chirilli:2012jd,Altinoluk:2014eka,Ducloue:2016shw}, partial results for inclusive dijet production in pA \cite{Iancu:2020mos}, and most recently inclusive dijet production in eA \cite{Caucal:2021ent}. Numerical results with NLO impact factor and NLL small-$x$ resummation have been obtained only for the structure functions \cite{Beuf:2020dxl}, and single inclusive particle production in pA \cite{Stasto:2013cha,Ducloue:2017mpb,Ducloue:2017dit,Liu:2020mpy} which compared well to data. We anticipate that in the next few years, more of these computations will be coupled to numerical routines and will provide precise quantitative results for the size of the NLO impact factors.
Another significant source of theoretical uncertainty lies in existing models used for the initial conditions of the small-$x$ evolution equations. The MV model \cite{McLerran:1993ni,McLerran:1993ka} is the most widely used framework to compute the initial conditions for the existing computations in the literature. This is in view of their simple numerical implementation relying on the Gaussian statistics of its color charge correlators \cite{Dumitru:2011vk,Iancu:2011nj}. Despite the fact that the MV model was conceived for the description of very large nuclei, it has been employed as the initial condition for protons and it has enjoyed great success in the phenomenological studies of HERA data (see also the variants MV$\gamma$ \cite{Albacete:2010sy} and MV$e$ \cite{Lappi:2013zma}). It remains to be seen if it provides a good description of less inclusive observables where non-Gaussian effects might play a larger role \cite{Dumitru:2011ax,Dumitru:2011zz,Giannini:2020xme}. A recent alternative approach for the initial conditions has been taken in \cite{Dumitru:2018vpr,Dumitru:2020fdh,Dumitru:2020gla,Dumitru:2021tvw} where the authors follow a perturbative approach to find the two-point, three-point and four-point function of color charge correlators inside the proton from the light-cone wave-functions of its valence quarks.
High energy Wilson line correlators are more naturally written in coordinate space, due to the diagonal nature of the scattering matrix in the eikonal approximation. This includes setting up their initial conditions as well as performing their small-$x$ evolution in coordinate space. However, most observables require their Fourier transform or convolutions to momentum space, making the relation between initial conditions and observables less transparent. Fortunately, in some special limits, it is possible to establish a clean factorization of the perturbatively calculable impact factors and the non-perturbative high energy correlators, which will make the connection of high energy correlators and their initial conditions more explicit. The paradigmatic example is the work in~\cite{Dominguez:2011wm} which established the connection between quadrupole and dipole operators to the WW gluon TMD and the dipole gluon TMD in the so called correlation limit. In the context of two particle production, this relation is physically realized in the near back-to-back production. This is an active area of research which has resulted in the development of improved TMD framework \cite{Kotko:2015ura,vanHameren:2016ftb,Petreska:2018cbf} and the CGC/TMD equivalence \cite{Altinoluk:2019fui,Altinoluk:2019wyu,Boussarie:2020vzf}. For comprehensive numerical studies both in pA and eA we refer the reader to \cite{Fujii:2020bkl,Boussarie:2021lkb}.
A further assumption of the saturation/CGC framework is the eikonal approximation, which is strictly valid at asymptotically high energies. Relaxing this assumption is necessary to access the physics of polarized measurements\cite{Kovchegov:2015pbl,Kovchegov:2016weo,Kovchegov:2016zex,Kovchegov:2017jxc,Kovchegov:2017lsr,Kovchegov:2018znm,Kovchegov:2020hgb,Adamiak:2021ppq,Cougoulic:2019aja,Cougoulic:2020tbc}. Efforts are also being carried out to study the effect of sub-eikonal contributions to various unpolarized observables \cite{Altinoluk:2014oxa,Altinoluk:2015gia,Altinoluk:2015xuy,Agostini:2019avp,Agostini:2019hkj,Altinoluk:2020oyd} with emphasis in non-trivial azimuthal correlations. These contributions might be relevant for precise EIC and RHIC phenomenology where the energies are much less compared to LHC or LHeC.
Powerful techniques such as the glasma graph approximation \cite{Armesto:2006bv,Dumitru:2008wn,Dumitru:2010iy,Kovchegov:2012nd,Kovchegov:2013ewa} suited for collisions where both hadrons/nuclei are treated as dilute objects are being extended to account for asymmetric dilute-dense scenarios \cite{Altinoluk:2015uaa,Altinoluk:2018hcu,Altinoluk:2018ogz,Altinoluk:2020wpf}. On the other hand, it remains challenging to make progress in analytically understanding nuclear collisions where the saturation scales of both colliding objects are large, and one must resort to complex numerical evaluations of the full Yang-Mills evolution \cite{Krasnitz:2003jw,Lappi:2003bi,Blaizot:2010kh,Schenke:2015aqa}. Efforts to quantify the effect of saturation corrections to the dilute projectile on multi-particle production have been made in \cite{Schlichting:2019bvy} by comparing the dilute-dense approximation to the full Yang-Mills simulation. In addition, recent analytical work in \cite{Chirilli:2015tea,Kovchegov:2018jun,Li:2021yiv,Li:2021zmf} accounting for the effect of multiple scattering and saturation in the field of the proton has been carried out. It is imperative to assess the impact of these contributions in multiparticle production and azimuthal correlations at the RHIC and the LHC .
\subsection{Experimental requirements}
The last twenty years have resulted in many experimental results which may have sensitivity to gluon saturation. Nonetheless, to this date we cannot unequivocally state that we have confirmed a gluon saturated state.
One pre-requisite for the next generation of pertinent experimental publications is the need to measure a diverse number of observables. As mentioned throughout this document, the capability to compare results at a number of center of mass energies (energy, $Q^{2}$ scan) as well as several atomic masses ($A$) is essential to obtain a clean physics picture at current and future colliders. Discovery of a gluon saturated state dictates better quantification of i)competing mechanisms, ii)regimes of validity and iii)transitions from dilute and dense gluon states to a truly gluon saturated state. Appropriate detectors equipping finer kinematic areas (small-$x$) are needed to discern a set of measurements sensitive to gluon saturation with better precision. All of these are vital requirements for the future experimental endeavours described in Section~\ref{Sec:EIC}.
We highlight that the use of electron beams in the foreseeable future introduces a number of several machine induced backgrounds~\cite{SynRad:2004} that need to be accounted for to ensure successful data-taking and data-analyses campaigns. Past and current experiments~\cite{Seidel:2004rs,Belle-IISVD:2020wwc, Bartel:1985yd} have demonstrated
that beam-related backgrounds can shut down an accelerator, force detectors to reduce bunch-crossing frequency and overall data rate to counteract beam-background interactions. Mis-identification and tracking efficiency biases using standard trigger, Monte Carlo and background rejection techniques can cripple even the cleanest theoretical probe. The lessons learned from past and present experiments need to be carefully carried over to future experiments. As of today there are a number of key experimental regions at the EIC and likely the LHeC which are susceptible to high backgrounds~\cite{AbdulKhalek:2021gbh}. Additionally, many of the physics signatures we have outlined demand i) reconstruction of full DIS events ii) detection of small cross-sections very close to the collider beam pipe or in regions that may be bombarded by parasitic particles iii) scan of kinematic regions (e.g. $\eta,\,y,\,\phi$). Besides assuring we have the most up-to-date technologies and coverage to deal with the collision and data taking rates, it is imperative that we evolve our particle detection techniques to cope with a new generation of diverse measurements. In the last decade, analyses of large data from high-energy nuclear experiments have been rapidly evolving to Artificial Intelligence techniques. Collider experiments located at the European Organization for Nuclear Research (CERN) have begun a massive effort to introduce and develop AI techniques in all of their physics experiments. The USA's Department of Energy (DOE) and other national science organizations (NSF, APS) have made interdisciplinary AI research, including at future colliders, a key effort that will support the nation’s long-term economic and national security ~\cite{DOE:AI}, it is becoming clear that the next standard in high energy nuclear physics computing will be based on AI~\cite{Feickert:2021ajf, Kasieczka:2021xcg, Amoroso:2020lgh}. Some of the techniques which AI could help in the long term to unambiguously measure gluon saturation can include:
\begin{enumerate}
\item Accurate description of physics and machine induced backgrounds. This requires an effort of open-sourced, cross-collaboration simulation packages that include theory, phenomenology studies as well as up-to-date machine background knowledge. Two principal machine backgrounds that we can learn from past experiments are synchrotron radiation and beam-gas interactions. Synchrotron radiation occurs when the trajectory of a charged particle is bent, synchrotron photons are emitted tangential to the particle’s path. More concretely, these backgrounds can affect tracking detectors and calorimeters by depositing energy leading to detector \emph{hits}. Ultimately this can also lead to a large number of ghost tracks and large detector occupancy effects. Beam-gas interactions on the other hand occur when proton or ion beam particles collide with residual gas. Ion beam interactions with gas cause beam particle losses and halo, which can reach the detectors. Addition of these backgrounds in future simulations is needed for detector design or AI-based data training techniques; as such these should be included in the next generation of DIS experiments.~\cite{Feickert:2021ajf}
\item Improved jet tagging capabilities which can disentangle jets that come from quarks, gluons, gluon-dense vs saturated gluon signatures.
Jet tagging refers to the reconstruction of streams of particles coming from the collision or displaced vertices with the flexibility of a loose event selection requirement. The classification of jets depends on the kinematic variables such as transverse momentum ($p_\perp$), pseudorapidity (rapidity) $\eta$($y$), azimuthal angle $\phi$, number of tracks, energy ($E$). We remind the reader that jets can be contaminated by many soft processes that are not correlated to the jet. We often rely on classification/regression tasks which give us an approximation of the background. A potential AI application which should build upon existing experiments and further developed could be to extract and study list of features using kinematic variables from simulations. The list of features could be used to form jet images or graphs in $\eta-\phi$ plane which will be used as an input of various AI-related algorithms to classify jet events from background events~\cite{Guest:2018yhq}.
\item Precisely identify particles: open and hidden charm mesons, direct photons, electrons all while minimizing biases. While standard cut and slice techniques have done a excellent job when the detectors are adequate and production cross-sections are large, many rare resonances or small cross-sections have suffered from these same methods and have yet reached statistical significance. While machine learning techniques are currently implemented for identification of rare particles in certain physics cases of nuclear experiments at accelerators, AI is at its infancy and has not replaced or considerably complemented standard particle identification methods at high energy nuclear experiments. Applying Machine Learning algorithms can give advantages in the signal to background ratios as strict cuts and slices on the variables are minimized or eliminated altogether. This however requires a dedicated computing effort to go beyond the standard ML methods used so far.
\end{enumerate}
In this document we have outlined a number of existing challenges, potential solutions as well as the high gains/rewards in unambiguously identifying a gluon saturated state. These can be summarized into being able to disentangle the features that characterize a truly saturated state from ones that may indicate the presence of competing effects or a gluon density which may be large but not necessarily saturated. This challenge is particularly well-suited for a new generation of tools and techniques at the theory and experimental level. It is also well suited for a new generation of cross-experiment and cross-field collaborations.
\section{Acknowledgments}\label{Sec:Ack}
We would like to thank Bj\"{o}rn Schenke, Ernst Sichtermann, Raju Venugopalan, for providing elucidating comments to our project. Astrid Morreale and Farid Salazar are respectively supported by i)the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract No.~${\rm KB-01-01-02\_2}$OPE. ii) the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract No.~DE-SC0012704, and the joint Brookhaven National Laboratory-Stony Brook University Center for Frontiers in Nuclear Science (CFNS).
\clearpage
\vspace{6pt}
\abbreviations{The following abbreviations are used in this manuscript:\\
\noindent
\begin{tabular}{@{}ll}
MDPI & Multidisciplinary Digital Publishing Institute\\
DOAJ & Directory of open access journals\\
TLA & Three letter acronym\\
LD & Linear dichroism
\end{tabular}}
|
1,314,259,996,736 | arxiv | \section{Introduction}\label{sec:Introduction}
Since the invention of turbo codes~\cite{Berrou93} and the rediscovery of low-density parity-check~(LDPC) codes~\cite{Gallager63}, many turbo/LDPC-like codes have been proposed in the past two decades.
Among them, the convolutional LDPC codes~\cite{Felstrom99}, recast as spatially coupled LDPC~{\color{black}(SC-LDPC)} codes in~\cite{Kudekar11}, exhibit a threshold saturation phenomenon and were proved to have better performance than their block counterparts.
In a certain sense, the terminology ``spatial coupling" is more general, as can be interpreted as making connections among independent subgraphs, or equivalently, as introducing memory among successive independent transmissions.
With this interpretation, braided block codes~\cite{Feltstrom09} and staircase codes~\cite{Smith12}, as the convolutional versions of (generalized) product codes, can be classified as spatially coupled codes. In~\cite{Moloudi14}, the spatially coupled version of turbo codes was proposed, whose belief propagation~(BP) threshold is also better than that of the uncoupled ensemble.
{\color{black}Recently, block Markov superposition transmission~(BMST)~\cite{Ma13,Ma15,Liang14c}
was proposed}, which can also be viewed as the spatial coupling of generator matrices of short codes.
{\color{black}The original BMST codes are defined over the binary field $\mathbb{F}_2$.
In~\cite{Ma15}, it has been pointed out that any code with fast encoding algorithms and soft-in soft-out~(SISO) decoding algorithms can be taken as the basic code.
For example, one can take the Hadamard transform~(HT) coset
codes as the basic codes, resulting in a class of multiple-rate codes with rates ranging from $1/2^p$ to $(2^p-1)/2^p$, where $p$ is a positive integer~\cite{Hu14,Liang15}.
Even more flexibly, one can use the repetition and/or single-parity-check~(RSPC) codes as the basic codes to construct a class of multiple-rate codes with rates ranging from $1/N$ to $(N-1)/N$, where $N>1$ is an integer~\cite{Hu15}.
It has been verified by simulation that the construction approach is applicable not only to binary phase-shift keying~(BPSK) modulation but also to bit-interleaved coded modulation~(BICM)~\cite{Liang14}, spatial modulation~\cite{Yang14}, continuous phase modulation~(CPM)~\cite{Liu15}, and intensity modulation in visible light communications~(VLC)~\cite{Xu15}.}
In this paper, we propose a procedure to construct codes over groups, which extends the construction of BMST-RSPC codes~\cite{Hu15} in the following two aspects.
First, we allow uncoded symbols occurring in the basic codes. Hence the encoding/decoding algorithms for the basic codes become simpler.
Second, we derive a performance union bound for the repetition codes with any given signal mapping, which is critical for designing good BMST codes without invoking simulations.
We will not argue that the BMST construction can always deliver better codes than other existing constructions.\footnote{Actually, compared with SC-LDPC codes, the BMST codes usually have a higher error floor.
However, the existence of the high error floor is not a big issue since it can be lowered if necessary by increasing the encoding memory.
}
Rather, we argue that the proposed one is more flexible in the sense that it applies to {\em any} given signal set~(of moderate size), {\em any} given~(rational) code rate and {\em any} target error performance~(of interest).
We start with constructing group codes, referred to as RUN codes, with any given rate by time-sharing between repetition~(R) codes and/or uncoded~(UN) transmission.
By transmitting the RUN codes in the BMST manner, we can have a class of good codes~(called BMST-RUN codes). The performance of a BMST-RUN code is closely related to the encoding memory and can be predicted analytically in the high signal-to-noise~ratio~(SNR) region with the aid of the readily-derived union bound.
Simulation results show that the BMST-RUN codes can approach the Shannon limits at any given target error rate (of interest) in a wide range of code rates over
\textcolor{black}{both} additive white Gaussian noise~(AWGN) channels {\color{black}and Rayleigh flat fading channels}.
{\color{black}The pragmatic reader may question the necessity to construct codes over high-order signal constellations, since bandwidth efficiency can also be attained by BICM with binary codes. However, in addition to the flexility of the construction, the BMST-RUN codes have the following competitive advantages.
\begin{itemize}
\item BMST-RUN codes can be easily designed to obtain shaping gain in at least two ways. One is designing codes directly over a well-shaped signal constellation, say, non-uniformly spaced constellation~\cite{Sun93}. The other is implementing Gallager mapping for conventional signal constellations~\cite{Ma04}.
In both cases, neither optimization for bit-mapping~(at the transmitter) nor iterations between decoding and demapping~(at the receiver) are required.
\item BMST-RUN codes can be defined over signal sets of any size, such as 3-ary pulse amplitude modulation~(3-PAM) and 5-PAM, which can be useful to transmit real samples directly~\cite{Yang12}.
\end{itemize}
}
{\color{black}The rest of this paper is organized as follows. In Section~\ref{sec:ReviewOfBMST}, we take a brief review of the BMST technique. In Section~\ref{sec:CodesOverGroups}, we discuss constructing group codes with any given signal set and any given code rate. In Section~\ref{sec:BMSToverGroups}, we propose the construction method of BMST-RUN codes and discuss the performance lower bound. In Section~\ref{sec:Examples}, we give simulation results and make a performance comparison between the BMST-RUN codes and the BMST-BICM scheme. In Section~\ref{sec:Conclusion}, we conclude this paper.}
{\color{black}
\section{Review of Binary BMST Codes}\label{sec:ReviewOfBMST}
Binary BMST codes are convolutional codes with large constraint lengths~\cite{Ma13,Ma15}. Typically,
a binary BMST code of memory $m$ consists of a short code~(called the \emph{basic code}) and at most $m+1$ interleavers~\cite{Liang14c}. Let $\mathcal{C}[n,k]$ be the basic code defined by a $k \times n$ generator matrix $\bm{G}$ over the binary field $\mathbb{F}_2$. Denote $\bm{u}^{(0)}, \bm{u}^{(1)}, \cdots, \bm{u}^{(L-1)}$ as $L$ blocks of data to be transmitted, where $\bm{u}^{(t)} \in \mathbb{F}^k_2$ for $0 \leq t \leq L-1$. Then, the encoding output $\bm{c}^{(t)} \in \mathbb{F}^n_2$ at time $t$ can be expressed as~\cite{Liang14c}
\begin{equation}
\bm{c}^{(t)} = \bm{u}^{(t)}\bm{G}\bm{\varPi}_0 + \bm{u}^{(t-1)}\bm{G}\bm{\varPi}_1 + \cdots + \bm{u}^{(t-m)}\bm{G}\bm{\varPi}_m,
\end{equation}
where $\bm{u}^{(t)}$ is initialized to be $\mathbf{0} \in \mathbb{F}^k_2$ for $t<0$ and $\bm{\varPi}_0, \cdots, \bm{\varPi}_m$ are $m+1$ permutation matrices of order $n$.
For $L \leq t \leq L+m-1$, the zero message sequence $\bm{u}^{(t)} = \mathbf{0} \in \mathbb{F}^k_2$ is input into the encoder for termination.
Then, $\bm{c}^{(t)}$ is mapped to a signal vector $\bm{s}^{(t)}$ and transmitted over the channel, resulting in a received vector $\bm{y}^{(t)}$.
At the receiver, the decoder executes the sliding-window decoding~(SWD) algorithm to recover the transmitted data~$\bm{u}^{(0)}, \cdots, \bm{u}^{(L-1)}$~\cite{Ma13,Ma15}. Specifically, for an SWD algorithm with a decoding delay $d$, the decoder takes $\bm{y}^{(t)}, \cdots, \bm{y}^{(t+d)}$ as inputs to recover $\bm{u}^{(t)}$ at time $t+d$, which is similar to the window decoding~(WD) of the SC-LDPC codes~\cite{Lentmaier10,Iyengar12,Iyengar13}.
The structure of the BMST codes also admits a two-phase decoding (TPD) algorithm~\cite{Liang14c}, which can be used to reduce the decoding delay and to predict the performance in the extremely low bit-error-rate~(BER) region.
As discussed in~\cite{Ma15}, binary BMST codes have the following two attractive features.
\begin{enumerate}
\item Any code~(linear or nonlinear) can be the basic code as long as it has fast encoding algorithms and SISO decoding algorithms.
\item Binary BMST codes have a simple genie-aided lower bound when transmitted over AWGN channels using BPSK modulation, which shows that the maximum extra coding gain can approach $10\log_{10}(m+1)$~dB compared with the basic code. {\color{black}The tightness of this simple lower bound in the high SNR region under the SWD algorithm has been verified by both the simulation and the extrinsic information transfer~(EXIT) chart analysis~\cite{Huang15}}.
\end{enumerate}
Based on the above two facts, a general procedure has been proposed for constructing capacity-approaching codes at any
given target error rate~\cite{Liang14c}.
Suppose that we want to construct a binary BMST code of rate $R$ at a target BER of $p_{\rm target}$. First, we find a rate-$R$ short code~$\mathcal{C}$ as the basic code. Then, we can determine the encoding memory $m$ by \begin{equation}\label{eq:ComputeMemory}
m = \left\lceil 10^{\frac{\gamma_{\rm target} - \gamma_{\lim}}{10}}-1 \right\rceil,
\end{equation}
where $\gamma_{\rm target}$ is the minimum SNR for the code $\mathcal{C}$ to achieve the BER $p_{\rm target}$, $\gamma_{\lim}$ is the Shannon limit corresponding to the rate $R$, and $\left\lceil x \right\rceil$ stands for the minimum integer greater than or equal to $x$. Finally, by generating $m+1$ interleavers uniformly at random, the BMST code is constructed.
With this method, we have constructed a binary BMST code of memory $30$ using the Cartesian product of the R code $[2,1]^{5000}$, which has a predicted BER lower than $10^{-15}$ within one dB away from the Shannon limit.
\section{RUN Codes over Groups}\label{sec:CodesOverGroups}
\subsection{System Model and Notations}
Consider a symbol set $\mathcal{M} = \{0, 1, \cdots, q-1 \}$ and an $\ell$-dimensional signal constellation $\mathcal{A} \subset \mathbb{R}^\ell$ of size $q$. The symbol set $\mathcal{M}$ can be treated as a group by defining the operation $u \oplus w = (u+w) \mod q$ for $u, w \in \mathcal{M}$ . Let $\varphi$ be a (fixed) one-to-one mapping $\varphi: \mathcal{M} \rightarrow \mathcal{A}$. Let $u \in \mathcal{M}$ be a symbol to be transmitted.
For the convenience of performance analysis, instead of transmitting $\varphi(u)$ directly, we transmit a signal $s = \varphi(u \oplus w)$, where $w$ is a sample of a uniformly distributed random variable over $\mathcal{M}$ and assumed to be known at the receiver.
The received signal $y = s + z$, where $+$ denotes the component-wise addition over $\mathbb{R}^\ell$ and $z$ is an $\ell$-dimensional sample from
a zero-mean white Gaussian noise process with variance~$\sigma^2$ per dimension.
The SNR is defined as
\begin{equation}\label{eq:uncoded}
{\rm SNR} = \frac{\sum_{s\in\mathcal{A}}\|s\|^2}{\ell\sigma^2q},
\end{equation}
where $\|s\|^2$ is the squared Euclidean norm of $s$.
In this paper, for a discrete random variable $V$ over a finite set $\mathcal{V}$, we denote its {\em a priori message} and {\em extrinsic message} as $P^a_{V}(v), v \in \mathcal{V}$ and $P^e_{V}(v), v \in \mathcal{V}$, respectively. A SISO decoding is a process that takes {\em a priori} messages as inputs and delivers extrinsic messages as outputs. We assume that the information messages are independent and uniformly distributed~(i.u.d.) over $\mathcal{M}$.
\subsection{Repetition~(R) Codes}\label{subsec:SystemModel}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{RepetitionSystemModel.eps}\\
\vspace{-0.0cm}
\caption{\color{black}A message $u$ is encoded into $\bm{v} = (u,\cdots,u)$ and transmitted over AWGN channels.}
\label{fig:SystemModel}
\vspace{-0.0cm}
\end{figure}
{\color{black}Fig.~\ref{fig:SystemModel} shows the transmission of a message $u$ for $N$ times over AWGN channels.}
\subsubsection{Encoding} The encoder of an R code $\mathcal{C}[N, 1]$ over $\mathcal{M}$ takes as input a single symbol $u \in \mathcal{M}$ and delivers as output an $N$-dimensional vector $\bm{v} = \left( v_0, \cdots, v_{N-1} \right) = \left( u, \cdots, u \right)$.
\subsubsection{Mapping} The $j$-th component $v_j$ of the codeword $\bm{v}$ is mapped to the signal $s_j = \varphi( v_j {\oplus} w_j )$ for $j=0,\cdots,N-1$, where $\bm{w} = (w_0,\cdots,w_{N-1})$ is a random vector sampled from an i.u.d. process over $\mathcal{M}$.
\subsubsection{Demapping} Let $\bm{y} = (y_0, \cdots, y_{N-1})$ be the received signal vector corresponding
to the codeword $\bm{v}$. The {\em a priori} messages input to the decoder are computed as
\begin{equation}\label{eq:channelAPP}
P^a_{V_j}\left(v\right) \propto \exp\left(-\frac{\|y_j-\varphi(v \oplus w_j)\|^2}{2\sigma^2} \right), v \in \mathcal{M}
\end{equation}
for $j=0,\cdots,N-1$.
\subsubsection{Decoding}
The SISO decoding algorithm computes the {\em a posteriori} messages
\begin{equation}
P^e_{U}(u) \propto \prod_{0 \leq \ell \leq N-1} P^a_{V_\ell}(u), u \in \mathcal{M}
\end{equation}
for making decisions and the extrinsic messages
\begin{equation}\label{eq:RUN_SISO_decoding}
P^e_{V_j}(v) \propto \prod_{0 \leq \ell \leq N-1, \ell \neq j} P^a_{V_\ell}(v), v \in \mathcal{M}
\end{equation}
for $j=0,\cdots,N-1$ for iteratively decoding when coupled with other sub-systems.
\subsubsection{Complexity} Both the encoding/mapping and the demapping/decoding have linear computational complexity per coded symbol.
\subsubsection{Performance} Let $\hat{u}$ denote the hard decision output. The performance is measured by the symbol-error-rate~(SER) ${\rm SER} \triangleq \Pr\{ \hat{U} \neq U \} = \sum_{u \in \mathcal{M}} \frac{1}{q} \Pr\{ \hat{U} \neq U | U = u \}$. Define $e=\hat{u} \ominus u$, where $\ominus$ denotes the subtraction under modulo-$q$ operation. Due to the existence of the random vector $\bm{w}$, the peformance is irrelevant to the transmitted symbol $u$. We define
\begin{equation}\label{eq:DeX}
D_{e}\left( X \right) = \sum_{w \in \mathcal{M}} \frac{1}{q} X^{\|\varphi(w) - \varphi(e \oplus w)\|^2}
\end{equation}
as the average Euclidian distance enumerating function~(EDEF) corresponding to the error $e$, where $X$ is a dummy variable. Then, the average EDEF $B^{(N)}\left(X\right)$ for the R code $\mathcal{C}[N,1]$ over all possible messages $u$ and all possible vectors $\bm{w}$ can be computed as
\begin{align}
&B^{(N)}(X) \nonumber \\
&= \sum_{e \in \mathcal{M}} \sum_{\bm{w} \in \mathcal{M}^N} \frac{1}{q^N} \sum_{u \in \mathcal{M} } \frac{1}{q} X^{\sum\limits_{j=0}^{N-1}\|\varphi(u \oplus w_j) - \varphi(u \oplus e \oplus w_j)\|^2} \nonumber \\
&= \sum_{e \in \mathcal{M}} (D_{e}(X))^N \triangleq \sum_{\delta} B_{\delta}^{(N)}X^{\delta^2},
\end{align}
where $B_{\delta}^{(N)}$ denotes the average number of signal pairs $(\bm{s}, \hat{\bm{s}})$ with Euclidean distance $\delta$, $\bm{s} = \left( \varphi(u \oplus w_0), \cdots, \varphi(u \oplus w_{N-1}) \right)$ and $\hat{\bm{s}} = \left( \varphi(\hat{u} \oplus w_0), \cdots, \varphi(\hat{u} \oplus w_{N-1}) \right)$. The performance {\color{black}under the mapping $\varphi$} can be upper-bounded by the union bound as
\begin{equation}\label{eq:RcodeUnionBound}
{\rm SER} = f_{\varphi,N} ({\rm SNR}) \leq \sum_{\delta > 0} B^{(N)}_{\delta}{\rm Q}\left( \frac{\delta}{2\sigma} \right),
\end{equation}
where ${\rm Q}\left( \frac{\delta}{2\sigma} \right)$ is the pair-wise error probability with ${\rm Q}\left(x\right)\triangleq\int_{x}^{+\infty}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{z^2}{2}\right)dz$.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{SignalSetsAndMappingsNew.eps}\\
\vspace{-0.0cm}
\caption{Examples of signal constellations and mappings.}
\label{fig:SignalSetsAndMappings}
\vspace{-0.0cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{ScramblingRUNexample.eps}\\
\vspace{-0.0cm}
\caption{Performances and bounds of RUN codes. The ``rate'' in the legend of this figure~(or other similar figures in this paper) refers to the code rate. A rate-$R$ code over a $q$-ary constellation has a spectral efficiency of $R \log_2(q)$ in bits per symbol, at which the Shannon limit is determined.}
\label{fig:PerformanceRUNcodes}
\vspace{-0.0cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{DifferentMapping.eps}\\
\vspace{-0.0cm}
\caption{Performances and bounds of R codes with 4-PAM under different mappings.}
\label{fig:DifferentMapping}
\vspace{-0.0cm}
\end{figure}
{\color{black}
From the above derivation, we can see that the performance bounds of the R codes are related to the mapping $\varphi$.
In this paper, we consider as examples the BPSK, the signal set $\{-1,0,+1\}$~(denoted as $3$-PAM), $4$-PAM, $8$-ary phase-shift keying ($8$-PSK) modulation, $16$-ary quadrature amplitude modulation~($16$-QAM), or $16$-PAM, which are depicted in Fig.~\ref{fig:SignalSetsAndMappings} along with mappings denoted by $\varphi_0, \cdots, \varphi_9$ as specified in the figure.
Fig.~\ref{fig:PerformanceRUNcodes} and Fig.~\ref{fig:DifferentMapping} show performance bounds for several R codes defined with the considered constellations.
From the figures, we have the following observations.
\begin{enumerate}
\item The performance gap between the code $\mathcal{C}[N,1]$ and the uncoded transmission, when measured by the SNR instead of $E_b/N_0$, is roughly $10\log_{10}(N)$~dB.
\item
Given a signal constellation, mappings that are universally good for all R codes may not exist.
For example, as shown in Fig.~\ref{fig:DifferentMapping}, $\varphi_2$ is better than $\varphi_3$ for rate $1/63$~($N=63$) but becomes worse for rate $1/7$~($N=7$).
\end{enumerate}
}
\subsection{Time-Sharing}\label{subsecTimeSharing}
With repetition codes over groups, we are able to implement code rates $\frac{1}{N}$ for any given integer $N \geq 1$. To implement other code rates, we turn to the time-sharing technique.
To be precise, let $R = \frac{P}{Q}$ be the target rate. There must exist a unique $N \geq 1$ such that $\frac{1}{N+1} < \frac{P}{Q} \leq \frac{1}{N}$.
Then, we can implement a code by time-sharing between the code $\mathcal{C}[N+1, 1]$ and the code $\mathcal{C}[N, 1]$, which is equivalent to encoding $\alpha P$ information symbols with the code $\mathcal{C}[N+1, 1]$ and the remaining $(1-\alpha)P$ symbols with the code $\mathcal{C}[N, 1]$, where $\alpha = \frac{1}{R}-N$ is the time-sharing factor.
Apparently, to construct codes with rate $R>\frac{1}{2}$, we need time-sharing between the code $\mathcal{C}[2,1]$ and the uncoded transmission.
For this reason, we call this class of codes as \emph{RUN codes}, which consist of the R codes and codes obtained by time-sharing between the R codes and/or the uncoded transmission.
We denote a RUN code of rate $\frac{P}{Q}$ as $\mathcal{C}_{\rm RUN}[Q, P]$.
Replacing in Fig.~\ref{fig:SystemModel} the R codes with the RUN codes, we then have a coding system that can transmit messages with any given code rate over any given signal set.
\subsubsection{Encoding} Let $\bm{u} \in \mathcal{M}^P$ be the message sequence. The encoder of the code $\mathcal{C}_{\rm RUN}[Q, P]$ encodes the left-most $\alpha P$ symbols of $\bm{u}$ into $\alpha P$ codewords of $\mathcal{C}[N+1, 1]$ and the remaining symbols into $(1-\alpha)P$ codewords of $\mathcal{C}[N, 1]$.
\subsubsection{Decoding} The decoding is equivalent to decoding separately $\alpha P$ codewords of $\mathcal{C}[N+1, 1]$ and $(1-\alpha)P$ codewords of $\mathcal{C}[N, 1]$.
\subsubsection{Complexity} Both the encoding/mapping and the demapping/decoding have the same complexity as the R codes.
\subsubsection{Performance} The performance of the RUN code of rate $R=\frac{P}{Q}$ is given by
\begin{equation}
{\rm SER} = \alpha \cdot f_{\varphi,N+1}\left( {\rm SNR} \right) + (1-\alpha) \cdot f_{\varphi,N}\left( {\rm SNR} \right),
\end{equation}
which can be upper-bounded with the aid of (\ref{eq:RcodeUnionBound}).
Performances and bounds of several RUN codes defined with
{\color{black}BPSK modulation, $3$-PAM, $4$-PAM, $8$-PSK modulation, or $16$-QAM}
are shown in Fig.~\ref{fig:PerformanceRUNcodes} and Fig.~\ref{fig:DifferentMapping}. We notice that the union bounds with BPSK modulation are the exact performances, while those with other signal sets are upper bounds to the performances. We also notice that the upper bounds become tight as the SER is lower than $10^{-2}$ for all other signal sets.
{\color{black}Not surprisingly, the performances of the RUN codes are far away from the corresponding Shannon limits~(more than $5$~dB) at the SER lower than $10^{-2}$.
}
\section{BMST over Groups}\label{sec:BMSToverGroups}
\subsection{BMST Codes with RUN Codes As Basic Codes}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{encoderclip.eps}
\vspace{-0.0cm}
\caption{Encoding structure of a BMST-RUN code with memory $m$.}
\label{fig:encoder}
\vspace{-0.0cm}
\end{figure}
We have constructed a class of codes called RUN codes with any given code rate over groups. However, the RUN codes perform far away from the Shannon limits, as evidenced by the examples in Fig.~\ref{fig:PerformanceRUNcodes} {\color{black}and Fig.~\ref{fig:DifferentMapping}}. To remedy this, we transmit the RUN codes in the BMST manner as inspired by the fact that, as pointed out in~\cite{Ma15}, any short code can be embedded into the BMST system to obtain extra coding gain in the low error-rate region. The resulted codes are referred to as BMST-RUN codes. More precisely, we use the $B$-fold Cartesian product of the RUN code $\mathcal{C}_{\rm RUN}[Q,P]$~(denoted as $\mathcal{C}_{\rm RUN}[Q,P]^B$) as the basic code. Fig.~\ref{fig:encoder} shows the encoding structure of a BMST-RUN code with memory $m$, where \fbox{\small RUN} represents the basic encoder, \fbox{$\Pi_1$}, $\cdots$, \fbox{$\Pi_m$} represents $m$ symbol-wise interleavers, \fbox{$+$} represents the superposition with modulo-$q$ addition, and \fbox{$\varphi$} represents the mapping $\varphi$.
Let $\bm{u}^{(t)} \in \mathcal{M}^{PB}$ and $\bm{v}^{(t)} \in \mathcal{M}^{QB}$ be the information sequence and the corresponding codeword of the code $\mathcal{C}_{\rm RUN}[Q,P]^B$ at time $t$, respectively. Then, the sub-codeword $\bm{c}^{(t)}$ can be expressed as
\begin{equation}\label{eq:BMSTRUNencoding}
\bm{c}^{(t)} = \bm{v}^{(t)} \oplus \bm{w}^{(t,1)} \oplus \cdots \oplus \bm{w}^{(t,m)},
\end{equation}
where $\oplus$ denotes the symbol-wise modulo-$q$ addition, $\bm{v}^{(t)} = \mathbf{0} \in \mathcal{M}^{QB}$ for $t<0$ and $\bm{w}^{(t,i)}$ is the interleaved version of $\bm{v}^{(t-i)}$ by the $i$-th interleaver $\Pi_{i}$ for $i=1,\cdots,m$. Then, $\bm{c}^{(t)}$ is mapped to the signal vector $\bm{s}^{(t)} \in \mathcal{A}^{QB}$ symbol-by-symbol and input to the channel. After every $L$ sub-blocks of information sequence, we terminate the encoding by inputting $m$ all-zero sequences $\bm{u}^{(t)}=\mathbf{0} \in \mathcal{M}^{PB}(L \leq t \leq L+m-1)$ to the encoder. The termination will cause a code rate loss. However, the rate loss can be negligible as $L$ is large enough.
\subsection{Choice of Encoding Memory}\label{subsec:BMSToverGroups}
The critical parameter for BMST-RUN codes to approach the Shannon limits at a given target SER is the encoding memory $m$, which can be determined by the genie-aided lower bound.
Essentially the same as for the binary BMST codes~\cite{Ma15}, the genie-aided bound for a BMST-RUN code can be easily derived by assuming all but one sub-blocks $\left\{ \bm{u}^{(i)}, 0 \leq i \leq L-1, i \neq t \right\}$ are known at the receiver. With this assumption, the genie-aided system becomes an equivalent system that transmits the basic RUN codeword $m+1$ times. Hence the performance of the genie-aided system is the same as the RUN code obtained by time-sharing between the code $\mathcal{C}[(N+1)(m+1),1]$ and the code $\mathcal{C}[N(m+1),1]$.
As a result, the genie-aided bound {\color{black}under a mapping $\varphi$} is given by
\begin{equation}
\begin{aligned}
&{\rm SER} = f_{\rm \scriptstyle BMST-RUN}({\rm SNR}, m) \geq f_{\rm \scriptstyle genie}({\rm SNR}, m)& \\
&= \alpha \!\cdot\! f_{\varphi,(N\!+\!1)(m\!+\!1)}\left( {\rm SNR} \right) \!+ \! (1\!-\!\alpha) \!\cdot\! f_{\varphi,N(m\!+\!1)}\left( {\rm SNR} \right),&
\end{aligned}
\end{equation}
which can be approximated using the union bound in the high SNR region.
{\color{black}
Given a signal set $\mathcal{A}$ of size $q$ with labeling $\varphi$, a rate $R=P/Q$ and a target SER $p_{\rm target}$, we can construct a good BMST-RUN code using the following steps.}
\begin{enumerate}
\item {\color{black}Construct the code $\mathcal{C}_{\rm RUN}[Q,P]^B$ over the modulo-$q$ group by finding $N$ such that $\frac{1}{N+1} < \frac{P}{Q} \leq \frac{1}{N}$ and determining the time-sharing factor $\alpha$ between the R code~$[N+1,1]$ and the R code~$[N,1]$.
To approach the Shannon limit and to avoid error propagation, we usually choose $B$ such that $QB \geq 1000$.}
\item Find the Shannon limit $\gamma_{\lim}$ under the signal set $\mathcal{A}$ corresponding to the rate $R$.
\item \label{step:chooseMemory}{\color{black}Find an encoding memory $m$ such that $m$ is the minimum integer satisfying $f_{\rm \scriptstyle genie}(\gamma_{\lim}, m) \leq p_{\rm target}$.}
\item Generate $m$ interleavers of size $QB$ uniformly at random.
\end{enumerate}
\subsection{Decoding of BMST-RUN Codes}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{decoderclip.eps}
\vspace{-0.0cm}
\caption{The unified (high-level) normal graph of a BMST-RUN code with $L=4$ and $m=2$.}
\label{fig:decoder}
\vspace{-0.0cm}
\end{figure}
A BMST-RUN code can be decoded by an SWD algorithm with a decoding delay $d$ over its normal graph, which is similar to that of the binary BMST codes~\cite{Ma15}. Fig.~\ref{fig:decoder} shows the unified (high-level) normal graph of a BMST-RUN code with $L=4$ and $m=2$. {The normal graph can also be divided into \emph{layers}{\color{black}, each of which consists of four types of nodes. These nodes represent similar constraints to those for binary BMST codes and have similar message processing as outlined below.}
\begin{itemize}
\item The process at the node \fbox{\small RUN} is the SISO decoding of the RUN codes as described in Section~\ref{subsec:SystemModel}.
\item The process at the node \fbox{$=$} can be implemented in the same way as the message processing at a generic variable node of an LDPC code (binary or non-binary).
\item The process at the node \fbox{$+$} can be implemented in the same way as the message processing at a generic check node of an LDPC code (binary or non-binary).
\item The process at the node \fbox{$\Pi$} is the same as the original one, which interleaves or deinterleaves the input messages.
\end{itemize}
{\color{black}
Upon the arrival of the received vector $\bm{y}^{(t)}$~(corresponding to the sub-block $\bm{c}^{(t)}$) at time $t$, the SWD algorithm takes as inputs the \emph{a posterior} probabilities~(APPs) corresponding to $\bm{C}^{(t)}$ and uses the APPs corresponding to $\bm{C}^{(t-d)}, \cdots, \bm{C}^{(t)}$ to recover $\bm{u}^{(t-d)}$, where the computation of APPs
is similar to (\ref{eq:channelAPP}). After $\bm{u}^{(t-d)}$ is recovered, the decoder discards $\bm{y}^{(t-d)}$ and slides one layer of the normal graph to the ``right" to recover $\bm{u}^{(t-d+1)}$ with $\bm{y}^{(t+1)}$ received.
}
\section{Examples of BMST-RUN Codes}\label{sec:Examples}
\begin{table}[t]
\caption{Construction Examples of BMST-RUN Codes over AWGN Channels\label{tab:MemoryRequired}}
\centering
\begin{tabular}{p{0.95cm}p{0.15cm}p{0.9cm}cp{0.3cm}p{0.5cm}rrl}
\hline
\hline
\scriptsize $\mathcal{A}$ & \scriptsize $\frac{P}{Q}$ & \scriptsize \scriptsize $\left(\frac{1}{N+1}, \frac{1}{N}\right)$ & \scriptsize $\alpha$ & \scriptsize $B$ & \scriptsize $p_{\rm target}$ & \scriptsize $\gamma_{\lim}$ (dB) & \scriptsize \color{black} $m$ & \scriptsize \color{black} $\varphi$$^{*}$\\
\hline
\scriptsize BPSK & \scriptsize $\frac{1}{8}$ & \scriptsize $\left(\frac{1}{9}, \frac{1}{8}\right)$ & \scriptsize $0$ & \scriptsize $1250$ & \scriptsize $10^{-5}$ & \scriptsize $-7.2$ & \scriptsize $11$ & \scriptsize $\varphi_0$ \\
\scriptsize BPSK & \scriptsize $\frac{2}{8}$ & \scriptsize $\left(\frac{1}{5}, \frac{1}{4}\right)$ & \scriptsize $0$ & \scriptsize $1250$ & \scriptsize $10^{-5}$ & \scriptsize $-3.8$ & \scriptsize $10$ & \scriptsize $\varphi_0$ \\
\scriptsize BPSK & \scriptsize $\frac{3}{8}$ & \scriptsize $\left(\frac{1}{3}, \frac{1}{2}\right)$ & \scriptsize $\frac{2}{3}$ & \scriptsize $1250$ & \scriptsize $10^{-5}$ & \scriptsize $-1.6$ & \scriptsize $11$ & \scriptsize $\varphi_0$\\
\scriptsize BPSK & \scriptsize $\frac{4}{8}$ & \scriptsize $\left(\frac{1}{3}, \frac{1}{2}\right)$ & \scriptsize $0$ & \scriptsize $1250$ & \scriptsize $10^{-5}$ & \scriptsize $ 0.2$ & \scriptsize $8$ & \scriptsize $\varphi_0$\\
\scriptsize BPSK & \scriptsize $\frac{5}{8}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{3}{5}$ & \scriptsize $1250$ & \scriptsize $10^{-5}$ & \scriptsize $1.8$ & \scriptsize $10$ & \scriptsize $\varphi_0$\\
\scriptsize BPSK & \scriptsize $\frac{6}{8}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{1}{3}$ & \scriptsize $1250$ & \scriptsize $10^{-5}$ & \scriptsize $ 3.4$ & \scriptsize $7$ & \scriptsize $\varphi_0$\\
\scriptsize BPSK & \scriptsize $\frac{7}{8}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{1}{7}$ & \scriptsize $1250$ & \scriptsize $10^{-5}$ & \scriptsize $ 5.3$ & \scriptsize $5$ & \scriptsize $\varphi_0$\\
\hline
\scriptsize $3$-PAM & \scriptsize $\frac{1}{7}$ & \scriptsize $\left(\frac{1}{8}, \frac{1}{7}\right)$ & \scriptsize $0$ & \scriptsize $300$ & \scriptsize $10^{-4}$ & \scriptsize $-4.3$ & \scriptsize $7$ & \scriptsize $\varphi_1$ \\
\scriptsize $3$-PAM & \scriptsize $\frac{2}{7}$ & \scriptsize $\left(\frac{1}{4}, \frac{1}{3}\right)$ & \scriptsize $\frac{1}{2}$ & \scriptsize $300$ & \scriptsize $10^{-4}$ & \scriptsize $-0.5$ & \scriptsize $6$ & \scriptsize $\varphi_1$ \\
\scriptsize $3$-PAM & \scriptsize $\frac{3}{7}$ & \scriptsize $\left(\frac{1}{3}, \frac{1}{2}\right)$ & \scriptsize $\frac{1}{3}$ & \scriptsize $300$ & \scriptsize $10^{-4}$ & \scriptsize $2.1$ & \scriptsize $6$ & \scriptsize $\varphi_1$ \\
\scriptsize $3$-PAM & \scriptsize $\frac{4}{7}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{3}{4}$ & \scriptsize $300$ & \scriptsize $10^{-4}$ & \scriptsize $4.4$ & \scriptsize $6$ & \scriptsize $\varphi_1$ \\
\scriptsize $3$-PAM & \scriptsize $\frac{5}{7}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{2}{5}$ & \scriptsize $300$ & \scriptsize $10^{-4}$ & \scriptsize $6.5$ & \scriptsize $5$ & \scriptsize $\varphi_1$ \\
\scriptsize $3$-PAM & \scriptsize $\frac{6}{7}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{1}{6}$ & \scriptsize $300$ & \scriptsize $10^{-4}$ & \scriptsize $8.8$ & \scriptsize $3$ & \scriptsize $\varphi_1$ \\
\hline
\scriptsize $4$-PAM & \scriptsize $\frac{1}{7}$ & \scriptsize $\left(\frac{1}{8}, \frac{1}{7}\right)$ & \scriptsize $0$ & \scriptsize $200$ & \scriptsize $10^{-4}$ & \scriptsize $-3.1$ & \scriptsize $9$ & \scriptsize $\varphi_3$ \\
\scriptsize $4$-PAM & \scriptsize $\frac{2}{7}$ & \scriptsize $\left(\frac{1}{4}, \frac{1}{3}\right)$ & \scriptsize $\frac{1}{2}$ & \scriptsize $200$ & \scriptsize $10^{-4}$ & \scriptsize $0.9$ & \scriptsize $8$ & \scriptsize $\varphi_3$ \\
\scriptsize $4$-PAM & \scriptsize $\frac{3}{7}$ & \scriptsize $\left(\frac{1}{3}, \frac{1}{2}\right)$ & \scriptsize $\frac{1}{3}$ & \scriptsize $200$ & \scriptsize $10^{-4}$ & \scriptsize $3.8$ & \scriptsize $6$ & \scriptsize $\varphi_3$ \\
\scriptsize $4$-PAM & \scriptsize $\frac{4}{7}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{3}{4}$ & \scriptsize $200$ & \scriptsize $10^{-4}$ & \scriptsize $6.3$ & \scriptsize $7$ & \scriptsize $\varphi_3$ \\
\scriptsize $4$-PAM & \scriptsize $\frac{5}{7}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{2}{5}$ & \scriptsize $200$ & \scriptsize $10^{-4}$ & \scriptsize $8.7$ & \scriptsize $5$ & \scriptsize $\varphi_3$ \\
\scriptsize $4$-PAM & \scriptsize $\frac{6}{7}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{1}{6}$ & \scriptsize $200$ & \scriptsize $10^{-4}$ & \scriptsize $11.2$ & \scriptsize $3$ & \scriptsize $\varphi_3$ \\
\hline
\scriptsize $8$-PSK & \scriptsize $\frac{1}{5}$ & \scriptsize $\left(\frac{1}{6}, \frac{1}{5}\right)$ & \scriptsize $0$ & \scriptsize $150$ & \scriptsize $10^{-4}$ & \scriptsize $-2.8$ & \scriptsize $6$ & \scriptsize $\varphi_5$ \\
\scriptsize $8$-PSK & \scriptsize $\frac{2}{5}$ & \scriptsize $\left(\frac{1}{3}, \frac{1}{2}\right)$ & \scriptsize $\frac{1}{2}$ & \scriptsize $150$ & \scriptsize $10^{-4}$ & \scriptsize $1.3$ & \scriptsize $6$ & \scriptsize $\varphi_6$ \\
\scriptsize $8$-PSK & \scriptsize $\frac{3}{5}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{2}{3}$ & \scriptsize $150$ & \scriptsize $10^{-4}$ & \scriptsize $4.7$ & \scriptsize $6$ & \scriptsize $\varphi_6$ \\
\scriptsize $8$-PSK & \scriptsize $\frac{4}{5}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{1}{4}$ & \scriptsize $150$ & \scriptsize $10^{-4}$ & \scriptsize $8.1$ & \scriptsize $4$ & \scriptsize $\varphi_6$ \\
\hline
\scriptsize $16$-QAM & \scriptsize $\frac{239}{255}$ & \scriptsize $\left(\frac{1}{2}, 1\right)$ & \scriptsize $\frac{16}{239}$ & \scriptsize $4$ & \scriptsize $10^{-3}$ & \scriptsize $12.7$ & \scriptsize $2$ & \scriptsize $\varphi_7$ \\
\hline
\scriptsize uniformly spaced $16$-PAM & \scriptsize $\frac{1}{2}$ & \scriptsize $\left(\frac{1}{3}, \frac{1}{2}\right)$ & \scriptsize $0$ & \scriptsize $250$ & \scriptsize $10^{-3}$ & \scriptsize $12.5$ & \scriptsize $5$ & \scriptsize $\varphi_8$ \\
\scriptsize non-uniformly spaced $16$-PAM~\cite{Sun93} & \scriptsize $\frac{1}{2}$ & \scriptsize $\left(\frac{1}{3}, \frac{1}{2}\right)$ & \scriptsize $0$ & \scriptsize $250$ & \scriptsize $10^{-3}$ & \scriptsize $12.0$ & \scriptsize $5$ & \scriptsize $\varphi_9$ \\
\hline
\end{tabular}
{\color{black}
\begin{tablenotes
\footnotesize
\item{*} The mappings in this table are the same as those specified in Fig.~\ref{fig:SignalSetsAndMappings}.
Notice that the shaping gain of the non-uniformly spaced $16$-PAM is about $0.5$~dB.
\end{tablenotes}
}
\end{table}
In this section, we present simulation results of several BMST-RUN codes over AWGN channels {\color{black}and Rayleigh flat fading channels}, where code parameters are shown in Table~\ref{tab:MemoryRequired}.
For all simulations, the encoder terminates every $L=1000$ sub-blocks and the decoder executes the SWD algorithm with a maximum iteration number $18$. Without specification, the decoding delay $d$ of the SWD algorithm is set to be $3m$.
\subsection{BMST-RUN Codes with One-Dimensional Signal Sets}\label{subsec:OneDim}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{BPSK-BMSTRUNcapacity.eps} \\
\vspace{-0.0cm}
\caption{The required SNRs to achieve the SER of $10^{-5}$ for the BMST-RUN codes with the codes $\mathcal{C}_{\rm RUN}[Q,P]^{1250}(\frac{P}{Q}=\frac{1}{8}, \cdots, \frac{7}{8})$ as basic codes defined with BPSK modulation.}
\label{fig:BPSKcapacity}
\vspace{-0.0cm}
\end{figure}
Consider BMST-RUN codes of rates $\frac{K}{8}(K=1,\cdots,7)$ defined with BPSK modulation to approach the Shannon limits at the SER of $10^{-5}$. Fig.~\ref{fig:BPSKcapacity} shows the required SNRs for the BMST-RUN codes to achieve the SER of $10^{-5}$. Also shown in Fig.~\ref{fig:BPSKcapacity} is the channel capacity curve with i.u.d. inputs. It can be seen that the gaps between the required SNRs and the Shannon limits are within $1$~dB for all considered rates.
{\color{black}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Z3Z-BMSTRUN.eps}\\
\vspace{-0.0cm}
\caption{Performances of the BMST-RUN codes with the codes $\mathcal{C}_{\rm RUN}[Q,P]^{300}$ $(\frac{P}{Q}=\frac{1}{7}, \cdots, \frac{6}{7})$ as basic codes defined with $3$-PAM.}
\label{fig:3AMsnrser}
\vspace{-0.0cm}
\end{figure}
Consider BMST-RUN codes of rates $\frac{K}{7}(K\!=\!1,\!\cdots\!,\!6)$ defined with $3$-PAM to approach the Shannon limits at the SER of $10^{-4}$. Fig.~\ref{fig:3AMsnrser} shows the SER performance curves for all codes together with their lower bounds and the corresponding Shannon limits. We can see that the performance curves match well with the corresponding lower bounds for all codes in the high SNR region. In addition, all codes have an SER lower than $10^{-4}$ at the SNR within $1$ dB away from the corresponding Shannon limits, which is similar to the BPSK modulation case.}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{16AM-BMSTRUN-shapingVSuniform.eps}\\
\vspace{-0.0cm}
\caption{\color{black}Comparison of the BMST-RUN code with the code $\mathcal{C}_{\rm RUN}[2,1]^{250}$ as the basic code defined with two distinct $16$-PAM constellations under the mapping $\varphi_8$ and $\varphi_9$ in Fig.~\ref{fig:SignalSetsAndMappings}.}
\label{fig:16AMsnrser}
\vspace{-0.0cm}
\end{figure}
{\color{black}Consider a rate-$\frac{1}{2}$ BMST-RUN code of memory $5$ defined
over two distinct $16$-PAM constellations, where one consists of uniformly spaced signal points (under the mapping $\varphi_8$ in Fig.~\ref{fig:SignalSetsAndMappings}) and the other consists of non-uniformly spaced signal points (under the mapping $\varphi_9$ in Fig.~\ref{fig:SignalSetsAndMappings}) as designed in~\cite{Sun93}.
The SER performance curves with a decoding delay $d=15$ together with the lower bounds and the Shannon limits are shown in Fig.~\ref{fig:16AMsnrser}. From the figure, we can see that the BMST-RUN code has an SER lower than $10^{-3}$ at the SNR about $1.0$ away from their respective Shannon limits for both uniformly spaced signal points and non-uniformly spaced signal points. In addition,
the BMST-RUN code with non-uniformly spaced signal points performs about $0.5$~dB better than that with uniformly spaced signal points and also has a lower error floor.}
\subsection{BMST-RUN Codes with Two-Dimensional Signal Sets}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{8PSK-BMSTRUN.eps}\\
\vspace{-0.0cm}
\caption{Performances of the BMST-RUN codes with the codes $\mathcal{C}_{\!\rm RUN\!}[Q\!,\!P]^{\!150\!}(\!\frac{P}{Q}\!=\!\frac{1}{5}, \!\cdots\!, \!\frac{4}{5}\!)$ as basic codes defined with $8$-PSK modulation.}
\label{fig:8PSKsnrser}
\vspace{-0.0cm}
\end{figure}
Consider BMST-RUN codes of rates $\frac{K}{5}(K\!=\!1,\!\cdots\!,\!4)$ defined with $8$-PSK modulation to approach the Shannon limits at the SER of $10^{-4}$. Fig.~\ref{fig:8PSKsnrser} shows the SER performance curves for all codes together with their lower bounds and the corresponding Shannon limits.
{\color{black}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{16QAM-BMSTRUNmodulo.eps}\\
\vspace{-0.0cm}
\caption{\color{black}Performance of the BMST-RUN code with the code $\mathcal{C}_{\rm RUN}[255,239]^{4}$ as the basic code defined with $16$-QAM, where the mapping is $\varphi_7$ in Fig.~\ref{fig:SignalSetsAndMappings}.}
\label{fig:16QAMsnrser}
\vspace{-0.0cm}
\end{figure}
Consider a BMST-RUN code of rate $\frac{239}{255}$ defined with $16$-QAM~(under the mapping $\varphi_7$ in Fig.~\ref{fig:SignalSetsAndMappings}) to approach the Shannon limit at the SER of $10^{-3}$, where an encoding memory $m=2$ is required. The SER performance curves with decoding delays $d=6$ and $20$ together with the lower bound and the Shannon limit are shown in Fig.~\ref{fig:16QAMsnrser}. Since a large fraction of information symbols~($\frac{223}{239}$) are uncoded in the basic code, a large decoding delay $d=10m=20$ is required to approach the lower bound. With the decoding delay $d=20$, the BMST-RUN code achieves the SER of $10^{-3}$ at the SNR about $1$~dB away from the Shannon limit.
{\color{black}From the above two examples, we can see that BMST codes with two-dimensional signal constellations behave similarly as they do with one-dimensional signal constellations.
}
}
{\color{black}
\subsection{Comparison with BMST-BICM}\label{subsec:Comparison}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{4AM-BMSTRUNvsBICM.eps}\\
\vspace{-0.0cm}
\caption{\color{black}Performance of the BMST-RUN codes with the codes $\mathcal{C}_{\rm RUN}[7,K]^{200}(K\!=\!1,\!\cdots\!,\!6)$ over the modulo-$4$ group and the BMST-BICM scheme with the codes $\mathcal{C}_{\rm RUN}[7,K]^{400}(K\!=\!1,\!\cdots\!,\!6)$ over $\mathbb{F}_2$ as basic codes, where both schemes are under $4$-PAM with the mapping $\varphi_3$ in Fig.~\ref{fig:SignalSetsAndMappings}.}
\label{fig:4AMsnrser}
\vspace{-0.0cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{4AM-BMSTRUNvsBICMcapacity.eps}\\
\vspace{-0.0cm}
\caption{\color{black}The required SNRs to achieve the BER of $10^{-4}$ {\color{black}over AWGN channels} for the BMST-RUN codes with the codes $\mathcal{C}_{\rm RUN}[7,K]^{200}(K\!=\!1,\!\cdots\!,\!6)$ over the modulo-$4$ group and the BMST-BICM scheme with the codes $\mathcal{C}_{\rm RUN}[7,K]^{400}(K\!=\!1,\!\cdots\!,\!6)$ over $\mathbb{F}_2$ as basic codes, where both schemes are under $4$-PAM with the mapping $\varphi_3$ in Fig.~\ref{fig:SignalSetsAndMappings}.}
\label{fig:4AMsnrcapacity}
\vspace{-0.0cm}
\end{figure}
\color{black}
The examples in the previous subsections suggest that the proposed construction is effective for a wide range of code rates and signal sets. Also, the SWD algorithm is near-optimal in the high SNR region.
Since binary BMST codes also have such behaviors and can be combined with different signal sets~\cite{Liang14}, we need clarify the advantage of BMST-RUN codes over groups. Some advantages have been mentioned in the Introduction. In this subsection, we will show that the BMST-RUN codes can perform better than the BMST-BICM scheme.
\color{black}
\color{black}
To make a fair comparison, we have the following settings.
\begin{itemize}
\item For the BMST-BICM scheme, the basic codes are the RUN codes $[7,K]^{400}(K\!=\!1,\!\cdots\!,\!6)$ over $\mathbb{F}_2$, while for the BMST-RUN codes, the basic codes are the RUN codes $[7,K]^{200}(K\!=\!1,\!\cdots\!,\!6)$ over the modulo-$4$ group. Such setting ensures that both schemes have the same sub-block length {\color{black}$2800$ in bits}.
\item Both the BMST-RUN codes and the BMST-BICM scheme use the $4$-PAM with the mapping $\varphi_3$ in Fig.~\ref{fig:SignalSetsAndMappings}.
\item For a specific code rate, the BMST-BICM scheme has the same encoding memory and the same decoding delay as the BMST-RUN code. The encoding memories are presented in Table~\ref{tab:MemoryRequired}, while the decoding delay is set to be $3m$ for an encoding memory $m$.
\end{itemize}
Since the performance of the BMST-BICM scheme can not be measured in SER, we compare the performance in BER.
Fig.~\ref{fig:4AMsnrser} shows the BER performance curves for both the BMST-RUN codes~(denoted as ``RUN") and the BMST-BICM scheme~(denoted as ``BICM") together with the Shannon limits.
Fig.~\ref{fig:4AMsnrcapacity} shows the required SNRs to achieve the BER of $10^{-4}$ for both the BMST-RUN codes and the BMST-BICM scheme together with capacity curve of $4$-PAM under i.u.d. inputs.
From these two figures, we have the following observations.
\begin{itemize}
\item With the same encoding memory and decoding delay, the BMST-RUN codes achieve a lower BER than the BMST-BICM scheme for all considered code rates.
\item The BMST-RUN codes perform better than the BMST-BICM scheme in the lower code rate region and have a similar performance as the BMST-BICM scheme in the high code rate region.
\end{itemize}
}
\subsection{\color{black}BMST-RUN Codes over Rayleigh Channels}\label{subsec:Rayleigh}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{4AM-BMSTRUN-Rayleigh-capacity-graymapping.eps} \\
\vspace{-0.0cm}
\caption{\color{black}The required SNRs to achieve the SER of $10^{-4}$ for the BMST-RUN codes with the codes $\mathcal{C}_{\rm RUN}[Q,P]^{200}(\frac{P}{Q}=\frac{1}{7}, \cdots, \frac{6}{7})$ as basic codes defined with 4-PAM modulation~(under the mapping $\varphi_3$ in Fig.~\ref{fig:SignalSetsAndMappings}) over Rayleigh flat fading channels.}
\label{fig:4AMRayleighcapacity}
\vspace{-0.0cm}
\end{figure}
{\color{black}It has been shown that BMST-RUN codes perform well over AWGN channels and are comparable to binary BMST codes with BICM. More interestingly and importantly, BMST construction is also applicable to other ergodic channels. Here, we give an example for fading channels as an evidence.}
{\color{black}
Consider BMST-RUN codes of rates $\frac{K}{7}(K=1,\cdots,6)$ defined with 4-PAM modulation~(under the mapping $\varphi_3$ in Fig.~\ref{fig:SignalSetsAndMappings}) over Rayleigh flat fading channels. To approach the Shannon limits at the SER of $10^{-4}$, \color{black}the required encoding memories for rates $\frac{1}{7},\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},$ and $\frac{6}{7}$ are $7,7,6,7,5,$ and $4$, respectively. Fig.~\ref{fig:4AMRayleighcapacity} shows the required SNRs for the BMST-RUN codes to achieve the SER of $10^{-4}$. Also shown in Fig.~\ref{fig:4AMRayleighcapacity} is the channel capacity curve with i.u.d. inputs. It can be seen that the gaps between the required SNRs and the Shannon limits are about $1$~dB for all rates, which is similar to the case for AWGN channels.}
\section{Conclusions}\label{sec:Conclusion}
In this paper, by combining the block Markov superposition transmission~(BMST) with {\color{black}the RUN codes over groups,} we have proposed a simple scheme called BMST-RUN codes to approach the Shannon limits at any target symbol-error-rate~(SER) with any given (rational) rate over any alphabet (of moderate size). We have also derived the genie-aided lower bound for the BMST-RUN codes. Simulation results have shown that the BMST-RUN codes have a similar behavior to the binary BMST codes and have good performance for a wide range of code rates over {\color{black}both} AWGN channels {\color{black}and Rayleigh flat fading channels}. {\color{black}Compared with the BMST with bit-interleaved coded modulation~(BMST-BICM) scheme, the BMST-RUN codes are more flexible, which {\color{black}can be combined} with signal sets of any size. In addition, with the same encoding memory, the BMST-RUN codes have a better performance than the BMST-BICM scheme under the same decoding latency.}
\section*{Acknowledgment}
The authors wish to thank Mr. Kechao Huang and Mr. Jinshun Zhu for useful discussions.
|
1,314,259,996,737 | arxiv |
\section{Introduction}
\label{sec:intro}
With the advancement of sensor-related technologies in the field of data science and big-data analytics, `veracity' is becoming one of the most important V's, along with velocity, variety and volume. In a recent article, \cite{luko14} mentioned how the data source, data collection technologies and data processing methodologies could induce bias, ambiguity and noise in real-world big-data applications. Practitioners often not only want to detect the noisy observations but also want to assign a reliability score to each of the observations, which efficiently indicates the `relative' magnitude of the associated noise with it. For example, studies conducted by \cite{evans97} show that dynamic display of veracity of the contents can help the map-users (e.g. in google maps) to take advantage of the reliability information in making decisions. Though there are some works on veracity analysis in media and communications (for example, see \citealt{conroy15}; \citealt{rendon18} etc.) reliability assessment in the analysis of spatial data is not common in literature. Standard geostatistical analysis of the corrupted observations without taking the noisy nature of the data into account can result in erroneous inference and prediction. In a recent study, \cite{chak} introduced a reliability metric, namely veracity score (VS), to assess the credibility of the varying-quality crowdsourced observations in a geostatistical setting. In their analysis of mobile-sensor-generated noisy weather data, the authors have used a `high-quality' reference data -- coming from the ground weather stations -- to define the veracity scores (VS) of the noisy observations.
However, often in practice, such high-quality reference data is not available. In this paper, we consider the VS when there is no reference data available.
Although \cite{chak} have proposed veracity scoring based on `local' summaries from the noisy data, statistical properties of the VS and corresponding VS-based estimators remain unexplored. In this paper, we investigate statistical properties of the VS-based methodology under fairly general conditions on the underlying spatial process. We assume an additive-multiplicative noise structure for the corrupted observations. In contrast to the more commonly used i.i.d. additive measurement errors (see e.g., \citealt{diggle10}), the underlying noise structure here is non-stationary due to the presence of the multiplicative component. Assuming a suitable spatial asymptotic framework (e.g. mixed-increasing domain, \citealt{hall94}), we establish the consistency of the VS-based regression parameter estimator. The key result used here is a simplified asymptotic representation of the VS based on an extension of the Ghosh-Bahadur representation of sample quantiles of irregularly spaced spatial data. In addition, we show that the MSE of the VS-based estimator does not depend on the noise variances associated with the `bad' observations asymptotically. This explains the empirical observation on robustness of the VS-based estimator against the noise inherent in crowdsourced spatial data as reported in \cite{chak}. We also considered asymptotic behavior of the MSE of the OLS estimator. We show that the MSE of the OLS estimator is bounded below by a quantity that involves the sum of the associated noise variances. In situations where the combined noise level of the `bad' observations is high, accuracy of the OLS estimator can be substantially inferior to the VS-based estimator.
Next, we evaluate finite sample properties of VS-based estimation though extensive simulations. As the proportion of noisy observations increases from $5\%$ to $20\%$, the relative efficiency (defined as ratio of MSEs) of the VS-based estimators of the regression parameters increases from $145\%$ to more than $300\%$ for a sample of size $500$ (see Table~\ref{tab:RegParamMSE}). Similar advantages of the VS-based estimators are noted at varying levels of the additive and multiplicative noise variances. We also carried out a comparative numerical analysis with the robust-REML method, a competing robust geostatistical technique proposed by \cite{kunsch11} (also, see \citealt{georob18}, \citealt{georobMan}). The VS-based technique outperforms the robust-REML method both in terms of scalability and statistical accuracy (see Table~\ref{tab:VSvsGR}). Finally, we applied the VS-based analysis on a real data set containing measurements of mass percentages of ash in coal seams. The VS-based methodology identified all of the outliers previously detected in \cite{cressie93} and \cite{georobMan}'s analysis of the coal ash data. More importantly, our analysis indicates the presence of spatial correlation (see Table~\ref{tab:coalashCovEst}) while the earlier analyses completely failed to capture the spatial dependence. Incorporation of the spatial dependence in the VS-based analysis led to comparable or much smaller point-wise prediction errors (indicated by the sizes of the circles in Figure~\ref{fig:re-vs-gr}b).
The organization of the rest of the paper is as follows. In Section~\ref{sec:vs}, we provide the background, introduce some notation, and briefly review the VS-based methodology. Section~\ref{sec:AsympVS} defines the theoretical framework, states the assumptions and the results regarding asymptotic properties of the VS-based as well as the OLS estimators. In Section~\ref{sec:SimStudy}, we summarize the simulation design and simulation results for evaluating the VS-based methodology. Section~\ref{sec:readl-dat-ex} provides the details of the real data example. Finally, in Section~\ref{sec:conclusion}, we summarize our findings and future directions to this work. Proofs of the main results of this paper are given in Appendix~\ref{sec:proofs}. A supplementary material of this paper is available which contains the proofs of the auxiliary results and additional simulation results.
\section{Veracity Score (VS) Methods}
\label{sec:vs}
In this section, we briefly review the VS-based method in geostatistics. Before that, here we provide the background and a brief recap of some of the preliminary notation introduced in \cite{chak}.
Let $\LP Y(\mbox{${\mathbf{s}}$}): \mbox{${\mathbf{s}}$} \in \mathcal{D} \subset \mbox{${\mathbb{R}^{2}}$} \right\}$ be the spatial process of our interest. We assume a spatial regression model, i.e. $\LP Y(\mbox{${\mathbf{s}}$}) \right\}$ has a decomposition of the form
\begin{equation}
\label{eq:LinMod}
\begin{split}
Y(\mbox{${\mathbf{s}}$}) = \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$})^\prime \mbox{$\boldsymbol{\beta}$} + \epsilon(\mbox{${\mathbf{s}}$}),
\end{split}
\end{equation} where $\mbox{${\mathbf{x}}$}(\cdot) = \left( x_1(\cdot),...,x_p(\cdot) \right)^\prime$ is the $p$-dimensional deterministic vector process of known covariates, $\mbox{$\boldsymbol{\beta}$}$ is the unknown regression parameter vector, and $\LP \epsilon(\mbox{${\mathbf{s}}$}) \right\}$ is a spatially correlated residual process. We assume that $\LP \epsilon(\cdot) \right\}$ is an intrinsically stationary process with an admissible parametric variogram function $
2\gamma(\mbox{${\mathbf{h}}$}; \mbox{$\boldsymbol{\theta}$}) = \text{Var}\LP \epsilon(\mbox{${\mathbf{s}}$}) - \epsilon(\mbox{${\mathbf{s}}$} + \mbox{${\mathbf{h}}$}) \right\}
$ where $\mbox{$\boldsymbol{\theta}$}$ is the covariance parameter of interest. Under second-order stationarity, it can be shown that $\gamma(\mbox{${\mathbf{h}}$}) = C(\boldsymbol{0}) - C(\mbox{${\mathbf{h}}$}) $ where $C(\mbox{${\mathbf{h}}$}) = \text{Cov}\left( \epsilon(\mbox{${\mathbf{s}}$}), \epsilon(\mbox{${\mathbf{s}}$} + \mbox{${\mathbf{h}}$}) \right)$ is the covariance function of the $\epsilon$-process. For more details about variograms and covariograms see Chapter 2 in \cite{cressie93}.
Now, instead of observing $\LP Y(\mbox{${\mathbf{s}}$}_1), \dots , Y(\mbox{${\mathbf{s}}$}_n) \right\}$ at locations (can be irregularly space) $\mathcal{S}_n = \LP \mbox{${\mathbf{s}}$}_1, \dots , \mbox{${\mathbf{s}}$}_n \right\}$, we observe corrupted observations $\LP Z(\mbox{${\mathbf{s}}$}_1), \dots , Z(\mbox{${\mathbf{s}}$}_n) \right\}$. We write the corrupted observations as,
\begin{equation} \label{eq:genMod_Z}
\begin{split}
Z(\mbox{${\mathbf{s}}$}_i) = \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$}_i)^\prime \mbox{$\boldsymbol{\beta}$} + w(\mbox{${\mathbf{s}}$}_i),
\end{split}
\end{equation} where $\LP w(\mbox{${\mathbf{s}}$}) \right\}$ is a mean-zero spatially correlated process, possibly nonstationary, which contains noise in addition to the small-scale local variations induced by $\epsilon(\mbox{${\mathbf{s}}$})$. For example, as mentioned by \cite{chak}, let the corruption is coming through an additive-multiplicative noise model, i.e.
\begin{equation} \label{eq:AddMultNoiseMod}
\begin{split}
Z(\mbox{${\mathbf{s}}$}_i) = \epsilon_{M_i} Y(\mbox{${\mathbf{s}}$}_i) + \epsilon_{A_i}(\mbox{${\mathbf{s}}$}_i),
\end{split}
\end{equation} where $\epsilon_{M_i}$ and $\epsilon_{A_i}$ are the randomm variables corresponding to the multiplicative and additive noise parts. Then the $w$-process can be written as $w(\mbox{${\mathbf{s}}$}_i) = \epsilon_{M_i}(\mu(\mbox{${\mathbf{s}}$}_i) - 1) + \epsilon_{M_i}\epsilon(\mbox{${\mathbf{s}}$}_i) + \epsilon_{A_i}$. In case there is no multiplicative component, $w(\mbox{${\mathbf{s}}$}_i) = \epsilon(\mbox{${\mathbf{s}}$}_i) + \epsilon_{A_i}$. Observe that existance of the multiplicative component make the noisy residuals $\LP w(\mbox{${\mathbf{s}}$}) \right\}$ dependent on the location $\mbox{${\mathbf{s}}$}$ and hence non-stationary in nature.
\subsection{Formulation of VS}
\label{subsec:Formulation-VS}
Denote a square $\delta$-neighborhood ($\delta > 0$) around location $\mbox{${\mathbf{s}}$}$ as $\mathcal{B}_{\delta}(\mbox{${\mathbf{s}}$})$, i.e. $\mathcal{B}_{\delta}(\mbox{${\mathbf{s}}$}) = (\mbox{${\mathbf{s}}$} - \delta, \mbox{${\mathbf{s}}$} + \delta]$, where the subtractions are component-wise. Then the VS of the observation $Z(\mbox{${\mathbf{s}}$}_i)$ is defined as \citep{chak},
\begin{equation}\label{eq:VerSc_woRef}
\begin{split}
V(\mbox{${\mathbf{s}}$}_i) = \phi\left( \frac{\lvert Z(\mbox{${\mathbf{s}}$}_i) - \mathcal{C}(\mathbf{Z}_i) \rvert}{\alpha + D(\mathbf{Z}_i)}\right),
\end{split}
\end{equation}
where $\phi : \mbox{${\mathbb{R}}$}^+\cup \LP 0 \right\} \to \mbox{${\mathbb{R}}$}^+ \cup \LP 0 \right\}$ is some non-increasing bounded above function, referred as \textit{veracity function}; $\alpha \geq 0$ is a regularity parameter, called the \textit{baseline deviation}; $\mathbf{Z}_i = \left( Z(\mbox{${\mathbf{s}}$}_{i_1}), \dots , Z(\mbox{${\mathbf{s}}$}_{i_{n(i)}})\right)^\prime$ where $\{ \mbox{${\mathbf{s}}$}_{i_1}, \dots , \mbox{${\mathbf{s}}$}_{i_{n(i)}} \}$ is the set of observation locations in the small $\delta$-neighborhood $\mathcal{B}_\delta(\mbox{${\mathbf{s}}$}_i)$; and finally, $\mathcal{C}(\mathbf{x})$ is a measure of central tendency and $D(\mathbf{x})$ is a measure of dispersion of the values in the vector $\mathbf{x}$. Clearly, the definition of VS given in Equation~\ref{eq:VerSc_woRef} measures the amount of deviation of the observation $Z(\mbox{${\mathbf{s}}$}_i)$ from the `local' summary $\mathcal{C}(\mathbf{Z}_i)$ relative to the `local' variation $D(\mathbf{Z}_i)$. The veracity function $\phi(\cdot)$ being a non-increasing function, larger the value of the scaled deviation $\frac{\lvert Z(\mbox{${\mathbf{s}}$}_i) - \mathcal{C}(\mathbf{Z}_i) \rvert}{\alpha + D(\mathbf{Z}_i)}$ lower is the VS and for lower values of the scaled deviation, VS will be higher.
For the measure of center and dispersion we have considered the following two choices that are widely used by practitioners:
\begin{enumerate}
\item $\mathcal{C}(\mathbf{Z}_i) = Q_2(\mathbf{Z}_i)$ and $D(\mathbf{Z}_i) = \text{IQR}(\mathbf{Z}_i) = Q_3(\mathbf{Z}_i) - Q_1(\mathbf{Z}_i)$, where $Q_j(\mbox{${\mathbf{x}}$})$ denotes the $j$-th quartile of the observations in $\mbox{${\mathbf{x}}$}$, $j \in \LP 1, 2, 3, 4 \right\}$;
\item $\mathcal{C}(\mathbf{Z}_i) = \bar{Z}_{i_.} = \frac{1}{n(i)}\sum_{j = 1}^{n(i)} Z(\mbox{${\mathbf{s}}$}_{i_j})$, and $D(\mathbf{Z}_i) = \text{sd}(\mathbf{Z}_i) = \sqrt{\frac{1}{n(i)-1}\sum_{j} \left( Z(\mbox{${\mathbf{s}}$}_{i_j}) - \bar{Z}_{i_.} \right)^2}$.
\end{enumerate} Clearly, the first set of choices are based on sample quantiles and are expected to be more robust and efficient as compared to the sample mean and standard deviation in the analysis of noisy data (for details, see \citealt{sen68}). In this paper, we refer the VS with choice (a) as the measure of location and scale as `Medain-VS' and denote it by $V^{(m)}(\mbox{${\mathbf{s}}$}_i)$. Similarly, for choice (b) we denote the VS as $V^{(a)}(\mbox{${\mathbf{s}}$}_i)$ ($(a)$ for average) and refer it as `Mean-VS'. For the veracity function we have adopted the choice used by \cite{chak}, i.e. $\phi(x) = \exp(-x)$.
\subsection{VS-based estimation in spatial regression}
\label{subsec:VS-based-est}
Under the setting discussed above, as the covariance parameter is unknown, the standard practice is to estimate the regression parameter using \textit{ordinary least squares} (OLS): $
\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{ols}} = \left( \mathbf{X}^\prime \mathbf{X} \right)^{-1} \mathbf{X}^\prime\mathbf{Z},
$ where $\mathbf{X} = \left( \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$}_1), \dots , \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$}_n) \right)^\prime$. Next, the residuals from the OLS fit are used in \textit{weighted least squares}-based variogram estimation (for details, see page 90, Chapter 2, \citealt{cressie93}) to obtain the covariance parameter estimator $\hat{\mbox{$\boldsymbol{\theta}$}}_{\text{wls}}$. Once the covariance parameter is estimated, one can try to improve the mean parameter estimates by using \textit{estimated generalized least squares} (EGLS) estimator, given by $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{egls}} = \left( X^\prime \Sigma^{-1}(\hat{\mbox{$\boldsymbol{\theta}$}}_{\text{wls}}) X \right)^{-1} X^\prime \Sigma^{-1}(\hat{\mbox{$\boldsymbol{\theta}$}}_{\text{wls}}) \mathbf{Z}$, where $\Sigma(\hat{\mbox{$\boldsymbol{\theta}$}}_{\text{wls}})$ is the estimated covariance matrix. But, this last stage of updating the OLS fit introduces additional variability due to the estimation of the covariance parameters and hence not necessarily preferable than the OLS estimator. Finally, the estimated mean and covariance structure is used to predict the process at a new location $\mbox{${\mathbf{s}}$}_0$ using \textit{best linear unbiased predictor} or kriging predictor (for details, see Chapter 3, \citealt{cressie93}).
The standard approach of estimation and kriging is not robust in nature and hence, may produce erroneous inference and prediction in case there are outliers in the data. To estimate the mean parameter $\mbox{$\boldsymbol{\beta}$}$ robustly, instead of using ordinary squared error loss, in VS-based estimation, weighted squared error loss is used with VS as the corresponding weights \citep{chak} as shown below:
\begin{equation}\label{eq:VSEstimator}
\begin{split}
\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}} &= \underset{\mbox{$\boldsymbol{\beta}$}}{\text{argmin}} \; \sum_{i = 1}^n V(\mbox{${\mathbf{s}}$}_i)\left( Z(\mbox{${\mathbf{s}}$}_i)- \mathbf{x}(\mbox{${\mathbf{s}}$}_i)^\prime \mbox{$\boldsymbol{\beta}$} \right)^2\\
&= \left( \mbox{${\mathbf{X}}$}^\prime \mathbf{D}_v \mbox{${\mathbf{X}}$} \right)^{-1}\mbox{${\mathbf{X}}$}^\prime \mathbf{D}_v \mathbf{Z},
\end{split}
\end{equation} where $\mathbf{D}_v = \text{diag}( V(\mbox{${\mathbf{s}}$}_1), \dots $ $, V(\mbox{${\mathbf{s}}$}_n) )$. Clearly, as the VS is expected to be smaller for `bad' observations, the VS-based estimator given in Equation~\ref{eq:VSEstimator} is expected to be affected less by the high noise associated with those. If the VS can capture the noisy observations perfectly, i.e. $V(\mbox{${\mathbf{s}}$}_i) \approx 0$ if $Z(\mbox{${\mathbf{s}}$}_i)$ is contaminated and $V(\mbox{${\mathbf{s}}$}_i) = 1$ otherwise, then $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}$ is approximately equal to the OLS estimator computed only from the `noiseless' observations in the data.
Depending on the versions of VS used to compute the VS-based estimator, we consider two versions of the VS-based estimator in this article: `Mean-VS', denoted by $\hat{\mbox{$\boldsymbol{\beta}$}}^{(a)}_{\text{vs}}$, where the sample mean and standard deviations are used to assess veracity and `Median-VS', denoted by $\hat{\mbox{$\boldsymbol{\beta}$}}^{(m)}_{\text{vs}}$, with sample median and IQR in the definition of VS. Though the focus of this work is mainly on the properties of the `Median-VS' regression estimator, in the following sections we put some remarks about the properties and asymptotic behaviors of the `Mean-VS' version as well.
The de-trended observations, i.e. $\hat{\epsilon}_{\text{vs}}(\mbox{${\mathbf{s}}$}_i) = Z(\mbox{${\mathbf{s}}$}_i) - \mathbf{x}(\mbox{${\mathbf{s}}$}_i)^\prime \hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}$ for $i \in \LP1,2,...,n\right\}$ -- unlike the case of dealing with high-quality observations -- are not free of noise and hence, it is not reasonable to use these directly in the covariance parameter estimation. \cite{chak} proposed a VS-based smoothing of the residuals:
\begin{equation}\label{eq:NewRes_wRef}
\begin{split}
\tilde{\epsilon}(\mbox{${\mathbf{s}}$}_i) = V(\mbox{${\mathbf{s}}$}_i)^q \hat{\epsilon}_{\text{vs}}(\mbox{${\mathbf{s}}$}_i) + (1 - V(\mbox{${\mathbf{s}}$}_i)^q)Q_2(\mathbf{Z}_i - \mathbf{X}_i \hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}),
\end{split}
\end{equation} where $\mathbf{X}_i = \left( \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$}_{i_1}),...,\mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$}_{i_{n(i)}}) \right)^\prime$ is the $n(i)\times p$ matrix of the covariates corresponding to the observations in $\mathcal{B}_{\boldsymbol{\delta}}(\mbox{${\mathbf{s}}$}_i)$; and $q \geq 0$ is another regularity parameter which determines the smoothness of the residuals defined in Equation~\ref{eq:NewRes_wRef}. Next, these VS-based smoothed residual vector $\tilde{\mbox{$\boldsymbol{\epsilon}$}} = \left( \tilde{\epsilon}(\mbox{${\mathbf{s}}$}_1), \dots , \tilde{\epsilon}(\mbox{${\mathbf{s}}$}_n) \right)^\prime$ is used in least squares based variogram model fitting (for example, see \citealt{cressie80}) to get the VS-based covariance parameter estimator $\hat{\mbox{$\boldsymbol{\theta}$}}_{\text{vs}}$. Finally, the prediction of the $\epsilon$-process at a new location $\mbox{${\mathbf{s}}$}_0$, denoted by $\tilde{\epsilon}(\mbox{${\mathbf{s}}$}_0)$, is obtained using ordinary kriging with the estimated covariance parameters and the VS-based smoothed residuals as discussed in \cite{chak}.
In the next section, we have focused on the asymptotic behavior of the VS and VS-based regression parameter estimator. Though investigation of asymptotic properties VS-based covariance estimation is beyond the scope of this paper, we have evaluated the same using simulations in Section~\ref{sec:SimStudy}.
\section{Asymptotic properties of the VS-based regression estimator}
\label{sec:AsympVS}
Before going to the main results of this section we need to introduce some notation, specify the theoretical framework and state the regularity conditions. From this section onward, by $C(\cdot), C, C_1, C_2,\dots$ we denote constants with respect to the sample size $n$. For $d \geq 2$, we denote the volume of a set $A \subset \mbox{${\mathbb{R}^{d}}$}$ as $\lvert A \rvert$, i.e., the Lebesgue measure of $A$ if it has nonzero volume and the cardinality of $A$ if $A$ is finite. For two sequences of positive reals $\LP a_n \right\}_{n = 1}^\infty$ and $\LP b_n \right\}_{n = 1}^\infty$ we say one is \textit{of the order} of another (denoted as $a_n \sim b_n$) if $a_n/b_n \to C > 0$.
\subsection{Model specification}
\label{subsec:ModelSpec}
As discussed before, the process of our interest is denoted as $\LP Y(\mbox{${\mathbf{s}}$}) : \mbox{${\mathbf{s}}$} \in \mathcal{D} \subset \mbox{${\mathbb{R}^{2}}$} \right\}$ and instead of observing realizations from the `true' process, we observe corrupted observations $\LP Z(\mbox{${\mathbf{s}}$}_1), \dots , Z(\mbox{${\mathbf{s}}$}_n) \right\}$ where, the observations locations $\LP \mbox{${\mathbf{s}}$}_1, \dots , \mbox{${\mathbf{s}}$}_n \right\}$ are possibly irregularly spaced. The results of this paper are specific to the spatial regression model defined in Equation~\ref{eq:LinMod}. For the zero mean second-order stationary residual process $\LP \epsilon(\mbox{${\mathbf{s}}$}) : \mbox{${\mathbf{s}}$} \in \mathcal{D}\right\}$, let us denote the marginal distribution function of $\epsilon(\mbox{${\mathbf{s}}$})$ as $F_\epsilon(\frac{x}{\sigma_\epsilon})$ and, the spatial dependence structure can be formulated as,
$$
\text{Cov}\left( \epsilon(\mbox{${\mathbf{s}}$}_i), \epsilon(\mbox{${\mathbf{s}}$}_j) \right) = \begin{cases} \sigma_\epsilon^2 \; \rho_{\epsilon} \left( \mbox{${\mathbf{s}}$}_i - \mbox{${\mathbf{s}}$}_j \right) \;\; &\text{if} \;\; \mbox{${\mathbf{s}}$}_i \neq \mbox{${\mathbf{s}}$}_j\\
\sigma_\epsilon^2 \;\; &\text{otherwise.} \end{cases}
$$ Here, $F_\epsilon(\cdot)$ is a distribution function of a zero mean unit-variance random variable and $\rho_\epsilon(\cdot)$ is a non-negative definite function on $\mbox{${\mathbb{R}^{2}}$}$ with $\rho_\epsilon(\mathbf{0}) = 1$. To preserve the identifiability, we assume that the supports of $\LP \epsilon_{M_i}: i \in \LP 1, \dots , n \right\} \right\}$ are contained in $[0,\infty)$.
We further assume that among $\LP Z(\mbox{${\mathbf{s}}$}_1), \dots , Z(\mbox{${\mathbf{s}}$}_n) \right\}$ only a `small' portion is `bad-quality data' and the corrupted observations are coming through the additive-multiplicative noise model given in Equation~\ref{eq:AddMultNoiseMod}. We formulate this assumption as follows.
\begin{equation}
\label{eq:error_dist}
\begin{split}
\epsilon_{M_i} \sim \begin{cases}
\Delta(1) \;\; &\text{if}\;\; i \in G_n\\
F_M\left( \frac{\cdot - 1}{\sigma_M} \right) \;\; &\text{o.w.}
\end{cases}\; ;\;\;
\epsilon_{A_i} \sim \begin{cases}
\Delta(0) \;\; &\text{if}\;\; i \in G_n\\
F_A\left( \frac{\cdot}{\sigma_A} \right) \;\; &\text{o.w.}\; ,
\end{cases}
\end{split}
\end{equation} where, $\Delta(x)$ denotes a degenerate distribution with point mass at $-\infty < x < \infty$; $G_n \subset \LP 1, \dots , n \right\}$ is a subset of indices; and $\sigma_M, \sigma_A$ are positive constants. With this model, if $i \in G_n$, we have no noise associated with the observation, i.e., $Z(\mbox{${\mathbf{s}}$}_i) = Y(\mbox{${\mathbf{s}}$}_i)$. If $i \notin G_n$, then $Z(\mbox{${\mathbf{s}}$}_i) = \epsilon_{M_i} Y(\mbox{${\mathbf{s}}$}_i) + \epsilon_{A_i}$, where $\epsilon_{M_i}$ and $\epsilon_{A_i}$ have positive variances. Also, we have taken $\LP \epsilon_{M_i}\right\}_{i = 1}^n$, $\LP \epsilon_{A_i}\right\}_{i = 1}^n$ and $\LP \epsilon(\mbox{${\mathbf{s}}$}_i) \right\}_{i = 1}^n$ are independent of each other. We further assume that the proportion of ``good'' observations is a constant (w.r.t. $n$) denoted by $q_e$, i.e., $\lvert G_n \rvert /n \approx q_e$, and $1 - q_e$ is the proportion of noisy observations in the data. Observe that, the variance corresponding to $\epsilon_{M_i}$, $\sigma_{M_i} = \sigma_M$ if $i \notin G_n$ and $\sigma_{M_i} = 0$ if $i \in G_n$. Similar observation can be made for the additive noise variances (denoted by $\sigma_{A_i}$) as well.
Under the specified model we can rewrite our observations as,
\begin{equation}
\begin{split}
Z(\mbox{${\mathbf{s}}$}_i) = \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$}_i)^\prime \mbox{$\boldsymbol{\beta}$} + w(\mbox{${\mathbf{s}}$}_i),
\end{split}
\end{equation} where $E(w(\mbox{${\mathbf{s}}$}_i)) = 0$, $\mathrm{Var}(w(\mbox{${\mathbf{s}}$}_i)) = \sigma_i^2$ and,
$$
\begin{aligned}
\sigma_i^2 &= \begin{cases}
\sigma_\epsilon^2 \;\; &\text{if}\;\; i \in G_n\\
\sigma_\epsilon^2 + \tau_i^2\;\; &\text{o.w.},
\end{cases}
\end{aligned}
$$ where, $\tau_i^2 \equiv \tau(\mbox{${\mathbf{s}}$}_i)^2 = (\mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$}_i)^\prime \mbox{$\boldsymbol{\beta}$})^2 \sigma_M^2 + \sigma_\epsilon^2\sigma_M^2 + \sigma_A^2$ is the additional noise variance associated with the corrupted observation at location $\mbox{${\mathbf{s}}$}_i$. Clearly, if $i \in G_n$, $\tau_i^2 = 0$. We denote the marginal distribution function of $w(\mbox{${\mathbf{s}}$}_i)$ as $F_i(x)$ for $i \in \LP 1, \dots , n\right\}$, for any $n \geq 2$. Clearly, if $i \in G_n$, $w(\mbox{${\mathbf{s}}$}_i) =\epsilon(\mbox{${\mathbf{s}}$}_i)$ and $F_i(x) = F_\epsilon(x/\sigma_\epsilon)$ and if $i \in G_n^c$, we denote the marginal distribution function of $w(\mbox{${\mathbf{s}}$}_i)$ as $F_2(\frac{x}{\sigma_i})$, where $F_2(\cdot)$ is again a distribution function of a centered and scaled random variable. Without loss of any generality, we take the baseline deviation parameter $\alpha$ in the definition of VS to be equal to $0$. The results of this paper can be straightforwardly extended for any other finite constant (w.r.t. $n$) $\alpha > 0$.
\subsection{Spatial framework and notations}
We take the sampling region to be $\mathcal{D} \equiv \mathcal{D}_n = \lambda_n [0,1]^2$, i.e. a 2-dimensional square region with area $\lambda_n^2$ where $\LP \lambda_n \right\}$ is a sequence of positive reals such that $\lambda_n \to \infty$ as $n \to \infty$ and
\begin{equation}
\label{eq:CondnLambda}
\begin{split}
\lambda_n^{-1} + n^{-1} \lambda_n^{2} \to 0.
\end{split}
\end{equation} Under condition (\ref{eq:CondnLambda}), the spatial asymptotic framework we consider is similar to the \textit{mixed increasing-domain} asymptotic framework used in \cite{hall94}, \cite{lahiri99}, etc.The first component of (\ref{eq:CondnLambda}) states the domain has to increase with the sample size, and the second component allows the possibility of infilling sampling points as well. The results stated here are not particular to the shape and position of the rectangular region $\mathcal{D}_n$. Our results can be extended to more general shapes using the asymptotic framework where the region is obtained by inflating a prototype region contained in $(-1/2, 1/2]^2$ with center at the origin by a scaling constant as discussed in \cite{lahiri02}.
Now, we introduce some notation required to state the regularity conditions, the results as well as the proofs of this paper. For any $A \subset \mbox{${\mathbb{R}^{2}}$}$ and a random field $\LP T(\mbox{${\mathbf{s}}$}): \mbox{${\mathbf{s}}$} \in \mbox{${\mathbb{R}^{2}}$} \right\}$ let us denote $\mathcal{F}_T(A) = \sigma \langle T(\mbox{${\mathbf{s}}$}) : \mbox{${\mathbf{s}}$} \in A \rangle$, the $\sigma$-field generated by the random variables $\LP T(\mbox{${\mathbf{s}}$}) : \mbox{${\mathbf{s}}$} \in A \right\}$. For any $d \geq 2$ and for any $\mathbf{x} \in \mbox{${\mathbb{R}^{d}}$}$ the $L_1$ and $L_2$ norms are denoted by $\lvert \mathbf{x} \rvert$ and $\lVert \mathbf{x} \rVert$ respectively. For any two sets $A$ and $B$ in $\mbox{${\mathbb{R}^{2}}$}$ we denote the the distance between them as $d(A,B) = \inf \LP \lvert \mbox{${\mathbf{s}}$} - \mbox{${\mathbf{s}}$}^\prime \rvert : \mbox{${\mathbf{s}}$} \in A, \mbox{${\mathbf{s}}$}^\prime \in B \right\}$. For any random field $\LP T(\mbox{${\mathbf{s}}$}) : \mbox{${\mathbf{s}}$} \in \mbox{${\mathbb{R}^{2}}$} \right\}$ we define the strong-mixing coefficient as
\begin{equation}
\label{eq:MixingCoef}
\begin{split}
\alpha_T (u, v) = \sup \LP \tilde{\alpha}_T(A, B) : \; d(A,B) \geq u; \;\; \lvert A \rvert \leq v, \;\; \lvert B \rvert \leq v \right\},
\end{split}
\end{equation} where the supremum is taken over the set of all 2-dimensional rectangles $A$, $B$ in $\mbox{${\mathbb{R}^{2}}$}$, $u > 0$, $v > 0$; and
$$
\tilde{\alpha}_T(A, B) = \sup \LP \lvert P(V_1 \cap V_2) - P(V_1) P(V_2) \rvert : V_1 \in \mathcal{F}_T(A), \; V_2 \in \mathcal{F}_T(B) \right\}.
$$ For a distribution function $F(\cdot)$, we define the inverse of it as $F^{-1}(p)= \text{inf}\LP x: F(x) \geq p \right\}$, for $p \in [0,1]$.
To definition of VS in Section~\ref{sec:vs} uses a square neighborhood $\mathcal{B}_{\delta}(\mbox{${\mathbf{s}}$}_i)$ around the spatial point $\mbox{${\mathbf{s}}$}_i$ with one side of $2\delta$ units. From this section we change the notation as: $\delta \equiv \delta_n$ and $\mathcal{B}_{\delta}(\cdot) \equiv \mathcal{B}_{\delta_n}(\cdot)$ to ensure that the size of the `local' neighborhood also varies with the sample size $n$. Let $M_{n(i)}^p$ be the smallest $p$-th sample quantile of $\LP w(\mbox{${\mathbf{s}}$}_{i_1}), \dots , w(\mbox{${\mathbf{s}}$}_{i_{n(i)}}) \right\}$, the residuals associated with the observations in the square-neighborhood $\mathcal{B}_{\delta_n}(\mbox{${\mathbf{s}}$}_i)$. We denote $\hat{F}_{n(i)}(x) = \LP \text{No. of $w(s_{i_j}) \leq x$}\right\}/n(i)$, the empirical distribution function of the realizations from the $w$-process in the neighborhood around $\mbox{${\mathbf{s}}$}_i$. Clearly, $M_{n(i)}^p = \hat{F}_{n(i)}^{-1}(p)$. By $\bar{F}_{n(i)}(x)$ we denote the distribution function defined as $\bar{F}_{n(i)}(x) = \left( \sum_{j = 1}^{n(i)} F_{i_j}(x) \right)/n(i)$, where $F_{i_j}$ is the distribution function of $w(\mbox{${\mathbf{s}}$}_{i_j})$. The smallest $p$-th quantile of the distribution function $\bar{F}_{n(i)}$ is denoted by $\xi_{n(i)}^p$, i.e. $\xi_{n(i)}^p = \bar{F}_{n(i)}^{-1}(p)$.
\subsection{Consistency of the VS-based regression parameter estimator}
\label{subsec:AsympCons}
Here we focus on consistency of the $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}^{(m)}$, the Median-VS version of VS-based estimator. The conditions we need for that are following.
\begin{itemize}
\item[(C.1)] Number of observations in any unit square in $\mathcal{D}_n$ is in between $[ C_1 \frac{n}{\lambda_n^2} , C_2 \frac{n}{\lambda_n^2} ]$ for some positive constants $C_1$ and $C_2$ such that $C_1 < C_2$.
\item[(C.2)] Number of non-noisy observations ($Z(\mbox{${\mathbf{s}}$}_i)$ such that $i \in G_n$) in any unit square of the sampling region is bounded below by $C_1 q_e n\lambda_n^{-2}$.
\item[(C.3)] The covariate process $\mathbf{x}(\mbox{${\mathbf{s}}$})$ is such that,
$\mathbf{x}(\mbox{${\mathbf{s}}$}) = \mathbf{x}_0(\lambda_n^{-1} \mbox{${\mathbf{s}}$})$ where $\mathbf{x}_0: [0,1]^2 \to \mbox{${\mathbb{R}^{p}}$}$ is a differentiable function with bounded partial derivatives in $(0,1)^2$ and $\frac{1}{n} \sum_{i=1}^n \lVert \mathbf{x}(\mbox{${\mathbf{s}}$}_i) \rVert^2 = O(1)$ for any set of $\LP \mbox{${\mathbf{s}}$}_1, \dots , \mbox{${\mathbf{s}}$}_n \right\} \subset \mathcal{D}_n$ and for any $n \geq 2$.
\item[(C.4)] $\LP \epsilon(\mbox{${\mathbf{s}}$}) : \; \mbox{${\mathbf{s}}$} \in \mbox{${\mathbb{R}^{2}}$} \right\}$ is a second-order stationary random field such that, $\int_{\mbox{${\mathbb{R}^{2}}$}} \; \lvert \rho_\epsilon \left( \mbox{${\mathbf{h}}$} \right) \rvert d\mbox{${\mathbf{h}}$} < \infty$; and for some $\kappa > 0$, $E \lvert\epsilon (\mathbf{0}) \rvert ^{2+\kappa} < \infty$ and $\alpha_{\epsilon}(u, v) \leq C u^{-\nu_1} v^{\nu_2}$ with $\nu_1 > 4/\kappa$ and $\nu_2 > 0$.
\item[(C.5)] $\delta_n^{-1} + \lambda_n^{-1} \delta_n + n^{-1/2} \lambda_n \delta_n^{-1} \to 0.$
\item[(C.6)] $\underset{n}{\text{sup}} \underset{i \in \LP1, \dots , n \right\}}{\text{sup}} \lvert \xi^p_{n(i)} \rvert < \infty$ for all $p \in \LP 0.25, 0.75 \right\}$.
\item[(C.7)] For all $i \in \LP 1, \dots , n \right\}$, $j \in \LP 1, \dots , n(i) \right\}$ and $p \in \LP 0.25, 0.5, 0.75 \right\}$, $F_{i_j}(\cdot)$ is absolutely continuous in an neighborhood of $\xi^p_{n(i)}$, i.e. $f_{i_j}(x) = \left( d/dx \right) F_{i_j}(x)$ exists in that neighborhood.
\item[(C.8)] For all $i \in \LP 1, \dots , n \right\},\;\;$ $0 < \underset{j \in \LP 1, \dots , n(i) \right\}}{\text{inf}}f_{i_j}(\xi^p_{n(i)}) \leq \underset{j \in \LP 1, \dots , n(i) \right\}}{\text{sup}}f_{i_j}(\xi^p_{n(i)}) < \infty$.
\item[(C.9)] $F_i^{-1}(0.5) = 0$ for all $i \in \LP 1, \dots , n \right\}$.
\end{itemize}
In a later remark we have discussed how the conditions would change if we have considered the Mean-VS estimator, $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}^{(a)}$. We now comment briefly on the conditions. (C.1) assumes that the sampling design has to be such that the number of observations in any unit square varies as the same rate as of $n/\lambda_n^2$, the `average' number of observations per unit square. In (C.2) we restrict the number of noisy observations in any unit square by $C_1(1 - q_e)n\lambda_n^{-2}$ which combining with (C.1) specifies that at most $1 - q_e$ proportion of the observations in an unit block can be noisy, i.e. corresponding noise variances (denoted by $\tau_i^2$) are non-zero. (C.3) puts regularity conditions on the covariates; it states that the covariate process is an `inflated' version of a `smooth' process $\mbox{${\mathbf{x}}$}_0(\mbox{${\mathbf{s}}$})$ in the prototype sampling region $[0,1]^2$, making $\mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$})$ a slowly varying process over the space. The other condition in (C.3) is standard in regression theory which puts a bound on the magnitude of the covariates as compared to the sample size. (C.4) states the required moment and mixing conditions on the residual process $\LP \epsilon(\mbox{${\mathbf{s}}$}) \right\}$. Standard Gaussian process with `nicely' decaying covariance functions, e.g. Expoenential, Mat\'ern, Gaussian, etc. satisfies this condition. (C.5) makes sure that the size of the $\delta_n$-neighborhoods used to compute the VS increases with $n$ resulting in increasing number of observations for VS computation. But, at the same time the second term of condition (C.5) states that the block has to be small enough as compared to the whole sampling region retaining the `local' feature of VS. The third term in (C.5) states that the rate at which the samples are infilling in the $\delta_n$-neighborhood has to go to $\infty$ as $n \to \infty$. Conditions (C.6) - (C.8) are standard assumptions needed for quantile consistency of non-i.i.d. random variables (for details, see \citealt{sen68} and \citealt{jkg71}). (C.9) is needed to make the $w(\mbox{${\mathbf{s}}$}_i)$'s marginal population median is equal to $0$.
We now state the consistency results of this paper. The following two propositions are the key to prove the rest of the results of our paper. In the following proposition we provide an asymptotic approximation of the Median-VS.
\begin{prop}
\label{prop:MedVSApprox}
Under conditions (C.1) to (C.9), for all $i \in \LP 1, 2, \dots , n \right\}$,
$$
V^{(m)}(\mbox{${\mathbf{s}}$}_i) = \exp\left( - \frac{\lvert w(\mbox{${\mathbf{s}}$}_i) \rvert}{\mathcal{I}_n(\mbox{${\mathbf{s}}$}_i)} \right) + O_p(a_n),
$$ where $\mathcal{I}_n(\mbox{${\mathbf{s}}$}_i) = \xi_{n(i)}^{0.75} - \xi_{n(i)}^{0.25}$, i.e. the IQR of the distribution function $\bar{F}_{n(i)}(x)$ and $a_n$ is a sequence of positive reals $\downarrow 0$ given by,
$
a_n = n^{-1/2} \lambda_n \delta_n^{-1} + \delta_n \lambda_n^{-1}.
$
\end{prop}
The proof of this proposition is based on the extension of the Ghosh-Bahadur representation of sample quantiles for irregularly spaced spatial data (Lemma~\ref{lem:quantile-consist} in Appendix~\ref{app-subsec:aux-lem}). Before we comment on the implication of Proposition~\ref{prop:MedVSApprox}, let us state the following proposition.
\begin{prop}
\label{prop:IqrBound}
Under conditions (C.1) to (C.9), for all $i \in \LP 1, \dots , n \right\}$
$$
\begin{aligned}
C_\epsilon^{(l)}(q_e) \leq \mathcal{I}_n(\mbox{${\mathbf{s}}$}_i) = \xi_{n(i)}^{0.75} - \xi_{n(i)}^{0.25} \leq C_\epsilon^{(u)}(q_e)
\end{aligned}
$$ where, $C_\epsilon^{(l)}(q_e) = \sigma_\epsilon \left( F_\epsilon^{-1}(\text{max}\LP 1 - (0.25/q_e), 0 \right\}) - F_\epsilon^{-1}(\text{min}\LP 0.25/q_e, 1 \right\}) \right)$ and\\ $C_\epsilon^{(u)}(q_e) = \sigma_\epsilon \left( F_\epsilon^{-1}(\text{min}\LP 0.75/q_e, 1 \right\}) - F_\epsilon^{-1}(\text{max}\LP 1 - (0.75/q_e), 0 \right\}) \right)$.
\end{prop}
From Proposition~\ref{prop:IqrBound} we conclude that the scaling term, $\mathcal{I}_n(\mbox{${\mathbf{s}}$}_i)$, in the Median-VS approximation in Proposition~\ref{prop:MedVSApprox} is bounded above by $C_\epsilon^{(u)}(q_e)$, which is a finite constant if $q_e > 0.75$, i.e. the proportion of noisy observations is less than $25\%$. Hence, under the assumption that $q_e > 0.75$, from Proposition~\ref{prop:MedVSApprox} we can see that if the noise variance associated with $w(\mbox{${\mathbf{s}}$}_i)$ -- denoted by $\tau_i^2$ -- is large, with higher probability $\lvert w(\mbox{${\mathbf{s}}$}_i) \rvert$ will take larger values making $\exp\left( - \lvert w(\mbox{${\mathbf{s}}$}_i) \rvert/ \mathcal{I}_n(\mbox{${\mathbf{s}}$}_i) \right)$ (from here denoted by $\tilde{V}^{(m)}(\mbox{${\mathbf{s}}$}_i)$) -- which is bounded above by $\exp\left( - \lvert w(\mbox{${\mathbf{s}}$}_i) \rvert/ C^{(u)}_\epsilon(q_e) \right)$ -- closer to $0$. Whereas, if $w(\mbox{${\mathbf{s}}$}_i)$ has less variance then with higher probability $\lvert w(\mbox{${\mathbf{s}}$}_i) \rvert$ will take values in a neighborhood of $0$ making $\tilde{V}^{(m)}(\mbox{${\mathbf{s}}$}_i)$ closer to $1$. Also, note that the lower bound $C^{(l)}_\epsilon(q_e) > 0$ if we have $q_e > 0.5$. Hence, given that the poportion of good observations in the data is more than $75\%$, Proposition~\ref{prop:MedVSApprox} along with Proposition~\ref{prop:IqrBound} asymptotically justifies that Median-VS will give low scores to `bad' observations and high scores to `good' observations.
Next, we use the approximation in Proposition~\ref{prop:MedVSApprox} to come up with a representation of the VS-based mean parameter estimator $\hat{\mbox{$\boldsymbol{\beta}$}}^{(m)}_{\text{vs}}$ in the following theorem.
\begin{theo}
\label{theo:VS-Mean-Prepresentation}
Under conditions (C.1) to (C.9),
\begin{equation}
\label{eq:MedVSRepTheo}
\begin{split}
\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}^{(m)} = \mbox{$\boldsymbol{\beta}$} + \left( \frac{1}{n} \mbox{${\mathbf{X}}$}^\prime E\left(\tilde{\mathbf{D}}_v\right) \mbox{${\mathbf{X}}$} \right)^{-1} \left( \frac{1}{n} \mbox{${\mathbf{X}}$}^\prime \tilde{\mathbf{D}}_v \mathbf{w} \right) + O_p \left( a_n \right),
\end{split}
\end{equation}
where $\tilde{\mathbf{D}}_v = \text{diag}\left( \tilde{V}^{(m)}(\mbox{${\mathbf{s}}$}_1), \dots , \tilde{V}^{(m)}(\mbox{${\mathbf{s}}$}_n) \right)$ and $\mathbf{w} = \left( w(\mbox{${\mathbf{s}}$}_1), \dots , w(\mbox{${\mathbf{s}}$}_n) \right)^\prime$.
\end{theo}
The proof of Theorem~\ref{theo:VS-Mean-Prepresentation} is given in Appendix~\ref{app-subsubsec:med-vs-reg-rep}. Now, we add some remarks regarding the representation of Median-VS-based estimator given in Equation~\ref{eq:MedVSRepTheo}.
\begin{rem}
\label{rem:Med-VS-representation}
The representation in Equation~\ref{eq:MedVSRepTheo} asymptotically approximates the estimation error in $\hat{\mbox{$\boldsymbol{\beta}$}}^{(m)}_{\text{vs}}$ by the random variable given by $\left( \frac{1}{n} \mbox{${\mathbf{X}}$}^\prime E\left(\tilde{D}_v\right) \mbox{${\mathbf{X}}$} \right)^{-1} \left( \frac{1}{n} \mbox{${\mathbf{X}}$}^\prime \tilde{D}_v \mathbf{w} \right).$ Observe that, if $w(\mbox{${\mathbf{s}}$}_i)$ are marginally symmetric around $0$, then $E\left( \tilde{V}(\mbox{${\mathbf{s}}$}_i) w(\mbox{${\mathbf{s}}$}_i) \right) = 0$. Moreover, as proved in Section~\ref{app-subsubsec:vs-ineq}, $E\left( \tilde{V}(\mbox{${\mathbf{s}}$}_i)^2 w(\mbox{${\mathbf{s}}$}_i)^2 \right) \leq \exp(-2)(C_\epsilon^{(u)}(q_e))^2$. Using this, in the next section we show that the MSE in VS-based estimation of the regression parameter is asymptotically not dependent on the high noise variances, i.e. $\tau_i^2$. On the contrary, the same is not true for the OLS estimator as, $E\left( w(\mbox{${\mathbf{s}}$}_i)^2 \right) = \sigma_\epsilon^2 + \tau_i^2$. This observation highlights the importance of the representation in Equation~\ref{eq:MedVSRepTheo}. We elaborately discuss the advantage of VS-based estimation in this regard in Section~\ref{subsec:AsympeffVSBased}.
\end{rem}
We use Theorem~\ref{theo:VS-Mean-Prepresentation} to prove the consistency of $\hat{\mbox{$\boldsymbol{\beta}$}}^{(m)}_{\text{vs}}$ in the following corollary. For that we need the a set of additional conditions as follows.
\begin{itemize}
\item[(C.10)] $\frac{1}{n}\mbox{${\mathbf{X}}$}^\prime \mbox{${\mathbf{X}}$} \to \mbox{${\mathbf{C}}$}_X \succ 0$.
\item[(C.11)] $E\left( \tilde{D}_v \mathbf{w} \right) = \mathbf{0}$.
\item[(C.12)] The proportion of non-noisy observations, i.e. $q_e$ is strictly greater than $0.75$.
\item[(C.13)] For any real number $a > 0$, $\psi_\epsilon(a) = \int_0^\infty e^{-x} P \left( \lvert \epsilon(\mbox{${\mathbf{s}}$}) \rvert < a x \right) dx > 0$.
\end{itemize}
We now briefly comment on the above conditions. Condition (C.10) is standard assumption needed for the consistency of the least squares based mean parameter estimators. (C.11) is trivially valid if the $w(\mbox{${\mathbf{s}}$}_i)$'s are marginally symmetric around $0$. (C.12) is needed as to get a finite upper-bound on the scaling component of $\tilde{V}^{(m)}(\mbox{${\mathbf{s}}$}_i)$, i.e. $I_n(\mbox{${\mathbf{s}}$}_i)$, we need the proportion of noisy observations to be less than $0.25$. (C.13) is required to ensure the invertibility of the matrix $\left( n^{-1} \mbox{${\mathbf{X}}$}^\prime E\left(\tilde{\mathbf{D}}_v\right) \mbox{${\mathbf{X}}$} \right)$. This condition is trivially true when the marginal distribution of $\epsilon(\mbox{${\mathbf{s}}$})$ has positive mass around a neighborhood of its center.
\begin{cor}
\label{cor:MedVSConsistent}
Under conditions (C.1) to (C.13),
$
\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}^{(m)} = \mbox{$\boldsymbol{\beta}$} + O_p \left( a_n \right).
$
\end{cor}
Corollary~\ref{cor:MedVSConsistent} not only proves the consistency of the Median-VS-based estimator, but also provides an order of rate of convergence in probability for the Median-VS-based estimator. We state the proof of Corollary~\ref{cor:MedVSConsistent} in Appendix~\ref{app-subsubsec:med-vs-consist-cor}. We now put some remarks on the consistency of the Mean-VS-based estimator.
\begin{rem}
\label{rem:mean-vs-consistency}
Similar consistency results of Mean-VS-based estimator $\left( \hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}^{(a)}\right)$ can also be proved but that requires a set of different conditions than that are stated in (C.1) - (C.9). As the sample standard deviation is used in the definition of Mean-VS we need $\frac{1}{n} \sum_{i=1}^n \lVert \mathbf{x}(\mbox{${\mathbf{s}}$}_i) \rVert^4 = O(1)$ as well as all the error components including $\epsilon(\mbox{${\mathbf{s}}$})$ must have finite fourth order moments. In addition, the mixing condition in (C.4) changes as we would require $\epsilon$-process to be fourth-order stationary and all the correlations of the form $\text{cor}\left( \epsilon^a(\cdot), \epsilon^b(\cdot + \mbox{${\mathbf{h}}$}) \right)$, for all $a,b \in \LP 1, \dots , 4 \right\}$, have to be integrable over the space. Conditions (C.5) - (C.9) are not needed as those are specific to the existence of the population quantiles used in the results of the Median-VS-based estimator.
\end{rem}
Though the consistency of the VS-based estimators has been proved in this section, that does not necessarily point out the advantages of the VS-based estimation over the standard methods, e.g. the OLS, when it comes to analyzing noisy spatial data. In the next subsection, we investigate the asymptotic mean squared errors of the VS-based as well as the OLS estimator. For the next section we only consider the Median-VS-based regression estimator ($\hat{\mbox{$\boldsymbol{\beta}$}}^{(m)}_{\text{vs}}$) and hence, for simplicity of nations, we denote it by $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}}$.
\subsection{Asymptotic efficiency of VS-based estimators}
\label{subsec:AsympeffVSBased}
From Theorem~\ref{theo:VS-Mean-Prepresentation}, we can write $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}} - \mbox{$\boldsymbol{\beta}$} = \mathbf{l}_n^{\text{vs}} + O_p(a_n)$, where the leading error term in the VS-based estimation is given by, $\mathbf{l}_n^{\text{vs}} = \left( n^{-1} \mbox{${\mathbf{X}}$}^\prime E\left(\tilde{\mathbf{D}}_v\right) \mbox{${\mathbf{X}}$} \right)^{-1} \left( n^{-1} \mbox{${\mathbf{X}}$}^\prime \tilde{\mathbf{D}}_v \mathbf{w} \right)$. On the other hand, due to linearity of the OLS estimator, $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{ols}} - \mbox{$\boldsymbol{\beta}$} = \mathbf{l}_n^{\text{ols}}$, where it can be easily shown that $\mathbf{l}_n^{\text{ols}} = \left( n^{-1} \mbox{${\mathbf{X}}$}^\prime \mbox{${\mathbf{X}}$} \right)^{-1} \left( n^{-1} \mbox{${\mathbf{X}}$}^\prime \mathbf{w} \right)$. Before stating the results of this section, we need to introduce a few more notations. It can be shown easily that $E\left( \lVert \mathbf{l}_n^{\text{ols}} \rVert^2 \right) = n^{-1} \text{tr}\left( \mbox{${\mathbf{H}}$}_n \boldsymbol{\Sigma}_w \right)$ (details are given in Appendix~\ref{app-subsubsec:ols-ineq}), where $\mbox{${\mathbf{H}}$}_n = n^{-1}\mathbf{X}(n^{-1}\mathbf{X}^\prime \mathbf{X})^{-2}\mathbf{X}^\prime$, $\boldsymbol{\Sigma}_w = \mathrm{Var}\left( \mathbf{w} \right)$. For any symmetric matrix $\mbox{${\mathbf{A}}$}$, by $\lambda_{\text{min}}\left( \mbox{${\mathbf{A}}$} \right)$, we denote the minimum eigenvalue of $\mbox{${\mathbf{A}}$}$. Let us introduce the following notations: $M_X = \underset{\mbox{${\mathbf{s}}$} \in \mathcal{D}_n}{\text{sup}} \underset{1 \leq j \leq p}{\text{max}} \; x^2_j(\mbox{${\mathbf{s}}$}) $; $C^{(0)} = 2^{-1}\text{tr}\left( \mbox{${\mathbf{C}}$}_X^{-1} \right)$; $C^{(1)} = 2 p \left(\lambda_{\text{min}}\left( \mbox{${\mathbf{C}}$}_X \right)\Rp^{-2} M_X$; and finally $\mathcal{M}_n(\varepsilon) = \LP 1 \leq i \leq n : \left( \mbox{${\mathbf{H}}$}_n \right)_{ii} > n^{-1} \varepsilon \right\}$, where by $\left( \mbox{${\mathbf{A}}$} \right)_{ij}$ we denote the $ij$-th element of the matrix $\mbox{${\mathbf{A}}$}$. Now, we have the required notation to state the main results of this section.
\begin{theo}
\label{theo:vs_ineq}
Let assumptions (C.1) to (C.13) hold true. Then, for positive constants $C_1(q_e)$, $C_2$ and $C_3$,
\begin{equation}
\label{eq:vs_ineq}
\begin{split}
E\left( \lVert \mathbf{l}_n^{\text{vs}} \rVert^2\right) \leq C_3 \cdot \left( n^{-1}\left( C_1(q_e) \right)^{2} + C_2\lambda_n^{-4}\right),
\end{split}
\end{equation} where $C_1(q_e) = \left( C_\epsilon^{(u)}(q_e)/\psi_\epsilon\left( C_\epsilon^{(l)}(q_e) \right)\Rp$ and $C_2$, $C_3$ are independent of the noise variance parameters ($\tau_i^2$ for $i \in G_n$, and $q_e$).
\end{theo}
\begin{theo}
\label{theo:ols_ineq}
Let assumptions (C.1) to (C.4) and (C.10) hold true. Then, for any $0 < \varepsilon < \text{min}\left( C^{(0)}, C^{(1)} \right)$ and large enough $n$,
\begin{equation}
\label{eq:ols_ineq}
\begin{split}
E\left( \lVert \mathbf{l}_n^{\text{ols}} \rVert^2 \right) > n^{-1}\varepsilon\left( \sigma_\epsilon^2 \left( C^{(0)} - \varepsilon \right)\left( C^{(1)} - \varepsilon \right)^{-1} + n^{-1}\sum_{i \in \mathcal{M}_n(\varepsilon)\cap G_n^c} \tau_i^2 \right) - C_1 \lambda_n^{-4},
\end{split}
\end{equation}
with $\lvert \mathcal{M}_n(\varepsilon) \rvert \geq \frac{C^{(0)} - \varepsilon}{C^{(1)} - \varepsilon}n$. Here $C_1$ is a positive constant not dependent on the error variance parameters ($\tau_i^2$ for $i \in G_n$, and $q_e$).
\end{theo}
\begin{cor}
\label{cor:ols_ineq}
Let assumptions (C.1) - (C.4) and (C.10) hold true. In addition, assume $\text{inf}_n \; \underset{1\leq i \leq n}{\text{min}} \lVert \mbox{${\mathbf{X}}$}[i,] \rVert^2 > 0$. Then, there exists an $\varepsilon > 0$ such that for large enough $n$,
\begin{equation}
\label{eq:ols_ineq_cor}
\begin{split}
E\left( \lVert \mathbf{l}_n^{\text{ols}} \rVert^2 \right) > n^{-1}\left( \varepsilon\sigma_\epsilon^2 + n^{-1}\sum_{i \in G_n^c} \tau_i^2 \right) - C_1 \lambda_n^{-4},
\end{split}
\end{equation} for some positive constant $C_1$ not dependent on the noise model parameters.
\end{cor}
The proofs of Theorem~\ref{theo:vs_ineq}, Theorem~\ref{theo:ols_ineq} and Corollary~\ref{cor:ols_ineq} are stated in Appendix~\ref{app-subsubsec:vs-ineq}, \ref{app-subsubsec:ols-ineq} and \ref{app-subsubsec:ols-ineq-cor} respectively. Next, we put some remarks on the above results.
\begin{rem}
From Theorem~\ref{theo:vs_ineq} we see that the MSE of the leading error term in the Median-VS-based regression estimation can be bounded above by some constant $C_3$ times $n^{-1}\left( C_1(q_e) \right)^{2} + C_2\lambda_n^{-4}$, which goes to $0$ as $n \to \infty$, and is dependent on the noise model parameters only through the proportion of the good observations $q_e$. Clearly, the MSE of the Median-VS-based regression estimator is not dependent on the noise variances (denoted by $\tau_i^2$) associated with the `bad' observations. Note that, $\psi_\epsilon(\cdot)$ is an increasing function. We have already mentioned that as $q_e$ increase, $C_\epsilon^{(u)}(q_e)$ decreases and $C_\epsilon^{(l)}(q_e)$ increases, i.e. $C_1(q_e)$ is a decreasing function of $q_e$. Hence, if the proportion of good observations increases, the upper bound in Equation~\ref{eq:vs_ineq} decreases, restricting the range of $E\left( \lVert \mathbf{l}_n^{\text{vs}} \rVert^2\right)$ from above. The other term, $C_2\lambda_n^{-4}$ has hardly any effect on $E\left( \lVert \mathbf{l}_n^{\text{vs}} \rVert^2\right)$ as $\lambda_n^{-4}$ is of much smaller order as compared to $n^{-1}$.
\end{rem}
\begin{rem}
The lower-bound for the MSE of the OLS estimator, as we can see in Theorem~\ref{theo:ols_ineq} and Corollary~\ref{cor:ols_ineq}, is dependent on the additional noise variances of the corrupted observations through the summation $n^{-1}\sum_{i \in G_n^c \cap \mathcal{M}_n(\varepsilon)} \tau_i^2$ and $n^{-1}\sum_{i \in G_n^c} \tau_i^2$ respectively. Clearly, if the noise variances corresponding to some of the observations are high, efficiency of the OLS estimator will be significantly affected. For example, consider the case where $\tau_i^2 \approx n^c$ for some constant $c > 0$. Then for a given $q_e$, the lower-bound in Equation~\ref{eq:ols_ineq_cor}, is \textit{of order} $n^{c-1}$. In Table~\ref{tab:RECompEx} we show the advantage of using VS-based estimator as opposed to the OLS one in terms of the asymptotic order of relative efficiency.
\begin{table}[h]
\centering
\caption{\small Comparison of orders of MSEs of OLS and VS under the example case: $\tau_i^2 \approx n^c$.}
\label{tab:RECompEx}
\resizebox{0.32\columnwidth}{.12\textwidth}{%
\begin{tabular}{|cc|cc|c|}
\toprule
\textbf{n} & \textbf{c} & \textbf{OLS-LB} & \textbf{VS-UB} & \textbf{RE-LB} \\
\hline
\multirow{3}{*}{100} & 0.1 & 0.016 & 0.010 & 1.585 \\
& 0.5 & 0.100 & 0.010 & 10.000 \\
& 0.8 & 0.398 & 0.010 & 39.811 \\ \hline
\multirow{3}{*}{500} & 0.1 & 0.004 & 0.002 & 1.862 \\
& 0.5 & 0.045 & 0.002 & 22.361 \\
& 0.8 & 0.289 & 0.002 & 144.270 \\ \hline
\multirow{3}{*}{1000} & 0.1 & 0.002 & 0.001 & 1.995 \\
& 0.5 & 0.032 & 0.001 & 31.623 \\
& 0.8 & 0.251 & 0.001 & 251.189 \\
\hline
\end{tabular}%
}
\end{table}
Here, we refer the order of the lower bound of the MSE of the OLS estimator by OLS-LB, the upper bound for the MSE of the VS-based estimator by VS-UB and, the lower bound for the relative efficiency of VS-based estimator to the OLS one (defines as MSE(VS)/MSE(OLS)) by RE-LB. Table~\ref{tab:RECompEx} clearly portrays the advantage of the VS-based estimation as compared to the OLS one: if the order of the noise variances associated with even a `small' portion of the observations is `high' , due to the robustness of the VS-based estimator against the noisy observations, there is a significant gain in efficiency relative to the OLS.
\end{rem}
\section{Simulation Study}
\label{sec:SimStudy}
In this section, we summarize our efforts to evaluate the VS-based estimator numerically. We first describe the simulation designs and then, we state the results and draw inference from them.
\subsection{Simulation Setup}
\label{subsec:SimSetup}
We use the following spatial regression model to simulate the realizations from the the `true' random field $\LP Y(\mbox{${\mathbf{s}}$}) \right\}$:
\begin{equation}
\label{eq:SimMod_LM_Y}
\begin{split}
Y(\mbox{${\mathbf{s}}$}_i) = \beta_0 + \left( \beta_x, \beta_y \right)^\prime \mbox{${\mathbf{s}}$}_i + \beta_h \; h(\mbox{${\mathbf{s}}$}_i) + \epsilon(\mbox{${\mathbf{s}}$}_i), \quad \text{for} \;\; i \in \LP 1, \dots , n \right\},
\end{split}
\end{equation} where, $\mbox{$\boldsymbol{\beta}$} = \left( \beta_0, \beta_x, \beta_y, \beta_h \right)^\prime$ is the vector of regression parameters; $h(\mbox{${\mathbf{s}}$})$ is a deterministic function of location $\mbox{${\mathbf{s}}$}$; and $\LP \epsilon(\mbox{${\mathbf{s}}$}) \right\}$ is a mean zero second-order stationary spatially correlated process. To define the function $h(\mbox{${\mathbf{s}}$})$ over the sampling region, we use the deterministic function $h(\mbox{${\mathbf{s}}$}) = H_1 \cdot \sum_{j=1}^{H_2} w_h(j) f(\mbox{${\mathbf{s}}$}; \; \boldsymbol{\mu}_j , \Sigma_j) + H_3$, where $f(\cdot; \boldsymbol{\mu}, \Sigma)$ denotes the bivariate normal density with mean $\boldsymbol{\mu}$ and covariance matrix $\Sigma$ and $\LP \left( \boldsymbol{\mu}_j, \Sigma_j \right) : 1 \leq j \leq H_2 \right\}$ is a fixed set of vectors and matrices. The choice of this function is motivated from the elevation function used in the simulations of \cite{chak}. The residual vector $\left( \epsilon(\mbox{${\mathbf{s}}$}_1), \dots , \epsilon(\mbox{${\mathbf{s}}$}_n) \right)^\prime$ are sampled from a mean zero Gaussian process with the Exponential covariance function given as $
C^{\text{exp}}(\mbox{${\mathbf{h}}$}; \boldsymbol{\theta}) = \sigma_\epsilon^2 \exp \left( \frac{\lVert \mbox{${\mathbf{h}}$} \rVert}{\rho} \right) + \tau^2 \mathbb{I}(d = 0)$ where, $\mathbb{I}(\cdot)$ denotes the indicator function. The covariance parameter vector of interest is $\boldsymbol{\theta} = \left( \tau^2, \sigma_\epsilon^2, \rho \right)^\prime$, where $\tau^2$ is the nugget effect, $\sigma_\epsilon^2, \rho$ are the partial sill and range parameters respectively (for details see \citealt{haskard07}; page 37 \citealt{gelfand10}).
To include noise in the varying-quality observations $\left( Z(\mbox{${\mathbf{s}}$}_1), \dots , Z(\mbox{${\mathbf{s}}$}_n) \right)^\prime$ using the additive-multiplicative noise structure given in Equation~\ref{eq:AddMultNoiseMod}, we use the following distributions for the additive and multiplicative components. Given a set of good observations $G_n$, for $i \in G_n$, the $\epsilon_{M_i}$ and $\epsilon_{A_i}$ are identically equal to $1$ and $0$, respectively. For $i \in G_n^c$, we take
\begin{equation}
\label{eq:error_dist_sim}
\begin{split}
\epsilon_{M_i} \underset{\text{indep.}}{\sim} 2 \times \text{Beta}(\alpha_{M},& \alpha_{M}), \quad \epsilon_{A_i} \underset{\text{indep.}}{\sim} N(0,\sigma_{A}^2).
\end{split}
\end{equation} Here, the variance associated with the multiplicative component of a noisy observation is given by $\sigma_{M}^2 = \frac{1}{2\alpha_{M} + 1}$. To select the set of `good' observations, i.e. $G_n$, we have arbitrarily selected $\lfloor q_e n \rfloor$ many (fixed) indices among $\LP 1, \dots , n \right\}$ prior the simulations. Note that, the choice of multiplicative error distribution in \ref{eq:error_dist_sim} restricts its realizations to be in $[0,2]$, and also, ensures that the multiplicative errors are symmetric around 1.
Estimation of the regression parameters through VS-based and the OLS method do not require the covariance parameters to be estimated. Whereas, the GLS estimator uses the covariance information which is often unknown in real data analysis. Hence, the covariance parameters are estimated from the de-trended spatial observations and then the estimated covariance parameters are used to compute GLS estimator. Then the updated mean estimates are used to compute the covariance parameters and we continue until the parameter estimates converge. This estimator is called the estimated GLS (referred as EGLS, for details see page 23 \citealt{cressie93}). In the following section, we compare the performance of VS-based (Mean-VS, Median-VS) and commonly used least squares based (OLS and EGLS) techniques for estimation of the regression parameters.
Though the theoretical analysis of the VS-based covariance parameter estimation is beyond the scope of this paper, in our simulation studies we evaluate the accuracy of the same to capture the dependence structure of the spatial process. The reasons are following. First, the boxplots of the regression parameter estimates in Section B.1 in the supplementary material suggest that the robust approach (robust regression with Huber's loss, \citealt{huber81}) has similar performance as compared to the VS-based methods. But, extension of the robust approach for estimation of the covariance parameters is not straightforward. Whereas, the VS computed from the data can be incorporated in the covariance analysis to reduce the effects of `bad' observations in the estimation. Hence, comparison of the VS-based technique of covariance estimation needs to be evaluated as compared to the other robust techniques of covariance parameter estimations. We consider the following two methods in that regard: the robust WLS based variogram model fitting by \cite{cressie80} (referred as WLS in Table~\ref{tab:CovParamMSE}) and the robust-REML technique proposed by \cite{kunsch11} and implemented by \cite{georob18}. Second, we want to justify the use of VS-based covariance analysis in the real data example in Section~\ref{sec:readl-dat-ex}.
For all the simulation results in this section we take $\mbox{$\boldsymbol{\beta}$} = \left( 70, 5, -2, -0.05 \right)^\prime$ and $\mbox{$\boldsymbol{\theta}$} = \left( 0, 6, 1\right)^\prime$. To investigate the performances of the different types of estimators with varying noise level, we consider the following set of values of the noise parameters: (a) $\sigma_A \in \LP 5, 50, 100 \right\}$, (b) $\alpha_M \in \LP 2, 0.5, 0.05 \right\}$, and (c) $ q_e \in \LP 0.95, 0.85, 0.75 \right\}$. The set of values for each of the above parameters are written in the order of increasing noise variance and proportion. For example, if the first values of all the three sets are considered then the additional noise variance of a corrupted observation at location $\mbox{${\mathbf{s}}$}$ is $0.2\left( \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$})^\prime \mbox{$\boldsymbol{\beta}$} \right)^2 + 26.2$, and the proportion of such observations is $5\%$; whereas the same variance will be $0.91\left( \mbox{${\mathbf{x}}$}(\mbox{${\mathbf{s}}$})^\prime \mbox{$\boldsymbol{\beta}$} \right)^2 + 10005.46$ if the last values of the sets are considered and the proportion of such noisy observations will be $25\%$. In the tables in the next section, when one of the noise model parameters are varying, the other parameters are fixed at the lowest noise level, i.e. if $\sigma_A$ is varying, then $\alpha_M$ is fixed at $2$ and $q_e$ is fixed at $0.95$.
\subsection{Simulation Results}
\label{subsec:SimResult}
Table~\ref{tab:RegParamMSE} shows the empirical mean square errors of the considered mean parameter estimators. In the final column of Table~\ref{tab:RegParamMSE}, the relative efficiency (RE) of the Median-VS-based regression estimator with respect to the OLS one is reported to emphasize the superiority of VS-based estimation in the analysis of noisy spatial data.
\begin{table}[h]
\centering
\caption{\small Performance of regression parameter estimators under additive-multiplicative noise model.}
\label{tab:RegParamMSE}
\resizebox{0.72\columnwidth}{.43\textwidth}{%
\begin{tabular}{|cc|cc|cc|c|}
\toprule
& \multicolumn{1}{c|}{} &
\multicolumn{2}{c|}{\textbf{VS-based MSE}} &
\multicolumn{2}{c|}{\textbf{LS-based MSE}} &
\textbf{RE(Med-VS, OLS)=}\\
\cline{3-6}
\textbf{$\sigma_A$} & $\mathbf{n}$ &
\textbf{Med-VS} &
\textbf{Mean-VS} &
\textbf{OLS} & \textbf{EGLS} & \textbf{MSE(OLS)/MSE(Med-VS)} \\
\hline
\hline
\multirow{3}{*}{5} & 500 & 1.807 & 1.959 & 3.237 & 5.620 & 1.791 \\
& 1000 & 1.166 & 1.184 & 1.662 & 3.416 & 1.425 \\
& 5000 & 0.567 & 0.574 & 0.746 & 1.436 & 1.317 \\ \hline
\multirow{3}{*}{50} & 500 & 1.806 & 1.906 & 4.783 & 14.862 & 2.648 \\
& 1000 & 1.134 & 1.165 & 2.534 & 6.112 & 2.234 \\
& 5000 & 0.701 & 0.693 & 1.055 & 2.114 & 1.505 \\ \hline
\multirow{3}{*}{100} & 500 & 2.350 & 2.905 & 12.938 & 36.208 & 5.505 \\
& 1000 & 0.969 & 1.102 & 7.817 & 13.895 & 8.067 \\
& 5000 & 0.618 & 0.595 & 1.618 & 6.064 & 2.619 \\ \hline
\toprule
& \multicolumn{1}{c|}{} &
\multicolumn{2}{c|}{\textbf{VS-based MSE}} &
\multicolumn{2}{c|}{\textbf{LS-based MSE}} &
\textbf{R.E.(Med-VS, OLS)=}\\
\cline{3-6}
\textbf{$\sigma_M$} & $\mathbf{n}$ &
\textbf{Med-VS} &
\textbf{Mean-VS} &
\textbf{OLS} & \textbf{EGLS} & \textbf{MSE(OLS)/MSE(Med-VS)} \\
\hline
\hline
\multirow{3}{*}{0.447} & 500 & 1.889 & 1.945 & 2.771 & 4.481 & 1.467 \\
& 1000 & 1.005 & 1.033 & 1.559 & 2.942 & 1.552 \\
& 5000 & 0.572 & 0.578 & 0.699 & 1.352 & 1.220 \\\hline
\multirow{3}{*}{0.707} & 500 & 2.219 & 2.253 & 4.098 & 13.248 & 1.847 \\
& 1000 & 1.063 & 1.117 & 2.685 & 6.031 & 2.525 \\
& 5000 & 0.647 & 0.641 & 0.919 & 2.527 & 1.420 \\ \hline
\multirow{3}{*}{0.953} & 500 & 2.221 & 2.422 & 6.182 & 20.996 & 2.784 \\
& 1000 & 1.201 & 1.206 & 3.578 & 9.048 & 2.978 \\
& 5000 & 0.669 & 0.671 & 1.476 & 4.746 & 2.206 \\ \hline
\toprule
& \multicolumn{1}{c|}{} &
\multicolumn{2}{c|}{\textbf{VS-based MSE}} &
\multicolumn{2}{c|}{\textbf{LS-based MSE}} &
\textbf{R.E.(Med-VS, OLS)=}\\
\cline{3-6}
\textbf{$q_e$} & $\mathbf{n}$ &
\textbf{Med-VS} &
\textbf{Mean-VS} &
\textbf{OLS} & \textbf{EGLS} & \textbf{MSE(OLS)/MSE(Med-VS)} \\
\hline
\hline
\multirow{3}{*}{0.95} & 500 & 2.121 & 2.160 & 3.080 & 5.697 & 1.452 \\
& 1000 & 1.121 & 1.110 & 1.447 & 2.527 & 1.291 \\
& 5000 & 0.742 & 0.741 & 0.836 & 1.281 & 1.126 \\ \hline
\multirow{3}{*}{0.9} & 500 & 1.783 & 2.026 & 4.274 & 8.825 & 2.397 \\
& 1000 & 1.129 & 1.177 & 2.362 & 5.475 & 2.092 \\
& 5000 & 0.634 & 0.635 & 0.833 & 2.288 & 1.314 \\ \hline
\multirow{3}{*}{0.8} & 500 & 2.017 & 2.351 & 6.074 & 15.545 & 3.011 \\
& 1000 & 1.087 & 1.267 & 3.567 & 9.290 & 3.283 \\
& 5000 & 0.550 & 0.565 & 1.105 & 4.618 & 2.010 \\ \hline
\toprule
\end{tabular}%
}
\end{table}
It is evident from Table~\ref{tab:RegParamMSE} that the VS-based estimation has better performance than the standard OLS and EGLS methods across all the cases. The increasing noise variance has very little effect on the performance of the VS-based estimation in terms of MSE. For instance, consider the case when $n = 500$. If the standard deviation of the associated additive noise increases from $5$ to $50$, the MSE of Median-VS-based estimator has hardly changed, whereas the MSE of the EGLS and OLS estimators have increased by approximately $164.4\%$ and $47.8\%$, in the respective order. The consistency of the VS-based estimator is also prominent from Table~\ref{tab:RegParamMSE} -- in each of the scenarios, as sample size increases, the MSE is going close to $0$. The relative efficiency column justifies the superiority of the VS-based estimator to the least squares-based estimators. For example, when $\sigma_A = 100$ and the sample size $n = 1000$, i.e. very high noise variance is associated with $5\%$ of the observations -- to achieve same accuracy as of the Median-VS-based estimators the OLS and the EGLS methods need approximately $8$ and $14.5$ times more observations. Also, given a sample size, the relative efficiency increases if the magnitude of the noise variances associated with the `bad' observations increase (controlled by $\sigma_M$ and $\sigma_A$) or the proportion of the `bad' observations increases. Note that, the advantage of the VS-based estimation for varying proportion of the non-noisy data (i.e. $q_e$) is not as `good' as in the case of varying additive noise variance $\sigma_A$. The reason is that the effect of the increasing noise variances on the MSE of the VS-based estimators is much less as cmpared to the increase of proportion of noisy observations, which we have established theoretically in Section~\ref{subsec:AsympeffVSBased}.
Observe that, in Table~\ref{tab:RegParamMSE}, the accuracy is hampered severely if instead of the OLS estimator EGLS is used to estimate the regression parameter. One of the possible reasons behind this is the inefficient estimation of the covariance parameters using robust weighted least squares technique.
\setlength{\tabcolsep}{12pt}
\begin{table}[h]
\centering
\caption{\small Performance of covaraince parameter estimators under additive-multiplicative noise model.}
\label{tab:CovParamMSE}
\resizebox{0.60\columnwidth}{.38\textwidth}{%
\begin{tabular}{|cc|cc|cc|}
\toprule
& \multicolumn{1}{c|}{} &
\multicolumn{2}{c|}{\textbf{psill MSE}} &
\multicolumn{2}{c|}{\textbf{range MSE}}\\
\cline{3-6}
\textbf{$\sigma_A$} & $\mathbf{n}$ &
\textbf{VS} &
\textbf{WLS} &
\textbf{VS} & \textbf{WLS}\\
\hline
\hline
\multirow{3}{*}{5} & 500 & 19.707 & 2058.760 & 2.471 & 3.099 \\
& 1000 & 1.255 & 1992.875 & 0.123 & 2.697 \\
& 5000 & 0.532 & 609.690 & 0.032 & 17.537 \\ \hline
\multirow{3}{*}{50} & 500 & 36.275 & 1334.724 & 4.549 & 0.580 \\
& 1000 & 1.800 & 1820.486 & 0.146 & 0.639 \\
& 5000 & 0.757 & 1163.299 & 0.046 & 11.401 \\ \hline
\multirow{3}{*}{100} & 500 & 45.653 & 682.913 & 9.477 & 0.502 \\
& 1000 & 1.675 & 913.053 & 0.186 & 0.506 \\
& 5000 & 0.933 & 1802.586 & 0.060 & 10.526 \\ \hline
\toprule
& \multicolumn{1}{c|}{} &
\multicolumn{2}{c|}{\textbf{psill MSE}} &
\multicolumn{2}{c|}{\textbf{range MSE}}\\
\cline{3-6}
\textbf{$\sigma_M$} & $\mathbf{n}$ &
\textbf{VS} &
\textbf{WLS} &
\textbf{VS} & \textbf{WLS}\\
\hline
\hline
\multirow{3}{*}{0.447} & 500 & 7.712 & 1996.900 & 0.928 & 0.757 \\
& 1000 & 1.401 & 1863.389 & 0.164 & 5.779 \\
& 5000 & 0.580 & 577.577 & 0.047 & 6.312 \\ \hline
\multirow{3}{*}{0.707} & 500 & 14.324 & 2045.811 & 2.097 & 13.983 \\
& 1000 & 1.663 & 1566.315 & 0.170 & 4.272 \\
& 5000 & 0.820 & 792.043 & 0.038 & 8.928 \\ \hline
\multirow{3}{*}{0.953} & 500 & 28.337 & 1578.990 & 3.621 & 54.341 \\
& 1000 & 1.507 & 1537.173 & 0.201 & 16.551 \\
& 5000 & 1.134 & 1185.805 & 0.091 & 1.645 \\ \hline
\toprule
& \multicolumn{1}{c|}{} &
\multicolumn{2}{c|}{\textbf{psill MSE}} &
\multicolumn{2}{c|}{\textbf{range MSE}}\\
\cline{3-6}
\textbf{$q_e$} & $\mathbf{n}$ &
\textbf{VS} &
\textbf{WLS} &
\textbf{VS} & \textbf{WLS}\\
\hline
\hline
\multirow{3}{*}{0.95} & 500 & 11.786 & 1803.304 & 1.302 & 2.062 \\
& 1000 & 1.230 & 2385.300 & 0.146 & 2.884 \\
& 5000 & 0.665 & 531.284 & 0.047 & 12.393 \\ \hline
\multirow{3}{*}{0.9} & 500 & 5.216 & 2318.057 & 0.906 & 5.762 \\
& 1000 & 1.292 & 1535.030 & 0.098 & 5.707 \\
& 5000 & 1.063 & 817.137 & 0.026 & 2.803 \\ \hline
\multirow{3}{*}{0.8} & 500 & 36.674 & 1745.520 & 1.865 & 0.658 \\
& 1000 & 6.300 & 1308.637 & 0.116 & 13.993 \\
& 5000 & 4.584 & 1428.831 & 0.049 & 0.790 \\ \hline
\toprule
\end{tabular}%
}
\end{table}
Hence, in Table~\ref{tab:CovParamMSE}, we have investigated the performance of the VS-based covariance estimation and compared it with that of the standard approach: robust weighted least squares (WLS) based variogram estimator. Clearly, the WLS approach has failed completely to estimate the covariance parameters -- partial sill (or psill, denoted by $\sigma_\epsilon^2$) and range (denoted by $\rho$). Due to the added noise variance in the data the optimization in the variogram fitting in WLS method is not stable. On the other hand, VS-based variogram estimation has reasonably good accuracy, given that the proportion of the noisy observations are not too large. Also, in most of the cases, if the noise model parameters are held fixed, the MSE of the VS-based estimators decreases, as the sample size $n$ increase, which justifies the consistency of the VS-based variogram model fitting technique. But, in the case where the proportion of noisy observations is large (e.g. $q_e = 0.8$, i.e. $20\%$ of the observations are noisy) the accuracy of the VS-based estimation is hampered. Clearly, the accuracy of the VS-based covariance estimation is more sensitive to the proportion of noisy observations in the data, rather than the magnitude of the noise variances -- the same fact that we have established theoretically for the VS-based mean estimation in Section~\ref{subsec:AsympeffVSBased}.
Next, in Table~\ref{tab:VSvsGR}, we briefly report the results of our comparative analysis of VS-based technique with another robust covariance estimation method in geostatistics -- robust REML proposed by \cite{kunsch11} and implemented in R-package \texttt{georob} by \cite{georob18} (referred as GR in Table~\ref{tab:VSvsGR}).
\begin{table}[h]
\caption{Comparison of VS-based cov. estimation with GR}\label{tab:VSvsGR}
\centering
\resizebox{.63\columnwidth}{.11\textwidth}{%
\begin{tabular}{|cc|cccc|}
\toprule
$\mathbf{\sigma_A}$ & $\mathbf{n}$ & \textbf{MSE($\hat{\sigma_\epsilon}^2$).VS} & \textbf{MSE($\hat{\sigma_\epsilon}^2$).GR} & \textbf{MSE($\hat{\rho}^2$).VS} & \textbf{MSE($\hat{\rho}^2$).GR} \\
\hline
\hline
\multirow{2}{*}{5} & 500 & 6.03 & 9.64e+22 & 1.38 & 3.95e+22 \\
& 1000 & 1.08 & 1.96e+23 & 0.11 & 1.77e+23 \\ \hline
\multirow{2}{*}{50} & 500 & 48.24 & 1.22e+23 & 7.14 & 1.38e+23 \\
& 1000 & 3.27 & 5.39e+22 & 0.68 & 1.75e+23 \\ \hline
\multirow{2}{*}{100} & 500 & 43.69 & 2.44e+22 & 9.44 & 9.19e+22 \\
& 1000 & 1.26 & 1.62e+21 & 0.18 & 2.40e+21 \\
\hline
\toprule
\end{tabular}%
}
\end{table}
The results are surprisingly in favor of the VS-based methodology. Clearly, the robust REML technique has completely failed to provide reasonable covariance estimates. We can not provide the results for all other choices of $n$ and noise model parameters because, the REML estimation in georob has huge time complexity. For example, for $\sigma_A = 5$ and $n = 1000$, on an average georob takes $145$ sec., whereas, VS-based estiamtion takes $2.5$ sec. -- i.e. georob takes on an average $58$ times more. Hence, both in terms of computational complexity and the accuracy of estimation, VS-based technique outperforms robust-REML method (as implemented in \citealt{georob18}).
In the next section, we are going to apply VS-based estimation technique on the coal ash data (\citealt{coalash}) and compare our results with that of \cite{georobMan}, where the author has implemented robust REML estimation on the same data set.
\section{Example: Coalash data}
\label{sec:readl-dat-ex}
In this section we consider the coal ash data (\citealt{coalash}), which reports measurement (in $\%$ mass) of ash present in coal seam in Pennsylvania. The data set is available in R-package \texttt{gstat} (\citealt{gstat}) and has been previously used to demonstrate robust geostatistical techniques by \cite{cressie93} and \cite{georobMan}. Let us denote the target process as $\LP Y(\mbox{${\mathbf{s}}$}) \right\}$ where $Y(\mbox{${\mathbf{s}}$})$ is the mass percentage (often referred as \textit{$\%$ mass}) of ash in the coal seam at location $\mbox{${\mathbf{s}}$}$. The observations plotted in Figure~\ref{fig:CoalashDat} are denoted by $\LP Z(\mbox{${\mathbf{s}}$}_1), \dots , Z(\mbox{${\mathbf{s}}$}_n) \right\}$ where $n = 208$. Some of these observations are corrupted due to measurement errors or other types of unknown noises.
We first compute the VS (Median-VS) of the observations and plot it spatially in Figure~\ref{fig:vs-coalash}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[trim={2.5cm 1cm 2.5cm 1.5cm}, width=.55\textwidth, height=0.15\textheight]{DataPlot}
\subcaption{Ash \% in coal seams}\label{fig:CoalashDat}
\end{subfigure}\hspace{2mm}%
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[trim={2.5cm 1cm 2.5cm 1.5cm}, width=.55\textwidth, height=0.15\textheight]{VS_DataPlot.pdf}
\subcaption{VS of the observations}\label{fig:vs-coalash}
\end{subfigure}
\caption{Spatial plots of the coalash data (a) and VS of the observations (b).}\label{fig:coalashDat-VS}
\end{figure} \cite{cressie93} identified a set of observations as outliers by investigating the deviation of the observations from their overall sample median. VS uses the similar idea -- instead of using the overall median, it assigns a score to each of the observations according to its deviation from the `local' median relative to the `local' variation. The observations that were identified as outliers by \cite{cressie93} and \citealt{georobMan} receive VS less than 0.18 (in a scale of $(0,1]$). As both in theoretical analysis and simulation studies we have established that VS-based estimation is hampered if proportion of noisy observations is significantly large, we check the proportion of observations which have VS less than $0.22$ -- which implies the deviation from local median is at least $1.51 \times$local-IQR. We find that approximately $12\%$ of the observations have `high' noise associated with it.
Once the VS of the observations has been computed we incorporate those score to estimate the mean and covariance structure of the process robustly. Through some explanatory data analysis similar to \cite{georobMan}, we consider the following spatial regression model for the target process $\LP Y(\mbox{${\mathbf{s}}$}) \right\}$,
\begin{equation}
\label{eq:coalash-sp-reg}
\begin{split}
Y(\mbox{${\mathbf{s}}$}) = \beta_0 + \beta_x s_x + \epsilon(\mbox{${\mathbf{s}}$}),
\end{split}
\end{equation} where $\mbox{$\boldsymbol{\beta}$} = \left( \beta_0, \beta_x \right)^\prime$ is the regression parameter, $s_x$ is longitude (i.e. $\mbox{${\mathbf{s}}$} = \left( s_x, s_y \right)^\prime$) and $\LP \epsilon(\mbox{${\mathbf{s}}$}) \right\}$ is a zero mean second order stationary process with Mat\'ern covariance and the covariance parameter vector is given by $\mbox{$\boldsymbol{\theta}$} = \left( \sigma_\epsilon^2, \rho, \eta^2, \nu \right)^\prime$ where $\sigma_\epsilon^2$ is the partiall sill, $\rho$ is the range, $\eta^2$ is the nugget and $\nu$ is the smoothness parameter (for details, see \citealt{haskard07}; \citealt{gelfand10}).
VS-based regression yields an estimate of $\hat{\mbox{$\boldsymbol{\beta}$}}_{\text{vs}} = \left( 11.071, -0.188 \right)^\prime$ for the regression parameter -- which suggests that the ash proportion decreases as we go from west to east.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[trim={3cm 1cm 1.5cm 0cm}, width=.5\textwidth, height=0.12\textheight]{hist_resid_vs}
\subcaption{}\label{fig:hist-resid-vs}
\end{subfigure}\hspace{2mm}%
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[trim={3cm 1cm 1.5cm 0cm}, width=.5\textwidth, height=0.12\textheight]{hist_resid_vs_smoothed}
\subcaption{}\label{fig:hist-resid-vs-smooth}
\end{subfigure}
\caption{VS-based smoothing of residuals: histogram of observed residuals from VS-based regression (a), histogram of smoothed residuals (b).}\label{fig:VS-Smoothing}
\end{figure} Next, the VS-based smoothing is implemented on the observed residuals from the VS-based regression and the effect of smoothing has been displayed through frequency plots in Figure~\ref{fig:VS-Smoothing}. Finally, the VS-based smooth residuals are used to estimate the covariance parameters through VS-based variogram model fitting. We summarize the details of VS-based covariance estimation in Figure~\ref{fig:coalashCovEst} and Table~\ref{tab:coalashCovEst}. Observe that, the range parameter estiamte ($0.486$) in Table~\ref{tab:coalashCovEst} is significantly greater than $0$, i.e. the VS-based analysis can capture the spatial correlation in the residual process, as opposed to the pure nugget model found in the analysis of \cite{georobMan}.\\
\begin{minipage}[c]{0.55\textwidth}
\centering
\resizebox{.6\columnwidth}{.18\textwidth}{%
\begin{tabular}{cc}
\hline
\hline
\textbf{Parameters} & \textbf{Estiamtes} \\
\hline
partial sill ($\sigma_\epsilon^2$) & 0.219 \\
range ($\rho$) & 0.486 \\
nugget ($\eta^2$) & 0.021 \\
smoothness ($\kappa$) & 5.00 \\
\hline
\hline
\end{tabular}%
}
\captionof{table}{\small{Estimated Mat\'ern parameters.}}
\label{tab:coalashCovEst}
\end{minipage}
\begin{minipage}[c]{0.45\textwidth}
\includegraphics[trim={1.5cm 1.5cm 1cm 0.2cm}, width=.95\textwidth, height = .19\textheight]{varioEst_vs}
\captionof{figure}{\small{Variogram estimation}}
\label{fig:coalashCovEst}
\end{minipage}
Next, we compare the VS-based analysis with the robust REML approach. To implement the robust REML methodology on the coal ash data the R-package \texttt{georob} (\citealt{georob18}) is used. We use leave-one-out cross-validation technique to compare the two approaches: for each of the observations in the coal ash data, we consider it as the test data and try to predict (kriging) it using all other observations without the test one. As, trying to predict a `bad' observation may lead to erroneous inference, from the test data set we exclude the observations marked as outliers by \cite{georobMan} (count $11$) in their analysis. But, the training data contains all the data points. Instead of using VS, we use the analysis done by \cite{georobMan} to remove the outliers from the test data so that the choice of test data in the cross-validated analysis is not biased towards the VS-based approach.
In Figure~\ref{fig:coalashPredDiag}, we have summarized the results of comparative analysis between VS-based and robust-REML approach.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[trim={1.5cm 1.5cm 2.5cm .5cm}, width=.55\textwidth, height=0.14\textheight]{ECDF_pred_err_vs}
\subcaption{}\label{fig:ecdf-vs-gr}
\end{subfigure}\hspace{2mm}%
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[trim={1.5cm 1.5cm 2.5cm .5cm}, width=.65\textwidth, height=0.14\textheight]{RE_VS_GR}
\subcaption{}\label{fig:re-vs-gr}
\end{subfigure}
\caption{Prediction comparison between VS and robust-REML: (a) - empirical c.d.f. of prediction errors and, (d) - relative efficiency in terms of margin of prediction errors of VS w.r.t. robust-REML.}\label{fig:coalashPredDiag}
\end{figure}
From the empirical distribution function plot in Figure~\ref{fig:ecdf-vs-gr} it is evident that the prediction errors corresponding to the robust REML predictor has higher mass in the tails than the VS-based predictions. Moreover, as we can see in Figure~\ref{fig:re-vs-gr}, the margins of prediction errors (ME), that is the half of the lengths of the prediction intervals, for the VS-based approach is either similar or smaller as compared to the robust REML for most of the test data points. To check the credibility of the VS-based prediction intervals we find that $98.5\%$ of the test data points were inside the $95\%$ VS-based prediction intervals. In terms of cross-validated mean squared prediction error (MSPE), the VS-based MSPE is $0.716$ whereas the robust-REML prediction has an MSPE of $0.743$, i.e. there is nearly $4\%$ gain in efficiency when VS-based analysis is carrired out in place of the robust REML approach. Moreover, in terms of the computational time VS-based approach has a huge advantage as compared to the robust REML, as shown in Table~\ref{tab:coalashCompTime}.
\begin{table}[ht]
\centering
\resizebox{0.35\columnwidth}{.1\textwidth}{%
\begin{tabular}{|c|cc|}
\hline
Computation Time & VS-based & robust-REML \\
\hline
Min. & 0.45 & 1.47 \\
1st Qu. & 0.54 & 1.99 \\
Median & 0.60 & 2.18 \\
Mean & 0.61 & 8.24 \\
3rd Qu. & 0.67 & 14.22 \\
Max. & 0.93 & 29.75 \\
\hline
\end{tabular}%
}
\caption{\small Time comparison between VS-based and robust-REML. Machine configuration: DELL R7425 Dual Processor AMD Epyc 32 core 2.2 GHz machines with 512GB RAM each running 64Bit Ubuntu Linux Version 18.04.}
\label{tab:coalashCompTime}
\end{table} Hence, for the analysis of larger spatial data sets, VS-based analysis has advantage over the robust REML approach.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we have investigated the large sample behavior of the veracity scoring technique proposed by \cite{chak} for irregularly spaced spatial data when there is no reference data available. Through the asymptotic approximation of the VS we have established that VS of an observation is expected to take `small' value (close to $0$) if the noise variance associated with the observation is high and, it is going to take larger values if the associated noise variance is less. Moreover, under the mixed-increasing domain spatial asymptotic framework and for a non-stationary noise model, we have sheded light upon the circumstances under which the weighted least squares regression estimator with veracity scores as the corresponding weights -- which are data-driven and hence, spatially correlated -- is asymptotically consistent. We have also provided an order on the rate of convergence in probability for the VS-based regression estimator. Next, we have established that the asymptotic mean squared error of the VS-based estimation is highly resistant to the magnitudes of the noise variances associated with the corrupted observations. But, the accuracy of the estimation can be hampered if the proportion of noisy observations increases. In addition, we have also considered the asymptotic MSE of the ordinary least squares estimator and showed how it depends on both the proportion of noisy observations as well as the magnitudes of the noise variances. We have provided empirical justifications of the advantages of the VS-based analysis through rigorous simulations. Finally, we have implemented the VS-based approach on the coal ash data and compared the results with that of robust REML technique -- which shows that the VS-based approach has advantage both in terms of prediction accuracy and computational time.
There are several future directions to this work. First, the theoretical properties of the VS-based covariance estimation and kriging under similar spatial asymptotic framework can be explored. Second, a version of VS for spatio-temporal data will be of high interest, especially for analyzing real-time noisy crowdsourced data. Third, the results of this paper can be extended for spatial processes on $\mbox{${\mathbb{R}^{d}}$}$ with $d \geq 3$. That way the proerties of the VS and the VS-based estimation will be extended for spatio-temporal data as well.
\section*{Acknowledgements}
This research is partially funded by National Science Foundation (NSF) grant DMS-1613192. The authors would like to thank Dr. Alyson Wilson for bringing the importance of this work to authors' attention and also for her insightful remarks during the course of this work.
|
1,314,259,996,738 | arxiv | \section{Introduction}
\section{Introduction}
Recently, the LHCb collaboration announced a measurement
of the difference between the time-integrated CP asymmetries in two singly Cabibbo
supressed (SCS) $D$ meson decay modes \cite{:2011in},
\begin{equation}
\begin{split}\label{DeltaACP1}
\Delta {\cal A}_{CP}&\equiv {\cal A}_{CP}(D\to K^+K^-) -
{\cal A}_{CP}(D\to \pi^+\pi^-) \\
&= (-0.82 \pm 0.21 \pm
0.11)\%,
\end{split}
\end{equation}
which has been confirmed by the CDF measurement~\cite{CDF-talk}
\begin{equation} \label{eq:CDFACP}
\Delta {\cal A}_{CP}= (-0.62 \pm 0.21 \pm
0.10)\%\,.
\end{equation}
The updated world average for the difference of the direct CP asymmetries is then $\Delta {\cal A}_{CP}^{\rm dir} = (-0.67 \pm 0.16)\%$
\cite{CDF-talk}. In the SM, CP violation (CPV) in SCS $D$ decays comes from the
interference of the tree and penguin amplitudes and is
parametrically suppressed by $\mathcal O(V_{cb} V_{ub}/V_{cs}
V_{us})\sim 10^{-3}$. However, the uncertainties on these order of magnitude estimates are large~\cite{Brod:2011re}.
In particular, the penguin contraction power corrections could be significantly
enhanced, possibly leading to the observed CP asymmetry.
The predictions for direct CPV in charm decays are notoriously
difficult to make, even if one is aiming at order of magnitude
estimates. For instance, in \cite{Golden:1989qx} and recently in \cite{Brod:2011re, Pirtskhalava:2011va, Feldmann:2012js, Brod:2012ud}
it was argued that large CP asymmetries can be expected in SCS $D$ decays.
In~\cite{Buccella:1994nf,Cheng:2012wr,Li:2012cf}, small CP asymmetries were obtained, while marginal consistency with measurements \eqref{DeltaACP1}, \eqref{eq:CDFACP} was found in \cite{Franco:2012ck}. At
the same time, there are viable new physics models (NP) that can enhance
$\Delta {\cal A}_{CP}$ and simultaneously avoid constraints from other
flavor violation searches, such as $D-\bar D$ mixing, as shown in
\cite{Feldmann:2012js,Grossman:2006jg,Isidori:2011qw,Wang:2011uu,Hochberg:2011ru,Chang:2012gn,Giudice:2012qq,Altmannshofer:2012ur,Chen:2012am,Hiller:2012wf}.
The aim of this work is to provide experimental tests that can
distinguish between a SM and NP origin for $\Delta {\cal A}_{CP}$.
The basic idea it to take advantage of the fact that the penguin
amplitudes are $\Delta I=1/2$ transitions. A prediction of
the SM is that CPV effects
are confined to the $\Delta I=1/2$
amplitude, to very good approximation (to be quantified below). Any observation of CPV effects due to the other possible isospin
reduced matrix element, namely the $\Delta I=3/2$ amplitude, would be a clear signal of new physics.
This insight can be used to experimentally search for NP in charm decays without relying on theoretical calculations. To do so we construct a set of CP asymmetry sum rules that will be obeyed if
the observed CP asymmetries are due to the SM, but violated if they
are due to NP. It is important to note that we can search in this way
only for a subset of NP models -- the ones that generate CPV in the $\Delta I=3/2$ reduced matrix elements. An example is a model with a single new
scalar field with nontrivial flavor couplings, as recently proposed in~\cite{Hochberg:2011ru} to explain the observed $\Delta {\mathcal A}_{CP}$. On the other hand, NP models that only
contribute through penguin $\Delta I=1/2$ operators would not violate
the derived sum rules and are thus much harder to distinguish from the SM
contributions to $\Delta {\cal A}_{CP}$. Examples are provided by flavor violating supersymmetric squark-gluino loops that mediate the
$c\to u g$ transition~\cite{Grossman:2006jg,Giudice:2012qq,Hiller:2012wf}.
In the derivation of the sum rules we use the isospin flavor symmetry
of QCD. Isospin symmetry is broken at ${\mathcal O}(1\%)$, which is
also the size of the CP asymmetries we are interested in. Thus special care is needed in order to avoid the introduction of
large errors in the sum rules. There are several
sources of isospin breaking that could modify the sum rules.
The $u$ and $d$ quark masses and electromagnetic interactions
break isospin in a CP conserving manner. Their effects are explicitly included in
the sum rules so that they cancel up to quadratic order in isospin breaking.
The electroweak penguin operators provide CP violating sources for isospin breaking.
However, their effects are suppressed by $\alpha/\alpha_S\sim {\mathcal O}(10^{-2})$ compared to
the leading CP violating but isospin conserving penguin contractions of the $Q_{1,2}$ operators, and can thus be safely neglected.
The paper is organized as follows. Our notation is introduced in Section \ref{Sec:Preliminaries},
the sum rules are derived in Section \ref{Sec:searching}, and we conclude in Section \ref{Conclusions}.
\section{Preliminaries}
\label{Sec:Preliminaries}
The CP-conjugate decay amplitudes for SCS
decays can be written as
\begin{equation}
\begin{split}
A_f (D \to f ) &= T_{f} +P_f e^{i(\delta^P_f-\gamma)}+A_f^{NP}e^{i(\delta^{NP}_f-\phi)},\\
\overline{A}_{\overline f}(\bar D \to \bar{f} ) &= T_{f}+P_f e^{i(\delta^P_f+\gamma)}+A_f^{NP}e^{i(\delta^{NP}_f+\phi)},
\end{split}
\end{equation}
where $T_{f} $ is the dominant SM ``tree" amplitude. It is proportional
to $V_{ud} V_{cd}^*$ and is taken to be real by convention.
The SM ``penguin" amplitude has magnitude $P_f $. It is CKM suppressed by ${\mathcal O}(V_{cb} V_{ub}/V_{cs} V_{us})$
compared to the tree amplitude. It carries the CKM weak phase
$\gamma=(67.3^{+4.2}_{-3.5})^\circ$~\cite{Charles:2011va} and the relative strong
phase $\delta_f $.
The NP amplitude has magnitude $A_{f}^{NP}$,
and carries a strong phase $\delta^{NP}$ and a weak phase $\phi$ relative
to the tree amplitude.
The direct CP asymmetry is given by
\begin{equation}\label{Adir.eq}
\begin{split}
{\cal A}_f^{\rm dir} &\equiv \frac{|A_f |^2-| \bar A_{\bar f }|^2}
{|A_f |^2 + | \bar A_{\bar f } |^2}= \\
& = 2 r_f^P \sin \gamma \sin \delta_f^P+2 r_f^{NP} \sin \phi \sin \delta_f^{NP},
\end{split}
\end{equation}
where $r_f^P\equiv P_f/T_f$, $r_f^{NP}\equiv A_f^{NP}/T_f$, and we have
neglected higher orders in $r_f^{P}$ and $r_f^{NP}$. The question we
are interested in is how one can distinguish between the SM
contributions to the direct CP asymmetries, proportional to $r_f^P$, and
the NP contributions to the direct CP asymmetries, proportional to
$r_f^{NP}$. To do so, we will utilize the transformation properties of
the SM and NP contributions under isospin.
We first review the transformation properties of the SM contributions under isospin. The effective $\Delta C=1$ weak
Hamiltonian, $H_{\rm eff}, $ is~\cite{Buchalla:1995vs}
\begin{equation}
\begin{split}
H_{\rm eff}^{\Delta C=1} &= \frac{G_F}{\sqrt{2}} \Big[\sum_{p=d,s}
V_{cp}^* V_{up} \left(C_1 Q_1^p + C_2 Q_2^p \right) \\
&-V_{cb}^* V_{ub} \sum_{i=3}^{6}
C_i Q_i + C_{8g} Q_{8 g} \Big]
+ {\rm h.c.} .
\end{split}
\end{equation}
The ``tree" operators are
\begin{equation}
\begin{split}
Q_1^p&=(\bar p c)_{V-A}(\bar u p)_{V-A}, \\
Q_2^p&=(\bar p_\alpha c_\beta)_{V-A} (\bar u_\beta p_\alpha)_{V-A}
\end{split}
\end{equation}
with summation over color indices $\alpha,\beta$ understood. The
QCD penguin operators are
\begin{equation}
\begin{split}
Q_{3,5}&=(\bar u c)_{V-A} \sum_{q=u,d,s} (\bar qq)_{V\mp
A},\\ Q_{4,6}&=(\bar u_\alpha c_\beta)_{V-A} \sum_{q=u,d,s} (\bar q_\beta
q_\alpha)_{V\mp A}, \\
Q_{8g} &= -\frac{g_s}{8\pi^2}\, m_c \bar u
\,\sigma_{\mu\nu}(1+\gamma_5) G^{\mu\nu} c\,,
\end{split}
\end{equation}
and we do not display the numerically further suppressed electroweak penguin operators.
The flavor structure of the tree operators for $D\to \pi\pi$ and $D_s\to
K\pi$ decays is $(\bar d c)(\bar u d)$. Thus they have both $\Delta I=3/2$
and $\Delta I=1/2$ components. The remaining operators, i.e.,
the $(\bar s c)(\bar u s)$ tree operators for $D\to KK$ decays and the penguin operators, are purely
$\Delta I=1/2$. Note that the penguin contraction contributions of the $(\bar d c)(\bar u d)$ tree operators to $D\to KK$
are $\Delta I =1/2$.
The NP models that contribute to the CP asymmetries in SCS $D$ decays can
be grouped in to two sets, (i) those in which the NP operators are purely $\Delta
I=1/2$, and (ii) those in which the NP operators also have $\Delta I=3/2$ components.
As we show below, one can use the isospin decomposition and the resulting sum
rules to search for the presence of $\Delta I=3/2$ NP
just using experimental information.
\section{Searching for new physics via isospin}
\label{Sec:searching}
We now derive CP asymmetry sum rules that can be used to probe for the
presence of $\Delta I=3/2$ NP contributions. Among the SCS decays,
the $D\to\pi\pi$, $D\to\rho\pi$, $D\to\rho\rho$, $D\to K\bar K\pi$ and $D_s\to K^{*}\pi$ modes
carry enough information for such tests. We discuss each
of them in turn.
\subsection{$D\to \pi\pi$ and $D\to \rho\rho$ decays}
The isospin decomposition of the $D^0\to \pi\pi$ decays is
\begin{subequations}\label{Apipi}
\begin{align}
A_{\pi^+\pi^-}&=\sqrt{2}{\cal A}_3+\sqrt2 {\cal A}_1,\\
A_{ \pi^0\pi^0}&=2 {\cal A}_3-{\cal A}_1,\\
A_{\pi^+\pi^0}&=3 {\cal A}_3,
\end{align}
\end{subequations}
where ${\cal A}_3$ and ${\cal A}_1$ are the reduced matrix elements
for the $\Delta I=3/2$ and $\Delta I=1/2$ Hamiltonians. The phase convention used is such that $(u,d)$, $(\bar d, -\bar u)$, $(D^+,D^0)$, $(K^+, K^0)$ and $(\bar K^0, K^-)$ form isospin doublets, while $(\pi^+,\pi^0,\pi^-)$ form a triplet. For the
$\bar D^0\to
\pi\pi$ system the isospin decomposition is similar to \eqref{Apipi},
with ${\cal A}_3$, ${\cal A}_1$ replaced by the CP conjugate matrix elements
$\bar {\cal A}_3$, $\bar {\cal A}_1$.
We decompose the reduced matrix elements into SM
and NP contributions, with magnitudes $A_k$ and $a_k$,
respectively,
\begin{equation} \label{eq:not-sep}
{\cal A}_k= A_k e^{i(\delta_k^A-\phi_k^A)}+a_ke^{i(\delta_k^a-\phi_k^a)}, \qquad k=1,3.
\end{equation}
By convention we can set the strong phase $\delta_3^A=0$. In the SM
the weak phase of the $\Delta I=3/2$ amplitude is
also zero to excellent approximation, so that
we can set $\phi_3^A=0$. Thus, in the SM the purely $\Delta I=3/2$ decay $D^+\to \pi^+\pi^0$ has
${\mathcal A}^{\rm dir}(D^+\to \pi^+\pi^0)=0$. However, the rate difference can be nonzero in the presence of NP, being given by
\begin{equation} \label{eq:dpipi}
|A_{\pi^+\pi^0}|^2-|\bar A_{\pi^-\pi^0}|^2=
36 a_3 A_3 \sin\phi_3^a \sin \delta_3^a.
\end{equation}
Note that the CP asymmetry is proportional to the $\Delta I=3/2$ NP
coefficient $a_3$.
Let us comment on the isospin
breaking effects that have been ignored in the decomposition of \eqref{Apipi}. The isospin breaking due to the $u$, $d$ quark masses and due to the electromagnetic interactions can be safely neglected since they are CP
conserving. Thus, they only modify ${\mathcal A}_{CP}(D^+\to \pi^+\pi^0)$ at
second order in small parameters. While ${\mathcal A}_{CP}(D^+\to
\pi^+\pi^0)\sim {\mathcal O}(r_f^{NP})$, the effect of isospin
breaking is ${\mathcal O}(\epsilon_I r_f^{NP, EWP} )$, where
$\epsilon_I$ is the typical size of isospin breaking. It is of order
$1\%$ and may be enhanced by at most a factor of a few. Similarly the
electroweak penguins can be neglected due to the small sizes of their
Wilson coefficients. Thus, we conclude that a measured nonzero CP
asymmetry in $D^+ \to \pi^+\pi^0$ would be a signal for $\Delta I=3/2$
NP.
Note that if a direct CP asymmetry is not found in $D^+\to \pi^+\pi^0$, this does
not mean that $\Delta {\mathcal A}_{CP}$ cannot be due to a new $\Delta I=3/2$ amplitude. It is
possible, for instance, that the strong phase difference $\delta_3^a$
between the NP and SM $\Delta I=3/2$ amplitudes is simply smaller
than the strong phase difference between the $\Delta I=3/2$ and $\Delta I=1/2$ amplitudes.
We therefore devise two more tests for the presence of new CP violating
phases in the $\Delta I=3/2$ operators. The first involves the sum of
rate differences
\begin{equation}\label{sum:rate:diff}
\begin{split}
|&A_{\pi^+\pi^-}|^2-|\bar A_{\pi^-\pi^+}|^2+|A_{\pi^0\pi^0}|^2-|\bar
A_{\pi^0\pi^0}|^2\\
&-\frac{2}{3}\big(|A_{\pi^+\pi^0}|^2-|\bar A_{\pi^-\pi^0}|^2\big) = 3 \left(|{\cal A}_1|^2- |\bar {\cal A}_1|^2\right).
\end{split}
\end{equation}
The important point is that this sum only depends on the $\Delta I=1/2$
amplitudes. Thus, if the sum is found to be nonzero this means that
there are $\Delta I=1/2$ contributions to the CP asymmetries. They
could be due to NP or they could be due to the SM. However, if the sum \eqref{sum:rate:diff} is found to
be zero, while the individual rate differences are nonzero, this would indicate
that the CP asymmetries are likely dominated by $\Delta I=3/2$ NP
contributions. This statement does come with a caveat. It would still be possible that, whereas the CPV weak phases are only present in the $\Delta I=1/2$ amplitude,
the strong phases between terms in ${\cal A}_1$ with different weak phases are small. In this case, ${\cal A}_{CP}(\pi^+\pi^-)$, and ${\cal A}_{CP}(\pi^0\pi^0)$ would be nonzero due to interference of the $\Delta I=1/2$ and $\Delta I=3/2$ amplitudes.
This possibility can be checked with more data if time dependent
$D(t)\to \pi^+\pi^-$ and $D(t)\to \pi^0\pi^0$ measurements become
available, or if there is additional information on relative phases
from a charm factory running on the $\Psi(3770)$ (for feasibility see, e.g. \cite{Bevan:2011up}). It amounts to measuring
the weak phase of the $\Delta I=3/2$ amplitude ${\cal A}_3$ via
generalized triangle constructions that also take isospin
breaking into account. From the isospin decomposition we have an isospin
sum rule
\begin{equation}
\begin{split}
\frac{1}{\sqrt2} &A_{\pi^+\pi^-}+A_{\pi^0\pi^0}-A_{\pi^+\pi^0}=A_{\rm break},
\end{split}
\end{equation}
and a similar sum rule for the CP-conjugate decays. The amplitude $A_{\rm break}$ is due to isospin breaking and is of order ${\mathcal O}(\epsilon_I A_{i})$.
It is equal in $D\to \pi\pi$ and $\bar D\to \pi\pi$ decays, i.e., $A_{\rm break}=\bar A_{\rm break}$, up to very small CP violating corrections which are down by an extra factor of $r_f\lesssim {\cal O}(0.01)$. One therefore has the following sum rule, valid even in the presence of isospin breaking,
\begin{equation}\label{triangle-rel}
\begin{split}
&\frac{1}{\sqrt2} A_{\pi^+\pi^-}+A_{\pi^0\pi^0} -\frac{1}{\sqrt2} \bar A_{\pi^-\pi^+}-\bar A_{\pi^0\pi^0}=\\
&~~3\big({\cal A}_3-\bar{\cal A}_3\big)=-6i a_3
e^{i\delta^a_3} \sin\phi^a_3,
\end{split}
\end{equation}
where in the last stage we use the fact that $A_3$ carries a negligible CP violating phase
in the SM. Note that isospin breaking in this relation has canceled (up to corrections quadratic in small parameters). Therefore, if
\begin{equation}\label{triangle-rel2}
\begin{split}
\frac{1}{\sqrt2} \big(A_{\pi^+\pi^-}- \bar A_{\pi^-\pi^+}\big)\ne -\big(A_{\pi^0\pi^0} -\bar A_{\pi^0\pi^0}\big),
\end{split}
\end{equation}
is found, this would mean there is CPV NP in the $\Delta I=3/2$ amplitude. The relative phases between the $A_{\pi^+\pi^-}$ and $\bar A_{\pi^-\pi^+}$ amplitudes and between the $ A_{\pi^0\pi^0}$ and $\bar A_{\pi^0\pi^0}$ amplitudes can be measured in entangled $\psi(3770)\to D\bar D$ decays.
In addition, the phase between the $A_{\pi^+\pi^-}$ and
$\bar{A}_{\pi^-\pi^+}$ amplitudes can be obtained from the time
dependent $D(t)\to \pi^+\pi^-$ decay. Similarly, the phase between
the $ A_{\pi^0\pi^0}$ and $\bar A_{\pi^0\pi^0}$ amplitudes can be
obtained from the time dependent $D(t)\to \pi^0\pi^0$ decay. The magnitudes of the amplitudes can be measured in their respective time integrated decays. We can thus form an experimental test. If
\begin{equation}\label{triangle-rel3}
\begin{split}
\frac{1}{\sqrt2} \big|A_{\pi^+\pi^-}- \bar A_{\pi^-\pi^+}\big|\ne \big|A_{\pi^0\pi^0} -\bar A_{\pi^0\pi^0}\big|,
\end{split}
\end{equation}
then a $\Delta I=3/2$ NP amplitude has been discovered.
While the above formalism has been written down for $D\to \pi\pi$ decays, it applies without changes to $D\to \rho\rho$ decays, but for each polarization amplitude separately. As long as the polarizations of the $\rho$ resonances are measured (or if the longitudinal decay modes dominate, as is the case in $B\to \rho\rho$ decays), the search for $\Delta I=3/2$ NP could be easier experimentally in $D\to \rho \rho$ decays.
\subsection{$D\to \rho\pi$ decays}
Another experimentally favorable probe is the isospin analysis of the $D\to \pi^+\pi^-\pi^0$ Dalitz plot in terms of the $D\to \rho\pi$ decays.
The isospin decomposition for $D^0$ decays is
\begin{align}
A_{\rho^+\pi^-}&={\cal A}_3+{\cal B}_3+\frac{1}{\sqrt2}{\cal A}_1+{\cal B}_1,\\
A_{\rho^0\pi^0}&=2{\cal A}_3-{\cal B}_1,\\
A_{\rho^-\pi^+}&={\cal A}_3-{\cal B}_3-\frac{1}{\sqrt2}{\cal A}_1+{\cal B}_1,
\end{align}
and for $D^+$ decays it is
\begin{align}
A_{\rho^+\pi^0}&={\frac{3}{\sqrt2}}{\cal A}_3-\frac{1}{\sqrt2}{\cal B}_3+{\cal A}_1,\\
A_{\rho^0\pi^+}&={\frac{3}{\sqrt2}}{\cal A}_3+\frac{1}{\sqrt2}{\cal B}_3-{\cal A}_1,
\end{align}
where ${\cal A}_{3}, {\cal B}_3$ are the $\Delta I=3/2$ amplitudes for $I=2,1$ final states, while ${\cal A}_1, {\cal B}_1$ are $\Delta I=1/2$ amplitudes for $I=1,0$ final states, and we have assumed that the $\Delta I=5/2$ amplitude is negligibly small (since these interactions are small and CP conserving they would introduce corrections to our result that are only quadratic in small parameters).
The $D\to \pi^+\pi^-\pi^0$ decay is dominated by the isospin 0 final state \cite{Gaspero:2008rs}, which means that the reduced amplitude ${\cal B}_1$ is expected to be the largest.
From the Dalitz plot for $D^0\to \pi^+\pi^-\pi^0$ one can measure the
relative phases of $A_{\rho^+\pi^-}, A_{\rho^0\pi^0}$ and
$A_{\rho^-\pi^+}$, as well as their magnitudes. The sensitivity to
phases comes from the overlaps of the $\rho$ resonances in the Dalitz
plot. This means that the magnitudes and phases (up to an overall
phase) of the reduced matrix elements ${\cal B}_1$, ${\cal A}_3$, and
${\cal B}_3+{\cal A}_1/\sqrt2$ are measurable. If the time dependent
Dalitz plot is measured then the relative phases between the ${\cal
A}_i$ and CP conjugate $\bar {\cal A}_i$ could be measured.
We first discuss CP asymmetry sum rules that can be obtained from time
integrated Dalitz plot measurements. We again employ a notation in which
the strong and weak phases of the SM and NP contributions appear explicitly, as in \eqref{eq:not-sep}. The notation we use is the straightforward generalization of \eqref{eq:not-sep}, with $\delta^{A}_{1,3}$, $\delta^{B}_{1,3}$ the SM strong phases, $\phi^{A}_{1,3}$, $\phi^{B}_{1,3}$ the SM weak phases, $A_{1,3}$, $B_{1,3}$ the magnitudes of the SM reduced amplitudes, while NP contributions are denoted by small letters, $A\to a$, $B\to b$. By convention, the strong phase of the
SM amplitude $A_3$ is taken to be zero, $\delta_3^A=0$. The weak phases
of the SM tree amplitudes $A_3$ and $B_3$ are also zero, $\phi_3^A=\phi_3^B=0$. There are two
combinations of measured amplitudes that are proportional to $\Delta
I=3/2$ amplitudes
\begin{equation}\begin{split}
A_{\rho^+\pi^0}+A_{\rho^0\pi^+}&=3 \sqrt2 {\cal A}_3,\\
A_{\rho^+\pi^-}+2A_{\rho^0\pi^0}+A_{\rho^-\pi^+}&=6{\cal A}_3.
\end{split}\end{equation}
A measurement of the second sum can be obtained from the $D^0\to \pi^+\pi^-\pi^0$ Dalitz plot.
If the related CP asymmetry
\begin{equation}
\begin{split}
|A_{\rho^+\pi^-}+&2A_{\rho^0\pi^0}\!+\!A_{\rho^-\pi^+}\!|^2-
|\overline{A}_{\rho^-\pi^+}+2\overline{A}_{\rho^0\pi^0}\!+\!\overline{A}_{\rho^+\pi^-}\!|^2
\\
=&36\big(|{\cal A}_3|^2-|\bar {\cal A}_3|^2\big)=144 A_3 a_3\sin\phi_3^a \sin\delta_3^a,\label{Arhopi-sumrule}
\end{split}
\end{equation}
is found to be nonzero, this would mean that the NP contribution $a_3$ is nonzero. If it is found to vanish, it could still be that this is due to the strong phase difference $\delta_3^a$ being vanishingly small.
Assuming that this is the case, i.e. that $\delta_3^a=0$, one can still test for the presence of $\Delta I=3/2$ CP violating NP. The weighted sum
\begin{equation}
\begin{split}\label{eq:sumrhopi}
&2\big(|A_{\rho^0\pi^0}|^2-|\bar A_{\rho^0\pi^0}|^2\big)+\\
&|A_{\rho^+\pi^-}+A_{\rho^-\pi^+}|^2-|\bar A_{\rho^+\pi^-}+\bar A_{\rho^-\pi^+}|^2\\
&=12 \big(|{\cal A}_3|^2-|\bar {\cal A}_3|^2\big)+6\big(|{\cal B}_1|^2-|\bar {\cal B}_1|^2\big),
\end{split}
\end{equation}
measures whether there is direct CP violation in the ${\cal A}_3$ or ${\cal B}_1$ reduced amplitudes. Let us assume that \eqref{Arhopi-sumrule}
is found to be vanishingly small, so that $|{\cal A}_3|=|\bar {\cal A}_3|$. If the sum \eqref{eq:sumrhopi} is found to be zero as well, while the individual CP asymmetries are nonzero, this would be a strong indication for $\Delta I=3/2$ NP. Again,as in the case of $\pi\pi$, there is a caveat, namely that it is possible that there is no direct CPV in ${\cal B}_1$ even though there are weak phases in ${\cal B}_1$. For instance, this would be the case if the strong phases for terms with different weak phases in ${\cal B}_1$ would be the same. The individual CP asymmetries would then be nonzero due to interference of ${\cal B}_1$ with the other amplitudes, rather than $\Delta I=3/2$ NP.
A definitive answer can be provided by another test that is directly sensitive to the weak phase of ${\cal A}_3$. This test is possible if the time dependent $D(t)\to \pi^+\pi^-\pi^0$ Dalitz plot is measured. In this case the relative phases between the $D^0\to \rho\pi$ and $\bar D^0\to \rho\pi$ amplitudes can be obtained (alternatively one could use time integrated entangled decays of $\psi(3770)$ at the charm factory).
The presence of a weak phase in ${\cal A}_3$ can then be determined from the following sum-rule
\begin{equation}\label{a3rhopi}
\begin{split}
&\big(A_{\rho^+\pi^-}+A_{\rho^-\pi^+}+2 A_{\rho^0\pi^0}\big)-\\
&\big(\bar A_{\rho^-\pi^+}+\bar A_{\rho^+\pi^-}+2 \bar A_{\rho^0\pi^0}\big)=\\
&~~~~~~~~6 \big({\cal A}_3-\bar{\cal A}_3\big)=-12i a_3
e^{i\delta^a_3} \sin\phi^a_3,
\end{split}
\end{equation}
where in the last stage we use the fact that the SM amplitude $A_3$ does not carry a
weak phase. Thus, a non-vanishing result for \eqref{a3rhopi} would provide definitive proof for $\Delta I=3/2$ NP. A similar sum rule for the CP asymmetries rather than the amplitudes
was given in \eqref{Arhopi-sumrule}. In that case the time
integrated Dalitz plot suffices to determine the sum
rule inputs.
\subsection{$D \to K \bar K \pi$ decays}
The isospin decomposition for the $D^0$ decays is
\begin{align}
A_{K^+ {\bar K^0} \pi^-}&={\cal B}_1-{\cal A}_1 +{\cal C}_3 +{\cal B}_3,\\
A_{K^+K^-\pi^0}&={\cal B}_1'+\frac{1}{\sqrt2}{\cal A}_1 +\sqrt2{\cal C}_3 +{\cal B}_3',\\
A_{K^0 {\bar K^0}\pi^0}&=-{\cal B}_1'+\frac{1}{\sqrt2}{\cal A}_1 +\sqrt2{\cal C}_3 -{\cal B}_3',\\
A_{K^0 K^-\pi^+}&=-{\cal B}_1-{\cal A}_1 +{\cal C}_3 -{\cal B}_3,
\end{align}
and for the $D^+$ decays it is
\begin{align}
A_{K^+{\bar K^0} \pi^0}&=\sqrt2 {\cal B}_1 +\frac{3}{\sqrt2}{\cal C}_3 -\frac{1}{\sqrt2}{\cal B}_3,\\
A_{K^+ K^-\pi^+}&=-{\cal B}_1+\sqrt 2 {\cal B}_1'+\frac{3}{2}{\cal C}_3 +\frac12 {\cal B}_3-\frac{1}{\sqrt2}{\cal B}_3',\\
A_{K^0 \bar K^0\pi^+}&=-{\cal B}_1-\sqrt 2 {\cal B}_1'+\frac{3}{2}{\cal C}_3 +\frac12 {\cal B}_3+\frac{1}{\sqrt2}{\cal B}_3',
\end{align}
where ${\cal B}_3, {\cal B}'_3$, ${\cal C}_3$ are $\Delta I=3/2$ amplitudes for $I=1,1,2$ final states with the two kaons in the $I=1,0,1$ isospin state, while ${\cal A}_1, {\cal B}_1$, ${\cal B}'_1$ are $\Delta I=1/2$ amplitudes for $I=0,1,1$ final states with the two kaons in $I=1,1,0$ isospin state.
The same results apply for $D\to K^*\bar K\pi$, $D\to K\bar K^*\pi$ and $D\to K^*\bar K^*\pi$ decays (and decays with a $\rho$ instead of a $\pi$ in the final state) with obvious replacements.
In the case of $D^+$ decays it is possible to construct a purely $\Delta I=3/2$ matrix element by summing only three decay amplitudes, while in the case of $D^0$ decays four amplitudes are needed. For this reason we only consider the $D^+$ decays. For instance, for $D^+$ decays to $K^*\bar K^*$ resonances we have for each polarization (to shorten the notation we do not show the polarizations explicitly)
\begin{equation}
\begin{split}
\sqrt2 &A_{K^{*+}{\bar K^{*0}} \pi^0}+A_{K^{*+} K^{*-}\pi^+}+A_{K^{*0} \bar K^{*0}\pi^+}= 6 \,{\cal C}_3.
\end{split}
\end{equation}
Thus, if the CP violating difference
\begin{equation}\label{testK*K*}
\begin{split}
&|\sqrt2 A_{K^{*+}{\bar K^{*0}} \pi^0}+A_{K^{*+} K^{*-}\pi^+}+A_{K^{*0} \bar K^{*0}\pi^+}|^2 \\
&-|\sqrt2 \bar A_{K^{*-}{ K^{*0}} \pi^0}+\bar A_{K^{*-} K^{*+}\pi^-}+\bar A_{\bar K^{*0} K^{*0}\pi^-}|^2,
\end{split}
\end{equation}
is found to be nonzero, this would mean that there is $\Delta I=3/2$ NP. The relative phases of the three amplitudes can be measured in the five-body decay $D^+\to K^0 K^- \pi^0\pi^+\pi^+$ and its CP conjugate. All three resonant decays, $D^+\to {K^{*+}{\bar K^{*0}} \pi^0}, D^+\to {K^{*+} K^{*-}\pi^+}$ and $D^+\to {K^{*0} \bar K^{*0}\pi^+}$ are part of this final state. The relative phases between the amplitudes can then be obtained from the overlaps of the resonances in the five body final state phase space.
A somewhat more complicated possibility is represented by the $D\to K \bar K^*\pi$ and $D\to K^* \bar K \pi$ decays. A test that is similar to \eqref{testK*K*} can be devised for each of the two sets of decays. If either one of the CP violating differences
\begin{equation}
\begin{split}\label{eq1sum}
&|\sqrt2 A_{K^{+}{\bar K^{*0}} \pi^0}+A_{K^{+} K^{*-}\pi^+}+A_{K^{0} \bar K^{*0}\pi^+}|^2 \\
&-|\sqrt2 \bar A_{K^{-}{ K^{*0}} \pi^0}+\bar A_{K^{-} K^{*+}\pi^-}+\bar A_{\bar K^{0} K^{*0}\pi^-}|^2,
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}\label{eq2sum}
&|\sqrt2 A_{K^{*+}{\bar K^{0}} \pi^0}+A_{K^{*+} K^{-}\pi^+}+A_{K^{*0} \bar K^{0}\pi^+}|^2 \\
&-|\sqrt2 \bar A_{K^{*-}{ K^{0}} \pi^0}+\bar A_{K^{*-} K^{+}\pi^-}+\bar A_{K^{*-} K^{+}\pi^-}|^2,
\end{split}
\end{equation}
is found to be nonzero, this would mean that there is $\Delta I=3/2$ NP.
In order to experimentally construct \eqref{eq1sum} or \eqref{eq2sum}, the magnitudes of the amplitudes and their relative phases need to be measured. To determine the relative phase differences a number of four body decays and their CP conjugates need to be measured. The phase difference between $A_{ K^{*+}K^-\pi^+}$ and $A_{ K^0\bar K^{0*} \pi^+}$ can be measured from the decay $D^+\to K^0K^- \pi^+\pi^+$ (the two amplitudes appear in \eqref{eq2sum} and \eqref{eq1sum}, respectively). The phase difference between $A_{K^{+}K^{*-}\pi^+}$ and $A_{K^{0*}\bar K^{0} \pi^+}$ can be measured from the decay $D^+\to K^+\bar K^0 \pi^-\pi^+$ (they appear in \eqref{eq1sum} and \eqref{eq2sum}, respectively). In order to completely fix all of the required phase differences, the decay $D^+\to K^0\bar K^0\pi^0\pi^+$ or the decay $D^+\to K^+ K^-\pi^0\pi^+$ also needs to be measured (as well as the CP conjugated decays of all the above mentioned modes). From the resonance overlaps in the decay $D^+\to K^0\bar K^0\pi^0\pi^+$, the relative phases of $A_{K^{*0}\bar K^{0} \pi^+}$, $A_{K^{*+}\bar K^{0} \pi^0}$ and $A_{K^{0}\bar K^{*0} \pi^+}$ can be obtained, so that \eqref{eq2sum} is fully determined. Similarly, from the decay $D^+\to K^+ K^-\pi^0\pi^+$ the relative phases of $A_{K^{+}\bar K^{*0} \pi^0}$, $A_{K^{*+} K^{-} \pi^+}$ and $A_{K^{+}K^{*-} \pi^+}$ can be obtained so that, \eqref{eq1sum} is fully determined.
\subsection{$D_s$ decays}
It is also possible to search for CP violation in $\Delta I=3/2$ amplitudes using $D_s^+\to K^*\pi$ decays. The isospin decomposition is
\begin{equation}
\begin{split}
A(D_s^+\to \pi^0 K^{*+})&=\sqrt2 {\cal A}_3 -{\cal A}_1,\\
A(D_s^+\to \pi^+ K^{*0})&={\cal A}_3 +\sqrt2 {\cal A}_1.
\end{split}
\end{equation}
The two decays can be measured from the common Dalitz plot for $D_s^+\to K_S\pi^+\pi^0$, which has $K^{*+}$ and $K^{*0}$ bands
that overlap with the $\rho^+$ band, while the two $K^*$ bands do no overlap directly. From the Dalitz plot analysis one can deduce the phase difference between the two amplitudes and construct the quantity
\begin{equation}
\sqrt2 A(D_s^+\to \pi^0 K^{*+})+A(D_s^+\to \pi^+ K^{*0})=3 {\cal A}_3.
\end{equation}
Direct CP violation in this sum, i.e.,
\begin{equation}
\begin{split}\label{DCPVDs}
&|\sqrt2 A(D_s^+\to \pi^0 K^{*+})+A(D_s^+\to \pi^+ K^{*0})|^2-\\
&|\sqrt2 A(D_s^-\to \pi^0 K^{*-})+A(D_s^-\to \pi^-{ \bar K^{*0}})|^2\ne 0,
\end{split}
\end{equation}
would necessarily be due to $\Delta I=3/2$ NP contributions. Additional information on the absolute value of $|A(D_s^+\to \pi^+ K^{*0})|$ can be obtained from the $D_s^+\to \pi^+ K^+\pi^-$ three body decay.
An analogous test using $D_s \to \rho K^*$ decays also exists, with expressions obtained from the above via the replacement $\pi\to \rho$ and valid for each polarization separately.
The relative phase between $A(D_s^+\to \rho^0 K^{*+})$ and $A(D_s^+\to \rho^+ K^{*0})$ can be measured from the four body decay $D_s^+\to \pi^+\pi^-K^+\pi^0$.
The absolute magnitude $|A(D_s^+\to \rho^0 K^{*+})|$ can be obtained from the more easily measured decay $D_s^+\to \pi^+\pi^- K_S \pi^+$, and can be used as a further constraint.
In most of the manuscript we kept the final states $K^0$ and $\bar K^0$ mesons explicit in the notation. When measurements are performed they will be part of the $K_S$ meson.
In checking for the presence of $\Delta I=3/2$ NP one thus needs to keep track of the CP violation in the neutral kaon system. This effect cannot be neglected as it generates CP
asymmetries of order $\epsilon_K$. However, this effect can be taken into
account explicitly by appropriately modifying the above sum rule equations and also by correcting for the time dependence efficiency for detecting the $K_S$~\cite{Grossman:2011zk}.
\section{Conclusions}
\label{Conclusions}
We have presented a set of isospin sum rules for CP asymmetries in singly Cabibbo suppressed $D$ decays that can be used to test for NP explanations of the measured $\Delta {\cal A}_{CP}={\cal A}_{CP}(D\to K^+K^-) - {\cal A}_{CP}(D\to \pi^+\pi^-) $ that originate from a $\Delta I=3/2$ matrix element. The simplest test only requires the measurement of ${\cal A}_{CP}(D^+\to \pi^+\pi^0)$. If this is found to be nonzero then one has discovered NP in the $\Delta I=3/2$ transition. The same is true if ${\cal A}_{CP}(D^+\to \rho^+\rho^0)\ne 0$ is found. Similar sum rules involving several $D\to\pi\pi, \rho \rho, \rho \pi, K^{(*)}\bar K^{(*)}\pi, K^{(*)}\bar K^{(*)}\rho$ or $D_s\to K^*\pi,K^*\rho$ decay amplitudes were also derived. The isospin sum rules \eqref{Arhopi-sumrule}, \eqref{eq1sum}, \eqref{eq2sum}, \eqref{DCPVDs} only require time integrated measurements, while the isospin sum rules \eqref{triangle-rel3}, \eqref{a3rhopi} need time dependent measurements. Generically, if this type of NP is responsible for the bulk of the measured $\Delta {\cal A}_{CP}$, then violations of the isospin sum rules at the order of $\sim {\mathcal O}(0.5\%)$ can be expected, while the sum rules would be zero in the SM, up to corrections that are second order in isospin breaking.
\section*{Acknowledgements}
We thank Brian Meadows and Michael D. Sokoloff for useful discussions. Y.~G. is supported in part by the NSF grant PHY-0757868 and by a grant
from the BSF. A.~K. is supported by DOE grant FG02-84-ER40153. This work was facilitated in part by the workshop "New Physics from Heavy Quarks in Hadron Colliders"
which was sponsored by the University of Washington and supported
by the DOE under contract DE-FG02-96ER40956.
|
1,314,259,996,739 | arxiv |
\section{Introduction}
\label{sec:intro}
Online learning has become an indispensable part of educational institutions. It has emerged as a necessary resource for students and schools all over the globe. The recent COVID-19 pandemic has made the transition to online learning even more pressing.
One very important aspect of online learning is the need to generate homework, test, and exam exercises
to aid and evaluate the learning progress of students~\cite{dunlosky2013improving}. Multiple choice questions (MCQs) are the most common form of exercises~\cite{gierl2017developing} in online education as they can easily be scored automatically.
However, the construction of MCQs is time consuming \cite{davis2009tools} and there is a need to continuously generate new (variants of) questions,
especially for testing, since students tend to share questions and correct answers from MCQs online (e.g., through social media).
The rapid digitization of educational resources opens up opportunities to adopt artificial intelligence (AI) to automate the process of MCQ construction. A substantial number of questions already exist in a digital format, thus providing the required data as a first
step toward building AI systems. The automation of MCQ construction could support both teachers and learners. Teachers could benefit from an increased efficiency in creating questions, in their already high workload. Students' learning experience could improve due to increased practice opportunities based on automatically generated exercises, \newtext{and if these systems are sufficiently accurate, they could power personalized learning~\cite{ma2014intelligent}}.
A crucial step in MCQ creation is the generation of distractors~\cite{liu2017automatic}. Distractors are incorrect options that are related to the answer to some degree. The quality of an MCQ heavily depends on the quality of distractors \cite{davis2009tools}. If the distractors do not sufficiently challenge learners%
, picking the correct answer becomes easy, ultimately degrading the discriminative power of the question. The automatic suggestion of distractors will be the focus of this paper.
Several works have already proposed distractor generation techniques for automatic MCQ creation, mostly based on selecting distractors according to their similarity to the correct answer.
In general, two approaches are used to measure the similarity between distractors and an answer: graph-based and corpus-based methods. \emph{Graph-based} approaches use the semantic distance between concepts in the graph as a similarity measure. In language learning applications, typically
WordNet~\cite{mitkov2009semantic, pino2008selection} is used to generate distractors,
while for factoid questions domain-specific (ontologies) are used to generate distractors~\cite{papasalouros2008automatic, faizan2018automatic, leo2019ontology, alsubait2014generating}.
In \emph{corpus-based methods}, similarity between distractors and answers has been defined as having similar frequency count~\cite{coniam1997preliminary}, belonging to the same POS class~\cite{goto2010automatic}, having a high co-occurrence likelihood~\cite{hill2016automatic}, having similar phonetic \chris{and}
morphological features~\cite{pino2008selection},
and being nearby in embedding spaces~\cite{kumar2015automatic,guo2016questimator,jiang2017distractor}. Other works such as~\cite{liu2017automatic, liang2017distractor,liang2018distractor,liu2016automatic}
use machine learning models to generate distractors by using a combination of the previous features and other types of information such as tf-idf scores.
While the current state-of-the-art in MCQ creation is promising, we see \thms{a number of limitations. First of all, existing models are often \emph{domain specific}.}
Indeed, the proposed techniques are tailored to the application and distractor types. In language learning, such as vocabulary, grammar or tense usage exercises, typically similarity based on basic syntactic \chris{and}
statistical information works well: frequency, POS information\chris{,} etc.\ In other domains, such as science, health, history, geography, etc., distractors should be selected on deeper understanding of context and semantics, and the current methods fail to capture such information.
The second limitation, \emph{language dependency}, is especially applicable to factoids. \thms{Models should be agnostic to language because facts do not change with languages. Moreover, building a new model for each language could be daunting task as it would require enough training data for each language.}
In this work, we study how the automatic retrieval of distractors can facilitate the efficient construction of MCQs. We use a high-quality large dataset of question, answer, distractor triples that are diverse in terms of language, domain, and type of questions. \thms{Our dataset was made available by a commercial organization active in the field of e-assessment (see \secref{sec:data}), and is therefore representative for the educational domain, with a total of 62k MCQ, none of them identical, encompassing only 92k different answers and distractors. Despite an average of 2.4 distractors per question, there is a large reuse of distractors over different questions. This motivates our premise to retrieve and \emph{reuse} distractors for new questions.} We make use of the latest data-driven Natural Language Processing (NLP) techniques to retrieve candidate distractors. We propose \emph{context-aware multilingual models} that are based on deep neural network models that select distractors by taking into account the context of the question. They are also able to handle variety of distractors in terms of length and type. We compare our proposed models to a competitive \emph{feature-based} baseline that is based on classical machine learning methods trained on several handcrafted features.
The methods are evaluated for distractor quality using automated metrics and a real-world user test with teachers. Both the automatic evaluation and the user study with teachers
indicate that the proposed context-aware methods outperform the feature-based baseline.
Our contribution can be summarized as follows:
\begin{itemize}
\item We built three multilingual Transformer-based distractor retrieval models that suggest distractors to teachers for multiple subjects in different languages. The first model (\secref{subsection:dqsim}) requires similar distractors to have similar semantic representations, while the second (\secref{subsection:qsim}) learns similar representations for similar questions, and the last combines the complementary advantages of
of these two models (\secref{subsection:dqsim}).
\item \rebuttal{We performed a user study with teachers to evaluate the quality of distractors proposed by the models, based on a four-level annotation scheme designed for that purpose.}
\item \rebuttal{The evaluation of our best model on in-distribution held-out data reveals an average increase of 20.4\% in terms of recall at 10, compared to our baseline model adapted from~\cite{liang2018distractor}.
The teacher-based annotations on language learning exercises
show an increase by 4.3\% in the fraction of good distractors among the top 10 results, compared to teacher annotations for the same baseline. For factoid questions, the fraction of quality distractors more than doubles w.r.t.~the baseline, with an improvement of 15.3\%.}
\item We \rebuttal{released\footnote{\url{https://dx.doi.org/10.21227/gnpy-d910} or \url{https://github.com/semerekiros/dist-retrieval}}} a test-set of educational questions of 6 subjects with 50 MCQs per subject and annotated distractors, and 77k size distractor vocabulary as benchmark to stimulate further research. The dataset, which is made by experts, contains multilingual and multi-domain
distractors.
\end{itemize}
The remainder of the paper is organized as follows: \secref{sec:relatedworks} describes the relevant work in MCQs in general and distractor generation in particular. \secref{sec:methodology} introduces the dataset, explains the details of the proposed methods and the evaluation setup of the user study with teachers. In \secref{sec:resultsanddiscussion}, the results of both the user study and automated evaluations is reported. And finally, in \secref{sec:conclusionandfuturework}, we present the conclusion, lines for future work, and limitation\chris{s} of our proposed models.
\section{Related work}
\label{sec:relatedworks}
\subsection{MCQs in Education}
Multiple choice questions (MCQs) are widely used forms of exercises that require students to select the best possible answer from a set of given options. They are used in the context of learning, and assessing learners' knowledge \chris{and}
skills.
MCQs are categorized as objective types of questions because they primarily deal with the facts or knowledge embedded in a text rather than subjective opinions~\cite{ch2018automatic}.
It has been shown that recalling information in response to a multiple-choice test question bolsters
memorizing capability, which leads to better retention of that information over time. It can also change the way information is represented in memory, potentially resulting in deeper understanding~\cite{butler2018multiple} of concepts.
An MCQ item consists of three elements:
\begin{itemize}
\setlength{\itemindent}{1em}
\item \emph{stem}: is the question, statement, or lead-in to the question.
\item \emph{key}: the correct answer.
\item \emph{distractors}: alternative answers meant to challenge students' understanding of the topic.
\end{itemize}
For example, consider the {\mcq} in the first row of \tabref{tab:annotationschemeexamples}: the stem of the \mcq{} is \emph{``Which inhabitants are not happy with Ethiopia's plans of the Nile?"}. Four potential answers
are given with the question. Among these, the correct answer is \emph{``Egyptians"}, which is the key. The
alternatives are the distractors.
\newtext{{\mcq}s are used in several teaching domains such as information technology~\cite{woodford2004using}, health~\cite{brady2005assessment,collins2006education}, historical knowledge~\cite{10.2307/40543353}, etc.\ They are also commonly used in standardized tests such as GRE and TOEFL. {\mcq}s are preferred to other question formats because }
they are easy to score, and students
\chris{can} also answer them relatively quickly since typing responses is not required. \newtext{Moreover, {\mcq}s enable a high level of test validity if they are drawn from a representative sample of the content areas that make up the pre-determined learning outcomes~\cite{collins2006education}.}
The most time-consuming and non-trivial task in constructing \mcq{} is distractor generation~\cite{davis2009tools,liang2018distractor}. Distractors should be plausible enough to force learners to put some thought before selecting the correct answer.
Preparing good multiple-choice questions is a skill that requires formal training~\cite{abdulghani2015faculty, naeem2012faculty}.
Moreover, several \mcq{} item writing guidelines are used by content specialists when they prepare educational tests.
These guidelines also include recommendations for developing and using distractors~\cite{haladyna1989taxonomy, haladyna2013developing,moreno2015guidelines}.
Despite these guidelines, inexperienced teachers may still construct poor MCQs due to lack of training and limited time~\cite{vyas2008multiple}.
Besides reducing teachers' workloads, the automation of the distractor generation could potentially correct some minor mistakes made by teachers.
For example, one of the rules suggested by~\cite{haladyna1989taxonomy} says: ``the length of distractors and the key should be about the same''.
Such property could be easily integrated in the automation process.
MCQs \chris{also} have drawbacks; they are typically used to measure lower-order levels of knowledge, and guesswork can be a factor in answering a question with a limited number of alternatives.
\newtext{Furthermore, because of a few missing details, learners' partial understanding of a topic may not be sufficient to correctly answer a question,
resulting in partial knowledge not being credited by {\mcq}s~\cite{butler2018multiple}.}
Nonetheless, \mcq{}s are still extensively utilized in large-scale tests since they are efficient to administer and easy to score objectively~\cite{gierl2017developing}.
\subsection{Distractor Generation}
Many strategies have been developed for generating distractors for a given question. The most common approach is to select a distractor based on its similarity to the key for a given question. Many researchers approximate the similarity between distractor and key according to WordNet~\cite{mitkov2006computer,mitkov2003computer,lin2007automatic}. WordNet~\cite{miller1990introduction} is a lexical database that groups words into sets
of synonyms, and concepts semantically close to the key are used as distractors. The usage of such lexical databases is sound for language or vocabulary learning but not for factoid type questions. We instead provide a more general approach that could be used for both tasks, and instead of only using the key as the source of information while suggesting distractors, we also make use of the stem.
For learning factual knowledge, several works rely on the use of specific domain ontology as a proxy for similarity.
Papasalouros~\textit{et al.}\ \cite{papasalouros2008automatic} employ several ontology-based strategies to generate distractors for MCQ questionnaires.
For example, they generate ``Brussels is a mountain" as a good distractor for an answer ``Everest is a mountain" because both concept \emph{City} and concept \emph{Mountain} share the parent
concept \emph{Location}.
Another very similar work by Lopetegui \textit{et al.}\ \cite{lopetegui2015novel} selects distractors that are declared siblings of the answer in
\chris{a} domain\chris{-}specific ontology.
The work by Leo~\textit{et al.}\ \cite{leo2019ontology} improves \chris{upon} the previous works by generating multi-word distractors from an ontology in the medical domain.
Other works that rely on knowledge bases
\chris{apply}
query relaxation methods\chris{,} where the queries used to generate the keys were slightly relaxed to generate distractors that share similar features with the key \cite{seyler2017knowledge,faizan2018automatic,stasaski2017multiple}. While the methods in these works are dependent on
their respective ontologies, we provide an approach that is ontology-agnostic and instead uses
contextual similarity between distractors and questions.
Another
\chris{line of works} for distractor generation
\chris{uses} machine-learning models.
Liu~\textit{et al.}\ \cite{liu2017automatic} use a regression model based on characteristics such as character glyph, phonological, and semantic similarity for generating distractors in Chinese. Liang~\textit{et al.}\ \cite{liang2018distractor} use two methods to rank distractors in the domain of school sciences.
\chris{The first method adopts} machine learning classifiers on manually engineered features (i.e., edit distance, POS similarity, etc.) to rank distractors.
\chris{The second uses} generative adversarial networks to rank distractors.
Our baseline method is inspired by their first approach but was made to account for the multilingual nature of our dataset by extending the feature set.
There have also been a number of works on
generating distractors in the context of machine comprehension~\cite{lai2017race}.
Distractor generation strategies that fall in this category assume access to a contextual resource such as a book chapter, an article or a wikipedia page where the MCQ was produced from.
The aim is then to generate a distractor that takes into account the reading comprehension text, and a pair \chris{composed} of \chris{the} question and its correct answer that originated from the text~\cite{gao2019generating,zhu2018hierarchical,chung2020bert}.
This line of work is incomparable to our work because we do not have access to an external contextual resource
the questions were prepared from.
In this paper, we focus on building one model that is able to suggest candidate distractors for teachers both in the context of language and factual knowledge learning.
Unlike previous methods, we tackle distractor generation with a multilingual dataset.
Our distractors are diverse both in terms of domain and language.
Moreover, the distractors are not limited to single words only.
\section{Methodology}
\label{sec:methodology}
In this section, we formally define distractor generation as a ranking problem; describe our datasets; describe in detail the feature-based baseline and proposed context-aware models including their training strategies \& prediction mechanisms.
\subsection{Task Definition: Distractor Retrieval}
\label{sec:taskdefinition}
We assume access to a
distractor candidate set $\mathcal{D}$ and a training MCQ dataset $\mathcal{M}$. \thms{Note that $\mathcal{D}$ can be obtained by pooling all answers (key\chris{s} and distractors) from $\mathcal{M}$ (as in our experimental setting), but could also be augmented, for example, with keywords extracted from particular source texts. We formally write
\chris{$\mathcal{M} = \left \{ (s_{i}, k_{i}, \mathcal{D}_i) | i=1, \ldots, N\right\}$.}
where for each item $i$ among all $N$ available MCQs, $s_{i}$ refers to the question stem, $k_{i}$ is the correct answer key, and $\mathcal{D}_{i} = \big\{d_{i}^{(1)},...,d_{i}^{(m_i)} \big\} \subseteq \mathcal{D}$ are the distractors in the MCQ linked to $s_{i}$ and $k_{i}$. }
The aim of the distractor retrieval task is to learn a point-wise ranking score $r_i(d): (s_{i}, k_{i}, d) \to \left [ 0, 1\right ]$ for all $d \in \mathcal{D}$, such that distractors in $\mathcal{D}_{i}$ are ranked higher than those in $\mathcal{D}\setminus \mathcal{D}_{i}$, when sorted according to the decreasing score $r_i(d)$. \\
This task definition resembles the metric learning \cite{kulis2013metric} problem in information retrieval. To learn the ranking function, we propose two types of models: feature-based models and context-aware \chris{n}eural \chris{n}etworks.
\begin{table}[t!]
\centering
\caption{The statistics of our dataset}
\label{tab:dataset}
\begin{tabular}{llll}
\hline
& \textbf{Train} & \textbf{Validation} & \textbf{Test} \\ \hline
\# Questions & 61758 & 600 & 500 \\
\# Distractors per question & 2.4 & 2.3 & 2.3 \\
Avg question length & 27.8 tokens & 28.1 tokens & 27.6 tokens \\
Avg distractor length & 2.2 tokens & 2.3 tokens & 2.1 tokens \\
Avg answer length & 2.2 tokens & 2.3 tokens & 2.2 tokens \\ \hline
Total \# distractors & 94,205 & - & - \\
Total \# distractors $\leq$ 6 tokens & 77,505 & - & - \\\hline
\end{tabular}
\end{table}
\subsection{Data}
\label{sec:data}
In this section, we describe our datasets, namely: \begin{enumerate*}[(i)]
\item \emph{Televic dataset}, a big dataset
\chris{that} we used to train our models.
\item \emph{Wezooz dataset}, a small-scale external test set used for evaluation.
\end{enumerate*}
\subsubsection{Televic dataset}
\label{sec:televicdataset}
this data is gathered through Televic Education's platform
assessmentQ.\footnote{\url{https://www.televic-education.com/en/assessmentq}} The tool is a comprehensive online platform for interactive workforce learning and high-stakes exams. It allows teachers to compose their questions and answers for practice and assessment. As a result, the dataset is made up of a large and high-quality set of questions, answers and distractors,
manually created
by experts in their respective fields. It encompasses a wide range of domains, subjects, and languages, without however any metadata on the particular course subjects that apply to the individual items.
We randomly divide our dataset into train/validation/test splits.
We discard distractors with more than 6 tokens \lucas{as they are very rare} and unlikely to be reused in different contexts. We keep questions with at least one distractor. \Tabref{tab:dataset} summarizes the statistics of our dataset.
The dataset contains around 62k MCQs in total. The size of the dataset is relatively large when compared to previously reported educational MCQ datasets such as SCiQ \cite{welbl2017crowdsourcing}, and MCQL \cite{liang2018distractor} which contain 13.7K and 7.1K MCQs respectively. On average, a question has more than 2 distractors, \rebuttal{and contains exactly one answer}. We use all the answer keys and distractors in the preprocessed dataset as the pool of candidate distractors (i.e., list of 77,505 filtered distractors) for proposing distractors for any new question.
The distractors in the dataset are not limited to single word distractors. More than 65\% of the distractors contain two or more words as can be seen in \figref{fig:answer_length}.
\begin{figure}[t]
\subfloat[]{%
\label{fig:answer_length}
\resizebox{0.5\textwidth}{!}{%
\begin{tikzpicture}
\begin{axis} [
axis x line*=bottom,
axis y line*=left,
ybar,
ymin=0,
xlabel={No of words per distractors},
ylabel={Frequency \%}, ]
\addplot coordinates {(1, 32.76389301059314) (4, 11.02150884481891) (5, 9.901552197979433) (2, 26.134472213978814) (3, 16.994180870417917) (6, 3.184392862211785)};
\end{axis}
\end{tikzpicture} %
}%
}
\subfloat[]{%
\label{fig:language_distribution}
\resizebox{0.5\textwidth}{!}{%
\begin{tikzpicture
\begin{axis} [
x tick label style={
/pgf/number format/1000 sep=},
axis x line*=bottom,
axis y line*=left,
ybar interval=0.5,
xmajorgrids=false,
ylabel={Frequency \%},
xlabel={Language\footnotemark},
ymin=0,
symbolic x coords={nl,fr,en,de, es,it,zh,pt,af,la},
x tick label style={anchor=north, text height=1.5ex, align=right}
]
\addplot coordinates {(nl, 56.271278118934745)
(fr, 28.688472429921408)
(en, 9.779184829297783)
(de, 1.8836106780362085)
(es, 1.1088485157020587)
(it, 0.7047631168665882)
(zh, 0.5488561519615641)
(pt, 0.194088262432785)
(af, 0.1702249514779344)
(la, 0.14636164052308379)
};
\end{axis}
\end{tikzpicture} %
}%
}
\caption{(a) distractor length in number of tokens and (b) language distribution for the Televic dataset.}
\end{figure}
\footnotetext{We used ISO 639-1:2002 standard for names of languages.}
The data stems from multiple languages. \Figref{fig:language_distribution} shows the language distribution as detected by an off-the-shelf language classifier.\footnote{We used the \emph{langid} language classifier: \url{https://github.com/saffsd/langid.py}}
Given that Televic is a Belgian company, more than 50\% of the questions are in Dutch, while French and English are the next most
common languages in the dataset.
Another dimension of the dataset is its domain diversity.
It comprises questions about language/vocabulary learning (e.g., French and English) and factoids \lucas{covering} subjects such as
Math, Health, History, Geography, and Sciences.
Besides material from secondary school education, it covers materials from assessment tasks for professionals such as training in hospitals or manufacturing firms.
The data is anonymized and contains no customer information.
Depending on the question type we observe different types of distractors. \begin{enumerate*}[(1)]
\item Factoid distractors:
names of people, locations, organizations, concepts, dates.
\item Distractors with numerical elements, such as multiples, factors, rounding errors, etc.\
\item Language distractors: spelling, grammatical, tense, etc.\
\end{enumerate*}
However, the proposed models are agnostic of the type and origin of the data, and the automated evaluation on the Televic test set contains a random sample covering the different question types and origins (see \secref{sec:automaticevaluation}).
\rebuttal{
Note that although our dataset is a real-world commercial dataset, it only contains single-answer \mcq{}s. However, the models we will put forward, could be readily extended towards multiple-answer \mcq{}s, if such data were available.}
\subsubsection{WeZooz dataset}
\label{sec:wezoozdataset}
this data is a small-scale test set of questions gathered from WeZooz Academy\chris{,}\footnote{\url{https://www.wezoozacademy.be/}}
\chris{which} is a Flanders-based company providing an online platform with digital teaching materials for secondary school students and teachers.
We selected four subjects; Natural sciences, Geography, Biology and History.
Each subject was made to contain a fixed list of 50 questions that were randomly selected, \thms{and augmented with distractor annotations by teachers for these respective subjects (see \secref{sec:experimentaldesign}).
Note that this is an \emph{external} test set, in the sense that the data distribution in the training set is not necessarily representative for this test set. This serves as a proof-of-concept for the general validity of our proposed method and models to specific use cases.
}
\subsection{Feature-based \thms{Distractor Scoring}}
We built a strong feature-based model as our baseline. Feature-based models are a class of machine learning models that require a pre-specified set of handcrafted features as input. We designed 20 types of features capturing similarity between questions, answers, and the collection of candidate distractors. Formally, given a triplet \chris{$(s,k, d)$} of question stem, key and distractor,
our feature-based model first maps the input into a 20-dimensional feature vector $\phi\left(s, k, d\right)\;\chris{\in}\;\mathbb{R}^{20}$, after which a classifier is trained to score the triplets according to compatibility of the question-answer-distractor combination. Our set of features can be segmented into four categories which are described below. A more detailed explanation of each feature can be found in \appref{appendix:featureset}.
\begin{enumerate}[(i)]
\item \emph{Morphological Features}: this category contains features that are related to the form and shape of words that occur in our $(s, k, d)$ triplets. This includes features such as edit distance, difference in token length, longest common suffix between $k$ \& $d$, \chris{etc.
\item \emph{Static embedding based features}: We trained a Word2Vec model \cite{mikolov2013efficient} on our dataset to learn static embeddings for the distractors. We treat distractors and answers attached to the same question as chunks sharing similar context. The objective is to learn a vector space in which their representations will also be closer. We leverage the embedding representations to extract several numerical features. For example, we calculate the cosine similarity and word mover's distance \cite{kusner2015word} between the embeddings of $d$ \& $k$.
\item \emph{Language Prior}: since our data is multilingual we also calculate the prior probability of the candidate distractor matching with the language of the question, and attach it to each feature vector.
\item \emph{Corpus-based Features}: this category contains features that are derived from the statistics of words in the corpus. It includes features that such as the frequency of a distractor in the dataset and the inverse document frequency of distractors.
\end{enumerate}
As classifier, we apply a \emph{logistic regression} model to distinguish feature representations of actual question-answer-distractor triplets, present in the training, from triplets for which the distractor components belong to different question-answer combinations, sampled randomly. During training, the model's parameters are set to output high scores for actual triplets while the model is penalized for predicting high scores for others.
\subsection{Context-aware Neural \thms{Distractor Scoring}}
\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth]{figures/qd-sim-for_preprint.pdf}
\caption{Our proposed context-aware distractor retrieval systems. For the \dsim{} model (i.e., left), distractor $d$ and concatenation of the stem $s$ \& key $k$ separated by \SEP{} are fed into the \underline{same} mBERT\textsubscript{(D-SIM)} encoder, and then their respective vector representations at \CLS{} are used as inputs to two \underline{different} dense layers that do not share parameters. The outputs of these dense layers, $h^{(d)}$ \& $h^{(sk)}$ are used to calculate the similarity between $d$ \& $s\SEP k$ using the dot product. Similarly for \qsim{} (i.e., right), two question stems $i$ \& $j$ are encoded separately using the \underline{same} mBERT\textsubscript{(Q-SIM)}, and their respective \CLS output vectors are fed into two \underline{different} dense layers (i.e., dense layer\textsubscript{(i)} \& dense layer\textsubscript{(j)}) to produce their corresponding representations $h^{(s_{i)}}$ \& $h^{(s_{j})}$. These are used to calculate their similarity between the two stems using dot product. The \dqsim{} model (i.e., top) linearly combines the two models using a merging layer with an $\alpha$ parameter. $(\star)$ denotes parameter reuse by the encoders.}
\label{fig:mbert}
\end{figure*}
Advanced context-aware neural models, unlike traditional feature-based models, do not require manual feature engineering. They have the ability to represent words depending on their semantic role and context in the considered text.
In this work, we primarily focus on such context-aware models called \emph{transformers} \cite{vaswani2017attention}, which provide rich representations, and proved to achieve state-of-the-art results for many tasks in NLP.
A transformer is a deep neural network
that uses a self-attention mechanism to assign importance weights to every part of the input sequence in how they are related to all other parts of the input. \thms{Transformers can scale to very large numbers of trainable parameters, stacked into very deep networks, which can still be trained very efficiently on parallel GPU hardware and thus learn from very large amounts of data.}
In NLP, such models are often trained on large unlabeled corpora to learn the inherent word and sentence level correlations (i.e., to model language structure) between varying contexts. This process is called \emph{pretraining}, and downstream NLP tasks usually rely on \thms{such a pretrained generic model to be \emph{finetuned} to their more specific} needs instead of training a new model from scratch. \thms{Leveraging the knowledge gained during a generic pretraining process to improve prediction effectiveness for a specific supervised learning task, is a form of \emph{transfer learning}~\cite{pan2009survey,goodfellow2016deep}}.
A common language task often used for pretraining transformer models called \emph{masked language modelling} (MLM) requires masking a portion of the input text and then training
a model to predict the masked tokens --- in other words, to reconstruct the
original non-masked input. BERT
{Bidirectional Encoder Representations from transformers})~\cite{devlin2018bert} is the most popular pretrained masked language model and has been widely used in many downstream tasks such as question answering \& generation, machine reading comprehension, and machine translation, by fine-tuning it using a labelled dataset that provides supervision signal.
In this work, we
\thms{present models to rank and retrieve distractors, based on such a pretrained transformer text encoder, which we finetuned by}
requiring similar distractors to have similar representations, and similar questions also to have similar representation. \thms{In the following paragraphs, we provide a detailed description of these models, }
visualized in \figref{fig:mbert}, followed by a description of the training procedure and the inference mechanism
\subsubsection{Distractor similarity based model (\dsim)}
\label{subsection:dsim}
We hypothesize that distractors co-occurring within the same MCQ item are semantically related \thms{through their link with the corresponding question stem and answer key.}
\thms{Following that hypothesis, the {\dsim} model is designed (and trained) to yield a similar vector representation for a given (stem, key) pair $(s_i, k_i)$, as each of the corresponding distractors $d_i$.
Following the same logic, all candidate distractors $d\in\mathcal{D}$ can then be scored in terms of their similarity (in representation space) with a given new (stem, key) pair, after which the top candidates are returned by the model as likely valid distractors.
}
\thms{We use the pretrained multilingual BERT (mBERT) encoder \cite{devlin2018bert}, \newtext{followed by a fully connected linear layer (i.e., dense layer)} to obtain initial representations for a (stem, key) pair, as well as for the distractors.
\newtext{We designed our model} in a bi-encoder setting}
inspired by \cite{guo2020multireqa}, \thms{and schematically shown on the left-hand side of \figref{fig:mbert}}.
\thms{The distractor $d$} is fed into the mBERT
encoder, and the output representation of the $[CLS]$ toke
\footnote{[CLS] is a special token that is prepended to the input, and its corresponding output representation is pretrained to represent the entire sequence that is used for classification tasks.}
\newtext{is used as an input to the dense layer. The output from the dense layer is taken as the corresponding representation $h_{d}$.} \delete{is taken as the corresponding representation $h_{d}$.}
\thms{The considered stem and key are concatenated into a single sequence of tokens\footnote{The often used [SEP] token is a special token known by the model, that separates input sentences.} as ``$s_i\;[SEP]\;k_i$'', which}
is fed into the \emph{same} mBERT encoder \thms{(i.e., with parameter reuse, as
indicated by the double arrow in \figref{fig:mbert}).}
Similar to the distractor embedding, we take the \CLS{} token representation \newtext{and feed it to the dense layer (i.e., \emph{different} dense layer with no parameter sharing), and take its output as the vector representation of the key-aware stem $h_{sk}.$} \delete{at the output side as the vector representation of the \thms{key-aware stem $h_{sk}$.}}
Finally, \thms{the similarity score between $(s_i, k_i)$ and $d$ is obtained as the dot product between their respective representations:}
\[r_i^{\text{\dsim}}(d) = h^{(sk)}_{i}\cdot h^{(d)}\]
During training, the encoder is fine-tuned to \thms{achieve higher scores for compatible stem/key and distractor combinations, and lower scores for incompatible ones (as described in \secref{sec:training} in more detail).}
\begin{table*}[t]
\centering
\scriptsize
\caption{{\qsim} training data examples.}
\label{tab:q-sim-example}
\begin{tabular}{l p{4cm}p{5cm}}
\toprule
\textbf{Distractor/Answer} & \textbf{Associated Questions}& \textbf{Description} \\ \midrule
\multirow{6}{2cm}{koolhydraten en vetten} & 1. Welke groepen voedingsstoffen leveren vooral energie? & \multirow{2}{5cm} {Factoid questions with multi-word distractor in Dutch. } \\
& 2. Welke voedselcomponenten kunnen stoffen leveren die zowel bij assimilatie als bij dissimilatie in cellen worden gebruikt?
\\ \midrule
\multirow{2}{4em}{surrounded} & 1. The guest house is \ldots\ on the countryside. & \multirow{2}{5cm}{A fill in the gap question for for English language learning.} \\
& 2. The valley was \ldots\ by forests. \\ \midrule
\multirow{2}{4em}{Marokko} & 1. Welk land is in 2011 gesplitst door het langdurig conflict in Darfur ? & \multirow{2}{5cm}{A combination of fill-in the gap and normal questions} \\
& 2. Rabat is de hoofdstad van \ldots \\ \bottomrule
\end{tabular}
\end{table*}
\subsubsection{Question similarity based model (\qsim)}
\label{subsection:qsim}
This model is based on the assumption that \thms{different questions that share one or more distractors or answer keys
are likely semantically related, such that}
their associated distractors could be used as good candidate distractors for one another. To accomplish this, we first rearrange the training data in such a way that these questions, sharing at least one distractor or key, are clustered together (see \tabref{tab:q-sim-example} for an example). Then, we train our \qsim{} model to produce similar representation for question stem pairs ($s_i, s_j$) that are in the same cluster.
The right-hand side of~\figref{fig:mbert} depicts the \qsim{} model, again based on a bi-encoder architecture.
\thms{The stem representation $h^{(s)}_i$ for a question \mcq$_i$ is again obtained through an mBERT encoder, \newtext{followed by a fully connected linear layer,} similarly to $h^{(sk)}_i$ but ignoring the question key.
The \qsim{} scoring function is defined as}
\[r_i^{\text{\qsim}}(d_{j}) = h_{i}^{(s)}\cdot h_{j}^{(s)}\]
\thms{and can be interpreted as follows. For a given question \mcq$_i$, its stem representation $h_i^{(s)}$ is compared through dot product similarity with the representation of any candidate distractor $d_j$ originating from a question \mcq$_j$. The particular representation of $d_j$ assumed in \qsim{} is in fact \mcq$_j$'s stem representation $h_j^{(s)}$.
Note that \qsim{} does not allow making a distinction in terms of score between different distractors from the same \mcq. Candidate distractors with the same score are considered equally likely according to \qsim{}, and ranked in an arbitrary order. }%
\rebuttal{Based on the intuition outlined above, more complex formulations for \qsim{} can be designed, for example with a feature characterizing the nature of the pairwise comparison (i.e., the actual answers of the considered questions, two of their respective distractors, or the answer for the one and a distractor for the other). However, given the already significant improvement of the presented basic \qsim{} formulation (see \secref{sec:automaticevaluation}), we chose to include only that model in our evaluation. In fact, its simple intuitive formulation makes it straightforward to explain to teachers, which is an important aspect in their trust in the model~\cite{khosravi2022explainable}.}
\subsubsection{Distractor and Question similarity model (\dqsim)}
\label{subsection:dqsim}
this model combines the previous two models using a merging layer \thms{(visualized on top of Fig.~\ref{fig:mbert})}, \rebuttal{based on the intuition that a well-chosen combined model may benefit from the complementary advantages of both individual models}. This \rebuttal{merging} layer
combines the outputs from \dsim{} and \qsim{} using a merging parameter \rebuttal{$\alpha$, to control the contribution of the individual models}.
We investigated
\rebuttal{empirical} score-based and rank-based merging strategies.
The score-based model \rebuttal{assumes a linear combination of}
both respective \rebuttal{\emph{scores}}
$r_i^{\text{\dsim}}$ and $r_i^{\text{\qsim}}$ from \dsim{} and \qsim{}, \rebuttal{in which their individual contribution is controlled by the hyperparameter $\alpha$:}
%
\[
r_i^{\text{\dqsimscore}}(d) = \alpha\,r_i^{\text{\dsim}}(d) + (1-\alpha)\,r_i^{\text{\qsim}}(d)
\]
\thms{The rank-based model combines the distractor \rebuttal{\emph{ranks}} $\rho_{i}^{\text{\dsim}}$ and $\rho_{i}^{\text{\qsim}}\in\{1, 2, 3, ..., N\}$ from \dsim{} and \qsim{} into the score}
\[
r_i^{\text{\dqsimrank}}(d) = \frac{\alpha}{\log \left ( \rho_{i}^{\text{\dsim}}(d) + 1 \right )} + \frac{1-\alpha}{\log \left ( \rho_{i}^{\text{\qsim}}(d) + 1 \right )}
\]
\newtext{This scoring function is based on weighted combination of inverse distractor rankings, \rebuttal{such that} high-ranked distractors have more weight. We use \rebuttal{logarithmic smoothing}
to avoid the potential contribution of low-ranked distractors from \rebuttal{vanishing too rapidly}.
}
\par
\subsection{Training}
\label{sec:training}
We use \emph{contrastive learning} as our training \rebuttal{strategy~\cite{sun2022dual}}. \rebuttal{Contrastive learning~\cite{mikolov2013efficient,chopra2005learning,smith2005contrastive}} is a machine learning technique that aims to learn representations of data by contrasting similar and dissimilar examples. It aims to bring similar instances closer together in the representation space by maximizing the similarity between their embeddings, while pushing dissimilar samples further apart by minimizing their similarity.
In a contrastive learning setting, it is often the case that similar \thms{example pairs} (i.e., also called positive examples) are
available explicitly in training datasets, whereas dissimilar or negative examples need to be sampled from an extremely large pool of instances.
\thms{For the \qsim{} model, a positive pair consists of two questions sharing at least one distractor, whereas for the \dsim{} model, we require similar representations for a given (stem, key) item and a distractor corresponding to the same \mcq.}
As a negative sampling strategy, we use in-batch negatives~\cite{karpukhin2020dense} while training our models. For \dsim, the in-batch negatives are gold-standard positive distractors for the other instances in the same batch. While for \qsim{}, the in-batch negatives are the positive questions that come from the other instances in the same batch. Reusing gold standard distractors or questions from the same batch as negatives \thms{makes training more efficient, compared to randomly sampling negatives for each positive pair in the batch.}
\thms{With the notation $r_i(d)$ (common in both \dsim{} and \qsim{}) for scoring \mcq$_i$ against distractor $d$, and by introducing the sigmoid function $\sigma(r)=1/(1+e^{-r})$, we can write the contrastive loss\rebuttal{\cite{sohn2016improved}} $\mathcal{L}_i$ to be minimized for \mcq$_i$ with matching distractors $d^+$ as follows:}
\thms{
\[
\mathcal{L}_i = -\sum_{d^+} \log \sigma\big(r_i(d^+)\big) - \sum_{d^-} \log \sigma\big(-r_i(d^-)\big)
\]
in which $r_i(d^+)$ denotes the score of a positive distractor for the considered question, and $r_i(d^-)$ the scores for the in-batch negatives (summed over the considered batch of training instances).
If the quantity $\sigma\big(r_i(d)\big)$ is interpreted as the probability that distractor $d$ is compatible with \mcq$_i$ (in the sense of model \dsim{} or \qsim{}), then minimizing the above loss term can be understood as maximizing the joint estimated probability of $d^+$ being compatible distractors for \mcq$_i$, and the in-batch negatives $d^-$ to be incompatible ones.
}
\subsection{Using the models for predictions}
\newtext{This section describes the inference mechanism for our models. Inference refers to using a trained model
to make predictions about new data.
For each of the models, the goal is inducing an ordering of all candidate distractors in response to a given question stem and answer key, such that the top ranked ones can be proposed as fitting distractors.}
For the \dsim{} model, since the \thms{considered (stem, key) pair and the distractor to be scored against it}
are independently fed to the network, the embeddings of the pool of distractors can be computed offline. \thms{The vector representation $h^{(sk)}$ of a given stem and its answer key is calculated, compared through the dot product with each of the pre-calculated distractor representations $h^{(d)}$, and these are then ordered according to decreasing score.}
Similarly, for the \qsim{} model, the pool of questions' embeddings is \thms{calculated offline and stored.
At run time, for a given question stem $s$, we compute its embedding $h^{(s)}$, score it against all pre-calculated stem representations for the \mcq{}s in the corpus, and rank the candidate distractors according to the decreasing score of their corresponding question stem. Note that we assign that same score to each of the distractors of a given stem (for use in \dqsimscore). We then rank all distractors according to decreasing scores (randomly ordering those with identical scores). }
Finally, \thms{once the scores for \dsim{} and \qsim{} are calculated for each candidate distractor, the \dqsim{} model can be evaluated directly, by ranking them according to the decreasing score $r^{\text{\dqsimscore{}}}$ or $r^{\text{\dqsimrank{}}}$.}
\section{Experimental Design}
\label{sec:experimentaldesign}
\newtext{This section describes the evaluation methodology
and the metrics we used to measure the quality of the generated distractors using the different methods described in \secref{sec:methodology}. \secref{sec:evaluation_setup} introduces our hypotheses and the experiments we designed to test them.
The automatic evaluation metrics we used are explained in \secref{sec:automated_metrics}.}
\subsection{Evaluation Setup}
\label{sec:evaluation_setup}
\newtext{In order to validate our models' theoretical effectiveness and practical applicability, we formulate the following three key hypotheses
\chris{, which we will test through experiments based on both automatic and human annotator evaluation:}
\begin{enumerate}[label=\textbullet~\textbf{Hypothesis~\arabic*}:, align=parleft, labelindent=\parindent,leftmargin=*,labelwidth=\widthof{Hypothesis~0}
itemindent=\widthof{Hypothesis~0},ref=Hypothesis~\arabic*]
\item \label{it:hypo1} \emph{Context-aware models generate better distractors when compared with feature-based models.}
\item \label{it:hypo2} \emph{Manual distractor quality scores are correlated with machine-generated distractor candidate rankings.}
\item \label{it:hypo3} \emph{Top-ranked machine-proposed distractor candidates are comparable in quality to expert-generated distractors, for a given question stem and answer key.}
\end{enumerate}
\chris{For \ref{it:hypo1}, we first of all set up a}
large-scale automatic evaluation experiment with the Televic dataset (see \tabref{tab:dataset}
\chris{.}
In addition, a focused small-scale automatic evaluation of context-aware and feature-based models was carried out on the WeZooz external data (see \secref{sec:wezoozdataset} for details) that contains several subjects.
\chris{We complemented that automatic evaluation with human evaluation, since hard comparison}
of ground-truth distractors with machine-generated distractors may not give the whole picture of accuracy.
\chris{Indeed, both for}
language learning and factual knowledge learning, {\mcq}s can have a potentially large set of viable distractors that are not included by the gold standard distractor set.
\chris{Thus,} automated metrics could flag a correctly proposed candidate distractor as wrong because of the scarcity of the gold standard dataset.
To avert this problem, many previous works asked human experts to judge the quality of the distractors that were generated by their systems~\cite{singh2013automatic, araki2016generating}.
\chris{Hence, we}
also invite\chris{d} teachers to provide their expert opinion, \thms{each focusing solely on a set of questions on their own subject of expertise.}
\thms{In the following paragraphs}, we explain the procedure we followed to set up
\chris{that} expert evaluatio
\chris{, which we will use in assessing all aforementioned Hypotheses 1--3.}
First, we prepared a small sample of test questions for language and factual knowledge learning.
For language learning, we used French and English.
These questions were randomly drawn from the held\chris{-}out test split of the Televic dataset introduced in \secref{sec:televicdataset}.
For the factoid type questions, we use the WeZooz dataset introduced in \secref{sec:wezoozdataset}.
Each of the subjects contains a fixed list of 50 questions.
Second, we applied the different trained models to rank distractors according to their relevance for each question in the test set.
We then kept the top-10 ranked candidate distractors for each of the models.
Finally, teachers were
shown distractor predictions
\chris{unified over} all models (i.e., duplicates were removed)
\chris{as well as} the provided gold-truth distractors for each test question \chris{(see the illustration provided}
in \figref{fig:screenshot} in \appref{appendix:userstudydetails}\chris{)}.
\chris{Note that the order of the unified list of distractors was randomized, to avoid introducing order bias.}
The teacher participants were explicitly instructed to rate each candidate distractor based on how much they thought it would help them if they were given the task of preparing distractors for that specific question.
Specifically, we asked them to annotate each distractor independently of the other distractors in the list\chris{,} based on a four-level annotation scheme that we designed to measure the quality of distractors.
\chris{Our scale} is closely related to the three-point \thms{evaluation scale}
proposed by \cite{araki2016generating}
(\Tabref{tab:annotationschemeexamples} shows examples of each category):
\begin{itemize}
\item \emph{True Answer}: specifies that the distractor partially or completely overlaps with the \thms{answer} key.
\item \emph{Good distractor}: specifies that the distractor is viable and could be used in an MCQ as is.
\item \emph{Poor distractor}: specifies that the distractor is on topic but could easily be ruled out by students. This could happen due to one or both of the following reasons.
\begin{itemize}
\item \emph{Poor meaning}: the distractor has poor meaning. For example, it is too general, \thms{although not completely off-topic}.
\item \emph{Poor format}: the distractor's format is different from the format of the answer \thms{key} and \thms{does}
not fit with the stem.
\end{itemize}
\item \emph{Nonsense distractor}: specifies that the proposed distractor is completely out of context.
\end{itemize}
Although the third category (i.e., poor distractor) implies that the proposed distractor is ineffective as is, a minor tweak \thms{may} result in a useful distractor. Furthermore, even if a significant change is required, it may inspire teachers
to create new effective distractors.
Using the annotations we gathered from the teachers, we tested
\chris{Hypotheses} 2 and 3.
For \emph{\ref{it:hypo2}},
\chris{we evaluated whether the higher ranked
distractors also have a higher perceived usefulness.}
This was done
by comparing the
\chris{human scoring of distractor candidates in the top-5 to that of those ranked 5--10:}
\chris{for a good distractor generation model, the top-5 should on average contain significantly more `good' ones.}
We designed a statistical analysis to test the null hypothesis that the rating distribution is not related to whether candidate distractors were ranked top-5 or 5--10.
We used Fisher's exact test\footnote{\semere{We also conducted a chi-square test and reached the same conclusions.}} to test this hypothesis.
For \emph{\ref{it:hypo3}}, we evaluated the extent to which the teachers perceived the system-generated distractor candidates as the ground-truth distractors. Again, we use Fisher's exact test to test the null hypothesis that the distribution of quality of distractors is not related to whether the distractors are human-generated or system-generated.
\subsection{Automated Metrics}
\label{sec:automated_metrics}
We use two groups of information retrieval metrics \lucas{to automatically evaluate} our systems:
\begin{enumerate*}[(1)]
\item Order-unaware metrics: Recall@$k$ and Precision@$k$\chris{,} which measure the fraction of gold-standard distractors that are in the top-$k$ distractors and the fraction of relevant distractors in the top-$k$ retrieved distractors, respectively.
\item Order aware metrics: mean reciprocal rank (MRR) and mean average precision (MAP)\chris{,} which \chris{respectively reflect}
\chris{how high the most relevant item is ranked in the list, and how high all relevant ones are ranked on average.}
\end{enumerate*}
\begin{table*}[ht]
\centering
\footnotesize
\caption{Annotation scheme examples
}
\label{tab:annotationschemeexamples}
\begin{tabular}{p{3cm} p{1.5cm}p{2.0cm}p{2.0cm}p{3cm}}
\toprule
\textbf{Question} & \textbf{Answer}& \textbf{Distractors} & \textbf{Category} & \textbf{Moderation} \\ \midrule
\multirow{4}{3cm}{Which inhabitants are not happy with Ethiopia's plans of the Nile?} & \multirow{4}{2em}{Egyptians} & 1. Itali & Poor format & because of wrong spelling. \\
& & 2. Kenyans & Good & - \\
& & 3. gypsies & Poor meaning & because
\\ \midrule
\multirow{3}{3cm}{My mum brought the washing in .... it was raining.} & \multirow{3}{2em}{because} & 1. until & Good & - \\
& & 2. since & True Answer & \\
& & 3. investigate & Nonsense & out of context
\\ \midrule
\multirow{4}{3cm}{How old was Beethoven when he died?} & \multirow{4}{5em}{56 years} & 1. 1.5v & Nonsense & out of context \\
& & 2. 60 years & Good & - \\
& & 3. 180 years & Poor meaning & humans cannot live 180 years. \\ \bottomrule
\end{tabular}
\end{table*}
\section{Results and Discussion}
\label{sec:resultsanddiscussion}
In this section, we provide evidence of the effectiveness of our context-aware models by reporting the experimental results and discussing the insights gained. \secref{sec:automaticevaluation} compares the baseline with our proposed context-aware models using reproducible automated metrics \chris{(\ref{it:hypo1})}.
\secref{sec:human_evaluation} discusses the user study results with experts \chris{(Hypotheses~1--3)}.
Note that all the \chris{numerical} results reported in this section are in percentage
\chris{points.}
\subsection{Automatic Evaluation}
\label{sec:automaticevaluation}
When considering the results of our automated evaluation based on the recovery of ground-truth distractors, it is essential to note that information about ground-truth distractors for a given item was never used during the model's training.
\Tabref{tab:results} shows the large-scale evaluation of the systems on the Televic \thms{test set}. \rebuttal{We report our results as the mean and standard deviation of five different runs of our models using five random seeds as shown in \tabref{tab:results}.}
All three context-aware models consistently outperform our \thms{feature based model (denoted `baseline')}
on all metrics.
\dqsim{} performs the best \thms{according to most} metrics,
\chris{confirming} that \qsim{} and \dsim{} have their own \thms{(complementary)} merits.
{\qsim} is better than {\dsim} at recovering ground truth distractors (i.e., Recall@10 of \rebuttal{82.3} compared to \rebuttal{76.0}), but inferior at ranking the best relevant distractor \rebuttal{at the top} in the list\chris{, which we conclude from the lower Precision@1}
\rebuttal{(40.4 vs.~44.9) and MRR (55.6 vs.~60.7) scores.
This is related to the nature of the {\qsim} model. The candidate distractors belonging to its best matching question would be put at the top of the returned distractors in a random order.
Our results show that {\dsim} is better at estimating the most likely distractor than {\qsim} is in finding a relevant question \emph{and} arriving with the relevant distractor on top after random ordering. However, the Precision@4 results show that {\qsim} has more success in identifying a question with good distractors, than {\dsim} has in detecting good distractors among its top 4 results. The other reported metrics (Recall@10, Precision@4, MAP) indicate the overall higher effectiveness of {\qsim} when looking further than only the top result.
}
\chris{In our MCQ generation setting, recall \thms{within the top 10 results} is the more important metric, since the presence of high quality distractors in the automatically generated list is more important than their correct ranking.}
\begin{table}[H]
\footnotesize
\begin{center}
\caption{Automatic ranking evaluation Full-ranking}
\label{tab:results}
\begin{tabular}{lllllll}
\toprule
\textbf{Models} &
\multicolumn{1}{l}{\textbf{R@10}} &
\multicolumn{1}{l}{\textbf{P@1}} &
\multicolumn{1}{l}{\textbf{P@4}} &
\multicolumn{1}{l}{\textbf{MAP}} &
\multicolumn{1}{l}{\textbf{MRR}} &
\\\midrule
Baseline & \rebuttal{71.3\std{1.2}} & \rebuttal{21.1\std{1.8}} & \rebuttal{23.7\std{0.5}} & \rebuttal{33.5\std{1.0}} & \rebuttal{43.9\std{1.9}}\\
\dsim & \rebuttal{76.0\std{0.7}} & \rebuttal{\textbf{44.9\std{0.5}}} & \rebuttal{24.4\std{0.8}} & \rebuttal{44.9\std{0.6}} & \rebuttal{60.7\std{1.3}} \\
\qsim & \rebuttal{82.3\std{0.5}} & \rebuttal{40.4\std{1.5}} & \rebuttal{35.9\std{0.9}} & \rebuttal{54.9\std{0.9}} & \rebuttal{55.6\std{1.1}}\\
\dqsim & \rebuttal{\textbf{91.7\std{0.6}}} & \rebuttal{41.9\std{0.8}} & \rebuttal{\textbf{38.2\std{0.7}}} & \rebuttal{\textbf{57.3\std{0.5}}} & \rebuttal{\textbf{62.8\std{0.4}}}\\
\bottomrule
\multicolumn{1}{c}{} & \\
\multicolumn{6}{l}{\footnotesize R:recall, P: precision, MAP: mean avg.~ precision,} \\
\multicolumn{6}{l}{\footnotesize MRR: mean reciprocal rank; evaluation on Televic test set.}
\end{tabular}
\end{center}
\end{table}
\Figref{fig:alpha} depicts the performance of {\dqsim} \chris{for the two merging strategies,} in terms of Recall@10
on the validation set described in \secref{subsection:dqsim}.
The linear combination of the scores outperforms the rank-based merging strategy.
The score-based strategy achieves the best performance at $\alpha =$ 0.8, giving more weight to the {\qsim} model. \newtext{This is reasonable given that the {\qsim} model outperforms the \dsim{} model on the recall metric.}
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[
legend style={draw=none},
axis lines = left,
title={DQ-SIM model curve},
xlabel={ $\alpha$},
ylabel={Recall@10},
xmin=0, xmax=1,
ytick={0.70,0.75,0.80,0.85,0.90,0.95,1.0},
legend pos=south east,
ymajorgrids=true,
xmajorgrids=true,
grid style=dashed,
]
\addplot[
color=blue,
mark=otimes*,
]
coordinates {
(0.0, 0.7529)
(0.1, 0.846)(0.2, 0.9049)(0.3, 0.9054)(0.4, 0.9017)(0.5, 0.8948)(0.6, 0.8924)(0.7, 0.8843)(0.8, 0.8739)(0.9, 0.8624)(1.0, 0.8452)
};
\addplot[
color=red,
mark=star,
]
coordinates {
(0.0, 0.7526)(0.1, 0.8587)(0.2, 0.8752)(0.3, 0.8909)(0.4, 0.8975)(0.5, 0.9075)(0.6, 0.9102)(0.7,0.9133)(0.8, 0.9166)(0.9, 0.9092)(1.0, 0.8474)
};
\node at (axis cs:0.0, 0.7526) [anchor=south west] {D-SIM limit};
\node at (axis cs:1.0, 0.8474) [anchor=north east] {Q-SIM limit};
\legend{DQ-SIM-rank, DQ-SIM-score}
\end{axis}
\end{tikzpicture}
\caption{Different $\alpha$ values for combining \qsim{} and \dsim{} models using rank and raw scores on the validation set}
\label{fig:alpha}
\end{figure}
\Tabref{tab:results_subjects} compares the baseline with \dqsim{} (i.e., \chris{the} best context-aware model according to \chris{the evaluation on the} Televic datase
) in a small-scale setting for all
\thms{four Wezooz dataset subjects as well as English and French from the Televic test set}.
Since
\chris{we want} to compare
\chris{models in terms of} their ability to rank relevant distractors higher in the list, we added the ground-truth distractors from all the subjects to the existing distractors pool.
Ideally, the best model would rank all the ground\chris{-truth} distractors high in the list.
Similar to the large-scale evaluation, {\dqsim} consistently outperforms the baseline for all subjects on both metrics.
Recall@10 and MAP are higher for the language category than
\chris{for factoid questions} because the test questions for the former come from the same distribution (i.e., Televic test questions) as \thms{the data the models were trained on (i.e., Televic train set)}.
On the other hand, the test data for the factoids come from a different distribution (i.e., WeZooz dataset) than the training dat
such that the evaluation for these subjects additionally measures the robustness of the model to a data distribution shift (i.e., its domain transfer abilities). \newtext{The \dqsim{} model is far more robust than the baseline.
\begin{table}[h!]
\begin{center}
\caption{Small scale Automatic ranking evaluation }
\label{tab:results_subjects}
\begin{tabular}{lccccc}
\toprule
\textbf{Models} &
\multicolumn{2}{c}{\textbf{Baseline}} &
\multicolumn{2}{c}{\textbf{\dqsim}} &
\\\cmidrule(lr){2-3} \cmidrule(lr){4-5}
& \multicolumn{1}{c}{\textbf{R@10}} &
\multicolumn{1}{c}{\textbf{MAP}} &
\multicolumn{1}{c}{\textbf{R@10}} &
\multicolumn{1}{c}{\textbf{MAP}}
\\\midrule
English\textsuperscript{*} & 60.1 & 33.6 & \textbf{98.3} & \textbf{85.8} \\
French\textsuperscript{*} & 46.6 & 17.7 & \textbf{81.1} & \textbf{61.1} \\ \midrule
Nat. Sciences & 24.3 & 7.7 & \textbf{74.3} & \textbf{37.3} \\
History & 14.3 & 3.4 & \textbf{62.2} & \textbf{35.7} \\
Biology & 30.6 & 7.6 & \textbf{72.0} & \textbf{41.8} \\
Geography & 32.3 & 12.1 & \textbf{61.5} & \textbf{34.4} \\
\bottomrule
\multicolumn{1}{c}{} \\
\multicolumn{5}{l}{\footnotesize{R: recall, MAP: mean avg.~precision; * denotes subject }} \\
\multicolumn{5}{l}{\footnotesize{is drawn from the Televic test set, while the rest are from WeZooz.}}
\end{tabular}
\end{center}
\end{table}
\subsection{Expert Evaluation}
\label{sec:human_evaluation}
Following the procedure introduced in \secref{sec:experimentaldesign}, a total of 12,723 ratings for distractor quality were gathered from the annotations by teachers (see \tabref{tab:annotationsdatadescription} for details of rating statistics).
These ratings come from the top\chris{-}10 ranked distractors for each of the four models, and the ground-truth distractors (i.e., all simultaneously presented and randomly shuffled). \thms{We retained the gold standard distractors in the lists to be annotated,}
because we wanted to investigate the agreement among teachers in creating distractors.
In the following subsections, we
study teachers' (dis)agreement on the quality of distractors, compare the various models using the evaluation from experts, and \thms{revisit Hypotheses~1--3 in light of these results.}
\vspace{\baselineskip}
\subsubsection{Inter-annotator agreement}
\label{sec:interannotatoragreement}
\chris{We adopt 2 strategies to assess inter-annotator agreement.}
First, we
\chris{analyze} how teachers rated the ground-truth distractors, which were made by other teachers who prepared the questions.
As can be seen from \tabref{tab:groundtruthconsistency}, in general, we find that 79\% of the actual distractors were deemed good, 11\% poor, followed by 7\% nonsense and 3\% true answers.
There is greater agreement between teachers in what is considered a good distractor on the factoids than for language \thms{learning exercises}
(83\% vs.\ 70\%).
\begin{table}[t]
\begin{center}
\caption{Inter-annotation agreement of ground-truth distractors (\%)}
\label{tab:groundtruthconsistency}
\begin{tabular}{lcccc}
\toprule
&
\multicolumn{1}{c}{\textbf{True Ans.}} &
\multicolumn{1}{c}{\textbf{Good }} &
\multicolumn{1}{c}{\textbf{Poor}} &
\multicolumn{1}{c}{\textbf{Nonsense }}
\\\midrule
Languages & 5 & 70 & 14 & 11 \\
Factoids & 2 & 83 & 9 & 6 \\
\midrule
Overall & 3 & 79 & 11 & 75 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
Second, we study the agreement of teachers by asking them to rate the same set of distractors using
\chris{our} four-level scale annotation schem
.
\chris{We selected the subjects}
English, from the languages category, and History, from factoids\chris{,}
for annotations by at least two teachers. \Tabref{tab:interannotation_jaccard} shows the inter-annotator agreement of teachers using the Jaccard similarity coefficient.
The Jaccard similarity measures similarity between two sets of data by calculating
\chris{what fraction of the union of those datasets is covered by their intersection.}
In our case, it is calculated as
the number of times the teachers agreed on a distractor category label (i.e., one of the four quality labels) divided by the total number of distractors that were annotated \chris{(by either annotator)} with that label.
In general,
\chris{we note} a higher agreement on what is considered a good distractor and a nonsense distractor.
Particularly, the overall agreement between the History teachers is higher than the English teachers.
This is in line with the higher agreement
\chris{for} factoid type questions discussed in the previous paragraph.
The Jaccard similarity is sensitive to small sample sizes.
For example, a total of only two distractors were rated `true answer' by the history teachers which yielded no similarity (i.e., a `0' in the first column in \tabref{tab:interannotation_jaccard}).
\chris{Calculating the inter-annotator agreement with the commonly used Cohen's kappa~\cite{mchugh2012interrater} value, we confirm aforementioned higher agreement for factoid questions than for language: Cohen's kappa is 29.3 among English teachers, which represents ``fair agreement'', and 40.5 among History teachers, indicating ``moderate agreement''.}
\chris{As a final metric to assess potential ambiguity in scoring distractors,}
\chris{we calculate conditional probabilities $P(X|Y)$ of having a second annotator assigning label $X$ given that a first one said $Y$.}
For example, unsurprisingly, the probability of rating a distractor `good' given that it was rated `nonsense' by
\chris{another} teacher and vice-versa was 6\% for English and 5\% for History.
This implies that the confusion in differentiating good distractors from nonsense distractors was minimal. Details are presented in \tabref{tab:conditionalprobraters} in \appref{appendix:userstudydetails}.
\begin{table}[t]
\begin{center}
\caption{Inter-annotation agreement of experts in terms of Jaccard similarity coefficent (\%)}
\label{tab:interannotation_jaccard}
\begin{tabular}{lccccc}
\toprule
\textbf{Subjects} &
\multicolumn{1}{c}{\textbf{True}} &
\multicolumn{1}{c}{\textbf{Good}} &
\multicolumn{1}{c}{\textbf{Poor}} &
\multicolumn{1}{c}{\textbf{Nonsense}} &
\multicolumn{1}{c}{\textbf{Overall}}
\\\midrule
English & 25.8 & 42.9 & 12.8 & 40.0 & 47.9\\
History & 0.0 & 43.6 & 24.3 & 59.7 & 57.7\\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\vspace{\baselineskip}
\subsubsection{Evaluation of models by experts}
\label{sec:model_comparison}
\Tabref{tab:results_expert_eval} shows the expert evaluation of distractors in terms of \emph{good distractor rate} (GDR@10) and \emph{nonsense distractor rate} (NDR@10).
GDR@10 is calculated as the percentage of distractors that were rated `good' among the top 10 ranked distractors for each model. Similarly, NDR@10 is calculated as the percentage of distractors that were rated `nonsense' among the top 10 ranked distractors for each model.
We are interested in reporting the NDR metric because \begin{enumerate*}[(i)]
\item it could be used to distinguish between good and bad systems, and
\item in a real-world scenario discarding a system with high NDR score could be helpful since \thms{the frequent occurrence of nonsense distractors may} scare away users by eroding their trust in \thms{the model}.
\end{enumerate*}
The reported metrics
are averages of all the subjects in each category. $\uparrow$ indicates larger values are better and $\downarrow$ indicates smaller values are better.
In general, context-aware models were rated better in proposing plausible distractors than the baseline model.
They also produced
\chris{fewer} nonsense distractors.
The {\dqsim} outperformed all the other models.
On average, 3 out of its top 10 proposed distractors were rated good distractors. Moreover, \thms{on average 5.5} distractors for languages and 5 for factoids were generally found on-topic (i.e., distractors rated as either good or poor distractors) for \dqsim.
The NDR@10 is lower for all models
\chris{for language subjects} than
\chris{for factoid questions}.
We hypothesize this is
because the test data for the language category comes from the same distribution the models were trained on
\begin{table}[t]
\begin{center}
\caption{Expert evaluation of distractors (\%)}
\label{tab:results_expert_eval}
\begin{tabular}{lcccc}
\toprule
\textbf{Models} &
\multicolumn{2}{c}{\textbf{Language learning}} &
\multicolumn{2}{c}{\textbf{Factoid learning }}
\\\cmidrule(lr){2-3} \cmidrule(lr){4-5}
& \multicolumn{1}{c}{\textbf{GDR@10 $\uparrow$}} &
\multicolumn{1}{c}{\textbf{NDR@10 $\downarrow$}} &
\multicolumn{1}{c}{\textbf{GDR@10 $\uparrow$}} &
\multicolumn{1}{c}{\textbf{NDR@10 $\downarrow$}}
\\\midrule
Baseline & 23.6 & 45.4 & 13.6 & 66.0 \\
\dsim & 25.9 & 45.2 & 15.0 & 64.8 \\
\qsim & 26.3 & 45.3 & 19.0 & 61.6 \\
\dqsim & \textbf{27.9} & \textbf{44.6} & \textbf{28.9} & \textbf{50.1} \\
\bottomrule
\multicolumn{1}{l}{} \\
\multicolumn{5}{l}{\footnotesize GDR: good distractor rate, NDR: nonsense distractor rate;} \\
\multicolumn{5}{l}{\footnotesize $\uparrow$: higher is better, $\downarrow$: lower is better; evaluation on WeZooz test set } \\
\end{tabular}
\end{center}
\end{table}
\vspace{\baselineskip}
\subsubsection{Discussion
\chris{of} key hypotheses}
\label{sec:hypotheses_testing}
\chris{We now discuss to what extent our experimental results confirm our aforementioned key Hypotheses 1--3.}
\ref{it:hypo1} states that the context-aware models generate better quality distractors than the feature-based models. As discussed in \secref{sec:automaticevaluation}, the automated evaluation shows that the context-aware models consistently outperform the feature-based model on the Televic and WeZooz datasets. The human evaluation in \secref{sec:model_comparison}
\chris{further confirms this} by demonstrating that distractors generated by context-aware models were rated higher in quality than those generated by feature-based models.
\newtext
\ref{it:hypo2} states that human distractor quality ratings are correlated with the automated candidate distractor rankings.}
To test this hypothesis, we collapsed the four ratings into two categories: \emph{plausible} (i.e., rated as good distractors) and \emph{less plausible} (i.e., rated as true answer, bad and nonsense distractors).
\Tabref{tab:contingency_table_hypo2} in \appref{appendix:userstudydetails} shows the
contingency table \chris{for Fisher's exact test} for
\chris{our} best model\chris{, i.e., } {\dqsim}.
\chris{The fraction of} top\chris{-}5 ranked distractors that received `good distractor' ratings (i.e., 30.3\%) is higher than
\chris{that for the ones ranked 5--10} (i.e., 21.6\%).
We found that this difference
is statistically significant. Indeed, the null hypothesis that the automatic ranking of distractors is unrelated to how teachers rated them is \thms{strongly rejected ($p=$1.7e-8)}.
\newtext
\ref{it:hypo3} asserts that the quality of top-ranked machine-generated distractors is comparable with human-made distractors.}
To test
\chris{this,} we compare the distribution of ratings of the ground-truth distractors (i.e., expert-generated distractors) with the distribution of ratings
\chris{for the} \dqsim{} model (i.e., system-generated distractors).
As
\chris{for \ref{it:hypo2},} we collapse the ratings into \emph{plausible} and \emph{less plausible} classes.
\Tabref{tab:contingency_table_hypo3} in \appref{appendix:userstudydetails} shows the
contingency table \chris{for Fisher's exact test, to compare the quality}
between system-generated and human-generated distractors.
The null hypothesis that the source of the distractor (i.e., human-generated or system-generated) is unrelated to the quality label assigned by the teachers\chris{,} is \thms{strongly rejected ($p<$1.e-10)}.
\thms{Indeed, }
the quality of the human-generated distractors was found to be better than the system-\thms{proposed} distractors.
\chris{Still, we believe system-generated distractors have value: given that they can be generated quickly and automatically, presenting them as suggestions --- rather than relying on a fully automated system --- seems a practically meaningful way of working, which could save teachers a significant amount of time (compared to purely creating a list of distractors without any assistance).}
\section{Conclusion and Future Work}
\label{sec:conclusionandfuturework}
This paper introduced and evaluated multilingual context-aware distractor retrieval models for reusing distractor candidates that can facilitate the task of MCQ creation. Particularly, we proposed three models: \begin{enumerate*}[(1)]
\item \chris{The} {\dsim} model that learns similar contextual representations for similar distractors,
\item \chris{The} \qsim{} model that requires similar questions to have similar representations, and
\item \chris{The} \dqsim{} model that linearly combines the previous two models benefiting from their respective strengths. \thms{Importantly, the \dqsim{} model showed a considerably reduced nonsense distractor rate, which we consider a useful asset in terms of trust in the model by teachers.}
\end{enumerate*}
\rebuttal{We also asked teachers to evaluate the quality of distractors using a four-level annotation scheme that we introduced.}
\rebuttal{As the result, }teachers \thms{considered} 3 out of 10 suggested distractors as high-quality, to be readily used.
\rebuttal{Additionally,} they found two more distractors to be within topic, albeit of
\chris{lower} quality, and \thms{useful as inspiration for}
teachers to \thms{come up with their own good} distractors. \rebuttal{Finally, we released a test consisting of 298 educational \mcq{}s with annotated distractors covering six subjects and a 77K distractor vocabulary to promote further research.}
In future work, we foresee three directions.
First, it is worth reiterating that the current work assumes access to a substantial pool of distractors.
Even though with such large item pools, it is expected that many options are available for an incoming newly written question, the current work is unable to generate a brand new distractor.
A possible solution
could be to employ pure generative models that can freely generate distractors.
Moreover, generative models could correct the `poor format' errors.
However, it has to be noted that such models require access to a context where the distractors and questions come from, such as a chapter of a book, Wikipedia article, etc.
A~second research direction is to extend the current work to a multimodal system that considers other sources of information\chris{, e.g., }
images that accompany MCQs in digital learning tools.
Finally, an area that we are currently investigating is how to
\chris{make sure the complete list of distractors in a single MCQ is sufficiently diverse:}
\chris{note that in}
the present study, we were only interested in retrieving a list of plausible distractors independent of each other.
However, typical MCQ distractors
\chris{should} not only \chris{be} plausible but also \thms{sufficiently} diverse.
\clearpage
\begin{ack}
\newtext{This work
\chris{was} funded by VLAIO (`Flanders Innovation \& Entrepreneurship') in Flanders, Belgium, through the \emph{imec-icon} project AIDA (`AI-Driven e-Assessment').
We would like to thank the partners Televic Education and WeZooz Academy for contributing data and use cases, as well as the ZAVO (`Zorgzaam Authentiek Vooruitstrevend Onderwijs') secondary school teachers for participating in the study.}
\end{ack}
|
1,314,259,996,740 | arxiv | \section{Introduction}\label{intro}
Power grids have long been a source of interesting optimization problems.
Perhaps best known among the optimization community are the unit commitment
problems and related
generator dispatching tasks, see \cite{hobbs}. However, recent
blackout events have renewed interest on problems related to grid
vulnerabilities.
A difficult problem that has been widely studied, the $N-K$ problem, concerns
the detection of small cardinality sets of lines or buses whose simultaneous
outage could develop into a significant failure event; see \cite{bie2}, \cite{pinar} and references therein. This is a hard combinatorial problem which,
unlike the typical formulations for the unit commitment problem, includes
a detailed model of flows in the grid. A different set of algorithmic
questions concern how to react to protect a grid when a significant event has taken place. This is
the outlook that we take in this paper.
In this context, the central modeling ingredient
is that
power grids display {\em cascading} behavior. A cascade is
the process by which components of the grid (especially, power lines) sequentially
become inoperative. In a catastrophic cascade this process accelerates (it 'snowballs') until
the grid collapses. A control action must take this multi-step behavior
into account because a myopic action taken at the start of the process so as to
immediately arrest the cascade may prove far from optimal.
The computation of an optimal control can be formulated as
a multi-stage mixed-integer programming problem;
ideally this should be a stochastic or robust formulation. Furthermore the
formulation will need to include an explicit model of the power flows. When
dealing with large-scale (or even medium-scale) grids it is likely that such
a formulation will prove extremely intractable. In addition, it is likely
that the prescribed control will call for counter-intuitive and possibly
impractical actions. See Section \ref{algorithms} and Appendix \ref{traditional}
In this paper, building on prior models for cascades, we consider an affine,
adaptive, distributive
control algorithm that is computed at the start of the cascade and deployed
during the cascade. The control sheds demand as a function of observations
of the state of the grid, with the objective of terminating the cascade with
a minimum amount of demand lost. The optimization problem handled at the
start of the cascade computes the coefficients in the affine control (one
set of coefficients per demand bus). The discussion of this approach
starts in Section \ref{adaptive1}; Section \ref{firstset} describes an
initial set of experiments with a simple form of affine control.
Most of our algorithms are first-order methods that compute
local optima (see Section \ref{firstorder});
however in a special case which is nevertheless
of interest we obtain
an exact algorithm that runs in polynomial-time (Section \ref{scalingproblem}). Algorithms that account for stochastics are discussed in Section \ref{stochasticoptimization}. We present numerical experiments
with parallel implementations of our algorithms, using as data
a snapshot of the U.S. Eastern Interconnect, with
approximately $15000$ buses and $23000$ lines.
\section{Notation}\label{notation}
In the {\em linearized} approximation to the power flow problem, we are
given a directed graph $G$ with $n$ buses and $m$ lines (denoted, respectively,
``nodes'' and ``arcs'' in traditional graph theoretic language). In addition
\begin{itemize}
\item A line $j$ is oriented from its {\em tail}, $t(j)$ to its {\em head}, $h(j)$. The orientation of any line is arbitrary and is simply used for notational
convenience. The set of lines is denoted by ${\cal A}$.
\item For each line $j$ we are given two positive quantities: its {\em flow limit} $u_{j}$ and its {\em reactance} $x_{j}$.
\item We are given a {\em supply-demand} vector $\beta \in {\D{R}^n}$ with the following interpretation. For a bus $i$, if $\beta_i > 0$ then $i$ is a {\em generator} (a source bus) while if
$\beta_i < 0$ then $i$ is a {\em load} (a demand bus) and in that case
$-\beta_i$ is the {\em demand} at $i$. The condition $\sum_i \beta_i = 0$ is
assumed to hold. For a generator bus $i$, we indicate by the constant $\tilde s_i$ the maximum supply of $i$. We denote by ${\cal G}$ denote the set of generators and by ${\cal D}$ the
set of demand buses.
\end{itemize}
The linearized power flow problem specifies a variable $f_{ij}$ associated
with each line $j$ and a variable $\phi_i$ associated with each bus
$i$. Denoting, for each bus $i$,
the set of lines oriented out of (into) $i$ by $\delta^{+}(i)$ (resp., $\delta^{-}(i)$), the power flow problem consists in finding a solution to the
following system of equations:
\begin{eqnarray}
\sum_{j \in \delta^{+}(i)} f_{j} - \sum_{(j)\in \delta^{-}(i)} f_{j} & = & \beta_i \ \ \forall \ \mbox{bus} \ i, \label{balance}\\
\phi_{t(j)} - \phi_{h(j)} - x_j f_{j} & = & 0 \ \ \ \forall \ \mbox{line} \ j . \label{ohm-eq}
\end{eqnarray}
These equations can simply be abbreviated as
$$ N f \ = \ \beta, \ \ \ N^T \phi - X f \ = \ 0,$$
\noindent
where $N$ denotes the bus-arc incidence matrix of $G$, and
$X = \mathop{\bf diag}\{x_{ij}\}$.
\begin{RE}\label{uniqueness}
It can easily be shown that system (\ref{balance})-(\ref{ohm-eq}) is feasible if and only if $\sum_{i \in K} \beta_i = 0$ for each
component (``island'') $K$ of $G$, and in that case the solution is unique
in the $f$ variables.
\end{RE}
We stress that the orientation of the lines is arbitrary,
consequently for a line $j$ we might have $f_{j} < 0$, indicating that
power flows in the reverse direction. As a final point we note that the flow limits $u_{ij}$
do not appear in (\ref{balance})-(\ref{ohm-eq}); consequently, it is possible
for the (unique) solution $f$ to exceed the flow limits, i.e. it is
possible that $|f_{j}| > u_{j}$ for certain lines $j$.
\section{Cascades}\label{cascades}
In this section we introduce our model of cascading failures of power grids, which draws from
the models in \cite{CarrerasCH02}, \cite{Carreras03},
\cite{CarrerasCH04}. A cascade starts with an initial event, for example the
removal of a number of lines, that alter and compromise the power grid.
Informally, this
is followed by a sequence of additional line outage events which are interspersed
with simple control mechanisms as the grid adjusts to decreased resources,
for example decreased generator capacity.
As a starting point, we have the following template:
\begin{center}
\fbox{
\begin{minipage}{0.9\linewidth}
\hspace*{.9in} \begin{TEMP}{GENERIC CASCADE TEMPLATE}\label{gencasc} \end{TEMP}
{\bf Input}: a power grid with graph $G$ (post-initiating event). Set $G^1 = G$.\\
{\bf For\ } $r = 1, 2, \ldots$ {\bf do\ } \\
\hspace*{.5in}(comment: round $r$ of the cascade)\\
\hspace*{.5in}{\bf 1.} Set $f^r = $ vector of power flows in $G^r$.\\
\hspace*{.5in}{\bf 2.} Set ${\cal O}^{r} = $ set of lines of $G^r$ that become outaged in round $r$. \\
\hspace*{.5in}{\bf 3.} Set $G^{r+1} = G^{r} - {\cal O}^r$. Adjust demands and supplies in $G^{r}$.\\
\end{minipage}
}
\end{center}
In this template, graph $G^1$ represents the grid we have after the initial
event that causes the cascade. To make the template complete, we must
specify a mechanism for determining the set ${\cal O}^r$ in Step 2, and how to
adjust demands and supplies in Step 3. We tackle the second issue first.
The supply/demand adjustment in Step 3 handles cases where
the removal of ${\cal O}^{r}$ from $G^{r}$ creates new
components in $G^{r+1}$. Any imbalance between supply and demand
in a component of $G^{r+1}$ must be corrected:
if, in a component of $G^{r+1}$, total supply exceeds total demand, then
supply must be reduced, and viceversa. Such actions constitute a form of
control. We will discuss this item in more detail in Section \ref{adaptive1}.
Now we turn to step Step 2. We will say that line $j$ is
{\em overloaded} in round $r$ if $|f^r_{j}| > u_{j}$.
Practical experience shows that overloaded lines are
likely to be come outaged. Two
ways to implement this observation are:
\begin{itemize}
\item [(F.1)] Simple deterministic rule: $j \in {\cal O}^r$ if
$|f^r_{j}|/u_{j} > 1$ (alternatively, if $|f^r_{j}|/u_{j} \ge 1$).
\item[(F.2)] Stochastic rule: for each line $j$ there is a function
$\gamma_{j}$ such that if $|f^r_{j}|/u_{j} > 1$ then
$j \in {\cal O}^r$ with probability $\gamma_{j} (|f^r_{j}|/u_{j})$. In
\cite{Carreras03}, the function $\gamma_{j}$ takes a fixed value $p$.
\end{itemize}
The two alternate forms of (F.1) are not strictly equivalent but we
assume that from a practical perspective they are essentially identical.
In any case, the use of (F.1) may not be desirable because it represents a
strict numerical criterion that may be difficult to implement using
standard numerical algorithms -- a possibility is to use for
$u_{j}$ a slightly larger value than the true flow limit. The constant
$\gamma_{j}$ version of rule (F.2) can be criticized in that we would expect that
higher overloads cause outages with higher probabilities; that is to say,
we would expect that $\gamma_{j}(t) \rightarrow 1$ as $t \rightarrow +\infty$.
The problem of choosing, and calibrating such functions $\gamma_{j}$ is
significant.
Both rules can additionally be criticized on the grounds that they are
memory-free; from a realistic perspective
a line that was highly overloaded
in round $r - 1$ should be more likely to be outaged in round $r$ than one
that was not. To address this issue, we assume that for any line $j$
we are given a parameter $0 \le \alpha_{j} \le 1$ and define quantities
$\tilde f^{r}_{j}$ by
\begin{eqnarray}
\tilde f^{r}_{j} & = & \alpha_{j} |f^{r}_{j}| \, + \, (1 - \alpha_{j}) \tilde f^{r-1}_{j}, \label{memory-1}
\end{eqnarray}
with $\tilde f^{0}_{j}$ set to the absolute value of the flow on $(j)$
prior to the incident that initiates the cascade. The $\tilde f^{r}_{j}$
quantities are then used instead of $|f^r_{j}|$ to obtain memory-dependent versions of rules (F.1) and (F.2). A variation of (\ref{memory-1}) is
\begin{eqnarray}
\tilde f^{r}_{j} & = & \alpha_{j} |f^{r}_{j}| \, + \, (1 - \alpha_{j}) |f^{r-1}_{j}|. \label{memory-2}
\end{eqnarray}
The choice of the parameters
$\lambda_{j}$ depends on the time scale of the cascade, but for robustness
purposes the $\lambda_{j}$ should be treated as noisy.
The memory-dependent versions of rules (F.1) and (F.2) can still give rise to non-smooth behavior and ill-conditioning: for example, solving the power
flow equations with different solvers can give rise to different cascades. In
order to lessen this difficulty, we introduce an additional detail in
choosing if a line becomes outaged.
\begin{center}
\fbox{
\begin{minipage}{0.9\linewidth}
\hspace*{.9in} \begin{RULE}{STOCHASTIC LINE OUTAGE}\label{stochout} \end{RULE}
\vskip 2pt
{\bf Parameters}: $0 \le \epsilon_r \le 1$ for each round $r$.\\
{\bf Notation}: refer to Template \ref{gencasc} and equation (\ref{memory-1}).\\
{\bf Application}: For a line $j$ in $G^r$:
\begin{eqnarray}
\mbox{if} && u_{j} < \tilde f^{r}_{j}, \ \ \mbox{then} \ \ j \in {\cal O}^r, \label{actual1} \\
\mbox{if} && (1 - \epsilon_r)u_{j} < \tilde f^{r}_{j} \le u_{j}, \nonumber \\
&& \mbox{then} \ \ j \in {\cal O}^r \ \ \mbox{with probability $\frac{1}{2}$}, \label{flipping}\\
\mbox{if} && \tilde f^{r}_{j} \le (1 - \epsilon_r)u_{j}, \ \ \ \mbox{then} \ \ \ j \notin {\cal O}^r. \label{actual3}
\end{eqnarray}
\vskip 2pt
\end{minipage}
}
\end{center}
\noindent The random choice in (\ref{flipping}) is an indirect way to incorporate some of the (poorly defined) ``noise'' mentioned above;
additionally, from a mathematical perspective,
it serves to smooth the cascade process.
Typically we would have $\epsilon_1 \le \epsilon_2 \le \ldots $,
indicating increasing uncertainty as the cascade progresses. If $\epsilon_r = 0$ for all $r$ we obtain the pure deterministic rule.
Rule \ref{stochout}, and extensions, will be used later in our numerical experiments.
\section{Control Algorithms}\label{algorithms}
We consider control algorithms designed to stop the cascade after a fixed
number of rounds with a maximum amount of total demand feasibly satisfied. In
developing such algorithms we assume that the cascade is initially
slow-paced so that significant computation is possible at time zero (immediately after the initiating event). While this may not be true for all cascades, it
was true in the case of the 2003 cascade in the Northeast U.S. and Canada
\cite{usc} with (arguably) on the order of one hour elapsing between subsequent
outages at the start. Thus, we assume that the algorithm is computed at time zero, with significant
information available as to the state of the grid; the algorithm will be
applied as the cascade progresses, with no further computation.
We assume, as a control requirement,
that there is a final round $R$ in the cascade
at the end of which no lines can be overloaded.
The computation of an optimal schedule for demand shedding can be stated as mixed-integer
optimization problem; see Appendix \ref{traditional}. However, it is not
clear that such an approach is either computationally feasible or even
desirable (see the discussion in the Appendix). In this paper
we will take a different outlook.
\subsection{Adaptive control}\label{adaptive1}
We focus on robust control algorithms that take as input
limited, real-time observations on the state of the grid
and which prescribe simple control
actions such as distributed load shedding (loss of demand). Our generic
cascade template (\ref{gencasc}) is modified as follows:
\begin{center}
\fbox{
\begin{minipage}{0.9\linewidth}
\hspace*{.9in} \begin{TEMP}{CASCADE CONTROL}\label{contcasc} \end{TEMP}
\vskip 5pt
{\bf Input}: a power grid with graph $G$. Set $G^1 = G$.\\
{\bf Step 0.} {\bf Compute} control algorithm.\\
{\bf For\ } $r = 1, 2, \ldots, R-1$, {\bf do\ } \\
\hspace*{.2in}(comment: controlled round $r$ of the cascade)\\
\hspace*{.2in}{\bf 1.} Set $f^r = $ vector of power flows in $G^r$.\\
\hspace*{.2in}{\bf 2.} {\bf Observe\ } state of grid (from state estimation).\\
\hspace*{.2in}{\bf 3.} {\bf Apply} control.\\
\hspace*{.2in}{\bf 4.} Set $g^r = $ vector of resulting power flows in $G^r$.\\
\hspace*{.2in}{\bf 5.} Set ${\cal O}^{r} = $ set of lines of $G^r$ that become outaged in round $r$. \\
\hspace*{.2in}{\bf 6.} Set $G^{r+1} = G^{r} - {\cal O}^r$. Adjust loads and generation in $G^{r}$.\\
\vskip 1pt
{\bf Termination} (round $R$). If any island of $G^R$ has line overloads, proportionally shed demand in that island until all line overloads are eliminated.\footnote{The criterion
of ``stability'' inherent in the termination step may obviously be incomplete
when using a more complete model of power flows than the linearized model.}
\vskip 6pt
\end{minipage}
}
\end{center}
In this template, steps 0, 2 and 3 are the only ones requiring controller
actions. The remaining steps are due to the physics of the grid or
underlying low-level automatic control steps. As discussed above, in this
paper we assume that the cascade allows enough time for significant
computation to take place in step 0. One could conceive of a variant of
the template where step 0 is carried out in advance of an initiating event,
thus obtaining a more general form of control. However, proceeding as in
the template resolves an exponential number of potential outcomes,
likely obtaining a simpler and faster step 0 and a more effective control
algorithm.
Having applied the control in a given round, a given (pre-existing)
component
of $G^r$ is likely to experience a supply/demand imbalance.
This condition must be removed, which we assume can be undertaken
through a low-level control mechanism. This is not a trivial assumption
and if the imbalance is large the rebalancing may be deemed impossible with
existing technology,
resulting in the loss of all demand in that component. It is
straightforward to implement such a ``maximum imbalance'' feature
in the above template; however for the
sake of simplicity we will assume that we can always rebalance supply and
demand by proportionally decreasing the output of each generator in a
given component (again, other rebalancing mechanisms are possible, giving rise to alternative
versions of the template). Having effected the rebalancing, a new set
of power flows will be instantiated: this is vector $g^r$ in Step 4. Steps 5
and 6 now follow as in the generic cascade template. In Step 6, some
of the (new) components of $G^{r+1}$ may have an excess of supply over demand
or the other way around, and again we make the assumption that this
excess is removed through a proportional scaling mechanism.
To make the above template complete, we need to describe the type of control
we have in mind, including what type of data observations it requires and
what kind of control actions it specifies. In terms of the last item,
many possibilities exist (including modifying the structure of the grid
by e.g. shutting down lines, in which case the terminology in Steps 4-6 is
not strictly correct) but in this paper we will focus on one type of action
which is feasible in practice: ``load shedding'' or the controlled loss of
demand.
In the linearized (DC) power flow models we have variables of just two types:
power flows and phase angles. In this paper we concentrate on control
algorithms that observe real-time quantities related to flows. Two such
quantities are:
\begin{itemize}
\item [(a)] The maximum overload: $\max_{j \in G^r} \{ |f^r_{j}|/u_{j} \}$.
\item [(b)] The maximum relative flow variability: $\max_{j \in G^{r-1}} \{ |f^r_{j} - f^{r-1}_{j}|/|f^{r-1}_{j}| \}$, where we assume the maximum is taken
over the lines with $f^{r-1}_{j} \neq 0$.
\end{itemize}
From a practical standpoint, a relevant issue is how accurately (a) or (b)
can be observed in real time, and whether such measurements can be disseminated
to all buses of the grid. We assume that in early rounds of a cascade this
is not a fundamentally difficult technological problem. Nevertheless, for
a given integer $\delta > 0$ we define the {\em radius-}$\delta$ version
of either (a) or (b), in which each bus of the grid is expected to perform
the measurement over all links within $\delta$ hops in the current topology. Note
that even if $\delta$ is large we are still constraining the measurement performed
by a bus $v$ to take place in the same component as $v$.
We will discuss this topic in greater depth below.
Putting aside this issue, we propose an affine control policy where at
each round $r < R$, each demand bus $v$ independently adjusts its demand
by making use of a (precomputed) triple $(c^r_v, b^r_v, s^r_v)$ of parameters.
\begin{center}
\fbox{
\begin{minipage}{0.9\linewidth}
\hspace*{.9in} \begin{PRO}{AFFINE CONTROL}\label{affinecontcasc} \end{PRO}
{\bf Input}: a power grid with graph $G$ (post-initiating event). Set $G^1 = G$.\\
{\bf 0.} {\bf Compute} triples $(c^r_v, b^r_v, s^r_v)$ for each $r < R$ and $v$.\\
{\bf For\ } $r = 1, 2, \ldots , R-1$, {\bf do\ } \\
\hspace*{.5in}(comment: controlled round $r$ of the cascade)\\
\hspace*{.5in}{\bf 1.} Set $f^r = $ vector of power flows in $G^r$,
and $d^r_v = $ the demand of any bus $v$. \\
\hspace*{.5in}{\bf 2.} For any demand bus $v$, let $\kappa^r_v$ be its data observation.\\
\hspace*{.65in} {\bf Apply control:} if $\kappa^r_v > c^r_v$, reset the demand of $v$ to $$\min\{1, \, [b^r_v + s^r_v (c^r - \kappa^r_v)]^+ \}~d^r_v.$$
\hspace*{.5in}{\bf 3.} Adjust generator outputs in each component of
$G^r$ so as to match demand.\\
\hspace*{.5in}{\bf 4.} Set ${\cal O}^{r} = $ set of lines of $G^r$ that become outaged as a result of the flows\\ \hspace*{.65in} instantiated in Step 4. \\
\hspace*{.5in}{\bf 5.} Set $G^{r+1} = G^{r} - {\cal O}^r$. Adjust demands and supplies in $G^{r}$.\\
{\bf Round R.} For any component $K$ of $G^R$, set
$\Psi^R_{K} \, \doteq \, \min\left\{ 1 \, , \, \max_{j \in K} \{ |f^R_{j}|/u_{j} \} \right\}$.\\
\hspace*{.8in}If $\Psi^R_{K} > 1$, then any bus $v$ of $K$ resets its demand to $d^R_v /\Psi^R_{K}$.
\end{minipage}
}
\end{center}
In Procedure \ref{affinecontcasc}, round $R$ handles cascade termination. Under the linearized power flow
the rescaling guarantees that no line overloads will exist.
Note that Step 0 requires the computation of $3(R-1)D$ parameters, where
$D$ is the number of demand buses. Special cases of the control are:
\begin{itemize}
\item [(1)] Time- or bus-independent control: for any demand bus $v$, $(c^r_v, b^r_v, s^r_v) = (c_v, b_v, s_v)$ for all $1 \le r < R$ and some triple $(c_v, b_v, s_v)$; or, for every demand bus $v$,
$(c^r_v, b^r_v, s^r_v) = (c^r, b^r, s^r)$ for each $1 \le r < R$, for
a certain triple $(c^r, b^r, s^r)$.
\item [(2)] Time-dependent, componentwise control. This is a control such that
for any given component $K$ of $G^r$, $(c^r_v, b^r_v, s^r_v)$ equals a fixed
triple $(c^r_K, b^r_K, s^r_K)$ for every $v \in K$.
\item [(3)] Segmented control. Let $(\Sigma_1, \Sigma_2, \ldots, \Sigma_H)$ be a
partition of the demand buses. Then we insist that for any round $r$,
$(c^r_v, b^r_v, s^r_v)$ takes a common value for all demand buses in a given
set $\Sigma_i$.
\end{itemize}
An example of type (3) is that where the $\Sigma_i$ are quantiles of the demand distribution. The resulting control is ``fair'' in that demands of
similar magnitudes are reduced by similar fractions. Controls of type (2),
in the case of the deterministic outage rule (F.1) can be explicitly
described a-priori in polynomial space. This follows because in any fixed round
$r$, $G^r$ will have at most $n$ components and these depend solely on the
structure of the control on rounds up to $r-1$. Thus in total at most $Rn$
distinct triples $(c^r_K, b^r_K, s^r_K)$ need to be specified.
Our primary focus are on algorithms and implementations for the most general version (time- and bus-dependent controls) of our approach, using the outage
rule (\ref{actual1})-(\ref{actual3}) with memory. This is given in
Section \ref{firstorder}. In Section \ref{scalingproblem} we will discuss a
special case where the optimal control can be efficiently computed.
\section{First set of experiments}\label{firstset}
To motivate our overall approach, we first present experimental results using special, simple cases of the control. The objective of these experiments is
to expose, first, the fact that in cases of severe contingencies
the application of control so as to immediately stop a cascade may be myopic
and could result in very suboptimal results. In other words, it may pay
off to allow some lines to become outaged. However, this clashes with the
intuitive notion that postponing action to much later in the cascade results
in increasing uncertainty (because from the perspective of an agent at the
start of the cascade, the later stages of the cascade should be more
uncertain). We want to measure the impact of uncertainty
in the timing of control, and to contrast it with the goal of avoiding
myopic, immediate action, as described above.
\subsubsection{The data}
In these experiments we used a snapshot of the U.S. Eastern Interconnect, with
approximately $15000$ buses, $23000$ lines, $2000$ generators and $6000$
load buses. The snapshot includes generator output levels, demands,
and line parameters.
Approximately $5000$ of the line flow limits were zero,
likely indicating a data error or missing data -- the minimum nonzero line
flow value was $3 \times 10^{-2}$. When the flow limit
of a line $j$ was equal to zero, we proceeded as follows,
where $f^0_j$ is the initial power flow value on line $j$:
\begin{itemize}
\item If $|f^0_j| \ge 10^{-6}$, we reset the flow limit to
$(1 + \gamma)|f^0_j|$, where $\gamma = 0.2$.
\item Otherwise, we reset the flow limit to $\beta = 10^{-4}$.
\end{itemize}
A
very small
number of lines had positive and large capacity, but nevertheless the
$|f^0_j|$ values were close (or identical) to the line flow limits; in
which case we increased the line flow limit by $25\%$.
The reader may wonder about the impact of these numerical choices. Based on
our limited testing with different choices of $\gamma$ and $\beta$,
the impact is very minor in terms of $\beta$. The same holds for $\gamma$
unless we choose much values (on the order of $10^2$) and even
then it is more a matter of degree than structure in the
cascades we will describe below.
Approximately $250$ of the lines had negative reactance values. While there
are valid reasons for the use of negative reactances, in the experiments below
we assumed data error and replaced each negative value by its absolute value.
\subsubsection{Methodology}\label{firstsetmethod}
In
all the experiments the same approach was employed: first, we interdicted
the grid according to a synthetic contingency, then we computed our affine
control, and finally we studied the behavior of this control.
To obtain contingencies we used the following methodology, which removes a
a set of $K$ random, high power flow lines from the grid, while preserving
connectivity. Here $K$ is a given, small integer, and as before $m$ is
the number of lines.
\begin{itemize}
\item [(1)] A spanning tree $T$ is computed.
\item [(2)] Let $\hat f$ denote the power flow vector corresponding to the
given demands and generator outputs. Renumber so that $|\hat f_1| \ge |\hat f_1| \ldots \ge |\hat f_m|$.
\item [(3)] Let $0 < \pi < 1$. Run steps (a) and (b), initialized with $S = \emptyset$, until stopping in (b): \\
\noindent For $j = 1, \ldots, m$, \\
\hspace*{.1in}{\bf (a)} If line $j \notin T$, then \\ \hspace*{.25in} with probability $\pi$ reset $S \leftarrow S \cup {j}$.\\
\hspace*{.1in}{\bf (b)} If $|S| = K$, stop.
\item[(4)] The set $S$ of lines is removed from the network, producing network $G$
in our cascade template.
\end{itemize}
We used values of $K$ ranging from $1$ to $50$,
and for $\pi$ we used values ranging from $.1$ to $.5$.
\subsubsection{The experiments}
We considered a case with $K = 2$ (two lines removed)
and $R = 20$ rounds. In the computation of
the moving averages of line overloads (eq. (\ref{memory-1})) we used
$\alpha = 0.9$.
First we consider the pure deterministic case of line outages, that is to
say we use line outage rule (\ref{stochout}) with $\epsilon_r = 0$
for all $r$. If no control is applied, then at the end of round of round $20$ the
{\em yield} (percentage of demand still being served) is $2.47 \%$.The
cascade is characterized by extremely high line overloads; see Table \ref{evolcomparo} (we will discuss implications of this below).
At the start of round 1, in fact, the maximum line overload is $40.96$, indicating that, likely, several lines with low flow limits are overloaded.
We computed the best control where
\begin{itemize}
\item [(i)] $c^r_v = b^r_v = 1$ for all $v$ and $r$.
\item [(ii)] $s^r_v = 0$ for all $v$ and $10 < r$. Thus, no control is applied
after round 10.
\item [(iii)] For each $1 \le r \le 10$, \underline{either} $s^r_v = 0.005$ for all
$v$, \underline{or} $s^r_v = 0$ for all $v$.
\end{itemize}
Thus we simply want to decide {\em when} to apply a control of a very simple
form. Further, we are restricted to applying control in the first half of
the cascade; this is done as protection against uncertainty in the later
rounds of the cascade
The rationale for the numerical values in (iii) is that $1 + 0.005*(1 - 40) \approx 0.80$, that is to
say, the application of this control in round 1 will ``only'' shed $20\%$ of
the demand.
We want to stress that the experiments in this section do not amount to a rigorous attempt at optimizing control. In fact, the control obtained through
(i)-(iii) is only near-optimal.
Instead we are trying to provide an
example of the difference between an adequate control and the no-control option,
and the questions that arise from the comparison. In particular, the amount $0.005$ was arrived at through a simple grid-search process.
\begin{table}[h]
\centering
\caption{{\bf \emph{Cascade evolutions}}}
\vskip 2 pt
\begin{tabular}{|c| r r r r| r r r r|}
\hline
\multicolumn{1}{|c|} {} & \multicolumn{4}{c|}{{\bf No control}} & \multicolumn{4}{c|}{{\bf c20}}\\
{\bf r} & \boldmath{$\kappa$} & {\bf O} & {\bf I} & {\bf Y} & \boldmath{$\kappa$} & {\bf O} & {\bf I} & {\bf Y} \\\hline \hline
\noalign{\smallskip}
{\bf 1} & 40.96 & 86 & 1 & 100 & 40.96 & 86 & 1 & 100 \\
{\bf 2} & 8.60 &187 & 8 & 99 & 8.60 & 165 & 8 & 96 \\
{\bf 3} & 55.51 & 365 &20 & 98 & 61.74 & 303 & 17 & 96 \\
{\bf 4} & 67.14 & 481 & 70 & 95 & 66.63 & 408 & 44 & 94 \\
{\bf 5} & 94.61 & 692 & 149 & 93 & 131.08 & 492 & 94 & 93 \\
{\bf 6} & 115.53 & 403 & 220 & 91 & 112.58 & 416 & 146 & 90 \\
{\bf 7} & 66.12 & 336 & 333 & 89 & 99.62 & 326 & 191 & 78 \\
{\bf 8} & 47.83 & 247 & 414 & 87 & 60.95 & 227 & 248 & 77 \\
{\bf 9} & 7.16 & 160 & 457 & 85 & 32.50 & 72 & 279 & 76 \\
{\bf 10} & 7.06 & 245 & 542 & 84 & 9.50 & 43 & 292 & 76 \\
{\bf 11} & 37.55 & 195 & 606 & 83 & 45.28 & 35 & 303 & 76 \\
{\bf 12} & 13.04 & 98 & 646 & 82 & 11.60 & 10 & 306 & 76 \\
{\bf 13} & 22.61 & 128 & 688 & 82 & 3.88 & 6 & 310 & 75 \\
{\bf 14} & 10.64 & 107 & 715 & 81 & 1.46 & 4 & 312 & 75 \\
{\bf 15} & 5.03 & 64 & 721 & 81 & 1.34 & 1 & 312 & 75 \\
{\bf 16} & 84.67 & 72 & 743 & 80 & 1.13 & 1 & 312 & 75 \\
{\bf 17} & 32.15 & 52 &756 & 80 & 1.38 & 2 & 312 & 75 \\
{\bf 18} & 6.50 & 43 & 763 & 80 & 1.26 & 1 & 312 & 75 \\
{\bf 19} & 9.97 & 85 & 812 & 80 & 0.99 & 0 & 312 & 75 \\
{\bf 20} & 32.34 & 39 & 812 & 2 & 0.99 & 0 & 312 & 75 \\
\hline
\end{tabular}
\label{evolcomparo}
\end{table}
In any case, the optimal control that satisfies conditions (i)-(iii)
(and which we shall refer to as {\bf c20} for future reference) attains
a termination yield of $75.2\%$, picks
rounds $2$ and $7$ to apply control. Note that
since the maximum line overload is high
in round $1$, c20 allows some lines to become outaged in round $1$.
This point is further elaborated in Table 1, where
``r'' indicates round and for each round, ``$\kappa$''
indicates maximum line overload at the start of the round, ``O'' is the number of lines outaged during the round, ``I''
is the number of islands at the end of the round and ``Y'' is
the (rounded) percentage
of demand
being delivered end of the round. {\bf We stress that we count {\em all} islands,
even those that consist of a single bus with no demand, and when computing the
maximum line overload we consider {\em all} lines, no matter how minor.}\\
\noindent {\bf Discussion.} We see that initially both cascades have extremely high line overloads, many line outages and large amounts of islanding. However,
under c20 after line 11
the overages are significantly smaller, and rapidly
decreasing, and after round 8 the number of new outages and islands is also much
smaller (and decreasing); both in spite of the fact that control is
last applied in round 7. Thus, effectively, the cascade has been ``stabilized''
under c20, long before the end of the time horizon.
The reader might wonder about the rapid decrease of yield from $80\%$ to
$2\%$ in the no-control case. This is due to the termination feature in
our cascades that requires all line overloads to be eliminated by the end
of the last round; since the no-control cascade has very high
maximum overload ($32.34$),
at the start of round $20$, the termination rule forces a drastic reduction in yield.
Nevertheless, in the no-control case,
the combination of comparatively high yield (up to
round 4), high number of line
outages, large line overloads and large amount of islanding suggest the possibility that many of the
outages involve unimportant lines, and likewise with many of the islands (though of course a $22\%$ yield loss should indicate a severe contingency). One wonders if somehow the no-control option {\em might} be attractive
if {\em enough} time (i.e., rounds) were available.
\begin{table}[h]
\centering
\caption{{\bf \emph{Further evolution of no-control cascade from Table \ref{evolcomparo}}}}
\vskip 2pt
\begin{tabular}{|c | r r r r r r r r|}
\hline
{\bf r} & {\bf 25} & {\bf 28} & {\bf 29} & {\bf 30} & {\bf 31} & {\bf 32} & {\bf 33} & {\bf 34} \\
\hline
\noalign{\smallskip}
{\bf O} & 21.63 & 2.00 & 5.70 &2.50 & 2.38 & 1.35 & 1.07 & 0.99 \\
{\bf Y} & 79 & 78 & 78 &78 & 78 & 78 & 78 & 78 \\
\hline
\end{tabular}
\label{table2}
\end{table}
To investigate these possibilities, we extended the no-control cascade. Table
\ref{table2} shows the results for selected rounds. We see that the no-control
approach finally yields stability by round 34, attaining yield $78\%$.
This is slightly better (but very close) to what c20 obtained in 20
rounds (and, furthermore, control action under
c20 was restricted to rounds 1-10).
Nevertheless, the no-control approach experiences significant line overloads
as late as round 32.
By maintaining high overloads into very late rounds, the no-control strategy
becomes more exposed to the unavoidable {\em uncertainty} that should be
taken into account when modeling cascades, and which we have up to now
ignored. We model noise by means of fault outage rule (\ref{stochout}). In the following set of tests we assume that
\begin{eqnarray}
\epsilon_r & = & 0.01 \, + 0.05* \lfloor r/10 \rfloor. \label{noise1}
\end{eqnarray}
Possibly, noise should be increasing at a faster rate than the
above formula stipulates (perhaps exponentially). However, the control
considered in Table \ref{evolcomparo} as well
as the no-control approach are both exposed to significant amounts of
noise after round 10; more so in the no-control case.
We would thus expect that under
rule (\ref{noise1}) the no-control approach will perform much more poorly.
To test these hypothesis, we ran 1000 simulations of cascades under rule (\ref{noise1}) for the no-control case and for control using c20. The results are
summarized as follows: using c20, the average yield is $42.90$ and the standard
deviation of yield is $27.47$, whereas using no control the average yield
is $7.96$ and the standard deviation is $9.33$. In other words, c20 proves
much more robust than the no-control strategy, which is not surprising given
the structure of rule (\ref{noise1}). A question that arises as a result is
whether c20 is in some sense optimally robust.
One way to investigate this question is to investigate controls that are
less exposed to uncertainty by restricting them to a shorter timeline, i.e.
by enforcing termination before round 20. For $T = 10, \, 15, \, 25$, we compute an optimal control required
to terminate by round $T$, and otherwise subject to rules
(i)-(iii), that is
$c^r_v = b^r_v = 1$ for all $v$ and $r$, $s^r_v = 0$ for all $v$ and $10 < r$,
and for each $1 \le r \le 10$, either $s^r_v = 0.005$ for all
$v$, or $s^r_v = 0$ for all $v$. We name these controls c10, c15 and c25, respectively.
Table \ref{table3} presents the comparisons between all the options we
have considered. In this table, ``DetY'' is the yield in the deterministic
case ($\epsilon_r = 0$ for all $r$), ``MaxY'' and ``MinY'' are the maximum
and minimum yields in all
the simulations (resp.), ``AveY'' is the average yield and ``StddY'' is the standard
deviation of yield.
\begin{table}[h]
\centering
\caption{{\bf \emph{Robustness comparison - 1000 runs using stochastic outage rule (\ref{stochout}) with noise as in (\ref{noise1}) }}}
\vskip 2pt
\begin{tabular}{|c | r r r r r|}
\hline
{\bf Option} & {\bf DetY} & {\bf MaxY} & {\bf MinY} & {\bf AveY} & {\bf StddY} \\
\hline
\noalign{\smallskip}
{\bf c10} & 37.49 & 39.05 & 0.00 & 11.81 & 11.84 \\
{\bf c15} & 72.44 & 71.85 & 0.00 & 33.94 & 22.51 \\
{\bf c20} & 75.19 & 76.30 & 1.17 & 41.90 & 27.47 \\
{\bf c25} & 77.23 & 42.34 & 1.38 & 11.99 & 10.97 \\
{\bf no control} & 77.75 & 36.04 & 0.00 & 7.96 & 9.33 \\
\hline
\end{tabular}
\vskip -5pt
\label{table3}
\end{table}
Control c20 emerges as superior over c15 and c10. This can be
explained as follows. Even though c15 and c10
are significantly less exposed to risk than c20,
they are also restricted to
operating, and terminating,
during a stage of the cascade characterized by extremely high line overloads.
Control c20, by being able to operate over 20 rounds, has ``more time''
while also avoiding the large uncertainty rounds 20 and higher.
For this reason, c20 is also
superior to c25 (their averages are separated by more than one standard
deviation).
One common feature
that emerges in controls c10, c15, c20 and c25 (not shown in the table) is that no control is taken in
round 1, and control is taken in round 2 (and in the cases of c10, c15 and c20, rounds 5 or 7).
We stress that (\ref{noise1}) is {\em one} categorization of noise. Using a different formula the outcome could be different, say c15 could prove best. However, the outlook we are taking here is that by computing {\em a} robust control
with respect to {\em some} rule such as (\ref{noise1}) we obtain a control
that remains robust (though possibly not optimally so) even if the model for
uncertainty were to be somewhat changed. And, in any case, computing a control
which is is somewhat robust should be better than completely ignoring uncertainty.
To explore these issues, we study the following model
\begin{eqnarray}
\epsilon_r & = & 0.01 \, + 0.005* r, \label{noise2}
\end{eqnarray}
which can be considered a smoothed version of (\ref{noise1}). Under this
model both c15 and c20 are exposed to more noise than c10, and more noise than under rule (\ref{noise1}). Consider Table \ref{table4}.
\begin{table}[h]
\centering
\caption{{\bf \emph{Robustness comparison - 1000 runs using stochastic outage rule (\ref{stochout}) with noise as in (\ref{noise2}) }}}
\vskip -5pt
\begin{tabular}{|c | r r r r r|}
\hline
{\bf Option} & {\bf DetY} & {\bf MaxY} & {\bf MinY} & {\bf AveY} & {\bf StddY} \\
\hline
\noalign{\smallskip}
{\bf c10} & 37.49 & 38.93 & 0.00 & 7.54 & 9.55 \\
{\bf c15} & 72.44 & 63.94 & 3.41 & 28.02 & 17.94 \\
{\bf c20} & 75.19 & 73.04 & 0.00 & 32.24 & 21.30 \\
{\bf c25} & 77.23 & 54.62 & 0.25 & 16.84 & 12.66 \\
{\bf no control} & 77.75 & 18.86 & 0.00 & 5.11 & 5.28 \\
\hline
\end{tabular}
\label{table4}
\end{table}
We see that c20 still appears superior to the other controls, though c15
is almost as good.\\
The above experiments do not amount to a full optimal
robust control computation. In Section \ref{gridstoch} we will return to these experiments from a stochastic optimization
perspective.
\section{Optimization methods}\label{methodology}
Given a control vector $(c,b,s)$, denote by $\tilde \Theta^R(c,b,s)$ the
final demand at termination of the $R$-round cascade controlled by $(c,b,s)$.
Our goal is to maximize $\tilde \Theta^R(c,b,s)$ over all controls. This is
a nonconcave, in fact very combinatorial, maximization problem \cite{boco}, \cite{luen}; it is very large (e.g. if $R = 10$ the $(c,b,s)$ vector has more than
$180000$ variables in the case of the Eastern Interconnect).
It is also important to incorporate stochastics.
In principle, the deterministic
case of our problem could be tackled using mixed-integer programming techniques,
and the stochastic version, using stochastic programming
\cite{lind}. Of course, one could choose a different formulation of the
cascade control problem than the one we chose (using a different kind of
control, for example). But any formulation will have to deal with
the combination of combinatorics in the network dynamics, multistage
behavior,
stochastics and very large size. In our opinion, this combination
places the problem outside the capabilities of current optimization
methodology, even in the deterministic case. We remind the reader that
we envision our control as being computed in real time and
we might only have one hour, or less, to do so.
Another point to stress is that nonconcavity in a maximization problem leads to
non-monotone behavior: in our case, just because a small change in control
leads to an improvement does not imply that a larger change will result in
greater improvement.
\subsection{First-order methods for the general case}\label{firstorder}
Here we describe a procedure to compute a control given by triples $(c^r_v, b^r_v, s^r_v)$, for each demand bus $v$ and each round $r$, and using a control law as in
Step 2 of algorithm (\ref{affinecontcasc}).
We will assume a semi-random outage rule as in (\ref{actual1})-(\ref{actual3}),
with memory, as in (\ref{memory-1}). As previously, the goal is to compute a control
that will maximize the expectation of the amount of demand being served by the end of round $R$, which as before
we denote by $\tilde \Theta^R(c,b,s)$. We stress that the control parameters we use
are state-independent; this is a design feature.
Toward this goal we will use an algorithm based on the following template:
\begin{center}
\fbox{
\begin{minipage}{0.9\linewidth}
\hspace*{.9in} \begin{PRO}{First-order algorithm}\label{stochgrad1} \end{PRO}
{\bf Input}: a control vector $(c,b,s)$.\\
{\bf For\ } $k = 1, 2, \ldots$ {\bf do\ } \\
\hspace*{.5in}{\bf 1.} Estimate $g = \nabla \tilde \Theta^R(c,b,s)$.\\
\hspace*{.5in}{\bf 2.} Choose ``step-size'' $\mu \ge 0$ and update control
to $(c,b,s) \, + \, \mu (g_c, g_b, g_s)$.\\
\hspace*{.5in}{\bf 3.} If $\mu$ is small enough, stop.
\end{minipage}
}
\end{center}
This is a common first-order (steepest-ascent) method.
In the deterministic case, Step 1 should be interpreted as a an approximate
rule since $\tilde \Theta^R$ is not differentiable (our stochastic outage rule
\ref{stochout} does smooth out the expectation). The vices of procedure
\ref{stochgrad1} are well known: even if $\tilde \Theta^R$ were smooth, its
nonconcavity implies that the steepest-ascent method may not converge to
a global optimum. And even if $\tilde \Theta^R$ were smooth and concave,
steepest ascent may zigzag or stall. See \cite{luen}.
In summary,
Procedure \ref{stochgrad1} should be viewed as a {\em local search} method
with which to explore the neighborhood of a solution. Finally, in our
setting the procedure could prove expensive, since each evaluation of $\tilde \Theta^R$
(including in the estimation of $\nabla \tilde \Theta^R$ through finite
differences) requires a cascade simulation, each round of which requires two
power flow computations in our setup.
On the positive side, however, the procedure is flexible enough to handle
(at increased computational cost) important features, such as more realistic
AC power flow models, or more complete renditions of low-level controls in
the operation of a power grid. Essentially, Procedure \ref{stochgrad1}
is an example of simulation-based optimization, i.e. it
only needs to have a ``black-box'' that computes the function $\tilde \Theta^R$.
An active research field that considers optimization under such assumptions
is that of {\em derivative-free optimization} (see \cite{derivfree})
and related methods that incorporate second-order information \cite{ipopt}.
In our estimation, these methodologies may not scale well to problems of
the size we consider. In forthcoming
work we will experiment on adaptations of these methodologies to our
problem.
When we consider a model that includes stochastics, the first-order method
can be viewed as a {\em stochastic gradients} algorithm (see \cite{robbinsmonro}, \cite{kushnerclark} -- an alternative methodology is
provided by bundle methods). In the stochastic gradients approach, a fixed {\em sample path} of the appropriate random variables is chosen in advance of
each gradient and step-length computation. In
Section \ref{stochasticoptimization} we will further discuss this approach.
Whether we use the stochastic setting or not, we cannot completely rely on
Procedure \ref{stochgrad1} as the sole optimization engine -- to repeat the
above, the resulting algorithm
would both be too slow and likely to get trapped in local
maxima. To help avoid these difficulties we rely on several heuristics
described later. In the next section we describe a special case of the
optimal control problem that can be efficiently solved.
\subsection{The optimal scaling problem}\label{scalingproblem}
In this section we describe an algorithm that computes an optimal time-dependent componentwise control under outage rule (F.1), without memory. Either version of rule (F.1) can be used; for simplicity of language we will use the first.
For brevity, we will refer to this as the {\em simple scaling setting}.
Our algorithm computes an optimal control
in time $O(m^{R-1}/(R-1)!)$ where as before $m$ is the number of lines.
\begin{RE} \label{triplestosingle} Consider an optimal control.
Let $1 \le r < R$ and let $K$ be a component of $G^r$ under the optimal
control. Then (at round $r$) we
will scale all demands in $K$ by a common multiplier $0 \le \lambda^r_K \le 1$ (defined
as in Procedure \ref{affinecontcasc}. Clearly, the control can
be equivalently defined by the values $\lambda^r_K \, (\le 1)$ rather than the triples
$(c^r_K, b^r_K, s^r_K)$, and we will use this convention below.\end{RE}
\begin{NO} Let $G$ be a graph, and let $\mu$ be a supply-demand vector on
$G$. We denote by $\hat f(G, \mu)$ the unique, feasible flow vector on
$G$ when $\mu$ is the supply-demand vector (see Remark \ref{uniqueness}).
\end{NO}
\noindent In what follows we assume that we have a given
supply-demand vector $\beta$. Let $R$ be the number of
rounds for the cascade. Our problem is to compute a control that maximizes
the total demand satisfied after $R$ rounds, assuming that at the start of
round 1, $\beta$ is the supply-demand vector. We will solve this as a
special case of a family of problems:
\begin{DE} For $t \ge 0$ real, denote by
$\Theta^{(R)}_G(t | \beta)$
the final total demand resulting from applying an optimal
control in an $R$-round cascade on graph $G$, where the initial supply-demand vector
is $t \beta$.
\end{DE}
We will show that $\Theta_G^{(R)}(t | \beta)$ is a nondecreasing
piecewise linear function of $t$ with at most $Rn$ pieces.
\begin{RE} \label{scalingflows} Let $\alpha$ be a supply-demand vector
on graph $G$. Let $t \ge 0$. Then
$\hat f(G, t \alpha) = t \hat f(G, \alpha)$.
\end{RE}
\begin{LE}\label{lemma1}
Let $\mu$ be a supply-demand vector. Suppose $G$ is connected.
Then
$\Theta_G^{(1)}(t | \mu)$ is a nondecreasing piecewise-linear function of $t$ with two pieces.
\end{LE}
{\em Proof.} Note that since $R = 1$, only Steps 1 and 2 in algorithm
(\ref{affinecontcasc}) will be executed. Further,
writing $\hat f = \hat f(G^1, \alpha)$,
when running (\ref{affinecontcasc}) starting
with the initial supply-demand vector $t \mu$, we will have
$f^1 = t \hat f$ in Step 1, and writing $\psi = \max_{j} |\hat f_{j} |/u_{j}$, we have that $\max_{j} |f^1_{j} |/u_{j} \, = \, t \psi$.
Denoting by $\tilde D$ the sum of demands implied by $\mu$ we have
as per our cascade termination criterion that the final total demand at the end of
$R = 1$ rounds will equal
\begin{eqnarray}
t \tilde D, && \mbox{if} \ \ t \, \le \, 1/\psi, \ \ \ \mbox{and} \\
\frac{t}{t \psi} ~ \tilde D \ = \frac{1}{\psi}~ \tilde D, && \mbox{otherwise}.\ \ \hspace*{.2in}\mbox{}\hspace*{\fill}\nolinebreak\mbox{$\rule{0.7em}{0.7em}$}
\end{eqnarray}
Now we turn to the general case with $R > 1$. We assume, without loss of
generality, that $G^1$ is connected. Let $\hat f = \hat f(G^1, \beta)$.
\begin{DE} A {\em critical point} is a real $\gamma > 0$, such that for
some line $j$, $\gamma \hat f_{j} = u_{j}$.
\end{DE}
Recall that we assume $u_j > 0$ for all $j$; thus let $0 < \gamma_1 < \gamma_2 < \ldots < \gamma_p$ be the set of all distinct
critical points. Here $0 \le p \le m$. Write $\gamma_0 = 0$ and
$\gamma_{p+1} = +\infty$.
\begin{DE}
For $1 \le i \le p$ let $F^i = \{ j \, \in \, {\cal A} \, : \, \gamma_h | \hat f_{j} | = u_{j} \}$.
\end{DE}
\noindent Now assume
that the initial supply-demand vector is $t \beta$ with $t > 0$ and
let $0 < \lambda^1 \le 1$ be the optimal multiplier used to scale demands
in round 1 (see Remark \ref{triplestosingle}). Write
\begin{eqnarray}
&& q = \mathop{\rm argmax}\{ h \, : \, \gamma_h < t \}. \label{defq}
\end{eqnarray}
Thus, $t \le \gamma_{q+1}$, and so $\lambda^1 t \le \gamma_{q+1}$. We stress that
these relationships remain valid in the boundary cases $q = 0$ and $q = p$.
\begin{NO} Let the index $i$ be such that $\lambda^1 t \in (\gamma_{i-1}, \gamma_{i}]$. \end{NO}
\noindent Note
that in Step 3 of round 1
of algorithm (\ref{affinecontcasc}) we will scale
all demands by $\lambda^1$, and since we assume $G^1$ is connected,
in Step 4 we will also scale all supplies by
$\lambda^1$. Thus, for any
$h \le i-1$, and any line $j \in F^h$, we have that after Step 4
the absolute value of the flow on $j$
is
\begin{eqnarray}
\lambda^1 \,t \, | \hat f_{j} | > \gamma_h \, | \hat f_{j} | = u_{j},
\end{eqnarray}
and consequently $j$ becomes outaged in round 1. On the other hand,
for any line $j \notin \cup_{h \le i-1} F^h$, the absolute value of the
flow on $j$ immediately after Step 4 is
\begin{eqnarray}
\lambda^1 \,t \, | \hat f_{j} | \le \gamma_{i} \, | \hat f_{j} | \le u_{j},
\end{eqnarray}
and so $j$ does not become outaged in round 1. In summary, the
set of outaged lines is $\cup_{h \le i-1} F^h$; in other
words, we obtain the same network $G^2 = G^1 \, - \, \cup_{h = 1}^{i-1} F^h$ for every
$t$ with $\lambda^1 t \in (\gamma_{i-1}, \gamma_{i}]$.
\begin{NO} For an index $j$, write ${\cal K}(j)$ = set of components of $G^1 \, - \, \cup_{h = 1}^{j} F^h$. \end{NO}
Let $H \in {\cal K}(i-1)$. Then, prior to Step 6 of round 1, the supply-demand vector for $H$
is precisely the restriction of $\lambda^1 t \beta$ to the buses of $H$, and when
we adjust supplies and demands in Step 6, we will proceed as follows
(where we use notation as in Section \ref{notation}):
\begin{itemize}
\item if $ \sum_{s \in {\cal D} \cap H} (-\lambda^1 t \beta_s) \, \ge \, \sum_{s \in {\cal G} \cap H} (\lambda^1 t \beta_s)$ then for each demand bus $s \in {\cal D} \cap H$ we will reset
its demand to
$$ -r \lambda^1 t \beta_s, \ \ \mbox{where} \ \ r = \frac{\sum_{s \in {\cal G} \cap H} (\lambda^1 t \beta_s)}{\sum_{s \in {\cal D} \cap H} (-\lambda^1 t \beta_s)} = -\frac{\sum_{s \in {\cal G} \cap H} (\beta_s)}{\sum_{s \in {\cal D} \cap H} (\beta_s)},$$
and we will leave all supplies in $H$ unchanged.
\item likewise, if
$ \sum_{s \in {\cal D} \cap H} (-t \lambda^1 \beta_s) \, < \, \sum_{s \in {\cal G} \cap H} (\lambda^1 t \beta_s)$ then the supply at each bus $s \in {\cal G} \cap H$ will be reset
to
$$ r \lambda^1 t \beta_s, \ \ \mbox{where} \ \ r = -\frac{\sum_{s \in {\cal D} \cap H} (\beta_s)}{\sum_{s \in {\cal G} \cap H}} (\beta_s),$$
but we will leave all demands in $H$ unchanged.
\end{itemize}
Note that in either case, in round 2 component $H$ will have a supply-demand vector of
the form $\lambda^1 t \beta^H$, where $\beta^H$ is a supply-demand vector.
Thus an optimal control on $H$, on rounds $2, \ldots, R$,
will yield a final total demand
\begin{eqnarray}
&& \Theta_H^{(R-1)}(\lambda_1 t | \hat \beta^H), \label{Hfinal}
\end{eqnarray}
which, inductively, is a nondecreasing function of $\lambda_1 t$, and therefore
is largest when
\begin{eqnarray}
&& \lambda^1 \ = \ \min\left\{1, \frac{\gamma_{i}}{t} \right\}.
\end{eqnarray}
\vspace{.1in}
\noindent {\bf Case 1.} Suppose $i \le q$. As noted above, by definition
(\ref{defq})
of $q$ we have that $\gamma_{i} \le \gamma_{q} < t$. Thus, the expression in (\ref{Hfinal})
is maximized when $\lambda_1 = \frac{\gamma_{i}}{t}$,
and we obtain final ($R$-round)
demand equal to
\begin{eqnarray}
&& \sum_{H \in {\cal K}(i-1)} \Theta_H^{(R-1)}(\gamma_{i} | \hat \beta^H), \label{fixedcase1}
\end{eqnarray}
which is independent of $t$.
\vspace{.1in}
\noindent {\bf Case 2.} Here $q < i$, and so $i = q+1$ by definition of $q$
and $\lambda^1 \le 1$. Thus (\ref{Hfinal}) is maximized by setting $\lambda^1 = 1$. The final demand equals
\begin{eqnarray}
&& \sum_{H \in {\cal K}(q)} \Theta_H^{(R-1)}(t | \hat \beta^H). \label{case2}
\end{eqnarray}
\noindent In summary, we have:
\begin{eqnarray}
\Theta^{(R)}_G(t|\hat \beta) \ = \ \max \left\{ \, \max_{1 \le i \le q} \left\{\sum_{H \in {\cal K}(i-1)} \Theta_H^{(R-1)}(\gamma_{i} | \hat \beta^H)\right\} \ , \ \sum_{H \in {\cal K}(q)} \Theta_H^{(R-1)}(t | \hat \beta^H) \, \right\}. \label{consolidated}
\end{eqnarray}
\begin{THM}\label{th2}
(i) $ \Theta^{(R)}_G(t|\hat \beta)$ is nondecreasing, piecewise-linear,
with at most
$$ \frac{m^{R-1}} {(R-1)!} ~ + ~ O\left(m^{\max\{1, R-2\}}\right)$$
breakpoints.\\
\noindent (ii) The optimal choice for $\lambda^1$ is $\lambda^1 = 1$ or
$\lambda^1 = \gamma_k/ t$ for some $k$.
\end{THM}
\noindent {\em Proof.} (i) By induction on $R$, starting from Lemma \ref{lemma1}. For the general step, consider the above discussion which assumes that
$\lambda^1 t \in (\gamma_{i-1}, \gamma_{i}]$. Then if Case 1 above holds,
we have that $ \Theta^{(R)}_G(t|\hat \beta)$ is constant. And if
Case 2 holds, then equation (\ref{consolidated}) applies. The form of
(\ref{consolidated}) guarantees that, inductively,
$ \Theta^{(R)}_G(t|\hat \beta)$ is nondecreasing piecewise-linear.
To analyze the number of breakpoints in $\Theta^{(R)}_G$, assume
first that $R = 2$. Consider the
effect of removing, {\em one at a time}, the lines of $\cup_{h = 1}^{p} F^h$.
Prior to its removal, each line $j$ has both ends in the same component $K$;
the
removal either creates two new components (if $j$ is a bridge of
$K$) or creates a new component (which differs from $K$ in that line $j$ is
not included). Thus the removal process can be represented as a binary tree
whose leaves correspond to the components of $G^1 \, - \, \cup_{h = 1}^{p} F^h$,
i.e. the members of ${\cal K}(p)$. Since these are disjoint there are at most
$n$ of them; since in a binary tree the number of degree three vertices is
at most the number of leaves we conclude that
$$ | \cup_{h = 1}^p {\cal K}(h) | \le m + n \le 2 m + 2.$$
Furthermore, let $H$ be a component in
$ \cup_{h = 1}^p {\cal K}(h)$. Define $h = \min \{ j \, : \, H \in {\cal K}(j) \}$
and $h' = \max \{ j \, : \, H \in {\cal K}(j) \}$. By Lemma \ref{lemma1}, it
follows that
$\Theta^{(1)}_H$ will contribute at most one breakpoint
to $\Theta^{(R)}_G$, and
that this breakpoint will occur for some $t$ with $\lambda^1 t \in [\, \gamma_h \, , \, \gamma_{h'} \,)$. The maximum in
(\ref{consolidated}) shows that for each $q$, one additional new breakpoint
is created. Thus, in total, $\Theta^{(R)}_G$ has at most $O(m)$
breakpoints and the result
is verified for $R = 2$.\\
\vspace{.1in}
In what follows we assume that $R \ge 3$. Suppose
$q = 0$ and thus $i = 1$. Since $\lambda_1 t < \gamma_1$, it follows that
no lines are outaged in round 1, i.e. $G^2 = G^1 = G$, and in subsequent rounds
no line will be overloaded. Thus, in this case, $\Theta^{(R)}_G(t|\hat \beta) = t \tilde D$ and there are no breakpoints. For $q > 0$ we proceed using (\ref{consolidated}). For each $H \in {\cal K}(q)$,
inductively,
$\Theta^{(R-1)}_H$ has at most
$$ \frac{m_H^{R-2}}{(R-2)!} \ + \ c \, m_H^{\max\{1, R-3\}} $$
breakpoints, where
$m_H$ denotes the number of lines in $H$ and $c \ge 0$ is a constant.
So (\ref{consolidated}) implies that subject to $i = q+1$, the
number of breakpoints in $\Theta^{R}_G$
is at most
\begin{eqnarray}
&& ~ 1 ~ + ~ \sum_{H \in {\cal K}(q)}\left[ \frac{m_H^{R-2}}{(R-2)!} \, + \, c \, m_H^{\max\{1, R-3\}}\right] \nonumber \\
&& \le \ 1 ~ + ~ \frac{\left( \, m \, - \, | \cup_{h = 1}^{q} F^h| \right)^{R-2}}{(R-2)!} ~ + ~ c \left( \, m \, - \, | \cup_{h = 1}^{q} F^h| \right)^{\max\{1,R-3\}} \nonumber \\
&& \le 1 ~ + ~ \frac{\left( \, m \, - \, q \right)^{R-2}}{(R-2)!} ~ + ~ c \left( \, m \, - \, q \, \right)^{\max\{1,R-3\}}
\label{sumconsolidated}
\end{eqnarray}
since $\left( \cup_{h = 1}^{q} F^h \right) \cap H = \emptyset$ for each
$H\in {\cal K}(q)$. Summing this expression over all $1 \le q \le p$, we
obtain that the total number of breakpoints is at most
\begin{eqnarray}
&& p ~ + ~ \sum_{q = 1}^p\left[ \frac{\left( \, m \, - \, q \right)^{R-2}}{(R-2)!} ~ + ~ c \left( \, m \, - \, q \right)^{\max\{1,R-3\}} \right] \nonumber \\
&& \le m ~ + ~ \sum_{q = 1}^m\left[ \frac{\left( \, m \, - \, q \right)^{R-2}}{(R-2)!} ~ + ~ c \left( \, m \, - \, q \right)^{\max\{1,R-3\}} \right] \nonumber \\
&& \le \frac{\left( \, m \, - \, 1 \right)^{R-1}}{(R-1)!} ~ + ~ O((m-1)^{R-2}) + ~ m ~ + ~ c \sum_{q = 1}^m \left( \, m \, - \, q \right)^{\max\{1,R-3\}}. \label{hew}
\end{eqnarray}
For $R = 3$ the last three terms in (\ref{hew}) are $O(m)$ and we are done as desired. For
$R > 3$, the last term in (\ref{hew}) equals
\begin{eqnarray}
&& c \frac{\left( \, m \, - \, 1 \right)^{R-2}}{R - 2} ~ + ~ O(m^{R-3}),
\end{eqnarray}
and again we conclude as desired for $c $ large enough.\\
\noindent (ii) This follows from the discussion leading to eq. (\ref{consolidated}). \rule{.1in}{.1in}
\vspace{.1in}
Part (ii) of Theorem \ref{th2} illustrates a weakness of the simple scaling
approach -- when applying an optimal control, at least one line becomes
fully loaded at each round. Such a strategy is likely non-robust. We
plan to address this issue in upcoming work; using the stochastic
outage (F.2) and computing an appropriate optimal control.
Despite the apparent shortcomings of the method, and of the simplicity of
the proposed control, the ability to compute a global optimum in polynomial
time (for fixed $R$) is a significant asset, especially as a starting point
for the simulation-based methods for the general problem that are
proposed below.
In forthcoming work we will
implement an appropriate version of the above algorithm; an relevant question
is whether the worst-case bound in Theorem \ref{th2} is attained using
realistic data.
\section{The algorithm}\label{thealgorithm}
Our algorithm implements Procedure \ref{stochgrad1} to implement
an affine control as in Template (\ref{affinecontcasc}),
repeated here for convenience.
The control specifies, for each round $r$ of the cascade and each demand
bus $v$, a triple $(c^r_v, b^r_v, s^r_v)$. At round $r < R$ of the cascade, each
demand bus $v$ observes the maximum line load $\kappa^r_v$ in the
component that $v$ currently belongs to. Then, where $d^r_v$ denotes the
current value of the demand at $v$,
\begin{eqnarray}
&& \mbox{if} \ \kappa^r_v > c^r_v, \ \mbox{demand at $v$ is reset to} \ \min\{1, \, [b^r_v + s^r_v (c^r_v - \kappa^r_v)]^+ \}~d^r_v. \label{reminder}
\end{eqnarray}
The ``normal'' case of such a control is that where $b^r_v = c^r_v = 1$, and
$s^r_v \ge 0$, which decreases demands in proportion to the maximum overload.
However, cases with $c^r_v \neq 1$ (delayed or proactive control) can
prove optimal. Setting $b^r_v < 1$ can result in nonsmooth (fixed-penalty)
controls. Finally, setting $s^r_v < 0$, though counterintuitive, can prove
optimal when non-monotone behavior occurs. \\
\noindent In order to initiate the gradient search method, we rely on
grid-search, a standard enumerative idea:
\vspace{.1in}
\noindent {\bf Grid search.} Here we fix $c^r_v = b^r_v = 1$ for all $r$ and
$v$, and $s^r_v = 0$ for all $v$ and all $2 < r < R$. Thus
the only remaining parameters are $s^r_v$ for all $v$ and $r =1, 2$.
We restrict the search to two values $\bar s^1$ and $\bar s^2$,
and insist that for all $v$ and $1 \le r \le 2$ we
have $s^r_v = \bar s^r$. In our current implementation,
this two-dimensional search,
in turn, is carried out one parameter at a time, as follows. Let $\tilde \kappa^1$
be the maximum line overload observed in the no-control cascade, during round
1. Assuming $\tilde \kappa^1 > 1$, then we enumerate all choices for
$\bar s^1$ of the form $\bar s^1 = (.1 + .008\, i)/(\tilde \kappa^1 - 1)$ for
$i = 0, 1, \ldots, 100$. In other words, this enumerates all controls where
in round 1 we scale demands by a factor of $.9 - .008 \,i$ for $i = 0, 1, \ldots, 100$. Let $\bar s^1_1 < \bar s^1_2$ be the two enumerated choices which
produce the highest and second highest $\tilde \Theta^R$ value. Then we
repeat the search in the interval $[\bar s^1_1 , \bar s^1_2]$ by enumerating
$\bar s^1 = \bar s^1_1 + i \, (\bar s^1_2 - \bar s^1_1)/100$ for $i = 0, 1, \ldots, 100$.
The value that produces that highest
$\Theta^R$ value is our final choice for $\bar s^1$. We fix this value and
now carry out the same type of search for $\bar s^2$.\\
\vspace{.1in}
We will see below that grid search can produce very good control vectors, but
which in general can be improved, sometimes significantly, by widening the
search. One can use the control computed by grid search to start the general
gradient search; however in high-dimensional cases even general gradient search
itself can be quite slow as each gradient estimation step could prove very
slow. This will not be the case if enough parallel computing resources
are available; however and we have found an additional step to be useful:
\vspace{.1in}
\noindent {\bf Segmented search}. As introduced in Section \ref{adaptive1},
consider
a fixed partition
$(\Sigma_1, \Sigma_2, \ldots, \Sigma_H)$ of the demand buses. We search,
using the first-order method,
for triples of the form $(\hat c^r_i, \hat b^r_i, \hat s^r_i)$ for
each $1 \le r < R$ and $1 \le i \le H$, so as to obtain the control where
for each $1 \le r < R$, and each demand bus $v$,
$(c^r_v, b^r_v, s^r_v) = (\hat c^r_i, \hat b^r_i, \hat s^r_i)$ if $v \in \Sigma_i$. In our implementation, the $\Sigma_i$ are {\em demand quantiles}. That
is to say, if $L$ is the number of demand buses, then $\Sigma_1$ contains
the $\lfloor H/L \rfloor$ buses with largest demand, $\Sigma_2$
contains the next $\lfloor H/L \rfloor$ buses with largest demand, and so on.
The advantage of this approach is that it considerably reduces the dimensionality of the problem, even if $H$ is chosen relatively large, such as
$H = 100$. In fact, the (segmented) first-order method
runs quite fast, and the approach in \cite{ipopt} might also be practicable.
Further, a segmented control is arguably 'fair' in that
it specifies, to some degree,
that similar buses are bound by similar control laws, though we stress that
when applying the control (\ref{reminder}) the actual demand reduction can
be very different for two buses in the same segment but in different components.
In the first set of experiments we have conducted, we have chosen $H = 50$ and
we fix $\hat b^r_i = 1$ for $1 \le r < R$ and $1 \le i \le H$. Thus, altogether, we have $2 \, H \, (R - 1) = 100 \, (R - 1)$ variables, still large but
much more manageable than full gradient search. In the experiments below,
we {\em forgo} full gradient search; however it would be straightforward to
follow up segmented search with full gradient search.
\subsection{Implementation details}\label{implementation}
The parallel implementation of our algorithm relies
on the familiar master-worker paradigm. Each worker performs computations of
the function $\tilde \Theta^R(c,b,s)$ for a given control $(c,b,s)$ whereas
the master carries out the gradient search algorithm. In the experiments
we report on here, we use the linear power flow model; linear programs are
solved using Cplex 12.0 \cite{cplex} and Gurobi 3.0 \cite{gurobi}. These solvers were used with all presolve options turned-off (this increased robustness).
Further, the flow component in the solution to
the linearized power flow system (\ref{balance})-(\ref{ohm-eq}) is invariant
under scaling of the $X$ vector; we scaled all reactances so that the largest
value was $100$ (also for solver robustness). Interprocess communication in
our algorithm uses Unix sockets. The computations below were performed on
three eight-core i7 machines with 48GB of RAM each.
\subsection{Second set of experiments}\label{experiments}
As before we use the Eastern Interconnect snapshot for our experiments.
Synthetic contingencies were developed by removing $K$, random, highly
loaded lines, as in Section \ref{firstsetmethod}.
\noindent Our first set of experiments, shown in Table 1, concern cascades
with $R = 4$ rounds. When applying rule (F.1), we used $\alpha = 0.55$.
As stated above, the segmented search was performed using $H = 50$ segments.
\begin{table}[h]
\centering
\caption{{\bf \emph{Performance of algorithm on 4-round cascades}}}
\vskip 10pt
\begin{tabular}{|c | r| r| r|}
\hline
{\bf K} & {\bf yield,} & {\bf yield,} & {\bf wallclock} \\
& {\bf no control} & {\bf control} & {\bf (sec)} \\
\hline
\hline
{\bf 1} & 90.04 & 95.03 & 268\\
{\bf 2} & 1.25 & 50.13 & 174\\
{\bf 5} & 32.94 & 81.05 & 214\\
{\bf 10} & 2.02 & 36.97 & 194\\
{\bf 20} & 1.64 & 27.84 & 220\\
{\bf 50} & 0.83 & 16.96 & 477 \\
\hline
\end{tabular}
\label{table1}
\end{table}
\noindent In this table, the columns headed 'yield' indicate the
{\em percentage} of total initial demand satisfied at the end of the cascade (without control, and using
the computed control), and 'wallclock' is the observed
parallel running time of the method. In each of these runs, the total number
of gradient steps was small, typically smaller than 5.
Note that in the
case $K = 1$ the interdiction has limited effect, but even so the control
is able to recover additional demand. In the case $K = 5$ the
demand loss in the no-control case is substantial, but so is the benefit
of the control. Finally, in the cases $K = 2, 10, 20, 50$ the network collapses
but the control sill recovers a significant amount of demand. More experiments of this type will be forthcoming.
In the next set of experiments we use the case $K = 50$ in Table \ref{table1}
to investigate in more detail
the behavior of the algorithm as $R$ increases. We used $\alpha = 0.5$ for
all these experiments. Note that keeping $\alpha$ constant but increasing
$R$ effectively considers cascades that take longer from a 'real time'
perspective, thereby giving more power to an agent applying control.
If, instead, we were to increase $R$ while also decreasing $\alpha$, thus
giving more weight to 'history', we would be able to model cascades that
last for a fixed period of time, but where the individual rounds encompass
shorter spans of time.
Table \ref{tablerounds} reports on the experiments. As before, 'yield'
is the percentage of demand satisfied at the end of the cascade, using
no control, the control obtained by grid-search, and the control obtained
by segmented gradient search (started at the control computed by grid search).
The two wallclock columns report, in seconds, the time used by grid- and
gradient-search. In the case of grid search, we report the time spent on each
of the two search steps (i.e., over rounds 1 and 2, respectively).
The column labeled 'grad steps' reports the number of
gradient steps.
\begin{table}[h]
\centering
\caption{{\bf \emph{Impact of increasing number of rounds on K = 50 case from Table \ref{table1}}}}
\vskip 10pt
\begin{tabular}{|c | r| r | r| r| r| r|}
\hline
{\bf R} & {\bf yield} & {\bf yield} & {\bf yield} & {\bf wallclock} & {\bf wallclock} & {\bf grad} \\
& {\bf no control} & {\bf grid} & {\bf gradient}& {\bf grid} & {\bf gradient} & {\bf steps} \\
\hline
\hline
{\bf 5} & 4.13 & 18.11 & 31.86 & 30 + 17 & 1340 & 7\\
{\bf 6} & 2.02 & 23.01 & 25.86 & 26 + 14 & 657 & 6\\
{\bf 7} & 2.25 & 25.10 & 25.98 & 33 + 15 & 434 & 3\\
{\bf 8} & 0.78 & 29.27 & 46.97 & 18 + 43 & 3151 & 10 \\
\hline
\end{tabular}
\label{tablerounds}
\end{table}
\noindent Next we will comment on Table \ref{tablerounds}.\\
\noindent {\bf Computational workload}
\noindent Consider the
case $R = 8$. Since we are using
$H = 50$ segments, we have altogether $100$ control variables $c^r_i$ and $s^r_i$ per round $r$.
Since there are $7$ rounds during which we will apply control, we have a total of
$700$ individual variables. Each partial derivative estimation requires
two simulations; thus in total each gradient estimation entails
$1400$ cascade simulations. Per iteration, the
step-size computations require $200$ additional cascade simulations; for
a total of $1600$ simulations per iteration of Procedure \ref{stochgrad1}.
The case $R = 8$ required
$10$ gradient iterations, and thus in total $16000$ simulations. Each 8-round
simulation (of the $15000$-bus Eastern
Interconnect, and using one core of the i7 CPU)
requires, on average, $4.5$ CPU seconds. This is primarily due to the
two power flow computations
per round, and linear solver data structure cleanup at the end of
the simulation (and to a much lesser degree, to graph algorithms used to
identify islands).
Thus in total the computation
of the $R = 8$ case required approximately $72000$ CPU seconds.
Since we have 24 worker cores, this
translates to approximately $3000$ wallclock seconds. The balance of time
with respect to the actual wallclock time in Table \ref{tablerounds} (i.e., $151$ seconds) is due to inter-process
communication and networking delays, and logging of statistics to disc by the master. On a per-simulation, per-core basis, this amounts
to $151*24/16000 \approx 0.22$ seconds, or roughly $5 \%$ as compared to
$4.5$ seconds total per simulation. \\
\noindent{\bf Grid-search vs gradient-search}
\noindent In several cases gradient search significantly improves on the grid
search solution. This is especially noticeable in the $R = 8$ case, and we
will examine this case in some detail.
First, the grid-search control we
computed in this case uses $(\bar c^1, \bar b^1, \bar s^1) = (1, 1, 0.00018)$,
and $( \bar c^r, \bar b^r, \bar s^r) = (1,1,0)$ for $r > 2$, effectively
limiting control to the first round. In contrast, the gradient-search control
we computed applies control as late as round 7
(which, as we have 8 rounds in total, is the last round for which we
compute a control as per our control template (\ref{affinecontcasc})) , and within earlier rounds
it applies different controls to different segments. In particular, in round
1 the gradient-search control uses the control vector $(0.95, 1, -0.0499)$
for segment 1 (the highest demand segment) as well as two other segments,
while for all other segments it uses $(1, 1, 0.00018)$. And in round 7
it uses $(1.1, 1, 0.05)$ for
segment 2, and $(0.95, 1, 0.05)$ for segment 3, while for all other segments
it uses $(1, 1, 0)$. Other controls different from $(1, 1, 0)$ are used
in rounds 2 and 4, while on rounds 5 and 6 it uses (1,1,0) throughout.
\begin{table}[h]
\centering
\caption{{\bf \emph{Controlled cascade evolutions}}}
\vskip 10 pt
\begin{tabular}{|c| r r r r| r r r r|}
\hline
\multicolumn{1}{|c|} {} & \multicolumn{4}{c|}{{\bf Grid-search}} & \multicolumn{4}{c|}{{\bf Gradient-search}}\\
{\bf Round} & \boldmath{$\kappa$} & {\bf faults} & {\bf comps} & {\bf yield} & \boldmath{$\kappa$} & {\bf faults} & {\bf comps} & {\bf yield} \\\hline \hline
{\bf 1} & 3.79 & 126 & 1 & 45.37 & 172.22 & 1629 & 32 & 60.72 \\
{\bf 2} & 33.49 & 32 & 1 & 45.37 & 97.44 & 1079 & 293 & 54.26 \\
{\bf 3} & 7.44 & 26 & 2 & 45.27 & 59.97 & 282 & 401 & 49.87 \\
{\bf 4} & 6.69 & 82 & 4 & 45.27 & 21.88 & 89 & 459 & 48.67 \\
{\bf 5} & 4.95 & 72 & 9 & 45.23 & 2.74 & 55 & 468 & 47.74 \\
{\bf 6} & 1.99 & 28 & 13 & 45.23 & 13.27 & 10 & 471 & 47.72 \\
{\bf 7} & 1.54 & 16 & 13 & 45.23 & 1.01 & 14 & 478 & 47.41 \\
{\bf 8} & 1.00 & 16 & 13 & 29.27 & 1.00 & 1 & 478 & 46.97 \\
\hline
\end{tabular}
\label{gridgradcomparo}
\end{table}
Table \ref{gridgradcomparo} compares the cascade evolution under both controls.
Here, we report the value, at the end of each round of: $\kappa$ (the maximum line overload); the number of components of the network; and the yield. 'faults'
is the number of line outages experienced during each round. Comparing the two evolutions, we see that the gradient-search control allows significantly more
outages in initial rounds (as well as much more islanding). Nevertheless, the
number of outages is (nearly) monotonically decreasing under gradient-search and
by round 5 it is smaller than under grid-search. Thus the gradient search
control appears to be maintaining high yield while carefully allowing outages
to take place. During round 7, gradient-search makes a large improvement on
the maximum line overload (from $13.27$ to $1.01$) but {\em without} losing
much yield; this is evidence of yet more sacrificial line outages.\\
\noindent {\bf Qualitative issues}
\noindent A comparison of
the entry in Table \ref{tablerounds} for $R = 5$ and those for $R = 6, 7$
might appear to indicate that the controls computed for $R = 6$ and $7$
are locally optimal,
because the
control
that achieves yield $31.86\%$ for $R = 5$ ``should be''
feasible for all $R \ge 5$.
While it is true that the controls in Table \ref{tablerounds}
can all be improved upon, the argument in the above paragraph is not
quite correct. Refer to our Cascade Control template \ref{contcasc}. The termination
step constitutes a last-recourse form of control -- if there are line
overloads at the start of the last round, loads are scaled so as to remove
the overloads, and in that case the cascade is considered terminated, regardless
of history (and of particular, of rule (\ref{memory-1}). We model
termination this way on purpose, so as to provide an agnostic termination
criterion that does not depend on numerical parameters of our model, in
particular, $\alpha$. Consider Table \ref{manykinds}.
\begin{table}[h]
\centering
\caption{{\bf \emph{Maximum line overload at end of each round for K = 50 case from Table \ref{tablerounds}}}}
\vskip 2pt
\begin{tabular}{|c | r r r r r|}
\hline
{\bf } & \boldmath$C_5$ & \boldmath$C_6$ & \boldmath$C_7$ & \boldmath$C_8$ & {\bf None} \\
\hline
\noalign{\smallskip}
{\bf 1} & 6.47 & 1.83 & 2.22 & 3.79 & 177.83 \\
{\bf 2} & 14.12 & 1.83 & 1.57 & 33.49 & 122.06 \\
{\bf 3} & 36.79 & 1.23 & 1.30 & 6.90 & 114.45 \\
{\bf 4} & 1.72 & 1.14 & 2.26 & 6.70 & 22.47 \\
{\bf 5} & & 0.99 & 1.18 & 59.33 & 45.43 \\
{\bf 6} & & & 1.08 & 1.98 & 40.33 \\
{\bf 7} & & & & 1.18 & 114.90 \\
\hline
\end{tabular}
\label{manykinds}
\end{table}
In this table, the columns labeled ``$C_k$'', for $k = 5, \ldots, 8$ represent
the controls in Table \ref{tablerounds}, whereas ``None'' means no control.
The table shows, for each round, the maximum line overload at the {\em end} of
that round, for each control option. We see that $C_5$ reaches the start of
the termination round, round $5$, with maximum overload $1.7232$; the
current yield at the start of round $5$ is $54.90 \%$ (not shown in the Table)
and most of the demand is in one island. Hence the termination step will
scale demands by $1/1.7232$ and yield will drop to $54.90/1.7232 \approx 31.86$,
as Table \ref{tablerounds} shows. As per our rules, this terminates
the cascade, although since $\alpha = 0.5$,
and because the end-or-round $3$ maximum overload is very large, the
maximum history-dependent line overload will be much larger than $1.732$
(it should be at least $0.5*36.79 = 18.40$. Hence control $C_5$, if implemented
in a cascade with $6$ or more rounds, will not result in a stable state by
the end of round $5$.
Another point that emerges from Table \ref{manykinds} is that
$C_5$ and $C_8$
tend to maintain higher line overloads late into the cascade -- this
is a severe cascade, and having more time to apply control pays off.
But by doing so $C_5$ and $C_8$ are likely less robust. Rather than performing
the same robustness analysis as in Section \ref{firstset}, we will next
consider the stability of the controls with respect to the $\alpha$
parameter in eq. (\ref{memory-1}) which in the above tests was set to
$0.5$.
This is a delicate issue, because the value of $\alpha$ is related to the
time duration of a round, and thus the structure of an optimal control
{\em should}
depend on $\alpha$ (in other words, how much time we have impacts what
kind of control we can apply). The question is how stable a control remains as
$\alpha$ is perturbed.
\begin{table}[h]
\centering
\caption{{\bf \emph{Stability of controls in Table \ref{tablerounds} as a function of $\alpha$}}}
\vskip 2pt
\begin{tabular}{|c | r r r r |}
\hline
\boldmath$\alpha$ & \boldmath$C_5$ & \boldmath$C_6$ & \boldmath$C_7$ & \boldmath$C_8$ \\
\hline
\noalign{\smallskip}
{\bf 0.45} & 1.49 & 25.05 &24.52 & 27.10 \\
{\bf 0.46} & 1.49 & 25.33 &24.52 &25.31 \\
{\bf 0.47} & 28.49 & 25.33 &24.52 & 25.31 \\
{\bf 0.48} & 28.47 &25.33 &24.52 & 26.08 \\
{\bf 0.49} & 28.47 & 25.33 &24.52 & 28.56 \\
{\bf 0.50} & 31.86 & 25.86 & 25.98 & 37.72 \\
{\bf 0.51} & 21.99 & 25.86 &25.98 & 34.11 \\
{\bf 0.52} & 20.99 & 25.86 &25.98 & 35.94 \\
{\bf 0.53} & 20.99 &25.86 &25.98 & 32.75 \\
{\bf 0.54} & 20.99 & 25.86 &25.98 & 32.75 \\
{\bf 0.55} & 20.99 & 25.86 & 25.98 & 31.83 \\
\hline
\end{tabular}
\label{alphadep}
\end{table}
In Table \ref{alphadep} we show the yields obtained by
running the $C_k$ controls from Table \ref{tablerounds} using their
respective numbers of rounds, but using different values for
$\alpha$. We see that in terms of the deviation from the nominal
case (i.e., $\alpha = 0.5$), $C_6$ and $C_7$ prove the most stable,
$C_8$ significantly less so and $C_6$ is very unstable. It is
still the case that $C_8$ remains best overall: this is due to
the severity of the cascade.
In Figure \ref{alphavsyield} we consider a broader experiment along the
same lines. For this experiment we used a control intended for the
$R = 8$, $K = 50$ case considered in previous sections, and computed
assuming $\alpha = 0.5$ (as per the memory-dependent rule (\ref{memory-1}) for
arc outages).
Figure \ref{alphavsyield} displays the actual yield as $\alpha$ is changed
away from $0.5$. We note that yield is relatively robust for $\alpha$ larger
than but close to $0.5$, but not if $\alpha$ is decreased.
We believe that this behavior is due to application of our (deterministic)
control results on lines becoming $100\%$ loaded. Several heuristics,
based on ``padding'' (increasing) or ``tightening'' (reducing)
the flow limits $u_j$, suggest themselves. However the greedy nature of
deterministic controls will remain a fundamental difficulty. Section
\ref{stochasticoptimization} discusses the computation of controls
under stochastics.
\begin{figure}[htb]
\centering
\begin{center}
\includegraphics[height=4in, angle=270]{alphayield}
\caption{yield as a function of alpha}\label{alphavsyield}
\end{center}
\end{figure}
\subsubsection{Additional tests}\label{additional}
\noindent The next set of experiments address basic conjectures that arise
in the context of the type of control that we consider:
\begin{itemize}
\item It is best to stop the cascade in the first round, i.e. to sufficiently
reduce demands in the first round so as to eliminate all line overloads.
\item It is best to apply control in the first round only, and ride out the cascade for the remaining rounds.
\end{itemize}
\noindent In fact it is a simple task to create small examples where both
conjectures above are proved wrong. Instead we explore these questions
using the Eastern Interconnect, with a random interdiction
of the type described above with $K = 50$, three rounds, and $\alpha = 1$ (no memory, and thus we obtain outage rule (F.1)). The results of this experiments are
as follows:
\begin{itemize}
\item Using no control, after three rounds $0.63 \%$ of the demand is satisfied.
\item Grid search produces a control with yield $45.46 \%$.
\item Starting from this control, and using segmented search with $50$ segments
improves yield to $50.02 \%$.
\end{itemize}
To gain a different perspective on this cascade, consider Table \ref{tableover},
which shows the distribution of line overloads at the start of the cascade.
\begin{table}[h]
\centering
\caption{{\bf \emph{Distribution of line overloads}}}
\vskip 10pt
\setlength{\extrarowheight}{3pt}
\begin{tabular}{|c || r | r| r| r| r| r| r| r| r| r| r| r| r|}
\hline
{\bf $\lceil$ overload $\rceil$} & 1505 & 58 & 48 & 32 & 22 & 19 & 11 & 7 & 6 & 5 & 4 & 3 & 2 \\
{\bf count} & 1 & 1 & 2 & 1 & 2 & 1 & 1 & 2 & 2 & 4 & 6 & 18 & 181 \\
\hline
\end{tabular}
\label{tableover}
\end{table}
Where $f^1$ denotes the flow vector at the start of the cascade, the table
indicates the quantity of lines $j$ whose (rounded-up) overload
$\lceil |f^1_j|/u_j \rceil$ equals a particular value. Thus, $181$ lines
$j$ are such that $1 < |f^1_j|/u_j \le 2$, $18$ lines $j$ are such
that $2 < |f^1_j|/u_j \le 3$, and so on. One line has overload greater than
$1504.93$. We will provide a more detailed analysis of this case in the near future;
however it is the case (as may seem plausible from the table) that in order
to stop {\em all} overloads in round 1 a drastic reduction of demands is
needed. We will instead describe the behavior of the optimal control
computed by grid search has
$(c^1_v, b^1_v, s^1_v) = (300.7, 1, 0.0004)$, and
$(c^2_v, b^2_v, s^2_v) = (1.26, 1, 0.62)$.
Thus, in round 1, all demands
will be scaled by a (approximately) factor of $1.0 + 0.0004*(300.7 - 1504.93) =
0.51831$. Considering Table \ref{tableover}, we see that in round 1 all lines
included in the columns with overload greater than $2$ will be outaged --
this is a total of $41$ lines, and in fact three more with overload close to
$2$ are outaged.
At the start of round 2, the maximum overload is approximately $1.36194$. Thus,
the control specifies that demands will be reduced by a factor of
$1.0 + 0.62*(1.26 - 1.36194) = 0.9368$. This does not completely remove
all overloads and an additional $4$ lines become outaged.
Finally, at the start of round 3, the maximum overload is $1.067891$. By the
rules of our cascade model, this overload is now removed by scaling down
all demands. Altogether, we obtain a yield of $0.51831*0.9368/1.067891 = 0.4547$, as stated above.
\section{Stochastic optimization}\label{stochasticoptimization}
To further motivate the need for stochastic modeling, we consider the
same setup as for the experiment in Figure \ref{alphavsyield}:
$R = 8$, $K = 50$ and $\alpha = 0.5$. We
simulated the behavior of the computed control
using the stochastic outage rule (\ref{actual1})-(\ref{actual3}), repeated
here for convenience. We are given a parameter $0 < \epsilon < 1$; if $\tilde f$ is a flow vector, then line $j$
is {\em not} outaged if $|\tilde f_j| < (1 - \epsilon)u_j$, it
{\em is} outaged if if $|\tilde f_j| > u_j$, and is outaged with probability
$1/2$ if $(1 - \epsilon)u_j \le \tilde f_j \le u_j$.
For various values of the tolerance $\epsilon$, we simulated the cascade
$10000$ times. For $\epsilon < 0.03$ little difference was observed with
the nominal (deterministic) setting in that the average yield was close
to (the deterministic yield of) $50 \%$. For $\epsilon = 0.03, 0.10$
and $0.20$ the results are displayed in Figure \ref{epsilon}.
\begin{figure}[htb]
\centering
\begin{center}
\includegraphics[height=5in, angle=270]{epsilon}
\caption{yield histogram under stochastic outages}\label{epsilon}
\end{center}
\end{figure}
The figure shows, for each value of the yield (and for each of the three
choices for $\epsilon$) how frequently that yield was observed. Note that
for $\epsilon = 0.03$ the distribution is essentially trimodal. This is typical
behavior and it points to a small number of critical lines,
which, in turn, result in a small number of cascade trajectories being overwhelmingly
likely. For $\epsilon = 0.20$
yields close to zero are observed, but, significantly, the average is positive.
In a stochastic setting, the yield $\Theta^R(c,b,s)$ resulting from
a control $(c,b,s)$ is a random variable.
Below we discuss different methodologies for computing a (locally)
optimal stochastic control, with the objective of maximizing
the expectation $\rm E( \Theta^R)$. Other merit criteria are also
of interest (and possibly better), such as a linear combination of expectation
and variance $\rm E(\Theta^R) - \lambda \, var(\Theta^R)$
($\lambda \ge 0$), or a Sharpe-ratio quantity $\rm E(\Theta^R)/var(\Theta^R)$.
The computational challenges inherent in any of these tasks are
significant. First, as displayed in Figure \ref{epsilon}, yield variances
can be extremely large. From a theoretical perspective, the number of samples
needed to obtain reliable estimates become inordinately large. Additionally,
there are subtle difficulties due to the non-monotonic behavior of power
flow systems (see \cite{bie2}), which is reminiscent of Braess' paradox \cite{braess}.
An example of this behavior is provide by our stochastic outage
rule (\ref{stochout}). Note that under this rule, a line is more likely
to become outaged than under the deterministic rule (i.e.,
when $(1 - \epsilon_r)u_{j} < \tilde f^{r}_{j} \le u_{j} $ the line may
become outaged in the stochastic setting; not so in the deterministic
setting). Nevertheless, one can produce cases where the deterministic yield
of a control is {\em smaller} than a sample yield of the same control under
rule \ref{stochout}. This phenomenon slows down convergence of our algorithms
and makes algorithm calibration difficult.
\subsection{Optimization methods through simulation}\label{gridstoch}
In either the first-order procedure (\ref{stochgrad1}) discussed above,
or in the grid-search setting, we obtain
a counterpart valid for a stochastic setting by replacing each (deterministic)
evaluation of a yield $\Theta^R(c,b,s)$ by an estimation of
$\rm E ~ \Theta^R(c,b,s)$.
This is the approach used in the next set of experiments, which parallel
those in Section \ref{firstset}. For convenience, we restate the setup for
these tests.
First, two random, though adversarially chosen
lines were removed from the grid. Next, we compute the best control such
that
\begin{itemize}
\item [(i)] $c^r_v = b^r_v = 1$ for all $v$ and $r$.
\item [(ii)] $s^r_v = 0$ for all $v$ and $10 < r$. Thus, no control is applied
after round 10.
\item [(iii)] For each $1 \le r \le 10$, \underline{either} $s^r_v = 0.005$ for all
$v$, \underline{or} $s^r_v = 0$ for all $v$.
\end{itemize}
This was done, in the deterministic setting, while setting the maximum number of rounds, $R$, to $10$, $15$
$20$ and $25$. In Section \ref{firstset}, we observed that the cascade is
characterized by high line overloads during the initial rounds; nevertheless
if ``enough'' rounds are allowed ($34$) then without
control the cascade stabilizes and produces approximately $78\%$ yield,
and this was slightly superior to what was obtained by the controls we
computed. In
summary, this case provides a good contrast between the (opposing) need to
``wait out'' an initially severe cascade on the one hand, with uncertainty
growth as we increase the number of rounds. Using the framework of stochastic outage rule (\ref{stochout}),
with
\begin{eqnarray}
\epsilon_r & = & 0.01 \, + 0.05* \lfloor r/10 \rfloor, \label{noise1prime}
\end{eqnarray}
we observed that the (deterministic) $20$-round control appeared superior.
In this section we will repeat these comparisons, except that now we will
compute controls of the form (i)-(iii) that (approximately) maximize
the expected yield. We compute such controls by modifying our
grid search: we evaluate a control vector by simulating $20$
cascades and computing the sample average yield. This can be a somewhat
coarse approximation because, depending on the model for noise,
many more than $20$ samples may be needed for an accurate answer since the
variance of yield can be high.
\begin{table}[h]
\centering
\caption{{\bf \emph{Stochastic grid search results in case from Section \ref{firstset}}}}
\vskip 10pt
\setlength{\extrarowheight}{3pt}
\begin{tabular}{|c || c | c| c| c|| c| c|}
\hline
{\bf R} & {\bf ave kR} & {\bf std kR} & {\bf ave kR} & {\bf std kR} & {\bf ave cR} & {\bf std cR}\\
& {\bf N = 20} & {\bf N = 20} & {\bf N = 1000} & {\bf N = 1000} & {\bf N = 1000} & {\bf N = 1000} \\
\hline
{\bf 20} & 55.78 & 11.88 & 50.23 & 18.57 & 41.90 & 27.47 \\
{\bf 15} & 48.85 & 10.03 & 41.32 & 13.43 & 33.94 & 22.51 \\
{\bf 10} & 37.16 & 10.74 & 28.65 & 15.13 & 7.54 & 9.55 \\
\hline
\end{tabular}
\label{tablestochgrid}
\end{table}
For $R = 10, 15, 20$, denote by kR the control computed by the algorithm.
Table \ref{tablestochgrid} shows the results of our experiments.
In the table, for each $R$,
'ave kR' and 'std kR' are the estimates of the average and standard
of the yield of kR using $N = 20$ and $N = 1000$ samples. Finally, 'ave cR' and 'std cR' are
the 1000-sample average and standard deviation of yield of the {\em deterministic} controls c20, c15, c10, computed in Section \ref{firstset} (as in Table \ref{table4}).
We see that the kR controls are uniformly superior to their cR counterparts,
sometimes by almost one standard deviation. We observe that
k20 is superior to k15, and much superior to k10. This parallels observations
made in Section \ref{firstset}.
\subsection{Stochastic gradients}
The stochastic gradients method is a well-known approach for solving
optimization problems with uncertainty. Because of the nonconcave nature
of the yield maximization problem, in our case it will amount to a local
search method. Furthermore, there are some difficulties that are
caused by the nonsmoothness in our models.
We will only outline here how we are approaching these difficulties.
The core step in the stochastic gradients approach is to (randomly)
sample a cascade, and, keeping the
cascade fixed, to compute the impact on yield of infinitesimally small
changes in the control parameters. This is followed by a line search
to optimize the step size. This basic step is repeated.
A difficulty that we encounter when we attempt to
make this outline more specific is that yield is not a differentiable
function of the control parameters, for several reasons, the main one being the
stochastic outage protocol (\ref{stochout}), which, while smoother than a
completely deterministic rule, is not smooth enough, due to its abrupt
transition between stochastic and deterministic regimes.
We modify rule (\ref{stochout}) so that the probability of a line
will outage is always strictly positive and strictly smaller than $1$; however
when the overload is larger than $1$ the outaging probability will be very
close to $1$, and when the overload is significantly less than $1$ the
outaging probability (while positive) will be very small. To this effect
consider a function
$$ F \, : \, \field{R}_+ \, \rightarrow \, [0,1), \ \ \ \mbox{with} \ \ F(0) = 0 \ \mbox{and} \ F(x) \rightarrow 1 \ \ \mbox{as} \ \ x \rightarrow +\infty,$$
where the convergence is very rapid. An example is $F(x) \, = \, 1 - e^{-Mx}$, for large $M > 0$. Likewise, consider a function
$$ G \, : \, [0,1] \, \rightarrow \, [0,1), \ \ \ \mbox{with} \ \ G(0) = 1 \ \mbox{and} \ G(x) \rightarrow 0 \ \ \mbox{as} \ \ x \rightarrow 1,$$
and again with rapid convergence. An example is $G(x) \, = \, e^{-Mx}$
for large $M > 0$.
Having chosen $F$ and $G$, we modify outage rule (\ref{stochout}) as follows.
At round $r$, and given a tolerance $0 \ge \epsilon_r < 1$,
line $j$ is outaged
\begin{eqnarray}
\mbox{with probability} && \left \{
\begin{array}{lll}
\frac{1}{2}G\left(1 - \tilde f^r_{j}/(1 - \epsilon_r) \right), & \mbox{if $\tilde f^r_{j} \le (1 - \epsilon_r)u_{j}$} & \\
& & \\
\frac{1}{2}, & \mbox{if} \ {(1 - \epsilon_r)u_{j} < \tilde f^{r}_{j} < u_{j}} & \label{smoothstoch} \\
& & \\
\frac{1}{2} \left(1 + F\left(\tilde f^{r}_{j}/u_{j} \ - \ 1 \right) \right), & \mbox{if $u_{j} \le \tilde f^{r}_{j}$}. & \\
\end{array} \right. \label{hehehe}
\end{eqnarray}
In other words, if $\tilde f^{r}_j > u_j$, the outage probability
is very large, but is bounded strictly away from $1$, and if
$\tilde f^{r}_j < (1 - \epsilon_r)u_j$ the outage probability is very small
but remains strictly positive. By choosing $F$ and $G$ appropriately we
obtain an outage model that is arbitrarily close to rule (\ref{stochout}).
Another source of nonsmoothness in our models is the general form our
control law in Step 2 of Procedure (\ref{affinecontcasc}). However, it is
easy to see that the law can be approximated (arbitrarily closely) using
a smooth control.
The computation of the (stochastic) gradient of the yield function at a
given control vector $(\bar c, \bar b, \bar s)$ can now be described. First,
we sample a random cascade under the control $(\bar c, \bar b, \bar s)$ and
outaging lines using rule (\ref{hehehe}). This produces a particular
sequence of lines that become outaged; i.e. at round $r$ a certain set
$S^r$ of lines is outaged, for $r = 1, 2, \ldots, R-1$.
Next, we compute the change in yield that results when we perturb the control
by a vector $(\epsilon^c, \epsilon^b, \epsilon^s)$ with infinitesimally
small entries, while still assuming
that set $S^r$ is the set of lines outaged at round $r$, for each $r$. This is a deterministic
computation; rule (\ref{hehehe}) guarantees that the given cascade structure
retains positive probability. This computation gives us the stochastic
gradient.
However, at this point we need to deal with the final reason that
the yield function is not smooth, and this is the demand/supply
adjustment in Step 3 of our generic cascade template (\ref{gencasc}) (or
in Step 6 of the cascade control template (\ref{contcasc})). If, at round
$r$, under control $(c,b,s)$ a certain component $K$ has equal demand and
supply, then no adjustment takes place.
However, even a small change in
the control that results in shedding less demand by round $r$ will not
result in an increase of yield, whereas the opposite change in control will
possibly result in a decrease in yield. Thus, in essence, a left derivative
is different from the right derivative; and moreover this is not a probability
zero event.
Nevertheless, it is still possible to adjust the stochastic gradients framework
so as to recover a valid first-order method. The resulting approach is
related to the classical Frank-Wolfe method \cite{berts}. In forthcoming work we will report on experiments with this approach.
\section{Upcoming work}
Our forthcoming work will focus on three areas: stochastics or robustness, in
particular concerning the optimal scaling problem in Section \ref{scalingproblem}, an investigation of game-theoretic aspects of the type of control
we study, and the use of AC power flow models.
With regards to the last point, a recent paper of Lavaei and Low
\cite{lavaeilow} may yield a robust solver for AC power flow systems under
severe contingencies. Even though the work in \cite{lavaeilow}
relies on semi-definite programming, in fact one of the algorithms can
be restated as a second-order conic program, which may be more efficient.
\section*{Acknowledgment}
We would like to thank Ian Dobson and Ian Hiskens for fruitful discussions,
and for making the Eastern Interconnect data available to us.
|
1,314,259,996,741 | arxiv | \section{Introduction}
Recently much effort has focused on the study
of orientifold compactifications of type II string theory
with space-time filling D-branes and background fluxes.
The reason is that these compactifications can lead to calculable
four-dimensional effective theories supporting string vacua relevant for
particle physics and cosmology \cite{reviewPP,reviewcosmo,FluxReviews}.
Particularly well controlled are warped type IIB Calabi-Yau orientifolds with
space-time filling D3 and D7 branes which yield a four-dimensional effective
theory with $\mathcal{N}=1$ supersymmetry \cite{GKP,FluxReviews}.
It was realized that in these compactifications
the inclusion of background fluxes and certain non-perturbative
corrections might lead to a stabilization of all unwanted scalar moduli
fields in a local vacuum \cite{KKLT}. This was
demonstrated for specific examples e.g.~in refs.~\cite{DDFGK,BBCQ,Lust1,Lust2} and strengthened the
believe in a vast landscape of supersymmetric and non-supersymmetric
string vacua \cite{FluxReviews}. In order to study these vacua
a precise knowledge of the $\mathcal{N}=1$ characteristic data of the four-dimensional
effective theory is of central importance. In particular, this includes the
understanding of perturbative and non-perturbative corrections
to the K\"ahler potential and the superpotential.
The aim of this work is to investigate the leading perturbative and
non-perturbative corrections for Calabi-Yau orientifolds with
O3 and O7 planes.
We first study the
$\alpha'$ corrections inherited from the underlying $\mathcal{N}=2$
theory which survive the large volume limit of the orientifold.
This includes the perturbative $\alpha'$ corrections discussed
in ref.~\cite{BBHL}. Moreover, we argue by using the results of refs.~\cite{GL1,GL2}
that also non-perturbative $\alpha'$ corrections involving the
NS-NS B-field can survive the large volume limit of the orientifold.
These corrections are generically present in compactifications in which the
B-field is not entirely projected out by the orientifold symmetry.%
\footnote{An example of a Calabi-Yau orientifold with non-vanishing B-field
moduli is presented in the second part of this paper. For other
examples which admit these additional moduli fields, see e.g.~ref.~\cite{Lust2}.}
The real B-field scalars combine with the scalars of the R-R
two-form $C_2$ into complex scalars $G^a$ through the combination $C_2-\tau B_2$,
where $\tau$ is the complex dilaton-axion \cite{GL1}.
The perturbative and non-perturbative $\alpha'$ corrections in the orientifold
large volume limit do not correct the $\mathcal{N}=1$ coordinates.
They do however contribute to the K\"ahler potential and
we will be able to determine these corrections explicitly
in terms of the topological invariants of the underlying Calabi-Yau manifold.
We will also study the non-perturbative superpotential generated by D3-instantons
wrapping a four-cycle in the Calabi-Yau manifold and show that it generically
depends on the scalars $\tau$ and $G^a$. In order to
do that, we implement the non-perturbative symmetries
inherited from the full type IIB string theory.
Type IIB string theory possesses a strong-weak duality known as S-duality.
This non-perturbative symmetry relates one type IIB theory with complex
string coupling $\tau$ to a dual type IIB string theory with string coupling $-1/\tau$.
Moreover, it exchanges the NS-NS and R-R two-forms and thus fundamental
strings with D1 branes. Together with the shifts in the axion, $\tau \rightarrow \tau+1$, the S-duality transformation
generates the discrete duality group $Sl(2,\mathbb{Z})$. In an $\mathcal{N}=1$ compactification
this group will generically be reduced further or broken completely by
to the non-trivial background geometry.
However, in the orientifold compactifications under consideration the complex dilaton $\tau$
does not vary over the compact six-dimensional geometry and appears as four-dimensional chiral
field \cite{GKP, FluxReviews}. In this limit we expect that a subgroup $\Gamma_S$
of the full $Sl(2,\mathbb{Z})$ duality is a symmetry of the four-dimensional theory in
analogy to refs.~\cite{FILQ,CFILQ}. Determining the transformations of the $\mathcal{N}=1$ coordinates under
$\Gamma_S$ as well as integral shifts of the NS-NS B-field allows us to
study the moduli dependence and symmetries of the K\"ahler potential and
superpotential in the orientifold large volume limit.
We begin by discussing the transformation properties of the K\"ahler potentials under
$\Gamma_S$ when $\alpha'$ corrections are included. In order for these to be invariant under
$\Gamma_S$ also contributions from D1 and D(-1) branes
have to be taken into account. In general, it is hard to compute these
corrections. We will however be able to discuss candidate
completions which reproduce the perturbative and non-perturbative
$\alpha'$ corrections and admit the desired transformation properties.
In order to obtain these solutions we will simply sum over images
of the $\alpha'$ corrections under the duality group following \cite{GG, RRSTV}.
This does however not guarantee that the result is the true non-perturbative
completion. Firstly, this analysis is only valid in the orientifold limit in which
the type IIB symmetry is not entirely broken by the vacuum and a discrete
group $\Gamma_S$ is preserved. Secondly, even though this symmetry group
ideally restricts the answer to be generated by a finite set of appropriately
transforming functions additional boundary conditions are needed to
fix the precise form of the duality invariant completion.\footnote{See \cite{BCOV,YY,Klemm, GKMW} for the discussion of
an analogous problem within topological
string theory.}
For corrections to the $\mathcal{N}=1$ K\"ahler potential this task
is even more involved, since the K\"ahler potential is not
protected by holomorphicity or non-renormalization theorems.
The application of string-string dualities such as heterotic-F-theory duality
might however help to compute these corrections explicitly as
argued, for example, in refs.~\cite{BM, HMS}.
One expects that modularity arguments are however more powerful
when arguing about the superpotential.
In $\mathcal{N}=1$ theories the superpotential is holomorphic and protected
against perturbative corrections. For the type IIB orientifold
setups the determination of the D3-instanton superpotential
is of central importance. However, its explicit form is
in general hard to determine \cite{Witten,DGW, CL}.
Nevertheless, by
combining holomorphicity and modular properties under the inherited
type IIB $Sl(2,\mathbb{Z})$ symmetry as well as shifts in the
NS-NS B-field the moduli dependence of the superpotential
in general large volume orientifolds can be discussed. In case
the complex dilaton $\tau$ varies over the internal space only a local
analysis of the superpotential can be performed \cite{Witten:1996hc,Ganor}. Here
our results are more restrictive due to the fact that $\tau,G^a$ do
not vary over the compact space.
We find that the
complex fields $G^a$ depending on the NS-NS and R-R two-form
moduli naturally arise through products of theta-functions
and modular forms with the complex
dilaton-axion $\tau$ as modular parameter. In the second part of the paper
we propose that this set of theta-functions can be determined for
a specific orientifold example.
The specific example we consider is an orientifold of the Enriques
Calabi-Yau. The underlying Calabi-Yau manifold is a $K3$ fibration of the form
$Y_E=(K3 \times T^2)/\mathbb{Z}_2$ \cite{Borcea,FHSV}, where the freely acting $\mathbb{Z}_2$ symmetry
yield a minus sign on the complex coordinate of $T^2$ and acts as the Enriques involution on the $K3$ surface \cite{SurfaceB}.
We will show that an appropriate definition of the
orientifold projection allows to explicitly determine the $\mathcal{N}=1$ four-dimensional
effective theory. Since the geometric moduli space of the
underlying $\mathcal{N}=2$ theory is not corrected by world-sheet instantons or perturbative
$\alpha'$ corrections the resulting $\mathcal{N}=1$ theory is particularly well controlled.
We will show that the $\mathcal{N}=1$ moduli space is a product of two cosets $\tilde \mathcal M_{\rm sk} \times \tilde \mathcal M_{\rm q}$.
The first factor $\tilde \mathcal M_{\rm ks}$ arises from the reduction of the
$\mathcal{N}=2$ special K\"ahler manifold containing the complex structure deformations of $Y_E$. It is itself a
special K\"ahler manifold and was studied intensively in the literature \cite{KM,GKMW}.
The reduction of the $\mathcal{N}=2$ quaternionic manifold leads to a K\"ahler manifold
$\tilde \mathcal M_{\rm q}$ of half its dimension. Remarkably,
$\tilde \mathcal M_{\rm q}$ can be identified with the original $\mathcal{N}=2$ special
K\"ahler manifold of complexified K\"ahler structure deformations $\mathcal M_{\rm ks}$ times
an $Sl(2,\mathbb{R})/U(1)$ factor.
In this identification half of the
NS-NS fields arising as real parts of coordinates on $\mathcal M_{\rm sk}$ are
replaced by R-R fields. The resulting $\mathcal{N}=1$ coordinates encode the
correct couplings to D(-1), D1 and D3 branes. Note however, that
this duality is not performed in the large volume coordinates on $\mathcal M_{\rm sk}$,
but rather at a special locus where also the volume of the K3 fiber can be
small.
The physics in the regime where the K3 fiber of the Enriques Calabi-Yau is
small was studied intensively in the underlying $\mathcal{N}=2$ theory. It was shown
in ref.~\cite{aspinwall} that at the limit were the K3 fiber is of Planck length the type II
theory undergoes a phase transition somewhat similar to the well-known
conifold transition. It was later argued in ref.~\cite{KM} that the light BPS
degrees of freedom at this locus are bound states of D4, D2 and D0
branes wrapped around specific four and two-cycles of $Y_E$.
The authors of \cite{KM} showed that the topological string theory on the Enriques
Calabi-Yau can be resummed to count the degeneracies of these
degrees of freedom. The leading contributions arise through a particular
holomorphic function $\Phi_{\rm B}$ known from the work of Borcherds \cite{borcherdsone, borcherds}
and Harvey, Moore \cite{HM,HM2}. Here we will employ the duality of the theory on
$\mathcal M_{\rm ks}$ at this special locus to the corresponding orientifold theory.
We propose that $\Phi_{\rm B}$ naturally arises in the $\mathcal{N}=1$ superpotential containing
the D3-instanton corrections proportional to $e^{i T_S}$, where $T_S$ contains the
volume of the K3 fiber. In accord with our general considerations,
the coefficients are indeed generalizations of theta functions
depending on the modular parameter $\tau$, the dilaton-axion, as well as the scalars
$G^a$ arising from the NS-NS
and R-R two-forms. The study of the Enriques orientifold exemplifies nicely
the interplay of holomorphicity and symmetry properties for the non-perturbative
superpotential.
This paper is organized as follows. In section \ref{orirev} we briefly review the effective
theory of type IIB orientifolds with O3 and O7 planes. We discuss the reduction
of an $\mathcal{N}=2$ theory defined by two general pre-potentials for complex structure
and K\"ahler structure deformations respectively. It is then shown in section \ref{largevolume}
that certain $\alpha'$ corrections survive in the large volume limit of the orientifold
and correct the K\"ahler potential in an explicitly calculable way.
The modular completion of these corrections by D(-1) and D1 brane contributions
is discussed in section \ref{Kaehlersymm}. In section \ref{generalsup} we turn to the discussion
of the non-perturbative superpotential generated by D3-instantons. We study its
transformations under the type IIB symmetries and argue for a moduli dependence
through generalizations of theta functions. In section \ref{EnriquesO} we present an explicit example
by introducing an orientifold of the Enriques Calabi-Yau manifold. We first summarize
some details about the $\mathcal{N}=2$ theory in section \ref{GeometryN=2}. The K\"ahler potential and
an interesting duality map is studied in \ref{effectiveN=1action}. Finally, in section \ref{EnriquesW} we propose
a particular non-perturbative superpotential counting degeneracies of D3, D1, D(-1)
bound states.
\section{Non-perturbative Corrections and Modularity}
In this section we discuss non-perturbative corrections and
the transformation properties of the $\mathcal{N}=1$ effective action
of type IIB string theory compactified on an orientifold background. We begin
with a brief review of the four-dimensional effective theory in section
\ref{orirev}. In section \ref{largevolume} we show that in the orientifold large volume
limit the perturbative and certain non-perturbative $\alpha'$ corrections
inherited from the underlying $\mathcal{N}=2$
theory correct the $\mathcal{N}=1$ K\"ahler potential.
We will argue that these corrections generically do not respect the
type IIB $Sl(2,\mathbb{Z})$ symmetries in section \ref{Kaehlersymm}. Since in the orientifold limit
a subgroup
$\Gamma_S$ of this symmetry group is expected to
be preserved we comment on modular completions of the K\"ahler potential.
Finally, in section \ref{generalsup} we analyze the transformation properties of the $\mathcal{N}=1$ complex
coordinates and constrain the D-instanton superpotentials to contain generalizations
of theta functions. This leads to a new moduli dependence of the superpotential
which is generic for many orientifold compactifications.
\subsection{Brief review of the effective action of type IIB orientifolds
\label{orirev}}
In this section we review the $\mathcal{N}=1$ effective supergravity theory arising
by compactification of type IIB supergravity on an orientifold background
following \cite{GKP,BBHL,GL1,GL2,TG}. We will focus on orientifold projections yielding O3 and O7 planes
and include the leading perturbative $\alpha'$ corrections \cite{BBHL} as well as the world-sheet instanton
corrections inherited from the underlying $\mathcal{N}=2$ theory \cite{GL2}. Since
there exists a number of reviews \cite{FluxReviews} on this topic we will keep our discussion brief.
In type IIB orientifolds with O3/O7 planes the orientifold projection takes
the form $(-1)^{F_L} \Omega_p \sigma$, where $F_L$ is the left fermion
number, $\Omega_p$ is the world-sheet parity reversal and $\sigma$ is
some geometric involutive symmetry of the background.
In order to preserve $\mathcal{N}=1$ supersymmetry
$\sigma$ has to be a holomorphic and isometric involution. It acts
non-trivially on the internal Calabi-Yau manifold $Y$ and leaves the four flat directions invariant.
For models with O3/O7 planes $\sigma$ acts on the K\"ahler form $J$ and holomorphic three form $\Omega$ of $Y$
as
\begin{equation} \label{JOmegatrans}
\sigma^* J = J \ , \qquad \sigma^* \Omega = - \Omega\ ,
\end{equation}
where $\sigma^*$ is the pull-back.
In order to remain in the spectrum the NS-NS and R-R fields have to transform as
follows under $\sigma^*$. The dilaton $\phi$, the axion $C_0$ as well as the
four-form $C_4$ are invariant under the action of $\sigma$, while the NS-NS two-form $B_2$ and R-R
two-form $C_2$ transform with a minus sign.
Type IIB Calabi-Yau orientifolds with O3/O7 planes have the following truncated $\mathcal{N}=1$ moduli
space:
\begin{equation} \label{modspace1}
\tilde \mathcal M_{\rm sk} \times \tilde \mathcal M_{\rm q}\ ,
\end{equation}
where $ \tilde \mathcal M_{\rm sk}$ is a special K\"ahler manifold inside the $\mathcal{N}=2$ special K\"ahler
manifold $\mathcal M_{\rm sk}$ and $\tilde \mathcal M_{\rm q}$ is a K\"ahler manifold inside the $\mathcal{N}=2$ quaternionic
manifold $\mathcal M_{\rm q}$. In the following we will describe the geometry of the moduli
space \eqref{modspace1} in more detail.
Let us start with some comments on the cohomology of the orientifold
theory and the reduction of $\mathcal M_{\rm sk}$. Since $\sigma$ is a holomorphic
involution the cohomology groups $H^{(p,q)}$ split
into two eigenspaces under the action of $\sigma^*$ as
$H^{(p,q)}=H^{(p,q)}_+ \oplus H^{(p,q)}_-$.
We denote the dimensions of $H^{(p,q)}_\pm$ by $h^{(p,q)}_\pm$.
The four-dimensional invariant spectrum is found by using a Kaluza-Klein
expansion in harmonic forms keeping only the fields which in addition
obey the correct transformations under $\sigma^*$. This induces a reduction
of the special K\"ahler manifold $\mathcal M_{\rm sk}$
for the orientifold setups.
Since $\sigma$ transforms the complex three-form $\Omega$
with a minus sign the complex structure deformations parametrized
by the elements of $H^{(2,1)}$ are reduced to $h^{(2,1)}_-$ complex
scalars $z^k$. It can be shown that these define a $h^{(2,1)}_-$ dimensional
special K\"ahler submanifold $\tilde \mathcal M_{\rm sk}$ of the original $\mathcal{N}=2$ moduli
space of complex structure deformations. The K\"ahler potential on $\tilde \mathcal M_{\rm sk}$
takes the well-known form
\begin{equation}
K_{\rm cs}(z,\bar z) =-\ln\big[ i \int_Y \Omega(z) \wedge \bar \Omega(\bar z)\big] \ ,
\end{equation}
where $\Omega(z^k)$ varies holomorphically over $\tilde \mathcal M_{\rm sk}$.
Recall that in the underlying $\mathcal{N}=2$ theory the complex scalars $z$ were
part of vector multiplets. In the orientifold reduction also $h^{(2,1)}_+$ of the
vectors survive. The gauge-kinetic coupling function is the second derivative
of the pre-potential of the underlying $\mathcal{N}=2$ special K\"ahler manifold $\mathcal M_{\rm sk}$
with respect to the complex structure deformations $z^\kappa$, which are then set to zero
in the orientifold scenario \cite{GL1}.
The reduction of the quaternionic space $\mathcal M_{\rm q}$ is slightly more involved.
Since $\sigma$ leaves the K\"ahler
form $J$ invariant and yields a minus sign
on the $B_2$ field we expand
\begin{equation} \label{JBexpansion}
J = v^\alpha \omega_\alpha \ ,\quad \alpha=1,\ldots,h^{(1,1)}_+\ , \qquad \qquad B_2 = b^a \omega_a\ , \quad a=1,\ldots,h^{(1,1)}_-\ ,
\end{equation}
where $\omega_\alpha$ is an integral basis of $H^2_+(Y,\mathbb{Z})$ and $\omega_a$ is an integral basis of $H^2_-(Y,\mathbb{Z})$.
The conditions \eqref{JBexpansion} defines a real subspace of the $h^{(1,1)}$ dimensional
space of complexified K\"ahler deformations $\mathcal M_{\rm ks}$ of $Y$. This is due to the fact that either the
real or the complex part of the complexified K\"ahler form survives:
\begin{equation} \label{complexJ}
-B_2 + iJ = t^A \omega_A = -b^a \omega_a + i v^\alpha \omega_\alpha\ .
\end{equation}
Let us now include the R-R forms. Invariance under the orientifold projection
enforces the expansions
\begin{equation} \label{CCexpansion}
C_2 = c^a \omega_a \ ,\qquad \qquad \quad C_4 = \rho_\alpha \tilde \omega^\alpha\ ,
\end{equation}
where $\omega_a$ was already introduced in \eqref{JBexpansion} and we have
denoted by $ \tilde \omega^\alpha$ an integral basis of $H^{4}_+(Y,\mathbb{Z})$ dual to $\omega_\alpha$.
Note that in \eqref{CCexpansion} we have only displayed the part of the expansion of $C_4$
which leads to four-dimensional scalars.\footnote{The vectors discussed in the previous
paragraph arise precisely in the expansion of $C_4$ into appropriate
three-forms.} Let us now define the even form
\begin{equation} \label{def-rho}
\rho = 1+t^A \omega_A - \mathcal{F}_{A} \tilde \omega^A + (2 \mathcal{F} - t^A \mathcal{F}_A) \epsilon\ ,
\end{equation}
where $\mathcal{F}$ is the pre-potential on $\mathcal M_{\rm ks}$ and $\mathcal{F}_{A}$ is its first derivative
with respect to $t^A$.
The orientifold effective theory including a general pre-potential $\mathcal{F}$
was derived in refs.~\cite{GL2,TG}.
It was shown there, that the complex coordinates on the K\"ahler manifold $\tilde \mathcal M_{\rm q}$ are obtained in the expansion
\begin{equation} \label{def-rhoc}
\rho_c \equiv e^{-B_2} \wedge C^{\rm RR} + i \text{Re}\big( C \rho \big)= \tau + G^a \omega_a - T_\alpha \tilde \omega^\alpha\ ,
\end{equation}
where $C^{\rm RR} = C_0 + C_2 + C_4$ and the function $C$ is identified with the dilaton $e^{-\phi}$.
The K\"ahler potential for the complex scalars $\tau,G^a,T_\alpha$ is then
shown to be
\begin{eqnarray} \label{Kqgeneral}
K_{\rm q}(\tau,G,T)& =& - 2 \ln \big[i \int_Y \big< C\rho,\overline{C\rho}\big>\big] \\
& =& - 2 \ln \big[i|C|^2 \big( 2(\mathcal{F} -\bar \mathcal{F})-(\mathcal{F}_\alpha +\bar \mathcal{F}_\alpha)(t^\alpha-\bar t^\alpha) \big) \big]\ , \nonumber
\eea
where we have inserted the even form $\rho$ defined in \eqref{def-rho} to evaluate the second equality.\footnote{%
\label{wedge-product}The anti-symmetric product between two even forms $\rho,\lambda$ is defined as the alternating wedge product
$\big<\rho,\lambda \big>= \rho_0 \wedge \lambda_6 - \rho_2 \wedge \lambda_4+\rho_4 \wedge \lambda_2 - \rho_6 \wedge \lambda_0$, where $\rho_p,\lambda_p$
are the $p$-form parts of $\rho,\lambda$.}
Note that $K$ is a function of the imaginary part $\text{Im}\rho_c= \text{Re}(C \rho)$ of $\rho_c$ only.
This implies that $K$ only depends on the combinations $\tau-\bar \tau$, $G^a-\bar G^a$ and $T_\alpha -\bar T_\alpha$.
For a general pre-potential $\mathcal{F}$ it is impossible to explicitly write $K$ as the function of $\tau,G^a,T_\alpha$.
This is due to the fact that one would need to express $\text{Im} (C\rho)$ as a function of $\text{Im} \rho_c=\text{Re}(C \rho)$
appearing in the $\mathcal{N}=1$ coordinates \eqref{def-rhoc}. This functional dependence is highly non-polynomial
and can only be determined explicitly in specific
examples.\footnote{This is equivalent to the problem of solving the attractor equations
for $\mathcal{N}=2$ black holes.}
Nevertheless, one can derive the K\"ahler metric
by using the underlying $\mathcal{N}=2$ special geometry \cite{GL2} or the work of Hitchin \cite{Hitchin} as
done in \cite{BG}.
So far we have determined the $\mathcal{N}=1$ kinetic terms of the scalar and vector
fields. Masses for these scalar fields can be
generated by a non-trivial superpotential or the presence of D-terms.
In the rest of the paper we will only discuss the inclusion of a superpotential.
In type IIB orientifolds with O3/O7 planes it can be generated by non-vanishing
R-R and NS-NS three-form flux $F_3$ and $H_3$ as well as non-perturbative corrections due
to D-instantons. It takes the form \cite{Witten,GVW,GKP,KKLT}
\begin{equation} \label{full_super}
W = \int_Y \Omega(z) \wedge \big(F_3 - \tau H_3 \big) + W_{\text{D-inst}}(\tau,z,G,T,\ldots)\ .
\end{equation}
The first term is the well-known Gukov-Vafa-Witten flux superpotential, while the
second term encodes the D-instanton effects. We will discuss the field dependence
and modular properties of $W_{\text{D-inst}}$ in section \ref{generalsup}. In order to do that
it is often convenient to also refer to the underlying F-theory description of
the orientifold setup. We therefore end this section with some
remarks on the F-theory embedding and four-dimensional symmetries.
Type IIB orientifolds with O3 and O7 planes arise as
a special limit of F-theory \cite{Vafa} compactified on particular four-dimensional
Calabi-Yau manifolds \cite{Sen}. These fourfolds have to admit an elliptic
fibration
\begin{equation} \label{elliptic}
Y_4 \rightarrow B_3\ ,
\end{equation}
where $B_3$ is some three-dimensional
base manifold. The complex structure of the torus fiber
corresponds to the complex dilaton $\tau$ introduced above.
In general $\tau$ can vary over the base $B_3$. This implies the
existence of a modular group $\Gamma_M$ associated to the elliptic fibration.
This group encodes the monodromies around the singular
points of the fibration and is a discrete subgroup of the torus
symmetry group $Sl(2,\mathbb{Z})$. The complete $Sl(2,\mathbb{Z})$ symmetry corresponds to
the non-perturbative symmetry of type IIB string theory. In the full F-theory compactification
it is reduced or broken due to the background geometry $Y_4$ \cite{Vafa, BKMT}. Roughly speaking,
the larger the modular group $\Gamma_M \in Sl(2,\mathbb{Z})$, the fewer symmetries survive in the
effective four-dimensional action.
In this paper we will entirely focus on the orientifold limit reviewed in this section \cite{GKP, FluxReviews}.
It was shown in \cite{Sen} that in this limit the base $B_3$ can
be obtained as a quotient of a Calabi-Yau manifold by an involution $\sigma$
as discussed above.
The singularities of elliptic fibration \eqref{elliptic} determine the location of the space-time filling
O7 planes and D7 branes. However, in the above orientifold limit, both the complex
dilaton as well as the fields $G^a$ do not vary over the base $B_3$,
but correspond to chiral fields in four space-time dimensions. In other
words, in this limit the monodromy group $\Gamma_M$ acts trivially on
$\tau,G^a$ and we expect that a subgroup $\Gamma_S \subset Sl(2,\mathbb{Z})$
survives as a symmetry of the effective action.
This symmetry posses stringent constrains on the $\mathcal{N}=1$ characteristic
data of the orientifold compactification in analogy to \cite{FILQ,CFILQ}. In the next sections
we discuss these conditions in detail. Clearly, a more general
analysis would consider the full F-theroy compactification and we hope to
return to this problem in forthcoming work. Let us just remark here, that
there is no known effective action of twelve-dimensional
F-theory. The four-dimensional $\mathcal{N}=1$ effective theory thus has
to be determined by an M-theory lift. More precisely, one compactifies
M-theory on the elliptically fibered fourfold $Y_4$ to obtain a three-dimensional
effective theory. This theory is then lifted to four-dimensions by growing
an extra non-compact dimension. The F-theory moduli thus arise from the
expansion of the M-theory fields, such as the three-form $C_M$,
into harmonics of $Y_4$. A detailed discussion of the derivation
of the effective action can be found, for example, in refs.~\cite{Mayr:1996sh,HL,TG}.
\subsection{Perturbative and non-perturbative $\alpha'$ corrections in the orientifold large volume limit \label{largevolume}}
In this section we simplify the discussion and work in the large volume limit of the orientifold $Y/\sigma$.
This implies that we consider the regime where $v^\alpha$ is large.
Note that this is not the same as demanding that all $v^A$ are large on the underlying Calabi-Yau
manifold, since $v^a=0$ in the orientifold setup.
In other words, the contributions depending on $t^a = -b^a$ are not necessarily
suppressed in the large volume limit of the orientifold. We therefore include
the non-pertubative $\alpha'$ corrections inherited from the underlying $\mathcal{N}=2$ theory.
More precisely, we obtain in this limit a
pre-potential of the form\footnote{%
Note that in general
$\mathcal{F}$ can also admit a cubic and linear term of the form $B_{AB} t^A t^B$, $A_A t^A$.
However, since $A_A,B_{AB}$ are always real it is easy to check
that they do not appear in the K\"ahler potential \eqref{Kqgeneral}. They only
correct the coordinates $T_\alpha$ and we will not consider these contributions
in the following.}
\begin{eqnarray} \label{simple_F}
\mathcal{F} &=& \mathcal{F}_{\rm class}+ \mathcal{F}_{\rm pert}+\mathcal{F}_{\rm b}\\
&=& - \tfrac{1}{3!} \mathcal{K}_{ABC} t^A t^B t^C - \tfrac{i}{2}\zeta(3)\chi + i \sum_{\beta \in H_2^-(Y,\mathbb{Z})} n_{\beta}^0 \ \text{Li}_3 (e^{i k_a t^a}) \ , \nonumber
\eea
where $k_a = \int_\beta \omega_a$ with $\omega_a$ being an integral basis of $H^2_-(Y,\mathbb{Z})$.
Let us discuss the three contributions in \eqref{simple_F} in turn.
The cubic term $\mathcal{F}_{\rm class}$ corresponds to the classical contribution and
we denote the triple intersections of the integral basis $\omega_A \in H^{2}(Y,\mathbb{Z})$ by
\begin{equation} \label{triple_inters}
\mathcal{K}_{ABC} = \int_Y \omega_A \wedge \omega_B \wedge \omega_C\ .
\end{equation}
Note that in the orientifold setup consistency requires that for the spilt $\omega_A=(\omega_\alpha,\omega_a)$
the following intersections have to vanish:
\begin{equation}\label{inter_constraints}
\mathcal{K}_{\alpha \beta a}=\mathcal{K}_{abc} = 0\ .
\end{equation}
In other words only the intersections $\mathcal{K}_{\alpha \beta \gamma}$ and $\mathcal{K}_{\alpha ab}$ with zero or two negative
indices can appear in \eqref{simple_F}.
The second term $\mathcal{F}_{\rm pert}$ in \eqref{simple_F} is proportional to the Euler characteristic $\chi = 2(h^{(1,1)}-h^{(2,1)})$ of $Y$.
It corresponds to an $(\alpha')^3$ perturbative correction of the effective action and was first considered
in orientifold setups in ref.~\cite{BBHL}.
The third term $\mathcal{F}_{\rm b}$ is inherited from
the non-perturbative $\alpha'$ corrections of the $\mathcal{N}=2$ pre-potential and was not discussed in the literature
so far. In the large volume limit
of the orientifold only the terms depending on the B-field moduli $t^a=-b^a$
survive in the third polylogarithm Li$_3(x)=\sum_{n>0} n^{-3} x^n$.
All other contributions are suppressed exponentially by the volume of the curves in $H_2^+(Y,\mathbb{Z})$.
In other words, only the terms proportional to the integer genus zero Gopakumar-Vafa invariants $n^0_{\beta}$ \cite{GV}
for a curve $\beta$ in the negative eigenspace $H_2^-(Y,\mathbb{Z})$ remain in the pre-potential.
They can be determined for many explicit examples of Calabi-Yau manifolds my using
mirror symmetry \cite{Mirrorbook}. However, we have
to make a cautionary remark on the convergence of the expansion \eqref{simple_F}. Since the polylogarithm
$ \text{Li}_3 (e^{i k_a t^a}) $ is bounded $\mathcal{F}_{\rm b}$ appears divergent when summing over all $\beta$.
This would be very generically the case if $\beta$ is not restricted to any sublattice in $H_2(Y,\mathbb{Z})$ since
the Gopakumar-Vafa invariants grow very rapidly. However, in the expression \eqref{simple_F} for $\mathcal{F}_{\rm b}$
we only sum over degrees $k_A$ which are of the form $k_A=(0,k_a)$, i.e.~vanish on the positive
eigenspace of the orientifold. There are indeed examples for which the $n_{\beta}^0$
truncates on such a sublattice $(0,k_a)$.\footnote{We are grateful to A.~Klemm for discussions on this
point.} More generally, in case $\mathcal{F}_{\rm b}$ is not finite this can be traced back to
the fact that we are actually working in the wrong coordinates $t^a$. Before restricting
to the orientifold limit $\text{Im} t^a\rightarrow 0$ the expression $\mathcal{F}_{\rm b}$ has to be resummed in terms
of dual coordinates valid around $\text{Im} t^a=0$. One is then able to implement the orientifold
projection with a finite $\mathcal{F}_{\rm b}$. In the following we will simply assume that $\mathcal{F}_{\rm b}$ is finite
when restricting our general considerations to appropriate specific examples.
In order to determine the $\mathcal{N}=1$ coordinates
we first insert the large volume pre-potential \eqref{simple_F} into the
definition \eqref{def-rho} of the even form $\rho$. Due to the presence
of the $\alpha'$ corrections $\mathcal{F}_{\rm pert} + \mathcal{F}_{\rm b}$ the classical
expression $\rho_{\rm class} = e^{-B_2+iJ}$ will receive non-trivial corrections.
However, it is easy to check that these corrections will not contribute
to the definition of the $\mathcal{N}=1$ coordinates $\tau,G^a,T_\alpha$
defined in \eqref{def-rhoc}.
A straightforward computation shows
that $\tau,G^a,T_\alpha$ are given in
terms of the real coordinates introduced in \eqref{JBexpansion} and \eqref{CCexpansion}
by
\begin{eqnarray} \label{def-tauG}
\tau &=& C_0 + ie^{-\phi}\ , \qquad \qquad G^a = c^a - \tau b^a\ ,\\
\label{def-T}
T_\alpha &=&\tfrac{1}{2} ie^{-\phi} \mathcal{K}_{\alpha \beta \gamma} v^\beta v^\gamma - \tilde \rho_\alpha-\frac{1}{2(\tau -\bar \tau)} \mathcal{K}_{\alpha ab} G^a (G-\bar G)^b\ ,
\eea
where $\tilde \rho_\alpha = \rho_\alpha - \frac12 \mathcal{K}_{\alpha ab} c^a b^b$.
These are precisely the coordinates introduced in ref.~\cite{GL1}.\footnote{In contrast to ref.~\cite{GL1} we rescaled the
coordinates $T_\alpha = \frac{2i}{3} T^{\rm ref.}_\alpha$ and identified $\tilde \rho_{\alpha} = \rho_\alpha^{\rm ref.}$.}
However, in contrast to the classical results the K\"ahler potential $K_{\rm q}$ is now corrected by the $\alpha'$
contributions encoded by $\mathcal{F}_{\rm pert}+\mathcal{F}_{\rm b}$ in \eqref{simple_F}.
Let us make this more precise and evaluate the K\"ahler potential for the large volume pre-potential \eqref{simple_F}.
Inserting $\mathcal{F}$ into the general expression \eqref{Kqgeneral} for $K_{\rm q}$ one derives
\begin{equation}
K_{\rm q} =- 2\ln \Big[e^{-2\phi} \big(\tfrac{1}{3!}\mathcal{K}_{\alpha \beta \gamma} v^\alpha v^\beta v^\gamma + 2\zeta(3) \chi - 4 \text{Im} \mathcal{F}_{\text{b}} \big) \Big]\ .
\end{equation}
In this expression the non-perturbative corrections inherited from the underlying
$\mathcal{N}=2$ theory take the form
\begin{eqnarray}
\text{Im} \mathcal{F}_{\text{b} }(\tau,G) &=& \tfrac{1}{2}\sum_{\beta\in H_2^-(Y,\mathbb{Z})} n^0_{\beta} \, \Big[ \text{Li}_3 \Big(e^{ i \frac{k_a (G^a-\bar G^a )}{\tau-\bar \tau}} \Big) + \text{Li}_3 \Big(e^{ -i \frac{k_a (G^a-\bar G^a )}{\tau-\bar \tau}} \Big)\Big]\ , \nonumber \\
\label{ImFb}
&=& \sum_{\beta\in H_2^-(Y,\mathbb{Z})} \sum^{\infty}_{n=1}\ \frac{n^0_{\beta}}{n^3}\ \cos \left( n \frac{k_a (G^a-\bar G^a )}{\tau-\bar \tau}\right)\ ,
\eea
where $k_a = \int_\beta \omega_a$ as in \eqref{simple_F}.
This implies that the moduli dependence on $\tau,G^a$ of both $\alpha'$ corrections to the K\"ahler potential can
be determined explicitly.
Rescaling the K\"ahler deformations $v^\alpha$ to the Einstein frame we can write
$K_{\rm q}$ into the form
\begin{equation}
\label{Kqgen}
K_{\rm q}= - \ln\big[ -i (\tau -\bar \tau) \big]- 2 \ln\Big[ V_{E} +
\tfrac{1}{(2i)^{3/2}} \big(\tau -\bar \tau \big)^{3/2} \big[2\zeta(3) \chi -4 \text{Im} \mathcal{F}_{\text{b}}\big] \Big]\ ,
\end{equation}
where $ V_{E}(\tau,G,T)$ is the Einstein frame volume of the Calabi-Yau orientifold and $\mathcal{F}_{\rm b}(\tau,G)$
is explicitly given in \eqref{ImFb}.
The large volume
K\"ahler potential \eqref{Kqgen} includes the special cases derived in refs.~\cite{GKP,BBHL,GL1}.
Here we were able to include the non-perturbative contribution $\mathcal{F}_{\rm b}(\tau,G)$
and have shown that they can be expressed as explicit functions in $G^a-\bar G^a$ and $\tau-\bar \tau$.
In the next section we will discuss the invariance of the general K\"ahler potential
\eqref{Kqgen} under the $Sl(2,\mathbb{Z})$ symmetry
of type IIB string theory as well as shifts in the B-field.
\subsection{Symmetries of the K\"ahler potential \label{Kaehlersymm}}
In this section we discuss the transformation properties of the K\"ahler potential under
dualities inherited from the ten-dimensional type IIB string theory. We will focus on the
$Sl(2,\mathbb{Z})$ symmetry of type IIB as well as shifts in the NS-NS two-form $B_2$.
Let us begin by discussing the symmetry of $K$ under shifts of the
NS-NS two-form $B_2$. More precisely, we will consider
\begin{equation} \label{B2shift}
B_2 \quad \rightarrow \quad B_2 + 2\pi \chi_2\ , \qquad \quad \chi_2=n^a \omega_a\ ,
\end{equation}
where $\chi_2$ is an integral two form in $H^{2}_-(Y_E,\mathbb{Z})$. For this transformation
we easily verify that the K\"ahler potential is invariant. The Einstein frame
volume $V_E$ in \eqref{Kqgen} is invariant due to its purely geometrical origin, while
the perturbative contribution from $\mathcal{F}_{\rm pert}$ is independent of $B_2$ and
hence trivially invariant. Only the non-perturbative corrections encoded by $\mathcal{F}_{\rm b}$
explicitly depend on $B_2$. However, $B_2$ only arises through the exponential
$\exp(-i\int_\beta B_2)$ which is invariant under integral shifts. We thus conclude that $K$ is indeed invariant
under \eqref{B2shift}.
In contrast, we will see in the next section that the $\mathcal{N}=1$ coordinates $G^a,T_\alpha$
transform non-trivially under the shifts \eqref{B2shift}. This will allow us to infer valuable
information about the moduli dependence of the D-instanton superpotential in \eqref{full_super}.
Let us turn to the symmetry inherited from the underlying type IIB theory.
Recall that type IIB string theory admits the discrete
symmetry group $Sl(2,\mathbb{Z})$. Denoting the ten-dimensional dilaton-axion
as $\tau= C_0 + i e^{-\phi}$ this group acts by modular transformations and rotates
the ten-dimensional NS-NS and R-R two-forms $B_2$ and $C_2$ into each other.
More explicitly, we have
\begin{equation} \label{modular-tau}
\tau \quad \rightarrow\quad \frac{a\tau+b}{c\tau +d} \ ,\qquad \qquad \left(\begin{array}{c} C_2\\ B_2\end{array} \right) \quad \rightarrow\quad \left(\begin{array}{c}a\,C_2+b\, B_2\\c\,C_2+d\, B_2 \\\end{array} \right)\ ,
\end{equation}
where the integer matrix {\footnotesize $\left(\begin{array}{cc}a&b\\ c& d \end{array}\right)$} is an element of $Sl(2,\mathbb{Z})$.%
\footnote{
Here we have been a bit sloppy with factors of $2\pi$, which however can be restored easily.}
These transformations include in particular the map $\tau \rightarrow -1/\tau$ which
inverts the string coupling and corresponds to the strong-weak duality known as S-duality.
Compactifying type IIB string theory on a Calabi-Yau orientifold background
can reduce the symmetry group $Sl(2,\mathbb{Z})$ to a subgroup $\Gamma_S$
as discussed at the end of section \ref{orirev}.
Let us now check how the K\"ahler potential and K\"ahler coordinates
transforms under modular transformations \eqref{modular-tau} in $\Gamma_S$.
We concentrate in the following on the large volume compactification characterized
by the $\alpha'$ corrected pre-potential \eqref{simple_F}.
Using the explicit expressions \eqref{def-tauG} and \eqref{def-T} for $G^a, T_\alpha$ we note that these $\mathcal{N}=1$
coordinates
transform under \eqref{modular-tau} as %
\footnote{For $T_\alpha$ to transform as in \eqref{tautransgen} we have used that $e^{-\phi/2} v^\alpha$ and
$\tilde \rho_\alpha$ are invariant under \eqref{modular-tau}. The combination $e^{-\phi/2} v^\alpha$ is precisely the invariant Einstein
frame K\"ahler structure deformation, while $\tilde \rho_\alpha$ arises in the expansion of an $Sl(2,\mathbb{Z})$ invariant $\tilde C_4$ with
field strength $F_5 = d \tilde C_4 -\frac12 dB_2 \wedge C_2 + \frac12 B_2 \wedge dC_2$. We have also
used that $(\tau -\bar \tau)^{-1} \rightarrow (c\tau +d)^2 (\tau -\bar \tau)^{-1} -c(c\tau+d)$.}
\begin{equation} \label{tautransgen}
G^a \ \rightarrow \ \frac{G^a}{c \tau + d}\ ,
\qquad \qquad T_\alpha\ \rightarrow\ T_\alpha + \frac{1}{2} \frac{c\ \mathcal{K}_{\alpha a b} G^{a} G^b}{c\tau+d}\ .
\end{equation}
where $a,b,c,d$ are the entries of an element of $ \Gamma_S$.
We next analyze how the perturbatively corrected K\"ahler potential \eqref{Kqgen} transforms under \eqref{tautransgen}.
It is very easy to evaluate the transformation properties of the first term in \eqref{Kqgen} since
\begin{equation}
(\tau -\bar \tau)^{-1} \quad \rightarrow \quad |c\tau +d|^2 (\tau -\bar \tau)^{-1}\ .
\end{equation}
We thus have to focus on the transformation of the combination
\begin{equation} \label{pertcorr}
V_{E}(\tau,G,T) + \tfrac{1}{(2i)^{3/2}} \big(\tau -\bar \tau \big)^{3/2} \big[2\zeta(3) \chi -4 \text{Im} \mathcal{F}_{\text{b}}(\tau,G)\big] \ .
\end{equation}
Clearly, the Einstein-frame volume $V_E$ is invariant under $\Gamma_S$, since it is
a purely geometric quantity. Note however, that invariance does not hold for the $\alpha'$
correction in \eqref{pertcorr}. This can be traced back to the fact that we did not include all
corrections relevant in this large volume limit. Analogously to the discussion in refs.~\cite{GG,RRSTV}
one can argue that also corrections due to D(-1) branes as well as the reduction of D1
instantons have to be included. These couple to
the complex dilaton $\tau$ and $G^a$ and can complete the $\alpha'$ correction in \eqref{pertcorr}
in a modular invariant form.
We propose that by including these contributions the large volume K\"ahler potential $K_{\rm q}$ takes the form
\begin{equation}
\label{Kqgenmod}
K_q= - \ln\big[ -i (\tau -\bar \tau) \big]- 2 \ln\Big[ V_{E} + \tfrac12 \chi\, f(\tau,\bar \tau) -4 g(\tau,\bar \tau,G,\bar G)\Big]\ ,
\end{equation}
and transforms under $\Gamma_S$ as
\begin{equation} \label{transK}
e^{K} \quad \rightarrow \quad |c\tau +d|^2 e^K\ .
\end{equation}
In general it is hard to determine the precise form of the modular invariant forms
$f(\tau,\bar \tau)$ and $g(\tau,\bar \tau,G,\bar G)$. In the remainder of this section
we will discuss some properties of $g,f$ as well as some candidate modular
completions. A calculation of
$f,g$ might be possible by restricting the class of Calabi-Yau manifolds to $K3$ fibrations
where heterotic-F-theroy duality can be applied.
In the following we will first discuss the modular invariant function $f(\tau,\bar \tau)$
in \eqref{Kqgen}. In order to do that, we recall that in ref.~\cite{GG} a similar problem
arose in the computation of the $R^4$ correction to the
ten-dimensional type IIB supergravity action. In this ten-dimensional setup,
an additional analysis of the properties of the $\tau$-dependent coefficient
$\hat f(\tau,\bar \tau)$ led to the identification
\begin{equation} \label{sumhatf}
\hat f(\tau,\bar \tau) = \sum_{(n,m) \in P} \frac{(\tau-\bar \tau)^{3/2}}{(2i)^{3/2}|m+n \tau|^3} \ ,
\end{equation}
where $P=\mathbb{Z}^2/(0,0)$ is a two-dimensional lattice without the origin.
This non-holomorphic Eisenstein series includes indeed the perturbative correction
in \eqref{pertcorr}, when $n=0$ in the sum \eqref{sumhatf}. Moreover, it is invariant under the
full group $Sl(2,\mathbb{Z})$ and hence a candidate modular completion of the K\"ahler
potential. It was also conjectured in ref.~\cite{RRSTV} that the function \eqref{sumhatf}
is the correct modular completion of the analog situation in the underlying $\mathcal{N}=2$
theory. In our setup one might want to restrict the sum in \eqref{sumhatf} only to orbits of
the subgroup $\Gamma_S$. However, in any case modularity together with the limit $n=0$
alone seems not sufficient to fix the form of
$f(\tau,\bar \tau)$ in \eqref{Kqgen}. Additional conditions such as the singularity
structure or the suppression of further mixed contribution
are needed to determine $f(\tau,\bar \tau)$ unambiguously.
This is in general hard and beyond the scope of this paper. For the general discussion
of the superpotential we will simply assume that such a modular completion exists, while
for our explicit example in section \ref{EnriquesO} we will find that $\chi=0$.
Let us also briefly discuss the modular completion $g(\tau,\bar \tau,G,\bar G)$
of the non-perturbative $\alpha'$ corrections inherited from $\mathcal{N}=2$.
The corrections we are missing in our computation are the D1 branes
dual to the world-sheets inducing the contribution $\mathcal{F}_{\rm b}$. More precisely,
we need to include the whole set of $(p,q)$ strings \cite{Schwarz:1995dk,Witten:1995im} to restore $\Gamma_S$ duality.
Again we are facing the problem that such corrections are hard to compute in general
and we can only discuss some candidate solution for $g$.
In ref.~\cite{RRSTV} the modular completion of the underlying $\mathcal{N}=2$
quaternionic geometry was conjectured to arise from
a summation over all $Sl(2,\mathbb{Z})$ images of the world-sheet instanton
corrections. In the orientifold limit this leads to the following definition
of a modular invariant $\hat g$
\begin{equation}
\hat g(\tau,\bar \tau,G,\bar G) = \sum_\beta n_{k_a} \sum_{(m,n)\in P} \frac{(\tau-\bar \tau)^{3/2}}{(2i)^{3/2}|n+m \tau|^3} \cos \Big((n+\tau m) \frac{k_a(G^a - \bar G^a)}{\tau-\bar \tau} - m k_a G^a \Big)\ .
\end{equation}
This sum encodes all images under $Sl(2,\mathbb{Z})$ of the world-sheet instanton corrections in $\text{Im} \mathcal{F}_{\rm b}$
divided by stabilizer group generated by shifts $\tau \rightarrow \tau +1$. In general one might also want
to restrict to orbits of the subgroup $\Gamma_S$.
It is not hard to check that $\hat g$ contains the contribution $\text{Im} \mathcal{F}_{\rm b}$ for $m=0$.
Once again we have to remark that even though $\hat g$ has the desired
properties, the true correction $g$ is expected to be more complicated.
It would thus be desirable to find independent ways to calculate
$g$ for specific setups. In the example of section \ref{EnriquesO} all non-perturbative
$\alpha'$ corrections will be absent such that no $g$ is inherited from $\mathcal{N}=2$.
Before moving on to the discussion of the superpotential,
let us compare the question of determining $f(\tau,\bar \tau)$
and $g(\tau,\bar \tau,G,\bar G)$ to a somewhat similar situation within
topological string theory on a Calabi-Yau threefold \cite{BCOV,YY, Klemm, GKMW}. The symmetry
group in this case is the target space duality group arising from the
monodromies around singularities in the moduli space. One
can thus attempt to parametrize the non-perturbative
corrections by modular forms of this duality group which form a finite ring.
Fortunately, the singularity structure for the topological string partition
function is often known and additional boundary conditions allow to
fix the precise modular forms encoding the non-perturbative corrections
at least up to a certain genus. These boundary conditions arise from the
singularities of the moduli space or through the application of
string-string dualities (see e.g.~\cite{Klemm,GKMW}).
One might thus hope that to redo a similar analysis
in the $\mathcal{N}=1$ theories discussed in this work. Clearly, one of the obstacles
is the non-holomorphicity of the K\"ahler potential as well as the presence
of additional perturbative corrections. For the holomorphic
$\mathcal{N}=1$ superpotential this situation is improved as we will discuss in the
next section.
\subsection{D-instanton superpotentials in type IIB orientifolds \label{generalsup}}
Let us now discuss the D-instanton superpotential arising in type
IIB orientifolds with O3/O7 planes. The instantons contributing
to the superpotential are typically Euclidean D3 branes wrapped
around special four-cycles inside the Calabi-Yau orientifold.
In order to give the precise conditions
when such a potential arises, one has to embed this orientifold setup into
an F-theroy compactification. These conditions have been investigated
first by Witten in \cite{Witten} and later refined for compactifications with
background fluxes \cite{zeromodes}. Since here our primary interest is the
definition of a symmetry invariant superpotential for a generic orientifold
compactification, we will directly go to the orientifold and assume that
these conditions are satisfied for the cycles under consideration.
In the type IIB orientifolds discussed in the previous sections the instanton
superpotential arises from specific Euclidean D3 branes. Let us consider such
a brane warp around a devisor $\Sigma$ in $Y/\sigma$.
We will pick the devisor such that it non-trivially contributes to
the superpotential.
Schematically these contributions are of the form
\begin{equation}
f(X^I)\ e^{-V_\Sigma + i \phi_\Sigma}
\end{equation}
where $V_{\Sigma}$ is the Einstein-frame volume of $\Sigma$ and $\phi_\Sigma$ is the integral
of the R-R four-form $C_4$ over $\Sigma$. The function $f(X^I)$ can depend on
other chiral multiplets in the spectrum and we will be the main focus of our
considerations. Before turning to the discussion of $f$, let us
first note that the form of the exponential is not yet exact, since
we are missing the coupling to the lower R-R forms and the B-field in the exponential.
Recall that the effective
action on the word-volume of the Euclidean D3 brane takes the
form
\begin{equation}
S^{D3}=iT_{D3} \int_{\mathcal{W}_4} d^4 \lambda e^{-\phi} \sqrt{\det\big(g-B_2+F\big)}+ T_{D3} \int_{\mathcal{W}_4} C^{\text{RR}} \wedge e^{-B_2 + F} \ ,
\end{equation}
where $C^{\text{RR}}=C_0 + C_2 + C_4$ are the Ramond-Ramond fields and $F$ is the fieldstrength on the
brane. The first and second term correspond to the Born-Infeld and Chern-Simons
coupling respectively. In order that the D-instanton preserves
supersymmetry it has to wrap a supersymmetric cycle.
Applying the standard calibration conditions for supersymmetric branes
we find that the correct couplings to the R-R forms and the B-field \cite{BBS}.
The correct superpotential contribution is thus proportional to
\begin{equation} \label{exponent}
\text{exp}\Big[ - \tfrac12 \int_\Sigma e^{-\phi} \big(J \wedge J- B_2 \wedge B_2\big) - i \int_\Sigma \big( C_4 - C_2\wedge B_2 +\tfrac12 C_0 B_2\wedge B_2\big) \Big]\ .
\end{equation}
Note that the first term under the first integral is $V_{\Sigma}$, since the K\"ahler form $J$ is evaluated in
the string-frame metric. The expression \eqref{exponent} is precisely exp$(-i\int \rho_c)$ with $\rho_c$ introduced
in \eqref{def-rhoc}. Thus we find that the generic superpotential is of the expected form
\begin{equation} \label{Dinst_superpot}
W_{\text{D-inst}} = \sum_\Sigma f_\Sigma(X^I) \ e^{i n^{\ \alpha}_\Sigma\, T_\alpha}\ ,\qquad \qquad n_{\Sigma}^{\ \alpha} = \int_\Sigma \tilde \omega^\alpha\ ,
\end{equation}
where $n_\Sigma^{\ \alpha}$ are integers for $\Sigma \in H_{4}(Y,\mathbb{Z})$ and $\tilde \omega^\alpha \in H^{4}_+ (Y,\mathbb{Z})$.
We are now in the position to discuss the moduli dependence of $f(X)$ in more detail.
So far we did not discuss the holomorphic function $f(X)$. In general, it can
depend on various other moduli $\{X^I\}$ of the orientifold or underlying F-theory compactification.
As in \eqref{elliptic} we denote the elliptically fibered fourfold corresponding to the orientifold by $Y_4$.
The moduli dependence of $f$ can arise from:
\begin{itemize}
\item[(a)] the complex structure deformations of $Y_4$: in the orientifold limit these
include the complex dilaton $\tau$ corresponding
to the complex structure of the elliptic fiber, the complex structure deformations of $Y/\sigma$
as well as the D7 brane moduli,
\item[(b)] the $h^{(2,1)}$ complex scalars arising in the expansion of $C_M$ in $H^{(2,1)}(Y_4)$: these
include the complex scalars $G^a$ as well as Wilson lines of the D7 brane,
\item[(c)] the complex coordinates $x^i$ labeling the position of space-time filling D3-branes in $Y_4$ or $Y/\sigma$.
\end{itemize}
In the following we will discuss $f(X)$ as a function of the complex dilaton $\tau$,
the moduli $G^a$ arising by expanding the type IIB NS-NS and R-R two-form.
An analysis of the dependence of $f(X)$
on the positions of the space-time filling D3-branes $x^i$ on $Y/\sigma$
can be found in \cite{Ganor,Baumann:2006th}.
It turns out that a direct computation of the function $f(X)$ is in
general very hard and involves the evaluation of appropriate determinants \cite{Witten}.
However, we can already learn much about $f$ by studying the transformation
properties of the superpotential and the K\"ahler potential under
shifts and modular transformations. This was already initiated in refs.~\cite{Witten:1996hc,Ganor} for
M- and F-theory compactifications were only a local analysis can be performed.
Here we will make this discussion very concrete for the type IIB orientifolds
studied in section \ref{orirev} and focus on its dependence on $\tau,G^a$
decomposing $f(X) = A_0 \Theta(\tau,G^a)$, with $A_0$ depending on the remaining
moduli. We thus write
\begin{equation} \label{superpotexp}
W_{\text{ D-inst}} = A_0 \sum_{\Sigma} \Theta_{\Sigma}(\tau,G) e^{in_{\Sigma}^{\ \alpha}\, T_\alpha }\ .
\end{equation}
Let us now investigate the transformation properties of the coefficients $\Theta_{\Sigma}(\tau,G^a)$ in more detail.
We will first discuss the duality transformations induced by modular changes
of the complex dilaton $\tau$ as given in \eqref{modular-tau}. In section \ref{Kaehlersymm} we have argued that
$e^K$ transforms as given in equation \eqref{transK} under modular transformations.
From this we conclude that the superpotential has to change as \footnote{%
In the following we will not include a possible phase. For a related discussion of the possibility
to include such a phase factor see, for example, refs.~\cite{FILQ, CFILQ}.}
\begin{equation} \label{transWundertau}
W \quad \rightarrow \quad (c\tau +d)^{-1} W \ .
\end{equation}
To see this, we note that the combination $e^{K}|W|^2$ has to be invariant since
it determines, for example, in the physical gravitino mass.
Equation \eqref{transWundertau} exactly states
that $W$ has to be a modular form of weight $-1$ under the duality group $\Gamma_S$.
Let us note that this is obviously true for the flux superpotential $W= \int \Omega\wedge (F_3 -\tau H_3)$ in \eqref{full_super}.
For the D-instanton superpotential \eqref{superpotexp} we will see momentarily, that this imposes constraints
on the functions $\Theta_{\Sigma}(\tau,G)$.
The second transformation we will consider are the shifts \eqref{B2shift}
in the NS-NS two-form $B_2$. More precisely,
let us transform the orientifold coordinates by $b^a \rightarrow b^a + 2\pi n^a$.
From the definitions \eqref{def-tauG} and \eqref{def-T} of the
coordinates $G^a,T_\alpha$ we deduce that
\begin{eqnarray} \label{Gshift}
G^a \quad &\rightarrow& \quad G^a - 2 \pi \tau n^a \ ,\\
T_\alpha \quad&\rightarrow& \quad T_\alpha - 2\pi \mathcal{K}_{\alpha ab} n^a G^b + 2 \pi^2 \tau \mathcal{K}_{\alpha ab} n^a n^b \nonumber \ .
\eea
As we have seen in section \ref{Kaehlersymm}, it is not hard to check that this is a symmetry
of the orientifold K\"ahler potential. Due to the invariance of the combination $e^K |W|^2$ we conclude that
$W$ can only transform by a trivial phase factor and is otherwise invariant.
Invariance of $W$ together with the fact that $T_\alpha$ transforms as in \eqref{Gshift}
restricts the coefficient functions $\Theta_{\Sigma}(\tau,G)$
of the instanton superpotential \eqref{superpotexp} as we will discuss next.
We can now infer the properties of the functions $\Theta_{\Sigma}(\tau,G)$
appearing in \eqref{superpotexp}. Our strategy is to use the fact that $W$
is a modular form of weight $-1$ but otherwise invariant under \eqref{modular-tau}, \eqref{tautransgen} and \eqref{Gshift}.
Since $e^{i n_\Sigma^{\ \alpha} T_\alpha}$ in \eqref{superpotexp} transforms non-trivially under these symmetries also $\Theta_{\Sigma}(\tau,G)$
has to transform in order to ensure the correct modular properties of $W$.
It turns out that the $\Theta$'s are
generalizations of the well-known theta functions, or more precisely
appropriate holomorphic Jacobi forms.\footnote{Holomorphicity here only means that
$\Theta_\Sigma(\tau,G)$ is independent of $\bar \tau,\bar G^a$ and
does not restrict the singularity structure.} To summarize their properties
we simplify our analysis and restrict our attention
to the case where only one $T\equiv T_{\alpha'}$ transforms non-trivially
under the above groups. In other words, we will
assume here that the only non-vanishing intersection with negative
indices is $\mathcal{K}_{\alpha' a b}=-C_{ab}$.
We also denote $n^{\ \alpha'}_{\Sigma}=n$. The Jacobi form $\Theta_n(\tau,G)$
then turns out to be of weight $-1$ and index $n$. In other words,
under the transformation \eqref{tautransgen}
this form transforms as
\begin{equation} \label{transtheta1}
\Theta_n(\tau,G) \ \rightarrow \ (c\tau+d)^{-1} \text{exp} \Big({\frac{n i}{2} \frac{c\ C_{a b} G^{a} G^b}{c\tau+d}}\Big)\ \Theta_n (\tau,G) \ ,
\end{equation}
which is consistent with the required transformation behavior \eqref{transWundertau}.
Also the transformation \eqref{Gshift} of $e^{inT}$ is cancelled by the corresponding Jacobi form
$\Theta_n$ since
\begin{equation} \label{transtheta2}
\Theta_n(\tau,G) \ \rightarrow \ \text{exp} \big(- 2\pi i n C_{ab} n^a G^b + 2 \pi^2 i n \tau C_{ab} n^a n^b \big)\ \Theta_n (\tau,G)
\end{equation}
under the transformation \eqref{Gshift}. Carefully restoring factors of $2\pi$
the transformations \eqref{transtheta1} and \eqref{transtheta2} are
exactly the transformation properties of
Jacobi forms. For only one
field $G^a$, the theory of Jacobi forms is
extensively reviewed by Eichler and Zagier in ref.~\cite{EZ}. The more general situation including vectors
$G^a$ is discussed, for example, in the work of Borcherds \cite{binfinite} (section 3).
Before turning to the example in the next section,
let us summarize some classical results about candidate Jacobi forms $\Theta_n$ \cite{EZ,binfinite}. In order to do that
we introduce the theta functions of weight $s/2$ and index $m$ by setting
\begin{equation}
\theta_{(m)\ L + r}(\tau, G) = \sum_{n_a \in L + r} e^{i \tau n^2/2} e^{m i G^a n_a}\ ,\qquad n^2=C^{ab} n_a n_b\ ,
\end{equation}
where $L$ is some positive definite rational lattice of dimension $s$, and
$r$ is some vector which admits an expansion in a basis of $L$
with rational coefficients.
It can be shown that any Jacobi form $\Theta_n$ can be written as
a sum of products of the theta functions $ \theta_{(m)\ L + r}$ and
modular forms $\tilde \eta(\tau)$. Heuristically, we can write
\begin{equation} \label{def-Thetan}
\Theta_n(\tau,G) = \sum \frac{\theta_{(n)}(\tau, G)}{\tilde \eta(\tau)}\ .
\end{equation}
This form is well known from various other perspectives. For example, it was shown
in \cite{Alvarez-Gaume:1987vm} that the partition function of a chiral boson on a genus one surface is of this form.
More importantly, also the partition function
of the M5 brane takes a form similar to \eqref{def-Thetan} as was first discussed in ref.~\cite{Witten:1996hc}.
This is no surprise, since we know that the F-theory lift of the D3 instantons are six-dimensional
branes. Analyzing F-theory from the M-theory point of view as mentioned at the end of
section \ref{orirev} these six-dimensional branes are M5 branes wrapped around four-cycles
in the base $B_3$ of \eqref{elliptic} as well as on the two-dimensional fiber.
Clearly, an important task is to explicitly find the correct Jacobi forms $\Theta_n(\tau,G)$ for
specific examples. One suspects that this problem is more tractable then determining the
modular corrections to the K\"ahler potential due to the holomorphicity of $W$ and the absence
of perturbative corrections. Ideally, one likes to use physical arguments, for example
on the singularity structure of $W$, to restrict the set of candidate Jacobi forms to a finite set.
Computing $W$ in a particular limit, e.g.~an orbifold limit, might then determine the
correct linear combination to appear in the full $W$.
In the next section, we will take a different route in the study of the Enriques orientifold.
We will use some intuition from the topological strings on the
Enriques Calabi-Yau to propose a candidate $W$ including non-trivial
Jacobi forms $\Theta_n$.
\section{D-instantons and the Enriques orientifold \label{EnriquesO}}
In this section we discuss one type IIB orientifold compactification in
more detail and illustrate some of the general story outlined in the previous section.
We construct an orientifold of the Enriques Calabi-Yau $Y_E$ and argue that
the quantum corrections are under particular control. It is also
shown how the $\mathcal{N}=1$ K\"ahler manifold $\tilde \mathcal M_{\rm q}$ inside the $\mathcal{N}=2$ quaternionic
space can be identified with the original special K\"ahler moduli
space times a $Sl(2,\mathbb{R})/U(1)$ factor. In this duality the new complex coordinates contain the
R-R fields as in \eqref{def-rhoc} and provide the correct couplings to
D-instantons. We use this identification to translate instanton
expansions known from topological string theory
on $Y_E$ to the corresponding physical orientifold setup.
This leads us to propose a specific D-instanton superpotential
for the Enriques orientifold.
\subsection{Enriques Calabi-Yau and counting of D(-1)-D1-D3 states \label{GeometryN=2}}
Let us begin by reviewing some basic facts about the Enriques Calabi-Yau $Y_E$
and its moduli space. The Enriques Calabi-Yau takes the form $Y_E = (K3 \times \mathbb{T}^2)/\mathbb{Z}_2$,
where the $\mathbb{Z}_2$ acts as an inversion of the complex coordinate of $\mathbb{T}^2$ and as the Enriques involution
on $K3$ \cite{Borcea,FHSV, SurfaceB}. $Y_E$ has holonomy group $SU(2) \times \mathbb{Z}_2$. This implies that type
II string theory compactified on the Enriques Calabi-Yau will lead to a four-dimensional
theory with $\mathcal{N}=2$ supersymmetry. Nevertheless, due to the fact that it does not have the
full $SU(3)$ holonomy of generic Calabi-Yau threefolds, various special properties of
$\mathcal{N}=4$ compactifications on $K3\times \mathbb{T}^2$ are inherited.
In order to discuss the moduli space of $Y_E$ we first need
to summarize the cohomology on this Calabi-Yau manifold.
We review in appendix \ref{EnriquesGeom} that the two-form and
three-from integral cohomologies can be identified
with the following lattices \cite{FHSV}
\begin{eqnarray} \label{EnriquesCohomology2}
H^{2}(Y_E,\mathbb{Z}) &\cong& \mathbb{Z} \oplus \Gamma^{1,1} \oplus \Gamma_{E_8}(-1)\ ,\\
\label{EnriquesCohomology3}
H^{3}(Y_E,\mathbb{Z}) &\cong& \big( \Gamma^{1,1} \oplus \Gamma_{E_8}(-1) \oplus \Gamma^{1,1}_g \big)\oplus \big( \Gamma^{1,1} \oplus \Gamma_{E_8}(-1) \oplus \Gamma^{1,1}_g \big)\ ,
\eea
where $\Gamma^{1,1}$ is a two-dimensional lattice with signature $(1,1)$ and inner product
{\footnotesize $\left( \begin{array}{cc} 0&1\\ 1&0\end{array}\right)$},
and $\Gamma_{E_8}(-1)$ has an inner product given by $-1$ times
the Cartan matrix of the exceptional group $E_8$.
We denote an integral basis $(\omega_A) = (\omega_S,\omega_i, \omega_a)$ of $H^{2}(Y_E,\mathbb{Z})$,
where $\omega_S$, $\omega_i$ and $\omega_a$ are basis elements of the three terms in
\eqref{EnriquesCohomology2} respectively.
We already defined the triple intersections $\mathcal{K}_{ABC}$ in \eqref{triple_inters}.
Using the relation to the underlying $K3\times \mathbb{T}^2$ one shows
that the only non-vanishing intersections are
\begin{equation} \label{Enriques_int}
\mathcal{K}_{S12} = \mathcal{K}_{S21} = 1\ ,\qquad \mathcal{K}_{Sa b} =-C_{a b}\ ,
\end{equation}
where in the appropriate basis the inverse $C^{ab}$ of $C_{ab}$ is the Cartan
matrix of $E_8$ as already mentioned before.
As in section \ref{orirev} we also introduce a basis $(\tilde \omega^A)=(\tilde \omega^S, \tilde \omega^i, \tilde \omega^a)$
of $H^{4}(Y_E,\mathbb{Z})$ dual to $\omega_A$. Finally, we will need to introduce
a real symplectic basis $(\alpha_A,\beta^A)$ of the third cohomology $H^{3}(Y_E,\mathbb{Z})$.
The explicit form \eqref{EnriquesCohomology2} and \eqref{EnriquesCohomology3} of the integral
cohomology of $Y_E$
allows us to read of the dimensions $h^{(p,q)}$ of the cohomologies $H^{(p,q)}(Y_E)$.
We find that
\begin{equation} \label{dimHodge}
h^{(1,1)}(Y_E)= h^{(2,1)}(Y_E)=11\ .
\end{equation}
This implies that the moduli spaces of complex structure deformations $\mathcal M_{\rm cs}$
as well as of K\"ahler structure deformations $\mathcal M_{\rm ks}$ are both complex eleven-dimensional.
Moreover, one shows that both of these spaces are the coset \cite{FHSV}
\begin{equation} \label{csks}
\mathcal M_{\rm cs/ks}\ =\ Sl(2,\mathbb{R})/U(1)\ \times\ O(10,2)/\big(O(10)\times O(2)\big)\ ,
\end{equation}
where $O(q,p,\mathbb{R})$ are orthogonal groups with values in the real numbers.
The identification $\mathcal M_{\rm cs} \cong \mathcal M_{ks}$ arises due to the fact that
the Enriques Calabi-Yau is self-mirror.
In a careful treatment one also finds that these cosets have to be divided by
the discrete symmetry group
\begin{equation}
O_{E}(\mathbb{Z})\ \equiv\ Sl(2,\mathbb{Z}) \times O(10,2,\mathbb{Z})\ ,
\end{equation}
which is a non-perturbative symmetry of string theory on $Y_E$.
The presence of this discrete factor is of central importance. All
functions on $\mathcal M_{\rm cs/ks}$ have to transform covariantly under
$O_{E}(\mathbb{Z})$ to be well defined.
Furthermore, note that after dividing by $O_E(\mathbb{Z})$
the identification \eqref{csks} is exact and receives no corrections
due to world-sheet instantons \cite{FHSV,mp}. As we will discuss next
this implies that the Enriques Calabi-Yau is a special example
with an exact pre-potential cubic in the moduli around the large volume
or large complex structure point.
To make this more precise we discuss the geometry of the moduli
space $\mathcal M_{\rm ks}$ in more detail. Clearly, due to the fact that $Y_E$
is self-mirror the geometry of $\mathcal M_{\rm cs}$ takes a similar form.
Compactifying Type II string theory on the Enriques Calabi-Yau yields an
effective four-dimensional theory with $\mathcal{N}=2$ supersymmetry.
In general, the $\mathcal{N}=2$ scalar moduli space consists of a special K\"ahler $\mathcal M_{\rm sk}$
times a quaternionic manifold $\mathcal M_{\rm q}$. For the Enriques Calabi-Yau both spaces
are cosets. Since we are interested in type IIB compactifications
we find that the complex structure deformations are the space
$\mathcal M_{\rm sk}$ while the K\"ahler structure deformations sit inside
the quaternionic space $\mathcal M_{\rm q}$.
One finds \cite{FHSV}
\begin{equation} \label{moduli_space}
\mathcal M_{\rm sk} = \mathcal M_{\rm cs}\ ,\qquad \qquad \mathcal M_{\rm q} = O(12,4)/ \big(O(12)\times O(4)\big)\ \supset\ \mathcal M_{\rm ks} \ .
\end{equation}
Note that $\mathcal M_{\rm sk}$ is exact and receives no perturbative corrections or corrections due
to world-sheet or D-instantons. In contrast, $\mathcal M_{\rm q}$ is in general perturbatively and non-perturbatively
corrected.
The geometry of the two moduli spaces in \eqref{moduli_space} is encoded by two cubic pre-potentials.
For $\mathcal M_{\rm sk}$ one finds around the large complex structure point
a pre-potential of the form \footnote{A more careful analysis reveals that there is a linear term $-z^S$ in $\tilde \mathcal{F}(z)$ \cite{KM}.
This term however does not appear in the K\"ahler potential and hence
not in any physical object discussed in the following.}
\begin{equation} \label{def-tildeF}
\tilde \mathcal{F}(z) = - z^S z^1 z^2 + \tfrac{1}{2} z^S C_{ab} z^a z^b \ .
\end{equation}
Due to the absence of world-sheet instanton corrections this potential is exact and can be transformed
and used at other points in the moduli space $\mathcal M_{\rm sk}$.
This special K\"ahler manifold encodes deformations of the complex structure through the holomorphic
$(3,0)$ form
\begin{equation} \label{Omega_periods}
\Omega(z) = X^K(z)\alpha_K - \tilde \mathcal{F}_K(z)\beta^K\ ,
\end{equation}
where $(\alpha_K,\beta^K)$ is a real symplectic basis of $H^{3}(Y_E,\mathbb{Z})$.
The periods of $\Omega$ are thus $(X^K,\tilde \mathcal{F}_{K})$,
where $\tilde \mathcal{F}_{K}$ is the derivative of $\tilde \mathcal{F}(z)$ with respect to $X^K$. In the
spacial coordinates $z$ above one has $z^S=X^S/X^0$, $z^i = X^i/X^0$ and $z^a = X^a/X^0$.
One can thus rewrite $\tilde \mathcal{F}_{K}$ as derivatives with respect to the coordinates $z$ \cite{N=2rev}.
The quaternionic manifold $\mathcal M_{\rm q}$ can be constructed by starting with the underlying special
K\"ahler manifold $\mathcal M_{\rm ks}(t)$.
The coordinates $t^A=(S,t^i,t^a)$ are the complexified K\"ahler structure deformations of $Y_E$
arising in the expansion of $-B_2+iJ$ into the two-form basis $\omega_A=(\omega_S,\omega_i,\omega_a)$.
The geometry of the special K\"ahler manifold is determined by the
pre-potential \footnote{As in \eqref{def-tildeF} we ignore a linear term in $S$ which can be absorbed into a
redefinition of the coordinates on $\mathcal M_{\rm q}$.}
\begin{equation} \label{def-Ft}
\mathcal{F}(t)= - S t^1 t^2+\tfrac{1}{2!} S C_{ab} t^a t^b\ .
\end{equation}
It is straightforward to derive the corresponding K\"ahler potential $ K_{\rm ks}(S,\bar S,t,\bar t) $.
In general,
$K_{\rm ks}$ can be obtained from the even form $\rho$ introduced in \eqref{def-rho}
by setting $K_{\rm ks} =-\ln i \big< \rho,\bar \rho\big>$ with wedge product defined in footnote \ref{wedge-product}.
Inserting \eqref{def-Ft} into this expression one evaluates
\begin{equation} \label{def-Y}
K_{\rm ks} = - \ln\big(i(S-\bar S) Y \big)\ , \qquad \quad Y = (t-\bar t)^1 (t-\bar t)^2 -\tfrac12 (t-\bar t)^a (t-\bar t)^b C_{a b} \ .
\end{equation}
The classical quaternionic geometry can be obtained from $\mathcal M_{\rm ks}$ by applying the c-map construction \cite{Ferrara:1989ik}.
Since our focus will be the orientifold scenario, we will not review the details here. Let us however note
that the quaternionic geometry is invariant under the K\"ahler transformations of $K_{\rm ks}$.
It is therefore naturally formulated in terms of the invariant combination $C\rho$, with $C$
proportional to the dilaton $ e^{-\phi}$. Note that $C$ and $\rho$ itself do transform under the K\"ahler
transformations $K_{\rm ks} \rightarrow K_{\rm ks} -f(t) -\bar f(\bar t)$ as
\begin{equation} \label{trans-Crho}
C \quad \rightarrow\quad e^{-f} \, C\ ,\qquad \qquad \rho\quad\rightarrow \quad e^f\, \rho\ ,
\end{equation}
where $f(t)$ is a holomorphic function of the moduli.
We will now go one step further and discuss a first set of
quantum corrections depending on the moduli of
$\mathcal M_{\rm ks}$. Following \cite{HM,HM2,KM} we will introduce a
functional $\Phi_{\rm B}$ which counts the leading degeneracies of D(-1), D1, D3
states on the Enriques fiber. Before recalling the precise form of these corrections
let us note that this investigation will not take place
in the large volume limit but rather at a second special locus of the
Enriques moduli space. At this locus also Euclidean D3 branes
wrapped around a the Enriques fiber are becoming light. To make this more precise, we will choose `dual'
coordinate $\mathcal{T}^1,\mathcal{T}^2,\mathcal{T}^a$ in which large $\text{Im} \mathcal{T}$ implies a small volume
of the $K3$.
The transformations from the large volume limit
to this special Enriques locus is given by
\begin{equation} \label{def-cT}
\mathcal{T}^2 = -\frac{1}{2 t^2}\ ,\qquad \mathcal{T}^1 = \frac{1}{t^2}\big( t^1 t^2 - \tfrac{1}{2} C_{ab} t^a t^b \big)\ ,\qquad \mathcal{T}^a = - \frac{1}{t^2} t^a\ .
\end{equation}
Under this change of coordinates we find that $Y$ defined in \eqref{def-Y} transforms as
\begin{equation}
2 Y = \frac{1}{2\, \mathcal{T}^2 \bar \mathcal{T}^2}\big[2 (\mathcal{T} -\bar \mathcal{T})^1 (\mathcal{T} -\bar \mathcal{T})^2 - (\mathcal{T}-\bar \mathcal{T})^a (\mathcal{T}-\bar \mathcal{T})^b C^D_{a b} \big]= \frac{1}{\mathcal{T}^2 \bar \mathcal{T}^2} Y_D
\end{equation}
where we have introduced $C^D_{ij} = C_{ij}$, $C^D_{a b} = \tfrac{1}{2} C_{a b}$ and defined $Y_D$.
In other words, defining the dual K\"ahler potential $K_D(S,\mathcal{T})$ as
\begin{equation} \label{def-KD}
K_D(S,\mathcal{T}) = - \ln \big(i(S-\bar S) Y_D \big) \ ,
\end{equation}
one finds that $K_{\rm ks}$ and $K_{D}$
differ only by a K\"ahler transformation.\footnote{%
One finds that $K_{\rm ks}=K_{D} - f -\bar f$,
where $f=- \ln \big(i\sqrt{2}\, \mathcal{T}^2 \big)$.}
From the coordinate definition \eqref{def-cT} one concludes that the
corresponding cohomology lattice is
\begin{equation} \label{BHM-reduction}
\Gamma^{1,1}_s \oplus \Gamma_{E_8}(-2) \cong H^{0}(E,\mathbb{Z}) \oplus H^{4}(E,\mathbb{Z}) \oplus \Gamma_{E_8}(-2)
\end{equation}
where $H^{0}(E,\mathbb{Z})$ and $ H^{4}(E,\mathbb{Z})$ are the zero and four cohomology of the Enriques
fiber. This can be seen as follows. The K\"ahler invariant combination to consider is
$C\rho$ with $C$ and $\rho$ transforming as in \eqref{trans-Crho}. One can thus remove the
overall factor of $1/t^2$ in the definitions \eqref{def-cT}. On the one hand this leads to $\mathcal{T}^2 \propto C$ such
that $\mathcal{T}^2$ scales the element in
$H^{0}(E)$. On the other hand $\mathcal{T}^1 \propto C(2 t^1 t^2 - C_{ab} t^a t^b)$ which is the
square of the complexified K\"ahler form and hence parametrizes $H^{4}(E)$. We also
see that the lattice \eqref{BHM-reduction} contains the self-dual lattice $ \Gamma_{E_8}(-2)$
which has intersection form $C^{D\, a b} = 2 C^{a b}$. The extra factor $2$ arises due
to the factor $1/2$ in the definition of $\mathcal{T}^2$. We will see in the next section
that the coordinates $\mathcal{T}^1,\mathcal{T}^2,\mathcal{T}^a$ have a second advantage, since they can be
identified with the $\mathcal{N}=1$ coordinates of the orientifold theory.
We are now in the position to recall a functional $\Phi_{\rm B}(\mathcal{T})$ counting
the leading degeneracies of Euclidean D(-1), D1, D3 branes on the Enriques fiber.
It was shown in refs.~\cite{borcherdsone, borcherds}, that for $\mathcal{T}^i,\mathcal{T}^a$ with $Y_D < -1$ one
defines a convergent functional
\begin{equation} \label{Borcherds-Form}
\Phi_{\rm B} (\mathcal{T}) = e^{i \mathcal{T}^1} \prod_{r \in \Pi^+} (1- e^{i r\cdot \mathcal{T}})^{ (-1)^{m+n} c_{\rm B} (r^2/2)}\ ,
\end{equation}
where $ r\cdot \mathcal{T} = n\mathcal{T}^1 + m\mathcal{T}^2 - C^D_{ab} r^a \mathcal{T}^b$ for vectors $r=(m,n,r^a)$ in the lattice \eqref{BHM-reduction}.
In the product \eqref{Borcherds-Form} we denote by $\Pi^+$ the set of positive
roots of the fake monster Lie superalgebra consisting of all nonzero
vectors $r$ with $r^2 = 2mn-C^D_{ab} r^a r^b \ge -2$ such that $m>0$, or $m=0$ and $n>0$.
The exponents $c_{\rm B}(r^2/2)$ are given via the modular form
\begin{equation} \label{coeff_funct}
\sum_n c_{\rm B}(n) q^n = \frac{\eta(q^2)^8}{\eta(q)^8 \eta(q^4)^8}\ ,\qquad \qquad r^2/2 =n\ ,
\end{equation}
where $\eta(q)$ is the standard eta function. It was argued in ref.~\cite{KM}
that $ \Phi_{\rm B} (\mathcal{T}) $ counts the degeneracies of D(-1), D1, D3 branes on the Enriques fiber. To show
this Klemm and Mari\~no \cite{KM} applied a similar argument as Gopakumar and Vafa \cite{GV}
by performing a Schwinger calculation including the light states at the moduli space locus
parametrized by $\mathcal{T}^i,\mathcal{T}^a$. The corresponding BPS particles are bound states of D3 branes wrapping the Enriques fiber,
D1 wrapped around the curves in the $E_8$ sublattice in \eqref{BHM-reduction} and D(-1) branes.
The leading degeneracies are counted by the lowest genus free-energies $\mathcal{F}^{(g)}$
of the topological string on $Y_E$. Since $\mathcal{F}^{(0)}$ is trivial for the Enriques Calabi-Yau
the first non-trivial contribution arises from a resummation of $\mathcal{F}^{(1)}$ which precisely
contains the holomorphic function $ \Phi_{\rm B} (\mathcal{T}) $. It is important to remark, that $ \Phi_{\rm B} (\mathcal{T}) $ has particularly
nice modular properties as we will discuss in section \ref{EnriquesW}. For contributions
from the higher $\mathcal{F}^{(g)}$ this is only the case if also a non-holomorphic dependence
is included. Therefore, we will propose in section \ref{EnriquesW} that $\Phi_{\rm B}$ might contain the leading
contribution to a holomorphic and modular superpotential of the orientifold theory on the Enriques
Calabi-Yau.
\subsection{Effective action for the Enriques orientifold \label{effectiveN=1action}}
In this section we study the effective four-dimensional $\mathcal{N}=1$ supergravity obtained
by compactifying type IIB supergravity on an orientifold of the Enriques Calabi-Yau $Y_E$.
In order to do this we first have to define an involution $\sigma$ on $Y_E$ and
investigate its action on the cohomology.
It was shown in refs.~\cite{Dolgachev, SurfaceB} that involutions on the Enriques surface
can be characterized by their action on the lattice \eqref{EnriquesCohomology2}.
In particular, there exist an involution acting with a minus sign on
the $\Gamma_{E_8}(-1)$ term in \eqref{EnriquesCohomology2}, while leaving the $\Gamma^{1,1}$
term invariant. We complete this involution by also inverting the $\mathbb{P}^1\cong T^2/\mathbb{Z}_2$ base
of the fibration. This keeps the volume form of $\mathbb{P}^1$ invariant. We thus find for the
second cohomology lattice \eqref{EnriquesCohomology2} the split
\begin{equation} \label{two_split}
H^{2}_+(Y_E,\mathbb{Z})\ \cong\ \mathbb{Z} \oplus \Gamma^{1,1}\ ,\qquad H^{2}_-(Y_E,\mathbb{Z})\ \cong\ \Gamma_{E_8}(-1)\ ,
\end{equation}
where $H^2_\pm$ are the plus and minus eigenspaces of $\sigma^*$.
An integral basis $\omega_A=(\omega_S,\omega_i,\omega_a)$ of $H^2(Y_E,\mathbb{Z})$ is introduced
by setting
\begin{equation} \label{omega_pm}
\omega_{\alpha}=(\omega_S,\omega_i) \in H^{2}_+(Y_E,\mathbb{Z}) \ ,\qquad \qquad \omega_a \in H^{2}_-(Y_E,\mathbb{Z})\ .
\end{equation}
This is consistent with the basis $\omega_A$ introduced in the previous section.
The non-vanishing triple intersections $\mathcal{K}_{Sij}$ and $\mathcal{K}_{Sab}$
where already given in \eqref{Enriques_int}. It is important to
note that the orientifold constraints \eqref{inter_constraints} are indeed satisfied, since
$\mathcal{K}_{abc}$, $\mathcal{K}_{a\alpha \beta}$ vanish for $\alpha,\beta$ running over $S,i$.
The odd cohomology $H^3(Y_E,\mathbb{Z})$ also splits into positive and negative eigenspaces
under the involution. In order to make this split explicit, we note that the above $\sigma$ can be
extended to the underlying $K3$ surface such that it acts with a minus sign on the $\Gamma_{E_8}(-1)$
terms in the second cohomology lattice $H^{2}(K3,\mathbb{Z})$ given in \eqref{K3cohom}, while keeping
the remaining terms invariant. This is of course consistent with the split of the two-cohomology
\eqref{two_split}. The third cohomology $H^3(Y_E,\mathbb{Z})$ of the Enriques Calabi-Yau
is obtained by wedging one-forms of the $T^2$ with two-forms of the $K3$ both
anti-invariant under the $\mathbb{Z}_2$ involution defining the Enriques Calabi-Yau. Also including the negative sign
of $\sigma$ on the two one-forms of $T^2/\mathbb{Z}_2$ we thus
find that \eqref{EnriquesCohomology3} splits as
\begin{eqnarray} \label{third-cohom}
H^{3}_+(Y_E,\mathbb{Z})& \cong& \Gamma_{E_8}(-1)\oplus \Gamma_{E_8}(-1)\ ,\\
H^{3}_-(Y_E,\mathbb{Z}) & \cong & \big(\Gamma^{1,1} \oplus \Gamma_g^{1,1}\big) \oplus \big(\Gamma^{1,1} \oplus \Gamma_g^{1,1}\big) \ . \nonumber
\eea
We are now in the position to discuss the reduction of the moduli spaces following
the general approach in section \ref{orirev}.
Let us first discuss the reduction of the $\mathcal{N}=2$ special K\"ahler manifold $\mathcal M_{\rm cs}$
spanned by the complex structure deformations $z^\alpha=(z^S,z^i)$ and $z^a$. From \eqref{JOmegatrans} we note
that the holomorphic three-form $\Omega$ is an element of the negative
eigenspace of $\sigma^*$. This implies that in the orientifold setup we have $z^a=0$ and
the expansion \eqref{Omega_periods} reduces to
\begin{eqnarray}
\Omega& =& X^0(\alpha_0 + {z^\alpha}\alpha_\alpha - \tilde \mathcal{F}_{z^\alpha} \beta^\alpha - (2\tilde \mathcal{F} - z^\alpha \tilde \mathcal{F}_{z^\alpha})\beta^0)\\
&=&X^0(\alpha_0 + {z^\alpha}\alpha_\alpha + z^1 z^2 \beta^S + z^S z^2 \beta^1 + z^S z^1 \beta^2 + z^S z^1 z^2\beta^0)\ , \nonumber
\eea
where $(\alpha_0,\alpha_\alpha,\beta^\alpha,\beta^0)$ is a real symplectic basis of $H^{3}_-(Y_E,\mathbb{Z})$ given in \eqref{third-cohom}.
The prepotential for this reduced special K\"ahler manifold $\tilde \mathcal M_{\rm sk}(z)$ is thus a function of the three moduli
$z^\alpha=(z^S, z^i)$ only and takes the form $\tilde \mathcal{F}(z^I)=- z^S z^1 z^2$.
The K\"ahler potential is evaluated explicitly to be of the form
\begin{equation}
K_{\rm cs} =-\ln\big[ i \int \Omega(z) \wedge \bar \Omega(\bar z)\big]
= -\ln\big[ i (z^S-\bar z^S)(z^1-\bar z^1)(z^2-\bar z^2)\big]\ ,
\end{equation}
where we have removed the fundamental period $X^{0}$ by a K\"ahler transformation.
The geometry of this reduced moduli space $\tilde \mathcal M_{\rm cs}$ has been studied intensively
in the literature \cite{KM,GKMW}. It can be shown that the mirror map takes a particularly simple
form due to the absence of world-sheet instantons. It respects the discrete target space
symmetry $Sl(2,\mathbb{Z})\times \Gamma(2) \times \Gamma(2)$ in the three coordinates $z^S,z^i$ and
can be given in terms of modular functions of these groups.
Note that in addition to the chiral multiplets just discussed, the projected Enriques theory
also admits $h^{(2,1)}_+=8$, $\mathcal{N}=1$ vector multiplets $A_{a}$. The gauge-kinetic coupling
function has to be holomorphic and is simply given by
\begin{equation}
f_{a b}(z) = -i C_{a b} z^S \ .
\end{equation}
The kinetic term for $A_{a}$ has coupling matrix $\frac12 \text{Re} (f_{a b})=\frac12 C_{a b} \text{Im} z^S$ and
is indeed positive definite for $ \text{Im} z^S >0$.
Let us now turn to the discussion of the K\"ahler moduli space $\tilde \mathcal M_{\rm q}$ inside the
quaternionic moduli space $\mathcal M_{\rm q}$. In \eqref{JBexpansion} and \eqref{CCexpansion} we already
specified the orientifold invariant expansions of the K\"ahler form $J$, the NS-NS two-form
$B_2$ and the R-R forms $C_2,C_4$.
In the basis introduced in \eqref{omega_pm} we can summarize these expansions as
\begin{equation}
J = v^S \omega_S + v^i \omega_i \ ,\qquad B_2 = b^a \omega_a\ ,\qquad C_2 = c^a \omega_a\ ,\qquad C_4 = \rho_S \tilde \omega^S+ \rho_i \tilde \omega^i\ ,
\end{equation}
where the basis $(\tilde \omega^S, \tilde \omega^i)$ of $H^4_+(Y_E,\mathbb{Z})$ is chosen to be dual to $(\omega_S,\omega_i)$.
The real scalar fields $v^a,\rho_a$ as well as $b^S,b^i,c^S,c^i$ have to vanish i.e.~are projected out by the orientifold.
The $\mathcal{N}=1$ coordinates on the K\"ahler manifold $\tilde \mathcal M_{\rm q}$ are obtained by
expanding the complex even form $\rho_c$ as in \eqref{def-rhoc}.
This implies that the coordinates $\tau,G^a$ are exactly as given in \eqref{def-tauG}.
The coordinates $T_{\alpha}=(T_S,T_i)$ take the same form as the large volume result
\eqref{def-T} due to the absence
of world-sheet instantons in the Enriques Calabi-Yau. Explicitly, one evaluates
\begin{eqnarray} \label{def-TE}
T_S &=&ie^{-\phi} v^1 v^2 - \tilde \rho_S+\frac{1}{2(\tau -\bar \tau)} C_{a b} G^a (G-\bar G)^b\ ,\\
T_i &=& \tfrac{1}{2}i e^{-\phi} v^S v^j- \rho_i\ ,\qquad \qquad i,j=1,2\ , \quad i\neq j\ , \nonumber
%
\eea
where $\tilde \rho_S=\rho_S -\tfrac{1}{2} C_{a b} c^a b^b$.
The $\mathcal{N}=1$ K\"ahler potential can be also deduced from our general
considerations in section \ref{orirev}. More precisely, one uses \eqref{def-TE} together with \eqref{Kqgeneral} or \eqref{Kqgen} to
evaluate
\begin{eqnarray} \label{def-KqE}
K_{\rm q} &=& - \ln\big[\tfrac{1}{4}i(T_1-\bar T_1) \big( 2 (T_S - \bar T_S) (\tau -\bar \tau) - C_{a b} (G-\bar G)^{a} (G-\bar G)^b \big]\nonumber \\
&&- \ln\big[-i(T_2 -\bar T_2)\big]\ .
\eea
This simple explicit form of $K_{\rm q}$ arises
due to the special form of the intersections \eqref{Enriques_int}
and the simple cubic pre-potential \eqref{def-Ft}.
Note that $K_{\rm q}$ is not corrected by $\mathcal{N}=2$
$\alpha'$ contributions, since these vanish identically
for the Enriques Calabi-Yau.
In particular, one notices that the perturbative $\alpha'$ corrections proportional to the
Euler characteristic $\chi(Y_E)$ vanish due to $\chi(Y_E)=2(h^{(1,1)}-h^{(2,1)})=0$.
We thus conclude that the $\mathcal{N}=1$ Enriques orientifold theory is particularly
well under control due to the simplicity of the underlying $\mathcal{N}=2$ theory.
The $\mathcal{N}=1$ moduli space $\tilde \mathcal M_{\rm q}$ is also a coset, which is evaluated to be
of the form
\begin{equation} \label{EcMq}
\tilde \mathcal M_{\rm q}
\ =\ Sl(2,\mathbb{R})/U(1)\ \times\ \Big( Sl(2,\mathbb{R})/U(1)\ \times\ O(10,2)/\big(O(10)\times O(2)\big) \Big)\ .
\end{equation}
Remarkably, we find that the original $\mathcal{N}=2$ special K\"ahler manifold $\mathcal M_{\rm ks}$ given in \eqref{csks}
arises as the second factor of $\tilde \mathcal M_{\rm q}$. Such a phenomenon was already
studied from a supergravity point of view in refs.~\cite{D'Auria:2004kx}. In the following we
will discuss this duality in more detail and make contact to the second parametrization
of $\mathcal M_{\rm ks}$ introduced in \eqref{def-cT}.
Let us now discuss the appearance of the factor $\mathcal M_{\rm ks}$ in \eqref{EcMq} in more
detail. Recall that we introduced in \eqref{def-cT} a special set of coordinates $S,\mathcal{T}^i,\mathcal{T}^a$
on $\mathcal M_{\rm ks}$. Imposing the orientifold constraints that in the large volume
coordinates we have $b^S=b^i=v^\alpha=0$ one shows that the $S, \mathcal{T}$ coordinates
truncate as
\begin{align} \label{dual_limit}
C\mathcal{T}^1\ & \rightarrow \ i e^{-\phi} \big(v^1 v^2 +\tfrac{1}{2} C_{ab} b^a b^b\big)\ , & C\mathcal{T}^2 \ &\rightarrow \ \tfrac12 i e^{-\phi}\ , \\
C\mathcal{T}^a \ & \rightarrow \ -i e^{-\phi} b^a\ , \qquad &CS \ & \rightarrow\ i v^2 v^S\ .\nonumber
\end{align}
In this evaluation $C$ was used in the gauge associated to the coordinates $\mathcal{T}^i,\mathcal{T}^a$.
It differs by a factor $2v^2$ from its large volume value $e^{-\phi}$ as imposed
by its transformation property \eqref{trans-Crho}.
We can now compare the orientifold truncations \eqref{dual_limit} with the definitions \eqref{def-tauG} and
\eqref{def-TE} of the $\mathcal{N}=1$ coordinates. The orientifold limit of the $CS,C\mathcal{T}^i,\mathcal{T}^a$ are
precisely the imaginary parts of $\tau,G^a,T_S, T_1$.
Viewing the $\mathcal{N}=1$ coordinates as analytic continuation
we can make the following
identifications
\begin{equation} \label{coord_id}
\mathcal{T}^1\ \rightarrow \ T_S\ , \qquad \quad \mathcal{T}^2 \ \rightarrow \ \tfrac12 \tau\ , \qquad \quad
\mathcal{T}^a \ \rightarrow \ G^a \ , \qquad \quad S \ \rightarrow\ 2 T_1\ .
\end{equation}
Using this map it is easy to check that also the $\mathcal{N}=1$ K\"ahler potential \eqref{def-KqE} for the
scalars $\tau,G^a,T_S, T_1$ can be identified with the K\"ahler potential $K_D$ on $\mathcal M_{\rm ks}$
given in \eqref{def-KD}. This clarifies the fact that the special K\"ahler manifold $\mathcal M_{\rm ks}$
arises as the second factor in the $\mathcal{N}=1$ moduli space \eqref{EcMq}.
In the next section we will discuss the holomorphic superpotential
and use the duality map \eqref{coord_id} to propose
explicit expression for $W$ arising from D3 instantons.
\subsection{The D-instanton superpotential \label{EnriquesW}}
In this section we propose a specific D-instanton superpotential for the
Enriques orientifold. Since our main focus is the dependence of
$W_{\text{D-inst}}$ on the moduli $\tau,G^a$ we will concentrate on the
contribution proportional to $e^{in T_S}$. As seen in \eqref{def-TE} only the complex
coordinate $T_S$ depends on the fields $G^a$ and hence shifts as discussed
in section \ref{generalsup}. The imaginary part of $T_S$ contains the volume
form of the Enriques fiber modded out by the orientifold involution $\sigma$.
If the corresponding four-cycle $\Sigma$ can be extended to the F-theory picture such that it
contributes to the D-instanton superpotential we expect a correction
of the form
\begin{equation} \label{Ins_potential}
W_{\text{D-inst}} = \sum_n \Theta_n (\tau,G^a) e^{in T_S}\ .
\end{equation}
In this expression we have also included multi-coverings of $\Sigma$ labeled by $n$. A priory
it is not clear that these will contribute and higher $\Theta_n$ might be zero.
We will now use our intuition from topological string theory on the
Enriques Calabi-Yau and conjecture a possible form of $W_{\text{D-inst}}$.
Recall that in section \ref{GeometryN=2} we introduced a specific function $\Phi_{\rm B}(\mathcal{T}^1,\mathcal{T}^2,\mathcal{T}^a)$
encoding the lowest order degeneracies of D3, D1, D(-1) bound states on the
Enriques Calabi-Yau. In such states, the D3 instanton wraps the Enriques fiber and
couples to the complex coordinate $\mathcal{T}^1$, while
the D1 branes wrap cycles in the $E_8$ lattice of the second cohomology and
couple to complex coordinate $\mathcal{T}^a$. The D(-1) couple to the complex
field $\mathcal{T}^2$ and appear in generic D3, D1, D(-1) bound states.
Note that these are also the states which can
appear in the instanton superpotential \eqref{Ins_potential}.
More precisely, using the map \eqref{coord_id} we identify the coordinates
$\mathcal{T}^2,\mathcal{T}^a$ with the orientifold coordinates $\tau,G^a$.
The fiber volume appears in $\mathcal{T}^1$ which is identified with $T_S$.
We now expand the function $\Phi_{\rm B}$ given in \eqref{Borcherds-Form}
in powers of $e^{inT_S}$ as
\begin{equation} \label{Phiexp}
\Phi_{\rm B}(T_S,\tfrac12 \tau,G^a)\ =\ \sum_n \theta_n(\tau,G^a)\, e^{inT_S} \ ,
\end{equation}
which defines the coefficients $\theta_n(\tau,G^a)$.
Our proposal is that the $G^a$ dependence of the D-instanton superpotential \eqref{Ins_potential}
arises through these functions $\theta_n(\tau,G)$.
In other words, the superpotential arising due to D3 instantons on the Enriques fiber should take the form
\begin{equation} \label{conjW}
W_{\text{D-inst}} = A_0 \sum_n \frac{c_n\ \theta_n(\tau,G^a)}{\eta^{10}(\tau)}\ e^{inT_S} \ ,
\end{equation}
where $\eta(\tau)$ is the standard eta-function and $c_n$ are appropriate numerical coefficients.
Unfortunately, without the complete F-theory picture we will not be able to check \eqref{conjW}
directly and details might change in an explicit analysis. However, making contact to the
discussion in section \ref{generalsup} we will discuss in the remainder of this section
that the $\theta_n$ have the correct properties to ensure that $W_{\text{D-inst}}$ is
a modular form of weight $-1$ in $\tau$. Moreover, also the
shifts of $T_S$ given in \eqref{tautransgen} and \eqref{Gshift} are appropriately canceled
by shifts of $\theta_n$ as needed for consistency.
Let us finish this section with some remarks on the properties of
the functions $\theta_n$ in \eqref{conjW}. These can be determined explicitly
by expanding the expression for $\Phi_{\rm B}$ in the product representation \eqref{Borcherds-Form}
or the corresponding sum representation \cite{borcherdsone,borcherds}. It was shown in ref.~\cite{borcherds}
that $\Phi_{\rm B}$ is an automorphic form of weight $4$. Following the arguments of \cite{EZ,binfinite}
one deduces that the coefficient functions $\theta_n$ are Jacobi forms of weight $4$ and index $n$, i.e.~transform
as given in \eqref{transtheta1} and \eqref{transtheta2} under modular transformations and B-shifts.
In fact, in ref.~\cite{binfinite} automorphic
forms similar to $\Phi_{\rm B}$ were constructed by combining
appropriate Jacobi forms with the exponential $e^{in T_S}$. The precise form
of $\theta_n$ is then determined by a lift of the modular coefficient functions
such as \eqref{coeff_funct}.
Instead of giving the explicit expressions for $\theta_n(\tau,G)$ we indirectly
check some of their properties through a differential equation which they obey.
In order to do that, we note that $\Phi_{\rm B}(\mathcal{T})$ satisfies a wave equation of the form \cite{borcherdsone,borcherds}\footnote{This is far from obvious in the product representation of $\Phi_{\rm B}$, but can be easily checked
when writing $\Phi_{\rm B}$ as a sum \cite{borcherdsone,borcherds}.}
\begin{equation}
2 \frac{\partial^2 \Phi_{\rm B}}{\partial \mathcal{T}^1 \partial \mathcal{T}^2} -
C_D^{a b} \frac{\partial^2 \Phi_{\rm B} }{\partial \mathcal{T}^a \partial \mathcal{T}^b} =0\ .
\end{equation}
This equation is readily translated into a condition on the functions $ \theta_n (\tau,G) $ in \eqref{conjW}.
One finds
\begin{equation}
\Big(i n \frac{\partial}{\partial \tau} - \tfrac{1}{2} C^{a b} \frac{\partial^2}{\partial G^a \partial G^b}\Big) \theta_n (\tau,G) = 0\ ,
\end{equation}
which is the higher-dimensional analog of the heat equation for theta-functions
on an appropriate lattice. It also indicates that $\theta_n(\tau,G)$ are Jacobi forms
as expected from the general discussion above.
Since $\Phi_{\rm B}$ and hence $\theta_n(\tau,G)$ are of weight $4$ we conclude that the inclusion
of the $\eta^{10}(\tau)$ factor ensures that $W_{\text{D-inst}}$ is of weight
$-1$ as needed for \eqref{transWundertau}.
To actually show that $\theta_n(\tau,G)$ and $\eta(\tau)$ appear in the correct
way in the conjectured superpotential \eqref{conjW} one might calculate
$W_{\text{D-inst}}$ in a specific limit. In particular, it would be interesting to derive $W_{\text{D-inst}}$
in the orbifold limit using its heterotic dual.
\section{Conclusions}
In this paper we discussed the symmetries and non-perturbative
corrections of the four-dimensional effective theory
arising in type IIB orientifolds with O3 and O7 planes.
We studied both the K\"ahler potential and
superpotential in the orientifold large volume limit for general $\mathcal{N}=1$ compactifications
and later concentrated on a specific orientifold of the Enriques Calabi-Yau.
In our general analysis we first discussed the $\mathcal{N}=1$ K\"ahler
potential including perturbative and non-perturbative $\alpha'$ corrections
inherited from the underlying $\mathcal{N}=2$ theory. A subset of the non-perturbative
$\alpha'$ corrections were shown to survive the
orientifold large volume limit, since they depend on the
scalars $G^a$ arising from the NS-NS and R-R two-forms. They
contribute to the K\"ahler potential in an explicitly calculable way, but do not
alter the $\mathcal{N}=1$ chiral coordinates.
It was argued that in order to
ensure duality invariance of the $\alpha'$ corrections to the
K\"ahler potential also contribution due to D(-1) and D1 branes have to
be taken into account. In general, it seems hard to determine these
corrections directly. We thus restrained ourselves to a brief discussion of
candidate modular completions proposed for the underlying $\mathcal{N}=2$
theory. It would be interesting to derive these corrections explicitly
by using heterotic-F-theory duality or be analyzing specific orbifold
examples. Already the inclusion of the $\alpha'$ corrections will lead
to interesting new phenomenological properties of these compactifications
and a study of explicit examples is desirable.
From a phenomenological point of view the two-form scalars $G^a$
have to be rendered massive in a vacuum.
We have shown that this can be achieved by a potential induced by
D3 instantons. More precisely, we have used the symmetries
of the orientifold theory to argue that the two-form scalars arise through
Jacobi forms in front of the D3 instanton contribution $e^{i n T}$ in the superpotential.
These are generalizations of the well known theta-functions and depend
on the dilaton-axion $\tau$ as modular parameter. Due to holomorphicity
and modular invariance one might hope that the set of candidate
Jacobi forms can be restricted to a finite set for a given example. Candidate
forms should appear in topological string theory on the underlying
Calabi-Yau manifold counting degeneracies of D1, D(-1) states on cycles which
become singular in the orientifold background. Additional boundary conditions
obtained in computations performed in specific limits of the theory might then
fix the precise form of the D-instanton superpotential.
In the finial part of the paper we studied a specific example. We considered an
orientifold of the Enriques Calabi-Yau. The kinetic terms of the
four-dimensional $\mathcal{N}=1$ effective theory are determined in terms
of a simple K\"ahler potential. We showed that the corresponding
moduli of bulk moduli fields is a product of cosets. Interestingly, the reduction
of the underlying quaternionic $\mathcal{N}=2$ geometry led to a K\"ahler manifold which can be identified
with the original deformation space of the complexified K\"ahler structure of the
underlying Calabi-Yau manifold times an $Sl(2,\mathbb{R})/U(1)$ factor. This duality
can be used in the study of the D-instanton superpotential on the Enriques Calabi-Yau.
We mapped Jacobi forms known from topological string theory
on the Enriques Calabi-Yau to the corresponding $\mathcal{N}=1$ orientifold. This lead to
a conjecture of a specific D3-instanton superpotential. Unfortunately, explicit
tests of this proposal are still missing and would involve a careful construction of
an F-theory realization of the Enriques scenario. It would be also interesting to
investigate other examples. Particularly, other K3 fibrations might allow to investigate
similar questions, which can then be tested using string-string dualities.
\section*{Acknowledgments}
I would like to thank Ian Ellwood, Jan Louis, Peter Mayr, Marcos Mari\~no, Liam McAllister,
Frank Saueressig, Sav Sethi and Gary Shiu for helpful
comments and discussions. I am particularly grateful to Albrecht Klemm for many
illuminating discussions and remarks on the draft.
This work was supported in part by NSF CAREER Award No. PHY-0348093,
DOE grant DE-FG-02-95ER40896, a Research Innovation Award and a Cottrell
Scholar Award from Research Corporation.
\bigskip \bigskip
|
1,314,259,996,742 | arxiv | \section{Introduction}
For an $r$-uniform hypergraph $\mathcal H$, the \emph{Tur\'an number} of $\mathcal H$, denoted $\ex(n, \mathcal H)$, is defined as the maximum number of edges an $n$-vertex $r$-uniform hypergraph can have without containing a copy of $\mathcal H$ as a subgraph. For ($2$-uniform) graphs, we have a fairly good understanding of Tur\'an numbers.
The first theorem proved about them is Mantel's theorem \cite{mantel1907problem}, which says that, for the triangle, we have $\ex(n, K_3)= \floor{n^2/4}$. This was generalised by Tur\'an \cite{turan1941extremal} who showed that $\ex(n,K_r) \approx (1-\frac{1}{r-1})\binom{n}{2}$.
For non-complete graphs we know less, and usually only know what the Tur\'an number of a graph is asymptotically, up to $o(n^2)$ terms.
Because of this, we study the \emph{Tur\'an density} of an $r$-uniform hypergraph $\mathcal H$, denoted $\pi(\mathcal{H})$, and defined as $\pi(\mathcal H)=\lim_{n\to \infty} \frac{\ex(n,\mathcal H)}{\binom n r}$. This limit is known to exist, and, moreover, it is clear that $\pi(\mathcal H)\in [0,1]$ for every $H$.
The Tur\'an densities of ($2$-uniform) graphs were completely determined by Erd\H{o}s and Stone~\cite{erdos1946structure}, who showed that every graph $H$ satisfies $\pi(H)=1-\frac{1}{\chi(H)-1}$.
The special case of the Erd\H{o}s--Stone theorem for bipartite graphs can be generalised to higher uniformities, as follows (see \cite{erdos1971extremal}): every $r$-partite $r$-uniform hypergraph $\mathcal{H}$ satisfies $\pi(\mathcal{H}) = 0$. (An $r$-uniform hypergraph $\mathcal{H}$ is said to be \emph{$r$-partite} if its vertices can be $r$-coloured so that every edge has one vertex of each colour.)
Nevertheless, in general, our understanding of Tur\'an numbers in higher uniformities is very limited, and there is only a small number of hypergraphs whose Tur\'an densities are known; see Keevash \cite{keevash11} for a comprehensive survey of the topic listing a number of such hypergraphs.
A notable, relatively early example is a result of de Caen and F\"uredi \cite{de2000maximum} showing that the Tur\'an density of the Fano plane is $3/4$ (see also \cite{furedi2005triple, keevash2005turan}). More recently, the impactful computer-assisted ``flag-algebra'' technique has been used to obtain a number of sharpest known upper bounds on Tur\'an densities (see \cite{baber2011new, keevash11, razborov2013flag} and the references therein).
Given the sporadicity of hypergraphs whose Tur\'an densities are known, it is unsurprising that there are many conjectures about Tur\'an densities of specific hypergraphs. The most famous of these is Tur\'an's conjecture \cite{turan1961research}, that the Tur\'an density of the \textit{tetrahedron} $K_4^{(3)}$ is $5/9$. Frankl and F\"uredi \cite{ff84} conjectured that the the Tur\'an density of the 3-edge subgraph of $K_4^{(3)}$ (usually denoted $K_4^-$) is $2/7$.
A particularly relevant conjecture for us concerns tight cycles. The \emph{$r$-uniform tight cycle} of length $\ell$, denoted $\mathcal C_{\ell}^r$, is defined to be the hypergraph with vertex set $\{1, \dots, \ell\}$ and hyperedges all sets of the form $\{x, x+1,\dots, x+r-1 \!\pmod {\ell}\}$. The following conjecture, usually attributed to Mubayi and R\"odl, appears for instance in \cite{falgas2012turan,mubayi2011hypergraph}.
\begin{conjecture}
\label{Conjecture_Mubayi_Rodl}
$\pi(\mathcal C_5^3)=2\sqrt 3-3$.
\end{conjecture}
The lower bound $\pi(\mathcal C_5^3) \geq 2\sqrt 3-3 \approx 0.464$ was found by Mubayi and R\"odl (see \Cref{ex:iterated-blow-up} below for a description of their example) and the best upper bound is due to Razborov~\cite{razborov20103}, who showed $\pi(\mathcal C_5^3)\leq 0.468$.
One basic reason why hypergraphs are more difficult than graphs is that the extremal $\mathcal H$-free hypergraphs can be much more complicated than the extremal graphs. In the 2-uniform case, the Erd\H{o}s--Stone theorem shows that all optimal graphs are close to being complete multipartite. For higher-uniformity hypergraphs there have been numerous papers discovering more complicated possible extremal hypergraphs, for instance~\cite{brown83,ff84,blm11}, as well as Conjectures~\ref{Conjecture_Mubayi_Rodl} and~\ref{conj:c5-minus}. For some hypergraphs, such as $K_4^{(3)}$, the conjectured extremal constructions are even non-unique and very different from each other~\cite{brown83,kostochka84,flaass88,razborov11}.
One class of extremal examples, which does not occur for graphs, is an ``iterated blow-up construction''. The conjectured extremal example for \Cref{Conjecture_Mubayi_Rodl} is an instance of such a construction.
\begin{example}[Iterated blow-up construction with no copies of $\mathcal{C}_{5}^3$] \label{ex:iterated-blow-up}
Consider nested vertex sets $V_1\supseteq \ldots \supseteq V_t$ with $|V_i|=x_i$. Let $\mathcal H(x_1, \ldots, x_t)$ be a 3-uniform hypergraph on the vertex set $V_1$, where $xyz$ is an edge whenever $x,y\in V_i$ and $z\in V_{i+1}$ for some $i$ (see Figure~\ref{Figure_extremal_hypergraph}).
We claim that there is no copy of $\mathcal{C}_5^3$. To see this, say that an edge with two vertices in $V_i \setminus V_{i+1}$ and one vertex in $V_{i+1}$ has \emph{type $i$}, and observe that if two edges $e$ and $f$ intersect in two vertices, they are of the same type. Thus, if $C = (u_1 \ldots u_5)$ is a cycle, then its edges all have the same type, say $i$. Without loss of generality, $u_1 \in V_i \setminus V_{i+1}$ and $u_2, u_3 \in V_{i+1}$. It follows that $u_4 \in V_i \setminus V_{i+1}$, and thus $u_5 \in V_{i+1}$. But then $u_4 u_5 u_1$ is not an edge of $\mathcal{H}(x_1, \ldots, x_t)$, a contradiction.
Thus $\pi(\mathcal C_5^3)\geq e(\mathcal H(x_1, \ldots, x_t))/\binom n3$ for all choices of $x_1, \ldots, x_t$. Let $f(n)$ denote the maximum number of edges that such a hypergraph on $n$ vertices can have i.e $f(n):=\max(e(\mathcal H(x_1, \ldots, x_t):x_t\leq\dots\leq x_1=n)$.
It is possible to show $\lim_{n\to \infty} f(n)/\binom n3=2\sqrt 3-3$ (see Section~\ref{sec:optimal} for details), which gives $\pi(\mathcal C_5^3)\geq 2\sqrt 3-3$.
Let $\mathcal{G}_n = \mathcal{H}(x_1, \ldots, x_t)$ for a choice of $n = x_1 \ge \ldots \ge x_t$ such that $e(\mathcal{G}_n) = f(n)$.
\end{example}
\begin{figure}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{center}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [draw opacity=0][fill={rgb, 255:red, 180; green, 180; blue, 180 } ,fill opacity=1 ] (407.01,155.57) -- (266,182.14) -- (266,129) -- cycle ;
\draw (100,53.15) .. controls (100,40.99) and (109.85,31.14) .. (122.01,31.14) -- (463,31.14) .. controls (475.15,31.14) and (485.01,40.99) .. (485.01,53.15) -- (485.01,246.14) .. controls (485.01,258.29) and (475.15,268.14) .. (463,268.14) -- (122.01,268.14) .. controls (109.85,268.14) and (100,258.29) .. (100,246.14) -- cycle ;
\draw (228,58.94) .. controls (228,47.45) and (237.31,38.14) .. (248.8,38.14) -- (453.21,38.14) .. controls (464.69,38.14) and (474.01,47.45) .. (474.01,58.94) -- (474.01,241.34) .. controls (474.01,252.83) and (464.69,262.14) .. (453.21,262.14) -- (248.8,262.14) .. controls (237.31,262.14) and (228,252.83) .. (228,241.34) -- cycle ;
\draw (310.01,59.81) .. controls (310.01,51.71) and (316.57,45.14) .. (324.68,45.14) -- (453.34,45.14) .. controls (461.44,45.14) and (468.01,51.71) .. (468.01,59.81) -- (468.01,240.47) .. controls (468.01,248.57) and (461.44,255.14) .. (453.34,255.14) -- (324.68,255.14) .. controls (316.57,255.14) and (310.01,248.57) .. (310.01,240.47) -- cycle ;
\draw (382.01,58.29) .. controls (382.01,54.34) and (385.21,51.14) .. (389.15,51.14) -- (451.86,51.14) .. controls (455.8,51.14) and (459.01,54.34) .. (459.01,58.29) -- (459.01,229.99) .. controls (459.01,233.94) and (455.8,237.14) .. (451.86,237.14) -- (389.15,237.14) .. controls (385.21,237.14) and (382.01,233.94) .. (382.01,229.99) -- cycle ;
\draw [draw opacity=0][fill={rgb, 255:red, 176; green, 176; blue, 176 } ,fill opacity=1 ] (405.01,94.82) -- (180.5,134.14) -- (180.5,55.5) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (343.01,155.07) -- (248,181.14) -- (248,129) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (405.01,206.07) -- (363,225.14) -- (363,187) -- cycle ;
\draw [draw opacity=0][fill={rgb, 255:red, 61; green, 61; blue, 61 } ,fill opacity=1 ] (333.01,93.07) -- (154,131.14) -- (154,55) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (263.01,91.57) -- (137,129.14) -- (137,54) -- cycle ;
\draw (131,193.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$V_{1}$};
\draw (251,194.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$V_{2}$};
\draw (318,197.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$V_{3}$};
\draw (419,195.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$V_{4}$};
\end{tikzpicture}
\end{center}
\caption{An illustration of the hypergraph $\mathcal H(x_1, x_2, x_3, x_4)$. }\label{Figure_extremal_hypergraph}
\end{figure}
Note that in the above construction, $\mathcal G_n$ has no tight cycles of lengths $\ell\equiv 1$ or $2 \!\pmod 3$ either. So it is plausible that Conjecture~\ref{Conjecture_Mubayi_Rodl} could be strengthened to say that $\pi(\mathcal C_\ell^3)=2\sqrt 3-3$ for all $\ell\geq 5$ with $\ell\equiv 1$ or $2 \pmod 3$ (notice that $\mathcal{C}_4^3 = K_4^{(3)}$, and there are known examples of $K_4^{(3)}$-free $3$-uniform graphs with density at least $5/9 > 2\sqrt{3} - 3$). The main result of our paper is to show that this is true for sufficiently large $\ell$.
\begin{restatable}{theorem}{thmSingleCycle}\label{thm:single-cycle}
Let $\ell$ be sufficiently large with $\ell\equiv 1$ or $2 \!\pmod 3$. Then $\pi(\mathcal C_\ell^3)=2\sqrt 3-3$.
\end{restatable}
To our knowledge, this is the first example of a Tur\'an density being determined where the extremal construction is an iterated blow-up construction, and could be a step towards Conjecture~\ref{Conjecture_Mubayi_Rodl}.
This is also one of the few examples of hypergraphs with irrational Tur\'an densities. Such hypergraphs were recently found by Yan and Peng~\cite{yp22}, as well as Wu~\cite{wu22}, motivated by the work of Chung and Graham~\cite{cg98}, Baber and Talbot~\cite{baber2011new}, and Pikhurko~\cite{pikhurko2014possible}.
We remark that Conjecure~\ref{Conjecture_Mubayi_Rodl} would imply Theorem~\ref{thm:single-cycle} via \Cref{theorem_blow_up} below (using the same argument as in the proof of Theorem~\ref{thm:single-cycle} in Section~\ref{sec:diameter}).
One of our main tools, which may be of independent interest, is a 3-uniform analogue of the statement ``a graph is bipartite if and only if it does not contain an odd cycle''; see \Cref{thm:good-colouring}. Thus, we characterise 3-uniform hypergraphs $\mathcal{H}$ which do not contain \textit{homomorphic images} of cycles $\mathcal{C}^{3}_\ell$ with $3 \nmid \ell$, in terms of certain colourings of $V(\mathcal{H})^2$, as explained in the proof overview.
Throughout the paper we will informally refer to 3-uniform cycles of length $\ell\equiv 1$ or $2 \pmod 3$ as ``odd cycles'', and we will often refer to $3$-uniform hypergraphs as 3-graphs.
\subsection*{Related results}
As we mentioned, there are very few hypergraphs with a known Tur\'an density, but let us state some recent results on Tur\'an-type problems for tight cycles.
A well-studied hypergraph parameter is the so-called \emph{uniform Tur\'an density}, the infimum over all $d$ for which any
sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at
least $d$ contains $\mathcal{H}$. This line of research was initiated by Erd\H{o}s and S\'os
\cite{erdHos1982ramsey} and, parallel to the classical Tur\'an densities, the motivating questions in the area are determining the uniform Tur\'an densities of the tetrahedron $K_4^{(3)}$ and its 3-edge subgraph $K_4^-$. The latter was found to be $1/4$ by Glebov, Kr\'a\v{l}, and Volec~\cite{gkv16} and later by Reiher, R\"odl, and Schacht~\cite{rrs18} with a different proof. In 2022, Buci\'c, Cooper, Kr{\'a}\v{l}, Mohr, and Munha Correia showed that for $\ell \geq 5$ and not divisible by 3, the uniform Tur\'an density of $\mathcal{C}_{\ell}^3$ is $\frac{4}{27}$~\cite{bckmm21}.
Another question that has attracted a lot of interest in the last few years is this: what is the extremal number of tight cycles (the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no tight cycles)? For $r=2$, the answer is of course $n-1$, but it turns out that the behaviour is rather different for $r\geq 3$. More specifically, after a series of results~\cite{hm19,janzer21,st22,letzter2021hypergraphs}, we know that the extremal number of tight $r$-uniform cycles lies between $\Omega\left(n^{r-1} \log n / \log \log n\right)$ and $O\left(n^{r-1} \log^5 n \right)$.
\section{Proof overview} \label{sec:overview}
For an $r$-uniform hypergraph $\mathcal{H}$, the \emph{$t$-blow-up} of $\mathcal{H}$, denoted $\mathcal{H}[t]$, is defined to be the $r$-uniform hypergraph with vertex set $V(\mathcal{H})\times [t]$ and edges all $r$-tuples $\{(x_1, i_1), \dots, (x_r, i_r)\}$ with $\{x_1, \dots, x_r\}\in E(\mathcal{H})$.
The starting point of our proof is the following theorem, which asserts that the blow-up of a hypergraph $\mathcal{H}$ has the same Tur\'an density as $\mathcal{H}$.
\begin{theorem}[\cite{keevash11}, Theorem 2.2] \label{theorem_blow_up}
Let $t$ be an integer and let $\mathcal{H}$ be an $r$-uniform hypergraph. Then $\pi(\mathcal{H}[t]) = \pi(\mathcal{H})$.
\end{theorem}
It shows that, rather than focusing on the Tur\'an density $\pi(\mathcal C_k^3)$ for an odd cycle $C_k^3$, we can instead work out the Tur\'an density of $\pi(\mathcal{H})$ for any hypergraph $\mathcal{H}$ whose blow-up $\mathcal{H}[t]$ contains $\mathcal C_k^3$ for some $t$. We refer to such hypergraphs $\mathcal{H}$ as pseudocycles, and they can be equivalently defined as follows.
\begin{definition} \label{def:pseudocycle}
A \emph{pseudocycle} of length $\ell$ in a 3-uniform hypergraph $\mathcal{H}$ is a sequence of (not necessarily distinct) vertices $v_1, \ldots, v_{\ell}$, such that for each $i \in [\ell]$, we have that $\{v_i, v_{i+1\!\pmod{\ell}}, v_{i+2\!\pmod{\ell}}\}$ is an edge of $\mathcal{H}$.
A \emph{pseudopath} of order $\ell$ is defined analogously.
\end{definition}
It is easy to show that for a hypergraph $\mathcal{H}$, the properties ``$\mathcal{H}[t]$ contains a $\mathcal C_k^3$ for some $t$'' and ``$\mathcal{H}$ contains a length $k$ pseudocycle'' are equivalent.
Thus, the next question is --- what is the maximum number of edges that a 3-uniform hypergraph can have without containing an odd pseudocycle? To understand our approach to this, consider the analogous question about graphs --- what is the maximum number of edges in a (2-uniform) graph with no odd circuits? This is easy to answer using Kotzig's Lemma --- a graph has no odd circuit if, and only if, it is bipartite. The maximum number of edges in an $n$-vertex bipartite graph is $\floor{\frac{n^2}{4}}$.
Our approach to the 3-uniform case is analogous to this. We first find the relevant generalisation of bipartite graphs, and then maximise the number of edges over this class of graphs. To define this generalisation, recall
that a graph is bipartite if, and only if, it has a proper 2-vertex-colouring. In our context, we will be colouring the shadow of a 3-uniform hypergraph. The \emph{shadow} of a hypergraph $\mathcal{H}$, denoted $\partial \mathcal{H}$, is the graph on vertices $V(\mathcal{H})$ whose edges are pairs $xy$ that are contained in an edge in $\mathcal{H}$.
\begin{definition} \label{def:good-col}
A \emph{good colouring} of a 3-uniform hypergraph $\mathcal{H}$ is a colouring of its shadow, such that each edge $xy$ in the shadow is either coloured blue or coloured red and given an orientation, and every edge $e$ in $\mathcal{H}$ can be written as $xyz$ where $xy$ and $xz$ are red and directed from $x$ and $yz$ is blue.
\end{definition}
The key first step of our proof is to show that the notion of ``good colouring'' is exactly equivalent to $\mathcal{H}$ not containing an odd pseudocycle.
\begin{restatable}{theorem}{thmGoodColouring}\label{thm:good-colouring}
A 3-uniform hypergraph $\mathcal{H}$ has a good colouring if, and only if, $\mathcal{H}$ has no pseudocycle of length $\ell$ with $3 \nmid \ell$.
\end{restatable}
\def m_{\mathrm{cherry}} {m_{\mathrm{cherry}}}
\def m_{\mathrm{good}} {m_{\mathrm{good}}}
This theorem is proved in Section~\ref{sec:good-col}.
Having established the above theorem, we next wish to maximise the number of edges in a hypergraph with a good colouring. To this end, we define a \emph{coloured graph} to be a complete graph whose edges are either coloured blue or coloured red and oriented.
A \emph{cherry} in a coloured graph $G$ is a triple $xyz$ such that $xy$ and $xz$ are red and directed from $x$ and $yz$ is blue. Denote by $c(G)$ the number of cherries in $G$.
Notice that if we have a good colouring of the shadow of $\mathcal{H}$ (and the remaining vertex pairs can be coloured arbitrarily), then all edges of $\mathcal{H}$ will be cherries in the resulting coloured graph. Thus, let $m_{\mathrm{cherry}}(n)$ be the maximum number of cherries in an $n$-vertex coloured graph. The quantity $m_{\mathrm{cherry}}(n)$ has been studied before by Falgas-Ravry and Vaughan \cite{falgas2012turan}, who used flag algebras to show that $\lim_{n\to \infty} m_{\mathrm{cherry}}/\binom n3=2\sqrt 3-3$. Huang \cite{huang2014maximum} worked on the area further and determined the maximum number of ``induced out-stars'' of size $t$ in an $n$-vertex coloured graph. The following special case of their results is relevant for us.
\begin{theorem}[Falgas-Ravry--Vaughan~\cite{falgas2012turan}; Huang~\cite{huang2014maximum}]
\label{theorem_intro_falgas_ravry}
Every coloured graph on $n$ vertices contains at most $f(n)$ cherries.
\end{theorem}
Combining Theorem~\ref{thm:good-colouring} and \Cref{theorem_intro_falgas_ravry} already yields the following weakening of Theorem~\ref{thm:single-cycle}.
\begin{restatable}{corollary}{corLongPseudocylces} \label{corollary_intro_long_pseudocycles}
If $\mathcal{H}$ is a $3$-uniform hypergraph on $n$ vertices which contains no pseudocycles of order $\ell$ with $3 \nmid \ell$, then $e(\mathcal{H}) \le f(n)$.
\end{restatable}
Thus, the next goal is to prove a version of the above corollary which holds when forbidding \emph{short pseudocycles}. This is done by controlling the diameter of the hypergraph $\mathcal{H}$.
\begin{definition}
The \emph{diameter} of a hypergraph $\mathcal{H}$ is the minimum $\ell$ such that the following holds: for every $x, y, z, w \in V(\mathcal{H})$ (where $x, y$ are distinct and $z, w$ are distinct) whenever there is a pseudopath from $xy$ to $zw$, there is such a pseudopath of order at most $\ell$.
\end{definition}
In Section~\ref{sec:diameter}, we show that, for $\ell\gg \epsilon^{-1}$, every 3-uniform hypergraph $\mathcal{H}$ contains a subhypergraph $\mathcal{H}'$ with $e(\mathcal{H}')\geq e(\mathcal{H})-\epsilon n^3$ such that $\mathcal{H}'$ has diameter at most $\ell$ (see Proposition~\ref{prop:small-diameter}). Then we show that in every 3-uniform hypergraph of diameter $\ell$, if there is some odd pseudocycle, then there is also an odd pseudocycle of length at most $4\ell$ (see Proposition~\ref{prop:diam-cyc}). Combining these with Corollary~\ref{corollary_intro_long_pseudocycles} shows the following
\begin{restatable}{corollary}{corWeak} \label{cor:weak-main}
Let $1/n \ll 1/\ell \ll \epsilon \ll 1$, and let $\mathcal{H}$ be an $n$-vertex hypergraph with no odd pseudocycles of length at most $\ell$. Then $e(\mathcal{H}) \le f(n) + \epsilon n^3$.
\end{restatable}
Note that this is still not strong enough to combine with Theorem~\ref{theorem_blow_up} to yield Theorem~\ref{thm:single-cycle}. The issue is that the length of the cycle $\ell$ depends on $\epsilon$ --- therefore, when combined with Theorem~\ref{theorem_blow_up}, we would only get that $\lim_{m\to \infty}\pi(\mathcal C_m^3)=2\sqrt 3-3$. To go further, we prove a ``stability version'' of Theorem~\ref{theorem_intro_falgas_ravry}. We show that if a coloured graph $D$ on $n$ vertices contains more than $f(n)-\epsilon n^3$ cherries, then $D$ must have a very constrained structure similar to the iterated blow-up construction (see Theorem~\ref{thm:stability} for the precise statement). Once we have this, we can obtain the following strengthening of Corollary~\ref{cor:weak-main}
\begin{restatable}{theorem}{thmPseudocycle} \label{thm:pseudocycles}
There exists $L > 0$ such that the following holds. If $\mathcal{H}$ is a $3$-uniform hypergraph on $n$ vertices which contains no pseudocycles of length $\ell$, with $\ell \leq L$ and $3 \nmid \ell$, then $e(\mathcal{H}) \le f(n) + O(1)$.
\end{restatable}
This theorem easily combines with Theorem~\ref{theorem_blow_up} in order to give our main result, Theorem~\ref{thm:single-cycle} (see \Cref{sec:diameter}).
\section{Finding a good colouring} \label{sec:good-col}
Recall that a \emph{pseudocycle} of order $m$ (or \emph{$m$-pseudocycle}) is a sequence $v_1 \ldots v_m$ such that $v_i v_{i+1} v_{i+2}$ is an edge for $i \in [m]$ (indices taken mod $3$). A \emph{pseudopath} of order $m$ is defined analogously.
A hypergraph is called \textit{tightly connected} if there is a pseudopath between any two edges. Given vertices $x, y, z, w$ (not necessarily distinct), a pseudopath from $xy$ to $zw$ (where $xy$ and $zw$ are ordered pairs) is a pseudopath whose first two vertices are $x$ and $y$ (in this order) and the last two vertices are $z$ and $w$.
The \emph{shadow} of a hypergraph $\mathcal{H}$, denoted $\partial \mathcal{H}$, is the graph on vertices $V(\mathcal{H})$ whose edges are pairs $xy$ that are contained in an edge in $\mathcal{H}$.
Recall that a \emph{good colouring} of a hypergraph $\mathcal{H}$ is a colouring of its shadow, such that each edge $xy$ in the shadow is either coloured blue or coloured red and given an orientation, and every edge $e$ in $\mathcal{H}$ can be written as $xyz$ where $xy$ and $xz$ are red and directed from $x$ and $yz$ is blue. Such an edge is called a \emph{cherry}, and the the vertex $x$ is called its \emph{apex}.
\begin{figure}[h!]
\centering
\includegraphics[scale = 1]{cherry.pdf}
\vspace{-.3cm}
\caption{A cherry $xyz$ with apex $x$}
\end{figure}
In this section, we will prove \Cref{thm:good-colouring}, restated here.
\thmGoodColouring*
It is easy to see that a hypergraph with a good colouring has no psedocycles of length $\ell$ with $3 \nmid \ell$, so the main effort will be put into proving the ``if'' direction.
Namely, we need to show that every hypergraph with no odd pseudocycles has a good colouring. Before specifying such a colouring, let us give some intuition. Any proper path (that is, with no repetitions) $v_1 \ldots v_k$ has a good colouring, and this colouring is unique given the colour of $v_1 v_2$ (see \Cref{fig:path} for the three good colourings of a path of order 9, and notice that each such colouring colours each edge in the shadow differently).
\begin{figure}[h!]
\centering
\includegraphics[scale = .9]{path1.pdf}
\hspace{.4cm}
\includegraphics[scale = .9]{path2.pdf}
\hspace{.4cm}
\includegraphics[scale = .9]{path3.pdf}
\caption{The three good colourings of a path of order $9$}
\label{fig:path}
\end{figure}
A proper cycle has a good colouring if and only if it is tripartite (i.e.~the number of vertices is divisible by 3). See \Cref{fig:cycle} for a good colouring of a cycle of length 18 and notice that every third vertex is an apex.
\begin{figure}[h!]
\centering
\includegraphics[scale = 1]{cycle.pdf}
\caption{A good colouring of a cycle whose length is divisible by $3$}
\label{fig:cycle}
\end{figure}
Moreover, if there is a path $P = xy \ldots yx$, then the order of $P$ uniquely determines the colour of $xy$. This fact will be used to construct a good colouring of our hypergraph $\mathcal{H}$ -- we will start from a specific pair $xy$ and extend the colouring uniquely along pseudopaths. The difficulty is to show that this colouring is well defined, so the actual colouring definition will involve some more formalism.
For a pseudopath $P = v_1 \ldots v_k$, define $\tilde{P}$ by
\begin{equation} \label{eqn:tilde}
\tilde{P} := v_{k-1} v_k v_{k-2} v_{k-1} v_{k-3} \dots v_4 v_2 v_3 v_1 v_2;
\end{equation}
note that $\tilde{P}$ is a pseudopath from $v_{k-1}v_k$ to $v_1 v_2$ of order $2k-2$ (because every vertex but $v_1$ and $v_k$ appears twice).
\begin{proof}[Proof of \Cref{thm:good-colouring}]
Whenever we talk about a path or cycle in this proof, we mean a pseudopath or pseudocycle.
As we said above, it is easy to show that a hypergraph with a good colouring has no odd cycles, so it suffices to show that if $\mathcal{H}$ has no odd cycles then $\mathcal{H}$ has a good colouring.
Note that we may assume that $\mathcal{H}$ is tightly connected.
Let $P_0$ be a shortest path with the property that its first two vertices are the same as the last two but in reversed order, if such a path exists. Write $P_0 := v_0 \ldots v_k$ and denote $x := v_0 = v_k$ and $y := v_1 = v_{k-1}$.
Define $\sigma$ as follows.
\begin{equation} \label{eq:sigma-def}
\sigma \modthree{2k}.
\end{equation}
Intuitively, $\sigma$ is defined so that if $P_0$ has a good colouring, then the apexes in this colouring are at index $\sigma \pmod{3}$.
If $P_0$ does not exist, define $\sigma = 2$.
Let $\{z,w\}$ be an edge in the shadow of $\mathcal{H}$, and let $P = xy v_2 v_3 \ldots v_k$ be a path from $xy$ whose last three vertices contain $z$ and $w$. Let $i_w \in \{k-2,k-1, k\}$ be the index of $w$ (namely, $v_{i_w} = w$), and define $i_z$ analogously. Define the index
$$\eta(P, \{z,w\}) = \begin{cases}
w, & i_w \modthree{\sigma}, \\
z, &i_z \modthree{\sigma},\\
*, &\text{otherwise}.
\end{cases}$$
In particular, this defines $\eta(xy, \{x, y\})$.
We claim that $\eta(P, \{z,w\})$ is independent of the choice of the path $P$.
\begin{claim} \label{claim:consistent}
Let $z, w \in V(\mathcal{H})$ be distinct.
Let $P = v_0 \ldots v_p$ and $Q = u_0 \ldots u_q$ be two paths
starting at $xy$ such that $z$ and $w$ are among their last three vertices. Then
$$\eta(P, \{z, w\}) = \eta(Q, \{z, w\}).$$
\end{claim}
\begin{proof}
Let $i_z \in \{p-2, p-1, p\}$ such that $v_{i_z} = z$, and define $j_z$ similarly with respect to $Q$.
It suffices to prove that $i_z \modthree{\sigma}$ if and only if $j_z \modthree{\sigma}$. For, then the same equivalence holds for $w$. But this implies that $\eta(P, \{z, w\}) = *$ if and only if $ \eta(Q, \{z, w\})= *$.
First, we modify $P$ so as to assume that $P$ ends with $zw$ or $wz$. If this is not the case, then up to swapping $z$ and $w$ we have that $P$ ends with either $zw*$ or $z*w$. In the former case remove the last vertex of $P$, and in the latter case append $z$ to $P$. It is easy to see that the statement of the claim holds for the original $P$ if and only if it holds for the modified path. Similarly, we may assume that $Q$ ends with $zw$ or $wz$.
Assume first that $P$ and $Q$ both end with $zw$. Then $\tilde{Q}$ (defined as in \eqref{eqn:tilde}) is a path from $zw$ to $xy$ of order $2(q+1)-2 = 2q$. Hence $v_2 v_3 \ldots v_{p-2} \tilde{Q}$ is a cycle, and by assumption its order is divisible by 3. That is, $p-3+2q \modthree{0}$, and thus $p \modthree{q}$. Since $i_z = p-1$ and $j_z = q-1$, this proves \Cref{claim:consistent}. The same argument holds when $P$ and $Q$ both end with $wz$.
Secondly, assume that $P$ is a path from $xy$ to $zw$ and $Q$ is a path from $xy$ to $wz$. Note that this case only arises if $P_0$ is defined, as $v_0 v_1 \ldots v_{p-2} zw u_{q-2}\dots u_0$ is a path from $xy$ to $yx$. Then consider the cycle $v_2 \ldots v_{p-2} zw u_{q-2} \ldots u_2 \tilde{P_0}$. This is indeed a cycle because $u_1 u_0 = yx$, $v_0 v_1 = xy$ and $\tilde{P_0}$ is a path from $yx$ to $xy$. The order of this cycle is $p-3+2+q-3+2k \modthree{p+q+2+\sigma}$, using~\eqref{eq:sigma-def}. Now substitute $p = i_z+1$ and $q = j_z$. We have $i_z + j_z +\sigma \modthree{0}$, so $i_z \modthree{\sigma}$ if and only if $j_z \modthree{\sigma}$.
\end{proof}
Note that for every edge $\{z,w\}$ in the shadow of $\mathcal{H}$ there is a path $P$ starting at $xy$ whose last three vertices contain $z$ and $w$. Indeed, as $\mathcal{H}$ is tightly connected, there is a path $Q$ such that $x$ and $y$ are among its first three vertices and $z$ and $w$ among its last three vertices. Using a modification as in the proof of \Cref{claim:consistent} we may assume that $Q$ starts with $xy$ or $yx$. If it starts with $xy$ we are done, and otherwise the reverse of the path $\tilde{Q}$ satisfies the requirements.
Given an edge $\{z,w\}$ in the shadow of $\mathcal{H}$, define $\eta(zw) = \eta(P, \{zw\})$, where $P$ is any path from $xy$ whose last three vertices contain $z$ and $w$ (which exists by the previous paragraph). This parameter is well defined by \Cref{claim:consistent}. Now define $\chi$ as follows: let $zw$ be blue if $\eta(zw) = *$, and let it be red and oriented away from $\eta(zw)$ otherwise.
Finally, we show that $\chi$ is a good colouring. To see this, consider an edge $uvw$ of $G$.
Let $P$ be a path from $xy$ whose last three vertices are $u, v, w$ (in some order); such a path exists by the paragraph above. Write $P:=xy v_2 v_3 \ldots v_{p-2}v_{p-1}v_p$, let $i \in \{p-2, p-1, p \}$ with $i \modthree{\sigma}$, and we may assume that $v_i =u$. Then $\eta(uv)=\eta(uw) =u$ and $\eta(vw) = *$, which implies that $uvw$ is a cherry with apex $u$.
\end{proof}
\begin{remark}
Our proof actually shows that if $G$ does not contain a path $P_0$ starting and ending with $xy$ and $yx$ respectively, then the graph is tripartite.
Notice that a good colouring of $\mathcal{H}$ can be extended from the shadow of $\mathcal{H}$ to $K_n$ with no restrictions. Thus, in what follows it will suffice to analyse colourings of complete graphs by blue edges and red oriented edges (we will call such graphs \emph{coloured graphs}).
\end{remark}
\section{Maximising the number of cherries} \label{sec:optimal}
The results of the previous section establish a connection between maximising the number of edges in an odd-pseudocycle-free hypergraph and maximising the number of \textit{cherries} in colourings of $K_n$ (formally defined below). It will turn out that both problems have the same extremal construction which yields the maximum $f(n)$. Recall that we have defined $f(n)$ as the maximum number of edges in a hypergraph $\mathcal{H}(x_1, \ldots, x_k)$ with $\sum_i{x_i} = n$. An explicit expression for $f$ is
\begin{equation} \label{eqn:f}
f(n) = \max_{k \ge 1} \max_{\substack{x_1, \ldots, x_k \ge 1 : \\ x_1 + \ldots + x_k = n}} \left\{\sum_{1 \le i < j \le k} \binom{x_i}{2} \cdot x_j\right\}.
\end{equation}
Equivalently, we have the recursive characterisation
\begin{align} \label{eqn:k}
\begin{split}
& f(1) = 0,\\
& f(n) = \max_{k \in [n-1]} \binom{k}{2} (n-k) + f(n-k) \,\,\text{ for $n \ge 2$}.
\end{split}
\end{align}
Write
\begin{equation} \label{eq:alphabeta}
\beta = \frac{3 - \sqrt{3}}{2} \approx 0.634 \qquad \text{and} \qquad
\alpha = \frac{\beta(1 - \beta)}{2(3 - 3\beta + 3\beta^2)} = \frac{\sqrt{3}}{3} - \frac{1}{2} \approx 0.077.
\end{equation}
The following proposition will be proved in \Cref{subsec:calculus}.
\begin{proposition} \label{prop:alpha}
$f(n) = \alpha n^3 + o(n^3)$.
\end{proposition}
We remark that the density of the corresponding hypergraph $\mathcal{H}_n$ is $6\alpha = 2\sqrt{3}-3$, as already noted by Mubayi and R\"odl~\cite{mubayi2002turan}.
As in Section~\ref{sec:overview}, we call a graph $G$ \emph{coloured} if it is a complete graph whose edges are either coloured blue or coloured red and oriented.
A \emph{cherry} in a coloured graph $G$ is a triple $xyz$ such that $xy$ and $xz$ are red and directed from $x$ and $yz$ is blue. Denote by $c(G)$ the number of cherries in $G$. Theorem~\ref{theorem_intro_falgas_ravry} states that $c(G) \leq f(n)$ for any $n$-vertex coloured graph $G$. Recall that this was originally proved by Falgas-Ravry and Vaughan \cite{falgas2012turan} (who used flag algebras and also proved a similar result for out-directed stars on four vertices) and by Huang \cite{huang2014maximum} (who used a symmetrisation argument, and proved a similar result for out-directed stars on $k$ vertices, for every $k \ge 3$). Nevertheless, we provide a proof, both for completeness and because we need most of the groundwork to prove a stability version of Theorem~\ref{theorem_intro_falgas_ravry}.
As mentioned in the proof overview (\Cref{sec:overview}), \Cref{corollary_intro_long_pseudocycles}, which is a weak version of our main result and is restated here, follows directly from \Cref{theorem_intro_falgas_ravry} (proved in the next section) and \Cref{thm:good-colouring} (proved in the previous section).
\corLongPseudocylces*
\section{Stability with symmetrisation} \label{sec:symm}
Most of the work in this section will go into proving the following lemma, providing a stability version of Theorem~\ref{theorem_intro_falgas_ravry}.
It will then be iterated to prove a stability result about cherries in coloured graphs; recall that $\beta = (3 - \sqrt{3})/2$ (see \eqref{eq:alphabeta}).
We point out that this stability result is somewhat similar to a general result due to Liu--Pikhurko--Sharifzadeh--Staden \cite{liu2020stability} which allows one to obtain stability versions of a class of extremal results that can be proved using a symmetrisation argument. However, while we indeed prove the extremal result in Theorem~\ref{theorem_intro_falgas_ravry} using a symmetrisation argument, the result in \cite{liu2020stability} does not apply to automatically convert it into a stability result.
\begin{lemma} \label{lem:ind-step}
Let $ 1/n \ll \varepsilon \ll 1$ and let $G$ be a coloured graph on $n$ vertices satisfying $c(G) \ge f(n) - \varepsilon^2 n^3$.
Then there is a coloured graph $G'$ on $V(G)$ satisfying: $c(G') \ge c(G)$; the graphs $G$ and $G'$ differ on at most $800\varepsilon^{1/2} n^2$ edges; moreover, there is a set $Q \subseteq V(G)$ satisfying $\big| |Q| - \beta n \big| \le 100\varepsilon n$; $Q$ is a blue clique in $G'$; and all other edges in $G'$ that are incident with $Q$ are red and oriented towards $Q$.
\end{lemma}
The proof consists of two main parts: first we show that $G$ has a blue almost-clique on a vertex set $Q'$ of size roughly $\beta n$. Then we show that most $(V \setminus Q', Q')$ edges are red and point towards $Q'$. In both parts, we make use of a ``symmetrisation procedure'' which builds blue cliques without decreasing the number of cherries.
A \emph{blue clone-clique} in a coloured graph $G$ is a set of vertices $Q$ such that $Q$ is a blue clique in $G$, and for any $v \notin Q$, either all edges between $v$ and $Q$ are blue, or they are all red and have the same orientation (namely, they all point towards $v$ or all point away from $v$). A \emph{full blue clone-clique} is a blue clone-clique $Q$ such that all $(V \setminus Q, Q)$ edges are red.
The symmetrisation procedure, which will be described in detail in the next section, receives as input a vertex $x$ in a graph $G$, and produces a graph $G'$ on the same vertex set, which has at least as many cherries as $G$ and has a full blue clone-clique $Q$ in $G'$ that contains $x$.
The symmetrisation procedure can be applied repeatedly to a coloured graph $G$ to find a coloured graph $G'$ with at least as many cherries as $G$, and whose vertices can be partitioned into full blue clone-cliques. Some calculus (detailed in \Cref{subsec:calculus}) will show that such a $G'$ contains a full blue clone-clique $Q'$ of size approximately $\beta n$.
To proceed we need two lemmas (\Cref{lem:Q-red,lem:Q-blue}; see \Cref{subsec:Q-lemmas}) that together tell us the following. Suppose that a symmetrisation procedure on $G$ resulted in a full blue clone-clique $Q$, of size approximately $\beta n$. Then (even before performing symmetrisation) almost all edges in $G[Q, V \setminus Q]$ are red and point towards $Q$, and almost all edges in $G[Q]$ are blue.
Applying these lemmas to the previously found blue clone-clique $Q'$, we conclude first that $G[Q']$ is almost fully blue.
We then show that there is a particular instance of the symmetrisation procedure that results in a graph $G'$ and full blue clone-clique $Q$ such that $Q$ and $Q'$ differ on only few vertices. Lemma~\ref{lem:Q-red} implies that almost all $G[Q', V \setminus Q']$ edges are red and point towards $Q'$. This essentially completes the proof. This part is detailed in \Cref{subsec:proof}.
In \Cref{subsec:iteration}, we iterate \Cref{lem:ind-step} to prove the following result.
\begin{theorem}\label{thm:stability}
Let $1/n \ll \varepsilon_1 \ll \varepsilon_2 \ll 1$.
Let $G$ be a coloured graph on $n$ vertices satisfying $c(G) \ge f(n) - \varepsilon_1 n^3$.
Then there exists a coloured graph $G'$ on the same vertex set, satisfying:
\begin{enumerate}[label = \rm(\alph*), ref = \rm (\alph*)]
\item \label{itm:stability-a}
$c(G') \ge c(G)$,
\item \label{itm:stability-b}
$G$ and $G'$ differ on at most $\varepsilon_2 n^2$ edges,
\item \label{itm:stability-c}
the vertices of $G'$ can be partitioned into $Q_1, \ldots, Q_t$ such that:
\begin{enumerate}[label = \rm(\roman*), ref = c\rm(\roman*)]
\item \label{itm:stability-c1}
$|Q_1| \ge \ldots \ge |Q_t|$,
\item \label{itm:stability-c2}
all edges in $Q_i$ are blue, for $i \in [t]$,
\item \label{itm:stability-c3}
all edges in $(Q_i, Q_j)$ are red and directed towards $Q_i$, for $1 \le i < j \le t$,
\item \label{itm:stability-c4} $\big||Q_i| - \beta \cdot |Q_i \cup \ldots \cup Q_t| \big| \le \varepsilon_2 n$ for $i \in [t]$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\def N^- {N^-}
\def N^+ {N^+}
In a coloured graph $G$, let $N^-_G(x)$ be the red in-neighbourhood of $x$ and let $N^+_G(x)$ be the red out-neighbourhood of $x$ (we sometimes omit the subscript $G$).
\subsection{The symmetrisation procedure} \label{subsec:symmproc}
Given $x \in V(G)$, the \emph{symmetrisation procedure} $S_G(x)$ (or $S(x)$ in short) builds a blue clone-clique containing $x$; see \Cref{fig:symmetrisation} for a detailed description. The result of the procedure depends on the choice of $x_{k+1}$ in step \ref{step:4}, but we suppress this dependence in the notation $S_G(x)$.
\begin{figure}[h!t!]
\begin{framed}
\begin{minipage}{.98\textwidth}
\begin{center}
{\bf The symmetrisation procedure $S_G(x)$}
\end{center}
{\bf Input:} a coloured graph $G$ on vertex set $V$ and a vertex $x \in V$.
\vspace{.2cm}
{\bf Output:} a graph $G'$ and a full blue clone-clique $Q$ in $G'$ containing $x$.
\vspace{.2cm}
{\bf The process:} the algorithm builds sequences $x_1, \ldots, x_t$ and $y_1, \ldots, y_t$ of vertices in $V$ and $G_1, \ldots, G_t$ of graphs on $V$, for some $t$, as follows.
\begin{enumerate}
\item
Set $x_1, y_1 := x$ and $G_1 := G$.
\item
Suppose $x_1, \ldots, x_k$ and $G_1, \ldots, G_k$ are given and $\{x_1, \ldots, x_k\}$ is a blue clone-clique in $G_k$.
\item \label{step:3}
If there are no vertices in $V \setminus \{x_1, \ldots, x_k\}$ whose edges to $\{x_1, \ldots, x_k\}$ are blue, put $Q := \{x_1, \ldots, x_k\}$ and $G' := G_k$, and return $G'$ and $Q$.
\item \label{step:4}
Otherwise, let $x_{k+1}$ be a vertex in $V \setminus \{x_1, \ldots, x_k\}$ which sends blue edges to $\{x_1, \ldots, x_k\}$ ($x_{k+1}$ can be chosen arbitrarily or judiciously).
\item
For $y \in V$, let $G_{k+1}(y)$ be the graph obtained by replacing $\{x_1, \ldots, x_{k+1}\}$ by $k+1$ copies of $y$ and letting the new vertices form a blue clique.
If $c(G_{k+1}(x_{k+1})) - c(G_k) \ge k \cdot (c(G_{k+1}(x_1)) - c(G_k))$, set $y_{k+1} = x_{k+1}$, and otherwise let $y_{k+1} = x_1$.
Define $G_{k+1} := G_{k+1}(y_{k+1})$, and return to step \ref{step:3}.
\end{enumerate}
\end{minipage}
\end{framed}\vspace{-0.5cm}
\caption{Description of the symmetrisation process $S_G(x)$}
\label{fig:symmetrisation}
\end{figure}
We now show that the procedure $S_G(x)$ does not decrease the number of cherries. In fact, we prove a stronger quantitative claim.
\begin{claim} \label{claim:nhoods}
Let $x_1, \ldots, x_t$, $y_1, \ldots, y_t$ and $G_1, \ldots, G_t$ be sequences produced by $S_G(x)$, let $k \in [t-1]$, and use $N^-(u)$ as a shorthand for $N^-_{G_k}(u)$. Then, one of the following holds.
\begin{enumerate}[label = \rm(\roman*)]
\item \label{itm:nhoods1}
$y_{k+1} = x_1$ and $c(G_{k+1})-c(G_k) \geq \frac{k+1}{4} \cdot \big| N^-(x_1) \,\triangle\, N^-(x_{k+1})\big|$,
\item \label{itm:nhoodsk}
$y_{k+1} = x_{k+1}$ and $c(G_{k+1})-c(G_k) \geq \frac{k(k+1)}{4} \cdot \big| N^-(x_1) \, \triangle \, N^-(x_{k+1})\big|$.
\end{enumerate}
In particular, $c(G_{k+1}) \ge c(G_k)$.
\end{claim}
\begin{proof}
Let $c(x_i)$ denote the number of cherries in $G_k$ containing $x_i$ and no other vertices in $x_1, \ldots, x_k$.
Recall that for $y \in \{x_1, \ldots, x_{k+1}\}$ the graph $G_{k+1}(y)$ is obtained from $G_k$ by replacing $x_1, \ldots, x_{k+1}$ by copies of $y$ that form a blue clique. Write $\Delta_j := c(G_{k+1}(x_j)) - c(G_k)$.
Then
\begin{equation*}
\Delta_j =
(k+1)c(x_j) - \sum_{i \in [k+1]} c(x_i) + \binom{k+1}{2}\big|N^-(x_j)\big| - \frac 12 \sum_{i_1 \neq i_2} \big|N^-(x_{i_1}) \cap N^-(x_{i_2})\big|.
\end{equation*}
Summing over $j \in [k+1]$, we obtain
\begin{align*}
\sum_j \Delta_j
& = \binom{k+1}{2}\sum_j \big|N^-(x_j)\big| - \frac{k+1}{2}\sum_{i \neq j}\big|N^-(x_i) \cap N^-(x_j)\big| \\
& = \frac{k+1}{2}\sum_{i \neq j} \big|N^-(x_j) \setminus N^-(x_i)\big|.
\end{align*}
In particular, since $\{x_1, \dots x_k\}$ is a blue clique-clone,
\begin{align*}
k \Delta_1 + \Delta_{k+1}
& = \frac{k(k+1)}{2} \cdot \left( \big|N^-(x_{k+1}) \setminus N^-(x_1)\big| + \big|N^-(x_{1}) \setminus N^-(x_{k+1})\big|\right) \\
& = \frac{k(k+1)}{2} \cdot \big|N^-(x_1) \,\triangle\, N^-(x_{k+1})\big|.
\end{align*}
Now, $\max(k \Delta_1, \Delta_{k+1})$ is at least one half of the RHS. Thus, if $y_{k+1} = x_{k+1}$ then $\Delta_{k+1} \ge k \Delta_1$ and so $\Delta_{k+1} = \max(k \Delta_1, \Delta_{k+1}) \ge \frac{k(k+1)}{4} \cdot \big|N^-(x_1) \,\triangle\, N^-(x_{k+1})\big|$, and if $y_1 = x_1$ then $k \Delta_1 > \Delta_{k+1}$ and so
$\Delta_1 = \max(k\Delta_1, \Delta_{k+1}) \geq \frac{k+1}{4} \cdot |N^-(x_1) \,\triangle\, N^-(x_{k+1})\big|$.
\end{proof}
\Cref{theorem_intro_falgas_ravry} follows easily from the above claim.
\def G_{\final} {G_{\final}}
\begin{proof}[Proof of Theorem~\ref{theorem_intro_falgas_ravry}]
Let $G$ be a coloured graph on $n$ vertices. Run the following process: starting with $G' = G$, as long as there is a vertex $x$ which is not in a full blue clone-clique in $G'$, run $S_{G'}(x)$ and replace $G'$ by the resulting graph. Let $G_{\final}$ be the graph $G'$ at the end of the process (notice that the process will indeed end, because $S_{G'}(x)$ keeps full blue clone-cliques intact). Then $c(G_{\final}) \ge c(G)$ by \Cref{claim:nhoods}, and the vertices of $G_{\final}$ can be partitioned into full blue clone-cliques $Q_1, \ldots, Q_t$; for convenience suppose that $|Q_1| \ge \ldots \ge |Q_t|$. Replace $G_{\final}$ by the graph $G_{\final}'$ obtained by directing the red edges between $Q_i$ and $Q_j$ towards $Q_i$, for $1 \le i < j \le t$. It is straightforward to verify that $c(G_{\final}') \ge c(G_{\final})$, as the number of cherries in $Q_i \cup Q_j$ is larger when the arcs in $(Q_i, Q_j)$ point towards the larger clique. Finally, denoting $q_i := |Q_i|$, observe that $c(G_{\final}') = \sum_{i \le j} \binom{q_i}{2}q_j \le f(n)$ (see \eqref{eqn:f}). Thus $c(G) \le f(n)$, as claimed.
\end{proof}
\subsection{Optimising the clique size} \label{subsec:calculus}
Before proceeding to analyse the symmetrisation procedure, we prove the following lemma regarding the structure of a graph whose vertices are partitioned into full blue clone-cliques, mostly using calculus; recall that $\alpha$ and $\beta$ are defined in \eqref{eq:alphabeta}.
\begin{lemma} \label{lem:theta}
Let $1/n \ll \varepsilon \ll 1$.
Let $G$ be a coloured graph on $n$ vertices whose vertices can be partitioned into full blue clone-cliques, and suppose that $c(G) \ge f(n) - \varepsilon^2 n^3$. Then $G$ has a full blue clone-clique $Q$ satisfying $\big| |Q| - \beta n \big| \le 100 \varepsilon n$.
\end{lemma}
Define a function $g:[0,1] \to \mathbb{R}$ as follows.
\begin{equation} \label{eqn:g}
g(x) = \frac{x(1-x)}{2(3-3x+x^2)}.
\end{equation}
It will be convenient to note the following equation.
\begin{equation} \label{eqn:g-var}
\left(1 - (1 - x)^3\right) \cdot g(x) = \frac{1}{2} \cdot x^2(1 - x).
\end{equation}
One can check that $g'$ is decreasing and $g'(\beta) = 0$, showing that
\begin{equation} \label{eqn:g-beta}
g(x) \le g(\beta) = \alpha \qquad \text{for $x \in [0,1]$}.
\end{equation}
We first prove \Cref{prop:alpha} regarding the value of $f(n)$.
\begin{proof}[Proof of \Cref{prop:alpha}]
We show that $f(n) \leq \alpha n^3$ by induction on $n$. This is true for $n=1$. Suppose that $f(m)\leq \alpha m^3$ for $m< n$.
Given $k \in [n-1]$ that maximises the LHS in \eqref{eqn:k}, write $x = k/n$. The recursive definition of $f$ implies that $ \frac{f(n)}{n^3} \leq \frac 12 \cdot x^2(1-x) + \alpha(1-x)^3$. Subtracting $\alpha$ and using \eqref{eqn:g-var}, we obtain
\begin{equation*}
\frac{f(n)}{n^3}- \alpha \leq \frac 12 \cdot x^2(1-x) - \alpha(1-(1-x)^3) = (1-(1-x)^3)(g(x)-\alpha) \le 0,
\end{equation*}
as required.
To verify that $f(n) \geq (\alpha + o(1))n^3$, set $x_i =\lfloor \beta (1-\beta)^i n \rfloor$ in~\eqref{eqn:f}.
\end{proof}
\begin{proof}[Proof of \Cref{lem:theta}]
Let $Q_1, \ldots, Q_t$ be the full blue clone-cliques in $G$, arranged in descending order according to their sizes. Let $G'$ be obtained from $G$ by orienting the $(Q_i, Q_j)$ (red) edges towards $Q_i$, for $1 \le i < j \le t$. As explained before and by assumption on $G$, $c(G') \ge c(G) \ge f(n) - \varepsilon^2 n^3$.
Notice that $|Q_1| \ge 0.01 n$, because otherwise $c(G) \le n^2|Q_1| \le 0.01 n^3 < f(n) - \varepsilon^2 n^3$ (recall that $f(n) \approx 0.077n^3$, by \Cref{prop:alpha}).
Write $|Q_1| = \theta n$. Then, using $f(n) = \alpha n^3 + o(n^3) = g(\beta) n^3 + o(n^3)$ (which follows from \Cref{prop:alpha} and the definition of $\alpha$ in \eqref{eq:alphabeta}),
\begin{equation*}
c(G') \le \binom{|Q_1|}{2}(n - |Q_1|) + f(n - |Q_1|)
\le \frac{1}{2} \theta^2 (1 - \theta)n^3 + g(\beta)(1 - \theta)^3n^3 + o(n^3).
\end{equation*}
Thus, using \eqref{eqn:g-var},
\begin{align*}
\varepsilon^2 \ge \frac{f(n) - c(G')}{n^3}
& \ge g(\beta) - g(\beta)(1 - \theta)^3 - \frac{1}{2}\theta^2(1 - \theta) + o(1) \\
& = \theta \cdot (3 - 3\theta + \theta^2) \cdot (g(\beta) - g(\theta)) + o(1) \\
& \ge 0.02 \cdot (g(\beta) - g(\theta)) + o(1).
\end{align*}
For the last inequality we used $\theta \ge 0.01$, which implies $\theta(3-3\theta+\theta^2) \ge 0.02$.
By bounding the $o(1)$ term by $\varepsilon^2/2$ and using \Cref{claim:calculus} below, we get
\begin{equation*}
100\varepsilon^2 \ge g(\beta) - g(\theta) \ge \min\{0.05(\beta - \theta)^2, 0.005\}.
\end{equation*}
Since $\varepsilon$ is very small, we get $100 \varepsilon^2 \ge 0.05(\beta - \theta)^2$, which implies $|\beta - \theta| \le 100\varepsilon$.
\end{proof}
\begin{claim} \label{claim:calculus}
For $x \in [0,1]$,
\begin{equation}
g(\beta) - g(x) \geq \min\{0.05(\beta-x)^2, 0.005\}.
\end{equation}
\end{claim}
\begin{proof}
We use the following facts that can be checked easily.
\begin{itemize}
\item
The function $g(x)$ is increasing on $[0,\beta]$ and decreasing on $[\beta, 1]$. In particular, its maximum is attained at $\beta$, and $g'(\beta) = 0$.
\item
$g(\beta) - g(0.5) \ge 0.005$.
\item
The second derivative $g''(x)$ (which is $\frac{-x(2x^2 - 9x + 9)}{(x^2 - 3x + 3)^3}$) is non-negative and decreasing on $[0,1]$. In particular $g''(x) \le g''(0.5) \le -0.4$ for $x \in [0.5, 1]$.
\item
By Taylor's expansion: $g(x) = g(\beta) + \frac{1}{2}g'(\beta)(x - \beta) + \frac{1}{6}g''(c_x)(x - \beta)^2$ for every $x \in [0,1]$ and some $c_x$ between $x$ and $\beta$.
\end{itemize}
By the first and second items (using $\beta > 0.5$), if $x \in [0, 0.5]$ then
\begin{equation*}
g(\beta) - g(x) \ge g(\beta) - g(0.5) \ge 0.005.
\end{equation*}
By the first, third and fourth items, if $x \in [0.5, 1]$, then
\begin{equation*}
g(\beta) - g(x) \ge \frac{0.4}{6}(x - \beta)^2 \ge 0.05(x - \beta)^2.
\end{equation*}
The two inequalities prove the claim.
%
\end{proof}
\subsection{Blue clone-cliques before and after symmetrisation} \label{subsec:Q-lemmas}
The next two lemmas show that if a symmetrisation procedure on $G$ produces a full blue clone-clique $Q$ of size apprximately $\beta n$, then almost all edges in $G[Q, V \setminus Q]$ are red and oriented towards $Q$ and almost all edges in $G[Q]$ are blue.
\begin{lemma} \label{lem:Q-red}
Let $1/n \ll \varepsilon \ll 1$.
Let $G = (V, E)$ be a coloured graph on $n$ vertices with at least $f(n) - \varepsilon^2 n^3$ cherries. Suppose that $G'$ and $Q$ are the output
of a procedure $S_G(x)$, and suppose that $|Q| \ge 0.55n$.
Then all but at most $10\varepsilon n^2$ edges in $G[Q, V \setminus Q]$ are red and directed towards $Q$.
\end{lemma}
\begin{proof}
\def \Vi{V_{\inn}}
\def \Vo{V_{\out}}
Set $U := V \setminus Q$, let $\Vi$ be the set of vertices $u$ in $U$ for which $uq$ is a red arc in $G'$ for every $q \in Q$, and let $\Vo := U \setminus \Vi$.
We will show that $\Vo$ is small, and that not many pairs incident to $\Vi$ were recoloured during the symmetrisation procedure $S_G(x)$.
First, we show $|\Vo| \leq 40\varepsilon^2 n$.
Let $G''$ be obtained from $G'$ by reorienting the edges in $G'[Q, \Vo]$ to point towards $Q$. Notice that the cherries in $G'$ that contain an edge in $(Q, \Vo)$ consist of one vertex in $Q$ and two in $\Vo$, and thus their number is at most $|Q|\binom{|\Vo|}{2}$. Also, every set consisting of two vertices in $Q$ and one in $\Vo$ is a cherry in $G''$ but not in $G'$. Thus, using $|Q| \ge 0.55 n$ which implies $|Q| - |\Vo| \ge 0.1n$,
\begin{align*}
c(G'') - c(G')
& \ge \binom{|Q|}{2}|\Vo| - \binom{|\Vo|}{2}|Q|
= \frac 12 |Q||\Vo|(|Q| - |\Vo|) \\
& \ge \frac{1}{2} \cdot \frac{n}{2} \cdot \frac{n}{10} \cdot |\Vo|
= \frac{n^2}{40} \cdot |\Vo|.
\end{align*}
Recall that $c(G) \ge f(n) - \varepsilon^2 n^3$ by assumption, $c(G') \ge c(G)$ by \Cref{claim:nhoods}, and $c(G'') \le f(n)$ by Theorem~\ref{theorem_intro_falgas_ravry}. Altogether, this implies $c(G'') - c(G') \le \varepsilon^2 n^3$ and thus $|\Vo| \le 40\varepsilon^2 n$, as claimed.
Let $R$ be the set of edges $qv$ in $(Q, V \setminus Q)$ that are red and oriented towards $Q$ in $G'$ but not in $G$.
We now upper-bound $|R|$. Notice that each such edge in $R$ was recoloured to a red arc oriented towards $Q$ at some point during $S_G(x)$ (possibly more than once).
Let $G = G_1, \ldots, G_{t} = G'$ be the graphs obtained during the symmetrisation process on $Q$ and let $x_1, \ldots, x_t$ be the corresponding sequence of vertices.
For each $v \in V$ and $k \in [t]$, let $A_k(v)$ be the set of ordered pairs $vq$ which changed to red arcs in step $k$ (so they were recoloured from $G_{k-1}$ to $G_k$).
We claim that $\sum_{k \geq \varepsilon n} \sum_{v \in \Vi} |A_k(v)| \leq 4 \varepsilon n^2$.
To see this, fix $k \geq \varepsilon n$ and consider the $k$-th step. If $y_k = x_1$, then $A_k(v) = \{vx_k\}$ for $v \in N^-(x_1) \setminus N^-(x_k)$ and $A_k(v) = \emptyset$ otherwise, where $N^-(\cdot)$ refers to the in-neighbourhood with respect to $G_{k-1}$. Thus, using \Cref{claim:nhoods}~\ref{itm:nhoods1},
\begin{equation*}
\sum_{v \in \Vi} |A_k(v)|
\leq \big|N^-(x_1) \setminus N^-(x_k)\big|
\leq \frac{4}{k} \cdot \big(c(G_k) - c(G_{k-1})\big).
\end{equation*}
If $y_k = x_k$, then $A_k(v) = \{vx_1, \ldots, vx_{k-1}\}$ for $v \in N^-(x_k) \setminus N^-(x_1)$ and $A_k(v) = \emptyset$ otherwise. Thus, by \Cref{claim:nhoods}~\ref{itm:nhoodsk},
\begin{equation*}
\sum_{v \in \Vi} |A_k(v)|
\leq (k-1) \cdot \big|N^-(x_1) \setminus N^-(x_k)\big|
\leq \frac{4}{k} \cdot \big(c(G_k) - c(G_{k-1})\big),
\end{equation*}
In either case, we get that for $k \geq \varepsilon n$,
\begin{equation*}
\sum_{v \in \Vi} |A_k(v)| \leq \frac{4}{\varepsilon n}\big(c(G_k) - c(G_{k-1})\big).
\end{equation*}
Summing over $k \geq \varepsilon n$, we obtain the required inequality
\begin{equation*}
\sum_{k \ge \varepsilon n} \sum_{v \in \Vi} |A_k(v)| \leq \frac{4}{\varepsilon n}\big(c(G') - c(G_{\varepsilon n})\big) \leq 4 \varepsilon n^2,
\end{equation*}
Where the last equality holds since $c(G') - c(G_{\varepsilon n}) \leq \varepsilon^2 n^2$.
Note that $|R| \le \varepsilon n^2 + \sum_{k \ge \varepsilon n} \sum_{v \in \Vi} |A_k(v)| \le 5\varepsilon n^2$.
In total, all but at most $(40 \varepsilon^2 + 5\varepsilon)n^2 \leq 10 \varepsilon n^2$ pairs in $(Q, V \setminus Q)$ are red and oriented towards $Q$.
\end{proof}
\begin{lemma} \label{lem:Q-blue}
Let $1/n \ll \varepsilon \ll 1$.
Let $G$ be a coloured graph on $n$ vertices with at least $f(n) - \varepsilon^2 n^3$ cherries. Suppose that $G'$ and $Q$ are the graph and full blue clone-clique produced by the procedure $S_G(x)$, and suppose that $0.55n \le |Q| \le 0.65n$.
Then all but $1200\varepsilon n^2$ edges in $G[Q]$ are blue.
\end{lemma}
\begin{proof}
Let $F$ and $F'$ be the graphs obtained from $G$ and $G'$ by colouring all $(Q, V \setminus Q)$ edges red and orienting them towards $Q$. Notice that $F'$ can be obtained from $F$ by colouring all edges in $Q$ blue.
We will first derive an upper bound on $c(F') - c(F)$. By \Cref{lem:Q-red}, the graphs $G$ and $F$ differ on at most $10\varepsilon n^2$ edges and thus $|c(G) - c(F)| \le 10\varepsilon n^3$. Similarly, $|c(G') - c(F')| \le 10\varepsilon n^3$ (the lemma is still applicable, as $S_{G'}(x)$ does not change the graph $G'$). By assumption on $G$ we also have $c(G') - c(G) \le \varepsilon^2 n^3$. Altogether,
\begin{equation} \label{eqn:upper}
c(F') - c(F) \le c(G') - c(G) + 20\varepsilon n^3 \le (\varepsilon^2 + 20\varepsilon)n^3 \le 30 \varepsilon n^3.
\end{equation}
\def d^+ {d^+}
We now obtain a lower bound on the same quantity.
Let $e$ be the number of red edges in $G[Q]$.
The number of cherries in $F$ that are not cherries in $F'$ is at most $\sum_{q \in Q} \binom{d^+(q)}{2}$, where $d^+(q)$ denotes the red in-degree of $q$ in $F[Q]$. Notice that
\begin{equation*}
\sum_{q \in Q} \binom{d^+(q)}{2} \le \frac{1}{2} \sum_{q \in Q}(d^+(q))^2 \le \frac 12 e|Q|,
\end{equation*}
because $d^+(q) \le |Q|$ and $e = \sum_q d^+(q)$.
On the other hand, the number of cherries in $F'$ that are not cherries in $F$ is exactly $e(n - |Q|)$. Thus,
\begin{equation} \label{eqn:lower}
c(F') - c(F)
\ge e(n - |Q|) - \frac 12 e|Q|
= e\cdot \big(n - \frac{3}{2}|Q|\big)
\ge \frac{en}{40},
\end{equation}
using $|Q| \le 0.65 n$.
By \eqref{eqn:upper} and \eqref{eqn:lower}, we have $e \le 1200 \varepsilon n^2$, as claimed.
\end{proof}
\subsection{Proof of \Cref{lem:ind-step}} \label{subsec:proof}
%
Finally, we start with the actual proof of \Cref{lem:ind-step}. The first step is to find a set $Q'$ of the right size almost all of whose edges in $G$ are blue.
\begin{lemma} \label{lem:beta-clique}
Let $1/n \ll \varepsilon \ll 1$.
Let $G$ be a coloured graph on $n$ vertices, satisfying $c(G) \ge f(n) - \varepsilon^2 n^3$.
Then there is a set $Q' \subseteq V(G)$ such that $\big||Q'| - \beta n\big| \le 100\varepsilon n$ and all but at most $1200\varepsilon n^2$ edges in $G[Q']$ are blue.
\end{lemma}
\begin{proof}
Similarly to the proof of Theorem~\ref{theorem_intro_falgas_ravry}, start with $G' = G$, and, as long as $G'$ has a vertex $x$ which is not in a full blue clone-clique, run the symmetrisation procedure $S_{G'}(x)$, and replace $G'$ by the resulting graphs. Denote by $G_{\final}$ the graph at the end of the process (as before, the process is guaranteed to end). Then the vertices of $G_{\final}$ can be partitioned into full blue clone-cliques $Q_1, \ldots, Q_t$.
Let $Q'$ be the vertex set of the largest clone-clique. By \Cref{lem:theta}, we have $\big||Q'| - \beta n\big| \le 100\varepsilon n$. In particular $|Q'| \in [0.55n, 0.65n]$.
Let $F_1$ be the graph created just before the symmetrisation procedure was started on an element of $Q'$, and let $F_2$ be the graph just after $Q'$ was built. Notice that $c(F_2) \ge c(F_1) \ge c(G) \ge f(n) - \varepsilon^2 n^3$.
By \Cref{lem:Q-blue}, all but at most $1200\varepsilon n^2$ edges in $F_1[Q']$ are blue. Notice that during the above process, the edges in $Q'$ remain untouched until right before a symmetrisation process is started on an element of $Q'$. It follows that all but at most $1200 \varepsilon n^2$ edges in $G[Q']$ are blue.
\end{proof}
Now, we can complete the proof by running a symmetrisation procedure that generates a blue clique $Q$ which will be close to $Q'$.
\begin{proof}[Proof of Lemma~\ref{lem:ind-step}]
Apply \Cref{lem:beta-clique} to find $Q'$ such that $\big||Q'| - \beta n\big| \le 100\varepsilon n$ and $G[Q']$ has at most $\delta^2 n^2$ red edges (with $\delta^2= 1200\varepsilon$).
\begin{claim}
We can run a symmetrisation procedure on $G$ which results in a graph $G'$ and a full blue clone-clique $Q$ satisfying $|Q' \setminus Q| \leq 3 \delta n$.
\end{claim}
\begin{proof}
Let $A$ be the set of vertices in $Q'$ with more than $\delta n$ red (in- or out-) neighbours in $G[Q']$. The bound on the number of red edges in $Q'$ gives $|A| < 2\delta n$. Define $Q'' := Q' \setminus A$.
We will run a symmetrisation procedure on $G$, but with a specific ordering of vertices.
We start with $x_1 \in Q''$ (chosen arbitrarily). Assuming that $\{x_1, \ldots, x_k\}$ are defined and contained in $Q''$, if possible we pick $x_{k+1}$ to also be in $Q''$ (we can do this as long as there is a vertex in $Q'' \setminus \{x_1, \ldots, x_k\}$ whose edges to $\{x_1, \ldots, x_k\}$ are blue). Once this is no longer possible, we continue with the symmetrisation procedure using an arbitrary order of vertices. Let $Q$ be the full blue clone-clique build by this procedure.
Let $k$ be largest such that $\{x_1, \ldots, x_k\} \subseteq Q''$. It is easy to see that throughout the procedure, until at least step $k$, every vertex in $Q''$ has at most $\delta n$ non-blue neighbours in $Q'' \setminus \{x_1, \ldots, x_k\}$. Thus $k \ge |Q''| - \delta n \ge |Q| - 3\delta n$, as otherwise we could find a suitable $x_{k+1}$ in $Q''$, contradicting the choice of $k$. It follows that $|Q' \setminus Q| \le 3\delta n$.
\end{proof}
Let $G'$ and $Q$ be as in the above Claim. We claim that $|Q| \le (\beta + 100\varepsilon)n$. Indeed, this follows from \Cref{lem:theta} by running symmetrisation procedures repeatedly, starting from $G'$, until the vertices can be partitioned into full blue clone-cliques (one of which is $Q$). It follows that $|Q \setminus Q'| \le 3\delta n + |Q| - |Q'| \le (3\delta + 200\varepsilon)n \le 5\delta n$. In particular, the number of red edges in $G[Q]$ is at most the number of red edges in $G[Q']$ plus the number of edges incident with $Q \setminus Q'$, which amounts to a total of at most $(\delta^2 + 5\delta)n^2 \le 10\delta n^2$ red edges in $G[Q]$.
By \Cref{lem:Q-red}, all but at most $10\varepsilon n^2$ edges in $G[Q, V \setminus Q]$ are red and oriented towards $Q$, and similarly for $G'[Q, V \setminus Q]$.
\def \Vi{V_{\inn}}
\def \Vo{V_{\out}}
Since $Q$ is a full blue clone-clique, the vertices in $V \setminus Q$ can be partitioned into $\Vi$ and $\Vo$, where $vq$ is a red arc for every $v \in \Vi$ and $q \in Q$ and $qv$ is a red arc for $v \in \Vo$ and $q \in Q$. Thus, by the previous paragraph and because $|Q| \ge n/2$, $|\Vo| \le 20\varepsilon n$.
Let $G''$ be obtained from $G'$ be reorienting all $(Q, V \setminus Q)$ edges towards $Q$. Then
\begin{equation*}
c(G'') - c(G')
\ge \binom{|Q|}{2}{|\Vo|} - \binom{|\Vo|}{2}|Q|
= |Q||\Vo| \cdot (|Q| - |\Vo|) \ge 0.
\end{equation*}
It follows that $c(G'') \ge c(G') \ge c(G)$. Moreover, $G''$ and $G'$ differ on at most $|\Vo|n \le 20\varepsilon n^2$ edges, and thus $G$ and $G''$ differ on at most $(20\varepsilon + 10\varepsilon + 10\delta) \le 20\delta n^2$ edges.
Since $G''$ has the required structure, this proves \Cref{lem:ind-step}.
\end{proof}
\subsection{Full stability result} \label{subsec:iteration}
\begin{proof}[Proof of \Cref{thm:stability}]
Let $\varepsilon_1 \ll \eta \ll \varepsilon_2$.
The idea is simply to iterate \Cref{lem:ind-step}.
We will find graphs $G_1, \ldots, G_s$ and sets $Q_1, \ldots, Q_s$, satisfying the following conditions, for $k \in [s]$ (for convenience set $G_0 := G$, $Q_0 := \emptyset$ and $V := V(G)$).
\begin{enumerate}[label = \rm(\arabic*)]
\item \label{itm:full-stab-1}
$G_k$ is a coloured graph on vertex set $V \setminus (Q_1 \cup \ldots \cup Q_{k-1})$.
\item \label{itm:full-stab-2}
$Q_k$ is a blue clique in $G_k$, all other edges incident with $Q_k$ in $G_k$ are red and point towards $Q_k$.
\item \label{itm:full-stab-3}
$\big||Q_k| - \beta|G_k|\big| \le \eta|G_k|$.
\item \label{itm:full-stab-4}
$G_k$ and $G_{k-1} \setminus Q_{k-1}$ differ on at most $\eta |G_{k}|^2$ edges.
\item \label{itm:full-stab-5}
$c(G_k) \ge c(G_{k-1} \setminus Q_{k-1})$.
\item \label{itm:full-stab-6}
$c(G_k \setminus Q_k) \ge f(|G_k \setminus Q_k|) - \varepsilon_1 n^3$.
\end{enumerate}
To see how such a sequence can be built, suppose that $G_1, \ldots, G_{k-1}$ and $Q_1, \ldots, Q_{k-1}$ are defined and satisfy the above conditions. If $|G_{k-1} \setminus Q_{k-1}| \le \eta n$, we stop the process and set $s := k-1$. Otherwise, we apply \Cref{lem:ind-step} to the graph $G_{k-1} \setminus Q_{k-1}$. Notice that by \ref{itm:full-stab-6} and the assumption on $|G_{k-1} \setminus Q_{k-1}|$, we have $c(G_k \setminus Q_k) \ge f(|G_k \setminus Q_k) - \varepsilon_1 \eta^{-3} |G_k \setminus Q_k|^3$. Since $\varepsilon_1 \eta^{-3} \ll \eta$, the lemma is applicable. The lemma produces a graph $G_k$ on vertex set $V(G_{k-1}) \setminus Q_{k-1} = V \setminus (Q_1 \cup \ldots \cup Q_{k-1})$ satisfying items \ref{itm:full-stab-1} to \ref{itm:full-stab-5}. It remains to verify \ref{itm:full-stab-6}.
Note that
\begin{align*}
c(G_k) = \binom{|Q_k|}{2} \cdot |G_k \setminus Q_k| + c(G_k \setminus Q_k).
\end{align*}
Also
\begin{align*}
c(G_k)
\ge c(G_{k-1} \setminus Q_{k-1})
& \ge f(|G_{k-1} \setminus Q_{k-1}|) - \varepsilon_1 n^3 \\
& = f(|G_k|) - \varepsilon_1 n^3 \\
& \ge \binom{|Q_k|}{2}|G_k \setminus Q_k| + f(|G_k \setminus Q_k|) - \varepsilon_1 n^3,
\end{align*}
where the last inequality follows from the definition of $f$. The two inequalities imply \ref{itm:full-stab-6}.
To finish, run a symmetrisation procedure on $G_s \setminus Q_s$ repeatedly, to obtain a graph $H$ whose vertices are partitioned into full blue clone-cliques $Q_{s+1}, \ldots, Q_t$ (arranged in decreasing size); the edges between any two of them point towards the larger clique; and $c(H) \ge c(G_t \setminus Q_t)$. Let $G'$ be the graph on vertex set $V$, such that $Q_1, \ldots, Q_t$ are blue cliques and the edges between any two of them are red and point towards the larger clique (note that $Q_1, \ldots, Q_t$ partition $V$).
To complete the proof of \Cref{thm:stability}, we need to show that properties \ref{itm:stability-a} to \ref{itm:stability-c} hold.
For \ref{itm:stability-a}, define $G_k'$ to be the graph on vertex set $V$, obtained from $G'$ by replacing $V \setminus (Q_1 \cup \ldots \cup Q_{k-1})$ by a copy of $G_k$ (this makes sense due to \ref{itm:full-stab-1}).
It is easy to see that $c(G_k') - c(G_{k-1}') = c(G_k) - c(G_{k-1} \setminus Q_{k-1}) \ge 0$ for $k \in [s]$, using \ref{itm:full-stab-5}. Similarly, $c(G') \ge c(G_s')$. Altogether, $c(G') \ge c(G_1') = c(G)$, as required for \ref{itm:stability-a}.
Before continuing, we derive an upper bound on $s$. By \ref{itm:full-stab-3} we have $|Q_k| \ge 0.55|G_k|$ for $k \in [s]$, so $|G_k| \le 2^{-(k-1)}n$. Since $|G_t| \le \eta n$, this implies that $s \le 2\log(1/\eta) \le \eta^{-1/2}$, say.
By \ref{itm:full-stab-4} we find that $G'$ and $G$ differ on at most $((s \eta + \eta)n^2 \le 2\eta^{1/2}n^2 \le \varepsilon_2 n^2$ edges. Property \ref{itm:stability-b} follows.
Notice that the estimate $|Q_k| \ge 0.55|G_k|$, which follows from \ref{itm:full-stab-3} implies $|Q_1| \ge \ldots \ge |Q_s|$. Thus \ref{itm:stability-c1} to \ref{itm:stability-c3} clearly hold. Finally, \ref{itm:stability-c4} holds trivially for $k > s$ and, for $k \le s$, it follows from~\ref{itm:full-stab-3} and $\eta \leq \varepsilon_2$.
\end{proof}
\section{Hypergraphs with no short odd pseudocycles} \label{sec:diameter}
In this section we leverage the stability result about cherries, \Cref{thm:stability}, and the connection between hypergraphs with no odd pseudocycles to good colourings (\Cref{thm:good-colouring}) to prove the following result regarding the structure of a dense hypergraph with no short odd pseudocycles. In case of cycles and pseudocycles, the \textit{length} (number of edges) and order (number of vertices) coincide, so, since there is no danger of confusion, we prefer the term \textit{length}. Given vertex sets $X_1, X_2, X_3 \subset V(\mathcal{H})$, an \textit{$X_1X_2X_3$-triple} in $\mathcal{H}$ is an (unordered) edge $x_1x_2x_3 \in E(\mathcal{H})$ with $x_i \in X_i$ for $i \in [3]$.
\begin{theorem} \label{thm:partition}
Let $n \gg \ell \gg 1$.
Let $\mathcal{H}$ be a $3$-uniform hypergraph on $n$ vertices which contains no odd pseudocycles of length at most $\ell$, and which maximises the number of edges under these conditions.
Then there is a partition $\{A, B\}$ of the vertices of $\mathcal{H}$ into non-empty sets such that all $AAB$ triples are edges of $\mathcal{H}$ (and there are no $AAA$ and $ABB$ triples).
\end{theorem}
By iterating the above result, we prove Theorem~\ref{thm:pseudocycles}, restated here, which gives an upper bound on the number of edges in a hypergraph with no short odd pseudocycles.
\thmPseudocycle*
\begin{proof}[Proof of Theorem~\ref{thm:pseudocycles} using \Cref{thm:partition}]
Let $L$ and $n_0$ be such that \Cref{thm:partition} holds for $\ell = L$ and $n \ge n_0$. Denote by $g(n)$ the maximum number of edges in an $n$-vertex hypergraph with no odd pseudocycles of length at most $L$. Then for every $n \ge n_0$ there exists $a_n \in [n-1]$ such that $g(n) \le \binom{a_n}{2}(n-a_n) + g(n-a_n)$. Iterating this and recalling the definition of $f(n)$ implies that $g(n) \le f(n) + \binom{n_0}{3}$.
\end{proof}
Recall that \Cref{thm:pseudocycles} is tight, up to the additive $O(1)$ error term, as evidenced by $\mathcal{H}(x_1, \ldots, x_k)$ for a suitable choice of $x_i$'s.
We next show how \Cref{thm:pseudocycles} implies our main result, \Cref{thm:single-cycle}, restated here.
\thmSingleCycle*
Recall that the \emph{$t$-blow-up} of an $r$-uniform hypergraph $\mathcal{H}$, denoted $\mathcal{H}[t]$, is the hypergraph with vertex set $V(\mathcal{H}) \times [t]$ and edges all $r$-sets $\{(x_1, i_1), \ldots, (x_r, i_r)\}$ such that $\{x_1, \ldots, x_r\} \in E(\mathcal{H})$.
For a family $\mathcal{F}$ of hypergraphs, we denote by $\mathcal{F}[t]$ the family of $t$-blow-ups of members of $\mathcal{F}$. Recall that \Cref{theorem_blow_up} (whose proof can be found in \cite{keevash11}) asserts that taking the $t$-blow-up of a hypergraph does not change its Tur\'an density. The following generalisation for finite families of hypergraphs can be proved similarly.
\begin{theorem}[\cite{keevash11}, Theorem 2.2] \label{thm:ex-blowup}
Let $s$ and $t$ be integers, and let $\mathcal{F}$ be a family of $r$-graphs with $|\mathcal{F}|\leq s$. Then $\pi(\mathcal{F}(t)) = \pi(\mathcal{F})$.
\end{theorem}
To prove \Cref{thm:single-cycle}, we will note that an odd cycle $C^{(3)}_m$ is contained in an \textit{$m$-blow-up} of any odd pseudocycle of length at most $m/2$, and apply the last theorem.
\begin{proof}[Proof of Theorem~\ref{thm:single-cycle} using \Cref{thm:pseudocycles}]
Let $m$ be an integer with $m \geq 2L$ and $3 \nmid m$, where $L$ is the constant from Theorem~\ref{thm:pseudocycles}. Recall that $f(n) = (2\sqrt{3} - 3 + o(1)) \binom n3$. Let $\varepsilon >0$ and let $\mathcal{H}$ be an $n$-vertex 3-uniform hypergraph with
$e(\mathcal{H}) \geq (2\sqrt{3} - 3 + \varepsilon) \binom n3$ and $n$ sufficiently large. We claim that $\mathcal{H}$ contains a copy of $C^{(3)}_m$.
Theorem~\ref{thm:pseudocycles} and Theorem~\ref{thm:ex-blowup} imply that $\mathcal{H}$ contains $F[m]$ for some $\ell$-pseudocycle $F$ with $\ell \leq L$ and $3 \nmid \ell$. It suffices to show that $F$ contains an $m$-pseudocycle, because then $C_m^{(3)}$ will be contained in $F[m]$. To see this, let $v_1 \dots v_\ell$ be an ordering of $V(F)$ such that $v_i v_{i+1} v_{i+2} \in E(F)$, with the indices taken modulo $\ell$.
In case $m \modthree{\ell}$, consider the sequence $$(v_1 v_2 v_3) ^{\frac{m-\ell}{3}}v_1 v_2 \dots v_\ell,$$
where $(v_1 v_2 v_3)^x$ stands for $x$ repetitions of the sequence $v_1 v_2 v_3$.
This is a sequence of order $m$ certifying that $F$ contains an $m$-pseudocycle.
Otherwise, if $m \modthree {2\ell}$, the same is certified for instance by the sequence
\begin{equation*}
(v_1 v_2 v_3) ^{\frac{m-2\ell}{3}} (v_1 v_2 \dots v_\ell) ^2. \qedhere
\end{equation*}
\end{proof}
All that remains now is to prove \Cref{thm:partition}. We will state and prove some preliminary results in the following subsection, and then prove the theorem in \Cref{subsec:proof-partition}.
\subsection{Preparation}
The \emph{diameter} of a hypergraph $\mathcal{H}$ is the minimum $\ell$ such that the following holds: for every $x, y, z, w \in V(\mathcal{H})$ (where $x, y$ are distinct and $z, w$ are distinct) whenever there is a pseudopath from $xy$ to $zw$, there is such a pseudopath of order at most $\ell$.
We have already shown that $n$-vertex hypergraphs with no odd pseudocycles have at most $f(n)$ edges. To prove the same for pseudocycles of bounded length, we will pass to a subhypergraph with bounded diameter, which is the purpose of the following two propositions.
\begin{proposition} \label{prop:diam-cyc}
Let $\mathcal{H}$ be a 3-uniform hypergraph of diameter $\ell \geq 4$. If $\mathcal{H}$ has an odd pseudocycle, then it has an odd pseudocycle of length at most $4 \ell$.
\end{proposition}
\begin{proof}
Let $C$ be the shortest odd pseudocycle in $\mathcal{H}$. Assuming that its length is at least $3\ell +4$, we may index it by $xy v_1 \ldots v_k ab u_1 \ldots u_t$ with $t \geq 2 \ell$, $k \geq \ell $. Note that the length of $C$ is $k+t+4 \not\equiv 0 \pmod{3}$.
Since $\mathcal{H}$ contains a pseudopath from $xy$ to $ab$, it also contains such a pseudopath $P = xy w_1 \ldots w_r ab$ with $r \leq \ell -4$.
The pseudocycle $xy w_1 \ldots w_r ab u_1 \ldots u_t$ is shorter than $C$, so it must not be odd, that is, $r+t+4 \modthree{0}$.
Now consider the pseudocycle $C_1 = v_1 \ldots v_k \tilde{P}$. Recall that $\tilde{P}$ is a $(2r+2)$-vertex pseudopath from $ab$ to $xy$ (see \eqref{eqn:tilde}), so $C_1$ is indeed a pseudocycle.
The length of $C_1$ is $k + 2r +6 \equiv k-r \equiv k+t+4 \not\equiv 0 \pmod{3}$. Noting that $k+2r +6 \leq k + 2\ell - 2 \le k + t$, this contradicts the minimality of $C$.
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\end{proof}
\begin{proposition} \label{prop:small-diameter}
Let $ 1/\ell \ll \varepsilon \ll 1$, and let $\mathcal{H}$ be an $n$-vertex hypergraph.
Then there is a subgraph $\mathcal{H}' \subseteq \mathcal{H}$ with $e(\mathcal{H}') \ge e(\mathcal{H}) - \varepsilon n^3$ whose diameter is at most $\ell$
\end{proposition}
\begin{proof}
First we form a subgraph $\mathcal{H}' \subseteq \mathcal{H}$ in which each vertex pair has codegree either $0$ or at least $\varepsilon n$, as follows. If there are vertices $u, v$ whose codegree in the \emph{current} hypergraph is smaller than $\varepsilon n$, delete all edges containing $uv$. Repeat this step until each pair has codegree degree either $0$ or at least $\varepsilon n$. Denote the resulting hypergraph by $\mathcal{H}'$. Observe that the number of deleted edges is at most $\varepsilon n \cdot \binom n2$ since the edges containing each pair were removed at most once. Hence $e(\mathcal{H}') \geq e(\mathcal{H})-\varepsilon n^3$.
Given ordered pairs $uv$ and $u'v'$ which have a pseudopath connecting them, let $P=uv x_0 x_1 \dots x_t u'v'$ be a shortest such pseudopath. For each $i$, let $B_i$ be the set of ordered pairs $ab$ such that $x_ix_{i+1}ab$ is a tight path. We claim that the sets $B_{10i}$ are mutually disjoint for $0\leq i < \frac{t}{10}$. Suppose not, and take $ab \in B_{10i} \cap B_{10j}$ for some $0 \le i<j < \frac{t}{10}$. Then $x_ix_{i+1}ab x_{j+1}a x_j x_{j+1}$ is a pseudopath with only five vertices between $x_i$ and $x_j$, which can be used to form a shorter pseudopath than $P$ connecting $uv$ and $u'v'$, contradiction.
Now since $|B_i| \geq \varepsilon^2 n^2/2$ for every $i$, using the codegree condition, we have
$$
\floor{\frac{t}{10}}\cdot \frac{\varepsilon^2 n^2}{2} \leq n^2,
$$
so $t \leq \frac{20}{\varepsilon^2}$.
Hence the diameter of $\mathcal{H}'$ is at most $\ell:=\frac{20}{\varepsilon^2}+4$, as required.
\end{proof}
As alluded to in \Cref{sec:overview}, we can already prove \Cref{cor:weak-main}, restated here, which is a weakening of \Cref{thm:pseudocycles}, with only an asymptotic upper bound, which depends on $\ell$, on the number of edges.
\corWeak*
This bound will be used in the proof of \Cref{prop:max-deg}. Note that the analogous bound on the extremal number of proper odd tight cycles follows from \Cref{thm:ex-blowup}.
\begin{proof}[Proof of \Cref{cor:weak-main}]
Assume the opposite, that $e(\mathcal{H}) \geq f(n) + \varepsilon n^3$. Applying Proposition~\ref{prop:small-diameter} with the parameters $\ell/4$ and $\varepsilon/2$, we obtain a hypergraph $\mathcal{H}' \subseteq \mathcal{H}$ with at least $f(n) + \varepsilon n^3 /2$ edges whose diameter is at most $\ell/4$. $\mathcal{H}'$ contains no odd pseudocycles of lenght at most $\ell$, so by \Cref{prop:diam-cyc}, it contains no odd pseudocycles. Hence we may apply Theorem~\ref{thm:good-colouring} to obtain a good colouring of $\partial \mathcal{H}'$ with $e(\mathcal{H}') > f(n)$ cherries, contradicting \Cref{theorem_intro_falgas_ravry}.
%
\end{proof}
The following proposition shows that hypergraphs with no short odd cycles whose number of edges is large are close to being regular.
\begin{proposition} \label{prop:min-deg}
Let $1/n \ll 1/\ell \ll \varepsilon \ll 1$, and let $\mathcal{H}$ be an $n$-vertex hypergraph with no odd pseudocycles of length at most $\ell$, which maximises the number of edges under these conditions. Then $d(u) \ge (3\alpha - \varepsilon)n^2$ for every vertex $u$.
\end{proposition}
\begin{proof}
Given vertices $u$ and $v$ in $\mathcal{H}$, consider the hypergraph $\mathcal{H}_{uv}$ obtained from $\mathcal{H}$ by removing all edges containing $v$ and then adding the edge $e - u + v$, for each edge $e$ that contains $u$ but not $v$. Observe that $\mathcal{H}$ has no odd pseudocycles of length at most $\ell$; indeed, if there were such a cycle then we could replace each instance of $v$ by $u$ to obtain an odd pseudocycle of the same length in $\mathcal{H}$ (whereby it is important that $\mathcal{H}_{uv}$ has no edges containing both $u$ and $v$), a contradiction. Since $e(\mathcal{H}_{uv}) \ge e(\mathcal{H}) - d(v) + d(u) - n$ and by maximality of $\mathcal{H}$, we have $d(v) \ge d(u) - n$. Since $u$ and $v$ were arbitrary, this implies that the maximum and minimum degrees of $\mathcal{H}$ differ by at most $n$. In particular, using $e(\mathcal{H}) \ge f(n) = \alpha n^3 + o(n^3)$, which follows from the maximality of $\mathcal{H}$ and \Cref{prop:alpha},
\begin{equation*}
\delta(\mathcal{H})
\ge \frac{3e(\mathcal{H})}{n} - n \ge \frac{3f(n)}{n} - n \ge (3 \alpha - \varepsilon)n^2.
\qedhere
\end{equation*}
\end{proof}
Next, we prove a stability version of the previous proposition.
\begin{proposition} \label{prop:max-deg}
Let $1/n \ll 1/\ell \ll \varepsilon_1 \ll \varepsilon_2 \ll 1$, and let $\mathcal{H}$ be an $n$-vertex $3$-uniform hypergraph with no odd pseudocycles of length at most $\ell$. If $e(\mathcal{H}) \ge f(n) - \varepsilon_1 n^3$ then $d(u) \le (3\alpha + \varepsilon_2)n^2$ for every vertex $u$.
\end{proposition}
\begin{proof}
Let $\mu = \sqrt{\varepsilon_1} \leq \varepsilon_2/10$.
Let $X$ be the set of vertices $x$ with $d(x) \le 3(\alpha + \mu)n^2$. Then $e(\mathcal{H}) \ge (n - |X|)(\alpha + \mu)n^2$. By \Cref{cor:weak-main} (and the properties of $f(n)$) we also have $e(\mathcal{H}) \le (\alpha + \varepsilon_1)n^3$.
Putting the two inequalities together, we get
\begin{align*}
& (\alpha + \varepsilon_1)n^3 \ge (n - |X|)(\alpha + \mu)n^2 \\
\Longrightarrow \quad
& |X| \ge \frac{(\alpha + \mu)n - (\alpha + \varepsilon_1)n}{\alpha + \mu} = \frac{\mu - \varepsilon_1}{\alpha + \mu} \cdot n \ge \mu n.
\end{align*}
Let $u$ be a vertex of maximum degree in $\mathcal{H}$, and let $X'$ be a subset of $X$ of size $t := \mu n$. We may assume $u \notin X'$ because otherwise $d_{\mathcal{H}}(u) \le (3\alpha + 3\mu)n^2 \le (3\alpha + \varepsilon_2)n^2$, as required.
Now consider the hypergraph $\mathcal{H}_1$ formed in two steps as follows. First, define $\mathcal{H}_0 = \mathcal{H} \setminus X'$; then $e(\mathcal{H}_0) \ge e(\mathcal{H}) - t \cdot 3(\alpha + \mu)n^2$ and $d_{\mathcal{H}_0}(u) \ge d_{\mathcal{H}}(u) - t n$. Second, let $\mathcal{H}_1$ be the hypergraph obtained by adding $|X'|$ copies of $u$ to $\mathcal{H}_0$. Then
\begin{align*}
e(\mathcal{H}_1)
& \ge e(\mathcal{H}_0) + t \cdot d_{\mathcal{H}_0}(u) \\
& \ge e(\mathcal{H}) - t \cdot 3(\alpha + \mu)n^2 + t \cdot (d_{\mathcal{H}}(u) - tn) \\
& \ge f(n) - \varepsilon_1 n^3 + t \cdot (d_{\mathcal{H}}(u) - tn - 3(\alpha + \mu)n^2) \\
& = f(n) - \varepsilon_1 n^3 + \mu n \cdot (d_{\mathcal{H}}(u) - (3\alpha + 4\mu)n^2).
\end{align*}
Notice that $\mathcal{H}_1$ has no odd pseudocycles of length at most $\ell$. Thus, by \Cref{cor:weak-main}, we have $e(\mathcal{H}_1) \le f(n) + \varepsilon_1 n^3$. Hence, using $\mu = \sqrt{\varepsilon_1} \leq \varepsilon_2 / 10$,
\begin{equation*}
d_{\mathcal{H}}(u)
\le (3\alpha + 4\mu)n^2 + (2\varepsilon_1/\mu)n^2
\le (3\alpha + \varepsilon_2)n^2,
\end{equation*}
as required.
\end{proof}
\subsection{The structure of odd-pseudocycle-free graphs} \label{subsec:proof-partition}
We now prove the main result in the section, \Cref{thm:partition}.
The starting point of the proof uses the relation between hypergraphs with no odd pseudocycles and good colourings of $K_n$, as well the stability result about cherries from the previous section, to conclude the following: there is a coloured graph $G$ with a nice structure such that almost all cherries in $G$ are triples in $\mathcal{H}$ and vice versa. This readily implies the existence of a partition $\{A, B\}$ of the vertices such that $|A| \approx \beta n$ and for almost every vertex $u$ in $\mathcal{H}$ the following holds: almost all vertices in $A$ are joined to almost all $A \times B$ pairs, and almost all vertices in $B$ are joined to almost all $A^{(2)}$ pairs. The main difficulty of the proof lies in showing that there is such a partition for which \textit{every} vertex in $A$ is joined to almost all pairs in $A \times B$, and similarly for vertices in $B$. This is achieved in \Cref{claim:structure} and the main idea is to compare several graphs obtained by modifying the triples containing a given vertex. Given a partition as above, to conclude the proof, we argue (using the fact that $\mathcal{H}$ has no short odd pseudocycles) that the number of $AAB$ ``non-edges'' exceeds the number of $AAA$ and $ABB$ edges, unless all of these numbers are 0. The maximality of $\mathcal{H}$ implies that all these numbers are indeed $0$, meaning that $\mathcal{H}$ has all $AAB$ edges and no $AAA$, $ABB$ edges.
\begin{proof}[Proof of \Cref{thm:partition}]
Let $\varepsilon_7 = 0.1$ and let $\varepsilon_1, \ldots, \varepsilon_6$, and $\ell$ satisfy
\begin{equation*}
0 < 1/\ell \ll \varepsilon_1 \ll \ldots \ll \varepsilon_7.
\end{equation*}
Let $\mathcal{H}'$ be a subgraph of $\mathcal{H}$ on the same vertex set with at least $e(\mathcal{H}) - \varepsilon_1 n^3$ edges, that has diameter at most $\ell/4$; such $\mathcal{H}'$ exists by \Cref{prop:small-diameter}. By \Cref{prop:diam-cyc}, $\mathcal{H}'$ has no odd pseudocycles, so by \Cref{thm:good-colouring}, there is a good colouring of $\partial\mathcal{H}'$.
Extending the good colouring of $\partial\mathcal{H}'$ arbitrarily to also cover vertex pairs which are not in the shadow, we obtain a coloured graph (recall that this is a complete graph whose edges are either blue or oriented and red) $G'$ on vertex set $V := V(\mathcal{H})$, such that every edge in $\mathcal{H}'$ is a cherry in $G'$.
By maximality of $\mathcal{H}$, we have $c(G') \ge e(\mathcal{H}') \ge e(\mathcal{H}) - \varepsilon_1 n^3 \ge f(n) - \varepsilon_1 n^3$.
Thus, by \Cref{thm:stability}, there is a graph $G$ satisfying \ref{itm:stability-a}--\ref{itm:stability-c} in \Cref{thm:stability} on vertex set $V$. That is, $G$ has at least as many cherries as $G'$, all but at most $\varepsilon_2 n^3$ cherries in $G$ are cherries in $G'$, and $V$ can be partitioned into sets $X_1, \ldots, X_k$ such that: $G[X_i]$ is blue for $i \in [k]$; $|X_i| = (\beta \pm \varepsilon_2 )n \cdot (|X_{i}| + \ldots + |X_k|)$ for $i \in [k]$; and all $X_i \times X_j$ pairs in $G$ are red and oriented towards $X_i$, for $1 \le i < j \le k$. Recall that $\beta=\frac{3 - \sqrt{3}}{2}$ was defined in \eqref{eq:alphabeta}.
Define $X_{>i} := X_{i+1} \cup \ldots \cup X_k$, and define $X_{\ge i}$ analogously.
Let $H$ be the subgraph of $G$ whose edges are either pairs in $X_i \times X_i$ that are in at least $(|X_{i+1}| + \ldots + |X_k|) - \varepsilon_3 n$ triples in $(X_i \times X_i \times X_{>i}) \cap E(\mathcal{H})$, or pairs in $X_i \times X_j$, where $i < j$, that are in at least $|X_i| - \varepsilon_3 n$ triples in $(X_i \times X_i \times X_j) \cap E(\mathcal{H})$.
Denoting the number of non-edges in $H$ by $\bar{e}(H)$, we have that the number of cherries in $G$ that are not edges in $\mathcal{H}$ is at least $\bar{e}(H) \cdot \varepsilon_3 n / 3$.
Recall that $e(\mathcal{H}') \ge e(\mathcal{H}) - \varepsilon_1 n^3 \ge f(n) - \varepsilon_1 n^3$ and that all edges in $\mathcal{H}'$ are cherries in $G'$. But $c(G') \leq f(n)$ (by~Theorem~\ref{theorem_intro_falgas_ravry}), so all but $\varepsilon_1 n^3$ cherries in $G'$ are edges in $\mathcal{H}'$ and thus in $\mathcal{H}$. Since there are at most $\varepsilon_2 n^3$ cherries in $G$ that are not cherries in $G'$, it follows that all but at most $(\varepsilon_1 + \varepsilon_2)n^3 \le 2\varepsilon_2 n^3$ cherries in $G$ are edges in $\mathcal{H}$.
Hence $\bar{e}(H) \cdot \varepsilon_3 n / 3 \le 2\varepsilon_2 n^3$, showing $\bar{e}(H) \le (6\varepsilon_2/ \varepsilon_3)n^3 \le \varepsilon_3 n^2$.
Let $k_0$ be the maximum $i$ such that $|X_i| \ge \varepsilon_4 n$.
Define subsets $X_i' \subseteq X_i$ as follows: if $i < k_0$ let $X_i'$ be the set of vertices in $X_i$ that have degree at least $|X_i| - \varepsilon_4 n$ in $H[X_i]$ and degree at least $|X_{>i}| - \varepsilon_4 n$ in $H[X_i, X_{>i}]$; if $i \ge k_0$, define $X_i' := \emptyset$.
Since $ (\varepsilon_4 n / 2)\sum_{i < k_0} |X_i \setminus X_i'| \le \bar{e}(H) \le \varepsilon_3 n^2$ and $|X_{\ge k_0}| \le 10 \varepsilon_4 n$ (using \ref{itm:stability-c4}), we have $$\sum_{i \in [k]} |X_i \setminus X_i'| \le 10\varepsilon_4 n + (2\varepsilon_3/\varepsilon_4)n \le 20 \varepsilon_4n.$$
Let $X := X_1' \cup \ldots \cup X_k'$ and $Y := V \setminus X$. We have seen that $|Y| \le 20\varepsilon_4 n \le \varepsilon_5 n$.
For $v \in V$, let $N(v)$ be the \emph{link} of $v$, namely the graph spanned by pairs $uw$ such that $uvw \in E(\mathcal{H})$.
Write $A := X_1'$ and $B := X \setminus X_1'$.
\begin{claim} \label{claim:structure}
One of the graphs $N(u)[A]$ and $N(u)[A, B]$ has at most $\varepsilon_6 n^2$ non-edges, for every $u \in V$.
\end{claim}
\begin{proof}
Let $\varepsilon_5 \ll \mu \ll \varepsilon_6$. Note that the claim holds for all $u \in X$, so it suffices to prove it for $u \in Y$. Fix such $u$.
Let $\mathcal{F}$ be the hypergraph on vertex set $X$ whose edges are all $X_i'X_i'X_j'$ triples with $1 \le i < j \le k_0$. We will construct two hypergraphs $\mathcal{F}_i^{+}$ (for $i \in \{1, 2\}$), that consist of $\mathcal{F}$ with one additional vertex $u_i$, which is a suitable modification of $u$, and that have no odd pseudocycles of length at most $\ell / 10$. We will argue that if both $N(u)[A]$ and $N(u)[A, B]$ have at least $\varepsilon_6 n^2$ non-edges then $d_{\mathcal{F}_i^+}(u_i) >(3\alpha + \mu)n^2$ for some $i \in [2]$, contradicting \Cref{prop:max-deg}.
Let $F_0$ be the graph on vertex set $X$ with edges $E(H) \cap E(N(u))$. Recall that vertices in $X_i'$ have at most $2\varepsilon_4 n$ non-neighbours in $H[X'_{>i}]$. Thus, using \Cref{prop:min-deg} for a lower bound on $d_{\mathcal{H}}(u)$, we have $e(F_0) \ge d_{\mathcal{H}}(u) - |Y| \cdot n - |X| \cdot 2\varepsilon_4 n \ge (3\alpha - 10\varepsilon_5)n^2$.
We modify $F_0$ as follows, while possible: remove each edge $xy$ satisfying: $x, y \in A$ and $x$ has degree $1$ in $A$; or $x \in A$, $y \in B$, and $x$ has degree $1$ into $B$ or $y$ has degree $1$ into $A$. Call the resulting graph $F$ and notice that $|E(F_0) \setminus E(F)| \le 2n$, implying that
\begin{equation} \label{eqn:deg-u}
e(F) \ge (3\alpha - 20 \varepsilon_5)n^2.
\end{equation}
Recall that $\mathcal{F}$ is the hypergraph on vertex set $X$ whose edges are all $X_i'X_i'X_j'$ triples with $1 \le i < j \le k_0$, and let $\mathcal{F}^+$ be the hypergraph obtained by adding the vertex $u$ to $\mathcal{F}$ along with all edges $uvw$ such that $vw \in E(F)$.
We argue that $\mathcal{F}^+$ has no odd pseudocycles of length at most $\ell/10$. To do so, we prove the following.
\begin{align} \label{eqn:F-to-H}
\begin{split}
&\text{Let $xy, vw \in E(H)$, and let $P$ be a pseudopath in $\mathcal{F}$ from $xy$ to $vw$ on $t$ vertices. Then} \\
&\text{there is a pseudopath $P'$ in $\mathcal{H}$ from $xy$ to $vw$ of order $t$ (if $t \in \{2, 3\}$) or $t + 3$ (otherwise).}
\end{split}
\end{align}
We prove \eqref{eqn:F-to-H} by induction on $t$. If $t = 2$ we can take $P' = P$. Suppose that $t = 3$, so $P = xyw$. Let $i_1, i_2, i_3$ be such that $x \in X_{i_1}'$, $y \in X_{i_2}'$ and $w \in X_{i_3}'$.
Since $xy \in E(H)$, we know that for almost every $a \in X_{i_3}'$ the following holds: $xya \in E(\mathcal{H})$ and $ya \in E(H)$; pick such an $a$ with $a \neq w$. Similarly, $yab$ and $ywb$ are edges in $\mathcal{H}$ for almost every $b \in X_{i_1}'$; pick such $b$. The path $xyabyw$ satisfies the requirements.
Next, suppose that $t = 4$, so $P = xyvw$. Let $i_1, i_2, i_3, i_4$ be such that $x \in X_{i_1}'$, $y \in X_{i_2}'$, $v \in X_{i_3}'$ and $w \in X_{i_4}'$. As $xy \in E(H)$, almost all $a \in X_{i_3}'$ satisfy $xya \in E(\mathcal{H})$ and $ya \in E(H)$; fix such $a$. Similarly, almost all $c \in X_{i_2}'$ satisfy $cvw \in E(\mathcal{H})$, $cv \in E(H)$ and $ac \in E(H)$; fix such $c$. Finally, almost every $b \in X_{i_3}'$ satisfies $yab, abc, bcv \in E(\mathcal{H})$; fix such $b$. Then $P' = xyabcvw$ satisfies the requirements.
Finally, suppose that $t \ge 5$, and write $P = v_1 \ldots v_t$, so $x = v_1$, $y = v_2$, $v = v_{t-1}$ and $w = v_t$. Let $i_j$ be such that $v_j \in X_{i_j}'$ for $j \in [t]$. As usual, since $xy = v_1 v_2 \in E(H)$, almost all $a \in X_{i_3}'$ satisfy: $v_2 a \in E(H)$ and $v_1 v_2 a \in E(\mathcal{H})$. Let $Q = v_2 a v_4 \ldots v_t$. Then $Q$ is a pseudopath in $\mathcal{F}$ of order $t-1$ that starts and ends with edges in $H$. By induction, there is a pseudopath $Q'$ in $\mathcal{H}$ from $v_2 a$ to $v_{t-1} v_t$ of order $t + 2$. Then we can take $P' = v_1 Q'$, completing the proof of \eqref{eqn:F-to-H}.
Now suppose that $C = v_1 \ldots v_t$ is a pseudocycle in $\mathcal{F}^+$, where $t \le \ell/10$. We need to show that $t$ is divisible by $3$. If $C$ does not go through $u$, then $C$ is in $\mathcal{F}$, implying that $t$ is indeed divisible by $3$. So we may assume that $C$ goes through $u$ at least once. This shows that $C$ can be written as $u P_1 u \ldots u P_k$, where $P_i$ is a pseudopath in $\mathcal{F}$ whose first two vertices and last two vertices form edges in $F$. It follows from \eqref{eqn:F-to-H} that for each $i \in [k]$ there is a pseudopath $P_i'$ in $\mathcal{H}$ whose first two vertices and last two vertices match those of $P_i$ and whose order satisfies $|P_i'| - |P_i| \in \{0, 3\}$. Then $C' := v P_1' v \ldots v P_k'$ is a cycle in $\mathcal{H}$ with $|C'| \le |C| + 3k \le 4|C| \le \ell$ and $|C'| \modthree{|C|}$. By the properties of $\mathcal{H}$, we have that $|C'|$ is divisible by $3$, implying that $|C|$ is divisible by $3$, as required.
Let $A_0$ and $A_1$ be the sets of vertices in $A$ incident with $AA$ and $AB$ edges in $F$, respectively (that is, $a_0 \in A_0$ if $F$ contains an edge $a_0x$ with $x \in A$).
We claim that $A_0$ and $A_1$ are disjoint. Indeed, if $a_1 \in A_0 \cap A_1$, then there are vertices $b_0 \in B$ and $a_0, a_2 \in A - a_1$ such that $a_0a_1b_0a_2$ is a path in $F$. Let $a_3$ and $b_1$ be arbitrary vertices in $A$ and $B$, respectively (distinct from previously chosen vertices). Then $a_0 a_1 u b_0 a_2 a_3 b_1$ is cycle of length 7 in $\mathcal{F}^+$, a contradiction.
Let $B_1$ be the set of vertices in $B$ incident with $AB$ edges in $F$. Using a similar argument to the above paragraph, we will show that $B_1$ is independent in $F$. Indeed, otherwise there is a path $a_1b_1b_2a_2$ in $F$. Choosing $a_3, a_4 \in A$ and $b_3 \in B$ to be arbitrary unused vertices, we obtain a cycle $a_1b_1ub_2a_2 a_3 b_3 a_4$ of length 8 in $\mathcal{F}^+$ and reach a contradiction.
Let $F_1$ and $F_2$ be graphs on vertex set $X$, defined as follows: $E(F_1) = A \times B$ and $E(F_2) = A^{(2)} \cup E(F[B])$. Now define $\mathcal{F}^+_i$ to be the graph obtained from $\mathcal{F}$ by adding a new vertex $u_i$ and edges $u_i vw$ such that $vw \in E(F_i)$, for $i \in [2]$. Thus $\mathcal{F}^+_i$ and $\mathcal{F}^+$ differ only on edges touching $u_i$ or $u$.
We claim that $\mathcal{F}^+_i$ has no odd pseudocycles of length at most $\ell/10$. Indeed, this is easy to see for $i = 1$, because we can think of $\mathcal{F}^+_1$ as obtained by extending $X_1'$ by one vertex. To see that this also holds for $i = 2$, notice that in $\mathcal{F}^+_2$, the $AAB$ and $BBB$ triples are in different strong components, so any pseudocycle $C$ in $\mathcal{F}^+_2$ is either a pseudocycle in $\mathcal{F}^+$ or consists only of edges containing exactly two vertices from $A$.
Notice that $e(\mathcal{F}^+_i) \ge c(G) - |Y|n^2 \ge f(n) - (\varepsilon_1 + \varepsilon_5) n^3 \ge f(n) - 2\varepsilon_5 n^3$, because all cherries in $G$ that do not touch $Y$ are edges in $\mathcal{F}$ and $c(G) \ge c(G') \ge f(n) - \varepsilon_1 n^3$. Using this lower bound and the fact that $\mathcal{F}^+_i$ has no odd pseudocycles of length at most $\ell/10$, \Cref{prop:max-deg} implies that $d_{\mathcal{F}^+_i}(u_i) \le (3\alpha + \mu)n^2$.
Since $d_{\mathcal{F}^+}(u) = e(F) \ge (3\alpha - 20\varepsilon_5)n^2$ (see \eqref{eqn:deg-u}), we have $e(F_i) - e(F) = d_{\mathcal{F}^+_i}(u_i)-d_{\mathcal{F}^+}(u) \le (\mu + 20\varepsilon_5)n^2 \le 2\mu n^2$ for $i \in [2]$.
To finish, suppose first that $|A_0| \ge |B_1|$. Recalling that $F$ and $F_1$ coincide on $B$, and that $F$ has no edges in $(A_1 \cup B_1) \times A_0$ or $A_1^{(2)}$, we have
\begin{align*}
2\mu n^2 \ge e(F_2)-e(F)
& \ge - |A_1||B_1| + |A_0||A_1| + \binom{|A_1|}{2} + \bar{e}(F[A_0]) \\
& \ge \frac{|A_1|^2}{2} + \bar{e}(F[A_0]) + O(n).
\end{align*}
It follows that $|A_1| \le 5\mu^{1/2} n$ and $\bar{e}(F[A_0]) \le 5\mu n^2$. Altogether $\bar{e}(F[A]) \le |A_1| \, n + \bar{e}(F[A_0]) \le 10 \mu^{1/2} n^2 \le \varepsilon_6 n^2$. Since $F[A] \subseteq N(u)[A]$, \Cref{claim:structure} is proved in this case.
Now we consider the remaining case, namely that $|A_0| \le |B_1|$. Let $B_0 = B \setminus B_1$, and recall that $F$ has no edges in $B_1^{(2)}$ or in $A_0 \times B_1$. Using $|A| \ge |B| = |B_0| + |B_1|$,
\begin{align*}
2\mu n^2 & \ge e(\mathcal{F}^+_1) - e(\mathcal{F}^+) \\
& \ge -\binom{|A_0|}{2} - \binom{|B_0|}{2} - |B_0||B_1| + |A||B_0| + |A_0||B_1| + \bar{e}(F[A_1, B_1]) \\
& \ge |A_0|(|B_1| - |A_0|) + |B_0|(|A| - |B_0| - |B_1|) + \frac{|A_0|^2}{2} + \frac{|B_0|^2}{2} + \bar{e}(F[A_1, B_1]) + O(n) \\
& \ge \frac{|A_0|^2}{2} + \frac{|B_0|^2}{2} + \bar{e}(F[A_1, B_1]) + O(n).
\end{align*}
Thus, we have $|A_0|, |B_0| \le 5 \mu^{1/2} n$ and $\bar{e}(F[A_1, B_1]) \le 5\mu n^2$.
This implies that $\bar{e}(F[A, B]) \le |A_0|\, n + |B_0|\, n + \bar{e}(F[A_1, B_1]) \le \varepsilon_6 n^2$, proving \Cref{claim:structure}.
\end{proof}
Let $A^*$ be the set of vertices $u$ such that $N(u)[A, B]$ has at most $\varepsilon_6 n^2$ non-edges, and let $B^* := V \setminus A^*$. Note that $A \subseteq A^*$, and by \Cref{claim:structure}, for every $u \in B^*$ the graph $N(u)[A]$ has at most $\varepsilon_6 n^2$ non-edges.
Let $t_1$ be the number of $A^*\AsA^*$ triples in $\mathcal{H}$, let $t_2$ be the number of $A^*B^*\Bs$ triples in $\mathcal{H}$, and let $s$ be the number of $A^*\AsB^*$ triples that are not edges in $\mathcal{H}$.
Let $\mathcal{H}^*$ be the hypergraph obtained from $\mathcal{H}$ by removing all $A^*\AsA^*$ and $A^*B^*\Bs$ triples and adding all missing $A^*\AsB^*$ triples. Then $\mathcal{H}^*$ has no odd pseudocycle of length at most $\ell$; this follows from observing that every pseudocycle in $\mathcal{H}^*$ is either a pseudocycle in $\mathcal{H}$ or each of its edges has exactly two vertices in $A^*$. Moreover, $e(\mathcal{H}^*) - e(\mathcal{H}) = s - (t_1 + t_2)$. By maximality of $\mathcal{H}$ we have $s \le t_1 + t_2$.
\begin{claim}
$t_1 \le \varepsilon_7 s$.
\end{claim}
\begin{proof}
Let $\varepsilon_6 \ll \mu \ll \varepsilon_7$.
We first show that for every distinct $u, v \in A^*$, there are at most $\mu n$ vertices $w \in A^*$ such that $uvw \in E(\mathcal{H})$.
Suppose there exist $u, v \in A^*$ violating this. Let $W$ be the set of vertices $w \in A^*$ such that $uvw \in E(\mathcal{H})$, so $|W|\geq \mu n$.
Consider the graph $(N(u) \cap N(v))[W, B]$; its edges are pairs $wb$ such that $w \in W$, $b \in B$, and $uwb, vwb \in E(\mathcal{H})$. This graph has at most $2\varepsilon_6 n^2$ non-edges, by \Cref{claim:structure}. Thus there exists $b \in B$ with at least $\frac 12 \mu n$ neighbours in the aforementioned graph; denote its set of neighbours by $W'$. Now, by \Cref{claim:structure}, $b$ is adjacent in $\mathcal{H}$ to all but at most $\varepsilon_6 n^2$ pairs in $W'$, so there exists a triple $w_1w_2b \in E(\mathcal{H})$ with $w_1, w_2 \in W'$. Thus $uvw_1bw_2$ is a pseudocycle of length 5, contradiction.
To finish the argument, we count the four-tuples
\begin{equation*}
Q := \{ \{u, v, w, z \}: u, v, w \in A^*, z \in B^*, uvw \in E(\mathcal{H}), uvz \notin E(\mathcal{H})\}
\end{equation*}
in two different ways. For each vertex $b \in B^*$ and $A^*\AsA^*$ triple $uvw \in E(\mathcal{H})$, at least one of the triples $uvb, uwb, vwb$ is not in $E(\mathcal{H})$, so $|Q| \geq t_1|B^*|$. On the other hand, it follows from the above paragraph that any $A^*\AsB^*$ triple $uvz \notin E(\mathcal{H})$ extends to at most $\mu n$ elements of $Q$, so $|Q| \leq s \mu n$. Hence
\begin{equation*}
t_1 \leq \frac{|Q|}{|B^*|} \le \frac{s \mu n}{|B^*|} \leq \varepsilon_7 s,
\end{equation*}
as claimed.
\end{proof}
\begin{claim}
$t_2 \le 2s/3$.
\end{claim}
\begin{proof}
Let $\varepsilon_6 \ll \mu \ll \varepsilon_7$.
To begin with, we show that if $uvw$ is an $A^* B^* B^*$ triple in $\mathcal{H}$ (with $u \in A^*$) then one of the pairs $uv$ and $vw$ is in at most $\mu n$ triples of form $A^*\AsB^*$ in $\mathcal{H}$. Fix an $A^*\AsB^*$ triple $uvw \in E(\mathcal{H})$.
Let $W'$ (resp.\ $V'$) be set of vertices $a \in A^*$ such that $uwa \in E(\mathcal{H})$ (resp.\ $uva \in E(\mathcal{H})$). Suppose that $|W'|, |V'| \geq \mu n$. Consider the graph $(N(v) \cap N(w))[W', V']$. By~\Cref{claim:structure}, this graph contains an edge $a_1a_2$, i.e.\ we have~$a_1a_2w$, $a_1a_2v \in E(\mathcal{H})$. By definition of $W'$ and $V'$, the triples $uwa_1$ and $uva_2$ are in $\mathcal{H}$. Hence $uwa_1a_2v$ is a cycle of length 5, contradiction.
Let $F$ be an auxiliary bipartite graph with parts $A^*$ and $B^*$ such that $uv$ is an edge of $F$ whenever (i) there is an $A^*B^*\Bs$ triple in $\mathcal{H}$ containing $uv$, and (ii) the number of $A^*\AsB^*$ triples containing $uv$ is at most $\mu n$. By the previous paragraph, each $A^*B^*\Bs$ triple in $\mathcal{H}$ contains an edge of $F$, so
\begin{equation*}
t_2 \leq |B^*| \cdot e(F) \leq 0.4n \cdot e(F).
\end{equation*}
Moreover, we claim that $d_F(v) \le \mu n$ for every $v \in B^*$. Indeed, by (ii), the graph $N(v)[A^*]$ has at least $d_F(v)(|A^*| - \mu n)/2$ non-edges. If $d_F(v)>\mu n$ then this quantity is larger than $2\varepsilon_6 n^2$, contradicting \Cref{claim:structure}. Also using (ii), we conclude that
\begin{equation*}
s \ge \sum_{v \in B^*} d_F(v) \cdot (|A^*| - d_F(v) - \mu n) \ge 0.6 n \cdot e(F).
\end{equation*}
It follows that $t_2 \le 2s/3$, as claimed.
\end{proof}
The last two claims, and the choice $\varepsilon_7=0.1$, say, show that $(t_1 + t_2) \le 0.8 s$. Since $s \le t_1 + t_2$ this implies that $t_1 = t_2 = s = 0$. That is, all $A^*\AsB^*$ triples are edges in $\mathcal{H}$ (and there are no $A^*\AsA^*$ or $A^*B^*\Bs$ edges). This proves \Cref{thm:partition}.
\end{proof}
\section{Open problems}
There are two natural extensions of our result. Firstly, one could prove~\Cref{Conjecture_Mubayi_Rodl}, or perhaps determine the density of $C_\ell^{(3)}$ for smaller values of $\ell$, say $\ell \leq 100$. Although we do not state our bound on $\ell$ explicitly, this would not be too cumbersome, since it is a polynomial in $\varepsilon_7$, and we set $\varepsilon_7 = 0.1$.
The second direction is determining the Tur\'an density of $r$-uniform tight cycles for $r \geq 4$. For this, we do not even know of a conjectured optimal construction. Moreover, our characterisation of odd-pseudocycle-free hypergraphs (\Cref{thm:good-colouring}) does not have an obvious extension, as the straightforward extension of Definition~\ref{def:good-col} is too strong
As mentioned in the introduction, there are many other specific 3-uniform hypergraphs for which determining the Tur\'an density would be very interesting. Let us point out one conjecture which is perhaps less well known, and which can be found for instance in \cite{mubayi2011hypergraph}.
\begin{conjecture} ~\label{conj:c5-minus}
Let $\mathcal{C}_5^{-}$ be the 3-uniform hypergraph obtained from the tight 5-cycle $\mathcal{C}_5^3$ by removing one edge. The Tur\'an density of $\mathcal{C}_5^{-}$ is $\frac 14$.
\end{conjecture}
As in our case, one conjectured extremal hypergraph is an iterated construction; one may take a complete 3-partite 3-uniform hypergraph and then repeat the same construction recursively within each of the three parts.
\subsection*{Acknowledgements}
We would like to thank Xizhi Liu, Yuejian Peng, and Oleg Pikhurko for bringing to our attention several important references.
|
1,314,259,996,743 | arxiv | \section{Introduction}\label{vved}
A distinctive feature of the gravity description in the framework of general relativity is a variety of its formulations, mostly geometric ones. Among the geometric formulations there are not only purely metric ones (as in the original General Relativity (GR)), but also others, which possess the existence of additional geometric structures that are not reducible to the metric. In many cases for the introduction of such structures one need to extend the formalism or even the physical essence of GR, e.~g., for the introduction of an orthogonal frame \cite{cartan2001} or an isometric embedding \cite{regge,statja18}.
However, there is a geometrical object which is \textit{a priori} not reducible to metric: a connection on the considered manifold.
In the original Einsteinian formulation the connection is present, but it is postulatively expressed through metric. One can thus raise a natural question: if one rejects the initial consistency of connection with metric, what the resulting theory would be like?
It is now well known that if the metric and connection are assumed to be independent variables, then in the assumption of the zero torsion the variation of the Einstein-Hilbert action with respect to metric leads to usual Einstein equations, and with respect to connection --- the expression for connection in the form of Christoffel symbols which are symmetric. The resulting theory is equivalent to GR. In the literature it is usually called the Palatini formalism (or Hilbert-Palatini formalism), although such a naming is not quite correct. In the paper by Palatini \cite{palatini} the variational principle for Einstein-Hilbert (EH) action was considered; but connection was not treated as an independent variable (a detailed historical survey of this question can be found in \cite{ferraris_pal}). For the first time it seems to have been done by Einstein \cite{ein_conn} in the attempt of unified field theory (UFT) construction. Note that his approach was more general than Palatini one: he assumed the \textit{ metric} is non-symmetric as well as the connection; and identified the antisymmetric part of the metric with the electromagnetic (EM) field tensor.
In the present paper we want to discuss a variant of gravity and electromagnetism unification which was somehow missed by Einstein and which seems far more natural. It appears when one assumes the independency of metric and connection, considering metric as symmetric and connection as non-symmetric, i.~e. all 64 components of connection is considered to be independent. Then it turns out to be possible to identify some components of the connection with the EM potential. During the XX century researchers closely approached this idea more than once (we give a short historical review of the corresponding publications in the section \ref{istor}), but for the first time it was fully embodied only in 1978 in the paper \cite{Krechet}. Moreover, the paper \cite{Krechet} remains practically unknown to the scientific community because of poor availability of the journal in which it was published. The main addition of our work to the previously known results is the natural method of coupling the connection with matter (in the form of a set of relativistic point particles), after which the theory takes the form of usual Einstein-Maxwell theory with matter.
In the section \ref{urav} we derive the field equations for the metric and connection using the variational principle for the standard EH action. This derivation is given here for ease of reading: although this interesting result was repeatedly obtained earlier by many authors, it is still not well known.
In the section \ref{mater} we consider a problem of including the matter (in the form of relativistic particles) in the variational principle. As we will show below, it is possible to organize non-minimal (i.~e. not reducible to the usual covariantization of derivatives) coupling of matter with connection that does not broke the general covariance. The field equations of the resulting theory are examined in the section~\ref{urdv}. The analysis shows that they reproduce Einstein-Maxwell equations.
The simplicity and naturalness of the constructed theory raise a question: why it had not been discovered by any of researchers who work in the field of the UFT: Einstein, Weyl, Eddindton etc.? The section~\ref{istor} is devoted to the review of early theories of geometrized electrodynamics and some recent works which are related
to the theory discussed here. In the last section~\ref{zakl} we briefly discuss the problems and perspectives of generalization of this theory.
\section{The affine-metric formulation of pure gravity}\label{urav}
Let us consider the affine-metric theory of gravity, assuming that the metric and connection are independent. The metric, as usual, is supposed to be symmetric, whereas connection is not (i.~e. the torsion is nonzero). In the present paper we choose the $(+,-,-,-)$ signature of spacetime. As an action of pure gravity (without matter) we choose the usual EH one
\begin{align}\label{1}
S_1=-\frac{1}{2\ka}\int\! d^4 x\, \sqrt{-g}\,g^{\n\be}\, R^{\m}\,\!_{\n\m\be},
\end{align}
where the Riemann-Christoffel curvature tensor is expressed only through independent connection $\Gamma^\alpha_{\mu\nu}$:
\begin{align}\label{s1}
R^{\mu}\,\!_{\nu\alpha\beta} = \partial_\alpha \Gamma^{\mu}_{\beta\nu}-\partial_\be \Gamma^{\mu}_{\al\nu}+
\Gamma^{\mu}_{\alpha\gamma} \Gamma^{\gamma}_{\beta\nu}-\Gamma^{\mu}_{\be\gamma} \Gamma^{\gamma}_{\al\nu}.
\end{align}
In \eqref{1} $g^{\n\be}$ is a metric, $g$ is its determinant and $\ka$ is an Einstein gravitational constant.
To fix the order of the indices of connection (which is not symmetric), we define an action of covariant derivative on an arbitrary vector as follows:
\disn{s2.0}{
D_\m u^\al=\dd_\m u^\al+\Gamma^\alpha_{\mu\nu}u^\n,
\nom}
i.~e. the first index of connection is the one that is related to the derivative. Note that while the connection remains independent, the Ricci tensor
\disn{s2.1}{
R_{\n\be}=R^{\m}\,\!_{\n\m\be}
\nom}
is not symmetric.
One can notice that the action \eqref{1} possesses not only diffeomorphic invariance, but also an additional invariance with respect to the so-called \qq{projective transformations} \cite{sandberg1975} (it is worth noting that such transformations were considered earlier by Einstein in \cite{ein1955-1})
\begin{align}\label{gauge}
{\Gamma'}^\alpha_{\mu\nu} = {\Gamma}^\alpha_{\mu\nu} + f_\mu \delta^\alpha_\nu
\end{align}
with the arbitrary function $f_\m$, i.~e. some gauge transformations.
Indeed, as one can see from \eqref{s1}, such a transformation adds an explicitly antisymmetric expression to the Ricci tensor
\begin{align}\label{ss1}
{R'}_{\n\be} = {R}_{\n\be} + \partial_\n f_\be - \partial_\be f_\n
\end{align}
(this result was obtained in \cite{ein1955-1}) which in \eqref{1} is contracted with symmetric metric and thus does not give a contribution. The action is a quadratic function of connection, therefore its symmetry with respect to transformations \eqref{gauge} leads to the fact that quadratic form which is contained in it is degenerate and has rank 60 while its dimension is 64 (64 is the number of independent components of connection $\Gamma^\alpha_{\mu\nu}$). Such a degeneracy of the action means that one can not find all the components of connection from the equation which is obtained from the action by varying it with respect to connection (note that in Palatini formalism with symmetric connection it is possible). A part of connection components must remain arbitrary. Let us prove it by a direct calculation.
Let us find the variation of the action \eqref{1} with respect to the connection. Using \eqref{s1} one can easily find that
\disn{s2}{
\de S_1=-\frac{1}{2\ka}\int\! d^4 x\, \sqrt{-g}\,g^{\n\be}\,
\ls D_\m \de\Gamma^{\mu}_{\beta\nu}-D_\be \de\Gamma^{\mu}_{\m\nu}+S^\ga{}_{\m\be}\de\Gamma^{\mu}_{\ga\nu}\rs,
\nom}
where a notation for the torsion was used:
\disn{s3}{
S^\al{}_{\m\n}=\Gamma^\al_{\m\n}-\Gamma^\al_{\n\m}.
\nom}
Let us introduce the quantity which determines the difference between the connection and its Riemannian part:
\disn{s4}{
C_\m{}^\al{}_\n=\Gamma^\al_{\m\n}-\bar\Gamma^\al_{\m\n},
\nom}
where $\bar\Gamma^\al_{\m\n}$ is a symmetric Riemannian connection expressed through metric (Christoffel symbols):
\begin{align}\label{s5}
\bar\Gamma^\alpha_{\mu\nu} = \frac{1}{2} g^{\alpha\beta}(\partial_\mu g_{\nu\beta}+\partial_\nu g_{\mu\beta}-\partial_\beta g_{\mu\nu}).
\end{align}
A so-called non-metricity can be easily expressed through $C_\m{}^\al{}_\n$
\disn{s6}{
D_\m g^{\n\be}=C_\m{}^{\n\be}+C_\m{}^{\be\n},
\nom}
as well as the torsion
\disn{s9.1}{
S^\al{}_{\m\n}=C_\m{}^\al{}_\n-C_\n{}^\al{}_\m.
\nom}
Note that using \eqref{s4} one can (up to a boundary term) write a relation
\disn{s7}{
\int\! d^4 x\, \sqrt{-g}\, D_\m u^\m=
\int\! d^4 x\, \sqrt{-g}\, \ls \bar D_\m u^\m+C_\m{}^\m{}_\n u^\n\rs=
\int\! d^4 x\, \sqrt{-g}\, C_\m{}^\m{}_\n u^\n,
\nom}
for the arbitrary vector $u^\m$, where $\bar D_\m$ is a Riemannian covariant derivative which contains only the Riemannian connection\eqref{s5}. Using this relation in the integration by parts as well as the formulas \eqref{s6} and \eqref{s9.1} one can perform a simple calculation and, dropping the boundary term, rewrite the variation \eqref{s2} in the following form:
\disn{s8}{
\de S_1=\frac{1}{2\ka}\int\! d^4 x\, \sqrt{-g}\ls
\ls C_\m{}^{\n\be}+C^{\n\be}{}_\m-C_\ga{}^\ga{}_\m g^{\n\be}\rs\de\Gamma^\m_{\be\n}-
C_\m{}^{\n\m}\de\Gamma^\be_{\be\n}\rs.
\nom}
The corresponding field equation is easy to obtain from it:
\disn{s9}{
C^{\m\n\be}+C^{\n\be\m}-C_\ga{}^{\ga\m} g^{\n\be}-C_\al{}^{\n\al}g^{\be\m}=0.
\nom}
Contracting the indices in multiple ways, one can easily notice that one way leads to the identity, whereas two others --- to the relations between different contractions of $C^{\m\n\be}$:
\disn{s10}{
C_{\ga\m}{}^{\ga}=C_\ga{}^\ga{}_\m=\frac{1}{4}C_{\m\ga}{}^\ga.
\nom}
However, the value of this contractions is not determined by \eqref{s9}, and remains arbitrary, as we mentioned above. Let us introduce the notation for it: $C_{\ga\m}{}^{\ga}\equiv \om A_\m$,
where $A_\m$ is an arbitrary vector field and $\om$ is a constant which will be fixed later.
Other components of $C^{\m\n\be}$ can be uniquely expressed through this quantity. To prove it, one need to consider a linear combination of three copies of \eqref{s9}, which differ by cyclic permutation of indices. As a result, one obtains
\disn{s11}{
C_\m{}^\n{}_\be=\om A_\m\de^\n_\be,
\nom}
and the final expression for the connection can be found using \eqref{s4}:
\disn{s12}{
\Gamma^\al_{\m\n}=\bar\Gamma^\al_{\m\n}+\om A_\m\de^\al_\n.
\nom}
As can be seen, field equation corresponding to variation of the action \eqref{1} with respect to connection leads to the fact that the connection coincides with the Christoffel symbols \eqref{s5} up to the gauge transformation \eqref{gauge}. Therefore some of components of connection $\Gamma^\al_{\m\n}$ corresponding to vector $A_\m$ can not been expressed through metric, remaining arbitrary. For the connection \eqref{s12} both the torsion and non-metricity are expressed through this vector:
\disn{s13}{
S^\al{}_{\m\n}=\om\ls A_\m\de^\al_\n-A_\n\de^\al_\m\rs,\qquad
D_\m g^{\n\be}=2\om A_\m g^{\n\be}.
\nom}
Spacetime with such non-metricity is called the Weyl-Cartan spacetime, and vector $-2\om A_\m$ is called the Weyl vector \cite{hehl}.
Note that for the connection of the form \eqref{s12}, i.~e. on-shell, $A_\m$ is simply connected with the traces of torsion an non-metricity tensors:
\disn{s14}{
A^\m=\frac{1}{3\om}S^{\ga\m}{}_\ga=\frac{1}{2\om}D_\ga g^{\ga\m},
\nom}
whereas in the curvature tensor \eqref{s1} is is contained as follows:
\disn{s14.1}{
R^\m{}_{\n\al\be}=\bar R^\m{}_{\n\al\be}+\om\de^\m_\n\ls\partial_\al A_\be - \partial_\be A_\al\rs,
\nom}
where $\bar R^\m{}_{\n\al\be}$ is a Riemannian expression for the curvature tensor which is constructed from \eqref{s5}.
Now let us find the variation of \eqref{1} with respect to metric. This task turns out to be quite simple since the action depends on the metric itself but not on its derivatives. After the variation we obtain
\begin{align}\label{ein}
R_{\mu\nu}+R_{\nu\mu}-R\, g_{\mu\nu} = 0,
\end{align}
where $R=g^{\al\be}R_{\al\be}$ is a scalar curvature.
The full system of field equations for the action \eqref{1} is a set of equations \eqref{ein} and \eqref{s12}.
If \eqref{s12} is satisfied, then it is easy to obtain from \eqref{s14.1} that
\disn{s15}{
R_{\mu\nu} = \bar R_{\mu\nu} + \om\ls\partial_\m A_\n - \partial_\n A_\m\rs,
\nom}
where $\bar R_{\mu\nu}$ is a Riemannian expression for the curvature tensor which is constructed from the Riemannian connection $\bar\Gamma^\al_{\m\n}$; therefore $\bar R_{\mu\nu}$ is symmetric and contains only metric. Making use of this fact, one can rewrite the equation \eqref{ein} in the usual form of the vacuum Einstein equations:
\disn{s16}{
\bar G_{\mu\nu}=0,
\nom}
where the Riemannian Einstein tensor $\bar G_{\mu\nu}$ is expressed through metric in a usual manner.
Thus for affine-metric formulation of pure gravity we see the satisfaction of vacuum Einstein equations for the metric, whereas the connection turns out to have its Riemannian value up to the gauge transformation \eqref{gauge} which leaves some degrees of freedom (DoF) of connection to be arbitrary. Such a theory can be treated \cite{sandberg1975} as equivalent to pure Einstein gravity if the vector field $A_\m$ is considered as a pure gauge DoF.
\section{ Interaction with relativistic particles}\label{mater}
Let us add the matter to the theory in the form of a set of relativistic point particles. First of all, let us add to the full action a standard term for relativistic particles with worldlines $x^\m_j(\ta)$ (index $j$ counts particles, $\ta$ parametrizes points of its worldlines) in a gravitational field defined by metric $g_{\m\n}$:
\begin{align}\label{s17}
S_2 = -\sum_j m_j \int\! d\ta\, \sqrt{\dot{x}_j^{\mu}(\ta)\dot{x}_j^\nu(\ta)g_{\mu\nu}(x_j(\ta))}.
\end{align}
Here $m_j$ is a mass of corresponding particle and $\dot{a}\equiv da/d\tau$.
If we restrict ourselves to a sum of \eqref{1} and \eqref{s17} in the action, then the forced acting on particles are equivalent to those in GR, and degrees of freedom corresponding to $A_\m$ (see \eqref{s12}) remain purely gauge. However, one can raise a question: what else local contributions containing gravitational and matter degrees of freedom one can add to the action assuming their invariance with respect to diffeomorphisms $x^\m\to x'^\m(x)$ as well as the reparametrizations of worldlines $\ta\to \ta'(\ta)$? It turns out that there is several possibilities to construct such contributions; one of them is the following:
\begin{align}\label{s18}
S_3 = -\sum_j \frac{q_j}{4\om} \int\! d\ta\, \dot{x}_j^{\be}(\ta) \Gamma^\m_{\be\m}(x_j(\ta)),
\end{align}
where $q_j$ are some arbitrary constants and the multiplier $1/(4\om)$ is written for the convenience of subsequent discussion. It is worth noting that in the case of Riemannian connection such a contribution is reduced to the addition of a full derivative with respect to $\tau$ to the Lagrangian due to the known feature of Riemannian connection:
\disn{s31}{
\bar\Gamma^\n_{\m\n}=\dd_\m\ln\sqrt{-g}.
\nom}
The choice of matter coupling with connection in the form \eqref{s18} leads to the most interesting results. The invariance of \eqref{s18} with respect to reparametrizations $\ta\to \ta'(\ta)$ is obvious, whereas its diffeomorphic invariance is present only up to a certain boundary terms and requires additional discussion.
Note that under the diffeomorphisms $x^\m\to x'^\m(x)$ the contracted connection transforms as
\begin{align}\label{part_conn}
{\Gamma}'^\m_{\be\m} = \frac{\partial x^\nu}{\partial x'^\be} \left( \Gamma^\m_{\nu\m} - \partial_\nu\ln\det\left| \frac{\partial x'}{\partial x} \right| \right),
\end{align}
so the quantity \eqref{s18} takes an increment
\begin{align}\label{diff_surf}
\Delta S_3 =
\sum_j \frac{q_j}{4\om} \int\! d\ta\, \dot{x}_j^{\be}(\ta) \partial_\be\ln\det\left| \frac{\partial x'}{\partial x} \right|=
\sum_j \frac{q_j}{4\om} \int\! d\ta\, \frac{d}{d\ta} \ln\det\left| \frac{\partial x'}{\partial x} \right|.
\end{align}
Such an increment in the form of the integral over full derivative does not affect the field equations of the theory. Its appearing after diffeomorphic transformations thus does not lead to breaking of general covariance of the field equations, as we explicitly show below.
Strictly speaking, the contribution \eqref{s18} to the action is not fully diffeomorphic invariant. It is invariant only with respect to a narrower group of coordinate transformations which are restricted by a condition
\disn{s19}{
\det\left| \frac{\partial x'}{\partial x} \right| \str{x^0\to\pm\infty} 1.
\nom}
It is worth noting that if the term \eqref{s18} is added to the full action, it is no longer invariant with respect to \eqref{gauge}. The invariance with respect to narrower group of gauge transformations (Einstein called them \qq{$\la$-transformations} \cite{ein1955-2}) remains:
\disn{s20}{
{\Gamma'}^\alpha_{\mu\nu} = {\Gamma}^\alpha_{\mu\nu} + (\dd_\mu \la) \delta^\alpha_\nu.
\nom}
The expression \eqref{s18} is strictly invariant with respect to \eqref{s20} (boundary terms do not appear) only in the additional assumption $\la \str{x^0\to\pm\infty} 0$ analogous to \eqref{s19} which again does not affect the invariance of the field equations.
One can notice that simultaneous coordinate change and transformation \eqref{s20} does not lead to the appearing of boundary terms if the condition
\disn{s21}{
\ln\det\left| \frac{\partial x'}{\partial x} \right|+4\la\str{x^0\to\pm\infty} 0.
\nom}
is satisfied. The Ricci tensor turns out to be invariant with respect to \eqref{s20} (but not to \eqref{gauge}), see \eqref{ss1}.
The theory corresponding to the choice of $S_1+S_2+S_3$ as a final form of action turns out to be self-inconsistent. Indeed, since the increment \eqref{s18} which was added to the original action \eqref{1} partially broke the gauge invariance with respect to \eqref{s20}, three of four degrees of freedom of $A_\m$ (see \eqref{s12}), which initially were gauge ones, become physical ones. However, they did not become dynamical ones since $S_1$ and $S_2$ do not depend on them and $S_3$ depends only linearly. Varying with respect to them leads to the equations
\begin{align}\label{s22}
\dot{x}_j^\be(\ta) = 0,
\end{align}
which are self-inconsistent because correct parametrization of the worldline means that this quantity cannot vanish. One can solve this problem by choosing the more complicated expression than EH term \eqref{1} for the action of gravity, so abovementioned degrees of freedom may become dynamical ones. In choosing such an expression one should keep in mind the following fact.
First of all, we note that in the framework of GR, when the only independent variable is metric, whereas connection is uniquely expressed through it as Christoffel symbols \eqref{s5}, the EH action is the only generally covariant action which is not lead to the higher derivatives in the field equations. However, for the theory considered here, in which the connection is independent, it is possible to construct some other invariants which possess such a property. To construct them one can use the curvature tensor \eqref{s1} as well as the torsion tensor \eqref{s3}.
In contrast with the case of Riemannian geometry corresponding to GR, when only one nontrivial contraction of curvature tensor exists (the one that leads to the Ricci tensor \eqref{s2.1}), in the case of independent connection there is one more nontrivial contraction:
\disn{s23}{
R^\m{}_{\m\al\be}=\dd_\al\Gamma^\m_{\be\m}-\dd_\be\Gamma^\m_{\al\m}.
\nom}
This antisymmetric tensor was introduced early in the Eddington theory (see below in the section \ref{istor}). According to Cartan's terminology it is called a \textit{segmentary curvature} or a \textit{homothety curvature}, see \cite{vizgin}. Let us choose a new term in addition to EH one \eqref{1} as an action of gravity in the form of quadratic expression with respect to contraction \eqref{s23}:
\disn{s24}{
S_4 = -\theta \int\! d^4 x\, \sqrt{-g}\,g^{\al\ga}g^{\be\de}
R^{\mu}{}_{\mu\alpha\beta} R^{\nu}{}_{\nu\ga\de},
\nom}
where $\te$ is an arbitrary constant.
The resulting form of the action for the considered theory is the following:
\disn{s25}{
S = (S_1+S_4)+S_2+S_3=
-\int\! d^4 x \,\sqrt{-g}\, \left(\frac{1}{2\ka}g^{\nu\be}R^\m{}_{\n\mu\be}+
\theta g^{\al\ga}g^{\be\de}
R^{\mu}{}_{\mu\alpha\beta} R^{\nu}{}_{\nu\ga\de}\right)-\ns-
\sum_j\int\! d\ta \ls m_j \sqrt{\dot{x}^{\mu}_j(\ta)\dot{x}^\nu_j(\ta) g_{\mu\nu}(x_j(\ta))} + \frac{q_j}{4\om} \dot{x}^{\be}_j(\ta)\Gamma^\m_{\be\m}(x_j(\ta))\rs.
\nom}
It is invariant with respect to diffeomorphisms as well as to gauge transformations \eqref{s20} restricted by the condition \eqref{s21}. To obtain the full set of field equations one need to vary this action with respect to all independent variables describing gravity ($g_{\m\n}(x)$ and $\Gamma^\al_{\m\n}(x)$) as well as the particles ($x^\m_j(\ta)$).
\section{The field equations}\label{urdv}
Firstly let us find the variation of the action \eqref{s25} with respect to the connection $\Gamma^\al_{\m\n}(x)$. Such a variation of the $S_1$ was already calculated in \eqref{s8}, whereas the term $S_2$ does not give a contribution. For $S_4$, assuming \eqref{s23}, we have
\disn{s26}{
\de S_4=4\te\int\! d^4 x \,\sqrt{-g}\,\ls \bar D_\al R^\n{}_\n{}^{\al\be}\rs \de\Gamma^\m_{\be\m},
\nom}
where $\bar D_\al$ is a Riemannian covariant derivative with the connection \eqref{s5}.
For the remaining term $S_3$ we obtain
\disn{s27}{
\de S_3=-\sum_j \frac{q_j}{4\om}\int\! d\ta\, \dot{x}^{\mu}_j\,\de\Gamma^\m_{\be\m}(x_j)=
-\frac{1}{4\om}\int\! d^4 x \,\sqrt{-g}\,j^\be \de\Gamma^\m_{\be\m},
\nom}
where we have introduced the quantity
\disn{s28}{
j^\m=\sum_j q_j \int\! d\ta\,\dot x^\m_j \de(x-x_j)\frac{1}{\sqrt{-g}}
\nom}
which is no other than four-current of relativistic particles if one interprets $q_j$ as their electric charges.
As a result we obtain a field equation which replaces \eqref{s9}:
\disn{s29}{
\frac{1}{2\ka}\ls C^{\m\n\be}+C^{\n\be\m}-C_\ga{}^{\ga\m} g^{\n\be}-C_\al{}^{\n\al}g^{\be\m}\rs+
\ls 4\te\bar D_\al R^\ga{}_\ga{}^{\al\be}-\frac{1}{4\om}j^\be\rs g^{\m\n}=0.
\nom}
Multiplying it on $g_{\m\n}$, one easily notices that it leads to the satisfaction of \eqref{s9} (and therefore \eqref{s12}) and
\disn{s30}{
4\te\bar D_\al R^\ga{}_\ga{}^{\al\be}-\frac{1}{4\om}j^\be=0.
\nom}
Substituting \eqref{s12} in \eqref{s23} and making use of \eqref{s31}, it is easy to notice that
\disn{s31.1}{
R^\ga{}_{\ga\al\be}=4\om\ls \dd_\al A_\be-\dd_\be A_\al \rs.
\nom}
As a result one can rewrite \eqref{s30} in the following form:
\disn{s32}{
\bar D_\al F^{\al\be}=4\pi j^\be,
\nom}
where
\disn{s33}{
F_{\al\be}=\dd_\al A_\be-\dd_\be A_\al,
\nom}
and the arbitrary constant $\om$ which was introduced above \eqref{s10} had now been fixed:
\disn{s33.1}{
\om=\frac{1}{16\sqrt{\pi\te}}.
\nom}
So the varying of action \eqref{s25} with respect to the connection leads to the expression for the connection \eqref{s12} as well as the equation \eqref{s32} which reproduces Maxwell equation in curved spacetime if one identifies the part $A_\m$ of connection DoFs with the EM potential. Note that under gauge transformations \eqref{s20}, which do not change the full action \eqref{s25}, the quantity $A_\m$ transforms as follows (according to \eqref{s12}):
\disn{s33.2}{
A'_\m=A_\m+\frac{1}{\om}\dd_\m\la,
\nom}
as EM potential should transform.
Now let us vary the action \eqref{s25} with respect to the particles
coordinates $x^\m_j(\ta)$, so contributions will be given only by $S_2$ and $S_3$ contributions.
They have the well-known form of the action of relativistic particles and action of its interaction with the EM potential of the form $\Gamma^\m_{\be\m}/(4\om)$ (if $q_j$ are charges).
So one can immediately write the known equations of motion appearing after the variation with respect to $x^\m_j(\ta)$:
\disn{s34}{
m_j u^\n_j \bar D_\n u_{j\al}=q_j F_{\al\n} u^\n_j,
\nom}
where we noticed that making use of \eqref{s12}, \eqref{s31} and \eqref{s33} leads to the equality
\disn{s35}{
\dd_\al \ls\frac{1}{4\om}\Gamma^\m_{\n\m}\rs-\dd_\n\ls\frac{1}{4\om}\Gamma^\m_{\al\m}\rs=\dd_\al A_\n-\dd_\n A_\al=F_{\al\n}.
\nom}
In the equation \eqref{s34} we have introduced a normalized 4-velocity vector of the relativistic particle
\disn{s36}{
u^\al_j=\frac{\dot x^\al_j}{\sqrt{\dot{x}^{\mu}_j\dot{x}^\nu_j g_{\mu\nu}(x_j)}}.
\nom}
The resulting expression \eqref{s34} obviously reproduces the equations of motion of relativistic particle in the gravitational field with metric $g_{\m\n}$ and EM field defined by the certain part of the connection treated as EM potential $A_\m$.
Let us finally vary the action \eqref{s25} with respect to metric $g_{\m\n}$. Contributions will be given by $S_1, S_2$ and $S_4$. The contribution of $S_1$ was calculated in
the section~\ref{urav} (see the left-hand side of \eqref{s16}). Merging it with contribution of $S_4$
(which can be easily calculated), we obtain
\disn{s37}{
\de (S_1+S_4)=\int\! d^4 x \,\sqrt{-g}\,\ls
\frac{1}{2\ka}\bar G^{\al\be}+2\te R^{\mu}{}_\mu{}^{\alpha\ga} R^{\nu}{}_\nu{}^\be{}_\ga-
\frac{\te}{2}g^{\al\be}R^{\mu}{}_\mu{}^{\ga\de} R^{\nu}{}_{\nu\ga\de}
\rs\de g_{\al\be}.
\nom}
The contribution of $S_2$
\disn{s38}{
\de S_2=-\sum_j \frac{m_j}{2} \int\! d\ta\, \sqrt{\dot{x}^{\mu}_j\dot{x}^\nu_j g_{\mu\nu}(x_j)}u^\al u^\be \de g_{\al\be}(x_j)=
-\frac{1}{2}\int\! d^4 x \,\sqrt{-g}\, T_\text{p}^{\al\be}\de g_{\al\be}
\nom}
is determined by the energy-momentum tensor of the relativistic particles
\disn{s39}{
T_\text{p}^{\al\be}=\sum_j m_j \int\! d\ta\, \sqrt{\dot{x}^{\mu}_j\dot{x}^\nu_j g_{\mu\nu}(x_j)}\de(x-x_j)\frac{1}{\sqrt{-g}}u^\al u^\be.
\nom}
As a result we obtain the field equations in the following form:
\disn{s40}{
\bar G^{\al\be}=\ka\ls T_\text{p}^{\al\be}-4\te\ls
R^{\mu}{}_\mu{}^{\alpha\ga} R^{\nu}{}_\nu{}^\be{}_\ga-\frac{1}{4}g^{\al\be}R^{\mu}{}_\mu{}^{\ga\de} R^{\nu}{}_{\nu\ga\de}\rs\rs.
\nom}
Using the corollary \eqref{s31.1} of the other field equations, the denotation \eqref{s33} and choosing of constant $\om$
\eqref{s33.1}, one easily notices that this equation is the Einstein one:
\disn{s41}{
\bar G^{\al\be}=\ka\ls T_\text{p}^{\al\be}+T_\text{EM}^{\al\be}\rs,
\nom}
where
\disn{s42}{
T_\text{EM}^{\al\be}=-\frac{1}{4\pi}\ls
F^{\alpha\ga} F^\be{}_\ga-\frac{1}{4}g^{\al\be}F^{\ga\de} F_{\ga\de}\rs
\nom}
reproduces the expression for the energy-momentum tensor of EM field with potential $A_\m$.
Therefore the full set of field equations corresponding to the action \eqref{s25} turns out to be the set of the equations \eqref{s32},\eqref{s34}, and \eqref{s41} together with the expression for the connection \eqref{s12}. These equations reproduce the Einstein-Maxwell equations if one identifies $A_\m$ in \eqref{s12} with the EM potential.
Such a natural unification of gravitation and electromagnetism appears if we restrict ourselves to the consideration of the matter in the form of classical relativistic point particles.
It is quite simple to generalize this consideration on the case of continuous media, whereas the generalization on the important case of matter fields (i.~e. spinor or even scalar electrodynamics) is a very non-trivial problem.
We will discuss it in the concluding section~\ref{zakl}. Now let us shortly review the ideas of classical UFTs related to the one that we described above.
\section{Historical notes}\label{istor}
The present section is devoted to the short discussion of the known UFTs in comparison with the approach described above. We do not consider extra dimensions because such theories are too much different from the above one. The main characteristics of the above theory which can be different in other theories are:
\begin{itemize}
\item the symmetricity of metric,
\item the non-symmetricity of connection (the presence of torsion),
\item the initial independence of metric and connection.
\end{itemize}
The literature concerning unified field theories is quite wide. We do not pretend to give a full comprehensive review here. Such reviews can be found, e.~g. in the monographs \cite{vizgin}, \cite{tonnelat} as well as in the reviews \cite{goenner1,goenner2}.
The most famous of the early attempts of the unification of gravity with electromagnetism is assumed to be Weyl theory which was firstly proposed in 1918 in \cite{weyl_1918} (see also \cite{weyl_raum}). The main idea of the Weyl approach is an introduction of an additional gauge invariance to the theory, namely the scaling invariance of metric (which is symmetric)
\begin{align}\label{is1}
{g'}_{\mu\nu}(x) = e^{2\lambda(x)} g_{\mu\nu}(x).
\end{align}
These transformations are often called conformal, which is not quite correct, since they (in contrast with conformal ones) are not a part of diffeomorphisms group and were postulated by Weyl besides of it. It is more convenient to call them \textit{Weyl transformations}.
The adjusting of the parallel transport rule with this new symmetry leads to the appearance of additional degrees of freedom $A_\m$ in the connection, which in Weyl theory takes the form
\disn{is2}{
\Gamma^\al_{\m\n}=\bar \Gamma^\al_{\m\n}+\frac{1}{2}\ls
A_\m\de^\al_{\n}+A_\n\de^\al_{\m}-A_\be g^{\be\al}g_{\m\n}\rs,
\nom}
(cf. \eqref{s12}), where $\bar \Gamma^\al_{\m\n}$ is, as above, the Riemannian connection \eqref{s5}. Note that in Weyl theory as well as in GR the connection is symmetric, in contrast with the above theory.
Under the gauge transformation \eqref{is1} additional DoFs are transformed as follows:
\disn{is3}{
A'_\be=A_\be-\dd_\be\ln\la,
\nom}
which allowed Weyl to interpret this quantity as an EM potential. The curvature tensor in Weyl theory, as well as in the above approach, splits into two terms:
\disn{is4}{
R^\m{}_{\n\al\be}=\bar R^\m{}_{\n\al\be}+\frac{1}{2}\de^\m_\n\ls\partial_\al A_\be - \partial_\be A_\al\rs,
\nom}
cf. \eqref{s14.1}.
However, the gauge invariance with respect to \eqref{is1} postulated by Weyl entailed two undesirable consequences. Firstly, the square of interval which was observable in GR, is no longer observable: \qq{only the ratios of the metric components have direct physical meaning}, and not these quantities themselves \cite{vizgin}. As a result, some additional \textit{ad hoc} methods of length measurements are required, see \cite{vizgin}. Secondly, the condition of Weyl-invariance of the action of the theory does not allow to use EH action \eqref{1} in it since EH action is not invariant with respect to \eqref{is1}. Instead, one has to use squared curvature tensor which leads to the poor agreement with the observables.
Another attempt to unify gravity with electromagnetism was made by Eddington \cite{eddington} in 1921. If we consider the Weyl approach as less \qq{strict} than GR one, since Weyl did not require the length conservation under a parallel transport while keeping the conservation of angles, than the Eddington approach is much less strict: he did not require the angles conservation as well, assuming that the connection is not related to metric at all. In this sense his approach is close to above one, but there is one essential difference: Eddington assumed the connection to be symmetric (as in the Weyl theory and GR), so in this sense his approach coincides with the Palatini one. In Eddington theory antisymmetric part of the Ricci tensor (cf. \eqref{s15}) and segmentary curvature \eqref{s23} do appear; due to the symmetricity of the connection they are related by the condition
\disn{is5}{
R_{\al\be}-R_{\be\al}=R^\m{}_{\m\al\be}=\dd_\al\Gamma^\m_{\be\m}-\dd_\be\Gamma^\m_{\al\m}.
\nom}
Note that from \eqref{s15} and \eqref{s31.1} one can see that in the above approach a similar but not quite equivalent condition is satisfied: antisymmetric part of the Ricci tensor is doubled.
The trace of the connection in the Eddington theory is identified with the EM potential (as in the section \ref{urdv}, see below \eqref{s33.1}), so the antisymmetric part of the Ricci tensor turns out to be the EM field tensor. The symmetric part of Ricci tensor, according to Eddington, is related to the metric by the condition (which is postulated without any variational principle)
\disn{is6}{
R_{\al\be}+R_{\be\al}=\La g_{\al\be},
\nom}
where $\La$ is a certain universal constant. The condition \eqref{is6} is called by Eddington \qq{natural world gauge}. The first and foremost significant drawback of the Eddington theory is the fact that field equations (including Einstein ones) are not derived from a variational principle, but instead are postulated using \qq{the \q{principle of identification} of the elements of the geometrical structure developed by him with fundamental physical quantities}, see~\cite{vizgin}.
It seems that the non-symmetric connection was firstly introduced by Schouten and Friedman in 1923-24 \cite{schouten,schouten_fr}. They suggest to consider \textit{half-symmetric} connection, which antisymmetric part has the form
\disn{is7}{
\Gamma^\al_{\m\n}-\Gamma^\al_{\n\m}=A_\m\de^\al_{\n}-A_\n\de^\al_{\m},
\nom}
whereas the Lagrangian of the theory is chosen to be the square root of Ricci tensor determinant (the simplest scalar density which can be constructed only from connection).
A similar idea was used much later in 1982 in \cite{ferraris1982} when the authors wrote the action of unified theory of gravitation and electromagnetism. However, they used only the symmetric part of the Ricci tensor and the whole action was far more complicated. Note that in this paper the role of the EM potential was played by the same components of the connection that in the above theory: the field equations entailed the satisfaction of the relation \eqref{s12} for the connection.
Needless to say, the man who most wanted to find the unified field theory was Einstein. One of the directions of his quest was quite close to the above approach. The difference between them is the Einstein's assumption of non-symmetricity of
metric\footnote{According to Goenner \cite{goenner1}, this idea was suggested to him by R.~F$\ddot{\text{o}}$rster, who thus repelled Einstein from the one of the most natural formulations of the unified theory.}.
Einstein had been working on such a generalization of GR since 1917 (see \cite{goenner1}); he wrote the first paper \cite{ein_conn} on this topic in 1925, whereas the last one \cite{ein1955-2} was written by him in 1955 shortly before his death and turned out to be his last paper ever.
In the approach developed by Einstein the action was chosen to be EH one \eqref{1} in which the metric and the connection are independent. Initially they are completely arbitrary, so the Einstein approach differs from the one that was described in section \ref{urav} only by the non-symmetricity of metric. As it was mentioned in the Introduction, Einstein in fact proposed the method which (for symmetric metric and connection) later became known as Palatini formalism.
It was shown in \cite{ein_conn} that among the appearing field equations a part of vacuum (i.~e. without current) Maxwell equations is present if one identifies the antisymmetric part of the metric with the EM field
tensor\footnote{Note that in the papers \cite{ein1955-1,ein1955-2} such an identification was no longer discussed, while suggested modification of GR is treated as \qq{an attempt to construct a theory of the total field by generalizing the equations of the purely gravitational field} rather than unification of gravity and electromagnetism.}.
The remaining equations are initially analyzed in the absence of EM field, and the manual elimination of torsion is performed to reproduce GR. For the general case Einstein performed an approximate analysis of the appearing equations, but he failed to construct a satisfactory UFT. The attempts of developing of theories with non-symmetric metrics had been made lately, e.~g. in \cite{Borchsenius1976} where it was noticed that in such a framework it is possible to identify the trace of the torsion with the EM potential (as in the section \ref{urav}, see~\eqref{s14}), and the action contained the interaction of this quantity with the current density.
Although Einstein paid a special attention to gradient transformations \eqref{s20} which he called \qq{$\la$-transformations} \cite{ein1955-2}, he did not connect them to gradient transformations of the EM potential (this occurred quite naturally in the above approach, see~\eqref{s33.2}). In the Einstein theory the action turns out to be invariant with respect to \eqref{s20} due to invariance of curvature tensor, but not with respect to more general transformations \eqref{gauge} since the metric is not symmetric and, according to \eqref{ss1}, its contraction with Ricci tensor remains dependent on the transformation parameter $f_\m$. If the metric is symmetric, then EH action is invariant with respect to \eqref{gauge} as well, which is discussed in the section \ref{urav}. The restriction of symmetry to \eqref{gauge} in the above approach happens only after the addition of the term \eqref{s18} in the action, which couples the connection to the matter.
Einstein was close to above approach once again: in the several papers, first of which was the paper \cite{ein_1923} written in 1923. In these papers he considered a certain implicit action constructed from symmetric and antisymmetric part of the Ricci tensor. It corresponds to the action of gravity $S_1+S_4$ that was used in the section \ref{mater}, in which the part $S_1$ is constructed from symmetric part of the Ricci tensor (in these papers Einstein assumed that both the metric and the connection is symmetric), while the part $S_4$ can be treated as constructed from antisymmetric one due to its coincidence \eqref{is5} with the segmentary curvature in the theory with symmetric connection. However, the condition of connection symmetricity made this UFT (which was eventually abandoned by Einstein) drastically different from the above one.
The approach proposed by Einstein in \cite{ein_1923} was generalized on the case of non-symmetric connection by young French mathematician Henri Eyraud in 1925\cite{eyraud}. Following Einstein, he assumed that the action somehow depends on symmetric and antisymmetric part of Ricci tensor, but in the presence of torsion the condition \eqref{is5} is no longer holding, so he failed to obtain the explicit form of the action ($S_1+S_4$) as well as Einstein himself. However, Eyraud obtained an expression for the non-symmetric connection in the form of \eqref{s12}, while a part of its degrees of freedom $A_\m$, which (considering \eqref{s31}) is the same that the trace of the connection $\Gamma^\al_{\m\al}$ up to the gradient transformation and a constant multiplier, was interpreted as the EM potential. The same expressions and interpretations were used in the paper by Straneo in 1931, although the expression \eqref{s12} was postulatively introduced there rather than derived from any variational principle.
Proceeding from the analyzed publications, we conclude that, strangely enough, the approach described in the section \ref{urav}-\ref{urdv} had not been discovered in the first half of XX century, at the time of the most intense search for UFT.
First paper in which the variant of such an approach was formulated is the 1978 paper \cite{Krechet}. However, it was published only in Russian in an obscure journal and thus has not been cited in the similar works. In \cite{Krechet} the authors start from EH action with symmetric metric and independent non-symmetric connection. Then they discover that the trace of the torsion (in fact it is the same that the part of the DoFs of connection corresponding to $A_\m$ in \eqref{s12}, see~\eqref{s14}) remains arbitrary \textit{on-shell}. Then they propose to supplement the action by the squared segmentary curvature $S_4$ \eqref{s24} and discover the coincidence of resulting expressions with the Einstein-Maxwell ones if the trace of the \textit{torsion} is identified with EM potential. In particular, they discover the possibility of interpretation of \qq{$\la$-transformations} (which are the symmetry transformations of the resulting theory) as gauge transformations of EM potential. Then the authors obtain the equations of motion of test particles \eqref{s34}, although they derive it from the \textit{ad hoc} condition (\qq{physical trajectories in the space-time} must have a least length among those along which the length of vector by parallel transport is conserved) rather than from the action, as it was done in section~\ref{urdv}. The particles is considered precisely as test ones, so backreaction (the appearing of the corresponding contributions in equations of motion \eqref{s32} and \eqref{s41}) is not discussed.
Almost simultaneously with the paper \cite{Krechet} (also in 1978) the Palatini formulation with non-symmetric connection was studied in the paper \cite{hehl}. However, the authors of \cite{hehl} did not make an attempt to use obtained result in the construction of UFT. Instead of it they studied the relation of the appearing theory (squared segmentary curvature was not added to the action) to the usual description of gravity in the framework of Riemannian geometry. This studies have been continued in the recent papers \cite{dadhich2012,bernal2017} which indicates a continuing interest to the idea of using non-symmetric connection. The possibility of addition of squared segmentary curvature to obtain a kind of Einstein-Maxwell theory that proposed in \cite{Krechet} was rediscovered independently in the paper \cite{tucker1995}. In the framework of the resulting theory the authors also considered a spherically symmetric solution corresponding to Reissner-Nordstr$\ddot{\text{o}}$m one.
This concludes our historical survey of the papers devoted to UFT which are related to the approach described in the sections \ref{urav}-\ref{urdv}.
\section{Conclusion}\label{zakl}
In the present paper we described a possible way of unification of gravity and electromagnetism, which had been somehow missed at the time of the most intensive search for such an unification in the first half of XX century. The idea of such way was proposed only in 1978 in the paper \cite{Krechet}.
It must be noted that in this framework the matter interacting with EM field can be defined only in \textit{purely classical sense}: as a set of point particles. The generalization on the case of ideal fluid (i.~e. continuous medium consisting on classical particles) is quite simple. For the description of such a matter one can choose a current $J^\m(x)$ as an independent variable, which is a vector density and satisfies the continuity equation $\dd_\m J^\m=0$. Various forms of the action of the ideal fluid based on this choice are examined in \cite{statja48}. The part of the action $S_3$ describing the interaction of the ideal fluid with the connection $\Gamma^\m_{\be\al}$ can be written by analogy with \eqref{s18} in the following form:
\disn{zz1}{
S_3=-\frac{q}{4\om}\int\! d^4 x\,J^\be\Gamma^\m_{\be\m},
\nom}
while the current $j^\m$ appearing in the equations of motion \eqref{s32} takes the form
\disn{zz2}{
j^\m=\frac{q}{\sqrt{-g}}J^\m
\nom}
instead of \eqref{s28}. In such a simple case all the matter has fixed (equal to $q$) charge-to-mass ratio, although one could introduce multiple types of matter with different values of such a ratio.
However, from the modern point of view the purely classical description of matter is obviously unsatisfactory as in the quantum theory the EM field interacts with complex (usually spinorial) matter fields rather than particles or continuous media. The EM potential ${\cal A}_\m$ turns out to be Abelian gauge field corresponding to local group of inner symmetry $U(1)$, and takes part in the definition of covariant derivative corresponding to that symmetry:
\disn{z1}{
\na_\m\psi=(\dd_\m+i {\cal A}_\m)\psi,
\nom}
i.~e. it multiplies on the imaginary unit. The lack of imaginary unit at the $A_\m$ in the expression for the connection \eqref{s12}
(assuming that the connection is real) means that $A_\m$ should be treated as a gauge field corresponding not to local phase transformations $U(1)$
\disn{z2}{
\psi'(x)=e^{-i\la(x)}\psi(x),
\nom}
but rather to local \emph{scale} transformations of tensorial matter
fields\footnote{
Note that it is not the presence of complexity itself in \eqref{z1} (which can be avoided by choosing $SO(2)$ as a real form of $U(1)$) is essential, but rather the fact that the quantity which \qq{elongates} the usual derivative is not reduced to simple scaling of matter fields as it occurs for scale transformations \eqref{z3}.}
\disn{z3}{
\psi'^\m(x)=e^{-\la(x)}\psi^\m(x),
\nom}
which are analogous to the scale transformations of the Weyl theory mentioned in the section ~\ref{istor} (see \eqref{is1}). A \qq{charge} of matter field in this case equals to difference between the number of its upper and lower indices (and is thus quantized \textit{on the classical level}).
Therefore the interpretation of $A_\m$ which presents in the real-valued expression \eqref{s12} as an EM potential in the field theory is possible if one somehow provides the appearance of imaginary unit at it. The efforts of researchers working on such an approach were mainly concentrated on the complexification of the connection to provide its interaction to the matter fields (see \cite{horie9409018th}, \cite{horie2} and the references therein). The consideration was usually performed in terms of frame bundles. It must be stressed that if the matter does not described in terms of fields, there is nowhere from which the global and certainly local $U(1)$ symmetry can arise since in the Lagrangian of classical electrodynamics with matter in the form of point particles there is no variables that can be affected by such transformations.
The question whether it is possible to obtain a satisfactory UFT in the framework of the above approach in the field-theoretic description of matter remains open. Nevertheless, the classical electrodynamics of charged particles appears in such a framework exceptionally simple, even simpler than in the most famous UFT, namely the Kaluza-Klein one \cite{wesson_kaluza}. In the Kaluza-Klein theory one needs to make a physical proposition about the existence of fifth dimension to obtain \qq{electromagnetic} degrees of freedom, whereas the above approach naturally appears when one \emph{eliminates} the postulate of the absence of torsion in the GR. This fact allows to put this approach on a par with the most famous classical UFTs and, in our opinion, assigns the sufficient historical and methodical value to it.
{\bf Acknowledgements.}
The work was supported by SPbSU grant N~11.38.223.2015.
One of the authors (A.~S.) thanks the Library of Russian Academy of Sciences for the assistance in working with sources.
|
1,314,259,996,744 | arxiv | \section{Introduction}
B\"acklund transformations and their associated nonlinear superposition principles, known as permutability theorems, play a central role in modern soliton theory as described in \cite{Gu-book, RogersSchief, RogersShadwick}, see also \cite{RogersAmes} for further applications. The importance of the combination of gauge and reciprocal transformations to link hierarchies of integrable systems is well-established (see, e.g., \cite{OevelRogers} and literature cited therein). Here, the concern is with the application of B\"acklund transformations to connect various non-Abelian equations to a~canonical KdV-type equation and to construct their recursion operators. Hierar\-chies of such nonlinear equations are considered wherein the unknown is an operator on a~Banach space. Such operator equations were originally introduced by Marchenko \cite{Marchenko} and further investigated and developed, in the framework of Banach space geometry, in \cite{Aden-Carl, Carl-Schiebold, Carl-Schiebold-1}.
The results comprised in this article represent the continuation of the study in \cite{CMS-2016a, Carillo:Schiebold:JMP2009, SIMAI2008,Carillo:Schiebold:JMP2011, Carillo:Schiebold:JNMP2012}. In particular, in \cite{Carillo:Schiebold:JMP2009, Carillo:Schiebold:JMP2011} the focus is on the operator potential Korteweg--de Vries, Korteweg--de Vries and modif\/ied Korteweg--de Vries equations, their connections via B\"acklund transformations and the construction of the recursion operators they admit.
One of the main advantages of connecting non-Abelian equations via B\"acklund transformations is that solutions can be transferred from one equation to another. In \cite{Carillo:Schiebold:JMP2009} operator-valued solutions (which can be interpreted as operator analogs of solitons) to the pKdV, KdV and mKdV hierarchies are constructed. In \cite{Carillo:Schiebold:JMP2011} suitable projection techniques are exploited to derive solution formulae to the corresponding scalar and matrix hierarchies. Note that the study of non-Abelian nonlinear evolution equations found its original interest in the case of matrix equations \cite{CalogeroDegasperis, LeviRB}. The results in the present article are valid for operator-valued functions. This level of generality permits to construct solutions to scalar and matrix equations that can be viewed as countable superposition of solitons, see also \cite{Schiebold-1, Schiebold-6dic3} for the connection between countable nonlinear superposition and Banach space geometry, and \cite{Schiebold-6dic2, Schiebold-6dic1} for further applications.
In \cite{Rogers:Carillo:1987}, the term {\it B\"acklund chart} was introduced to indicate the net of B\"acklund transformations connecting dif\/ferent evolution equations. In \cite{Fuchssteiner:Carillo:1989a} a wide B\"acklund chart which includes {\it scalar} 3rd order Abelian evolution equations is constructed. It connects, further to the KdV and the modif\/ied KdV equations, in particular the KdV singularity manifold equation, also known as as UrKdV or Schwarz-KdV \cite{Depireux, Schiff, Weiss, Wilson}, and the KdV interacting soliton equation \cite{Fuchssteiner1987}. The connections among the equations in this Abelian B\"acklund chart are applied to f\/ind, or recover, the recursion operators admitted by all these nonlinear evolution equations. Moreover, other structural properties such as the Hamiltonian, and bi-Hamiltonian, structure of these equations are also obtained via the B\"acklund chart \cite{Fuchssteiner:Carillo:1989a}. The present study is concerned about the extension of the {B\"acklund chart} in \cite{Carillo:Schiebold:JMP2009} to obtain the non-Abelian analog of the links established in the scalar case \cite{Fuchssteiner:Carillo:1989a}. Precisely, in addition to the pKdV, KdV and mKdV equations already connected in \cite{Carillo:Schiebold:JMP2009}, a dif\/ferent version of the non-Abelian modif\/ied KdV and two non-Abelian equations, respectively, analogs of the KdV singularity manifold and of the {KdV interacting soliton} equation, are all linked together via B\"acklund transformations. New, in the present B\"acklund chart, are the inclusion of a second modif\/ied KdV equation, denoted as amKdV, for {\it alternative} mKdV equation \cite{KK, Olver:Sokolov}, new are also the non-Abelian {KdV singularity manifold} equation and the non-Abelian KdV interacting soliton equation.
All the non-Abelian nonlinear evolution equations in the B\"acklund chart admit a~recursion operator. The recursion operators of some of them, such as the pKdV, KdV and mKdV \cite{Carillo:Schiebold:JMP2009, Olver:Sokolov, Schiebold2010} are known. We recover, via the established connections, the recursion operators admitted by the non-Abelian amKdV, given in \cite{Olver:Sokolov}. Then, the recursion operators of the non-Abelian KdV singularity manifold and of the KdV interacting soliton equations, both new, are constructed. Furthermore, the hereditariness of all the obtained recursion operators is proved combining links via B\"acklund transformations \cite{FokasFuchssteiner:1981, Fuchssteiner1979}, with the hereditariness of non-Abelian KdV recursion operator~\cite{Schiebold2010}. Finally, since all the nonlinear third order non-Abelian evolution equations admit hereditary recursion operators, according to \cite{FokasFuchssteiner:1981, Fuchssteiner1979}, all the links in the B\"acklund chart can be extended to the corresponding whole hierarchies.
The material is organized as follows. The opening Section~\ref{sec4} is devoted to the B\"acklund charts connecting nonlinear evolution equations which generalize, to the operator level, the pKdV, KdV and mKdV equations \cite{Carillo:Schiebold:JMP2009,Carillo:Schiebold:JMP2011}. Sections~\ref{s mkdvs} and~\ref{s new} are devoted to the construction of a novel B\"acklund chart. In Section~\ref{s mkdvs}, the B\"acklund chart is extended to incorporate also the non-Abelian amKdV equation; its recursion operator is constructed from the known recursion operators of the mKdV equations; f\/inally, the connection between the two dif\/ferent non-Abelian modif\/ied KdV equations is provided.
In Section~\ref{s new}, the novel B\"acklund chart is further enlarged. Then, the recursion operators of the KdV interacting soliton and of the KdV singularity manifold equations are constructed. In addition, in Section~\ref{s new}, a M\"obius type invariance exhibited by the non-Abelian KdV singularity manifold equation is established.
The subsequent Section~\ref{section5} is devoted to the proof of hereditariness of all the recursion operators in the previous sections. The hereditariness of all the recursion operators in the B\"acklund chart guarantees that the links can be transferred to whole hierarchies, relating their corresponding members. In the closing Section~\ref{rems}, further to some remarks on the interest of the present study, open problems and perspectives suggested by our new results are brief\/ly outlined. The article is complemented with an Appendix where some needed def\/initions, such as the def\/inition of B\"acklund transformation~\cite{FokasFuchssteiner:1981}, adopted throughout, as well as known results obtained in the case of Abelian nonlinear evolution equations~\cite{Fuchssteiner:Carillo:1989a} are comprised.
\section{Non-Abelian B\"acklund charts}\label{sec4}
In \cite{Carillo:Schiebold:JMP2009}, the recursion operators admitted by the non-Abelian potential KdV, KdV and modif\/ied KdV equations were shown to be related by B\"acklund transformations. Consider f\/irst the operator potential KdV equation (pKdV)
\begin{gather}\label{pkdv}
W_t = W_{xxx}+{3} W_x^2,
\end{gather}
where the unknown $W$ is a function whose values are bounded linear operators on some Banach space\footnote{Here capital letters are used to emphasize that the unknown is an operator acting on a Banach space.}. It admits the recursion operator
\begin{gather} \label{pKdV recursion operator}
\hat \Psi(W) = D^2 + A_{W_x} + D^{-1}A_{W_x}D + D^{-1}C_{W_{x}}D^{-1}C_{W_{x}},
\end{gather}
where $D$ denotes the derivative with respect to $x$, and $C_T$, $A_T$ denote the commutator and anti-commutator with respect to~$T$, namely,
\begin{gather*}
C_T S:= [T,S] \equiv TS-ST, \qquad A_T S:= \{T,S\}\equiv TS+ST.
\end{gather*}
For earlier occurrences of recursion operators in the non-Abelian setting we refer to \cite{CalogeroDegasperis2,Fuchssteiner:Chowdhury,GKS,Olver:Sokolov}. Consider next the operator Korteweg--de Vries equation (KdV)
\begin{gather} \label{nc-kdv}
U_t = U_{xxx}+3 \{U,U_{x}\},
\end{gather}
which admits the recursion operator
\begin{gather}\label{kdv-recop}
\Phi(U) = D^2 + 2A_U + A_{U_x}D^{-1} + C_UD^{-1}C_UD^{-1}.
\end{gather}
The recursion operators (\ref{pKdV recursion operator}) and (\ref{kdv-recop}) are linked via the B\"acklund transformation
\begin{gather*
B_1\colon \ U-W_x=0.
\end{gather*}
In fact, the transformation operator is $\Pi_{B_1}=D^{-1}$, and $\hat\Psi(W)=D^{-1}\Phi(U)D$. Hence, the non-Abelian pKdV (\ref{pkdv}) and KdV~(\ref{nc-kdv}) equations can also be written as
\begin{gather*}
W_t = \hat \Phi(W) W_x \qquad \text{and} \qquad U_t = \Phi(U) U_x.
\end{gather*}
Finally, the KdV equation (\ref{nc-kdv}) is related to the operator modif\/ied KdV equation (mKdV)
\begin{gather}\label{mkdv}
V_t = V_{xxx}-3 \big\{V^2 ,V_{x}\big\}
\end{gather}
via the Miura transformation
\begin{gather}\label{Mnc}
M\colon \ U + V_{x} + V^{2}= 0.
\end{gather}
Hence, the Miura transformation (\ref{Mnc}) allows to obtain the recursion operator $\Psi(V)$ of the mKdV (\ref{mkdv}) from the recursion operator $\Phi(U)$ of the KdV~(\ref{kdv-recop}) via
\begin{gather*
\Psi(V)=\Pi_M \Phi(U) \Pi_M^{-1}, \qquad \text{where} \quad \Pi_M = -(D+A_V)^{-1}.
\end{gather*}
The latter implies
\begin{gather*}
\Psi(V) = \big(D-C_VD^{-1}C_V\big)\big(D-A_VD^{-1}A_V\big)
\end{gather*}
(see \cite{Carillo:Schiebold:JMP2009} for details), which can be equivalently written as
\begin{gather}\label{mkdv-recop}
\Psi(V) = (D-C_V)D^{-1}(D+C_V) (D-A_V)D^{-1}(D+A_V) .
\end{gather}
The following B\"acklund chart summarizes the links among the non-Abelian pKdV (\ref{pkdv}), KdV (\ref{nc-kdv}), and mKdV (\ref{mkdv}) equations:
\begin{gather*
\boxed{W_t = W_{xxx} + 3 W^2_x}\buildrel B_1 \over{\text{\textendash\textendash}}
\boxed{U_t = U_{xxx} + 3 \{ U,U_x\}}\buildrel M \over{\text{\textendash\textendash}}\boxed{V_t = V_{xxx}- 3 \big\{ V^2,V_{x}\big\}}\,.
\end{gather*}
When the respective recursion operators are applied to the above equations iteratively, the B\"acklund chart can be extended to the corresponding hierarchies, and the connections can be summarized in
\begin{gather}\label{BCNH3}
\boxed{W_{t} = [\hat \Phi(W)]^{n} W_x}\buildrel B_1 \over{\text{\textendash\textendash}}
\boxed{U_{t} = [\Phi(U)]^{n} U_{x}}\buildrel M \over{\text{\textendash\textendash}}\boxed{V_t = [\Psi(V)]^n V_{x}}\,.
\end{gather}
Note that (\ref{BCNH3}) is the natural non-Abelian counterpart of the corresponding part of the B\"acklund chart (\ref{BC1*}) introduced in~\cite{Fuchssteiner:Carillo:1989a}.
\section{On the non-Abelian mKdV equations}\label{s mkdvs}
A distinguished feature of the B\"acklund chart studied in this article is that it proceeds via two versions of the non-Abelian mKdV equation. The link between those versions is considered in this section. The alternative non-Abelian mKdV equation (amKdV)
\begin{gather}\label{mkdv2}
\widetilde V_t = \widetilde V_{xxx} + 3[\widetilde V,\widetilde V_{xx}] - 6 \widetilde V\widetilde V_x\widetilde V
\end{gather}
was f\/irst described in \cite{KK}, where the Lax pair formulation and the inverse scattering problem were studied. In contrast to the non-Abelian mKdV \eqref{mkdv} studied in the previous section, \eqref{mkdv2} does not admit a Miura transformation \cite{Olver:Sokolov}. Note also that, in the matrix case, \eqref{mkdv} is invariant under both $V\rightarrow V^*$ and $V\rightarrow -V$, whereas \eqref{mkdv2} is only invariant under $\widetilde V\rightarrow -\widetilde V^*$ ($V^*$ denoting the transpose of $V$).
As already pointed out in \cite{A}, these two versions of the mKdV equation are linked by the gauge transformation $\widetilde V=G^{-1}VG$ with $G_x=VG$. Consider the B\"acklund transformation
\begin{gather}
B_2\colon \ G_x-VG =0 , \label{Bmkdvs1} \\
B_3\colon \ G_x-G\widetilde V =0 . \label{Bmkdvs2}
\end{gather}
Obviously, subsequent application of these B\"acklund transformations links \eqref{mkdv} to \eqref{mkdv2}.
The recursion operator $\widetilde{ \Psi}(\widetilde{V})$ of \eqref{mkdv2} is
\begin{gather} \label{psitilde}
\widetilde \Psi\big(\widetilde V\big) = \big(D+2C_{\widetilde V}\big) \big(D-2R_{\widetilde V}\big) \big(D+C_{\widetilde V}\big)^{-1} \big(D+2L_{\widetilde V}\big) D\big(D+C_{\widetilde V}\big)^{-1} .
\end{gather}
It was f\/irst given in \cite{GKS}, where it was derived using Lax representation. Here we use the B\"acklund link between the mKdV and the amKdV equation to give an alternative derivation together with a more conceptual formulation of the operator itself.
To this end, we introduce the derivation
\begin{gather} \label{der}
\mathbb{D}:=D+C_{\widetilde V}.
\end{gather}
\begin{thrm} \label{t1} On use of the derivation \eqref{der}, the recursion operator \eqref{psitilde} of the amKdV equation can be written as
\begin{gather*}
\widetilde \Psi\big(\widetilde V\big) =
\big(\mathbb{D}+C_{\widetilde V}\big)\big(\mathbb{D}-A_{\widetilde V}\big)\mathbb{D}^{-1}
\big(\mathbb{D}+A_{\widetilde V}\big)\big(\mathbb{D}-C_{\widetilde V}\big)\mathbb{D}^{-1} .
\end{gather*}
\end{thrm}
\begin{rem} Recall that two operators $T$ and $S$ are called \emph{related} if they are of the form $T=AB$ and $S=BA$. In~\cite{Carillo:Schiebold:JMP2009}, relatedness of the non-Abelian KdV and mKdV recursion operators on the image of the Miura transform was exploited to derive solutions of the non-Abelian mKdV hierarchy from solutions of the non-Abelian KdV hierarchy.
Here we observe that $\Psi(V)$ and $\widetilde{\Psi}(\widetilde{V})$ are related in a generalized sense: If $P(D,V) = (D-C_V)D^{-1}$ and $Q(D,V) = (D+C_V)(D-A_V)D^{-1}(D+A_V)$, then
\begin{gather*}
\Psi(V) = P(D,V) Q(D,V), \qquad \widetilde{\Psi}\big(\widetilde{V}\big) = Q\big(\mathbb{D},\widetilde V\big) P\big(\mathbb{D},\widetilde V\big).
\end{gather*}
\end{rem}
Before proving Theorem \ref{t1}, we observe some crucial properties of the derivation \eqref{der}.
\begin{prop} \label{p1} Let $V$ and $\widetilde V$ be related to each other via the B\"acklund transformations~$B_2$ and~$B_3$ with some intermediate invertible function $G$. Then the derivation~\eqref{der} is the result of conjugation of~$D$ with $K_G^{-1}$, i.e.,
\begin{gather*} \mathbb{D} = K_{G} D K_G^{-1} , \end{gather*}
where $K_G$ denotes the conjugation operator, $K_G T = G^{-1}TG$.
Moreover, it holds:
\begin{itemize}\itemsep=0pt
\item[$a)$] $K_G(D \pm C_V) K_G^{-1} = \mathbb{D} \pm C_{\widetilde V}$,
\item[$b)$] $K_G (D \pm A_V) K_G^{-1} = \mathbb{D} \pm A_{\widetilde V}$.
\end{itemize}
\end{prop}
\begin{proof} As an example, we verify $K_G D K_G^{-1} = \mathbb{D}$ and $K_G C_V K_G^{-1} = C_{\widetilde V}$.
Indeed,
\begin{gather*}
K_G D K_G^{-1} T = G^{-1} \big(GTG^{-1}\big)_x G = G^{-1} \big( G_xTG^{-1}+GT_xG^{-1} - GTG^{-1}G_xG^{-1} \big) G\\
\hphantom{K_G D K_G^{-1} T}{}= G^{-1} G_xT + T_x - TG^{-1}G_x = \widetilde V T + T_x - T \widetilde V = \big(D + C_{\widetilde V}\big)T = \mathbb{D} T.
\end{gather*}
Analogously,
\begin{gather*}
K_G C_V K_G^{-1} T = G^{-1} \big[V,GTG^{-1}\big] G = G^{-1} \big( VGTG^{-1}-GTG^{-1}V \big) G \\
\hphantom{K_G C_V K_G^{-1} T}{} = G^{-1} VGT - TG^{-1}VG = \big[G^{-1}VG,T\big] = \big[\widetilde V, T\big] = C_{\widetilde V}T.\tag*{\qed}
\end{gather*}
\renewcommand{\qed}{}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t1}] On use of the B\"acklund transformations $B_2$, $B_3$, the amKdV recursion operator $\widetilde \Psi(\widetilde V)$ is related to the mKdV recursion operator $\Psi(V)$ via
\begin{gather} \label{rec with pi}
\widetilde{\Psi}(\widetilde{V}) = \Pi \Psi(V) \Pi^{-1}
\end{gather}
with $\Pi=\Pi_3\Pi_2$, where
\begin{gather*}
\Pi_2 = -(B_2)_{\vphantom{\widetilde V}G}^{-1}(B_2)_{\vphantom{\widetilde V}V}^{\vphantom{-1}}, \qquad \Pi_3= -(B_3)_{\widetilde V}^{-1}(B_3)_{\vphantom{\widetilde V}G}^{\vphantom{-1}}.
\end{gather*}
Let $L_V$ and $R_V$ denote left and right multiplication by $V$, then $C_V T=(L_V - R_V)T$ and $A_V T=(L_V + R_V)T$. Now it is straightforward to verify $(B_2)_G = (D-L_V)$, $(B_2)_V = -R_G$. This leads to the transformation operator $\Pi_2 = (D-L_V)^{-1} R_G$. Similarly, $(B_3)_G = (D-R_{\widetilde{V}})$, $(B_3)_{\widetilde{V}} = -L_G$ leading to $\Pi_3 = L_{G^{-1}} (D-R_{\widetilde{V}})$. Hence,
\begin{gather*}
\Pi = L_{G^{-1}} \big(D-R_{\widetilde{V}}\big) (D-L_V)^{-1} R_G.
\end{gather*}
Next, using the identities $(D-L_V) R_G = R_G (D-C_V)$ and $(D-R_{\widetilde{V}}) R_G = R_G D$, which are straightforward to check, it follows
\begin{gather}
\Pi = L_{G^{-1}} \big(D-R_{\widetilde{V}}\big) \big((D-L_V)^{-1} R_G \big) = L_{G^{-1}} \big( \big(D-R_{\widetilde{V}}\big) R_G \big) (D-C_V)^{-1}
\nonumber \\
\hphantom{\Pi}{} = L_{G^{-1}} R_G D (D-C_V)^{-1} = K_{G} D (D-C_V)^{-1}. \label{pi}
\end{gather}
Hence, the amKdV recursion operator $\widetilde \Psi(\widetilde V)$ is obtained from the recursion operator $\Psi(V)$ of the mKdV equation as stated in~\eqref{mkdv-recop} via
\begin{gather*}
\widetilde{ \Psi}\big(\widetilde{V}\big) \stackrel{\eqref{rec with pi}, \eqref{pi}}= \big( K_{G} D (D-C_V)^{-1} \big) (D-C_V)D^{-1}(D+C_V)(D-A_V)\\
\hphantom{\widetilde{ \Psi}\big(\widetilde{V}\big)\stackrel{\eqref{rec with pi}, \eqref{pi}}=}{} \times D^{-1}(D+A_V) \big( (D-C_V) D^{-1} K_{G^{-1}} \big) \\
\hphantom{\widetilde{ \Psi}\big(\widetilde{V}\big)}{} = K_{G^{-1}} (D+C_V)(D-A_V)D^{-1}(D+A_V) (D-C_V) D^{-1} K_G \\
\hphantom{\widetilde{ \Psi}\big(\widetilde{V}\big)}{} = (\mathbb{D}+C_{\widetilde V})(\mathbb{D}-A_{\widetilde V})\mathbb{D}^{-1}
(\mathbb{D}+A_{\widetilde V})(\mathbb{D}-C_{\widetilde V})\mathbb{D}^{-1} ,
\end{gather*}
where the last step requires Proposition~\ref{p1}.
\end{proof}
To conclude this section, for the reader's convenience we give a direct verif\/ication that the mKdV and the amKdV equation are related via $\widetilde{V} = G^{-1} G_x$ and $V=G_xG^{-1}$. More precisely, we show
\begin{itemize}\itemsep=0pt
\item[a)] $V_t = \Psi(V)V_x \Longleftrightarrow \widetilde{V}_t = \widetilde \Psi(\widetilde{V})\widetilde{V}_x$,
\item[b)] $ \widetilde \Psi(\widetilde{V})\widetilde{V}_x$ constitutes the right-hand side of \eqref{mkdv2}.
\end{itemize}
To prove a), recall that $\widetilde \Psi(\widetilde V) = \Pi \Psi(V) \Pi^{-1}$ where $\Pi = K_G D (D-C_V)^{-1} $, compare \eqref{rec with pi}, \eqref{pi} in the proof of Theorem~\ref{t1}. Noting that $\widetilde V = K_G V$ implies $\widetilde{V}_x = K_G V_x$, since $(G^{-1}G_x)_x = G^{-1} (G_xG^{-1})_x G$ (observe that $\widetilde{V}_{xx} = K_G V_{xx}$ does not hold). Hence
\begin{gather}
\Pi^{-1} \widetilde{V}_x \stackrel{\eqref{pi}}{=} (D-C_V) D^{-1} K_G^{-1} \widetilde{V}_x = (D-C_V) D^{-1} V_x = (D-C_V) V = V_x. \label{h1}
\end{gather}
Furthermore, $\widetilde{V}_t = (G^{-1}G_x)_t = G^{-1}G_{xt} - G^{-1} G_t G^{-1} G_x = G^{-1} (G_{xt} G^{-1} - G_t G^{-1} G_x G^{-1}) G = K_G D(G_t G^{-1})$. This shows
\begin{gather}
\Pi^{-1} \widetilde{V}_t = (D-C_V) D^{-1} K_G^{-1} \widetilde{V}_t = (D-C_V) \big(G_t G^{-1}\big) = V_t . \label{h2}
\end{gather}
Now a) follows from $ \widetilde \Psi(\widetilde V)\widetilde V_x = \Pi \Psi(V) \Pi^{-1} \widetilde V_x \stackrel{\eqref{h1}}{=} \Pi \Psi(V) V_x = \Pi V_t \stackrel{\eqref{h2}}{=} \widetilde V_t $.
To prove b), note f\/irst that $\widetilde V_x = D\widetilde V = (D+C_{\widetilde V}) \widetilde V = \mathbb{D} \widetilde V$. Similarly, $(\mathbb{D}-C_{\widetilde V})\widetilde V = \mathbb{D} \widetilde V$, and then $(\mathbb{D}+A_{\widetilde V})\mathbb{D} \widetilde V = \mathbb{D}^2 \widetilde V + \{\widetilde V, \mathbb{D}\widetilde V\} = \mathbb{D} (\widetilde V^2+\mathbb{D}\widetilde V)$. Using Theorem~\ref{t1}, we hence get
\begin{gather*}
\widetilde{\Psi} \big(\widetilde{V}\big) \widetilde{V}_x
= \big(\mathbb{D}+C_{\widetilde V}\big) \big(\mathbb{D} - A_{\widetilde V}\big)\mathbb{D}^{-1}
\big(\mathbb{D}+A_{\widetilde V}\big) \big(\mathbb{D}-C_{\widetilde V}\big) \mathbb{D}^{-1} \mathbb{D} \widetilde V \\
\hphantom{\widetilde{\Psi} \big(\widetilde{V}\big) \widetilde{V}_x}{}
= \big(\mathbb{D}+C_{\widetilde V}\big) \big(\mathbb{D} - A_{\widetilde V}\big)\mathbb{D}^{-1}
\big(\mathbb{D}+A_{\widetilde V}\big) \mathbb{D} \widetilde V = \big(\mathbb{D}+C_{\widetilde V}\big) \big(\mathbb{D} - A_{\widetilde V}\big)
\big( \mathbb{D}\widetilde V+\widetilde V^2 \big) \\
\hphantom{\widetilde{\Psi} \big(\widetilde{V}\big) \widetilde{V}_x}{}
= \big(\mathbb{D}+C_{\widetilde V}\big) \big( \mathbb{D}^2 \widetilde V - 2 \widetilde V^3 \big)
= \mathbb{D}^3 \widetilde V + \big[\widetilde V,\mathbb{D}^2 \widetilde V\big] - 2 \mathbb{D} \big(\widetilde V^3\big) .
\end{gather*}
Finally, evaluation of $\mathbb{D}=D+C_{\widetilde V}$ gives the right-hand side of the amKdV equation.
\section{A new non-Abelian B\"acklund chart}\label{s new}
In the last two sections a non-Abelian generalization of the B\"acklund chart linking pKdV, KdV and mKdV equations was presented, and extended to include the amKdV equation \eqref{mkdv2}. The aim of the present section is a further extension to also include non-Abelian analogues of the KdV singularity manifold equation and the KdV interacting soliton equation. To the best of the authors' knowledge, the non-Abelian equations presented in this sections are new. Together with the links presented in the previous section, the resulting non-Abelian B\"acklund chart\footnote{To facilitate the comparison with the Abelian case, a brief overview of the results in~\cite{Fuchssteiner:Carillo:1989a} is provided in the Appendix.} we obtain can be summarized as follows:
\begin{gather}
\mbox{\scriptsize $\boxed{\mbox{pKdV}(W)} \, {\buildrel B_1\over{\text{\textendash\textendash}}} \,
\boxed{\mbox{KdV} (U)} \, {\buildrel M \over{\text{\textendash\textendash}}} \,
\boxed{\mbox{mKdV}(V)} \, {\buildrel B_2, B_3 \over{\text{\textendash\textendash}}} \, \boxed{\mbox{amKdV}(\widetilde V)}
{\buildrel B_4\over{\text{\textendash\textendash}}} \, \boxed{\mbox{KdV sing.}(\phi)} \, {\buildrel B_5 \over{\text{\textendash\textendash}}} \
\boxed{\mbox{int.\, soliton KdV}(S)}\, .$}\label{BC-nc}
\end{gather}
Moreover, the corresponding recursion operators are derived. Note that this chart extends to the respective hierarchies. In conclusion, it is shown that all recursion operators are hereditary. Finally, it is well-known that the KdV singularity manifold equation is invariant under the full M\"obius group, and it is shown that a generalized property holds for its non-Abelian counterpart.
To begin with, we introduce a non-Abelian analogue of the KdV singularity manifold equation (KdV sing.)
\begin{gather}\label{nc-KdVsing}
\phi_t = \phi_x \{ \phi; x \} ,
\end{gather}
where $\{ \phi; x \}$ denotes the following non-Abelian version of the Schwarzian derivative
\begin{gather*}
\{ \phi; x \} := \big( \phi_x^{-1}\phi_{xx} \big)_x - \frac{1}{2} \big( \phi_x^{-1}\phi_{xx} \big)^2.
\end{gather*}
\begin{rem}\quad
\begin{enumerate}\itemsep=0pt
\item[a)] To the best of the authors' knowledge, equation \eqref{nc-KdVsing} is new. For earlier non-Abelian ver\-sions of \eqref{nc-KdVsing} we refer to~\cite{Svinolupov}, see also~\cite{AF}. The dif\/ference to our approach is the non-Abelian interpretation of terms of the form $1/u_x^2$.
\item[b)] In the scalar case there is a very satisfying group theoretic explanation of the relation between the KdV sing.\ (also called UrKdV), mKdV and KdV equations~\cite{Wilson}, see also~\mbox{\cite{Depireux, Schiff}}. It seems to be an interesting problem to f\/ind an appropriate extension of these ideas to the non-Abelian setting. Departing from the present results, a f\/irst major dif\/f\/iculty is that our link between~\eqref{mkdv} and~\eqref{mkdv2} (which reduce to the same scalar equation) is not realized by an explicit mapping $V=F(\widetilde V)$.
\end{enumerate}
\end{rem}
The KdV sing.\ equation \eqref{nc-KdVsing} exhibits invariances analogous to its scalar counterpart. Note that the following theorem implies invariance under the full M\"obius group in the scalar case.
\begin{thrm} \label{t2} Let $A$, $B$, $C$, $D$ be constant operators. Then the non-Abelian KdV singularity manifold equation~\eqref{nc-KdVsing} is invariant under the transformations
\begin{gather*}
\phi \mapsto (\phi A+B)^{-1} (\phi C +D), \qquad \phi \mapsto (C\phi+D) (A\phi+B)^{-1} ,
\end{gather*}
provided that $A$ and $D-BA^{-1}C$ are invertible.
In other words, if $\phi$ is an invertible solution of \eqref{nc-KdVsing}, then also $(\phi A+B)^{-1} (\phi C +D)$ and $(C\phi+D) (A\phi+B)^{-1} $ are solutions of~\eqref{nc-KdVsing}.
\end{thrm}
\begin{rem} The condition that $D-BA^{-1}C$ is invertible can be understood as a generalization of the scalar condition $ad-bc\not=0$ in the case $a\not=0$. Note however, that this is not the only possibility to guarantee the invariances of Theorem~\ref{t2}. For example, the assumption that $B$ and $D-AB^{-1}C$ are invertible is also suf\/f\/icient.
\end{rem}
The main ingredient of the proof of Theorem \ref{t2} is the following observation.
\begin{prop} \label{p2} The non-Abelian KdV singularity manifold equation \eqref{nc-KdVsing} is invariant under the transformation $\phi\mapsto\phi^{-1}$.
\end{prop}
\begin{proof} To start with we calculate the Schwarzian derivative of $\psi:=\phi^{-1}$.
Since $\psi_x =-\phi^{-1}\phi_x\phi^{-1}$, we get $\psi_{xx} = -\phi^{-1}\phi_{xx}\phi^{-1} +2\phi^{-1}\phi_x\phi^{-1}\phi_x\phi^{-1}$. Hence
\begin{gather*}
\psi_x^{-1}\psi_{xx} = \phi\phi_x^{-1}\phi_{xx}\phi^{-1} - 2 \phi_x\phi^{-1},
\end{gather*}
and therefore
\begin{gather*}
\{\psi; x\} = \big(\psi_x^{-1}\psi_{xx}\big)_x - \frac{1}{2} \big(\psi_x^{-1}\psi_{xx}\big)^2 \\
\hphantom{\{\psi; x\}}{} = \phi_{xx}\phi^{-1} + \boxed{ \phi \big(\phi_x^{-1}\phi_{xx} \big)_x \phi^{-1} }
- \phi\phi_x^{-1}\phi_{xx}\phi^{-1}\phi_x\phi^{-1} - 2\phi_{xx}\phi^{-1} + 2\phi_x\phi^{-1}\phi_x\phi^{-1} \\
\hphantom{\{\psi; x\}=}{}
\boxed{ - \frac{1}{2} \phi \big(\phi_x^{-1}\phi_{xx}\big)^2 \phi^{-1} }
+ \phi \phi_x^{-1}\phi_{xx} \phi^{-1}\phi_x\phi^{-1} + \phi_{xx} \phi^{-1} - 2\phi_x\phi^{-1}\phi_x\phi^{-1}.
\end{gather*}
All terms with exception of the boxed ones cancel out, hence $\{\psi;x\} = \phi \{\phi, x\} \phi^{-1}$. As a result,
\begin{gather*}
\psi_t = -\phi^{-1} \phi_t \phi^{-1} \stackrel{\eqref{nc-KdVsing}}{=} -\phi^{-1} \big( \phi_x \{ \phi;x \} \big) \phi^{-1}
= \big({-} \phi^{-1}\phi_x \phi^{-1} \big) \big( \phi \{ \phi;x \} \phi^{-1} \big) = \psi_x \{ \psi; x\} ,
\end{gather*}
which completes the proof.
\end{proof}
The following invariance is straightforward to verify.
\begin{prop} \label{p3} The non-Abelian KdV singularity manifold equation \eqref{nc-KdVsing} is invariant under the transformation $\phi\mapsto A\phi B + C$ where $A$, $B$, $C$ are constant operators, provided that~$A$ and~$B$ are invertible.
\end{prop}
\begin{proof}[Proof of Theorem \ref{t2}] Let $\phi$ be a solution of~\eqref{nc-KdVsing}. By Proposition~\ref{p3}, also $\phi A+B$ solves~\eqref{nc-KdVsing} since~$A$ is invertible. Then, by Proposition~\ref{p2}, the inverse $\psi=(\phi A+B)^{-1}$ is a solution of~\eqref{nc-KdVsing}. Applying Proposition~\ref{p3}, $\psi(D-BA^{-1}C)+A^{-1}C = (\phi A+B)^{-1}(D-BA^{-1}C+(\phi A+B)A^{-1}C) = (\phi A+B)^{-1}(\phi C+D)$ is a solution of~\eqref{nc-KdVsing}.
\end{proof}
Finally, we introduce the non-Abelian KdV interacting soliton equation (int.\ soliton KdV)
\begin{gather}\label{intS}
S_t = \left( S \left( \big(S^{-1}S_x\big)_x - \frac{1}{2} \big(S^{-1}S_x\big)^2 \right) \right)_x .
\end{gather}
Again, to the best of the authors' knowledge, this equation is new.
Analogously as in Section~\ref{s mkdvs}, it can be verif\/ied that the amKdV \eqref{mkdv2}, KdV sing.~\eqref{nc-KdVsing} and int.\ soliton KdV~\eqref{intS} equations are related by the B\"acklund transformations
\begin{gather}
B_4\colon \ 2 \phi_x {\widetilde V} - \phi_{xx} =0 , \label{B-tildeVphi} \\
B_5\colon \ \phi_x-S =0 . \label{BSphi}
\end{gather}
\begin{rem} Note that the counterpart of the transformation $\widetilde{V} = \frac{1}{2}\phi_x ^{-1}\phi_{xx}$ from \eqref{B-tildeVphi}, namely, $\widehat{V} = -\frac{1}{2}\phi_{xx}\phi_x ^{-1}$, also maps solutions $\phi$ of the KdV sing.\ equation~\eqref{nc-KdVsing} to solutions $\widehat{V}$ of the amKdV equation~\eqref{mkdv2}.
\end{rem}
Both the recursion operators $\Upsilon(\phi)$ of \eqref{nc-KdVsing} as well as $\chi(S)$ of \eqref{intS} can be constructed on use of the B\"acklund links (\ref{B-tildeVphi}), (\ref{BSphi}).
\begin{thrm} \label{t rec ops} Define $N(T) = \frac{1}{2} T^{-1}T_x$. Then the recursion operators $\Upsilon (\phi)$ of KdV sing.~\eqref{nc-KdVsing} and $\chi(S)$ of~\eqref{intS} are given by
\begin{gather}
\Upsilon (\phi) = L_{\phi_x} \mathbb{D}^{-1} \big(\mathbb{D}-A_{N(\phi_x)}\big) \big(\mathbb{D}-C_{N(\phi_x)}\mathbb{D}^{-1} C_{N(\phi_x)}\big)
\big(\mathbb{D}+A_{N(\phi_x)}\big) L_{\phi_x^{-1}} , \label{rec sing} \\
\label{rec int} \chi(S)= \textstyle L_{S} \big(\mathbb{D}-A_{N(S)}\mathbb{D}^{-1}A_{N(S)}\big)
\big(\mathbb{D}-C_{N(S)}\mathbb{D}^{-1} C_{N(S)}\big) L_{S}^{-1} .
\end{gather}
Note that $\mathbb{D} = D+C_{N(\phi_x)} = D+C_{N(S)} $.
\end{thrm}
\begin{proof}Indeed, from the B\"acklund links we have that the recursion operators satisfy
\begin{alignat*}{3}
&\Upsilon (\phi) =\Pi_4 \widetilde{\Psi} \big(\widetilde{V}\big)\Pi_4^{-1}, \qquad && \Pi_4 = - (B_4)_{\phi}^{-1} (B_4)_{\widetilde{V}}, & \\
& \chi(S)= \Pi_5 \Upsilon (\phi) \Pi_5^{-1}, \qquad && \Pi_5 = - (B_5)_S^{-1} (B_5)_{\phi} .&
\end{alignat*}
It is straightforward to calculate $(B_4)_{\widetilde V} = 2L_{\phi_x}$, $(B_4)_{\phi} = -(D-2R_{\widetilde V})D$. Therefore, we get
\begin{gather*}
\Pi_4 = 2 D^{-1}\big(D-2R_{\widetilde{V}}\big)^{-1} L_{\phi_x} .
\end{gather*} Similarly $(B_5)_{S} = -I$, $(B_5)_{\phi} = D$, and hence $\Pi_5=D$. Consequently,
\begin{gather*}
\Pi_5 \Pi_4 = 2 \big(D-2R_{\widetilde{V}}\big)^{-1} L_{\phi_x} = 2 \big(\mathbb{D}-A_{\widetilde{V}}\big)^{-1} L_{S} .
\end{gather*}
Using the identity $(\mathbb{D}-A_{\widetilde V})L_S = L_S(\mathbb{D}+C_{\widetilde V})$, which can be checked directly using only the product rule and $\widetilde V = \frac{1}{2}S^{-1}S_x$, we f\/ind
\begin{gather*}
\Pi_5 \Pi_4 = 2 L_S \big(\mathbb{D} + C_{\widetilde{V}}\big)^{-1} .
\end{gather*}
From this and Theorem \ref{t1}, we get
\begin{gather*}
\chi (S) = (\Pi_5 \Pi_4) \widetilde{\Psi}\big(\widetilde{V}\big) (\Pi_5 \Pi_4 )^{-1}
= L_S \big(\mathbb{D}+C_{\widetilde{V}}\big)^{-1} \widetilde{\Psi}\big(\widetilde{V}\big) \big(\mathbb{D}+C_{\widetilde{V}}\big)L_S^{-1} \\
\hphantom{\chi (S)}{}
= L_S \big(\mathbb{D}-A_{\widetilde{V}}\big) \mathbb{D}^{-1} \big(\mathbb{D}+A_{\widetilde{V}}\big)
\big(\mathbb{D}-C_{\widetilde{V}}\big)\mathbb{D}^{-1} \big(\mathbb{D}+C_{\widetilde{V}}\big) L_{S^{-1}} .
\end{gather*}
The claim for $\chi(S)$ follows from observing that $(\mathbb{D}-A_{\widetilde{V}}) \mathbb{D}^{-1} (\mathbb{D}+A_{\widetilde{V}}) = \mathbb{D} -A_{\widetilde{V}}\mathbb{D}^{-1}A_{\widetilde{V}} = \mathbb{D} - A_{N(S)} \mathbb{D}^{-1} A_{N(S)} $ and $(\mathbb{D}-C_{\widetilde{V}}) \mathbb{D}^{-1} (\mathbb{D}+C_{\widetilde{V}}) = \mathbb{D} -C_{\widetilde{V}}\mathbb{D}^{-1}C_{\widetilde{V}} = \mathbb{D} - C_{N(S)} \mathbb{D}^{-1} C_{N(S)} $. Finally,
\begin{gather*}
\Upsilon (\phi) = \Pi_5^{-1} \chi(S) \Pi_5 = D^{-1} L_S \big( \mathbb{D} - A_{N(S)} \mathbb{D}^{-1} A_{N(S)} \big)\big(\mathbb{D} -
C_{N(S)} \mathbb{D}^{-1} C_{N(S)}\big) L_S^{-1} D.
\end{gather*}
Since $DL_{S}^{-1} = L_S^{-1}D - L_S^{-1}L_{S_x} L_S^{-1}$, the identity $L_S^{-1}D = (D+L_{S^{-1}S_x})L_S^{-1} = (D+2L_{N(S)})L_S^{-1} = (\mathbb{D}+A_{N(S)})L_S^{-1} $ holds, and thus
\begin{gather*}
\Upsilon (\phi) = L_S \big(\mathbb{D}+A_{N(S)}\big)^{-1} \big( \mathbb{D} - A_{N(S)} \mathbb{D}^{-1} A_{N(S)} \big)\big(\mathbb{D} -
C_{N(S)} \mathbb{D}^{-1} C_{N(S)}\big) \big(\mathbb{D}+A_{N(S)}\big) L_S^{-1} \\
\hphantom{\Upsilon (\phi)}{}
= L_{S} \mathbb{D}^{-1} \big(\mathbb{D}- A_{N(S)}\big) \big(\mathbb{D}-C_{N(S)}\mathbb{D}^{-1} C_{N(S)}\big) \big(\mathbb{D}+A_{N(S)}\big) L_S^{-1},
\end{gather*}
which is the claim for $\Upsilon (\phi) $ upon substituting $S=\phi_x$.
\end{proof}
In the scalar case, the recursion operator for the KdV sing.\ equation \eqref{nc-KdVsing} is well-known, see, e.g.,~\cite{Fuchssteiner:Carillo:1989a}, where it appears in the form
\begin{gather*}
\Upsilon(\phi) = \phi_x D^{-1} \big(D^3 + 2D{\sf U}+2{\sf U}D \big) \phi_x^{-1}.
\end{gather*}
with ${\sf U}=\frac{1}{2} (\phi_{xx}/\phi_x)_x - \frac{1}{4} (\phi_{xx}/\phi_x)^2$. To facilitate comparison to the non-Abelian case, we state the following reformulation of \eqref{rec sing}.
\begin{cor} The recursion operator $\Upsilon (\phi)$ of the KdV sing.\ equation \eqref{nc-KdVsing} in Theorem~{\rm \ref{t rec ops}} can be written as
\begin{gather*}
\Upsilon (\phi) = L_{\phi_x} \mathbb{D}^{-1} \big( \mathbb{D}^3 + \mathbb{D} A_{\sf U} + A_{\sf U} \mathbb{D} + C_{\sf U} \mathbb{D}^{-1} C_{\sf U}
\big) L_{\phi_x^{-1}} ,
\end{gather*}
where ${\sf U} = ( N(\phi_x) )_x - ( N(\phi_x) )^2$.
\end{cor}
\begin{proof} The corollary is an immediate consequence of Proposition \ref{factor}, the fact that $N(\phi_x) = \widetilde{V}$ and $\widetilde{V}_x= \mathbb{D}\widetilde{V}$.
\end{proof}
\begin{rem} Note that in the scalar case ${\sf U} = v_x -v^2$, i.e., $\sf U$ stems from a Miura transformation of a solution~$v$ to the mKdV equation. In the non-Abelian setting, we have ${\sf U} = \widetilde{V}_x -\widetilde{V}^2$, which means that~$\sf U$ stems from a (non-Abelian) Miura transformation of a~solution $\widetilde{V}$ to the amKdV equation~\eqref{mkdv2}.
\end{rem}
\begin{prop} \label{factor} Let $\sf D$ be a derivation, and $\sf U$, $\sf V$ related by the Miura transform ${\sf U} = {\sf D}{\sf V}-{\sf V}^2$. Then the following factorization holds:
\begin{gather*}
{\sf D}^3 + A_{\sf U} {\sf D} + {\sf D} A_{\sf U} + C_{\sf U} {\sf D}^{-1} C_{\sf U}
= ({\sf D}-A_{\sf V})\big({\sf D}-C_{\sf V}{\sf D}^{-1} C_{\sf V}\big)({\sf D}+A_{\sf V}) \\
\hphantom{{\sf D}^3 + A_{\sf U} {\sf D} + {\sf D} A_{\sf U} + C_{\sf U} {\sf D}^{-1} C_{\sf U}}{}
= ({\sf D}-A_{\sf V}) ({\sf D} \pm C_{\sf V}) {\sf D}^{-1} ({\sf D} \mp C_{\sf V}) ({\sf D}+A_{\sf V}).
\end{gather*}
\end{prop}
\begin{proof}The proof of the proposition is completely analogous to the proof of the factorization of the non-Abelian KdV recursion operator on the image of a~Miura transformation in \cite[Proposition~14]{Carillo:Schiebold:JMP2009}. The crucial observation is that due to the product rule the identities
\begin{gather*}
{\sf D} A_T = A_T {{\sf D}} + A_{{\sf D}T} , \qquad {\sf D} C_T = C_T {{\sf D}} + C_{{\sf D}T}
\end{gather*}
hold for any derivation ${\sf D}$.
\end{proof}
{\sloppy There are close algebraic relations between the non-Abelian recursion operators of the amKdV, KdV sing.\ and int.\ soliton KdV equations.
}
\begin{cor} For $\widetilde V =\frac{1}{2} \phi_x^{-1} \phi_{xx}$ and $S=\phi_x$, the recursion operators in Theorem~{\rm \ref{t1}} and the recursion operators in~\eqref{rec sing},~\eqref{rec int} are pairwise related. More precisely, it holds
\begin{gather*}
\widetilde \Psi\big(\widetilde V\big) = P_1P_2= Q_1 Q_2, \qquad \chi(S) = Q_2 Q_1 = R_1 R_2, \qquad \Upsilon(\phi) = P_2P_1 = R_2 R_1,
\end{gather*}
where
\begin{alignat*}{3}
& P_1 = \big(\mathbb{D}+C_{\widetilde V}\big) \big(\mathbb{D}+A_{\widetilde V}\big) L_{\phi_x}^{-1}, \qquad &&
P_2 = L_{\phi_x} \mathbb{D}^{-1} \big(\mathbb{D}-A_{\widetilde V}\big) (\mathbb{D}- C_{\widetilde V}) \mathbb{D}^{-1}, & \\
& Q_1 = \big(\mathbb{D}+C_{\widetilde V}\big) L_{\phi_x}^{-1} , \qquad && Q_2 = L_{\phi_x} \big(\mathbb{D}-A_{\widetilde V}\mathbb{D}^{-1} A_{\widetilde V}\big)\big(\mathbb{D}-C_{\widetilde V}\big) \mathbb{D}^{-1} ,& \\
& R_1 = L_{\phi_x} \big(\mathbb{D}+A_{\widetilde V}\big) L_{\phi_x}^{-1}, \qquad && R_2 = L_{\phi_x} \mathbb{D}^{-1} \big(\mathbb{D}-A_{\widetilde V}\big) \big(\mathbb{D} - C_{\widetilde V}\mathbb{D}^{-1}C_{\widetilde V}\big)L_{\phi_x}^{-1}.&
\end{alignat*}
\end{cor}
\section{Hereditariness and hierarchies}\label{section5}
The links obtained in the last two sections are summarized in
\begin{gather*}
\boxed{\mbox{mKdV}(V) } \ {\buildrel B_2, B_3 \over{\text{\textendash\textendash}}} \
\boxed{\mbox{amKdV}(\widetilde V)} \ {\buildrel B_4\over{\text{\textendash\textendash}}} \
\boxed{\mbox{KdV sing.}(\phi)}\ {\buildrel B_5 \over{\text{\textendash\textendash}}} \
\boxed{\mbox{int.\ soliton KdV}(S)}\,,
\end{gather*}
where the B\"acklund transformations $B_2$, $B_3$, $B_4$ and $B_5$ are given in \eqref{Bmkdvs1}, \eqref{Bmkdvs2}, \eqref{B-tildeVphi} and \eqref{BSphi}, respectively. Applying the respective recursion operators, the B\"acklund chart extends to the whole hierarchies as follows
\begin{gather} \label{BC-nc part 2 H}
\boxed{V_t = [\Psi(V)]^n V_{x}} \buildrel B_2, B_3 \over{\text{\textendash\textendash}}
\boxed{\widetilde V_t = [\widetilde \Psi(\widetilde V)]^n \widetilde V_{x}}
\buildrel B_4 \over{\text{\textendash\textendash}}
\boxed{\phi_t = [\Upsilon(\phi)]^n \phi_{x}} \buildrel B_5 \over{\text{\textendash\textendash}}
\boxed{S_t = [\chi(S)]^n S_x}\, .
\end{gather}
As known from the scalar case, hereditariness is a crucial property of recursion operators \cite{Fuchssteiner1979, Magri}. Unfortunately its direct verif\/ication often requires involved computations, in particular in the non-Abelian case. The following proposition uses the B\"acklund links established in the present article to avoid computations by reducing the proof of hereditariness to the hereditariness of the non-Abelian KdV recursion operator, which is proved in~\cite{Schiebold2010}.
\begin{prop} \label{p4} Each of the recursion operators $\Psi$ from the B\"acklund chart \eqref{BC-nc} has the following properties:
\begin{enumerate}\itemsep=0pt
\item[$a)$] $\Psi$ is hereditary,
\item[$b)$] $\Psi$ is a strong symmetry for all equations of the hierarchy generated by $\Psi$.
\end{enumerate}
\end{prop}
\begin{proof}
In \cite[(32)]{Schiebold2010} it is verif\/ied that the non-Abelian KdV recursion operator \eqref{kdv-recop} satisf\/ies the identity
\begin{gather*} [D, \Phi (U)] = \Phi'(U)[U_x] , \end{gather*}
implying that $\Phi(U)$ is a strong symmetry (recursion operator in the sense of~\cite{Olver}) for the trivial member $U_{t}=U_x$ of the non-Abelian KdV hierarchy \cite{Fuchssteiner1979}. Moreover, the main result in \cite{Schiebold2010} is that $\Phi(U)$ is hereditary. Hence $\Phi(U)$ is a strong symmetry for all equations of the non-Abelian KdV hierarchy \cite{Fuchssteiner1979}. As shown in \cite{FoFu2}, the properties of a) and b) are preserved under B\"acklund transformations.
\end{proof}
Each of the hierarchies in \eqref{BC-nc part 2 H} is of the form
\begin{gather} \label{hier}
U_t = [\Phi(U)]^n U_x .
\end{gather}
We may rewrite the right-hand side of \eqref{hier} as
\begin{gather*} [\Phi(U)]^n U_x = X_n(U) ,\end{gather*}
where $X_n$ is a vector f\/ield on the space of $x$-dependent operator-valued functions (see \cite{Schiebold2010} for details). The main consequence of Proposition~\ref{p4} is
\begin{cor} Let $X_1, X_2,\ldots $ be the vector fields of one of the hierarchies in \eqref{hier}. Then we have
\begin{gather*} [X_m,X_n] = 0 \end{gather*}
for all $m,n =1,2,\ldots$.
\end{cor}
The proof uses arguments explained in \cite[Section VI]{Schiebold2010}.
\section{Remarks, perspectives and open problems} \label{rems}
This section collects some remarks on the results previously presented together with some perspectives study and open problems. The chain of B\"acklund transformations we obtained, represents a not at all trivial generalization to the operator level of the corresponding one~\cite{Fuchssteiner:Carillo:1989a} which links the scalar pKdV, KdV, mKdV, KdV interacting soliton and KdV singularity mani\-fold hierarchies. Furthermore, it generalizes the non-Abelian B\"acklund chart in \cite{Carillo:Schiebold:JMP2009} since it connects further non-Abelian hierarchies. Specif\/ically, new non-Abelian equations, and, then, the corresponding hierarchies, arise, such as the amKdV, in~\eqref{mkdv2}.
\bigskip\noindent{\bf Remarks}
\begin{itemize}\itemsep=0pt
\item The two B\"acklund charts, respectively, in the Abelian \cite{Fuchssteiner:Carillo:1989a} (see Appendix) and the non Abelian case, connect the pKdV, KdV, mKdV, KdV-singularity manifold and Interacting Soliton KdV equations or their non-Abelian analogs.
\item Known the recursion operator of a nonlinear evolution equation then all the other nonlinear evolution equations linked to it via a B\"acklund chart admit a recursion operator. The latter can be constructed in the Abelian as well as in the non-Abelian case.
\item All nonlinear evolution equations in the same B\"acklund chart share all the structural pro\-perties which are preserved under B\"acklund transformations as soon as a~single equation, in it, enjoys them. Remarkable is the hereditariness of the recursion operators~\cite{Schiebold2010}.
\item Also in the non-Abelian operator case, given the hereditary recursion operators, the B\"acklund chart can be extended to the corresponding generated hierarchies. Again, the B\"acklund chart relates the corresponding members of each one of the involved hierarchies of nonlinear evolution equations.
\item Even if there are similarities between the Abelian scalar case and the non-Abelian operator case, in the second case the structure is richer. Indeed, two distinct non-Abelian mKdV equations appear: the non-Abelian mKdV and an {\it alternative} mKdV equations which do coincide in the Abelian case. Correspondingly, when commutativity is assumed, combination of the B\"acklund transformations $B_2$ and $B_3$ produces the identity transformation and, hence, the Abelian B\"acklund chart~\cite{Fuchssteiner:Carillo:1989a} is recovered.
\item A similar behavior can be observed also when the Cole--Hopf link connecting Burgers equation to linear heat equation is extended to the non-Abelian case \cite{SIMAI2008, Carillo:Schiebold:JNMP2012}. The heat equation is connected to two dif\/ferent Burgers equations, termed Burgers and mirror Burgers equations in \cite{CMS-2016a, Ku}. Then, recursion operators and the corresponding hierarchies follow from the Cole--Hopf link, which can be regarded as a special case of B\"acklund transformation.
\end{itemize}
\noindent{\bf Perspectives and open problems}
\begin{itemize}\itemsep=0pt
\item We expect a similar situation when the 5th order nonlinear evolution equations which appear in the B\"acklund chart in \cite{BS1, Rogers:Carillo:1987b} are extended to the non-Abelian case. This study is currently under investigation and we are devising also computer aided routines to check the algebraic properties of the recursion operators. Indeed, already in the Abelian case, the computations involved are very long and complicated.
\item Furthermore, if a nonlinear evolution equation admits a Hamiltonian and bi-Hamiltonian structure, related to the recursion operator \cite{[12], {Fuchssteiner:Carillo:1990a}, Benno-Walter, Magri}, then all nonlinear evolution equations in the same B\"acklund chart admit a Hamiltonian and bi-Hamiltonian structure.
\item The approach can be extended also when nonlinear evolution equations in $(2+1)$ dimensions are considered, namely, the unknown function is supposed to depend on two space variables rather than on a single space variable. Thus, in \cite{walsan1}, the Kadomtsev--Petviashvili (KP), modif\/ied Kadomtsev--Petviashvili (mKP) and $(2+1)$-dimensional Harry Dym equations, which represent, in turn, the $(2+1)$-dimensional analog of KdV, mKdV, and Harry Dym equations are all connected via B\"acklund transformations. Notably, the connection among their $(1+1)$-dimensional corresponding equations \cite{walsan2} follows on imposing suitable constraints to the $(2+1)$-dimensional ones.
\end{itemize}
|
1,314,259,996,745 | arxiv | \section*{References}
|
1,314,259,996,746 | arxiv | \section{Introduction}
The semiclassical description of
black-hole (BH) radiation \cite{hawk1} suggests that an
initial pure state evolves into a final mixed thermal state \cite{hawk2}.
A transition prom a pure to a mixed state is incompatible with
unitarity of quantum mechanics (QM), which constitutes the famous
BH information puzzle. The attempts to restore unitarity can be divided
into two types (for reviews, see, e.g., \cite{har}).
In the first type, the black hole does not evaporate
completely, but ends in a Planck-sized remnant that contains the information
missing in the Hawking radiation. The problem is that such a light object
should contain a huge amount of information, which seems unphysical.
In particular, light objects that may exist in a huge number of different
states should have a huge probability for creation in various physical
processes, which, however, is not seen in experiments.
A variant of the remnant scenario is the creation of a baby-universe not observable from our universe, but such an idea remains rather speculative.
In the second type, the black hole evaporates completely, but the radiation is not exactly thermal. Instead, there
are some additional subtle correlations among radiated particles.
It is argued that this requires a sort of nonlocality not present
in standard quantum field theory (QFT), suggesting that quantum gravity
should contain some new nonlocal features.
The most promising candidate for a consistent theory
of quantum gravity is string theory. Indeed,
it provides new insights on BH thermodynamics
(see, e.g., \cite{hor,peet} for reviews). In particular, it provides
a unitary description of BH radiation and offers a
microscopic explanation of the BH entropy proportional to the surface.
It also contains some nonlocal features that might explain
the desired deviation from exact thermality.
Nevertheless, the theoretical description of the mechanism
of BH radiation in string theory
(see, e.g., \cite{das})
seems completely different from that in the conventional semiclassical
theory, so it remains difficult to see where exactly
the semiclassical analysis fails. Thus, it would be desirable to understand
a generic property that a large class of models of quantum gravity,
including string theory, should
possess in order to save the unitarity of BH radiation.
The aim of this paper is to find such a generic resolution of the BH information puzzle, without using any explicit model of quantum gravity.
We find that neither a new sort of nonlocality (for the case of complete
evaporation) nor a huge amount of information in a light remnant
(for the case of a remnant scenario) is needed. In fact, we find that
no new unexpected property of physical laws is required. Instead, the standard
rules of QM applied to black holes
in a generic and intuitively appealing manner turn out to be sufficient.
\section{Physical insights}
\subsection{Pure thermal states and decoherence}
First, let us observe that a thermal distribution of particles is not
necessarily incompatible with a possibility that these particles are
in a pure state. For a simple example, consider a single
quantum harmonic oscillator
(with the frequency $\omega$) in the state
$ |\psi\rangle = \sum_n f_{\omega,n}|n\rangle $,
where $n=0,1,2,\ldots$ and
\begin{equation}\label{e2.1}
f_{\omega,n}=\sqrt{1-e^{-\beta\omega}} e^{-\beta\omega n/2} .
\end{equation}
Clearly, $|\psi\rangle$ is a pure state. Yet, the probabilities of
different energies $E=\omega n$ are proportional to $e^{-\beta E}$,
which corresponds to a thermal distribution with the temperature
$T=1/\beta$.
The density matrix $\rho=|\psi\rangle \langle\psi|$ can be written as
\begin{equation}\label{rho}
\rho=\sum_n |f_{\omega,n}|^2 |n\rangle \langle n| +
\sum_{n\neq n'} f_{\omega,n}f_{\omega,n'} |n\rangle \langle n'| .
\end{equation}
The first (diagonal) term represents the usual mixed thermal state. The second
(off-diagonal) term is responsible for the additional correlations steming from the fact that the state is pure. When a simple system (in this case, the single harmonic oscillator) interacts with an environment with a large number of
unobserved degrees of freedom, then, in practice, the presence of the second term is unobservable. Thus, for all practical purposes,
the state can be described by the first term only. In QM this is known as the
phenomenon of decoherence (for a review see, e.g., \cite{schlos}).
Thus, decoherence provides a mechanism for an effective transition
from a pure to a mixed state
\begin{equation}\label{decoh}
|\psi\rangle \langle\psi| \stackrel{\rm decoher}{\longrightarrow}
\sum_n |f_{\omega,n}|^2 |n\rangle \langle n| .
\end{equation}
It does not involve any violation of unitarity at the fundamental
level.
\subsection{The role of negative-frequency particles in semiclassical
and fully quantum black holes}
As we shall see, the observations above will play a role in our resolution
of the BH information paradox. Indeed,
the role of decoherence in BH thermodynamics has already been discussed in
\cite{kief}. Nevertheless, decoherence is not the main part of our resolution.
To see the true origin of the BH information puzzle, we start from the fact
that standard semiclassical analysis based on the Bogoliubov transformation
describes Hawking radiation as particle creation in which the initial
vacuum $|0\rangle$ transforms to a squeezed state \cite{bd}
\begin{equation}\label{e1}
|0\rangle \stackrel{\rm squeeze}{\longrightarrow}
|\psi\rangle_{\rm squeeze} ,
\end{equation}
where
\begin{equation}\label{e2}
|\psi\rangle_{\rm squeeze}=\prod_{\omega}\sum_n f_{\omega,n}(M)
|n_{-\omega}\rangle \otimes |n_{\omega}\rangle ,
\end{equation}
and, for massless uncharged spin-0 particles, $f_{\omega,n}(M)$ are given by (\ref{e2.1}) with $\beta\equiv 8\pi M$, where $M$ is the BH mass.
The product is taken over all possible positive values of $\omega$.
The state $|n_{\omega}\rangle$ represents on outgoing state containing
$n_{\omega}$ particles, each having frequency $\omega$, so that their
total energy is $E=\omega n_{\omega}$. Similarly, $|n_{-\omega}\rangle$
represents $n_{-\omega}$ ingoing particles, each having {\em negative} frequency
$-\omega$.
In our notation, the direct product $\otimes$ separates the
inside states on the left from the outside states on the right.
At this level the total energy is not yet conserved, as
the energy of the negative-frequency states is also positive,
in the sense that the sign of their energy is the same as that of
the interior matter determining the BH mass $M$.
The conservation of energy is provided by another mechanism, namely
by renormalization of the energy-momentum tensor implying a
flux of negative energy across the horizon into the
black hole \cite{bd}. The overall effect is that the BH mass
decreases, such that the total energy is conserved.
However, owing to the creation of negative-frequency particles that carry
information, the information content of the black hole {\em increases}
despite the fact that its mass decreases. Does it contradict the
first law of BH thermodynamics? Not necessarily, if the BH entropy
proportional to the BH surface (and thus to $M^2$) is interpreted
merely as the part of BH information that is available to the outside
observer. However, string theory suggests a very different interpretation
of BH entropy -- the entropy associated with counting of
the internal degrees of freedom of the black hole, independent on the
knowledge of an outside observer. Thus, from the string-theory point
of view, the information carried by the negative-frequency particles should be {\em unphysical}. Indeed, the physical mechanism of BH radiation in string
theory does not rest on the Bogoliubov transformation, and hence
does not lead to creation of particles in the BH interior \cite{das}.
Thus, our idea is to modify the semiclassical description of particle creation,
in a manner that removes the negative-frequency particles from physical
states.
For states $|n_{-\omega}\rangle$ we find convenient to
introduce a negative effective ``renormalized"
energy $E=-\omega n_{-\omega}$, without changing the information content
of these states.
This makes energy conserved already
at the level of (\ref{e2}), making the analysis simpler.
The product over $\omega$ shows that states of the form
$|n_{\omega}\rangle |n_{\omega'}\rangle \cdots$ with total
energies $E=\omega n_{\omega} + \omega' n_{\omega'} +\ldots$ also appear.
Thus, it is convenient to rewrite (\ref{e2}) as a sum over energy eigenstates
$|\pm E,\xi\rangle$
\begin{equation}\label{e3}
|\psi\rangle_{\rm squeeze}=\sum_E\sum_{\xi}d_{E,\xi}(M)
|-E,\xi\rangle \otimes |E,\xi\rangle ,
\end{equation}
where $\xi$ labels different states having the same outside or
inside energy $\pm E$, and the sum is taken over non-negative values
of $E$.
The coefficients $d_{E,\xi}$ can be expressed in terms of $f_{\omega,n}$,
but the explicit expression will not be needed here.
The squeezed state (\ref{e2}) is a pure state and the transition
(\ref{e1}) is unitary \cite{gris}. Consequently, the density matrix
constructed from (\ref{e3}) is pure.
However, an outside observer cannot observe the
inside states, so the density matrix describing the knowledge of the
outside observer is given by tracing out the inside degrees of freedom
of the total density matrix. Applying this to
(\ref{e3}), one obtains
\begin{equation}\label{out}
\rho_{\rm out}=\sum_E\sum_{\xi}|d_{E,\xi}(M)|^2 \,
|E,\xi\rangle \langle E,\xi| ,
\end{equation}
which is a mixed state.
However, we have argued that the negative-energy states are not physical,
which means that the mixed thermal state (\ref{out})
is obtained by tracing out over unphysical degrees of freedom. Hence,
this mixed thermal state may also be unphysical. A physical density matrix
should be obtained by tracing out over physical (but unobserved)
degrees of freedom. The difference between unphysical and unobserved
degrees is in the fact that the former cannot be observed even in principle,
by any observer.
The unphysical negative-energy particles can be intuitively viewed
as virtual particles analogous to those appearing in Feynman diagrams
of conventional perturbative QFT. They cannot exist as final measurable
states. Instead, they must be {\em absorbed} by physical states. In our case,
the physical object that should absorb them is the black hole.
To give a precise description of this process of absorption, one should
invoke a precise microscopic theory that presumably includes a
quantum theory of gravity as well. Nevertheless, the essential features of such
an absorption can be understood even without a precise microscopic theory.
For simplicity, we study uncharged and unrotating black holes. Thus,
we assume that a black hole with a mass $M$ can be described by a
quantum state $|M;\chi_M\!>$, where $\chi_M$ labels different BH states
having the same mass $M$. We assume that the number of different states
increases with $M$ and that there is only one state with mass $M=0$,
i.e., that $|0;\chi_0\!>=|0\!>$. In particular, such an assumption
is consistent with string theory asserting that entropy
of the internal BH degrees of freedom is proportional to the surface,
i.e., to $M^2$. It is also consistent with a more naive possibility
that the entropy is proportional to the volume, i.e., to $M^3$.
In fact, proportionality of entropy to the surface rests
on the validity on the Einstein equation, while thermal particle creation
from a horizon
is a much more general phenomenon \cite{viss}. As our analysis will not
depend on validity of the Einstein equation, we will not be able to specify
the exact number of states with mass $M$. For our purposes, it is sufficient
to assume that the absorption of negative-energy particles takes a generic form
\begin{equation}\label{e4}
|M;\chi_M\!> |-E,\xi\rangle \stackrel{\rm absorp}{\longrightarrow}
|M-E;\chi_{M-E}\!> .
\end{equation}
Such a form is dictated by energy conservation, which, indeed, is consistent
with the first law of BH thermodynamics.
Note that the left-hand side of (\ref{e4}) has a larger number
of different states than the right-hand side. Consequently, the operator
governing the absorption (\ref{e4}) is not invertible, and thus cannot be
unitary. Nevertheless, the overall unitarity is not necessarily violated.
To see why, note that, although the squeezing (\ref{e1}) is
described by a formally unitary operator, it is not unitary on the
{\em physical} Hilbert space (because the physical Hilbert space
does not contain the unphysical negative-energy particles).
Thus, neither the squeezing (\ref{e1}) nor the absorption (\ref{e4}) are
physical processes by themselves. What is physical is their composition
\begin{equation}\label{e5}
|M;\chi_M\!> \rightarrow \sum_E\sum_{\xi}d_{E,\xi}(M)
|M-E;\chi_{M-E}\!> \otimes |E,\xi\rangle .
\end{equation}
Thus, if the initial state is $|\Psi_0\rangle=|M;\chi_M\!>$,
then we have a physical transition
$|\Psi_0\rangle \rightarrow |\Psi_1\rangle$, where
$|\Psi_1\rangle$ is the right-hand side of (\ref{e5}).
The physical process (\ref{e5}) is expected to be unitary.
(An explicit verification
of unitarity requires a more specific model of quantum gravity.)
In fact, one may forget about the virtual subprocesses (\ref{e1}) and (\ref{e4})
and consider (\ref{e5}) as the only directly relevant physical process.
Indeed, the process of BH radiation in a more advanced theory of quantum gravity may not be based on a Bogoliubov transformation at all, so it may not
be formulated in terms of creation of virtual negative-energy particles
appearing in (\ref{e1}), but directly in terms of physical processes
of the form of (\ref{e5}). In fact, this is exactly what occurs
in string theory \cite{das}.
Note also that in (\ref{e4}) we assume that the right-hand side
does not depend on $\xi$. This reflects on the right-hand side
of (\ref{e5}) in the fact that the new BH state does not depend on the
state of radiation $\xi$. This means that there is no correlation between
radiated particles and BH interior, except for the trivial correlation
expressing the fact that total energy must be conserved.
The absence of such correlations is expected also from a
more general view of the semiclassical description
of particle creation \cite{niksc}.
As we shall see,
this destruction of the (unphysical) information contained in the
negative-energy particles on the left-hand side of (\ref{e4})
makes the remnant scenario viable, by removing the unwanted huge information
that otherwise would have to be be present in a light remnant.
Nevertheless, later we also discuss a possibility to relax the
assumption that the nontrivial correlation between exterior
radiation and BH interior is completely absent.
\section{The process of radiation -- unitary evolution
and the role of wave-function collapse}
Now the analysis of further steps of the process of BH radiation is
mainly technical. After (\ref{e5}), the remaining BH state radiates
again, now at a new larger temperature corresponding to the new smaller
BH mass $M-E$. Thus, the next step $|\Psi_1\rangle \rightarrow |\Psi_2\rangle$
is based on a process analogous to (\ref{e5})
\begin{eqnarray}\label{e6}
|M-E;\chi_{M-E}\!> \rightarrow \sum_{E'}\sum_{\xi'}d_{E',\xi'}(M-E)
\nonumber \\
\times |M-E-E';\chi_{M-E-E'}\!> \otimes |E',\xi'\rangle ,
\end{eqnarray}
so
\begin{eqnarray}\label{e7}
|\Psi_2\rangle = \sum_E\sum_{E'}\sum_{\xi}\sum_{\xi'}
d_{E,\xi}(M) d_{E',\xi'}(M-E) \nonumber \\
\times |M-E-E';\chi_{M-E-E'}\!> \otimes |E,\xi\rangle |E',\xi'\rangle .
\end{eqnarray}
Repeating the same process $t$ times, we obtain
\begin{eqnarray}\label{e8}
|\Psi_t\rangle & = & \sum_{E_1}\cdots\sum_{E_t}
\sum_{\xi_1}\cdots\sum_{\xi_t} \nonumber \\
& \times &
d_{E_1,\xi_1}(M) \cdots d_{E_t,\xi_t}(M-E_1- \cdots -E_{t-1}) \nonumber \\
& \times &
|M-{\cal E};\chi_{M-{\cal E}}\!> \otimes |E_1,\xi_1\rangle \cdots |E_t,\xi_t\rangle ,
\end{eqnarray}
where ${\cal E}=\sum_{t'=1}^{t}E_{t'}$.
(A continuous description of evolution labeled by a continuous time parameter $t$ is also possible, but this does not change our main conclusions.)
States with the same energy ${\cal E}$ can be grouped together,
so we can write
\begin{equation}\label{e10}
|\Psi_t\rangle=\sum_{\cal E} \sum_{\Xi} |M-{\cal E};\chi_{M-{\cal E}}\!> \otimes
D^{(t)}_{{\cal E},\Xi} |{\cal E},\Xi\rangle ,
\end{equation}
where $\Xi=\{ \xi_1,\ldots, \xi_t \}$ and the coefficients
$D^{(t)}_{{\cal E},\Xi}$ can be expressed in terms of
$d_{E_{t'},\xi_{t'}}$.
Note that,
for any finite $t$, $|\Psi_t\rangle$ contains contributions
from all possible BH masses $M'=M-{\cal E}$. At first sight,
it seems to imply that the unitary evolution (\ref{e10}) prevents the
black hole from evaporating completely during a finite time $t$.
Nevertheless, this is not really true. To see why, it is instructive
to consider a simpler quantum decay $|a\rangle\rightarrow|b\rangle$
in which the unitary evolution usually implies an exponential law
$|\psi(t)\rangle=\sqrt{1-e^{-\Gamma t}}|b\rangle + \sqrt{e^{-\Gamma t}}|a\rangle$.
For any finite $t$, there is a finite probability $e^{-\Gamma t}$
that the decay has not yet occurred. Nevertheless, a wave-function
collapse associated with an appropriate quantum measurement implies
that at each time the particle will be found either in the state
$|a\rangle$ or $|b\rangle$. Analogously, if the BH mass
$M'$ is measured at time $t$, the wave-function collapse implies
\begin{equation}\label{e11}
|\Psi_t\rangle \stackrel{\rm measure}{\longrightarrow}
|M-{\cal E};\chi_{M-{\cal E}}\!> \otimes
N_{\cal E} \sum_{\Xi} D^{(t)}_{{\cal E},\Xi} |{\cal E},\Xi\rangle ,
\end{equation}
where $N_{\cal E}$ is the normalization factor,
$N_{\cal E}^{-2}=\sum_{\Xi} |D^{(t)}_{{\cal E},\Xi}|^2$.
Now the black hole is in a definite pure state
$|M-{\cal E};\chi_{M-{\cal E}}\!>$ and the outside particles are in a
definite pure state
$N_{\cal E} \sum_{\Xi} D^{(t)}_{{\cal E},\Xi} |{\cal E},\Xi\rangle$.
(More realistically, the measurement uncertainty $\Delta M'$ is smaller
for smaller $M'$, so the outside particles are closer to a pure state
when $M'$ is smaller.) For example,
it is conceivable that some quantum mechanism might prevent transitions
(\ref{e5}) for $M-E<M_{\rm min}$ (where $M_{\rm min}$ is a hypothetic
minimal possible BH mass).
In this case, (\ref{e11}) may correspond
to a transition to a BH remnant with a mass $M-{\cal E}=M_{\rm min}$.
Such a BH remnant is not correlated with the radiated particles
(except for the correlation implied by energy conservation)
and the information content of the remnant is determined only by its mass.
The absence of such correlations is a consequence of the assumption
that the right hand-side of (\ref{e4}) does not depend on $\xi$.
This assumption could also be relaxed by allowing that at least
some different $\xi$'s may correspond to different BH states.
In this case, the BH state in (\ref{e11}) would also depend
on $\Xi$, so it would not sit in front of the sum over $\Xi$,
which would imply that neither the black hole nor the radiation
is in an exactly pure state, but that there is a small correlation
between them. Nevertheless, the maximal amount of possible correlation is restricted by the smallness of the BH mass. In particular, if
$M_{\rm min}=0$, then (\ref{e11}) may correspond to a complete
evaporation of the black hole, in which case the BH state
$|0\!>$ must be unique, implying that the final state of radiation
must be a pure state
$N_M \sum_{\Xi} D^{(t)}_{M,\Xi} |M,\Xi\rangle$.
\section{Discussion -- thermality, apparent nonunitarity, and
the origin of nonlocality}
We have seen that, under reasonable assumptions, the BH radiation is in a
pure state whenever the BH mass is measured exactly. Does it mean that
the BH radiation is not really thermal? Actually not.
Instead, the situation is analogous
to that in the discussion around Eqs.~(\ref{e2.1})-(\ref{decoh}).
For example, if the BH mass is measured after the first step
(\ref{e5}), then the radiation collapses to a pure state
equal (up to an overall normalization factor) to
$\sum_{\xi} d_{E,\xi}(M) |E,\xi\rangle$. This state is obtained from
$\prod_{\omega}\sum_n f_{\omega,n}(M)|n_{\omega}\rangle$
by rewriting it as a sum of products and retaining only those states
the total energy of which is equal to $E$. The density matrix
of such a pure state takes a form analogous to (\ref{rho}).
Due to the decoherence induced by the interaction with the environment,
in practice such a state can be effectively described by a mixed state
analogous to (\ref{decoh}). From (\ref{e2.1}) we see that it is a thermal
mixed state. More precisely, as the total energy $E$ is exactly
specified, while the number of particles is specified only in average,
this is a thermal state corresponding to a grand microcanonical ensemble.
By contrast, the thermal state (\ref{out}) (in which both total energy and number
of particles are specified only in average) corresponds to a
grand canonical ensemble.
At the end, let us recall that our resolution of the BH information
puzzle involves 4 different types of seemingly nonunitary evolutions.
The process of squeezing (\ref{e1}) is formally unitary \cite{gris},
but it is not unitary on the physical space. It is allways accompanied
with another nonunitary virtual process (the absorption
of negative-energy particles) Eq.~(\ref{e4}), which together are combined
into a physical unitary process (\ref{e5}). This represents the core of
our resolution of the BH information puzzle. The third nonunitary process
is the wave-function collapse (\ref{e11}). The exact meaning of
the collapse depends on the general interpretation of QM that one adopts.
In particular, in some interpretations (e.g., many-world interpretation
and the Bohmian interpretation) a true collapse does not really exist,
making QM fully consistent with unitarity. Finally, the fourth
nonunitary process is the phenomenon of decoherence (\ref{decoh}),
which corresponds only to an effective violation of unitarity, not a fundamental
one.
Finally note that, although our analysis allows a complete BH evaporation
without a true violation of unitarity, no new nonlocal mechanism
has been involved. The only new mechanism is the absorption
(\ref{e4}), which, however, occurs only inside the black hole,
thus not violating locality. Some nonlocal mechanisms {\em are} involved
in our analysis, namely quantum entanglement and quantum
wave-function collapse, but these are standard nonlocal aspects of QM.
\section*{Acknowledgements}
The author is grateful to T.~Jacobson for valuable discussions
on Hawking radiation and BH information.
This work was supported by the Ministry of Science of the
Republic of Croatia under Contract No.~098-0982930-2864.
|
1,314,259,996,747 | arxiv | \section{Introduction}\label{sec:intro}
Bayesian updating provides a principled and coherent approach to inference for probabilistic models \cite{robert2007bayesian}, but is predicated on the model class being true. That is, given a generative model $F_\theta(x)$ parametrized by a finite-dimensional parameter $\theta$, then for some parameter value $\theta_0 \in \Theta$ it is that $x \sim F_{\theta_0}(x)$. In reality all models are false. If the data is simple and small, and the models are sufficiently rich, then the consequences of model misspecification may not be severe. However, data is increasingly being captured at scale, both in terms of the number of observations as well as the diversity of data modalities. This poses a risk in conditioning on an assumption that the model is true.
In this paper we discuss a scalable approach to Bayesian nonparametric learning (NPL) from models without the assumption that $x \sim F_{\theta_0}(x)$. To do this we use a nonparametric prior for $F_0$ that is centred on a model but does not assume the model to be true.
A concentration parameter, $c$, in the nonparametric prior quantifies trust in the baseline model and this is subsequently reflected in the relative posterior influence given to the model-based inference for parameter $\theta$, compared to a purely empirical approach. In particular, $c \rightarrow \infty$ recovers the standard model-based Bayesian update while $c \rightarrow 0 $ leads to a Bayesian bootstrap estimator for the object of interest.
Our method provides both a means of regularizing a nonparametric update and correcting misspecified or approximate posterior inferences. This may be useful in a number of situations, including:
\begin{itemize}
\item [{[S0]}] Nonparametric regularization: where we wish to use a Bayesian NP approach but we would like to include a regularization term, centered on a conventional model, that induces stability and parametric structure on the problem.
\item [{[S1]}] Model misspecification: where we have used a parametric Bayesian model and we are concerned that the model may be misspecified.
\item [{[S2]}] Approximate posteriors: where for expediency we have used an approximate posterior, such as in Variational Bayes (VB), and we wish to account for the approximation.
\item [{[S3]}] Direct updating of the prior: where the parameters have meaning but little trust is placed in the validity of the generative likelihood, such as for classification trees that define piecewise constant predictions regardless of the smoothness of the actual decision boundaries.
\item [{[S4]}] Information sharing under privacy considerations: where we might be willing to share pseudo-data generated from a posterior model, but not the actual training data or model.
\item [{[S5]}] Direct updating from utility-functions: where the sole purpose of the modelling task is to perform some action or take a decision under a well-specified utility function.
\end{itemize}
Our work builds upon previous ideas including \cite{Newton1994} who introduced the weighted likelihood bootstrap (WLB) as a way of generating approximate samples from the posterior of a well-specified Bayesian model. \cite{Lyddon2018} highlighted that the WLB in fact provides an exact representation of uncertainty for the model parameters that minimize the Kullback-Leibler (KL) divergence, $d_{\text{KL}}(F_0,F_\theta)$, between the unknown data-generating distribution and the model likelihood $f_\theta(x)$, and hence is well motivated regardless of model validity. These approaches however do not allow for the inclusion of prior knowledge and do not provide a Bayesian update as we do here.
A major underlying theme behind our paper, and indeed an open field for future research, is the idea of obtaining targeted posterior samples via the maximization of a suitably randomized objective function. The WLB randomizes the log-likelihood function, effectively providing samples which are randomized maximum likelihood estimates, whereas we randomize a more general objective function under a NP posterior. The randomization takes into account knowledge associated with the parametric model.
\section{Foundations of Nonparametric Learning}\label{sec:foundations}
We begin with the simplest scenario, namely [S1], concerning a possibly misspecified model before moving on to more complicated situations. All of what follows can be considered from a viewpoint of NP regularization and hence we don't explicitly deal with [S0] from herein.
\subsection{Bayesian updating of misspecified models}
Suppose we have a parametric statistical model, $\cF_\Theta = \{ f_\theta(\cdot); \ \theta \in \Theta ) \}$, where for each $\theta \in \Theta \subseteq \Re^p$, $f_\theta : \cX \rightarrow \Re$ is a probability density. The conventional approach to Bayesian learning involves updating a prior distribution to a posterior through Bayes' theorem. This approach is well studied and well understood \cite{Bernardo2006}, but formally assumes that the model space captures the truth. We will derive a posterior update under weaker assumptions.
Suppose that $\cF_\Theta$ has been selected for the purpose of a prediction, or a decision, or some other modelling task. Consider the thought experiment where the modeller somehow gains access to Nature's true sampling distribution for the data, $F_0(x)$, which does not necessarily belong to $\cF_\Theta$. How should they then update their model?
With access to $F_0$ the modeller can simply request an infinite training set, ${x}_{1:\infty} \stackrel{iid}{\sim} F_0$,
and then update to the posterior ${\pi}(\theta | {x}_{1:\infty})$. Under an infinite sample size all uncertainty is removed and for regular models the posterior concentrates at a point mass at $\theta_0$, the parameter value maximizing the expected log-likelihood, assuming that the prior has support there.
\begin{eqnarray*}\label{eq:inf_sample}
\theta_0 = \argmax_{\theta\in\Theta} \lim_{n\to\infty} n^{-1} \sum_{i=1}^{n} \log f_\theta(x_i) = \argmax_{\theta\in\Theta} \int_{\cX} \log f_\theta(x)\,d F_0.
\end{eqnarray*}
It is straightforward to see that ${\theta}_0$ minimizes the KL divergence from the true data-generating mechanism to a density in $\cF_\Theta$
\begin{equation}\label{eq:theta_0}
\theta_0 = \argmax_{\theta \in \Theta} \int_\cX \log f_\theta(x) dF_0(x) = \argmin_{\theta\in\Theta} \int_\cX \log \frac{f_0(x)}{f_\theta(x)} dF_0(x).
\end{equation}
This is true regardless of whether $F_0$ is in the model space of $\cF_\Theta$ and is well-motivated as the target of statistical model fitting \cite{Akaike1981a,burnham2003model,Walker2013,Bissiri2016}.
Uncertainty in the value of $\theta_0$ flows directly from uncertainty in $F_0$. In reality, of course $F_0$ is unknown, but being ``Bayesian'' we can place a prior on it, $\pi(F)$, for $F \in \cF$, that should reflect our honest uncertainty about $F_0$. Typically the prior should have broad support unless we have special knowledge to hand, which is a problem with a parametric modelling approach that only supports a family of distribution functions indexed by a finite-dimensional parameter. The Bayesian NP literature provides a range of priors for this sole purpose, see for example \cite{hjort2010bayesian}. Once a prior for $F$ is chosen, the correct way to propagate uncertainty about $\theta$ comes naturally from the posterior distribution for the law ${\cal L}[\theta(F) | x_{1:n}]$, via ${\cal L}[F | x_{1:n}]$, where $\theta(F) = \argmax_{\theta\in\Theta} \int \log f_\theta(x) dF(x)$. The posterior on the parameter of interest is then captured in the marginal by treating $F$ as a latent probability measure,
\begin{equation}\label{eq:f_marg}
\tilde{\pi}(\theta | x_{1:n} ) = \int_\cF \pi(\theta, dF | x_{1:n} ) = \int_\cF \pi(\theta | F) \pi(dF | x_{1:n} )
\end{equation}
where $\pi(\theta | F)$ assigns probability $1$ to $\theta=\theta(F)$. We use $\tilde{\pi}$ to denote the NP update to distinguish it from the conventional Bayesian posterior, noting that in general the nonparametric posterior will be different to the standard Bayesian update, $\tilde{\pi}(\theta | x_{1:n}) \not\equiv \pi(\theta | x_{1:n})$, as they are conditioning on different prior states of knowledge. In particular, as stated above, $\pi(\theta | x_{1:n})$ assumes that $F_0 \in \cF_\Theta$.
\subsection{An NP prior using a MDP}
For our purposes, the mixture of Dirichlet processes (MDP) \cite{Antoniak1974} is a convenient vehicle for specifying prior beliefs $\pi(F)$ centered on parametric models\footnote{The MDP should not to be confused with the Dirichlet process mixture model (DPMM) \cite{Lo1984}.}.
The MDP prior can be written as
\begin{equation}\label{eq:MDP}
[F \mid \theta] \sim \text{DP}(c, f_\theta(\cdot) ) ; \qquad \theta \sim \pi(\theta).
\end{equation}
This is a mixture of standard Dirichlet processes with mixing distribution or hyper-prior $\pi(\theta)$, and concentration parameter $c$.
We write this as $F \sim \text{MDP}(\pi(\theta), c, f_\theta(\cdot) )$.
The MDP provides a practical, simple posterior update.
From the conjugacy property of the DP applied to (\ref{eq:MDP}), we have the conditional posterior update given data $x_{1:n}$, as
\begin{equation}\label{eq:MDP_posterior_condl}
[F \mid \theta, x_{1:n}] \sim \text{DP}\left( c+n, \ \frac{c}{c+n}f_\theta(\cdot) + \frac{1}{c+n} \sum_{i=1}^n \delta_{x_i}(\cdot) \right)
\end{equation}
where $\delta_x$ denotes the Dirac measure at $x$. We see the representation of $c$ as an effective sample size governing the trust in $f_{\theta}(x)$. Thus the marginal posterior distribution for ${\cal L}[F | x_{1:n}]$ can be written as
\begin{equation}\label{eq:mdp-marg}
\pi(dF \mid x_{1:n} ) = \int_\Theta \pi(dF \mid \theta, x_{1:n} ) \, \pi(\theta \mid x_{1:n} ) \, d\theta,
\end{equation}
i.e.
\begin{equation}\label{eq:MDP_posterior_full}
[F | x_{1:n}] \sim \text{MDP}\left( \pi(\theta | x_{1:n}), \ c+n, \ \frac{c}{c+n} f_\theta(\cdot) +\frac{1}{c+n}\sum\limits_{i=1}^n \delta_{x_i}(\cdot) \right).
\end{equation}
The mixing distribution $\pi(\theta | x_{1:n})$ coincides with the parametric Bayesian posterior \cite{Antoniak1974}, $\pi(\theta | x_{1:n}) \propto\prod_{i=1}^n~f_{\theta}(x_i)~\pi(\theta)$, assuming there are no ties in the data, although as noted above it does not follow that the NP marginal $\tilde{\pi}(\theta | x_{1:n})$ is equivalent to the parametric Bayesian posterior $\pi(\theta | x_{1:n}) $.
We can see from the form of the conditional MDP (\ref{eq:MDP_posterior_condl}) that the sampling distribution of the centering model, $f_\theta(x)$, regularizes the influence of the data $\sum_{i=1}^n \delta_{x_i}(\cdot)$. The resulting NP posterior (\ref{eq:mdp-marg}) combines the information from the posterior distribution of the centering model $\pi(\theta | x_{1:n})$ with the information in the empirical distribution of the data. This leads to a simple and highly parallelizable Monte Carlo sampling scheme as shown below.
\subsection{Monte Carlo conditional maximization}
The marginal (\ref{eq:f_marg}) facilitates a Monte Carlo estimator for functionals of interest, $G = \int_\Theta g(\theta) \tilde{\pi}(\theta | x_{1:n} ) d \theta$, by sampling
$\pi( \theta, dF | x_{1:n})$ jointly from the posterior,
\begin{eqnarray}\label{eq:FF_marg}
\int_\Theta g(\theta) \tilde{\pi}(\theta | x_{1:n} ) d \theta & \approx & \frac{1}{B} \sum_{i=1}^{B} g(\theta^{(i)}) \nonumber \\
\theta^{(i)} ~ = ~ \theta(F^{(i)}) & = & \argmax_{\theta \in \Theta} \int_\cX \log f_{\theta}(x) dF^{(i)}(x) \label{eq:FF_marg1} \\
F^{(i)} & \sim & \pi(dF | x_{1:n}). \label{eq:FF_marg2}
\end{eqnarray}
This involves an independent Monte Carlo draw (\ref{eq:FF_marg2}) from the MDP marginal followed by a conditional maximization of an objective function (\ref{eq:FF_marg1}) to obtain $\theta(F)$. This Monte Carlo conditional maximization (MCCM) sampler is highly amenable to fast implementation on distributed computer architectures. The resulting estimator has $1/B$ order of variance independent of the dimension of $\theta$ and, unlike MCMC, there are no issues of convergence nor need for a user-defined burn-in phase.
However, we do need to sample from the standard Bayesian posterior directly. This makes the approach particularly attractive to fast, tractable, approximations for $\pi(\theta | x_{1:n})$, such as a variational Bayes (VB). The NP update corrects for the approximation in a computationally efficient manner, leading to an exact posterior with optimal properties as shown below.
\subsection{A more general construction}\label{sec:general}
So far we have assumed, hypothetically, that:
\begin{enumerate}[(i)]
\item the modeller is interested in learning about $\theta_0=\argmax_\theta\int\log f_\theta(x)dF_0(x)$, rather than $\alpha_0 = \argmax_\alpha \int u(x, \alpha) dF_0(x)$ more generally, for a utility function $u(x, \alpha)$.
\item the mixing distribution $\pi(\theta | x_{1:n})$ of the MDP posterior is constructed from the same centering model that defines the target parameter, $\theta_0 = \argmax_\theta \int \log f_\theta(x) dF_0(x)$.
\end{enumerate}
Both of these assumptions can be relaxed. For the latter case, it is valid to center the MDP on a tractable baseline posterior $\pi(\gamma | x_{1:n})$ and still learn about $\theta_0$ in (\ref{eq:theta_0}) through the marginal $\tilde{\pi}(\theta | x_{1:n} )$ as in (\ref{eq:f_marg}) obtained via $\theta(F)$ and
\begin{equation}[F | x_{1:n}] \sim \text{MDP}\left( \ \pi(\gamma | x_{1:n}), \ c+n, \ \frac{c}{c+n} f_\gamma(\cdot) + \frac{1}{c+n} \sum_i\delta_{x_i} \right).
\end{equation}
For the former, we can use the mapping $\alpha(F) = \argmax_\alpha \int u(x, \alpha) dF(x)$ to derive the NPL posterior $\tilde{\pi}(\alpha | x_{1:n} ) = \int \pi(\alpha | F) \pi(dF | x_{1:n} )$ on actions or parameters maximizing some expected utility under a model-centered MDP posterior.
This highlights a major theme of the paper in the idea of obtaining posterior samples, $\pi(\alpha | x_{1:n})$, via maximization of a suitably randomized objective function. In generality the target is $\alpha_0 = \argmax_{\alpha} \int u(x, \alpha) dF_0(x)$, obtained by maximizing an objective function, and the randomization arises from the uncertainty in $F_0$ through $\pi(F | x_{1:n})$ that takes into account the information, and any misspecification, associated with the parametric centering model.
\subsection{A Posterior Bootstrap Algorithm}\label{Subsec:posterior bootstrap}
We will use the general construction of Section \ref{sec:general} to describe a sampling algorithm. We assume we have access to samples from the posterior mixing distribution, $\pi(\gamma | x_{1:n})$, in the MDP. In the case of model misspecification, [S1], if the data contains no ties, this is simply the parametric Bayesian posterior under $\{f_{\gamma}(x), \pi(\gamma)\}$, for which there is a large literature of computational methods available for sampling - see for example \cite{robert2005}. If there are ties then we refer the reader to \cite{Antoniak1974} or note that we can simply break ties by adding a new pseudo-variable, such as $x^* \sim N(0, \epsilon^2)$ for small $\epsilon$.
The sampling algorithm, found in Algorithm \ref{algo:mdp-training}, can be thought of as a mixture of Bayesian bootstraps. After a sample is drawn from the mixing posterior, $\pi(\gamma | x_{1:n})$, a posterior pseudo-sample is generated, $x_{1:T} \stackrel{iid}{\sim} f_{\gamma^{(i)}}(x)$, and added to the dataset, which is then randomly weighted. The parameter under this implicit distribution function is then computed as the solution of an optimization problem.
\begin{figure
\begin{algorithm}[H]
\SetAlgoLined
\KwData{ Dataset $x_{1:n} = (x_1,\ldots,x_n)$. \\
Parameter of interest $\alpha_0 = \alpha(F_0) = \argmax_\alpha \int u(x, \alpha) dF_0(x)$. \\
Mixing posterior $\pi(\gamma | x_{1:n})$, concentration parameter $c$, centering model $f_\gamma(x)$.\\
Number of centering model samples $T$.}
\Begin{
\For{$i = 1,\ldots,B$}{
Draw centering model parameter $\gamma^{(i)} \sim \pi(\gamma | x_{1:n} )$\;
Draw posterior pseudo-sample $x_{(n+1):(n+T)}^{(i)} \stackrel{iid}{\sim} f_{\gamma^{(i)}}$\;
Generate weights $(w^{(i)}_1,\ldots,w^{(i)}_n,w^{(i)}_{n+1},\ldots,w^{(i)}_{n+T}) \sim \text{Dirichlet}(1,\ldots,1,c/T,\ldots,c/T)$\;
Compute parameter update\\ $\,\qquad \tilde{\alpha}^{(i)} = \argmax_\alpha \left\{ \sum\limits_{j=1}^{n} w_j^{(i)} u(x_j, \alpha) + \sum\limits_{j=1}^{T} w_{n+j}^{(i)} u(x_{n+j}^{(i)}, \alpha) \right\};$
}
Return NP posterior sample $\{\tilde{\alpha}^{(i)} \}_{i=1}^B$.
}
\caption{The Posterior Bootstrap}\label{algo:mdp-training}
\end{algorithm}
\end{figure}
Note for the special case of correcting model misspecification [S1], we have $\gamma \equiv \theta$, $f_\gamma(\cdot) \equiv f_\theta(\cdot)$, $\pi(\gamma | x_{1:n}) \equiv \pi(\theta | x_{1:n})$, $\alpha \equiv \theta$, $u(x, \alpha) \equiv \log f_\theta(x)$, so that the posterior sample is given by
$$
\tilde{\theta}^{(i)} = \argmax_\theta \left\{ \sum\limits_{j=1}^n w_j^{(i)} \log f_\theta(x_j) + \sum\limits_{j=1}^T w_{n+j}^{(i)} \log f_\theta(x_{n+j}^{(i)}) \right\}.
$$
where $w^{(i)} \sim \text{Dirichlet}(\cdot)$ following Algorithm \ref{algo:mdp-training} and $x_{(n+1):(n+T)}^{(i)}$ are $T$ synthetic data observations drawn from the parametric sampling distribution under $\theta^{(i)}$ which itself is drawn from $\pi(\theta | x_{1:n})$.
We leave the concentration parameter $c$ to be set subjectively by the practitioner, representing faith in the parametric model. Some further guidance to the setting of $c$ can be found in Appendix \ref{app:setting_c}.
\subsection{Adaptive Nonparametric Learning: aNPL}
Instead of the Dirichlet distribution approximation to the Dirichlet process, a stick-breaking procedure \cite{Sethuraman1991} with a termination threshold set as a proportion, $\epsilon \in (0,1)$, of the expected model weight, $\frac{c}{c+n}$, has a number of desirable properties. This procedure entails following the usual DP stick-breaking construction for the model component of the MDP posterior, by repeatedly drawing $\text{Beta}(1,c)$-distributed stick breaks, but terminates when the unaccounted for probability measure $\prod_j (1 - v_j)$, multiplied by the average mass assigned to the model, $\frac{c}{c+n}$, drops below some threshold $\epsilon$ set by the user. This adaptive nonparametric learning (aNPL) algorithm is written out in full for the model misspecification setting [S1] in Appendix \ref{app:posterior_bootstrap_misspec}.
One advantage of this approach is that a number of theoretical results then hold, as for large enough $n$, under this adaptive scheme the parametric model is in effect `switched off', and in effect the MDP with $c=0$ is used to generate posterior samples when $n$ passes some user-controlled value. This is an interesting notion in itself. For small samples, we prefer the regularization that our model provides, though as $n$ grows the average probability mass assigned to the model decays like $(c+n)^{-1}$, as seen in (\ref{eq:MDP_posterior_condl}). In the adaptive version, we agree a hard threshold at which point we discard the model entirely and allow the data to speak for itself. We set this point at a level such that we are a priori comfortable that there is enough information in the sample alone with which to quantify uncertainty in our parameter of interest. For example, $\epsilon = 10^{-4}$ and $c=1$ only utilises the centring model for $n < 10,000$. Further, we could use this idea to set $c$, as the quantity is determined if a tolerance level, $\epsilon$, and a threshold $n$ over which the parametric model would be discarded, are provided by the practitioner.
\subsection{Properties}\label{sec:properties}
NPL has a number of desirable properties.
\paragraph{Honesty about correctness of model.} Uncertainty in the data-generating mechanism is quantified via a NP update that takes into account the model likelihood, prior, and concentration parameter $c$. Uncertainty about model parameters flows from uncertainty surrounding the data-generating mechanism, $F_0(x)$.
\paragraph{Incorporation of prior information.} The prior for $\theta$ is naturally incorporated as a mixing distribution for the MDP. This is in contrast to a number of Bayesian methods with similar computational properties but that do not admit a prior \cite{Newton1994,Chamberlain2003}.
\paragraph{Parallelized bootstrap computation.}
As shown in Section \ref{Subsec:posterior bootstrap}, NPL is trivially parallelizable through a Bayesian bootstrap and can be coupled with misspecified or approximate models to deliver highly scalable and exact inference.
\paragraph{Consistency.} Under mild regularity, all posterior mass concentrates in any neighbourhood of $\theta_0$ as defined in (\ref{eq:theta_0}), as the number of observations tends to infinity. This follows from an analogous property of the DP (see, for example \cite{hjort2010bayesian}).
\paragraph{Standard Bayesian inference is recovered as $\boldsymbol{c \to \infty}$.}
This follows from the property of the DP that it converges to the prior degenerate at the base probability distribution in the limit of $c\rightarrow\infty$.
\paragraph{Non-informative learning with $\boldsymbol{c =0}$.}
If no prior or model is available, setting $c=0$ recovers the WLB. This has an exact interpretation as an objective NP update \cite{Lyddon2018}, where the asymptotic properties of the misspecified WLB were studied. A result of relevance to us is presented in the following theorem.
\vspace{-5mm}
\begin{quotation}
\begin{theorem}\label{theorem1}
\normalfont
Let $\tilde{\theta}_n$ be a WLB sample of a parameter defined in (\ref{eq:theta_0}) or via aNPL, given $n$ observations $x_{1:n}$, and let $P_{c=0 \,}
be its probability measure. Under regularity conditions, for any Borel set $A \subset \mathbb{R}^p$, as $n \rightarrow \infty$ we have
\begin{equation*}
P_{c=0} \left\{ n^{1/2} \left( \tilde{\theta}_n - \hat{\theta}_n \right) \in A \mid x_{1:n} \right\} \ \rightarrow \ P( z \in A),
\end{equation*}
a.s. $x_{1:\infty}$, where $z \sim N_d\{ \, 0, \, J( \theta_0)^{-1} I(\theta_0) J(\theta_0)^{-1} \, \}$, with
\begin{align*}
I(\theta) &= \int \nabla \log f_\theta(x) \nabla \log f_\theta( x)^T \,dF_0(x),\\
J(\theta) &= - \int \nabla^2 \log f_\theta( x) \,dF_0(x),
\end{align*}
where $\nabla$ is the gradient operator with respect to $\theta$, and $\hat{\theta}_n$ is the MLE.
\end{theorem}
\begin{proof}
See \cite{Lyddon2018}, Theorem 1.
\end{proof}
\end{quotation}
In fact the theorem presented in \cite{Lyddon2018} is for a general utility function, consistent with point (i) of Section \ref{sec:general}. The covariance matrix $J( \theta_0)^{-1} I(\theta_0) J(\theta_0)^{-1}$ is a well-known quantity in the robust statistics literature, sometimes called the sandwich covariance matrix \cite{Cox1961, Huber1967, White1982a}. It's the asymptotic covariance matrix for a maximum likelihood estimator under potential misspecification; if well specified this coincides with the Bayesian posterior, but in general does not. \cite{Muller2013} showed that replacing a Bayesian posterior covariance matrix with the sandwich covariance matrix can provide a reduction in frequentist risk, asymptotically, for some decision problems. We will see next that for large samples the misspecified Bayesian posterior distribution is predictively suboptimal as well.
\paragraph{A superior asymptotic uncertainty quantification to Bayesian updating.}
A natural way to compare posterior distributions is by measuring their predictive risk, defined as the expected KL divergence of the posterior predictive to $F_0$. We consider only the situation where there is an absence of strong prior information, following \cite{Shimodaira2000,Fushiki2005}.
We say that predictive $\pi_1$ asymptotically dominates $\pi_2$ up to $o(n^{-k})$ if for all distributions $q$ there exists a non-negative and possibly positive real-valued functional $K(q(\cdot))$ such that:
\begin{equation*}
\E_{x_{1:n}\sim q} \,d_{\text{KL}}(q(\cdot), \pi_2(\,\cdot \mid x_{1:n} ) ) - \E_{x_{1:n} \sim q}\, d_{\text{KL}}(q(\cdot), \pi_1(\,\cdot \mid x_{1:n}) ) \,=\, K(q(\cdot)) + o(n^{-k}).
\end{equation*}
We have the following theorem about the asymptotic properties of the MDP with $c=0$, which covers aNPL, as the model component of aNPL is ignored for large enough $n$.
\vspace{-5mm}
\begin{quotation}
\begin{theorem}
\normalfont
The MDP posterior predictive with $c=0$ asymptotically dominates the standard Bayesian posterior predictive up to $o(n^{-1})$.
\end{theorem}
\begin{proof}
In \cite{Fushiki2005} the bootstrap predictive is shown to asymptotically dominate the standard Bayesian predictive up to $o(n^{-1})$. In Theorem 1 of \cite{Fushiki2010}, the predictive of the MDP with $c=0$ (equivalently called the Bayesian bootstrap) and the bootstrap predictive are shown to be equal up to $o_p(n^{-3/2})$. A Taylor expansion argument can show that the predictive risk of the MDP has the same asymptotic expansion up to $o(n^{-1})$ as that of the bootstrap. Thus Theorem 2 of \cite{Fushiki2005} can be proven with the MDP predictive in place of the bootstrap predictive. Thus the predictive of the MDP with $c=0$ must also dominate the standard Bayesian predictive.
\end{proof}
\end{quotation}
\section{Illustrations}\label{sec:illustrations}
\subsection{Exponential family, [S1]}
Suppose the centering model is an exponential family with parameter $\theta$ and sufficient statistic $s(x)$,
\begin{equation*}\label{eq:exp_family}
\cF_\Theta = \left\{ f_\theta(x) = g(x) \exp \left\{ \theta^T s(x) - K(\theta) \right\}; \ \theta \in \Theta \right\}.
\end{equation*}
Under assumed regularity, by differentiating under the integral sign of (\ref{eq:theta_0}) we find that our parameter of interest must satisfy $\E_{F_0} s(x) = \nabla_\theta K(\theta_0)$. For a particular $F$ drawn from the posterior bootstrap, the expected sufficient statistic is
\begin{equation*}
\nabla_\theta K(\tilde{\theta}^{(i)}) = \lim_{T\to \infty} \left\{ \sum\limits_{j=1}^n w_j^{(i)} s(x_j) + \sum\limits_{j=n+1}^{n+T} w_j^{(i)} s(x_j^{(i)}) \right\}.
\end{equation*}
with $\tilde{\theta}$ the posterior parameter value, and weights $w_{1:(n+T)}$ arising from the Dirichlet distribution as set out in Algorithm \ref{algo:mdp-training}, and $x_j^{(i)} \sim f_\theta(\cdot)$, with $\theta$ drawn from the parametric posterior. This provides a simple geometric interpretation of our method, as (random) convex combinations of (randomly-weighted) empirical sufficient statistics and model sufficient statistics from the parametric posterior. The distribution of the random weights is governed by $c$ and $n$ only. Our method generates stochastic maps from misspecified posterior samples to corrected NP posterior samples, by incorporating empirical information in the data over and above that captured by the model.
\subsection{Updating approximate posteriors [S2]: Variational Bayes uncertainty correction}
Variational approximations to Bayesian posteriors are a popular tool for obtaining fast, scalable, but approximate Bayesian posterior distributions \cite{bishop2006pattern, Blei2017}.
The approximate nature of the variational update can be accounted for using our approach. Figure \ref{fig:bishop_gaussian} shows a mean-field normal approximation $q$ to a correlated normal posterior $p$, an example similar to one from \cite{bishop2006pattern}, Section 10.1. We generated 100 observations from a bivariate normal distribution, centered at $(\frac{1}{2},\frac{1}{2})$, with dimension-wise variances both equal to $1$ and correlation equal to $0.9$, and independent normal priors on each dimension, both centered at $0$ with variance $100$. Each posterior plotted is based on $10,000$ posterior samples.
By applying the posterior bootstrap with a VB posterior (VB-NPL) in place of the Bayes posterior, we recover the correct covariance structure for decreasing prior concentration $c$, and the effect of the VB centering model is clear. If instead of $d_{\text{KL}}(q,p)$ we use $d_{\text{KL}}(p,q)$ as the objective, as it is for expectation propagation, the model posterior uncertainty may be overestimated, but is still corrected for using the posterior bootstrap.
\begin{figure}[t]
\centering
\includegraphics[scale=1]{Bishop_95_17jun18.pdf}
\caption{Posterior 95\% probability contour for a bivariate Gaussian, comparing VB-NPL with $c\in\{1,10^2,10^3,10^4\}$ (red, orange, green, blue respectively) to the known Bayes posterior (grey dashed line) and a VB approximation (black dashed line).}
\label{fig:bishop_gaussian}
\end{figure}
We demonstrate this in practice through a VB logistic regression model fit to the Statlog German Credit dataset, containing 1000 observations and 25 covariates (including intercept), from the UCI ML repository \cite{Dua:2017}, preprocessing via \cite{Fernandez-Delgado2014}. An independent normal prior with variance $100$ was assigned to each covariate, and 1000 posterior samples were generated for each method. We obtain a mean-field VB sample using automatic differentiation variational inference (ADVI) in Stan \cite{Kucukelbir2015a}. In Fig. \ref{fig:beta_21_22} we show that the NP update effectively corrects the VB approximation for small values of $c$. Of course, local variational methods can provide more accurate posterior approximations to Bayesian logistic posteriors \cite{Jaakkola1997}, though these too are approximations, that NP updating can correct.
When generating pseudo-samples for use in the posterior bootstrap, both features and classes are required. Given a set of features and a parameter draw from the mixing distribution, classes are generated according to the probability specified by the logistic distribution. For the features, in this example (and the example in Section \ref{subsec:BRF}) we repeatedly re-use the features of the dataset. In some applications additional features, independent but identically distributed as those in the dataset, may be available. In this case, these could instead be the basis of our pseudo-dataset.
\paragraph{Comparison with Bayesian logistic regression.} The conventional Bayesian logistic regression assumes the true log-odds of each class is linear in the predictors, and would use MCMC for inference \cite{Polson2013}. The MCMC samples, shown as points in Fig. \ref{fig:beta_21_22}, show a good match to the NPL update but MCMC requires a user defined burn-in and convergence checking. The runtime to generate 1 million samples by MCMC (discarding an equivalent burn-in), was 33 minutes, compared to 21 seconds with NPL, using an m5.24xlarge AWS instance with 96 vCPUs; a speed-up of 95 times. Additionally NPL has provably better predictive properties, as detailed in Section \ref{sec:properties}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{vb_logit_scatter_sb.pdf}
\caption{Posterior contour plot for $\beta_{22}$ vs $\beta_{21}$, for VB-NPL (green) and VB (blue), for three different values of the concentration parameter $c$. Scatter plot is a sample from a Bayesian logistic posterior (red) via Polya-Gamma scheme.}
\label{fig:beta_21_22}
\end{figure}
\subsection{Directly updating the prior [S3]: Bayesian Random Forests, using synthetic generated data [S4]}\label{subsec:BRF}
Random forests (RF) \cite{Breiman2001} is an ensemble learning method that is widely used and has demonstrated excellent general performance \cite{Fernandez-Delgado2014}. We construct a Bayesian RF (BRF), via NP learning with decision trees, under a prior mixing distribution (scenario [S3] in Section \ref{sec:intro}). This enables the incorporation of prior information, via a prior prediction function, in a principled way that is not available to RF. The step-like generative likelihood function arising from the tree partition structure does not reflect our beliefs about the true sampling distribution; the trees are just a convenient compression of the data. Because of this we simply update the prior in the MDP by specifying $\pi(\gamma | x_{1:n}) = \pi(\gamma)$. Details of our implementation of BRF can be found in Appendix \ref{app:brf}.
\paragraph{Information sharing with pseudo-observations [S4].} We emulate the situation whereby two analysts, each with a separate dataset, wish to share information, but a privacy restriction prevents direct accessing of both datasets.
To test our method, we split our dataset equally into an external training, internal training and test dataset. We compared an RF trained on the internal data to two different BRFs. The first is a non-informative BRF, obtained by setting $c=0$, and trained on the internal dataset only. The second is an informative BRF (so $c>0$), trained on an augmented dataset containing the internal dataset and pseudo-samples containing information from the external dataset, suitably weighted according to Algorithm \ref{algo:mdp-training}. These pseudo samples are comprised of the external features, along with class labels that are predictions from a non-informative BRF that has been trained on the external dataset. We assume that the posterior and predictive adequately preserves privacy. As a benchmark, we compared these methods to an RF trained on the combined internal and external datasets, a setup that incorporates all of the data but breaches the privacy constraints. See Figure \ref{fig:diagram-brf} for a pictorial description of the four methods. We compared the test accuracy, over 100 repetitions, for a number of values of $c$, generating 10,000 prior pseudo-samples for the BRF method with $c>0$, with all forests containing 100 trees.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.4]{brf-flow-30may.pdf}
\caption{Depiction of the four methods tested in our BRF example.}\label{fig:diagram-brf}
\end{figure}
Boxplots comparing our methods for 13 datasets from the UCI ML repository \cite{Dua:2017} can be found in Fig. \ref{fig:boxplots}. For small $c$, we find that BRF and RF have similar performance, but as $c$ increases, more weight is given to the externally-trained component and we find that BRF outperforms RF.
The best performance of our BRF tends to occur when $c$ is set equal to the number of samples in the external training dataset, in line with our intuition of the role of $c$ as an effective sample size. A figure demonstrating this can be found in Appendix \ref{app:brf}. Overall, the informative BRF provides a performance boost over RF; in fact the BRF accuracy is close to that obtained by the RF trained on both the internal and external datasets but without the need to share the data or the model.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.85]{box_all_10000_10June.pdf}
\caption{Boxplot of classification accuracy minus that of RF, for 13 UCI datasets and our 3 other methods.}
\label{fig:boxplots}
\end{figure}
\subsection{Direct updating of utility functions [S5]: population median}
We demonstrate inference for a population median, under a misspecified Gaussian model, with parameter of interest $\alpha_0 = \argmin_\alpha \int \vert \alpha-x \vert dF_0(x)$, and an MDP prior centered at a $\cN(\theta,1)$ with prior $\pi(\theta) = \cN(0,10^2)$. We use the posterior bootstrap to generate posterior samples that incorporate the prior model information with that from the data. Figure \ref{fig:posterior_median} presents histograms of posterior medians given a sample of $20$ observations from a skew-normal distribution with mean $0$, variance $1$ and median approximately $-0.2$. The left-most histogram is sharply peaked at the sample median but does not have support outside of $(x_{\min}, x_{\max})$. As $c$ grows smoothness from the parametric model is introduced to a point where the normal location parameter is used.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.85]{posterior_median2.pdf}
\caption{Posterior histogram for median (left to right) $c=0,20,80,1000$. Right-most: posterior expected loss as a function of observation $x$. Dotted line shows the loss to the sample median.}
\label{fig:posterior_median}
\end{figure}
\section{Discussion}\label{sec:discussion}
We have introduced a new approach for scalable Bayesian nonparametric learning (NPL) for parametric models that both facilitates prior regularization via a baseline model, and corrects for model misspecification by incorporating an empirical component that has greater influence as the number of observations grows. A concentration parameter $c$ encodes subjective beliefs on the validity of the model; $c=\infty$ recovers Bayesian updating under the baseline model, and $c=0$ ignores the model entirely. Under regularity conditions, asymptotically, our method closely matches parametric Bayesian updating if the posited model were indeed true, and will provide a superior predictive if the model is misspecified. Additionally, our construction admits a trivially parallelizable sampler without the drawbacks of common MCMC-based samplers.
Our approach can be seen to blur the boundaries between Bayesian and frequentist inference. Conventionally, the Bayesian approach conditions on data and treats the unknown parameter of interest as if it was a random variable with some prior on a known model class. Whereas, the frequentist approach treats the object of inference as a fixed but unknown constant and characterizes uncertainty through the finite sample variability of an estimator targetting this value. Here we randomize an objective function (an estimator) according to a Bayesian nonparametric prior on the sampling distribution leading to a quantification of subjective beliefs on the value that would be returned by the estimator under an infinite sample size. In Appendix \ref{app:models} we provide a review of the foundations of statistical models in this context.
At the heart of our approach is the notion of Bayesian updating via randomized objective functions through the posterior bootstrap. The posterior bootstrap acts on an augmented dataset, comprised of data and posterior predictive synthetic-samples, under which randomized maximum likelihood estimators provide a well-motivated quantification of uncertainty without assuming much about the data-generating mechanism. The precursor to this is the weighted likelihood bootstrap, which utilized a simpler form of randomization that ignored prior information. Our approach provides scope for quantifying uncertainty for more general machine learning models by randomizing their objective function suitably.
\newpage
|
1,314,259,996,748 | arxiv | \section{Introduction}
It is now a well established fact that at zero temperature and
sufficiently high densities quark matter is a color superconductor
\cite{barrois,cs} (see also Alford and Rischke contributions at this
workshop). The study starting from first principles was done in
\cite{weak,PR-sp1,weak-cfl}. At chemical potentials much higher than
the masses of the quarks $u$, $d$ and $s$, the favored state is the
so-called Color-Flavor-Locking (CFL) state, whereas at lower values
the strange quark decouples and the relevant phase is called
two-flavor color superconducting (2SC).
An interesting possibility is that in the interior of compact
stellar objects (CSO) some color superconducting phase might exist.
In fact we recall that the central densities for these stars could
be up to $10^{15}$ g/cm$^{3}$, whereas the temperature is of the
order of tens of keV. However the usual assumptions leading to prove
that for three flavors the favored state is CFL should now be
reviewed. Matter inside a CSO should be electrically neutral and
should not carry any color. Also conditions for $\beta$-equilibrium
should be fulfilled. As far as color is concerned, it is possible to
impose a simpler condition, that is color neutrality, since in
\cite{Amore:2001uf} it has been shown that there is no free energy
cost in projecting color singlet states out of color neutral states.
Furthermore one has to take into account that at the interesting
densities the mass of the strange quark is a relevant parameter. All
these three effects:
\begin{enumerate}
\item mass of the strange quark,
\item$\beta$-equilibrium,
\item color and electric neutrality
\end{enumerate}
imply that the radii of the Fermi spheres of the quarks that would
pair are not of the same size, thus creating a problem with the
usual BCS pairing. Let us start from the first point. Suppose to
have two fermions of masses $m_1=M$ and $m_2=0$ at the same chemical
potential $\mu$. The corresponding Fermi momenta are \begin{equation}
p_{F_1}=\sqrt{\mu^2-M^2},~~~p_{F_2}=\mu.\end{equation} Therefore the radius of
the Fermi sphere of the massive fermion is smaller than the one of
the massless particle. If we assume $M\ll \mu$ the massive particle
has an effective chemical potential
\begin{equation}\mu_{\rm{eff}}=\sqrt{\mu^2-M^2}\approx \mu-\frac{M^2}{2\mu},\end{equation}
and the mismatch between the two Fermi spheres is
\begin{equation}\delta\mu\approx\frac{M^2}{2\mu}.\label{eq:3}\end{equation} This shows that
the quantity $M^2/(2\mu)$ behaves as a chemical potential. Therefore
for $M\ll\mu$ the mass effects can be taken into account through
the introduction of the mismatch between the chemical potentials of
the two fermions given by eq. (\ref{eq:3}). This is the way we
will follow in our study.
Now let us discuss the $\beta$-equilibrium. If electrons are present
(as generally required by electrical neutrality) chemical potentials
of quarks of different electric charge are different. In fact, when
at the equilibrium for the process $d\to ue\bar\nu$ we have \begin{equation}
\mu_d-\mu_u=\mu_e.\end{equation} From this condition we get that for a quark
of charge $Q_i$ the chemical potential $\mu_i$ is given by \begin{equation}
\mu_i=\mu+Q_i\mu_Q,\end{equation} where $\mu_Q$ is the chemical potential
associated to the electric charge. Therefore \begin{equation}\mu_e=-\mu_Q.\end{equation}
Notice also that $\mu_e$ is not a free parameter since it is
determined by the neutrality condition \begin{equation} Q=-\frac{\partial \Omega}{\partial
\mu_e}=0.\end{equation} At the same time the chemical potentials associated to
the color generators $T_3$ and $T_8$ are determined by the color
neutrality conditions \begin{equation}\frac{\partial \Omega}{\partial \mu_3}= \frac{\partial
\Omega}{\partial \mu_8}=0.\label{eq:1}\end{equation}
We see that the general there is a mismatch between the quarks that
should pair according to the BCS mechanism for $\delta\mu=0$.
Therefore, in general, the system will go to a normal phase, since
the mismatch, as we shall see, tends to destroy the BCS pairing, or
a different phase will be formed. In the next Sections we will
explore some of these possible phases.
\section{Pairing Fermions with Different Fermi Momenta}
In order to discuss the pairing of fermions with different Fermi
momenta let us review the gap equation for the BCS condensate. The
condensation phenomenon is the key feature of a degenerate Fermi gas
with attractive interactions. Once one takes into account the
condensation the physics can be described using the Landau's idea of
quasi-particles. In this context quasi-particles are nothing but
fermionic excitations around the Fermi surface described by the
following dispersion relation \begin{equation} \epsilon(\vec
p,\Delta_0)=\sqrt{\xi^2+\Delta_0^2},\end{equation} with \begin{equation} \xi=E(\vec
p)-\mu\approx \frac{\partial E(\vec p)}{\partial \vec p}\Big|_{\vec
p={\vec p}_F}\cdot (\vec p-{\vec p}_F)={\vec v}_F\cdot(\vec p-{\vec
p}_F)\label{xi},\end{equation} and $\Delta_0$ the BCS condensate. The
quantities ${\vec v}_F$ and $(\vec p-{\vec p}_F)$ are called the
Fermi velocity and the residual momentum respectively. A easy way to
understand how the concept of quasi-particles comes about in this
context is to study the gap equation at finite temperature. For
simplicity let us consider the case of a four-fermi interaction. The
euclidean gap equation is given by \begin{equation}
1=g\int\frac{d^4p}{(2\pi)^4}\frac 1{p_4^2+|\vec
p\,|^2+\Delta^2_0}.\end{equation} From this expression it is easy to get the
gap equation at finite temperature. We need only to convert the
integral over $p_4$ into a sum over the Matsubara frequencies \begin{equation}
1=gT\int\frac{d^3
p}{(2\pi)^3}\sum_{n=-\infty}^{+\infty}\frac{1}{((2n+1)\pi
T)^2+\epsilon^2(\vec p,\Delta_0)}.\end{equation} Performing the sum we get \begin{equation}
1=\frac{g}2\int\frac{d^3p}{(2\pi)^3}\frac{1-n_u-n_d}{\epsilon(\vec
p,\Delta_0)}.\label{gap_T_finite}\end{equation} Here $n_u$ and $n_d$ are the
finite-temperature distribution functions for the excitations
(quasi-particles) corresponding to the original pairing fermions \begin{equation}
n_u=n_d=\frac 1{e^{\epsilon(\vec p,\Delta_0)/T}+1}.\end{equation} At zero
temperature ($n_u=n_d\to 0$) we find (restricting the integration to
a shell around the Fermi surface)\begin{equation}
1=\frac{g}2\int\frac{d\Omega_p\, p_F^2\,
d\xi}{(2\pi)^3}\frac{1}{\sqrt{\xi^2+\Delta_0^2}}.\end{equation} In the limit of
weak coupling we get \begin{equation} \Delta_0\approx 2\,\bar\xi\,
e^{-2/(g\rho)},\label{BCS}\end{equation} where $\bar\xi$ is a cutoff and \begin{equation}
\rho=\frac{p_F^2}{\pi^2 v_F}\end{equation} is the density of states at the
Fermi surface. This shows that decreasing the density of the states
the condensate decreases exponentially. From a phenomenological
point of view, one determines the coupling $g$ requiring that the
same four-fermi interaction, at zero temperature and density, gives
rise to a constituent mass of the order of $400~MeV$. From this
requirement, using values for $\mu\approx 400\div 500~MeV$
(interesting for the physics of compact stellar objects), one
obtains values of $\Delta_0$ in the range $20\div 100~MeV$. However,
since at very high density it is possible to use perturbative QCD,
one can evaluate the gap from first principles \cite{weak}. The
result is \begin{equation} \Delta_0\approx 2b\mu e^{-3\pi^2/\sqrt{2}g_s},\end{equation}
with \begin{equation} b\approx 256\pi^4\left(2/N_f\right)^{5/2} g_s^{-5}.\end{equation} It
is interesting to notice that from Nambu-Jona Lasinio type of models
one would expect a behavior of the type $\exp(-c/g_s^2)$ rather than
$\exp(-c/g_s)$. This is due to an extra infrared singularity from
the gluon propagator. Although this result is strictly valid only at
extremely high densities, if extrapolated down to densities
corresponding to $\mu\approx 400\div 500~MeV$, one finds again
$\Delta_0\approx 20\div 100~MeV$.
We start now our discussion considering a simple model with two
pairing quarks, $u$ and $d$, with chemical potentials
\begin{equation}\mu_u=\mu+\delta\mu,~~~~\mu_d=\mu-\delta\mu,\end{equation} and no further
constraints. The gap equation has the same formal expression as
given in eq. (\ref{gap_T_finite}) for the BCS case, but now
$n_u\not=n_d$ \begin{equation} n_{u,d}=\frac 1{e^{(\epsilon(\vec
p,\Delta)\pm\delta\mu)/T}+1}.\end{equation} In the limit of zero temperature
we obtain \begin{equation}
1=\frac{g}2\int\frac{d^3p}{(2\pi)^3}\frac{1}{\epsilon(\vec
p,\Delta)}\left(1-\theta(-\epsilon-\delta\mu)-\theta(-\epsilon+\delta\mu)
\right)\label{gap}.\end{equation} The meaning of the two step functions is that
at zero temperature there is no pairing when $\epsilon(\vec
p,\Delta)<|\delta\mu|$. In other words the pairing may happen only
for excitations with positive energy. However, the presence of
negative energy states, as in this case, implies that there must be
gapless modes. When this happens there are blocking regions in the
phase space, that is regions where the pairing cannot occur. The
effect is to inhibit part of the Fermi surface to the pairing giving
rise a to a smaller condensate with respect to the BCS case where
all the surface is used. In the actual case the gap equation at
$T=0$ has two different solutions (see for instance ref.
\cite{Casalbuoni:2003wh}) \begin{equation} a)~~~
\Delta=\Delta_0,~~~~~~b)~~~\Delta^2=2\delta\mu\Delta_0-\Delta_0^2,\end{equation}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.5\textwidth]{fig1.eps}
\caption{The two solutions of the gap equation with a mismatch
$\delta\mu$. The continuous line is the BCS solution, the dashed
one is called the Sarma solution.
}\label{fig:1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\textwidth]{fig2.eps}
\caption{The spectrum of quasi-particles for different values of the mismatch $\delta\mu$.}
\label{fig:2}
\end{center}
\end{figure}
where $\Delta_0$ is the BCS solution of the gap equation for
$\delta\mu=0$ The free energy of the two solutions are given by \begin{eqnarray}
a)&&~~~\Omega(\delta\mu)=\Omega_0(\delta\mu)-\displaystyle\frac \rho
4(-2\delta\mu^2+\Delta_0^2),\nn\\ b)&&~~~\Omega(\delta\mu)=\Omega_0(\delta\mu)
-\displaystyle\frac \rho
4(-4\delta\mu^2+4\delta\mu\Delta_0-\Delta_0^2),\end{eqnarray}} \newcommand{\nn}{\nonumber
with $\Omega_0(\delta\mu)$ the free energy for unpaired fermions.
For two massless fermions $p_F=\mu$ and $v_F=1$ and
$\rho={\mu^2}/{\pi}$. The two solutions are illustrated in Fig.
\ref{fig:1}. We see that the solution a) is always favored with
respect to the solution b) (called the Sarma phase
\cite{sarma:1963sa}). Furthermore the BCS phase goes to the normal
phase at \begin{equation}\delta\mu_1=\frac{\Delta_0}{\sqrt{2}}.\end{equation} This point is
called the Chandrasekhar-Clogston (CC) point \cite{Chandrasekhar}
(denoted by CC in Fig. \ref{fig:1}). Ignoring for the moment that in
this case, after the CC point the system goes to the normal phase,
we notice that the gaps of the two solutions coincide at
$\delta\mu=\Delta_0$. This is a special point, since in presence of
a mismatch the spectrum of the quasi-particles is modified as
follows\begin{equation} E_{\delta\mu=0}=\sqrt{(p-\mu)^2+\Delta^2}\to
E_{\delta\mu}=\left|\delta\mu\pm\sqrt{(p-\mu)^2+\Delta^2}\,\right|.\end{equation}
Therefore for $|\delta\mu|<\Delta$ we have gapped quasi-particles
with gaps $\Delta\pm\delta\mu$ (see Fig. \ref{fig:2}). However, for
$|\delta\mu|=\Delta$ a gapless mode appears and from this point on
there are regions of the phase space which do not contribute to the
gap equation (blocking regions). The gapless modes are characterized
by \begin{equation} E(p)=0\Rightarrow p=\mu\pm\sqrt{\delta\mu^2-\Delta^2}.\end{equation}
Since the energy cost for pairing two fermions belonging to Fermi
spheres with mismatch $\delta\mu$ is $2\delta\mu$ and the energy
gained in pairing is $2\Delta$, we see that the fermions begin to
unpair for \begin{equation} 2\delta\mu\ge 2\Delta.\end{equation} These considerations will
be relevant for the study of the gapless phases when neutrality is
required.
\section{The g2SC Phase}
The g2SC phase \cite{huang:2003ab} has the same condensate as the
2SC \begin{equation}\langle
0|\psi_{aL}^{\alpha}\psi_{bL}^\beta|0\rangle=\Delta\epsilon^{\alpha\beta
3}_{ab 3},~~~\alpha,\beta\in SU_c(3),~~~a,b\in SU(2)_L,\end{equation} and
technically, it is distinguished by 2SC due to the presence of
gapless modes starting at $\delta\mu=\Delta$. In this case only two
massless flavors are present (quarks $u$ and $d$) and there are 2
quarks ungapped $q_{ub}, q_{db}$ and 4 gapped $q_{ur}$, $q_{ug}$,
$q_{dr}$, $q_{dg}$, where the color indices
$1,2,3$ have been identified with $r,g,b$ (red, green and blue).
The difference with the usual 2SC phase is that color
and electrical neutrality are required:\begin{equation} \frac{\partial \Omega}{\partial\mu_e}=\frac{\partial
\Omega}{\partial\mu_3}=\frac{\partial \Omega}{\partial\mu_8}=0.\end{equation} This creates a
mismatch between the two Fermi spheres given by \begin{equation}
\delta\mu=\frac{p_F^d-p_F^u}2=\frac{\mu_d-\mu_u}2=\frac{\mu_e}2.\end{equation}
Furthermore the gap equation must be satisfied \begin{equation} \frac{\partial
\Omega}{\partial\Delta}=0.\end{equation}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\textwidth]{fig3.eps}
\caption{The plane $(\mu_e,\Delta)$ showing the lines of the solutions
of the gap equation (continuous) and to the neutrality condition (dashed). The common solution
is marked by a black dot.
}\label{fig:3}
\end{center}
\end{figure}
The solutions to these equations are plotted
in the plane $(\mu_e,\Delta)$ in Fig. \ref{fig:3}. In this figure we
see the two branches of solutions of the gap equation corresponding
to the BCS phase and to the Sarma phase (compare with Fig.
\ref{fig:1}). Therefore the solution to the present problem belongs
to the Sarma branch. In \cite{huang:2003ab} it is also shown that
the solution is a minimum of the free energy following the
neutrality line. On the other hand this point is a maximum following
the appropriate line $\mu_e=\rm{const.}$. We see that the
neutrality conditions promote the unstable phase (Sarma) to a stable
one. However this phase has an instability connected to the Meissner
mass of the gluons \cite{huang_instability1}. In this phase the
color group $SU_c(3)$ is spontaneously broken to $SU_c(2)$ with 5 of
the 8 gluons acquiring a mass; precisely the gluons 4,5,6,7,8. At
the point $\delta\mu=\Delta$ where the 2SC phase goes into the g2SC
one, all the massive gluons have imaginary mass. Furthermore the
gluons 4,5,6,7 have imaginary mass already starting at
$\delta\mu=\Delta/\sqrt{2}$, that is at the Chandrasekhar-Clogston
point, see Fig. \ref{fig:4}. This shows that both the g2SC and the
2SC phases are unstable. The instability of the g2SC phase seems to
be a general feature of the phases with gapless modes
\cite{alford_wang}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\textwidth]{fig4.eps}
\caption{Plot of $m_M^2/m_g^2$ vs. $\Delta/\delta\mu$.
Here $m_g^2=\mu^2 g^2/(3\pi^2)$.
The long-dashed line corresponds to the gluons 4,5,6,7,
whereas the short-dashed one to the gluon 8.}\label{fig:4}
\end{center}
\end{figure}
\section{The gCFL phase}
The gCFL phase is a generalization of the CFL phase which has been
studied both at $T=0$ \cite{Alford:2003fq,Alford:2004hz} and
$T\not=0$ \cite{gCFL_2}. The condensate has now the following form
\begin{equation} \langle
0|\psi_{aL}^\alpha\psi_{bL}^\beta|0\rangle=\Delta_1\epsilon^{\alpha\beta
1}\epsilon_{ab1}+\Delta_2\epsilon^{\alpha\beta
2}\epsilon_{ab2}+\Delta_3\epsilon^{\alpha\beta 3}\epsilon_{ab3}.\end{equation}
The CFL phase corresponds to all the three gaps $\Delta_i$ being
equal. Varying the gaps one gets many different phases. In
particular we will be interested to CFL, to g2SC characterized by
$\Delta_3\not=0$ and $\Delta_1=\Delta_2=0$ and to the gCFL phase
with $\Delta_3>\Delta_2>\Delta_1$. Notice that, in the actual
context, the strange quark is present also in the g2SC phase but
unpaired. The matrix of the condensates in the color ($r,g,b$) and
flavor ($u,d,s$) space is given below:\begin{equation}
\begin{array}{|c|ccccccccc|} \hline
& ru & gd & bs & rd & gu & rs & bu & gs & bd \\ \hline
ru & & \Delta_3 & \Delta_2 & & & & & & \\
gd & \Delta_3 & & \Delta_1 & & & & & & \\
bs & \Delta_2 & \Delta_1 & & & & & & & \\
rd & & & & & -\Delta_3 & & & & \\
gu & & & & -\Delta_3 & & & & & \\
rs & & & & & & & -\Delta_2 & & \\
bu & & & & & & -\Delta_2 & & & \\
gs & & & & & & & & & -\Delta_1 \\
bd & & & & & & & & -\Delta_1 & \\ \hline
\end{array}
\end{equation} In flavor space the gaps $\Delta_i$ correspond to the following
pairings \begin{equation} \Delta_1\Rightarrow ds,~~~\Delta_2\Rightarrow us,~~~
\Delta_3\Rightarrow ud.\end{equation} The mass of the strange quark is taken
into account by shifting all the chemical potentials involving the
strange quark as follows: \begin{equation}\mu_{\alpha s}\to \mu_{\alpha s}
-\frac{M_s^2}{2\mu}.\end{equation} It has also been shown in ref. \cite{alford}
that color and electric neutrality in CFL require \begin{equation}
\mu_8=-\frac{M_s^2}{2\mu},~~~\mu_e=\mu_3=0.\end{equation} At the same time the
various mismatches are given by \begin{equation}
\delta\mu_{bd-gs}=\frac{M_s^2}{2\mu},~~~\delta\mu_{rd-gu}=\mu_e=0,~~~
\delta\mu_{rs-bu}=\mu_e-\frac{M_s^2}{2\mu}.\end{equation} It turns out that in
the gCFL the electron density is different from zero and, as a
consequence, the mismatch between the quarks $d$ and $s$ is the
first one to give rise to the unpairing of the corresponing quarks.
This unpairing is expected to occur for \begin{equation}
2\frac{M_s^2}{2\mu}>2\Delta~~
\Rightarrow~~\frac{M_s^2}{\mu}>2\Delta.\end{equation} This has been
substantiated by the calculations in a NJL model modeled on one
gluon-exchange in \cite{Alford:2004hz}. The results for the gaps are
given in Fig. \ref{fig:5}. We see that the transition from the CFL
phase, where all gaps are equal, to the gapless phase occurs roughly
at $M_s^2/\mu =2\Delta$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\textwidth]{fig5.eps}
\caption{The behavior of the gap parameters in gCFL. The
parameters has been chosen in such a way that
$\Delta_{0}=25~MeV$ and $\mu=500~MeV$ \cite{Alford:2004hz}. The vertical line at
$M_s^2/\mu\approx 130 ~MeV$ marks the transition from the gCFL
phase to the normal one.
}\label{fig:5}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\textwidth]{fig6.eps}
\caption{ We give here the free energy of the various phases with
reference to the normal phase \cite{Alford:2004hz}, named unpaired in the figure.
\label{fig:6}}
\end{center}
\end{figure}
In Fig. \ref{fig:6} we show the free energy of the various phases
with reference to the normal phase. The CFL phase is the stable one
up to $M_s^2/\mu\approx 2\Delta$. Then the gCFL phase takes over up
to about 130 $MeV$ where the system goes to the normal phase. Notice
that except in a very tiny region around this point, the CFL and
gCFL phases win over the corresponding 2SC and g2SC ones. The thin
short-dashed line represents the free energy of the CFL phase up to
the point where it becomes equal to the free-energy of the normal
phase. This happens for $M_s^2/\mu\approx 4\Delta$. This point is
the analogue of the Chandrasekhar-Clogston point of the two-flavor
case.
The gCFL phase has gapless excitations and, as a consequence, the
chromomagnetic instability discussed in the case of the g2SC phase
shows up here too. This has been shown in
\cite{MeissnerCFL_1,MeissnerCFL_2}. The results of ref.
\cite{MeissnerCFL_1} are given in Fig. \ref{fig:7} for the various
gluon masses.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1\textwidth]{fig7.eps}
\caption{The figure shows, for the gCFL case, the masses of the gluons
1,2,3, 8 (left panel) and 4,5,6,7 (right panel) vs. $M_s^2/\mu$.
\label{fig:7}}
\end{center}
\end{figure}
The existence of the chromomagnetic instability is a serious problem
for the gapless phases (g2SC and gCFL) but also for the 2SC phase,
as we have discussed previously. A way out of this problem would be
to have gluon condensation. For instance, if one assumes
artificially $\langle A_\mu^3\rangle$ and $\langle A_\mu^8\rangle$
not zero and with a value of about 10 $MeV$ it can be shown that the
instability disappears \cite{MeissnerCFL_1}. Also, very recently in
\cite{miransky}, it has been shown the possibility of eliminating
the chromomagnetic instability in the 2SC phase through a gluonic
phase. However it is not clear if the same method can be extended to
the gapless phases.
Another interesting possibility has been considered in three papers
by Giannakis and Ren, who have considered the LOFF phase, that is a
nonhomogeneous phase first studied in a condensed matter context
\cite{LOFF1,LOFF2} and then in QCD in
\cite{Alford:2000ze,Bowers:2002xr} (for recent reviews of the LOFF
phase, see \cite{Casalbuoni:2003wh,Bowers:2003ye}). The results
obtained by Giannakis and Ren in the two-flavor case are the
following:
\begin{itemize}
\item The presence of the chromomagnetic instability in g2SC is
exactly what one needs in order that the LOFF phase is
energetically favored \cite{Ren1}.
\item The LOFF phase in the two-flavor case has no chromomagnetic
instabilities (though it has gapless modes) at least in the weak
coupling limit \cite{Ren2,Ren3}.
\end{itemize}
Of course these results make the LOFF phase a natural candidate for
the stable phase of QCD at moderate densities. In the next Sections
we will describe the LOFF phase in its simplest version and a very
simple approach to the problem with three flavors.
\section{The LOFF Phase}
According to the authors of refs. \cite{LOFF1,LOFF2} when fermions
belong to different Fermi spheres, they might prefer to pair
staying as much as possible close to their own Fermi surface. When
they are sitting exactly at the surface, the pairing is as shown in
Fig. \ref{fig:8}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.4\textwidth]{fig8.eps}
\caption{Pairing of fermions belonging to two Fermi spheres of
different radii according to LOFF. \label{fig:8}} \end{center}
\end{figure}
We see that the total momentum of the pair is ${\vec p}_1+{\vec
p}_2=2\vec q$ and, as we shall show, $|\vec q\,|$ is fixed
variationally whereas the direction of $\vec q$ is chosen
spontaneously. Since the total momentum of the pair is not zero the
condensate breaks rotational and translational invariance. The
simplest form of the condensate compatible with this breaking is
just a simple plane wave (more complicated possibilities will be
discussed later) \begin{equation} \langle\psi(x)\psi(x)\rangle\approx\Delta\,
e^{2i\vec q\cdot\vec x}.\label{single-wave}\end{equation} It should also be
noticed that the pairs use much less of the Fermi surface than they
do in the BCS case. In fact, in the case considered in Fig.
\ref{fig:8} the fermions can pair only if they belong to the circles
in figure. More generally there is a quite large region in momentum
space (the so called blocking region) which is excluded from
pairing. This leads to a condensate generally smaller than the BCS
one.
Let us now consider in more detail the LOFF phase. For two fermions
at different densities we have an extra term in the hamiltonian
which can be written as \begin{equation}
H_I=-\delta\mu\sigma_3\label{interaction},\end{equation} where, in the original
LOFF papers \cite{LOFF1,LOFF2} $\delta\mu$ is proportional to the
magnetic field due to the impurities, whereas in the actual case
$\delta\mu=(\mu_1-\mu_2)/2$ and $\sigma_3$ is a Pauli matrix acting
on the two fermion space. According to refs. \cite{LOFF1,LOFF2} this
favors the formation of pairs with momenta $\vec p_1=\vec k+\vec
q,~~~\vec p_2=-\vec k+\vec q$. We will discuss in detail the case of
a single plane wave (see eq. (\ref{single-wave})). The interaction
term of eq. (\ref{interaction}) gives rise to a shift in $\xi$ (see
eq. (\ref{xi})) due both to the non-zero momentum of the pair and to
the different chemical potentials \begin{equation} \xi=E(\vec p)-\mu\to E(\pm\vec
k+\vec q)-\mu\mp\delta\mu\approx \xi\mp\bar\mu,\end{equation} with \begin{equation}
\bar\mu=\delta\mu-{\vec v}_F\cdot\vec q.\end{equation} Notice that the previous
dispersion relations show the presence of gapless modes at momenta
depending on the angle with $\vec q$. Here we have assumed
$\delta\mu\ll\mu$ (with $\mu=(\mu_1+\mu_2)/2$) allowing us to expand
$E$ at the first order in $\vec q/\mu$ (see Fig. \ref{fig:8}).
The gap equation for the present case is obtained simply from eq.
(\ref{gap}) via the substitution \begin{equation}\delta\mu\to\bar\mu.\end{equation} By
studying eq. (\ref{gap}) one can show that increasing $\delta\mu$
starting from zero, we have first the BCS phase. Then at
$\delta\mu=\delta\mu_1$ there is a first order transition to the
LOFF phase \cite{LOFF1,Alford:2000ze}, and at
$\delta\mu=\delta\mu_2>\delta\mu_1$ there is a second order phase
transition to the normal phase \cite{LOFF1,Alford:2000ze}. We start
comparing the grand potential in the BCS phase to the one in the
normal phase. Their difference is given by \begin{equation} \Omega_{\rm
BCS}-\Omega_{\rm
normal}=-\frac{p_F^2}{4\pi^2v_F}\left(\Delta^2_0-2\delta\mu^2\right),\end{equation}
where the first term comes from the energy necessary to the BCS
condensation, whereas the last term arises from the grand potential
of two free fermions with different chemical potential. We recall
also that for massless fermions $p_F=\mu$ and $v_F=1$. We have again
assumed $\delta\mu\ll\mu$. This implies that there should be a first
order phase transition from the BCS to the normal phase at
$\delta\mu=\Delta_0/\sqrt{2}$ \cite{Chandrasekhar}, since the BCS
gap does not depend on $\delta\mu$. The situation is represented in
Fig. \ref{fig:9}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.9\textwidth]{fig9.eps}
\caption{The grand potential (left panel) and the condensates of the
BCS and LOFF phases vs. $\delta\mu$ (right panel). \label{fig:9}}
\end{center}
\end{figure}
In order to compare with the LOFF phase one can expand the gap
equation around the point $\Delta=0$ (Ginzburg-Landau expansion) to
explore the possibility of a second order phase transition
\cite{LOFF1}. The result for the free energy is \begin{equation} \Omega_{\rm
LOFF}-\Omega_{\rm normal}\approx
-0.44\,\rho(\delta\mu-\delta\mu_2)^2.\end{equation} At the same time, looking
at the minimum in $q$ of the free energy one finds \begin{equation} qv_F\approx
1.2\, \delta\mu.\label{q}\end{equation}
We see that in the window
between the intersection of the BCS curve and the LOFF curve in Fig.
\ref{fig:9} and $\delta\mu_2$, the LOFF phase is favored. Also at
the intersection there is a first order transition between the LOFF
and the BCS phase. Furthermore, since $\delta\mu_2$ is very close to
$\delta\mu_1$ the intersection point is practically given by
$\delta\mu_1$. In Fig. \ref{fig:9} we show, in the right panel, the
behaviour of the condensates. Although the window
$(\delta\mu_1,\delta\mu_2)\simeq(0.707,0.754)\Delta_0$ is rather
narrow, there are indications that, considering the realistic case
of QCD \cite{Leibovich:2001xr}, the window opens up. Such opening
occurs also for different crystalline structures than the single
plane wave considered here \cite{Bowers:2002xr,Casalbuoni:2004wm}.
\section{The LOFF phase with three flavors}
In the last Section we would like to illustrate some preliminary
result about the LOFF phase with three flavors. This problem has
been considered in \cite{casalbuoni_loff3} under various simplifying
hypothesis:
\begin{itemize}
\item The study has been made in the Ginzburg-Landau
approximation.
\item Only electrical neutrality has been required and the
chemical potentials for the color charges $T_3$ and $T_8$ have
been put equal to zero (see later).
\item The mass of the strange quark has been introduced as it was
done previously previously for the gCFL phase.
\item The study has been restricted to plane waves, assuming the
following generalization of the gCFL case:
\begin{equation} \langle\psi^\alpha_{aL}\psi^\beta_{bL}\rangle=\sum_{I=1}^3\Delta_I(\vec x)
\epsilon^{\alpha\beta I}\epsilon_{ab I},~~~\Delta_I(\vec x)=\Delta_I
e^{2i\vec q_I\cdot\vec x}\end{equation}
\item The condensate depends on three momenta, meaning three lengths
of the momenta $q_i$ and three angles. In \cite{casalbuoni_loff3}
only four particular geometries have been considered: 1) all the
momenta parallel pointing upward the $z$-axis, then 2), 3) and 4)
are obtained by inverting respectively the momentum $\vec q_1$,
$\vec q_2$ and $\vec q_3$.
\end{itemize}
Under the previous hypothesis the free energy (with reference to the
normal state) has the expansion \begin{equation} \Omega -\Omega_{normal}=
\sum_{I=1}^3\left(\frac{\alpha_I}{2}\,\Delta_I^2 ~+~
\frac{\beta_I}{4}\,\Delta_I^4 ~+~ \sum_{I\neq
J}\frac{\beta_{IJ}}{4}\,\Delta_I^2\Delta_J^2 \right) ~+~
O(\Delta^6)\end{equation} with \begin{equation}\alpha_I(q_I,\delta\mu_I)=-
\frac{4\mu^2}{\pi^2} \left(1 -
\frac{\delta\mu_I}{2q_I}\log\left|\frac{q_I+\delta\mu_I}{q_I-\delta\mu_I}\right|
- \frac{1}{2}\log\left|\frac{4(q_I^2
-\delta\mu_I^2)}{\Delta_0^2}\right|\right)\end{equation} \begin{equation}
\beta_I(q_I,\delta\mu_I)=\frac{\mu^2}{\pi^2}\frac{1}{q_I^2-\delta\mu_I^2}\end{equation}
\begin{equation} \beta_{12}=-2\frac{\mu^2}{\pi^2} \int\frac{d{\bf
n}}{4\pi}\,\frac{1}{(2{\bf q_1}\cdot{\bf
n}+\mu_s-\mu_d-i0^+)\,(2{\bf q_2}\cdot{\bf n}+\mu_s-\mu_u-i0^+)}\end{equation}
and the other $\beta_{IJ},~I\not = J$ obtained by the exchange \begin{equation}
12\rightarrow 23,~ \mu_s\leftrightarrow\mu_d,~~~12\rightarrow 13,~
\mu_s\leftrightarrow\mu_u\end{equation} The $\delta\mu_I$ are obtained from
\begin{equation}\mu_u=\mu-\frac 2 3 \mu_e,~~\mu_d=\mu+\frac 1 3\mu_e,~~
\mu_s=\mu+\frac 1 3\mu_e-\frac{M_s^2}{2\mu}\label{eq:6.7}\end{equation} In
particular the coefficients of $\Delta_I^2$ are the same as for LOFF
with two flavors. Therefore the minimization with respect to the
$|\vec q_I|$'s leads to the same result as in eq. (\ref{q}) \begin{equation}
|\vec q_I|=1.2 \delta\mu_I\label{eq:55}\end{equation} Then, one has to minimize
with respect to the gaps and to $\mu_e$ in order to require
electrical neutrality. It turns out that the configurations 3 and 4
have an extremely small gap. Furthermore for $M_s^2/\mu$ greater
than about 80 $MeV$ one has a solution with $\Delta_1=0$ and
$\Delta_2=\Delta_3$. In this case the configurations 1` and 2 have
the same free energy. The results for the free energy and for the
gap of this solution are given in Fig. \ref{fig:11} and
\ref{fig:12}. In this study, the following choice of the parameters
has been made: the BCS gap, $\Delta_0=25~MeV$, and the chemical
potential $\mu=500~MeV$. The values are the same discussed
previously for gCFL in order to allow for a comparison of the
results.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.5\textwidth]{fig11.eps}
\caption{The gap for LOFF with three flavors vs. $M_s^2/\mu$. The
line corresponds to the most favored solution, that is to the
configurations 1 and 2. \label{fig:11}}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.5\textwidth]{fig12.eps}
\caption{The free energy of the most favored configurations (1 and
2) considered for LOFF with three flavors vs.
$M_s^2/\mu$.\label{fig:12}}\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.55\textwidth]{fig13.eps}
\caption{Comparison of the free energy of the various phases with
the LOFF phase with three flavors.} \label{fig:13} \end{center}
\end{figure}
We are now in the position to compare these results with the ones
obtained in \cite{Alford:2004hz} for the gCFL phase. The comparison
is made in Fig. \ref{fig:13}. Ignoring the chromomagnetic
instabilities of the gapless phases and of 2SC we see that LOFF
takes over with respect to gCFL at about $M_s^2/\mu=128~MeV$ and
goes over to the normal phase for $M_s^2/\mu\approx 150~MeV$.
However, since the instability exists it should be cured in some
way. The results for the LOFF phase, assuming that also for three
flavors the chromomagnetic instability does not show up, say that it
could be the LOFF phase to takes over the CFL phase before the
transition to gCFL. For this it is necessary that the window for the
LOFF phase gets enlarged. However, in \cite{Casalbuoni:2004wm} it
has been show that for structures more general than the plane wave
the windows may indeed becomes larger. If define the window for the
single plane wave as $(\delta\mu_2-\delta\mu_1)/\delta\mu_2$ (see
the previous Section) we would get 0.06. The analogous ratio in
going from one to three plane waves goes to about
$(150-115)/150=.23$, with a gain of almost a factor 4. On the other
hand, in \cite{Casalbuoni:2004wm} it has been shown that considering
some of the crystalline structures already taken in exam in
\cite{Alford:2000ze}, as the face centered cube or the cube the
windows becomes about $(1.32-0.707)/1.32=0.46$ with a gain of about
7.7 with respect to the single plane wave. If these gains would be
maintained in going from two to three flavors with the face centered
cube structure, one could expect a gain from 4 to 7.7 with an
enlargement of the window between 88 and 170 $MeV$, which would be
enough to cover the region of gCFL (which is about 70 $MeV$).
\begin{figure}[h]\begin{center}
\includegraphics[width=.9\textwidth]{fig14.eps}
\caption{The chemical potential for gCFL (left panel) and LOFF
(right panel) vs. $M_s^2/\mu$. \label{fig:14}}\end{center}
\end{figure}
At last we want to comment about the approximation in neglecting the
color neutrality condition and assuming $\mu_3=\mu_8=0$. In Fig.
\ref{fig:14} we show the chemical potentials
$\mu_e,~\mu_3,~\mu_8$for the gCFL phase in the left panel, and
$\mu_e$ for the LOFF phase in the right panel. We can make two
observations: first of all, in the region of interest where LOFF
dominates over gCFL the behaviour of $\mu_e$ in the two phases is
pretty much similar, and $\mu_3,~\mu_8\ll\mu_e$ for gCFL. This
suggests that also in the LOFF case $\mu_3$ and $\mu_8$ are small.
Second and more important the result of \cite{casalbuoni_loff3}
shows that $\mu_e\approx M_s^2/(4\mu)$ as for the case of 3 color
and 3 flavor unpaired quarks \cite{alford}. As can be seen from eq.
(\ref{eq:6.7}) this coreesponds to a symmetrical spli of the $s$ and
$d$ Fermi surfaces around the $u$ Fermi surface. Therefore ${\bf
q}_2$ = ${\bf q}_3$ and the gaps $\Delta_2$ and $\Delta_3$ must
coincide. At the same time the separation of the $d$ and $s$
surfaces is the double and therefore $\Delta_1=0$. The unpaired
quarks have also $\mu_3=\mu_8=0$. Also, from Fig. \ref{fig:11} we
see that in our approximations the transition from the LOFF to the
normal phase is very close to be continuous. Since we expect also
the chemical potentials to be continuous at the transition point
very close to the critical point we should have $\mu_3=\mu_8=0$ also
on the LOFF side. This means the color neutrality condition should
be $\mu_3=\mu_8=0$ in the neighborhood of the transition. Therefore
we expect the determination of the point $M_s^2/\mu=150~MeV$ to be
safe. On the other hand, the requirement of color neutrality could
change the intersection point with gCFL. Nevertheless, since the
critical point for LOFF is higher than the one of gCFL, for
increasing $M_s$ the system must to go into the LOFF phase.
|
1,314,259,996,749 | arxiv | \section{Introduction}
\label{introduction}
The low-inclination, inner main belt asteroid population is an important source region for near-Earth objects (NEOs) and terrestrial planet impactors in the past, present and future \citep{scholl1991,morbidelli2003}. This region, which includes the Nysa-Polana complex, poses a challenge for family identification and analysis techniques, because it is dynamically very crowded, with many families overlapping in proper orbital element space. Recent efforts have used newly available color information from the Sloan Digital Sky Survey's Moving Objects Catalog \citep[SDSSMOC\footnote{http://www.astro.washington.edu/users/ivezic/sdssmoc/},][]{ivezic2002} and albedo information from the Wide-field Infrared Survey Explorer \citep[WISE/NEOWISE\footnote{http://irsa.ipac.caltech.edu/Missions/wise.html},][]{wright2010,mainzer2011} to augment the orbital databases in an effort to further distinguish families on the basis of their unique reflectance properties \citep{parker2008,masiero2013,carruba2013,milani2014,walsh2013,dykhuis2014}.
Analysis of the taxonomically diverse and dynamically overlapping families within the Nysa-Polana complex has suffered from a long and complicated history of identification and re-identification, often with changes in membership, taxonomy, and nomenclature. In order to place the current work in context, we first review the history of these identifications and the rationales that led to each. The various names assigned to each family structure throughout the literature are listed in Table~\ref{table1}. In the interest of connecting our work to future efforts, we also identify the Nysa-Polana region with the Family Identification Number (FIN) of 405, assigned to the region by Nesvorn{\'y} et al. and Masiero et al., both submitted. In this classification scheme, the Hertha1 family would correspond to 405a, the Polana family to 405b, and the Eulalia1 family to 405c. The Hertha2 and Eulalia2 families would await further confirmation from additional data.
The Nysa-Polana complex was first identified as a single family by \citet{brouwer1951}. \citet{zellner1977} subsequently distinguished two family structures in the region, and suggested an origin scenario that attempted to include both of the largest neighboring objects (44) Nysa and (135) Hertha as fragments from the same parent body. \citet{williams1979} confirmed the existence of two families, labeled as ``W-24'' and ``W-160.'' With the addition of reflectance property data from the Eight-Color Asteroid Survey (ECAS), \citet{tedesco1982} discovered several objects in the so-called ``Nysa'' family to be the otherwise rare ``F'' spectral type (using the taxonomy scheme developed by the authors for the ECAS). This spectral dissimilarity led \citet{bell1989} to conclude that the ``Nysa'' family members were unrelated to the E-type (44) Nysa, and should rather be associated with the F-type (142) Polana, labeling (44) Nysa an interloper in that family. \citet{kelley1994} suggested an origin scenario involving partial differentiation in attempt to link the E-type (44) Nysa and M-type (135) Hertha as ``parents'' of the F-type family, based on a shared silicate absorption feature near 0.9~$\mu$m.
\input{./Tables/summary_table_tolatex.txt}
{\scriptsize Abbreviations: Ze77 = \citet{zellner1977}; Wi79 = \citet{williams1979}; B89 = \citet{bell1989}; Za95 = \citet{zappala1995}; \\C01 = \citet{cellino2001}; MD05 = \citet{mothe-diniz2005}; D12 = \citet{delbo2012}; Wa13 = \citet{walsh2013}; \\M14 = \citet{milani2014}. }
\caption{\footnotesize Summary of the various names of Nysa-Polana complex families used in the literature, showing sources that modified the family nomenclature. See the text for a more detailed summary of each literature source. The first column shows the five structures of the Nysa-Polana complex, using the naming conventions of the present work. The spectral types in column 2 are broad categories only, meant to reflect the colors observed by the SDSS; the individual literature sources often use subcategories or parallel categories (such as Xc, Xk, E, F, B, etc.). Ditto marks indicate that the Eulalia1 structure, defined in Section \ref{other_dark}, was typically joined to the Polana structure until its distinction in the work of Wa13. The question mark expresses low confidence reported by Wa13. MD05 lumped the low-$e_\text{P}$ portion of our Hertha1 structure together with our Hertha2 structure, retaining the name Mildred for the higher-$e_\text{P}$ portion of Hertha1. }
\label{table1}
\end{table}
\citet{zappala1995} used two different clustering techniques to identify family members in the Nysa-Polana region, and found two overlapping families that the clustering methods failed to reliably distinguish. The Polana family was labeled as a possible subsequent collision on a Nysa family member.
\citet{cellino2001} undertook a spectroscopic campaign in an attempt to clarify membership in the region. They found three objects with X-type taxonomy among the F-type Polana objects and S-type Hertha objects. They renamed the Hertha objects as the ``Mildred'' family, based on the dissimilarity between the M-type (135) Hertha and the S-type family previously associated with it.
Further observations and dynamical analysis by \citet{mothe-diniz2005} and \citet{alvarez-candal2006} identified three more objects as X-types, including (44) Nysa and (135) Hertha. \citet{mothe-diniz2005} separated the Mildred family into two groups based on their eccentricities, and noted that the lower-$e_\text{P}$ group contained a mix of X- and S-type objects, while the higher-$e_\text{P}$ group contained primarily S-type objects only. They combined the lower-$e_\text{P}$ portion of the Mildred family with what we now call the Hertha2 family to produce a ``Hertha'' family, while the higher-$e_\text{P}$ component retained the name ``Mildred.'' In addition, they renamed the Polana family as ``McCuskey.'' More recently, \citet{delbo2012} tested a new classification algorithm on the region, and identified three groups of objects based on reflectance properties, confirming the presence of S-, X- and C-type spectra in the Nysa-Polana region.
\citet{walsh2013} used the recent data available from WISE/NEOWISE to study the dark (B- and C-type) objects in the Nysa-Polana complex, which until then had been considered a single family (usually called ``Polana''). They identified the structures of at least two old families, which they associated with (142) Polana and (495) Eulalia. A third structure at lower eccentricities was identified with (112) Iphigenia, although with low confidence in its significance or its independence from the Eulalia family.
The most recent analyses \citep{masiero2013,carruba2013,milani2014} incorporated major results from large reflectance surveys, primarily WISE and SDSS, and used automated clustering methods applied to this extended parameter space. In general, they confirmed the presence of at least one low-albedo (B- or C-type) collisional family and at least one high-albedo (S-type) collisional family in this region. \citet{milani2014} identified the first cluster in Table~\ref{table1} with a different parent asteroid, (3583) Burdett.
Here, in addition to seeking clusters in the dynamical and reflectance parameter space, we seek signatures of dynamical evolution that distinguish multiple families within a single cluster. Specifically, by considering cluster distributions in $a_\text{P}-H$ space, the distinct ``V-shape'' patterns of Yarkovsky evolution can be recognized. This technique was demonstrated recently for the ``Eulalia'' and ``New Polana'' families by \citet{walsh2013}, and was also successfully applied in our earlier analysis of the large and diffuse Flora family, which dominates the inner main belt \citep{dykhuis2014}. In addition, for the Hertha1 family, we extract collisional parameter information from the slope of the family's $a_\text{P}$-$e_\text{P}$ distribution.
In Section \ref{overview}, we give an overview of the geography of the Nysa-Polana region in proper elements and reflectance properties. Then, we separate the objects on the basis of their $a^*$ color parameter \citep[as defined by][]{ivezic2001}, and study first the objects with $a^* > -0.015$ (Section \ref{high_astar}), which we show to be one of two families likely associated with (135) Hertha (designated the Hertha1 family, see Table~\ref{table1}). Next we study the objects with $a^* < -0.015$ (Section \ref{low_astar}), identifying four structures and their likely parent objects.
\section{Overview of the region}
\label{overview}
The orbital distribution of asteroids in the Nysa-Polana region (2.1~AU $< a_\text{P} <$ 2.5~AU, $0.11 < e_\text{P} < 0.22$, and $0.02 < \sin i_\text{P} < 0.07$) with reliable proper orbital element data is shown in Fig.~\ref{01_orbital}. The catalog of synthetic proper elements is the same as that used in \citet{dykhuis2014}, computed using the Orbit9 software available from the Asteroids Dynamic Site (AstDyS)\footnote{http://hamilton.dm.unipi.it/astdys/}. The blue and red boxes in Fig.~\ref{01_orbital_a} roughly represent the regions associated with the low-albedo (C-type) and higher-albedo (S-type) groups, respectively, that have typically been identified in the region. Asteroid (142) Polana lies within the former range, (135) Hertha in the latter, and (44) Nysa in neither.
\begin {center}
EDITOR: PLACE FIGURE \ref{01_orbital} HERE
\end {center}
Figs.~\ref{01_orbital_b} and \ref{01_orbital_c} show the distributions in proper semimajor axis, $a_\text{P}$, vs. absolute magnitude, $H$, of the objects from the sub-regions marked in Fig.~\ref{01_orbital_a}. A ``V'' shape in $a_\text{P}-H$ space is generally attributed to the Yarkovsky effect, which yields a size-dependent spread in semimajor axes for a collisional family. The signatures of several overlapping V's are evident in Figs.~\ref{01_orbital_b} and \ref{01_orbital_c}, suggesting the presence of multiple families.
The boundary of each V-shaped Yarkovsky envelope can be described by the limiting distance in semimajor axis $\Delta a_\text{P}$ from the family's central semimajor axis location, which is generally assumed to be the proper semimajor axis of the largest remnant of the family. The distance is described by the relation \citep[from][]{vokrouhlicky2006}:
\begin{equation}
\Delta a_\text{P} = C \cdot 10^{H/5}.
\label{Cparam}
\end{equation}
\noindent The $C$ parameter (which for convenience we later refer to in units of mAU = $10^{-3}$~AU) is directly related to the family's dynamical evolution due to the Yarkovsky effect. Families with an older age will show more semimajor axis evolution, and reach higher boundary values of $C$ than their younger counterparts; thus the boundary of the $C$ parameter distribution for a given family can be used as a simple means to estimate the age, assuming appropriate calibration of the Yarkovsky semimajor axis drift rates and estimation of the semimajor axis dispersion due to the original collision ejection field.
The largest objects in the Nysa-Polana region --- (44) Nysa, (135) Hertha, and (142) Polana --- all lie quite close in semimajor axis to the Mars 1:2 mean-motion resonance (dotted line in Figs.~\ref{01_orbital_b} and \ref{01_orbital_c}, located at $a_\text{P}$ = 2.419~AU). The effects of this resonance were studied in detail by \citet{gallardo2011}, and must be taken into account in studies of the families in the inner main belt, as discussed in Section \ref{high_astar}.
Color information for the 8396 Nysa-Polana region objects that have SDSS color data (but not necessarily albedo data) are plotted in Fig.~\ref{02_reflectance}. Two clusters of objects are apparent in this plot, roughly associated with the S-type (high $a^*$ color) and C-type (low $a^*$ color) spectral categories. In Section \ref{high_astar}, we consider the cluster with $a^* > -0.015$, which is a recognizable dynamical family. The objects with $a^* < -0.015$ represent multiple dynamical families, as shown in Section \ref{low_astar}. This multiplicity is already indicated by the presence of a number of X-type objects in the region with $a^*$ colors intermediate between the S- and C-type ranges; in Fig.~\ref{02_reflectance} they appear as the subtle grouping between $-0.1 < a^* < -0.0$.
\begin {center}
EDITOR: PLACE FIGURE \ref{02_reflectance} HERE
\end {center}
\section{High-$a^*$ objects: The Hertha1 family}
\label{high_astar}
\subsection{Family identification and characterization}
\label{identify}
The proper element distribution for the Nysa-Polana region objects from Fig.~\ref{02_reflectance} with $a^* > -0.015$ is shown in Figs.~\ref{03_high_astar} and \ref{04_ae}. These figures suggest that the high-$a^*$ structure (within the red box in Fig.~\ref{03_high_astar}) is associated dynamically with parent object (135) Hertha. We designate this cluster as the ``Hertha1'' family (cf. Table~\ref{table1} for previous designations in the literature), with the ``1'' added because (135) Hertha is also associated with a smaller cluster of objects (``Hertha2,'' described in Section \ref{low_astar}).
\begin {center}
EDITOR: PLACE FIGURE \ref{03_high_astar} HERE
\end {center}
The distribution within the red box in Fig.~\ref{03_high_astar} could be interpreted as two separate clumps around $e_\text{P}$, $\sin i_\text{P}$ of (0.170, 0.045) and (0.185, 0.040). These two clumps were named the ``Hertha'' and ``Mildred'' families, respectively, by \citet{mothe-diniz2005}, cf. Table~\ref{table1}. However, the $a_\text{P}$-$e_\text{P}$ distribution of the objects within the red box (Fig.~\ref{04_ae}) shows a continuous and uniform trend of lower $e_\text{P}$ with increasing $a_\text{P}$, consistent with the ejection field from a single large-scale collision event. Moreover, as we show below, the Yarkovsky V shape is similarly continuous within and between the two populations. Thus we interpret the structure within the red box in Fig.~\ref{03_high_astar} as a single collisional family, Hertha1.
\begin {center}
EDITOR: PLACE FIGURE \ref{04_ae} HERE
\end {center}
The absolute magnitude distribution of the objects of the Hertha1 family yields information about its dynamical evolution under the influence of the Yarkovsky effect. The $a_\text{P}-H$ distribution plotted in Fig.~\ref{05_color} shows the V shape characteristic of a collisional family evolved under the Yarkovsky effect. The rightmost edge of the V shape is truncated by the Jupiter 3:1 mean motion resonance (represented by the blue dotted line), and the objects at high eccentricities and low semimajor axes are affected by encounters with Mars, modifying the distribution from an ``ideal'' V shape. The low-$a_\text{P}$, low-$e_\text{P}$ objects evident in Fig.~\ref{04_ae} are here shown to be primarily interlopers in the family, included in this sample only on the basis of their inclusion within the family ranges in $e_\text{P}$ and $\sin i_\text{P}$ and $a^*$.
\begin {center}
EDITOR: PLACE FIGURE \ref{05_color} HERE
\end {center}
In principle, the age of the family can be interpreted from the distribution of $C$ parameters of the family members (Equation \ref{Cparam}), the outer boundary of which describes the V-shaped envelope in $a_\text{P}$ vs. $H$. However, the correlation between $e_\text{P}$ and $a_\text{P}$ among the Hertha1 family objects (Fig.~\ref{04_ae}) means that the center of the Yarkovsky V shifts to lower $a_\text{P}$ for larger eccentricities. This complicates the identification of the Yarkovsky envelope that best fits the boundary of the family in semimajor axis; none of the example Yarkovsky envelopes plotted in Fig.~\ref{05_color} match the boundary well. The full semimajor axis width in Fig.~\ref{05_color} is due to the skewed $e_\text{P}$ vs. $a_\text{P}$ distribution in Fig.~\ref{04_ae}, which is related to the initial collision rather than the subsequent Yarkovsky drift.
This skew must be removed prior to interpretation of the family's age via the observed Yarkovsky spread in $a_\text{P}$. To do this, we need to adjust each member's semimajor axis by an amount
\begin{equation}
a_\text{P}' - a_\text{P} = \frac{1}{m}\cdot (e_\text{P} - e_{\text{P,0}}),
\label{aprime}
\end{equation}
\noindent where $m$ represents the slope of the $a_\text{P}$ vs. $e_\text{P}$ distribution, and $e_{\text{P,0}}$ is the family center in eccentricity. Given appropriate values for $m$ and $e_{\text{P,0}}$, we can apply a shear transformation to the distribution of objects to remove the eccentricity dependence and determine the Yarkovsky-induced spread in $a_\text{P}$.
In order to find the slope $m$ of the observed distribution, we must take into account the effects on the shape of the distribution due to the various resonances in and around the Nysa-Polana region (Fig.~\ref{04_ae}). The Mars-crossing region removes low-$a_\text{P}$, high-$e_\text{P}$ Hertha1 objects; the Jupiter 3:1 mean-motion resonance removes high-$a_\text{P}$, low-$e_\text{P}$ objects; lastly, the Mars 1:2 mean-motion resonance depletes family members both below $a_\text{P} = 2.43$~AU and above $e_\text{P} = 0.195$ as they drift outward in semimajor axes due to the Yarkovsky effect and exit the region before drifting back.
The only ``pristine'' edge of the family that remains untouched by resonances (and thus still reflects the semimajor axis dependence of $e_\text{P}$ from the original collision) is the lower left edge, at $0.15 < e_\text{P} < 0.185$, where the sloped boundary of the family is visible (Fig.~\ref{04_ae}). We use this boundary to estimate the slope of the family; specifically, we find the boundary at each value of eccentricity within the range $0.15 < e_\text{P} < 0.185$, using a stepsize of $\delta e = 0.001$ and an eccentricity width of $\Delta e = 0.01$. The latter parameter specifies the window in eccentricity which defines the sample of objects used to find the boundary at each eccentricity step. The boundary is defined for this purpose as the point of maximum increase in density of objects, or a maximum in the derivative of the density function (see Appendix A.1 of \citet{dykhuis2014} for discussion of the kernel density estimation and its derivatives). A linear fit to the boundary values yields a slope of $m = (-0.50 \pm 0.04)$~AU$^{-1}$, represented by the dotted line in Fig.~\ref{07_gauss}.
The slope $m$ is a signature of the semimajor axis and eccentricity dispersion of the Hertha1 family that resulted from the original collision. In particular, $m$ is determined by the true anomaly, $f$, of the parent asteroid at the time of its disruption. This relationship is described by Gauss' equations, which give the change in orbital elements for a collision fragment that experiences a velocity impulse $\Delta v$ \citep[see, e.g.][]{zappala2002}. Gauss' equations describe the behavior of the collision fragments in unperturbed (osculating) elements; in the Hertha region, the proper elements show similar behavior.
We plot the proper element distributions for several different values of $f$ in Fig.~\ref{07_gauss} \citep[cf.][Fig.~1]{nesvorny2002}, along with their corresponding slopes, $m$. Each ellipse represents an isotropic ejection field with $\Delta v = 285$~m/s, with $\Delta v$ chosen to approximate the eccentricity range of the family, estimated by the noticeable decrease in number density near $e_\text{P}$ = 0.15 and $e_\text{P}$ = 0.205. The ellipses were generated from 100 test particles, with osculating orbits calculated via the Gauss equations, and proper elements calculated via two-million-year integrations using Orbit9. The effects of the Mars 1:2 mean-motion resonance at $a_\text{P} = 2.419$~AU are observed in the proper element calculations. The lowest possible value for $m$ occurs at $f = 180\ensuremath{^\circ}$, where $m = -0.50$~AU$^{-1}$. Thus, assuming the original ejection velocity field was fairly isotropic, it appears the collision that formed this family occurred when the parent body was near aphelion, with a true anomaly within the range $170\ensuremath{^\circ} < f < 190\ensuremath{^\circ}$, calculated using the uncertainty on the slope as determined from the boundary values above.
\begin {center}
EDITOR: PLACE FIGURE \ref{07_gauss} HERE
\end {center}
In order to apply the correction (Eq. \ref{aprime}) to remove the effect of the initial slope in the ($e_\text{P}$,$a_\text{P}$) distribution, we need to evaluate $e_{\text{P,0}}$ in addition to $m$. The value $e_{\text{P,0}}$ is the proper eccentricity at the family's center, which we take to be the barycenter of the population. To minimize the influences of the nearby resonances on the barycenter, we consider only the largest objects (12.0~mag $< H <$ 14.0~mag), which have experienced the least amount of Yarkovsky semimajor axis dispersion toward significant resonances. We further restrict the sample to 2.31~AU $< a_\text{P} <$ 2.5~AU, 0.136 $< e_\text{P} <$ 0.23, and 0.033 $< \sin i_\text{P} <$ 0.051, in order to conservatively include all Hertha1 family members while largely eliminating members of the nearby Massalia and Flora families. We also removed from the sample all objects with albedos lower than 0.14, which are likely to be associated with the overlapping darker families discussed in Section \ref{low_astar}. This does not remove all of the contamination from those families, due to the incompleteness of the WISE database; however, such contamination would likely be primarily from smaller objects, and thus would have little effect on the determination of the barycenter of the Hertha1 family. Lastly, we removed asteroid (135) Hertha itself from the barycenter consideration, due to the uncertainty surrounding its inclusion in the family.
The barycenter of the sample described above is $a_{\text{P,0}} = 2.412 \pm 0.015$~AU and $e_{\text{P,0}} = 0.177 \pm 0.003$, with the uncertainty dominated by the exclusion of (135) Hertha from the sample. The corrected $e_\text{P}$ vs. $a_\text{P}'$ distribution, with $a_\text{P}'$ calculated using Eq. \ref{aprime} with $m = -0.50$~AU$^{-1}$ and $e_{\text{P,0}} = 0.177$, is shown in Fig.~\ref{08_aprime_e}.
\begin {center}
EDITOR: PLACE FIGURE \ref{08_aprime_e} HERE
\end {center}
\subsection{Post-collision history of the family}
\label{interpret}
The corrected semimajor axis dispersion of the Hertha1 family records information about the family's post-collision evolution under the Yarkovsky effect. In general, the most useful Yarkovsky drift information is preserved in the outer edges of a family, specifically the extent of drift at each size range, as parameterized by $C$. The age of the family is obtained from the value of $C$ for which Eq. \ref{Cparam} gives curves in $a_\text{P}'-H$ space that best fit the outer edges of the distribution. However, the Jupiter 3:1 resonance at high $a_\text{P}$ and the Mars-crossing region at low $a_\text{P}$ have significantly sculpted the boundaries of the Hertha1 family (Fig.~\ref{08_aprime_e}). The former renders the high-$a_\text{P}$ (right) edge of the family unreliable; the Mars-crossing region affects the low-$a_\text{P}$ (left) edge for $e_\text{P} > 0.185$. Accordingly, in order to determine an appropriate boundary $C$ value, we consider only the objects with $e_\text{P} < 0.185$, and fit only the lower-$a_\text{P}$ edge of their distribution in $a_\text{P}'-H$ space.
To determine the value of $C$ that best fits that edge, we calculate for each asteroid the value of $C$ that corresponds to its $a_\text{P}'$ and $H$ values (Eq. \ref{Cparam}). Fig.~\ref{09_cprime} shows the distribution of these $C$ values, with the curves representing the kernel density estimate (KDE) for the distribution along with its derivatives (see Appendix A.1 of \citet{dykhuis2014} for an explanation of the KDE and its derivatives). The best-fit edge of the family is the $C$ value where there is a maximum in the second derivative of the KDE (the blue curve in Fig.~\ref{09_cprime}).
\begin {center}
EDITOR: PLACE FIGURE \ref{09_cprime} HERE
\end {center}
There are comparable maxima in the second derivative at $C = -0.045$~mAU and $-0.037$~mAU; however, inspection of the distribution in $a_\text{P}'-H$ space (Fig.~\ref{10_aprime_H}) shows that the latter boundary would lie well inside the dense population, not at its edge. By contrast, $C = -0.045$~mAU (black curve in Fig.~\ref{10_aprime_H}) does indeed fit the outer edge of the distribution quite well. The uncertainty in the $C$ boundary ($\pm 0.005$~mAU, represented by the dashed curves in Fig.~\ref{10_aprime_H}) is dominated by the uncertainty in the $a_{\text{P,0}}$ center of the family. The $C$ boundary defining the edge of the family is much tighter for the corrected $a_\text{P}'-H$ distribution than it was for the uncorrected $a_\text{P}-H$ distribution plotted earlier in Fig.~\ref{05_color} ($C = \pm0.045$~mAU vs. $C = \pm0.075$~mAU).
\begin {center}
EDITOR: PLACE FIGURE \ref{10_aprime_H} HERE
\end {center}
The value of $C = 0.045$~mAU fits the leftmost edge of the distribution quite well for the larger objects ($H <$ 17~mag), but fails for the smallest objects ($H >$ 17 mag), which is not surprising for a couple of reasons. First, the observational bias in the database affects the completeness of the family sample for objects fainter than magnitude 16.5 \citep{bottke2014}. Second, the effects of ``variable'' or ``stochastic'' YORP on the drifting family members can cause their spin rates to random walk toward the typical YORP end states (mass shedding or non-principal axis rotation), which applies a size-dependent ``brake'' to the smallest objects \citep{bottke2013,bottke2014}. Stochastic YORP simulations of the Eulalia, ``New Polana'' and Erigone families by these authors demonstrated behavior similar to what is observed here.
The calculation of the age of the family from the observed Yarkovsky semimajor axis dispersion requires an estimate of the initial semimajor axis spread due to the original collision itself. This estimate is typically difficult to obtain, but in the case of the Hertha1 family, we can place an upper limit on the collision spread from the $a_\text{P}-e_\text{P}$ distribution. As shown in Section \ref{identify}, the slope of $m = -0.50$~AU$^{-1}$ indicates a collision near aphelion, which would result in minimal initial semimajor axis dispersion in $a_\text{P}'$. The maximum expected value of dispersion is $\delta a_{ej} = 0.005 \pm 0.008$~AU, which represents the width of the proper semimajor axis dispersion for a collision at true anomaly $f = 180\ensuremath{^\circ}$, with uncertainty derived from the uncertainty in $f$ (which results from the uncertainty in the slope $m$).
The semimajor axis dispersion due to the original collision is thus expected to be a small fraction of the Yarkovsky dispersion, resulting in an overestimation of the boundary $C$ parameter of less than 0.003 mAU, which is less than the uncertainty on the boundary $C$ parameter (0.005 mAU). Hence the initial semimajor axis dispersion has little effect on our calculation of the age from the Yarkovsky drift.
We use the inner boundary of the $C$ parameter distribution to determine the age of the family via the relation \citep{dykhuis2014}:
\begin{equation}
\label{yarcora3}
t = \frac{1329 \cdot |C| \cdot {a_{\text{P,0}}}^2}{c_Y \sqrt{p_\text{V}}(1-p_\text{V})\cos\epsilon}
\end{equation}
\noindent where $t$ is the time (in My) since the collision, $a_\text{P,0}$ is the time-averaged proper semimajor axis of the family members (in AU), $\epsilon$ is the obliquity, and $C$ is given in AU. The parameter $c_Y$ contains information about the asteroid's material properties (thermal conductivity, specific heat, material density), which are not known \emph{a priori}. As in \citet{dykhuis2014}, we adopt the value for $c_Y$ obtained from measurements of the precession rates of the Karin family, which yield a $c_Y$ parameter in the range of 0.0026 AU$^3$km~My$^{-1} < c_Y <$ 0.0035~AU$^3$km~My$^{-1}$.
With values of $C = \pm0.045$~mAU, albedo $p_\text{V} = 0.25$, $a_{\text{P,0}} = 2.412$~AU, and $c_Y$ = 0.00305~AU$^3$km~My$^{-1}$, we find an age for the Hertha1 family of $300^{+60}_{-50}$~My. The error in this estimate is dominated by the uncertainty in $c_Y$.
The age estimate also does not account for possible systematic errors due to the assumption of similar material properties between the members of the Hertha1 and Karin families. In particular, differences in bulk densities would affect the $c_Y$ parameter appropriate to both families. The average bulk densities for the two families are difficult to estimate, due to the lack of available density data for the family members. The only Hertha family object with density data is (135) Hertha itself \citep{carry2012}; its taxonomic type of X$_k$ makes it a poor standard for comparison with the S-type Karin family. The Karin family itself has members with density measurements, but the Koronis family, of which Karin is a subfamily, has density measurements for two of its members, (243) Ida and (720) Bohlinia; their densities are 2.35 $\pm$ 0.29 and 2.74 $\pm$ 0.56 g/$\text{cm}^3$, respectively \citep{carry2012}. These are both consistent with the averages for the S-type class, and thus are assumed here to be consistent with the unknown Hertha family average.
\section{Low-$a^*$ objects: The Hertha2, Polana and Eulalia families}
\label{low_astar}
We now consider the objects in the Nysa-Polana region with $a^* < -0.015$. The ($e_\text{P}$, $\sin i_\text{P}$) distribution of these objects is plotted in Fig.~\ref{11_low_astar}. Several structures appear within the blue box; the diffuse grouping near (142) Polana and the dense grouping near $e_\text{P} = 0.145$ and $\sin i_\text{P} = 0.047$ are here referred to as the Polana and Eulalia1 families, respectively (Table~\ref{table1}). The combined families had long been recognized as a single ``Polana'' family until their distinction in the work of \citet{walsh2013}.
\begin {center}
EDITOR: PLACE FIGURE \ref{11_low_astar} HERE
\end {center}
Comparison of Fig.~\ref{11_low_astar} with Figs.~\ref{01_orbital} and \ref{03_high_astar} shows that the Hertha1 family, roughly bounded by the red box, has been removed. However, a smaller, dense cluster of low-$a^*$ objects lies near the location of asteroid (135) Hertha, within the range of the now-removed Hertha1 family. We refer to this cluster as Hertha2.
\subsection{The Hertha2 family}
\label{hertha2}
The Hertha2 cluster lies very close to asteroid (135) Hertha in $a_\text{P}$ (Fig.~\ref{12_hertha2_aH}), as well as in $e_\text{P}$ and $\sin i_\text{P}$, suggesting a genetic relationship. In fact, the barycenter of the objects within the cyan box in Fig.~\ref{11_low_astar} (those plotted in Fig.~\ref{12_hertha2_aH}) is at $a_\text{P} = 2.426$~AU, nearly equal to the value for (135) Hertha itself, $a_\text{P} = 2.429$~AU. The cluster displays the characteristic V-shape in $a_\text{P}-H$ indicative of Yarkovsky evolution, further identifying the Hertha2 structure as a collisional family. The V-shape envelope corresponds to $C = \pm0.012$~mAU (significantly narrower than that observed for the Hertha1 family, $C$ = 0.045~mAU, Section \ref{high_astar}). This tighter V could represent either a younger age or a higher average density for the Hertha2 family; the implications are discussed further in Section \ref{discussion}.
\begin {center}
EDITOR: PLACE FIGURE \ref{12_hertha2_aH} HERE
\end {center}
The Hertha2 family also appears as a distinct cluster in reflectance properties. Fig.~\ref{13_hertha_region} shows the color distribution similar to Fig.~\ref{02_reflectance}, but restricted to the objects within the same dynamical range as the Hertha2 family (2.1~AU $< a_\text{P} <$ 2.5~AU, $0.1690 < e_\text{P} < 0.1812$, and $0.0420 < \sin i_\text{P} < 0.0471$). The Hertha2 family forms a diffuse grouping centered around $a^* = -0.07$; the members of the large Hertha1 family that reside in the same dynamical space are distinct from the Hertha2 members, clustering instead near $a^* = 0.13$.
\begin {center}
EDITOR: PLACE FIGURE \ref{13_hertha_region} HERE
\end {center}
\subsection{Other dark families}
\label{other_dark}
Other family structures among the dark asteroids in the Nysa-Polana region can be teased apart by considering the $a_\text{P}-H$ distribution in the context of the Yarkovsky evolution model, an approach utilized by \citet{walsh2013}. The $a_\text{P}-H$ distribution of the low-$a^*$ objects in the Nysa-Polana region is shown in Fig.~\ref{14_walsh}. This figure includes for context the cyan boundary $C$ parameter lines for the Hertha2 family (Fig.~\ref{12_hertha2_aH}), and plots new $C$ parameter lines for the two distinct families identified by \citet{walsh2013} (cf. Fig.~2 of that work), which are centered on (142) Polana and (495) Eulalia. These two families are visible in Fig.~\ref{14_walsh}; we refer to them as the Polana and Eulalia1 families, respectively. The latter is one of two structures that we identify with (495) Eulalia (the second is discussed below).
\begin {center}
EDITOR: PLACE FIGURE \ref{14_walsh} HERE
\end {center}
Fig.~\ref{11_low_astar} demonstrates a phenomenon that was noticed first by \citet{walsh2013}: (495) Eulalia is significantly offset from its nominal family in $e_\text{P}$ ($e_\text{P} = 0.12$ vs. the family average of $e_\text{P} = 0.145$). \citet{walsh2013} modeled the long-term evolution of (495) Eulalia, and demonstrated that proximity to the Jupiter 3:1 resonance results in variability of the asteroid's proper eccentricity between 0.11 and 0.15 over 500 Myr.
In order to explore whether any other asteroids might be related to (495) Eulalia, we modify the range under consideration to $0.05 < e_\text{P} < 0.2$ and $0.0 < \sin i_\text{P} < 0.08$, and replot the distribution of low-$a^*$ objects in this new range (Fig.~\ref{16_new_eulalia}). A diffuse grouping appears near $e_\text{P} = 0.11$ and $\sin i_\text{P} = 0.035$, indicated roughly by the magenta box in Fig.~\ref{16_new_eulalia}. This grouping was noted by \citet{walsh2013} after removal of the dense Eulalia1 cluster, and was tentatively associated with (112) Iphigenia (indicated by the green diamond).
\begin {center}
EDITOR: PLACE FIGURE \ref{16_new_eulalia} HERE
\end {center}
Fig.~\ref{17_eulalias} shows the $a_\text{P}-H$ distribution of the objects in the green and magenta boxes in Fig.~\ref{16_new_eulalia}, that is, the objects of the Eulalia1 and the putative Iphigenia cluster. The latter objects show similar structure to the Eulalia1 objects, and can be described by the same Yarkovsky $C$ envelope. A preliminary best fit $C$ envelope for each is at $C = \pm0.075$~mAU (slightly different from that found by \citet{walsh2013}, $C = \pm0.092$~mAU). Note that the two groups display a distinct offset in semimajor axes ($\delta a_\text{P} \sim 0.035$~AU); this offset, coupled with the identical boundary $C$ envelopes for the two clusters, leads us to designate the second cluster as ``Eulalia2'' and interpret it as possible evidence for an anisotropic ejection field resulting from the Eulalia1 collision event, rather than associate it with (112) Iphigenia. The implications of the structures of these two clusters are discussed further in Section \ref{discussion_eulalias}.
\begin {center}
EDITOR: PLACE FIGURE \ref{17_eulalias} HERE
\end {center}
The semimajor axis offset between these two populations was not discussed by \citet{walsh2013}, and indeed is not apparent in the corresponding figure in that work (their Fig.~20). Walsh et al. relied on the hierarchical clustering method (HCM) to identify groupings of objects for comparison. While the HCM is ideal for identifying clusters in multidimensional space, it is less ideal for comparison of neighboring families, as it tends to include in each sample objects from the neighboring family or families, which muddies the comparison. Instead, we select for comparison two distinct sets of objects (within the green and magenta boxes in Fig.~\ref{16_new_eulalia}) for which we expect minimal contamination between sets, which enables us to identify the small offset in $a_\text{P}$ among the Eulalia1 and Eulalia2 structures.
\section{Discussion}
\label{discussion}
We have identified five distinct clusters of asteroids within what has generally been called the Nysa-Polana orbital region of the asteroid belt. Two of these clusters appear to be associated with collisions on (135) Hertha; two other clusters appear to be associated with (495) Eulalia; one is associated with (142) Polana. None of these is associated with (44) Nysa.
This refined interpretation of the dynamical structure of the Nysa-Polana region has been made possible by the availability of color and albedo information from the SDSS and WISE projects, which have increased the dimensionality of data available for cluster analyses beyond simply the proper orbital element distribution. The analysis of the clusters within the Nysa-Polana region depends particularly upon this increased dimensionality, due to the overlap within the region of multiple large families with varied reflectance properties.
Other recent work has explored clusters within this newly extended parameter space using an automated approach based on the hierarchical clustering method \citep{masiero2013,milani2014,carruba2013}. Our approach differs from theirs in that we (like \citet{walsh2013}) have also incorporated the effects of Yarkovsky semimajor axis drift, which results in recognizable patterns in the relationship between semimajor axis spread and asteroid size (equivalently, the absolute magnitude), specifically the characteristic V shape in $a_\text{P}$ vs. $H$. In addition, we use the slope of the $a_\text{P}-e_\text{P}$ correlation among the Hertha1 family members as a constraint on the orbit configuration of the parent object at the time of the collision, enabling an improved evaluation of the Yarkovsky semimajor axis dispersion of the family independent of the collisional dispersion.
\subsection{Hertha1 and Hertha2}
\label{discussion_herthas}
The collision that formed the Hertha1 family $300^{+60}_{-50}$~Ma (Section \ref{high_astar}) distributed ejecta across nearly the entire span of the inner main belt in semimajor axis, depositing material directly into Mars-crossing orbits at the time of the event. Under the influence of the Yarkovsky effect, the Hertha1 family has since provided a steady stream of material of various sizes into both the Mars-crossing region and the Jupiter 3:1 resonance. Hence, it is plausible that the Hertha1 family has been a source population for significant impact events of various magnitudes on Earth in the past 300 million years, a time span which includes the event hypothesized to be connected with the Permian-Triassic extinction \citep[251.4 Ma,]{raup1982,jin2000}.
The weak Mars 1:2 resonance affects the semimajor axis and eccentricity distribution of the Hertha1 family \citep{gallardo2011}. The effect of this resonance is most noticeable in the dearth of objects in a depleted zone with $a_\text{P} < 2.43$~AU and $e_\text{P} > 0.195$ in Fig.~\ref{04_ae}, compared for example with the density of objects just below ($e_\text{P} < 0.19$) in the same figure. When objects drifting from the family center toward higher semimajor axes reach the resonance, it diffuses their eccentricities, resulting in higher average eccentricity of the resonant population. Thus, for objects starting at low eccentricities, the boost moves them to a higher-$e_\text{P}$ section of the Hertha1 family; for objects starting at high eccentricities, the boost may remove them from the family altogether, hence the paucity of blue dots in the depleted zone. This paucity demonstrates also the effects of the combined Yarkovsky and YORP effects on the Hertha1 family; objects that would otherwise have drifted back and forth as a result of YORP cycles, continually re-filling the depleted zone, have instead been removed from the family to higher eccentricities.
The presumed parentage and nomenclature of the Hertha1 family has varied throughout the literature (Table~\ref{table1}). Unlike the small, low-$a^*$ Hertha2 family, whose objects show reflectance properties consistent with X-type spectra and thus similar to (135) Hertha itself, the large Hertha1 family shows average reflectance properties consistent with S-type spectra. S-type objects would not generally be expected to originate from an X-type parent body, given the standard picture of S-type objects originating from chondritic, undifferentiated parent objects. However, \citet{weiss2013} suggested that some chondritic bodies may originate from the unmelted crusts of partly differentiated planetesimals. Moreover, \citet{thomas2014} have suggested that the Massalia family may have olivine-rich objects among its predominantly S-type members. Accordingly, it is plausible that the X-type (135) Hertha is associated with the S-type Hertha1 family.
In the following subsections, we consider four scenarios for the origin of the Hertha1 and Hertha2 families that might explain their different reflectance properties. None of the scenarios explain the observations completely; therefore the actual origin scenario may be a combination of these or other effects.
\subsubsection{Scenario 1: Unrelated families}
\label{unrelated}
The possibility that both Hertha1 and Hertha2 originated from two unrelated collisions on different parent objects, one of which (Hertha1) was catastrophically disrupted, cannot be ruled out here. One approach to understanding the type of collision expected to have formed the Hertha1 family involves comparison of the observed size-frequency distribution (SFD) of the family members with models of SFDs produced by catastrophic and cratering collision events. However, the incompleteness of the reflectance property databases, as well as the depletion of primarily smaller objects due to the nearby large resonances, have introduced significant bias into the family's observed SFD. Future efforts will explore the connection between the Hertha1 family and (135) Hertha in detail, including numerical integrations to determine the possible evolution of the proper orbital elements of (135) Hertha \citep[similar to the method used for (495) Eulalia by][]{walsh2013}. In the meantime, the close dynamical proximity of both Hertha1 and Hertha2 families to the asteroid (135) Hertha remains strong evidence that they share this common parent, and warrants discussion of additional origin scenarios.
\subsubsection{Scenario 2: Single collision on a partially differentiated parent}
\label{core}
A single collision on a partially differentiated planetesimal could produce, in principle, a family of objects with varied material properties. The tighter Yarkovsky V of the Hertha2 family ($C = \pm0.012$~mAU vs. $C = \pm0.045$~mAU for the Hertha1 family) would appear to suggest a younger age, and hence a separate collision. However, a single collision could still explain the two different Yarkovsky V's if the tighter V reflects a decreased Yarkovsky semimajor axis drift rate instead of a younger age, with the objects of the Hertha2 family drifting outward more slowly after the original collision.
A slower drift rate for the Hertha2 objects could result from their lower average albedo, $p_\text{V}$. The small numbers and small sizes of the Hertha2 family objects complicate the determination of the average albedo; however, it would have to be $\sim 0.014$ to account for the reduced semimajor axis spread, which is implausibly low (asteroid (135) Hertha has $p_\text{V} = 0.15$). Thus, the relatively narrow V of the Hertha2 family cannot be accounted for by lower albedo.
A higher average bulk density might alternatively explain the slow drift. The Yarkovsky drift rates scale inversely with density, so to account for the different V shape relative to the Hertha1 S-type family (with $\rho_\text{S} = 2.7$~g/cm$^3$), the average density that would be required for the Hertha2 objects would need to be 8.6~g/cm$^3$. For comparison, the bulk density of Hertha is $\rho_\text{H} = 5.23$~g/cm$^3$, and the average for the $\text{X}_\text{k}$-type objects is $\rho_{\text{X}_\text{k}} = 3.8$~g/cm$^3$ \citep{carry2012}. The larger density could be achieved if the family were composed primarily of metal objects; however, the family members' apparent association with the lower-density (135) Hertha does not imply that this is the case. Thus, density differences probably cannot explain the reduced spread of the Hertha2 family, and a more recent formation event seems the most plausible explanation.
Further evidence for a separate formation event comes from the observed $a_\text{P}-e_\text{P}$ correlation among the Hertha1 family members' orbits (Section \ref{03_high_astar}), and the lack of one among the Hertha2 family members. If the Hertha2 members formed in the same collision as the Hertha1 members, they would be expected to share this correlation. Thus the most likely explanation is that two separate collision events occurred.
\subsubsection{Scenario 3: Two collisions on a partially differentiated parent}
\label{subsequent}
A series of two collisions occurring on a partially differentiated parent could excavate families with varied material properties from different regions in the parent body. In the case of the Hertha families, an initial collision into a parent body's unmelted, chondritic crust could form the S-type Hertha1 family, leaving a large X-type remnant, (135) Hertha. A subsequent impact onto (135) Hertha could then create the X-type Hertha2 family.
A collision of magnitude sufficient to create the Hertha1 family might be expected to produce more than one remnant Hertha-like X-type among its fragments. The nearest object with reflectance properties similar to (135) Hertha is the large E-type (44) Nysa, which could have conceivably come from the original Hertha1 impact, although it is significantly displaced from the family in proper inclination. \citet{zellner1977} and \citet{kelley1994} attempted to link (44) Nysa and (135) Hertha in various differentiation structures, and \citet{rivkin1995,rivkin2000} demonstrated that the spectra of both objects share a hydration feature that could suggest a common origin. Nevertheless, the issue of whether a first impact could have cleanly removed the S-type material leaving only one X-type remnant for subsequent impact remains uncertain.
\subsubsection{Scenario 4: Two collisions with shock darkening or space weathering effects}
\label{shock_darkening}
The effects of shock darkening (through the presence of impact melt) can modify the apparent reflectance properties of asteroids without changing their basic material properties \citep{britt1994}. In the case of the Hertha families, the impact that formed the Hertha1 family could produce remnants such as (135) Hertha with a significant amount of shock-darkened material, resulting in the later excavation of material with X-type appearance during the Hertha2 formation event.
\citet{reddy2014} demonstrated the effects of shock darkening among laboratory samples of LL chondrite material mixed with impact melt, and matched the resulting spectra with observations of the X-type Baptistina family members. In appearance, the difference in $a^*$ color between the Hertha1 and Hertha2 families ($-0.07$ vs. 0.13) is strikingly analogous to that of the nearby Flora and Baptistina populations: The Baptistina members lie in nearly the same place in $a^*$ and $i-z$ color space as the Hertha2 members, as seen by comparison of the location of the Hertha2 family in Fig.~\ref{13_hertha_region} with the location of the Baptistina family in a similar plot \citep[e.g., Figs. 3e and 3f of][]{dykhuis2014}; and both the Flora and Hertha1 populations show reflectance properties typical of S-type objects.
Given the results of \citet{reddy2014} for the Baptistinas, it is worth considering the possibility that the X-type appearance of (135) Hertha and the Hertha2 family members results from shock darkening, i.e. the presence of significant impact melt matrix surrounding fragments of the original S-type material. In this case, a plausible origin hypothesis is that the impact that formed the Hertha1 family left behind a shock-darkened fragment, (135) Hertha, which experienced a later collision that formed the darkened Hertha2 family.
Another explanation for the difference in $a^*$ between the two Hertha families could be the effects of space weathering, which raises the $a^*$ value of exposed asteroid surfaces over time, resulting in higher $a^*$ for older families. In this case, the older Hertha1 family would be expected to show higher average $a^*$ values than the Hertha2 family, as observed.
Consideration of the surface properties of the four subfamilies of the large, S-type Koronis family \citep{molnar2011} raises issues regarding both the shock-darkening and the space-weathering hypotheses. These subfamilies range in age from 5.8~My to 300~My, and display a linear trend in $a^*$ from 0.02 for the youngest family to 0.07 for the oldest \citep{molnar2011,dykhuis2014}. This range of $a^*$ values is consistent with the color change expected as a result of space weathering, based on a study of various other families by \citet{willman2010}. The difference in $a^*$ between the two Hertha families is four times greater, so space weathering is unlikely to be the key factor in explaining this difference. If, instead, shock darkening is the dominant factor in the different appearance of the Hertha families, it raises the question of why similar effects have not led to more diversity among the Koronis families. The answer may simply lie in the unknown specific details of the collisions responsible for each family group.
\subsection{Eulalia1 and Eulalia2}
\label{discussion_eulalias}
The low-albedo families in the Nysa-Polana region are thought to be the most likely source families for asteroid (101955) Bennu, the target of the OSIRIS-REx sample return mission \citep{campins2010,walsh2013,bottke2014}. \citet{bottke2014} determined the probability that Bennu originated from what we call the Polana family to be 70\%, with the probability of an origin from what we call the Eulalia1 family of 30\%. The Eulalia2 objects identified here have similar orbital and reflectance properties, and might also be a candidate source population for Bennu. The possibility of a collisional connection between the Eulalia1 and Eulalia2 clusters, discussed below, could help further constrain the dynamics of the Bennu source region.
The Eulalia1 and Eulalia2 clusters show very similar structure in $a_\text{P}-H$ space, and can be roughly fit with $C$ envelopes that match, except for a semimajor axis offset of about $\delta a_\text{P} \sim 0.035$~AU (Section \ref{other_dark}, Fig.~\ref{17_eulalias}), with Eulalia1 centered at 2.46~AU and Eulalia2 centered at 2.495~AU. These values bracket the current $a_\text{P}$ value for (495) Eulalia ($a_\text{P}$ = 2.4867~AU). The Eulalia1 and Eulalia2 clusters also bracket (495) Eulalia in $e_\text{P}$ and $\sin i_\text{P}$ (Fig.~\ref{16_new_eulalia}). \citet{walsh2013} noted that the long-term evolution of (495) Eulalia ranges between $e_\text{P}$ values of 0.11 to 0.15; this range is contained within the larger range of the combined Eulalia1 and Eulalia2 families.
In light of these dynamical coincidences (matching $C$ envelopes, bracketing in each orbital dimension), it is plausible that the Eulalia1 and Eulalia2 clusters were originally part of the same collisional family, centered near the current position of (495) Eulalia. In this case, the twin ``lobes'' of the family protruding in opposite directions in $a_\text{P}, e_\text{P}$ and $\sin i_\text{P}$ could be explained by a non-isotropic ejecta pattern. In this scenario, due to the parent object's proximity to the Jupiter 3:1 resonance, a significant number of the original objects would have been deposited directly into the resonance (though not beyond it, see the discussion in Section 2.5 of \citet{walsh2013}). The resonance would then have dramatically reduced the number of objects in the high-$a_\text{P}$, low-$e_\text{P}$, low-$\sin i_\text{P}$ lobe of the family. Subsequent Yarkovsky drift would then have removed another half of the remaining family (the prograde objects) via drift into the Jupiter 3:1.
The above scenario could be tested dynamically, by modeling the best fit values of the $C$ boundaries and $a_\text{P,0}$ for the Eulalia1 and Eulalia2 clusters to confirm the similarity of $C$ for the two and determine their offset in semimajor axis. This modeling is necessary to confirm the significance of the observed offset, as the sample of Eulalia2 members (from the magenta box in Fig.~\ref{16_new_eulalia}) is fairly diffuse and thus might be expected to contain a significant number of background objects. In addition, orbital integrators that incorporate the Yarkovsky and YORP effects could be used to simulate and reproduce the observed $a_\text{P}-H$ distributions of the family from a single collision.
Lastly, spectral comparison of the reflectance properties of the Eulalia1 and Eulalia2 structures can also test the above origin scenario. The SDSS and WISE databases provide reflectance data for a large number of asteroids in the region; however, the completeness of these databases is only around 20\% and 10\%, respectively. Future surveys such as the Gaia space mission of the European Space Agency\footnote{sci.esa.int/gaia/} and the Large Synoptic Survey Telescope (LSST)\footnote{www.lsst.org/lsst/} are expected to significantly enhance these databases in the near future, providing opportunities for further analysis. The Polana and Eulalia1 and Eulalia2 structures are all similar in the wavelength range available to the SDSS, and a survey of 72 objects in the region shows spectral homogeneity in the NIR wavelengths as well (N. Pinilla-Alonso, in preparation). Future work in this area might possibly distinguish or further connect the Eulalia1 and Eulalia2 structures.
\section* {ACKNOWLEDGMENTS}
{\scriptsize The authors are grateful to Miroslav Broz and an anonymous reviewer for their comments, which significantly improved the quality of this manuscript. We acknowledge the use of data from the Sloan Digital Sky Survey, and thank the Sloan team and its sponsors (see http://www.sdss.org). In addition, this publication makes use of data products from the Wide-field Infrared Survey Explorer and NEOWISE, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. We thank Bill Bottke, Dante Lauretta and the OSIRIS-REx science team for their guidance and advice. In addition, we thank Nathaniel Dykhuis for his assistance with the statistical analysis. MJD was supported by an NSF Graduate Research Fellowship, Award No. DGE-1143953, with additional support from the OSIRIS-REx Dynamical Evolution Working Group.
\bibliographystyle{plainnat}
|
1,314,259,996,750 | arxiv | \section{Introduction}
Since the Bose-Einstein condensation (BEC) was realized in trapped
dilute atomic gases at ultra-low temperature,
theoretical studies of the BEC have been rapidly developing.
In the early stage of study,
ultra-cold alkali atoms such as $^{87}$Rb and $^{23}$Na
were mainly used for a formation of the BEC.
Many theoretical analyses were performed with
the Gross-Pitaevskii (GP) equation,
in which the two-body interaction takes a delta-function form.
In fact, the two-dimensional GP equation is suitable particularly
for the study of the BEC made of alkali atoms in a cylindrical trap,
where the s-wave scattering is dominant.
Thanks to recent experimental developments,
non-alkali atoms and molecules were also cooled down to form the BEC.
Such an example is seen in the condensate of Cr atoms \cite{Crexpt}.
The inter-atomic potential in the BEC made of Cr atoms cannot
be approximated exclusively by the delta function because of
the strong dipole moment carried by a Cr atom \cite{CRS05,ZZ05}.
Therefore, the dipole-dipole interaction (including a tensor force)
could be also responsible for the many-body dynamics, which involves
the d-wave scattering in the BEC.
Another example is the condensate of molecules \cite{BEC-BCS}, where
the delta function is not appropriate for the intermolecular interaction
because of the anisotropic nature of the interaction.
The BEC can be rotated
with the state-of-the-art experimental techniques,
such as the laser spoon \cite{Ke}.
Owing to these techniques,
it was demonstrated that the BEC undergoes the quantum phase transition
to vortex states.
The ``cranked'' GP equation was applied to the analysis of the rotating BEC,
and it successfully explained the quantum phase transition, including
the triangular lattice of vortices \cite{KMP00,Triangle}.
In the early stage of the study of the rotating BEC,
the two-dimensional GP equation was mainly
used for theoretical analyses \cite{GP2},
because the BEC was only rotated about the fixed axis.
Recently observed phenomena, such as
precession \cite{expt-tilt-1,expt-tilt-2} and bending \cite{bendexpt}
of vortices, require the rotational axis to move around
in a time-dependent manner, with respect to a certain coordinate frame.
These phenomena attract much interest in terms of the three-dimensional
spatial structures of the vortex states.
Hence, the three-dimensional GP equation is now being applied to
various vortex states \cite{GP3-1,GP3-2,GP3-3,GP3-4,GP3-5}.
In addition, a topological technique
was developed to produce vortices with spin 2 or 4 ($\hbar$),
by reversing the magnetic field of the trap \cite{BVortexRb,BVortexNa}.
(Below, we take the unit for angular momentum to be $\hbar=1$.)
The formation mechanism of such vortices
requires three-dimensional motion of the vortices.
In these situations, the total angular momentum vector needs to be treated
in a three-dimensional manner.
In this way, everytime new experimental progresses are achieved,
new physical situations are created, to
which the original GP equation cannot be applied in a naive way.
The cranked Hartree-Fock-Bogoliubov (cranked HFB) method,
which has been used to describe rotational states of atomic nuclei
\cite{3dCHFB,3dCHFB2}, could be a useful and powerful approach
in order to deal with these new situations.
Rotating BEC systems in a trap has been analyzed
with the GP equation, or the exact diagonalization method
using a set of the truncated basis.
Mottelson was the first to discuss the ``yrast'' structure \cite{BM75}. Here,
the yrast states mean the lowest-energy states for given angular momentum.
He proposed a scenario \cite{Mottelson-1}
that the quadrupole excitation is dominant when
the total angular momentum $L$ is much less than
the particle number $N$ ($L<<N$), and that
all the bosonic particles will occupy the p state
when the total angular momentum becomes equal to the particle number.
This situation can be interpreted as a creation of a vortex state.
This prediction was numerically verified by himself and his collaborators
using the two-dimensional GP equation \cite{Mottelson-2}.
Bertsch and Papenbrock also verified this prediction
using the diagonalization of the two-dimensional model Hamiltonian
\cite{Bertsch-1,Bertsch-2}.
Further detailed analysis was performed by Nakajima et. al.\cite{Nakajima}.
In this paper,
we apply the cranked HFB method to a simple schematic model,
where bosonic particles interact weakly through the
repulsive quadrupole-quadrupole interaction.
This model is too simple to describe the detailed structure of
realistic systems.
But, in limited situations, the quadrupole-quadrupole interaction
becomes a phenomenologically valid interaction that can reflect physically
essential properties of the realistic interaction.
For example, our model can describe low-energy rotational
excitations of weakly interacting alkali atoms, as discussed by
Mottelson \cite{Mottelson-1}.
Then, using the density matrix of the yrast states, as well as
its eigenvalues and eigenstates, we compare our results with the other methods.
Creation of a vortex state is also discussed within the framework of our model.
In Sections II and III, we present how the cranked HFB theory is extended
so as to calculate not only fermionic systems but also bosonic ones.
Unlike the GP equation, we do not assume the inter-atomic
potential to be the delta function. Also, we do not
suppose an {\it a priori} existence of the condensate,
which is the essential assumption in the GP equation.
The cranked HFB theory is a constrained mean-field theory,
and the value and direction of the total angular momentum vector
are controlled in the calculation.
With this method,
it is expected that we can numerically analyze
not only structure of dilute many-boson systems in a trap,
but also superfluidity produced by ultra-cold many-fermion systems in a trap.
In the present study, we focus on the study of weakly interacting Bose systems,
and an application of the cranked HFB theory to a simple model is presented
in Sections IV and V.
\section{Cranked Hartree-Fock-Bogoliubov Theory}
We describe the cranked HFB method that can be applied to rotating particles
interacting two-body interactions.
This method was originally proposed for a description of nuclear rotation
\cite{3dCHFB,3dCHFB2}, but we extend the method to deal with
not only fermions but bosons.
Let $c_\alpha^\dagger, c_\alpha$ be the creation and annihilation
operators of the single particle state
$\langle\xi|\alpha \rangle =\psi_\alpha (\xi)$, where $\xi$
represents the real-space coordinates,
spin coordinates and nuclear spin of particles.
The creation and annihilation operators of quasi-particle
$a^\dagger_i, a_i$ is given by the
Bogoliubov-Valatin transformation \cite{Bogo,Valatin},
\begin{eqnarray}
a_i^\dagger &=& \sum_\alpha
U_{\alpha i} c_\alpha^\dagger +V_{\alpha i} c_\alpha, \\
a_i &=& \sum_\alpha
U_{\alpha i}^* c_\alpha +V_{\alpha i}^* c_\alpha^\dagger.
\label{BV-trans}
\end{eqnarray}
The operators $a_i,c_\alpha$ obey the following commutation rule,
\begin{eqnarray}
\label{comm-rule}
[ a_i, a_j^\dagger ]_{\pm} &=& \delta_{ij}, \
[ a_i^\dagger, a_j^\dagger ]_{\pm} =[ a_i, a_j ]_\pm = 0,\\ \nonumber
\left[c_{\alpha}, c_{\beta}^{\dag}\right]_{\pm} &=& \delta_{\alpha \beta}, \
[ c_{\alpha}^\dagger, c_{\beta}^\dagger ]_{\pm} =[ c_{\alpha}, c_{\beta} ]_\pm = 0,
\end{eqnarray}
where the upper sign ($+$) applies to fermions and the lower ($-$) to bosons.
To satisfy the commutation rule (\ref{comm-rule}),
we need the following relations.
\begin{eqnarray}
U^\dagger U \pm V^\dagger V =1, U^T U^* \pm V^T V^* =1, \\
U^\dagger V^* \pm V^\dagger U^* =0, V^T U \pm U^T V =0.
\end{eqnarray}
Based on the variational principle, the $U$ and $V$ are determined.
The variational ansatz is chosen to be
\begin{equation}
|\Phi\rangle = N_f \exp(\sum_{\alpha \beta}
\frac{1}{2} f_{\alpha \beta} c^\dagger_\alpha c^\dagger_\beta) |0 \rangle,
\quad
f_{\alpha \beta} = \sum_i V_{\alpha i} ({U^*}^{-1})_{i\beta},
\label{HFBansatz}
\end{equation}
where, $N_f$ is a normalization constant and $|0\rangle$ is the true vacuum.
The variational state $|\Phi\rangle$ corresponds to the vacuum
in the quasi-particle basis, that is, $a_i |\Phi \rangle =0$.
The many-body Hamiltonian including a two-body interaction
$V(\xi_1,\xi_2)$ is generally written as
\begin{equation}
\label{hamiltonian}
\hat{H} =
\sum_{\alpha \beta} H^0_{\alpha \beta}c^\dagger_\alpha c_\beta
+\frac{1}{4}
\sum_{\alpha \beta \gamma \delta}
{\cal V}_{\alpha \beta \gamma \delta}
c^\dagger_\alpha
c^\dagger_\beta
c_\delta
c_\gamma,
\end{equation}
where ${\cal V}_{\alpha\beta\gamma\delta}$ is given by
\begin{widetext}
\begin{equation}
{\cal V}_{\alpha \beta \delta \gamma}
= \langle \psi_\alpha(\xi_1) \psi_\beta(\xi_2) |V(\xi_1,\xi_2)|
\psi_\gamma(\xi_1) \psi_\delta(\xi_2)) \rangle
\mp \langle \psi_\alpha(\xi_1) \psi_\beta(\xi_2) |V(\xi_1,\xi_2)|
\psi_\delta(\xi_1) \psi_\gamma(\xi_2)) \rangle.
\end{equation}
\end{widetext}
The one-body part $H^0_{\alpha \beta}$ includes the kinetic energy and
the spherical confinement potential.
Using Wick's theorem, we can represent
the Hamiltonian (\ref{hamiltonian}) by
the total energy $E$ and the one-body Hamiltonian $\hat{h}$, as
\begin{widetext}
\begin{eqnarray}
\hat{H} &=& E + \hat{h} +\frac{1}{4}
\sum_{\alpha \beta \gamma \delta}
{\cal V}_{\alpha \beta \gamma \delta}
: c^\dagger_\alpha
c^\dagger_\beta
c_\delta
c_\gamma
:,\\
E&=&
\sum_{\alpha \beta} H_{\alpha \beta}^0
\langle c_\alpha^\dagger c_\beta \rangle
+
\frac{1}{4}\sum_{\alpha \beta \gamma \delta}
{\cal V}_{\alpha \beta \gamma \delta}
(
\langle c_\delta c_\gamma \rangle \langle c^\dagger_\alpha c^\dagger_\beta \rangle
+\langle c^\dagger_\alpha c^\dagger_\beta \rangle \langle c_\delta
c_\gamma \rangle
+2 \langle c^\dagger_\alpha c_\gamma \rangle \langle c^\dagger_\beta
c_\delta \rangle
),\\
\label{onebody}
\hat{h} &=&
\sum_{\alpha \beta} H_{\alpha \beta}^0
: c_\alpha^\dagger c_\beta :
+
\frac{1}{4}\sum_{\alpha \beta \gamma \delta}
{\cal V}_{\alpha \beta \gamma \delta}
(
\langle c_\delta c_\gamma \rangle : c^\dagger_\alpha c^\dagger_\beta :
+\langle c^\dagger_\alpha c^\dagger_\beta \rangle :c_\delta c_\gamma:
+2 \langle c^\dagger_\alpha c_\gamma \rangle : c^\dagger_\beta c_\delta:
),
\end{eqnarray}
\end{widetext}
where $:\cdots:$ is the normal order product with respect to
$a,a^\dagger$ and an abbreviated expression is introduced for
an expectation value,
$\langle {\cal O} \rangle=\langle \Phi |{\cal O}| \Phi \rangle$.
The HFB wavefunction $|\Phi \rangle$ is determined through
the variational principle with constraints $\hat{C}_n$,
\begin{equation}
\delta \langle \Phi | \hat{H} - \sum_n \mu_n \hat{C}_n | \Phi \rangle =0,
\label{cond-vari}
\end{equation}
where $\mu_n$ is a Lagrange multiplier.
In this method,
three components of the total angular momentum
and the particle number are constrained.
Further constraints are imposed on the following quadrupole operators,
$
\hat{B}_1 = \sqrt{\frac{15}{2\pi}} yz,
\hat{B}_2 = \sqrt{\frac{15}{2\pi}} zx,
\hat{B}_3 = \sqrt{\frac{15}{2\pi}} xy$,
in order to fix the intrinsic coordinate axes of the system
along the principal axes of the quadrupole moments.
Therefore, we have seven constraints in our calculations.
\begin{eqnarray}
\langle \Phi | \hat{J}_x | \Phi \rangle &=& J_x,
\langle \Phi | \hat{J}_y | \Phi \rangle = J_y,
\langle \Phi | \hat{J}_z | \Phi \rangle = J_z,\\
\langle \Phi | \hat{B}_1 | \Phi \rangle &=& 0,
\langle \Phi | \hat{B}_2 | \Phi \rangle = 0,
\langle \Phi | \hat{B}_3 | \Phi \rangle = 0,\\
\langle \Phi | \hat{N} | \Phi \rangle &=& N,
\end{eqnarray}
In Eq.(\ref{cond-vari}), these constraints are represented
with $\hat{C}_{n}$'s as
$
\hat{C}_1=\hat{J}_x,
\hat{C}_2=\hat{J}_y,
\hat{C}_3=\hat{J}_z,
\hat{C}_4=\hat{B}_1,
\hat{C}_5=\hat{B}_2,
\hat{C}_6=\hat{B}_3,
\hat{C}_7=\hat{N}.
$
In particular, the term $-\sum_{n=1}^{3}\mu_n\hat{C}_n$ has been
called the ``cranking term'' in nuclear high-spin physics, because
it simulates the effect of the Coriolis force in the rotating mean-field
system.
\section{Method of Steepest Descent}
As mentioned earlier, we determine the HFB states,
following the variational principle.
By multiplying a unitary operator
to an arbitrary initial HFB state $|\Phi\rangle$,
another HFB state $|\Phi'\rangle$ is obtained.
This transformation is considered as a variational procedure
with respect to the matrices $U$ and $V$ of the Bogoliubov-Valatin
transformation, Eq.(\ref{BV-trans}).
The transformation is iterated until a local minimum is found
to satisfy Eq.(\ref{cond-vari}).
Now, let us explain how the unitary operator is given within our framework.
First of all,
from the extended Thouless theorem \cite{Thouless,Onishi},
the unitary transformation of the HFB state is expressed as
\begin{equation}
|\Phi' \rangle = \exp(\hat{d}) |\Phi \rangle,
\end{equation}
where
$\hat{d}$ is an anti-Hermitian operator
$\hat{d}=-\hat{d}^\dagger$, which is generally expressed as,
\begin{equation}
\hat{d} = \frac{1}{2}\sum_{ij}\left(
d_{ij}a_i^\dagger a_j^\dagger -d_{ij}^* a_j a_i
\right).
\end{equation}
A quasi-particle basis is then transformed in the following way.
\begin{equation}
\left(
\begin{array}{c}
a_i'^\dagger\\
a_i'
\end{array}
\right)
=
\left(
\begin{array}{c}
e^{-\hat{d}} a_i^\dagger e^{\hat{d}}\\
e^{-\hat{d}} a_i e^{\hat{d}}
\end{array}
\right)
=\exp({\pm{\cal D}})^T
\left(
\begin{array}{c}
a_i^\dagger\\
a_i
\end{array}
\right),
\end{equation}
where
\begin{equation}
{\cal D} =
\left(
\begin{array}{cc}
0 & d\\
d^*& 0
\end{array}
\right),
\end{equation}
and $d$ is a matrix representation of $\hat{d}$.
We choose the anti-Hermite operator $\hat{d}$ as,
\begin{equation}
\hat{d}=[ \eta \hat{r} +\hat{s}]^{\rm a},
\end{equation}
where we define $[\hat{\cal O}]^{\rm a}=
\frac{1}{2}[\hat{\cal N},\hat{\cal O}]$ and
the quasi-particle number operator is given
as $\hat{\cal N}=\sum_i a_i^\dagger a_i$.
An operator $\hat{s}$ and the single-particle Routhian $\hat{r}$ are
respectively defined as
\begin{eqnarray}
\hat{s} &=& \sum_n \delta_n \hat{C}_n,\\
\hat{r} &=& \hat{h}-\sum_n \mu_n \hat{C}_n.
\end{eqnarray}
Then, these parameters $\eta$, $\delta_n$, and $\mu_n$ are determined through
a minimization of $\langle \Phi' | \hat{H} | \Phi' \rangle$
under the constraints $\langle \Phi | \hat{C}_i | \Phi \rangle =c_i$.
The parameters $\delta_n$ and $\mu_n$ are evaluated by expanding
$\langle \Phi' | \hat{C}_i | \Phi' \rangle$
up to the first order in $\delta_n$ and $\mu_n$. That is,
\begin{eqnarray}
\delta_k &=& \sum_i L_{ki}^{-1}(c_i -\langle \Phi |\hat{C}_i |\Phi \rangle),\\
\mu_k &=& \sum_i L_{ki}^{-1}\langle \Phi | [\hat{C}_i,[\hat{h}]^a]|\Phi \rangle,
\end{eqnarray}
where $L_{ki}=
\langle \Phi | [ \hat{C}_i,[ \hat{C}_k ]^a ] | \Phi \rangle$.
Whereas, the parameter $\eta$ is determined
from a minimization condition for $E=\langle \Phi' | \hat{H} | \Phi' \rangle$,
through expanding $E$ up to the second order in $\eta$.
As a consequence, we have
\begin{equation}
\eta=
-\frac{
\langle \Phi | [\hat{H},[\hat{r}]^a] | \Phi \rangle
+\langle \Phi | [[\hat{H},[\hat{r}]^a],[\hat{s}]^a] | \Phi \rangle
}
{
\langle \Phi | [[\hat{H},[\hat{r}]^a],[\hat{r}]^a] | \Phi \rangle
}.\end{equation}
To check the convergence for the self-consistency in the calculation,
it is convenient to define the norm of $d$
as $|d|=\sqrt{\sum_{ij}\frac{1}{2}|d_{ij}|^2}$.
In our calculations, a criterion for the convergence is given whether
$|d|$ is less or greater than $\epsilon=1.0\times10^{-7}$.
When $|d| < \epsilon$, we judge that the convergence is numerically achieved.
Otherwise, the iteration for the self-consistency continue
until $|d|$ meets the above condition.
\section{A schematic model}
To examine the convergence procedure of our method (3D-cranked HFB)
for the bosonic case,
let us consider a toy model, which is similar to
a realistic system of dilute ultra-cold Bose gases confined by
an isotropic harmonic oscillator potential $V(r)=\frac{1}{2}M \omega^2 r^2$,
where $M$ represents the atomic mass.
We choose the single-particle states to be the harmonic oscillator states,
$\langle\bm{r}|\alpha \rangle =
\langle\bm{r}|c^\dagger_{\alpha} |0\rangle = R_{n_\alpha l_\alpha}(r)
i^{l_\alpha} Y_{l_\alpha m_\alpha}(\theta \phi)$, that is, a product of the
Laguerre polynomial and the spherical harmonics.
Let us denote this basis as $\alpha \equiv (n_{\alpha},l_{\alpha},m_{\alpha})$.
Mottelson discussed that the quadrupole correlation is important
in the low angular momentum region \cite{Mottelson-1}.
In accordance with his proposition,
the following Hamiltonian is considered in our calculations.
\begin{eqnarray}
\hat{H}_{\rm model}&=&
\hat{H}_0
+\frac{1}{2}\kappa \sum_{\mu=-2}^{2}(-)^\mu \hat{Q}_{-\mu} \hat{Q}_{\mu}
+g \hat{P}^\dagger \hat{P} \label{modelhamilt},\\
\hat{H}_0&=&
\sum_\alpha (2n_\alpha +l_\alpha + \frac{3}{2})\hbar \omega \delta
c^\dagger_\alpha c_\alpha,\\
\hat{Q}_{\mu} &=& \sum_{\alpha \beta}
\langle \alpha | 2 r^2 C_\mu^{(2)} | \beta \rangle
c_\alpha^\dagger c_\beta, \\
\hat{P} &=&
\sum_\alpha \sqrt{2l_\alpha +1}
\langle l_\alpha m_\alpha l_\alpha -m_\alpha | 00 \rangle
c_{\bar{\alpha}} c_\alpha,
\end{eqnarray}
where $ c_{\bar{\alpha}}$ is the annihilation operator corresponding to
the state $\bar{\alpha}=(n_\alpha, l_\alpha, -m_\alpha)$,
and
$C^{(k)}_{\kappa}(\Omega)$ is related to the spherical harmonics through
$C^{(k)}_\kappa(\Omega)=\sqrt{\frac{4\pi}{2k+1}}Y_{k \kappa}(\Omega)$.
This Hamiltonian should be considered
as a simple model for weakly interacting dilute atomic gases.
The parameter $\kappa$ represents strength of the quadrupole-quadrupole
interaction,
and $\kappa$ is positive in this work, to treat the repulsive two-body
interaction.
The last term of Eq. (\ref{modelhamilt}) is called the pairing interaction.
\cite{Ring-Schuck}
The parameter g represents the
strength of the pairing
interaction. In the present calculation, we set g to be zero, for the
sake of simplicity.
The one-body Hamiltonian (\ref{onebody}) in this model is given as,
\begin{equation}
\hat{h}_\text{model}= \hat{H}_0 +\kappa \sum_\mu (-)^\mu
\langle \hat{Q}_{-\mu} \rangle \hat{Q}_\mu
+g \langle \hat{P}^\dagger \rangle \hat{P}.
\label{eq-scf}
\end{equation}
The oscillator energy for the isotropic harmonic oscillator states,
$\hbar \omega$, is set to 1.9 (meV).
The quadrupole-quadrupole
interaction $\kappa/A^2$ is repulsive and its strength is set to
0.1, 0.5 and 1.0 (meV/$\mu$m$^4$), where $A$ is the mass number of an atom.
The single-particle model space used in this calculation are
the harmonic oscillator states of the
0s, 1s, 0d, 2s, 1d, 0g, 0p, 1p, 0f, 2p, 1f and 0h states.
To prepare the initial state, we use the deformed quadrupole mean field.
\begin{widetext}
\begin{equation}
\hat{h}_{\rm deform} = \hat{H}_0
- \frac{2}{3} M\omega^2 r^2
\left(
\beta\cos\gamma C^{(2)}_0(\Omega)
+\frac{1}{\sqrt{2}}\beta\sin\gamma(C^{(2)}_2(\Omega)
+C^{(2)}_{-2}(\Omega))
\right).
\label{eq-deform}
\end{equation}
\end{widetext}
The quadrupole parameters $(\beta,\gamma)$ are useful measures
to think about the shape of the many-body system.
$\beta$ is a measure for elongation or stretching,
while $\gamma$ for triaxiality or deviation from axial symmetry.
For example, a spherical shape has $\beta=0$ and
nuclear superdeformation has typically $\beta \simeq 0.6$.
The triaxial parameter $\gamma$ gives axial shapes
when $\gamma=0^{\circ}$ and $\gamma=60^{\circ}$.
The former shape corresponds to the so-called ``prolate''shape,
which is similar to a kiwifruit,
while the latter to ``oblate'' shape, similar to a mandarin orange.
By definition, a triaxial shape is invariant with respect to an
operation: $\gamma \rightarrow \gamma+120^{\circ}$.
We diagonalize the Hamiltonian (\ref{eq-deform}) for
$(\beta,\gamma)=(0.01,0)$ and obtain
the single-particle states. The initial state is created
as a Hartree-Fock state, that is, all the single-particle levels are
occupied below the Fermi level.
Since the initial state is symmetric under rotation about the $z$-axis,
collective rotation about the $z$-axis is suppressed.
In other words, we cannot crank the state around the symmetry axis.
To induce angular momentum, we initially set the constraints of
the total angular momentum to $(J_x, J_y, J_z)=(0.05, 0, 0)$.
As a next step, we tilt the angular momentum vector to
$(J_x, J_y, J_z)=(0, 0, 0.1)$.
With this procedure, we can increase $J_z$ up to 20,
with a step, $\Delta J_z = 0.05$.
\section{Results and discussions}
Figure \ref{fig:energy} represents the total energy $E$
as a function of the total angular momentum $L$.
Four lines are plotted in this figure.
One dotted line represents $\hbar\omega (L +\frac{3}{2}N)$
and the other three lines are calculated with the different
quadrupole-quadrupole interaction strengths,
which are $\kappa/A^2=0.1, \ 0.5$ and $1.0$ (meV/$\mu$m$^4$).
Despite the different values for $\kappa$,
These lines are nearly identical.
In other words, $E$ is almost independent of $\kappa$.
We also find that $E$ increases almost in proportion to $L$,
that is, $E\propto L$.
This result can be explained from a microscopic point of view.
As $L$ is increased, single-particle excitations are induced
one by one through the two-body interaction, so as to
satisfy the angular momentum constraints. In other words,
$\Delta E = \hbar\omega \Delta L$, where $\hbar\omega$ is the
single-particle energy spacing of the isotropic harmonic oscillator.
Due to the quantum statistics for bosons,
this excitation mode can continue until the number of particles occupying
the ground state becomes zero.
This linear behavior is already noticed by other authors \cite{Bertsch-1}.
However, with a careful look at our numerical result,
there is a slight deviation from the linearity
in the the total energy $(\hbar\omega(L+\frac{3}{2}N))$.
This small deviation is caused by deformation of the mean field,
which has a role to mix the single-particle orbits.
The evolution of the quadrupole deformation in response to rotation
is discussed below.
Figure \ref{fig:betagamma} shows
the deformation parameters $\beta$ and $\gamma$.
The strengths of the interaction are set in the same manner as in Figure 1.
These deformation parameters are self-consistently calculated as
\begin{eqnarray}
\beta &=& \frac{3\kappa}{M\omega^2}\sqrt{\langle Q_{20}\rangle^2 +2
\langle Q_{22} \rangle^2},\\
\gamma &=& \tan^{-1} \frac{\langle Q_{22}\rangle}{\sqrt{2}\langle Q_{20} \rangle}.
\end{eqnarray}
The unit for the quadrupole moment ($M\omega^2/\kappa$) is derived
from the consistency at $L=0$
between the one-body Hamiltonian (\ref{eq-scf}) and
the deformed mean-field Hamiltonian (\ref{eq-deform}).
The figure shows that the deformation parameters
do not depend on the interaction strength very much,
although $\gamma$ shows minor differences at low spin.
When the total angular momentum is small ($L\alt 1$),
the mean field has a almost spherical shape.
This is because the trapping potential is spherical and
the present two-body interaction is repulsive.
As $L$ is increased, $\beta$ increases gradually.
In the small angular momentum region ($L < 5$ ),
$\gamma$ is not $180^{\circ} (\equiv 60^{\circ})$.
(See the right panel of Figure \ref{fig:betagamma}.)
This result means that the shape of the mean field is not axial-symmetric,
but triaxial.
When the quadrupole-quadrupole interaction becomes stronger,
the deformation tends to prefer a more triaxial-deformed shape.
However, any of the three cases ends up with the oblate shape
($\gamma=180^{\circ}$) at high angular momentum ($L\agt 5$).
A reason for this tendency can be explained as the following:
When the quadrupole-quadrupole interaction is strong,
the harmonic oscillator states having the different magnetic quantum numbers
become more mixed through the interaction.
As a result, the magnetic quantum number is no longer a good quantum number.
This is nothing but axial symmetry breaking, or an emergence of triaxiality.
It should be noted, however, that $\gamma$ is substantial only at low
angular momentum, where $\beta$ is very small.
In other words, when the elongation is small ($\beta\simeq0$),
the triaxial degree of freedom is irrelevant
in terms of a deviation from a spherical shape.
That is, in our calculation,
the shape of the mean field can be regarded to be almost spherical
in the small $L$ region.
On the other hand, for the higher angular momentum region ($L > 5$),
$\gamma$ is almost constant to be $180^{\circ}$, meaning
that the mean field becomes an oblate shape.
The condition for the BEC of weakly interacting bosonic atoms in a trap
is given by $Nv/\hbar\omega << 1$ \cite{Mottelson-1},
where $v$ is an expectation value of the two-body interaction,
while $\hbar\omega$ represents the single-particle level spacing.
In our calculation, $Nv$ corresponds to an expectation value
of the quadrupole-quadrupole force, that is, $\kappa
(
\langle Q_{20} \rangle^2
+2\langle Q_{22} \rangle^2
)
$.
The ratio $Nv/\hbar\omega$
is then estimated to be $4.4\times 10^{-3} \beta A^2/\kappa$.
According to our calculation, deformation is up to $\beta\alt 1$,
so that the ratio is of order of $10^{-4}$ to $10^{-5}$ for our
three choices of the interaction strength ($\kappa/A^2=0.1, \ 0.5, \ 1$
meV/$\mu$m$^4$).
This result means that our calculations can be regarded as
a weakly interacting many-boson system.
Figure \ref{fig:occupation} shows the occupation probability,
$\rho_{\alpha \alpha}$ for $\kappa/A^2=0.1$ meV/$\mu$m$^4$, where
$\rho_{\alpha \beta}$ is the density matrix defined as
\begin{equation}
\rho_{\alpha \beta} = \sum_{i} V_{\alpha i}^* V_{\beta i}.
\end{equation}
($V$ is a matrix appearing in the Bogoliubov-Valatin transformation,
Eq.(\ref{BV-trans}).)
Although the occupation probabilities for
$\kappa/A^ = 0.5, 1$ meV/$\mu$m$^4$ are not plotted in Figure
\ref{fig:occupation},
we have calculated these occupation probabilities and found that
they are almost same as that of $\kappa/A^2 = 0.1$ meV/$\mu$m$^4$.
This result indicates that the wave-function does not strongly
depend on the strength of the quadrupole-quadrupole interaction.
At low $L$, the $(0s0)$ states are the major component in the HFB state.
The higher the total angular momentum, the more the $(0d2)$ state
admixes with the $(0s0)$ state.
The $(0g2)$ component is also mixed at $L\simeq 2N=20$,
while the $(0s0)$ component vanishes.
This result suggests that
the yrast state changes its structure gradually.
Figure \ref{fig:eigen} shows the eigenvalue
$\nu_a$ of the density matrix $\rho_{\alpha \beta}$ ,
as a function of $L$.
Only the plot for $\kappa/A^2 = 0.1$ (meV/$\mu$m$^4$) is displayed
because the occupation is almost independent of $\kappa/A^2$.
The largest eigenvalue in the figure
is equal to the total particle number $N=10$, and
this situation happens only at $L=0$ and $2N (=20)$
In these cases, all the particles occupy only one single-particle state,
which can be regarded as the condensate state.
On the other hand,
between $L=0$ and $2N$ ($0<L<2N=20$),
the particles are shared by the two states $\psi^{\text{A}}$ and
$\psi^{\text{B}}$, where $\nu_{\rm A} + \nu_{\rm B} = N = 10$.
This result indicates that most of the yrast states are non-condensates,
but a mixture of two single-particle components, $\psi^{\text{A}}$ and
$\psi^{\text{B}}$.
In Figure \ref{fig:component},
the eigenstates $\psi^{\text{A}}$ and $\psi^{\text{B}}$ are decomposed
into the single-particle basis, and their components
are displayed in terms of probability $(v^a_\alpha)^2$,
where $|\psi^a\rangle = \sum_\alpha v^a_\alpha |\alpha \rangle$.
First, as shown in the right panel of the figure,
the state $\psi^{\text{A}}$ at $L=0$
is found to have a condensate structure into the $(0s0)$ state,
which is consistent with the occupation number calculation
shown in Figure \ref{fig:occupation}.
As $L$ is increased,
the $(0d0)$ component starts to mix with the $s$ component
although the contribution from the $d$ state is minor.
This mixture is consistent with the growth of the quadrupole deformation,
as shown in the left panel of Figure \ref{fig:betagamma}.
When $L\alt 5$, axial symmetry is broken around the cranking axis,
as shown in Figure \ref{fig:betagamma}. In this situation, the whole system
rotates in a collective manner, which is consistent with Mottelson's model
claiming that the yrast structure is dominated by the collective quadrupole
excitation at low $L$. Such a collective rotation was actually
observed experimentally at lower rotational frequency
before vortices are formed \cite{MIT}.
However, the linear dependence of $E$ on $L$ at low $L$ seen in
Figure \ref{fig:energy} implies that the collective mode is
not the major mode in our model, but that the single-particle excitations are.
Next, in the right panel of Figure \ref{fig:component},
the state $\psi^{\text{B}}$ is decomposed into the d and g states.
This is because the quadrupole-quadrupole interaction does not mix the states
having different parity, so that the particles in the $s$ state
can not be excited to the $p$ state, but the $d$ or $g$ states.
The major component is the $(0d2)$ state when angular momentum is small
($L\alt 10$), whereas the $(0g2)$ state starts to mix
in the high angular momentum region ($L > 10$),
due to the onset of deformation in the mean field.
Although the $(0g4)$ state is included already at $L\simeq 0$,
this component can be considered as a minor component
because the occupation $\nu_B$ is nearly 0 in this region.
According to the previous discussion, at $L=2N$,
the many-body state goes into the condensate in the state $\psi^{\text{B}}$,
which is a mixture of the $(0d2)$ and $(0g2)$ states.
Both of these $(0d2)$ and $(0g2)$ states have angular dependence
$\sin^2 \theta$, so that the density along the $z$-axis vanishes
for these states. Therefore, the condensate at $L=2N$ is considered as
a quantized vortex state carrying 2 ($\hbar$).
The HFB solutions obtained for the intermediate $L$ values ($0<L<20$)
are quite different from the solutions obtained with the GP equation.
This is because the GP equation assumes
an {\it a priori} existence of the condensate for the whole range of $L$,
which can be expressed as a Hartree state, $(a^{\dag}_0)^N|0\rangle$.
Although our HFB ansatz includes this condensate state as a special case,
the mathematical form for the HFB state is generally more complicated
to allow a linear combination form of multiple Hartree states,
in accordance with Eq. (\ref{HFBansatz}).
If the condensate is realized at any $L$,
all the particles should occupy the one single-particle state that
is generally expressed by a liner combination of the basis states,
such as the $s$, $d$ and $g$ states.
As shown in Figure \ref{fig:eigen},
the yrast structure changes smoothly from the $\psi^{A}$-dominant states
to the $\psi^{B}$-dominant state, as $L$ is increased from 0 to $2N$.
Considering that the latter state at $L=2N$ corresponds to a
vortex state, a formation of the vortex starts as a shallow dent in the center
at small $L$.
As the amplitude of $\psi^{\text{B}}$ becomes larger for increasing $L$,
the depth of the dent becomes deeper.
Finally at $L=2N$, a complete vortex is formed, where density becomes zero.
This mechanism of the vortex formation is very different from the one
derived from the GP equation.
In the calculations using the GP equation \cite{KMP00},
a vortex enters from the ``outside''
of the system due to a continuity of the many-body wave function.
This sort of process needs a odd-number multipolarities, such as a dipole
($\lambda=1$) and an octupole ($\lambda=3$) correlations, which are
missing from our present model.
These higher multipole correlations may play an important role
also in a formation of vortex lattices, which violates rotational symmetry
of the system. In the present framework, vortices appear always
in the center of the system due to the axial symmetry possessed by the system.
To allow multiple vortices to appear away from the center,
we need, at least, a mixing between these states of
$(0s0)$, $(0d2)$ and $(0g2)$ to break the symmetry.
This situation is realized only when $\langle Q_{2\pm 2}\rangle$,
which is contained in the single-particle Hamiltonian, Eq.(\ref{eq-scf}),
is non-zero.
In other words, the deformation parameter $\gamma$ should not be equal
to $0^{\circ}$ or $180^{\circ}$ to allow triaxial deformation bringing
anisotropy to the system. However, in the present calculations,
$\gamma$ takes the value of $180^{\circ}$ in a wide range of angular momentum
($5 <L <20$). Consequently, the vortex lattice is not produced in
our model.
\section{Summary}
Extending the 3D-cranked Hartree-Fock-Bogoliubov method,
we have performed the numerical calculation for a rotating many-boson system
interacting through a weak and repulsive interaction,
trapped inside an isotropic harmonic oscillator potential.
Unlike the Gross-Pitaevskii equation, our calculation does not assume
the existence of the condensate {\it a priori}, but
general many-body states in the framework of the HFB method.
We applied the method to a simple model where the two-body interaction
is chosen to be a separable type called the quadrupole-quadrupole interaction.
Parity is conserved for this interaction, so that only
$\Delta l =2$ excitations is allowed through
the quadrupole-quadrupole interaction.
First of all, at $L=0$, our calculation shows that
the HFB state is interpreted as a condensate into $(0s0)$.
As increasing the total angular momentum $L$ from 0 to 20,
the particles in the yrast state transfer from $\phi^A$ to $\phi^B$.
which have the magnetic quantum numbers $M_z=0$ and $M_z=2$, respectively.
At low $L$, triaxial deformation is formed to allow collective rotation
around the cranking axis. However, the linearity $E\propto L$ implies
that the major excitation mode is still single-particle excitations.
At higher $L$, the system becomes oblate,
that is, axial symmetric around the rotating axis. Angular momentum is
thus produced by migrating from $s$ state to $d$ state.
These single-particle excitations are induced from the $s$ to
the $d$ state through the quadrupole-quadrupole interaction,
in agreement with the prediction by Mottelson.
Finally, at $L=2N$, the HFB state becomes another condensate,
in which all the particles occupy the single state expressed
by a linear combination of the $(0d2)$ and $(0g2)$ states.
This state can be interpreted
as a vortex state having angular momentum $2 (\hbar)$.
In this way, the yrast structure changes gradually and smoothly
in our framework.
This result can be accounted by the following two effects:
One is our choice of the HFB ansatz which allows a linear combination
form of multiple Hartree states. The other is the finite-number effect of
the total particle number ($N=10$).
These effects surely needs further investigations in the future studies.
We are currently proceeding to extend
our programming code to deal with more realistic inter-atomic potentials, and
plan to examine the effect of the pairing interaction in the HFB framework.
\section{Acknowledgment}
This work is supported by EPSRC with a grant EP/C520521/1.
|
1,314,259,996,751 | arxiv | \section*{Supplemental material}
In this supplement, we provide the results of runs using non-fluctuating optical Glauber initial conditions. The hydrodynamical runs used here are precisely the same as those used in Ref.~\cite{Brambilla:2022ynh}, which made use of the anisotropic hydrodynamics formalism~\cite{Martinez:2010sc,Alqahtani:2017mhy,Alqahtani:2020paa}. The simulation parameters for the evolution of the bottomonium wave function and the number of trajectories sampled were the same as were used for the IP-Glasma fluctuating initial condition runs presented in the main body. Note that, compared to Ref.~\cite{Brambilla:2022ynh}, we considered 200,000 trajectories instead of 20,000 trajectories.
In Fig.~\ref{fig:raavsnpart-smooth}, we present the smooth hydrodynamical initial condition results for $R_{AA}$ as a function of $N_{\rm part}$. We note that the central value and bands found with 200,000 trajectories are slightly lower than those obtained with 20,000 trajectories in Ref.~\cite{Brambilla:2022ynh}. We have confirmed that this change is consistent with the statistical uncertainties associated with the lower number of trajectories used in Ref.~\cite{Brambilla:2022ynh}. In Fig.~\ref{fig:raavspt-smooth}, we present our results obtained for $R_{AA}$ as a function of $p_T$ using optical Glauber initial conditions. Finally, in Fig.~\ref{fig:v2-smooth}, we present our results obtained for $v_2$ as a function of centrality and $p_T$ using optical Glauber initial conditions.
\begin{figure}[ht]
\centering
\includegraphics[width=0.45\linewidth]{raavsnpart-SkappaTau.pdf} \hspace{5mm}
\includegraphics[width=0.45\linewidth]{raavsnpart-SgammaTau.pdf}
\vspace{-3mm}
\caption{The nuclear suppression factor $R_{AA}$ for the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ states as a function of $N_{\rm part}$ obtained with optical Glauber initial conditions. The left panel shows variation of $\hat\kappa$ and the right panel shows variation of $\hat\gamma$. The experimental measurements shown are from the ALICE~\cite{Acharya:2020kls}, ATLAS~\cite{ATLAS5TeV}, and CMS~\cite{Sirunyan:2018nsz,CMSupsilonQM2022} collaborations.}
\label{fig:raavsnpart-smooth}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.45\linewidth]{raavspt-SkappaTau.pdf} \hspace{3mm}
\includegraphics[width=0.45\linewidth]{raavspt-SgammaTau.pdf}
\vspace{-3mm}
\caption{The nuclear suppression factor $R_{AA}$ for the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ states as a function of $p_T$ obtained with optical Glauber initial conditions. The bands and experimental data sources are the same as Fig.~\ref{fig:raavsnpart-smooth}.}
\label{fig:raavspt-smooth}
\end{figure*}
\begin{figure}[ht]
\centering
\includegraphics[width=0.475\linewidth]{v21svscent-g-k_smooth.pdf} \hspace{3mm}
\includegraphics[width=0.425\linewidth]{v21svspt-gamkapS.pdf}
\vspace{-2mm}
\caption{The anisotropic flow coefficient $v_{2}$ as a function of centrality (left) and transverse momentum (right) obtained with optical Glauber initial conditions. We show the $\hat\gamma$ variation in blue and the $\hat\kappa$ variation in red and compare to experimental data from the ALICE and CMS collaborations \cite{ALICE:2019pox,CMS:2020efs}.}
\label{fig:v2-smooth}
\end{figure}
\end{widetext}
\end{document}
|
1,314,259,996,752 | arxiv |
\section{Introduction}
Recently, large pre-trained LMs have been proven pivotal in programming-related tasks~\cite{wang2021codet5, chen2021codex, hendrycksapps2021, lu2021codexglue, papineni2002bleu}\footnote{Detailed related work is presented in Appendix \ref{app:related_work}}. Program synthesis aims to generate a code given the natural language description of a problem. Programming requirements in these problems vary in terms of complexity from a 3-5 line simple function to multiple functions that use advanced data structures. However, LMs such as Codex show below-par performance on the long and complicated programming questions. We observe that the natural language description of the program becomes long and complicated when there is superfluous information (see section ~\ref{sec:HumanSummaries}). The goal of adding this information to the description is to make it more understandable to humans. However, we find that this information confuses the model in understanding a task\footnote{See example in Appendix ~\ref{app:example_fake}}. We propose that removing the excess information and providing the model with the exact specifications of the problem can improve the performance of the LMs.
To remove excess information\footnote{Instructions for creating summaries given in Appendix \ref{app:instructions}}, we summarize the descriptions of the program in such a way that it does not lose important specifications. We use the APPS dataset \cite{hendrycksapps2021} and CodeContests dataset \cite{li2022competition} which are a collection of coding problems from different online sources and create a meta-dataset consisting of human and synthesized summaries
We perform all experiments using the GPT-based Codex model \cite{chen2021codex} on the proposed meta-dataset and show that the summarized version of complicated questions improves strict accuracy by 8.13\% on the APPS dataset and 11.85\% on CodeContests. From our analysis, we can see significant improvement for introductory (9.86\%) and interview (11.48\%) related programming questions. However, it shows improvement by a small margin ($\sim 2\%$) for competitive programming questions. Considering that automatic evaluation of a program does not reward for partial correctness, we perform qualitative evaluation on our meta-dataset and find that original questions often confuse models in understanding the underlying problem, as models latch on to some spurious words in the text (e.g. the word `list' in question makes the model design a list even though the underlying problem is on graphs). We further analyze model performance on different types of summaries (i.e., basic, expert, and synthetic) and provide instruction-design principles that can help future research on prompting in program synthesis.
\section{Method}
\subsection{Dataset}
We use the APPS \cite{hendrycksapps2021} and CodeContests \cite{li2022competition} datasets to create summaries. We crowd-sourced the creation of human summaries. The result was 373 human summaries for APPS and 80 summaries for CodeContests along with and 8663 synthetic summaries using both datasets. Table \ref{table:AllProblemNums} shows the statistics of the generated summaries.
\input{all_num_probs}
\subsubsection{Human Generated Summaries}
\label{sec:HumanSummaries}
For the APPS and CodeContests human-generated summaries, the crowd worker reads and understands the original questions, then creates summaries in two steps\footnote{Instructions for creating summaries are in Appendix \ref{app:instructions}}. First, we create a basic summary of the given problem and remove any information that is repeated and any hypothetical information without concrete instructions. For example, if the problem constructs a fake company or situation,
we replace the fake situation with direct instructions. Full example is included in Appendix \ref{app:example_fake}. Second, we create an expert summary of the problem. To create this, we further summarize the first summary. This expert summary includes the absolute minimum information for an expert to understand the problem. We would not expect a novice to understand these prompts. An example of expert summaries is given in Appendix \ref{app:expert}.
\subsubsection{Synthetic Summaries}
We have generated synthetic summaries of program descriptions using jumbo (178B), large (7.5B) Studio21 model \cite{J1WhitePaper}, GPT-3 Davinci model (175B) \cite{brown2020language} and PEGASUS model \cite{zhang2019pegasus}. To generate a summary, we provide these models with a few examples in the in-context learning setup \cite{brown2020language} from the human-generated summaries. For the few-shot examples, we use expert-level summaries.
\paragraph{Studio21} We use five examples with the large model, and three examples with the jumbo model\footnote{Examples are included in Appendix \ref{app:prompts}}. For both models, we use a temperature of $0.3$, and topP of $1$. For the format of our prompt, we use De-Jargonizer template\footnote{\url{https://studio.ai21.com/}} with a change to their header as shown in Appendix \ref{app:prompts}. We create a total of $7,505$ synthetic summaries using these models.
\paragraph{GPT-3} We use three examples for GPT-3 model. We empirically set temperature to $0.05$, topP to $1$, frequency penalty to $0.01$, presence penalty to $0.05$. To generate prompts, we followed their tl;dr template\footnote{\url{https://beta.openai.com/playground/p/default-tldr-summary?model=text-davinci-001}} as shown in Appendix \ref{app:prompts}. We create $785$ synthetic summaries using this model.
\paragraph{PEGASUS} We use the PEGASUS model \cite{zhang2019pegasus} to create program summaries for the same set of problems that were summarized by humans. We choose this model because it was trained specifically for abstractive summarization.
\input{main_results}
\subsection{Model}
We use OpenAI Codex to build baselines and the proposed approach.
\paragraph{Baseline} To create a baseline, we have used original program descriptions given in the datasets as prompts for the Codex model.
\paragraph{Proposed Approach} We have used summaries of original program descriptions given in the datasets as prompts for the Codex model.
\section{Experimental Setup}
\label{sec:exp-setup}
All the experiments are performed using the $davinci-codex$ \cite{chen2021codex} model provided through OpenAI\footnote{Implementation and parameters details in Appendix \ref{app:codex}}. At inference time, we use a modified version of the evaluation code\footnote{\url{https://github.com/hendrycks/apps/blob/main/eval/test_one_solution.py}} provided by \citet{hendrycksapps2021}. This evaluation code has four different outputs for each test case: (1) \textbf{-2}: the code has a syntax error and can not run, (2) \textbf{-1}: the code is syntactically correct but has a run time error, (3) \textbf{0}: the code runs without any errors but fails the test case, and (4) \textbf{1}: the code runs without any error and passes the test case. Similar to \citet{chen2021codex}, we implement a timeout for the code at inference time. If a test case takes more than $4$ seconds to run then we throw an exception and count that test case as a $-1$.
\paragraph{Experiments} To show effectiveness of the proposed approach, we have performed three different experiments using human generated summaries:
\begin{enumerate}[itemsep=-1ex]
\item All problems from basic and expert summaries are used at inference time. We term this experiment All Problems (AP).
\item We eliminate problems that perform worse\footnote{Definition of the worst problem is given in Appendix \ref{app:worst_probs}}
for either basic or expert summaries. We term this experiment Either Worst Problem Removal (EWPR).
\item We eliminate problems that perform worse
for both basic and expert summaries. We term this experiment Both Worst Problem Removal (BWPR).
\end{enumerate}
\paragraph{Motivation behind EWPR and BWPR}
If a summary caused every test case to perform worse then it's likely the crowd worker produced a faulty summary. To mitigate the effect of outliers in the dataset, we use the EWPR method to remove such problems. Another hypothesis is that every problem benefits from some level of summarization (i.e., basic or expert). To measure this, we use the BWPR method. From Table \ref{tab:number-of-probs} results, we identify that only 1 problem had both summaries (basic and expert) preform worse.
\paragraph{Metric} In \cite{ProgSynthesisLargeLangModels}, they show that the BLEU metric \cite{papineni2002bleu} does not correlate well with synthesis performance. Thus, we use Strict Accuracy (SAcc) as our evaluation metric for all experiments (see Appendix \ref{app:accuracy}).
\section{Results and Analysis}
\input{synthetic_results}
\subsection{Human Generated Summaries}
\label{sec:human-results}
From Table \ref{tab:main-results}, we can observe that both the summary-based models show on average superior performance compared to baseline. In particular, when calculating results for every problem, basic and expert summary-based models outperform baseline by 4.34\% and 5.15\% on average for APPS dataset, respectively. Further analysis shows that the expert summary-based model shows improved performance by $\sim1\%$ compared to the basic summary-based model.
On the CodeContests dataset \cite{li2022competition}, we show an average improvement of $11.88\%$ in terms of SAcc. For this dataset, we did not separate the problems by difficulty. This is because the problems come from different sources and have different scales of difficulty. Thus, we did not report the SAcc when weighted by difficulty in Table \ref{tab:main-results}.
Our analysis shows that many problems where the basic summary would fail, however, the expert summary would succeed and vice-versa. Thus, we choose the best summary for each problem after evaluating both summaries and then calculate the results for the best summaries. Table \ref{tab:take-best} shows results when taking the best summary for each problem for APPS dataset. We observe a 9.86\%, 11.48\%, and 1.91\% increase on SAcc for introductory, interview, and competition level problems, respectively.
\subsection{Synthetic Summaries}
Table \ref{tab:synthetic-results} and \ref{tab:code-contests-synthetic-results} show the results for baseline, synthetic summaries generated by GPT-3, Studio21 and PEGASUS in terms of SAcc for two experiments. For the AP experiment, we can observe that the performance of the baseline outperforms synthetic summary-based models. However, the proposed model shows an average similar performance compared to the baseline for the EWPR experiment. Moreover, Appendix \ref{app:abb_syn_results} shows the results for top 500 and top 1000 summaries from GPT-3 and Studio21, respectively.
\subsection{Analysis}
\paragraph{Why does eliminating the worst problems help?} From Tables \ref{tab:main-results}, we can observe that EWPR and BWPR have improved performance compared to AP for both human and synthetically generated summaries. By analyzing the summarized worst problems, we notice a difference in the summarization style which shows that these summaries are outliers and do not match the distribution of the other summaries. This can cause a problem in synthesizing a good program since the model loses important information. Hence, we believe that eliminating the worst problems improves model performance.
\paragraph{Is there any possible bias in the meta-dataset?} Recent studies shows that bias propagates in human-annotated datasets \cite{geva-etal-2019-modeling, parmar2022don}. Given that our summaries are also human-generated, there will be some bias in the dataset.
Some details that are critical to one person can be trivial to others. In the context of generating expert summaries, assumptions about expert knowledge can vary.
This bias causes drift in the dataset and hinders the model's performance. Similar to \citet{mishra2021cross}, we can provide a template for what is expected from the summary generator to reduce bias.
\paragraph{Why is competition accuracy low?}
We believe that these problems require multi-hop reasoning, even after summarization, which is still a challenge for language models.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{EMNLP 2022/pos_smaller.png}
\caption{(Top plot) Mean frequency of POS for problems where programs where the generated by both the original and summarized prompt pass all test cases, and (Bottom plot) mean frequency of POS for problems where the summary passes all test cases and the original did not. The blue bar represents the mean of the entire dataset. analyzed only the top 11 most occurring POS. The plot shows that higher number of nouns degrade model performance.}
\label{fig:pos-error-analysis}
\end{figure}
\paragraph{Impact of POS on Accuracy}
In the top plot of Figure \ref{fig:pos-error-analysis}, we observe that frequency of $nouns$ and $propernouns$ for problems that passed all test cases is lower than the entire dataset. In the bottom plot, we observe that the frequency for $nouns$ and $propernouns$ is higher for the original question (which had < 100\% accuracy on the test cases) and lower for the summary (which had 100\% accuracy on the test cases). Thus, we can see that number of nouns degrades performance. We also see in the bottom chart that overuse of punctuation can be detrimental to performance.
From the results in Figure \ref{fig:pos-error-analysis} we see results of nouns affecting performance along with excessive punctuation.
Additional detailed analysis is presented in Appendix \ref{app:analysis}.
\section{Conclusion}
This paper introduces a summarization-based approach for efficient program synthesis. Experimental results show that the proposed approach improves the performance of the Codex model by on average $\sim8\%$ across various levels of programming questions provided by the APPS and $\sim11\%$ on the CodeContests. Further, this paper proposes a meta-dataset consisting of $\sim450$ human-generated basic and expert-level summaries as well as $\sim8k$ synthetically generated summaries by GPT-3 and Studio21; this can be helpful for future research on writing better instructions for the program synthesis. We show that program synthesis models benefit from concise prompts, hence, we believe that less number of high-quality instances are better than more low-quality data instances.
\paragraph{Future Extensions} The decomposition of prompts has been shown to improve accuracy \cite{mishra-etal-2022-reframing, patel2022question}; splitting up the summarization task the resulting summary can potentially result in higher accuracy for the Codex model in future. Additionally, the PEGASUS model could be used in conjunction with other models to perform the detailed algorithm outlined in Appendix \ref{app:instructions}.
\section*{Limitations}
Our summary-based approach shows improved performance on program synthesis models, however, it shows competitive performance on synthetic summaries. We believe that the generation of high-quality summaries can improve performance, hence, designing efficient prompts to improve synthetic summaries can be the scope of further research. Furthermore, human-generated summaries show competitive performance on competition-level problems. These problems require reasoning with multiple logical leaps and knowledge of advanced algorithms and data structures. Hence, exploring new techniques for summarization can be a future research direction. In addition, this work only analyzes the codex model, hence, exploring the effect of summarization on other program synthesis models can be interesting.
\section{Introduction}
Recently, large pre-trained LMs have been proven pivotal in programming-related tasks~\cite{wang2021codet5, chen2021codex, hendrycksapps2021, lu2021codexglue, papineni2002bleu}\footnote{Detailed related work is presented in Appendix \ref{app:related_work}}. Program synthesis aims to generate a code given the natural language description of a problem. Programming requirements in these problems vary in terms of complexity from a 3-5 line simple function to multiple functions that use advanced data structures. However, LMs such as Codex show below-par performance on the long and complicated programming questions. We observe that the natural language description of the program becomes long and complicated when there is superfluous information (see section ~\ref{sec:HumanSummaries}). The goal of adding this information to the description is to make it more understandable to humans. However, we find that this information confuses the model in understanding a task\footnote{See example in Appendix ~\ref{app:example_fake}}. We propose that removing the excess information and providing the model with the exact specifications of the problem can improve the performance of the LMs.
To remove excess information\footnote{Instructions for creating summaries given in Appendix \ref{app:instructions}}, we summarize the descriptions of the program in such a way that it does not lose important specifications. We use the APPS dataset \cite{hendrycksapps2021} and CodeContests dataset \cite{li2022competition} which are a collection of coding problems from different online sources and create a meta-dataset consisting of human and synthesized summaries
We perform all experiments using the GPT-based Codex model \cite{chen2021codex} on the proposed meta-dataset and show that the summarized version of complicated questions improves strict accuracy by 8.13\% on the APPS dataset and 11.85\% on CodeContests. From our analysis, we can see significant improvement for introductory (9.86\%) and interview (11.48\%) related programming questions. However, it shows improvement by a small margin ($\sim 2\%$) for competitive programming questions. Considering that automatic evaluation of a program does not reward for partial correctness, we perform qualitative evaluation on our meta-dataset and find that original questions often confuse models in understanding the underlying problem, as models latch on to some spurious words in the text (e.g. the word `list' in question makes the model design a list even though the underlying problem is on graphs). We further analyze model performance on different types of summaries (i.e., basic, expert, and synthetic) and provide instruction-design principles that can help future research on prompting in program synthesis.
\section{Method}
\subsection{Dataset}
We use the APPS \cite{hendrycksapps2021} and CodeContests \cite{li2022competition} datasets to create summaries. We crowd-sourced the creation of human summaries. The result was 373 human summaries for APPS and 80 summaries for CodeContests along with and 8663 synthetic summaries using both datasets. Table \ref{table:AllProblemNums} shows the statistics of the generated summaries.
\input{all_num_probs}
\subsubsection{Human Generated Summaries}
\label{sec:HumanSummaries}
For the APPS and CodeContests human-generated summaries, the crowd worker reads and understands the original questions, then creates summaries in two steps\footnote{Instructions for creating summaries are in Appendix \ref{app:instructions}}. First, we create a basic summary of the given problem and remove any information that is repeated and any hypothetical information without concrete instructions. For example, if the problem constructs a fake company or situation,
we replace the fake situation with direct instructions. Full example is included in Appendix \ref{app:example_fake}. Second, we create an expert summary of the problem. To create this, we further summarize the first summary. This expert summary includes the absolute minimum information for an expert to understand the problem. We would not expect a novice to understand these prompts. An example of expert summaries is given in Appendix \ref{app:expert}.
\subsubsection{Synthetic Summaries}
We have generated synthetic summaries of program descriptions using jumbo (178B), large (7.5B) Studio21 model \cite{J1WhitePaper}, GPT-3 Davinci model (175B) \cite{brown2020language} and PEGASUS model \cite{zhang2019pegasus}. To generate a summary, we provide these models with a few examples in the in-context learning setup \cite{brown2020language} from the human-generated summaries. For the few-shot examples, we use expert-level summaries.
\paragraph{Studio21} We use five examples with the large model, and three examples with the jumbo model\footnote{Examples are included in Appendix \ref{app:prompts}}. For both models, we use a temperature of $0.3$, and topP of $1$. For the format of our prompt, we use De-Jargonizer template\footnote{\url{https://studio.ai21.com/}} with a change to their header as shown in Appendix \ref{app:prompts}. We create a total of $7,505$ synthetic summaries using these models.
\paragraph{GPT-3} We use three examples for GPT-3 model. We empirically set temperature to $0.05$, topP to $1$, frequency penalty to $0.01$, presence penalty to $0.05$. To generate prompts, we followed their tl;dr template\footnote{\url{https://beta.openai.com/playground/p/default-tldr-summary?model=text-davinci-001}} as shown in Appendix \ref{app:prompts}. We create $785$ synthetic summaries using this model.
\paragraph{PEGASUS} We use the PEGASUS model \cite{zhang2019pegasus} to create program summaries for the same set of problems that were summarized by humans. We choose this model because it was trained specifically for abstractive summarization.
\input{main_results}
\subsection{Model}
We use OpenAI Codex to build baselines and the proposed approach.
\paragraph{Baseline} To create a baseline, we have used original program descriptions given in the datasets as prompts for the Codex model.
\paragraph{Proposed Approach} We have used summaries of original program descriptions given in the datasets as prompts for the Codex model.
\section{Experimental Setup}
\label{sec:exp-setup}
All the experiments are performed using the $davinci-codex$ \cite{chen2021codex} model provided through OpenAI\footnote{Implementation and parameters details in Appendix \ref{app:codex}}. At inference time, we use a modified version of the evaluation code\footnote{\url{https://github.com/hendrycks/apps/blob/main/eval/test_one_solution.py}} provided by \citet{hendrycksapps2021}. This evaluation code has four different outputs for each test case: (1) \textbf{-2}: the code has a syntax error and can not run, (2) \textbf{-1}: the code is syntactically correct but has a run time error, (3) \textbf{0}: the code runs without any errors but fails the test case, and (4) \textbf{1}: the code runs without any error and passes the test case. Similar to \citet{chen2021codex}, we implement a timeout for the code at inference time. If a test case takes more than $4$ seconds to run then we throw an exception and count that test case as a $-1$.
\paragraph{Experiments} To show effectiveness of the proposed approach, we have performed three different experiments using human generated summaries:
\begin{enumerate}[itemsep=-1ex]
\item All problems from basic and expert summaries are used at inference time. We term this experiment All Problems (AP).
\item We eliminate problems that perform worse\footnote{Definition of the worst problem is given in Appendix \ref{app:worst_probs}}
for either basic or expert summaries. We term this experiment Either Worst Problem Removal (EWPR).
\item We eliminate problems that perform worse
for both basic and expert summaries. We term this experiment Both Worst Problem Removal (BWPR).
\end{enumerate}
\paragraph{Motivation behind EWPR and BWPR}
If a summary caused every test case to perform worse then it's likely the crowd worker produced a faulty summary. To mitigate the effect of outliers in the dataset, we use the EWPR method to remove such problems. Another hypothesis is that every problem benefits from some level of summarization (i.e., basic or expert). To measure this, we use the BWPR method. From Table \ref{tab:number-of-probs} results, we identify that only 1 problem had both summaries (basic and expert) preform worse.
\paragraph{Metric} In \cite{ProgSynthesisLargeLangModels}, they show that the BLEU metric \cite{papineni2002bleu} does not correlate well with synthesis performance. Thus, we use Strict Accuracy (SAcc) as our evaluation metric for all experiments (see Appendix \ref{app:accuracy}).
\section{Results and Analysis}
\input{synthetic_results}
\subsection{Human Generated Summaries}
\label{sec:human-results}
From Table \ref{tab:main-results}, we can observe that both the summary-based models show on average superior performance compared to baseline. In particular, when calculating results for every problem, basic and expert summary-based models outperform baseline by 4.34\% and 5.15\% on average for APPS dataset, respectively. Further analysis shows that the expert summary-based model shows improved performance by $\sim1\%$ compared to the basic summary-based model.
On the CodeContests dataset \cite{li2022competition}, we show an average improvement of $11.88\%$ in terms of SAcc. For this dataset, we did not separate the problems by difficulty. This is because the problems come from different sources and have different scales of difficulty. Thus, we did not report the SAcc when weighted by difficulty in Table \ref{tab:main-results}.
Our analysis shows that many problems where the basic summary would fail, however, the expert summary would succeed and vice-versa. Thus, we choose the best summary for each problem after evaluating both summaries and then calculate the results for the best summaries. Table \ref{tab:take-best} shows results when taking the best summary for each problem for APPS dataset. We observe a 9.86\%, 11.48\%, and 1.91\% increase on SAcc for introductory, interview, and competition level problems, respectively.
\subsection{Synthetic Summaries}
Table \ref{tab:synthetic-results} and \ref{tab:code-contests-synthetic-results} show the results for baseline, synthetic summaries generated by GPT-3, Studio21 and PEGASUS in terms of SAcc for two experiments. For the AP experiment, we can observe that the performance of the baseline outperforms synthetic summary-based models. However, the proposed model shows an average similar performance compared to the baseline for the EWPR experiment. Moreover, Appendix \ref{app:abb_syn_results} shows the results for top 500 and top 1000 summaries from GPT-3 and Studio21, respectively.
\subsection{Analysis}
\paragraph{Why does eliminating the worst problems help?} From Tables \ref{tab:main-results}, we can observe that EWPR and BWPR have improved performance compared to AP for both human and synthetically generated summaries. By analyzing the summarized worst problems, we notice a difference in the summarization style which shows that these summaries are outliers and do not match the distribution of the other summaries. This can cause a problem in synthesizing a good program since the model loses important information. Hence, we believe that eliminating the worst problems improves model performance.
\paragraph{Is there any possible bias in the meta-dataset?} Recent studies shows that bias propagates in human-annotated datasets \cite{geva-etal-2019-modeling, parmar2022don}. Given that our summaries are also human-generated, there will be some bias in the dataset.
Some details that are critical to one person can be trivial to others. In the context of generating expert summaries, assumptions about expert knowledge can vary.
This bias causes drift in the dataset and hinders the model's performance. Similar to \citet{mishra2021cross}, we can provide a template for what is expected from the summary generator to reduce bias.
\paragraph{Why is competition accuracy low?}
We believe that these problems require multi-hop reasoning, even after summarization, which is still a challenge for language models.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{EMNLP 2022/pos_smaller.png}
\caption{(Top plot) Mean frequency of POS for problems where programs where the generated by both the original and summarized prompt pass all test cases, and (Bottom plot) mean frequency of POS for problems where the summary passes all test cases and the original did not. The blue bar represents the mean of the entire dataset. analyzed only the top 11 most occurring POS. The plot shows that higher number of nouns degrade model performance.}
\label{fig:pos-error-analysis}
\end{figure}
\paragraph{Impact of POS on Accuracy}
In the top plot of Figure \ref{fig:pos-error-analysis}, we observe that frequency of $nouns$ and $propernouns$ for problems that passed all test cases is lower than the entire dataset. In the bottom plot, we observe that the frequency for $nouns$ and $propernouns$ is higher for the original question (which had < 100\% accuracy on the test cases) and lower for the summary (which had 100\% accuracy on the test cases). Thus, we can see that number of nouns degrades performance. We also see in the bottom chart that overuse of punctuation can be detrimental to performance.
From the results in Figure \ref{fig:pos-error-analysis} we see results of nouns affecting performance along with excessive punctuation.
Additional detailed analysis is presented in Appendix \ref{app:analysis}.
\section{Conclusion}
This paper introduces a summarization-based approach for efficient program synthesis. Experimental results show that the proposed approach improves the performance of the Codex model by on average $\sim8\%$ across various levels of programming questions provided by the APPS and $\sim11\%$ on the CodeContests. Further, this paper proposes a meta-dataset consisting of $\sim450$ human-generated basic and expert-level summaries as well as $\sim8k$ synthetically generated summaries by GPT-3 and Studio21; this can be helpful for future research on writing better instructions for the program synthesis. We show that program synthesis models benefit from concise prompts, hence, we believe that less number of high-quality instances are better than more low-quality data instances.
\paragraph{Future Extensions} The decomposition of prompts has been shown to improve accuracy \cite{mishra-etal-2022-reframing, patel2022question}; splitting up the summarization task the resulting summary can potentially result in higher accuracy for the Codex model in future. Additionally, the PEGASUS model could be used in conjunction with other models to perform the detailed algorithm outlined in Appendix \ref{app:instructions}.
\section*{Limitations}
Our summary-based approach shows improved performance on program synthesis models, however, it shows competitive performance on synthetic summaries. We believe that the generation of high-quality summaries can improve performance, hence, designing efficient prompts to improve synthetic summaries can be the scope of further research. Furthermore, human-generated summaries show competitive performance on competition-level problems. These problems require reasoning with multiple logical leaps and knowledge of advanced algorithms and data structures. Hence, exploring new techniques for summarization can be a future research direction. In addition, this work only analyzes the codex model, hence, exploring the effect of summarization on other program synthesis models can be interesting.
|
1,314,259,996,753 | arxiv | \section{Introduction}
Protoplanetary disks form during the gravitational collapse of rotating cloud cores and consist of molecular hydrogen, dust grains of various sizes, and chemical species of multiple complexities. While molecular hydrogen constitutes the main mass reservoir of the disk, dust plays an important role in determining the disk's thermal balance and supplying the planet formation process with building material. Chemical species, in particular water and carbon-based molecules, are the main prerequisites of life and also play a key role in the process of dust growth \citep[e.g.][]{2016ApJ...821...82O}. According to \cite{2013Sci...341..630Q}, the fronts where volatile species condense (snowlines) play an important role in the planet formation process. They define the location where dust coagulation is favoured because of an increase in the stickiness of icy grains.
The position of the snowlines for the most abundant volatiles may also have an influence on the elemental and molecular composition of planetesimals and planets. The position of the respective snowlines for different chemical species determine at what radii which molecules freeze onto dust grains. This influences the chemical composition of icy mantels of dust grains at specific regions in the disk. As the dust grains are the seeds of planet formation, the chemical composition of their icy mantels also influences the chemical composition of forming planetesimals and subsequently, planets. In the case of gas giants, the composition of the atmosphere is also linked to the position of the different snowlines, and therefore to their location on the disk. For instance, the carbon to oxygen ratio, a defining characteristic of the atmosphere, is set by the amount of gas phase oxygen and carbon a gas giant accretes from the disk. The accreted amount of oxygen and carbon-bearing molecules in the gas phase is dependent on the location on the disk where they freeze out (snowline).
Thus, planet formation efficiencies, their core-, and atmosphere compositions are strongly linked to the snowlines of volatile species.
Until recently, most numerical studies of protoplanetary disks were limited to the solar metallicity case. However, the detection of a growing number of exoplanets has revealed that the frequency of the giant planet occurrence depends on the metallicity of the host star \citep{2005A&A...437.1127S,2014prpl.conf..691B}. In addition, studies of protostars and pre-main-sequence stars in the Magellanic Clouds indicate that mass accretion rates are inversely proportional to metallicity \citep{2013ApJ...775...68D}. This finding seems to be corroborated by \citet{2010ApJ...723L.113Y,2016AJ....151...50Y} who inferred shorter lifetimes to low-metallicity disks in the outer regions of the Galaxy. However, this trend could also be explained by higher rates of disk photoevaporation due to a more diluted medium that allows a stronger radiation field \citep{2018ApJ...857...57N}. Another example of these studies is the work performed by \cite{2016ApJ...827..126B}. By observing the chemical abundances of HII regions in 12 nearby dwarf galaxies, the study suggests that there are different chemistry cycles in non-solar metallicity environments.
This evidence suggests that disk evolution and planet formation in protoplanetary disks may proceed differently in low metallicity environments.
The chemical evolution and composition of protoplanetary disks have been extensively studied in the context of solar-metallicity disks \citep[see a review by][]{2014prpl.conf..363P,2013ChRv..113.9016H}.
Among various compounds that are found in protoplanetary disks, volatile species with low sublimation temperatures are best studied because they are central to the process of planet formation.
In this work, we use the radiation thermo-chemical code P{\tiny RO}D{\tiny I}M{\tiny O}\; \citep{2009A&A...501..383W} to study the impact of low metallicity on important molecules (e.g. H$_2$O, CO) on protoplanetary disks.
When we refer to metallicity in this study, we mean the values of element abundances and the gas-to-dust ratio. To simulate a lower metallicity we reduce the element abundance of metals together with the amount of dust. Then we analyze the resulting temperature structure, radiation field, and chemical abundances of various molecules in different metallicities. The formation and destruction reactions of chemical species are also analyzed. Except for ionized molecules, we track the position of the respective snowlines. For a deeper analysis of the chemical abundances, we perform a comparison of the vertical column density between the models with lower metallicity and an artificially scaled-down reference model. Although it is expected that the behaviour of the column densities as a function of the metallicity may not be linear, we use the comparison with scaled-down models to be able to define the regions of the disk where the discrepancy is largest and to quantify the effect of non-linearity. We used an evolved, externally irradiated Class II disk for our models.
In Sect.\ref{method} we describe the gas and dust structure of the disk in the models as well as the chemical network and element abundances used for this work. We also explain how we applied the modifications in the disk models to simulate lower metallicities in the disks. Our results are presented in Sect.\ref{Results}. In Sect.\ref{Limitations of model} we discuss some aspects of the used model and the results and in Sect.\ref{Summary and conclusions} we present our conclusions and describe the following steps for this work.
\section{Method}\label{method}
We employed the radiation thermo-chemical code P{\tiny RO}D{\tiny I}M{\tiny O}\; \citep{2009A&A...501..383W,2010A&A...510A..18K,2011MNRAS.412..711T} to model the chemical composition of young protoplanetary disks with distinct metallicities. P{\tiny RO}D{\tiny I}M{\tiny O}\; includes the radiative transfer of stellar and background radiation along with gas and dust thermal balance to provide the temperature structure and the local radiation field of the disk. The chemical abundances are then calculated using steady-state chemistry employing a chemical reaction network used in \cite{2017A&A...607A..41K}. Various heating and cooling processes are also considered to solve consistently for the gas temperature and the chemistry. The implementation of X-rays and X-ray chemistry is according to \cite{2011A&A...526A.163A}.
\subsection{{Metallicities and initial abundances}}\label{lower metallicity}
\begin{table}[]
\centering
\caption{Elements included in this study, and their abundances in model Z1 on the scale log$n_{\mathrm{H}}$ =12 with $\epsilon$ as the abundances relative to H$_2$ (middle column) and their masses in amu (right column). The values are taken from \cite{2016A&A...586A.103W}}.
\begin{tabular}{c|c|c}
\hline
\hline
element & 12+log $\epsilon $ & $m$ [amu] \\
\hline
H & 12.00 & 1.0079 \\
He & 10.984 & 4.0026 \\
C & 8.14 & 12.011 \\
N & 7.90 & 14.007 \\
O & 8.48 & 15.999 \\
Ne & 7.95 & 20.180 \\
Na & 3.36 & 22.990 \\
Mg & 4.03 & 24.305 \\
Si & 4.24 & 28.086 \\
S & 5.27 & 32.066 \\
Ar & 6.08 & 39.948 \\
Fe & 3.24 & 55.845 \\
\hline
\end{tabular}
\label{table:el_abun}
\end{table}
We use different element abundances and dust-to-gas ratios to study the impact of lower metallicities on the chemical composition of the disk, focusing on water and several most abundant carbon-bearing molecules. We considered three model realizations of protoplanetary disks with three distinct metallicities $Z$.
In all cases the mass of the central star, stellar effective temperature, stellar luminosity and X-ray luminosity are fixed and set equal to $M_\ast=0.7~M_\odot$, $T_\ast=4000$~K, $L_\ast=1.0~L_\odot$ and $L_X=2.6\times10^{-4}~L_\odot$.
We assume a solar metallicity value of Z$_{\odot} = 0.02$ which is in agreement with the value of Z$_{\odot} = 0.0196 \pm 0.0014$ from \cite{2016ApJ...816...13V}. The abundances for our reference model are the abundances used for standard TTauri disk in the P{\tiny RO}D{\tiny I}M{\tiny O}\; code \citep{2016A&A...586A.103W}. {As a result, our reference model Z1 has a metallicity value of Z1$= 0.017$ and has, therefore, a value of Z1$=0.85\, Z_{\odot}$.} The element abundances corresponding to model Z1 are shown in Table \ref{table:el_abun}. The models are named according to their disk metallicity. Model Z01 and Z001 correspond to a metallicity of 0.1 and 0.01 (10\% and 1\% of the reference metallicity Z1 respectively). Thus, the metallicities of our models lie in the $0.85\times10^{-2}- 0.85\ Z_{\odot}$ limits. In order to produce models with different metallicities, we performed two modifications to the initial setup of the models:
\begin{table}[!h]
\caption{Variations of the model used in this study. The columns represent the disk gas mass, metallicity, and dust-to-gas ratio.}
\label{table:metallicities}
\centering
\begin{tabular}{l |c|c|c}
\hline
\hline
Model & $M_{\rm disk}$ [$M_{\sun}$] & Z [$Z_{\sun}$] & dust-to-gas ratio\\
\hline
Z1 & $3.0\times10^{-2}$ & $8.5\times10^{-1}$ & $1.0\times10^{-2}$ \\
Z01 & $3.0\times10^{-2}$ & $8.5\times10^{-2}$ & $1.0\times10^{-3}$\\
Z001& $3.0\times10^{-2}$ & $8.5\times10^{-3}$ & $1.0\times10^{-4}$\\
\hline
\end{tabular}
\end{table}
\begin{enumerate}
\item {We reduced the initial abundances of 13 of the 15 elements (all except H and He) by a factor of 10 and 100 for the gas and the dust.}
\item We reduced the dust-to-gas mass ratio in the input parameters by the same factors (models Z01 and Z001, respectively).
\end{enumerate}
This produced the above-mentioned three models (from herein referred to as Z1, Z01, and Z001) with the reference, 1/10 and 1/100 of reference metallicity, respectively. The disk mass, the metallicity relative to the solar value, and the dust-to-gas ratio are displayed in Table \ref{table:metallicities}.
Further variations of disk and star properties with metallicity will be considered in follow-up studies. The grid resolution is 150 points in the horizontal direction and 100 in the vertical direction. The details of the initial disk's physical structure, dust content, and chemical composition are provided below.
\subsection{The disk model}\label{disk model}
The disk model we use for this study assumes certain simplifications in its physical properties such as the star, the disk geometry, dust settling, and others. The set of assumptions is taken to reproduce the most commonly observed multi-wavelength properties of Class II protoplanetary disks. The basic structural properties of the model are described in detail in ~\cite{2016A&A...586A.103W}. The chemical network used in this study is introduced and explained in \cite{2017A&A...607A..41K}. In section \ref{ssec:gas disk structure}, \ref{ssec:dust disk structure}, and \ref{ssec:chemical model}we provide a brief overview of this model.
\subsubsection{Gas disk structure}\label{ssec:gas disk structure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.48\textwidth]{plots/disk_str.pdf}
\caption{Gas density structure (top) and radial surface density profile (bottom) of the reference model Z1.}
\label{fig:dens_struc_fiducial}
\end{figure}
We use a fixed parameterized density structure for the disk which is based on viscous evolution models. For further details see~\cite{2016A&A...586A.103W}.
The gas density structure is an axisymmetric flared (2D) function of the radius and the vertical height of the disk ($r$ and $h$, respectively) and it is given by
\begin{equation}\label{eq:dens}
\rho(r,z) = \frac{\Sigma(r)}{\sqrt{2\pi}\cdot h(r)}\exp{\left ( -\frac{z^{2}}{2h(r)^{2}}\right )}\quad [\text{g cm}^{-3}].
\end{equation}
Here, $\Sigma(r)$ is the radial surface density profile of the disk. We assume a simple power-law distribution with a tapered outer edge
\begin{equation}\label{eq:sigma}
\Sigma(r) = \Sigma_{0}\left (\frac{r}{R_{\rm in}}\right )^{-\pmb{\lambda}}\exp{\left (-\left (\frac{r}{R_{\rm tap}}\right )^{2-\pmb{\lambda}}\right )}\quad [\text{g cm}^{-2}].
\end{equation}
Here, $R_{\rm in}$ is the inner disk radius and $R_{\rm tap}$ is the characteristic radius. The constant $\Sigma_0$ is determined from the specified disk mass $M_{\mathrm{disk}} = 2\pi\int \Sigma (r)r\,dr$ and Eq.\ref{eq:sigma}. The vertical scale height $h(r)$ is given by a radial power law
\begin{equation}\label{eq:scaleh}
h(r) = H_{100}\left (\frac{r}{100\ \mathrm{au}}\right )^{\beta},
\end{equation}
where $H_{100}$
is the disk scale height at $r = 100$~au and has the value of 10 au and $\beta$ is the flaring power index. Fig.\ref{fig:dens_struc_fiducial} shows the density structure and the radial surface density. The parameters for the disk density structure are listed in Table \ref{table:1}.
\subsubsection{Dust disk structure}\label{ssec:dust disk structure}
For the fiducial model Z1 with the reference metallicity of Z1$=0.85\,Z_{\odot}$ we assumed a dust-to-gas mass ratio of 0.01 in the disk. The respective gaseous element abundances are listed in Table \ref{table:el_abun}and are taken from \cite{2017A&A...607A..41K}.
We take dust growth into account by using a dust size distribution with a minimum and maximum dust grain sizes of $a_{\rm min}=0.05\:\rm{\mu m}$ and $a_{\rm max}=3000\:\rm{\mu m}$, respectively. We use a simple power-law for the dust size distribution $f(a) \propto a^{-p}$ with the canonical value for interstellar grains of $p = 3.5$
\citep{1977ApJ...217..425M}. The dust composition consists of a mixture of $60 \%$ amorphous laboratory silicate, $15 \% $ amorphous carbon, and $25 \%$ vacuum. The vacuum represents the porosity of the dust grains. We use a distribution of hollow spheres with a maximum hollow volume ratio $V_{\mathrm{hollow}}^{\mathrm{max}} = 0.8$. All the relevant dust properties are listed in Table \ref{table:1} and are taken from \cite{2016A&A...586A.103W}.
In this work, we use dust settling using the method of \cite{1995Icar..114..237D}. The dust composition and dust size distribution are constant throughout the entire disk. Furthermore, the dust composition, its maximum and minimum sizes, and the slope of the dust size distribution are assumed to be identical for all considered metallicities. This may be an oversimplification, but the process of dust growth in low-metallicity disks is still poorly known.
\begin{table}
\caption{Main parameters model Z1.}
\label{table:1}
\centering
\begin{tabular}{l|c|c}
\hline
\hline
Quantity & Symbol & Value \\
\hline
stellar mass & $M_{*}$ & 0.7 $M_{\sun}$ \\
stellar effective temp. & $T_{*}$ & 4000 K \\
stellar luminosity & $L_{*}$ & 1.0 $L_{\sun}$ \\
X-ray luminosity & $L_{X}$ & 2.6$\times10^{-4} L_{\sun}$ \\
\hline
disk gas mass & $M_{\mathrm{disk}}$ & 0.03 $M_{\sun}$ \\
disk inner radius & $R_{\mathrm{in}}$ & 0.07 au \\
disk tapering-off radius & $R_{\mathrm{tap}}$ & 100 au \\
column dens. pow. ind. & \pmb{ $\lambda$} & 1.0 \\
reference scale height & $H_{100}$ & 10 au \\
flaring power index & $\beta$ & 1.15 \\
\hline
outer radius & $R_{\mathrm{out}}$ & 600 au \\
\hline
dust-to-gas mass ratio & $\delta$ & 0.01 \\
min. dust particle radius & $a_{\mathrm{min}}$ & 0.05 $\mu$m \\
max. dust particle radius & $a_{\mathrm{max}}$ & 3000 $\mu$m \\
dust size dist. power ind. & $a_{\mathrm{pow}}$ & 3.5 \\
Max. hollow volume ratio & $V_{\mathrm{hollow}}^{\mathrm{max}}$ & $80\%$ \\
dust composition$\ ^{a}$ & $\mathrm{Mg_{0.7} Fe_{0.3} SiO_{3}}$ & $60\%$ \\
(volume fractions) & amorph.carbon & $15\%$\\
& vacuum & $25\%$\\
\hline
cosmic ray H ion. rate & $\zeta_{\mathrm{CR}}$ & $1.7\times10^{-17}\mathrm{s^{-1}}$ \\
strength of interst. FUV & $\chi^{\mathrm{ISM}}$ & $1^{b}$ \\
\hline
distance & d & 140 pc \\
\hline
\end{tabular}
\tablefoot{ $^{(a)}$ Optical constants are from ~\cite{1995A&A...300..503D}. and ~\cite{1996MNRAS.282.1321Z}, BE-sample). $^{(b)} \chi^{ISM}$ is given in units of the Draine field (~\cite{1996ApJ...468..269D}; ~\cite{2009A&A...501..383W}).}
\end{table}
\subsubsection{Chemical model}\label{ssec:chemical model}
After modelling the radial and vertical physical structure of the disk, P{\tiny RO}D{\tiny I}M{\tiny O}\; calculates a detailed continuum and line radiative transfer as well as the heating and cooling balance and the gas and surface chemistry of the disk. The net formation rate of a chemical species $i$ is calculated with the following equation:
\begin{equation}\label{eq:chem}
\begin{split}
\frac{\mathrm{d}n_{i}}{\mathrm{d}t} = \sum_{jkl}R_{jk\to il}(T_{g})\ n_{j}n_{k} + \sum_{jl} \left( R^{\rm ph}_{j\to il}+R^{\rm cr}_{j\to il}\right)n_{j} + ... \\
- n_{i}\left( \sum_{jkl}R_{il\to jk}\ n_{l} + \sum_{jk} \left( R^{\rm ph}_{i\to jk}+R^{\rm cr}_{i\to jk}\right) + ...\right)
\end{split}
\end{equation}
Here, the first row represents all the formation reactions that produce species $i$ and the second row all the destruction reactions that destroy species $i$. $R_{jk\to il}$ indicates the two-body gas-phase reaction rate between two reactants ($j$ and $k$), forming two products ($i$ and $l$). $R^{\mathrm{ph}}_{j\to il}$ stands for the photo-reaction rate. This rate depends on the local strength of the UV radiation field. $R^{\mathrm{cr}}_{j\to il}$ indicates a cosmic ray-induced reaction rate. We only show three kinds of reactions in Eq.\eqref{eq:chem}. The dots indicate that there are other kinds of reactions involved (freeze-out, X-ray chemistry, etc.). The technical details regarding the chemistry are fully described in \cite{2009A&A...501..383W}.
We assume kinetic chemical equilibrium (steady-state chemistry) for this work. This means we have $\frac{\mathrm{d}n_{i}}{\mathrm{d}t} = 0 $ for the left side of Eq.~\eqref{eq:chem} for the steady-state model. We obtain i = 1...N$_{\mathrm{sp}}$ non-linear equations with j = 1...N$_{\mathrm{sp}}$ unknown particle densities $n_{j}$, where $N_{\rm sp}$ is the number of considered species. The chemistry of the disk is computed by solving Eq.~\eqref{eq:chem} using a globally convergent Newton-Raphson method as a steady-state solver \cite{2009A&A...501..383W} The steady-state chemistry approach has also been used and compared with time-dependent chemistry in \cite{2016A&A...586A.103W}.
The chemical network used in this work is based on the one used in \cite{2017A&A...607A..41K}. We use additionally several surface chemistry reactions described in \cite{2020A&A...635A..16T} and Thi et al (in prep). The surface chemistry network~\cite[]{2020A&A...635A..16T} is based in \cite{1993MNRAS.261...83H}.
The total amount of species is 250 and 5790 chemical reactions, respectively.
The modifications to the element abundances done for this work will be explained in detail in the next section.
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.99\hsize}
\includegraphics[width=\hsize]{plots/sdd_sdg.pdf}
\end{subfigure}
\caption{Surface density in the midplane for dust (black) and gas (blue) as a function of radius. The models Z1 (solid line), Z01 (dashed line), and Z001 (dotted line) are shown.}
\label{fig:sdd}
\end{figure}
\subsection{Surface density}
Fig.~\ref{fig:sdd} shows the resulting gas and dust surface density after the mentioned changes in the models have been applied. It shows a decrease in the dust surface density for lower metallicities. Note that we keep the gas surface density and disk mass constant in all our models.
\section{Results}\label{Results}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.99\hsize}
\includegraphics[width=\hsize]{plots/sed.pdf}
\end{subfigure}
\caption{Spectral energy distribution of the different disk models and stellar spectrum. The models Z1 (solid line), Z01 (dashed line), and Z001 (dotted line) are shown. A lower metallicity leads to a decrease in the flux density for the SED.}
\label{fig:sed}
\end{figure}
\begin{figure*}[!ht]
\centering
\begin{subfigure}{1.0\textwidth}
\includegraphics[width=\textwidth]{plots/x_cont_Td_Tg.pdf}
\end{subfigure}
\caption{
Contour plots of the models with different metallicities. The left, middle and right columns of the plot represent the models Z1, Z01, and Z001. Z01 and Z001 with 1/10 and 1/100 of the reference metallicity, respectively. The top, middle, and bottom rows correspond to the dust temperature, gas temperature, and the difference between the gas and dust temperature relative to the gas temperature. The decreasing metallicity leads to an increase in both dust and gas temperature but with the gas temperature increasing stronger (bottom row). The black dashed line represents the surface corresponding to a visual extinction of A$_{v} = 1$.}
\label{fig:cont_t}
\end{figure*}
\subsection{Spectral energy distribution (SED)}
Fig.~\ref{fig:sed} shows the impact of the metallicity in the models on the Spectral Energy Distribution (SED). For the calculation of the SED, a distance to the source of 140 pc is assumed.
It is clear that for reduced dust-to-gas ratio the flux density of the disk also decreases. This is a consequence of the lesser amount of dust that a model with reduced metallicity has and is shown clearly in Fig. \ref{fig:sdd} where the dust surface density is displayed.
At shorter wavelengths, the emission of the disk comes from the part where it is optically thick. Therefore it does not vary significantly if a fraction of dust is removed from it. At $ \lambda < 1\ \mu m$ the SED flux density is dominated by the stellar emission, which remains unchanged for the three models as the star parameters are not being changed. This suggests that low metallicity disks are more difficult to observe at longer wavelengths. At shorter wavelengths (e.g.near-infrared), they are as easy to observe as disks with higher metallicity. The impact that lower metallicity has on the stellar parameters is a subject that will be covered in future studies.
\subsection{Dust and gas temperature}
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/diff_temps.pdf}
\end{subfigure}
\caption{Contour plots that show the difference between the temperatures of the model Z1 and the Z01 model (top panel) and the Z001 model (bottom panel). The r.h.s column shows the difference in dust temperature and the l.h.s column the difference in the gas temperature. The red color represents the regions where the model with lower metallicity has a higher temperature. The white and blue colors represent the regions where both temperatures are similar and where the Z1 model has a higher temperature respectively.}
\label{fig:diff_temps}
\end{figure}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.99\textwidth}
\includegraphics[width=\textwidth]{plots/diff_UVfield.png}
\end{subfigure}
\caption{UV radiation field for the Z1 model (left panel), model Z01 (middle panel), and Z001 (right panel), in units of the Draine field. The dashed contour lines correspond to the tick values in the color bar. The decreasing metallicity enables the UV radiation to get to the midplane at a smaller radius.}
\label{fig:diff_rad}
\end{figure*}
The first result we analyzed in the produced models was the influence of a lower metallicity on the gas and dust temperatures. {For a description of the properties of model Z1 we separate the structure of the disk into three radial zones and three vertical layers following \cite{2013ChRv..113.9016H}. The radial structure consists of three zones:
\begin{itemize}
\item \textit{The inner zone}: it is located closest to the star and extends up to $\approx 1$ au and has a temperature above 100 K.
\item \textit{The middle zone}: it is located between 1 and 30 au. The temperatures in this zone are in the range of 100 (middle and upper parts of the disk) $\geq T \geq 20$ K (midplane).
\item \textit{The outer zone}: it starts at 30 au and extends to the edge of the disk. The temperature in this zone is below 20 K.
\end{itemize}
The vertical structure from bottom to top is as follows:
\begin{itemize}
\item \textit{The midplane}: The coldest vertical layer of the disk. Here the chemical reaction is predominantly gas grain interaction and dust surface chemistry.
\item \textit{The rich molecular layer}: The temperature in this layer lies between 20 and 100 K. It is warm enough to keep some species in the gas phase (e.g. CO). It is also deep enough that the molecules are mostly shielded from photodissociation. The conditions in this layer allow rich molecular chemistry to take place.
\item \textit{The photon-dominated layer}: The hottest layer is characterized by temperatures between 100 and 5000 K. X-rays and UV radiation are the main drivers of the chemistry. Most of the species here are in atomic form and very often ionized.
\end{itemize}}
Fig.~\ref{fig:cont_t} displays dust and gas temperatures and their difference in all the models. The figure shows that the decrease in metallicity produces higher gas and dust temperatures.
{As already mentioned, the top layer of the disk (photon-dominated layer) is characterised by gas temperatures of $\rm{T}_{\rm{gas}} > 1000\,\rm{K}$.}
Both gas and dust increase in temperature as the metallicity decreases, mostly because the radiation is able to penetrate the disk more deeply with less shielding effect by the dust mass in the environment. This also affects the thermal accommodation process between dust and gas and hinders the dust to reach the same temperatures as the gas.
The middle layers (rich molecular layers) show a greater temperature for the gas for a small radius but a greater dust temperature for radii greater than 100 au. In this part of the disk, the collisional exchange of energy is not enough to make both dust and gas temperatures equal. Gas and dust are thermally decoupled and depending on the position in this layer,the dust or gas will be warmer. For a radius smaller than 100 au this region is being directly hit by the stellar high energy radiation and the dominant heating processes are chemical heating followed by heating by the formation of H$_{2}$ on dust and heating by collisional de-excitation of H$_{2}$. These UV-driven heating processes elevate the gas temperature. For radii beyond 100 au, we have regions where the dust is warmer than the gas. These regions of the disk are shielded more effectively from the stellar radiation so that the heating of the gas by direct radiation (mainly X-rays) is not efficient. Additionally, OI and other line cooling processes of the gas dominate this region which leads to dust being warmer than gas. As the metallicity decreases, the region where dust is warmer than gas shrinks because the radiation is not being shielded as strongly and the gas heats up again.
Finally, for all three models with different metallicity, we observe that the gas and dust temperatures are in equilibrium in the midplane of all models. This section of the disk has a relatively low temperature and high density. This means that the energy exchange rate by collisions between dust and gas particles is frequent enough to reach thermalization. Thus, the gas and dust temperature become the same through a process called thermal accommodation \cite[]{2009A&A...501..383W}. As metallicity decreases, the dust density of these deep layers also decreases and the thermal accommodation becomes less efficient. This leads to a shrinking of the region (white area) where dust and gas have the same temperature.
We note that a general trend of increasing temperature with decreasing metallicity is true for the type of models considered in this work, which are often referred to as passive disks. For passive disks, the only heating source is stellar radiation and cosmic rays. In the case of active disks, the hydrodynamical processes provide additional heating via viscous and compressional heating, which operate predominantly in the disk's midplane. The active disk tends to decrease its temperature with a lower metallicity as the heat in a more diluted disk (characterized by lower opacity) can escape more easily and thus cool the disk (\cite{2020A&A...641A..72V}).
In order to have a closer look at the impact of the different metallicity on the gas and dust temperature of the disk, we compare the temperature of the lower metallicity models with model Z1.
Fig.~\ref{fig:diff_temps} shows the influence of a lower metallicity over the gas and dust temperature by displaying 2D plots of the relative temperature difference between the model Z1 and the ones with the reduced metallicity. A decreased metallicity leads to an increase in the gas and dust temperature in the disk. This effect is stronger in the middle layers of the disk but it also takes place in the midplane beyond a radius of 20 au for the Z01 model and a radius of 4 au for the Z001 model.
\subsection{Radiation field }
\begin{figure*}[!h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/3xabun_h2o.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/3xabun_ch4.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/3xabun_co.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/3xabun_co2.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/3xabun_hcn.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/3xabun_cn.png}
\end{subfigure}
\caption{Contour plots of the abundances. From left to right and top to bottom: H$_{2}$O, CH$_{4}$, CO, CO$_{2}$, HCN, and CN are represented relative to the total hydrogen nuclei number density. The dashed contour lines describe the tick values on the colorbar. The left column corresponds to model Z1. The middle column corresponds to the model Z01 and the left column to the model Z001. The upper row represents the gas species, the lower row the ice species, and the bottom row the total abundances. With decreasing metallicity the cavity in the midplane for all gas-phase species shown here shrinks (Z01) and vanishes (Z001). In the case of CH$_{4}, $ a second cavity in the upper layers of the disk grows as metallicity decreases.}
\label{fig:abuncont}
\end{figure*}
In Fig.~\ref{fig:diff_rad} the UV radiation ($91.2\ \rm{nm} <\lambda < 250\ \rm{nm}$) field in Draine units is shown.
The contour lines in the plots show that for the reduced metallicity models the radiation has higher values along the midplane of the disk. For the Z1 model, the radiation value of $10^{-1.5}$ reaches the midplane at a radius greater than 100 au. For the Z01 model, the same radiation value is reached at a radius around 10 and 20 au and for the Z001 model, this value is reached at a radius around 1 au.
This confirms that models with lower metallicity allow the radiation of the star to penetrate deeper regions of the disk. In this case, this is simply the effect of reducing the dust mass. The same effect would be achieved by using unchanged elemental abundances. The effect of a stronger radiation field in the disk on the respective disk chemistry is presented in sections \ref{ssec:Abundances}, \ref{ssec:Detection of the snowline} and \ref{ssec:Vertical column densities of chemical species}.
\subsection{Spatial distribution of abundances} \label{ssec:Abundances}
In this section, we present the impact of metallicity on the abundances of various molecules relative to hydrogen. The ice and gas abundances of H$_2$O, CH$_4$, CO, CO$_2$, HCN, CN, HCO$^+$ and N$_2$H$^+$ are displayed as a function of the radius and the height of the disk in Fig.~\ref{fig:abuncont} and Fig. \ref{fig:abuncont_2}. These species were selected because they are a product of reactions involving the two most abundant molecules in the gas phase (H$_2$ and CO) and because their molecular lines are known tracers of physical and chemical disk properties. For instance, lines of H$_{2}$O, CO in the IR regime can give us information about the temperature in the disk \cite[]{2013ChRv..113.9016H}. CN, HCN lines in the sub-mm and mm regime serve as photochemistry tracers {\cite[]{2013ChRv..113.9016H}}. HCO$^+$ and N$_{2}$H$^+$ lines in the same regime are used to trace the ionization \cite[]{2013ChRv..113.9016H}.
{In the study performed by \cite{2021arXiv211204930H} one of the main findings was that the abundances of CH$_4$ on cooler hot Jupiters can be used to link the composition of a planet's atmosphere to its formation location.
Additionally, even though CO is still considered to be the best gas mass tracer, CO$_{2}$ and H$_2$O can potentially be used as alternative mass tracers as well \cite[]{2017ApJ...849..130M}}. We chose to take a deep look into these specific molecules also because they play an important role as the building blocks for complex carbon-based molecules which are also the building blocks for organic life.
The impact of metallicity on the abundances can be described as follows. On the one hand, as the metallicity decreases the relative abundances of ice and gas species decrease the entire disk. This general decrease is the result of the reduction of the elemental abundances mentioned in section \ref{lower metallicity}. On the other hand, the dust reduction applied to obtain lower metallicities affects specific regions of the disk.
We focus on the impact on specific regions of the disk:
\begin{enumerate}
\item The midplane shows a cavity for the gaseous species that shrinks with decreasing metallicity. The ice species are also affected by this and exhibit a shrinking of their spatial distribution. This is a consequence of the increase of the input energy by a stronger radiation field in the disk as the metallicity decreases. The stronger radiation field desorbs or dissociates ice-phase species and enhances the formation reaction of some gas-phase species. This leads to a depletion of icy species in the location of the midplane cavity and an increase in the abundance of the respective gaseous species.
\item The total abundances displayed in the bottom row in the panes of Fig. \ref{fig:abuncont} also exhibit a cavity in the midplane that shrinks with decreasing metallicity. In the case of H$_2$O, the total abundance does not show a midplane cavity for any metallicity but a general decrease in all the disks is noticeable. Apart from H$_2$O, the behavior of the midplane cavity for the total abundances mirrors the replenishment of gaseous species as the metallicity decreases.
\end{enumerate}
We decided to make a simple comparison of the Z01 and Z001 models with models where we artificially reduced the chemical abundances of the Z1 model by a factor of 10 and 100, respectively. We do this comparison to show the non-linear behavior of the chemistry when reducing the metallicity. We named the models with the artificially reduced column density as scaled-down models.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/abun_hco+_n2h+.png}
\end{subfigure}
\caption{{Same as Fig. \ref{fig:abuncont} but for HCO$^+$ and N$_2$H$^+$. We only show the gaseous abundances because HCO$^+$ and N$_2$H$^+$ do not freeze-out in our models.}}
\label{fig:abuncont_2}
\end{figure}
\subsection{Vertical column densities}\label{ssec:Vertical column densities of chemical species}
In this section a detailed analysis of the vertical column density of the above-mentioned species is provided along with the chemical reactions responsible for the formation and destruction of the molecules studied in this work. We analyze the behavior of the total vertical column density of the already mentioned species. The ice species are followed by a $\#$ symbol.
Figs. \ref{fig:column_H2O} - \ref{fig:column_HCO+_N2H+} show the total vertical column density of the mentioned chemical species. We again made a simple comparison of the Z01 and Z001 models with models where we artificially reduced the vertical column density of the Z1 model by a factor of 10 and 100, respectively. The red shaded area in the figures highlights the cases where the column density is higher than the simple scaled-down of the Z1 model abundances and the green shaded area where it is lower.
For each molecule, we describe the general trends of the column density. We also give in Table \ref{table:form_destr} the respective changes in the formation and destruction reactions for the molecules for different metallicities. A more detailed display of the behavior of the molecules in a frozen and gaseous state is given in Appendix \ref{appendix:a}. The relevant reactions are given in a detailed form in Table \ref{tab:react_eq} in the Appendix as well.
For completeness, we also show the impact of the dust reduction and the elemental reduction of the metals separately in Appendix \ref{appendix:b}.
\subsubsection{H$_{2}$O}\label{cd_h2o}
\begin{figure}[!h]
\centering
\includegraphics[width=0.99\hsize]{plots/columnH2O.pdf}
\caption{Total vertical column density of H$_{2}$O for the models. The top row shows models Z1, Z01, and Z001 (solid, dashed, and dotted lines respectively). The middle row shows the difference between the Z01 model (black dashed line) and the values by reducing the vertical column density of Z1 manually by a factor of 10 (red dashed line). The red and green areas show the cases where the vertical column density is higher or lower than the simple scaled-down values. The bottom row shows the same plots as the middle row but for the Z001 model and the Z1 model reduced by a factor of 100.}
\label{fig:column_H2O}
\end{figure}
The top panel of Fig. \ref{fig:column_H2O} shows a total column density decrease with radius for all three models. However, there are small local drops that are located in different radii for each model. The total column densities (gas + ice) shown in the middle and bottom panels have equal or slightly lower values than the scaled-down values. The reason for the lower than {the scaled-down values} lies in the stronger radiation field and efficient photodissociation of $\mathrm{H_2O\#}$ (eq. \ref{eq:d_H2O_Z01_Z001}) in the deeper parts of the disk. It is the consequence of the radiation of the star being able to enter deeper regions of the disk as the metallicity of the model decreases. For Z1, the main destruction reaction for $\mathrm{H_2O\#}$ in the disk region of underabundance is thermal desorption, which does not affect the total column density. Table~\ref{table:form_destr} shows that for lower metallicities (Z01 and Z001) photodissociation takes over as the main destruction reaction because the radiation field is stronger in that part of the disk. $\mathrm{H_2O\#}$ is therefore not only being sublimated by the rising temperatures but is also being dissociated by photons that are able to reach the disk midplane (\cite{2006JChPh.124f4715A}). Particularly for Z001 the small drop beyond 2 au starts at the same radius where a great part of the UV radiation is not being strongly shielded and reaches the midplane for that model (see Fig. \ref{fig:diff_rad}). The resulting effect is a more efficient depletion of all H$_2$O as the metallicity decreases. This is shown clearly in Fig. \ref{fig:integ_ratio} where the total ratios $\mathcal{R}_{\mathrm{H_2O}} < 1$.
\begin{comment}
The destruction reaction takes place when $\mathrm{H_2O\#}$ on a dust grain is dissociated by a photon and $\mathrm{OH\#}$ and H$\#$ are formed
\begin{equation}\label{eq:d_H2O_Z01_Z001}
\begin{split}
\mathrm{H_{2}O\# + photon \rightarrow OH\# + H\#}.
\end{split}
\end{equation}
\end{comment}
\subsubsection{CH$_4$}\label{cd_ch4}
The three models show a decrease that starts at 1 au and continue until a radius of 20 au, where a strong {increase follows (top row of Fig. \ref{fig:column_CH4}). The decrease from 1 au to 20 au is due to the cavity for the total CH$_4$ that is visible in Fig. \ref{fig:abuncont}}. The middle and bottom rows show an underabundance inside $0.4$~au for the lower metallicity models. This is mainly due to the enhanced depletion by photodissociation of gaseous CH$_{4}$ caused by a stronger radiation field.
It is also shown that roughly between 0.8 and 8 au the lower metallicity models have higher values than scaled-down values. The overabundance of Z01 and Z001 between 0.8 and 8 au is the effect of the larger formation rates that the lower metallicity models have near the midplane and around a radius of 1.5 au. For instance, the CH$_4$ formation rates in that region for the Z1, Z01 and Z001 model are 3.44 x$10^{-13}$, 3.84 x$10^{-11}$ and 1.21 x$10^{-7}$ cm$^{-3}$s$^{-1}$ respectively. The main destruction rates of CH$_{4}$ also increase with decreasing metallicity but not enough to counteract the enhanced formation of CH$_{4}$ (see Table~\ref{table:form_destr}).
Beyond 20 au the three models follow the scaling-down by one and two orders of magnitude very closely and have either equal or slightly lower values than the scaled-down values.
The {total molecular amount} (Fig. \ref{fig:integ_ratio}) shows that the Z01 and Z001 models have a lower than 1 ratio in comparison to the scaled-down values. Thus, the overabundance is present in regions of the disk where the total abundance of CH$_4$ is very low and it does not compensate for the underabundance in the parts of the disk with a radius greater than 25 au.
\begin{figure}[!h]
\centering
\includegraphics[width=0.99\hsize]{plots/columnCH4.pdf}
\caption{Same as Fig. \ref{fig:column_H2O} but for CH$_{4}$.}
\label{fig:column_CH4}
\end{figure}
\subsubsection{CO}\label{cd_co}
\begin{figure*}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{plots/columnCO.pdf}
\end{subfigure}\hspace{0.02\textwidth}
\begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{plots/columnCO2.pdf}
\end{subfigure}
\par\bigskip
\begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{plots/columnHCN.pdf}
\end{subfigure}\hspace{0.02\textwidth}
\begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{plots/columnCN.pdf}
\end{subfigure}
\caption{Same as Fig, \ref{fig:column_H2O} but for {CO}, CO$_{2}$, HCN and CN.}
\label{fig:column_CO_CO_CH4_HCN}
\end{figure*}
CO shows a very similar trend to H$_2$O with the very important exception that the magnitudes of the differences to the scaled-down values are greater than for H$_2$O. For all three models, the total column density decreases with increasing radius. Z1 model has a strong bump for a radius between 0.1 and 1 au. This feature is absent or much less pronounced in the case of lower metallicity models. The reason for the existence of the mentioned bump is the different formation reactions of CO for the three different models. In that part of the disk, the reactant for the most important formation reaction of CO is gaseous H$_2$O. This reactant has an abundance that is orders of magnitude lower for models Z01 and Z001, so it is to be expected that CO will be much less abundant in that region for models {Z01} and Z001. This will be explained in more detail below in section \ref{det_bump}. As a result, CO is less abundant in the lower metallicity models than the scaled-down values in the vicinity of the bump.
Apart from that bump, models Z01 and Z001 show a CO overabundance along the complete disk. This overabundance increases as the metallicity decreases. Table \ref{table:form_destr} shows that the formation reaction for CO in that region changes with decreasing metallicity and also that the respective reaction rate increases. For example, the most important formation reaction of CO in the Z1 model is dissociative recombination of H$_2$CO$^+$ while for the Z01 and Z001 models the formation reaction is the neutral-neutral reaction of H and HCO and photodissociation of CO$_2$, respectively. We can see in Fig. \ref{fig:integ_ratio} that CO is overabundant relative to the scaled-down values as a result of the enhanced formation reactions. Table \ref{table:integ_ratios} confirms this overabundance with ratios for the gaseous state of 4.78 and 16.92 for models Z01 and Z001 respectively.
This is an important finding as CO is often used for observations. The resulting column density seen in Fig. \ref{fig:column_CO_CO_CH4_HCN} for models Z01 and Z001 does not show differences of order of magnitude compared to model Z1. Thus, if observations were performed on a sub-solar metallicity region, it would be incorrect to expect lower CO abundances than in the solar metallicity case. This assumption would then result in an underestimation of CO column density by an order of magnitude.
\subsubsection{CO$_2$}\label{cd_co2}
A similar bump to the CO case for small radii is also present here for model Z1. This bump is a consequence of the bump in the CO case. The origin of this bump will also be explained in more detail in Section \ref{det_bump}. Apart from that, all three models show a drop that reaches its minimum value at around 2 au. This drop is consistent with the location of the midplane cavity for this molecule. From 2 au on all models show an increase of the column density with radius until it starts to drop again towards the outer disk radius. All models show a similar increase that drops again at the end of the disk's radius.
This decrease and rise of the total column density around 2 au follow the depletion of gas phase CO$_2$ due to ion-neutral reactions (model Z1 and Z01) and photodissociation (model Z001). On the other hand, CO$_2\#$ is also being depleted in that part of the disk due to photodissociation (model Z1) and photodesorption (models Z01 and Z001). It is only beyond that radius where CO$_2\#$ builds up and is responsible for the increase of the total column density.
The comparison between the Z01 and Z001 case (middle and bottom panel for CO$_{2}$ in Fig. \ref{fig:column_CO_CO_CH4_HCN} shows an underabundance in the region where the bump is placed.
Between the already mentioned bump and the outer parts of the disk, CO$_{2}$ shows an overabundance. This overabundance present in the Z01 model decreases in size in model Z001. In the part of the disk with overabundance, we see again a change in the formation reaction type and in the reaction rate as metallicity decreases. For instance, the formation reaction changes from thermal desorption to photon desorption to the neutral-neutral reaction of OH and CO for models Z1, Z01, and Z001 respectively.
There is an underabundance in the outer parts of the disk that becomes more noticeable and extends into the disk with decreasing metallicity. This increase in size and magnitude in the underabundance as the metallicity decreases leads to the resulting ratios in Fig. \ref{fig:integ_ratio} for CO$_2$ in the sense that while the Z01 model shows a net overabundance, the Z001 model has a net underabundance.
\subsubsection{HCN}\label{cd_hcn}
The trend shown in the top panel for HCN {in Fig. \ref{fig:column_CO_CO_CH4_HCN}} exhibits a bump in the Z1 model for small radii. Apart from that, all models have a depression that reaches a minimum value at around 6 au and then show an increase beyond that radius. The values for a radius smaller than 2 au are caused by the amount of gaseous HCN while the values beyond 9 au by the amount of HCN\#.
In the middle and bottom panel, models Z01 and Z001 show an overabundance for different parts of the disk (for very small radii, for a radius between 0.9 and 9, and for a region around 35 au). The region with the most overabundance is located around 1.8 au and Table \ref{table:form_destr} confirms the rising of the respective reaction rates as the metallicity is reduced. The formation reaction also changes. For models Z1 and Z01, it is the dissociative recombination of HCNH$^+$ and for model Z001 it is the neutral-neutral reaction of CN and C$_2$H$_2$. The outer regions of the disk show an underabundance that increases for the Z001 model. Then Z001 model also shows an underabundance between 9 and 20 au. This underabundance is the result of the depletion of HCN\# due to the greater values for the respective destruction reaction as metallicity decreases (green shaded rows for HCN\# in Table \ref{table:form_destr}. Although the underabundance seems to increase as the metallicity is reduced, we see in Fig. \ref{fig:integ_ratio} that the ratios to the scaled-down values are still greater than 1 for the complete disk.
\begin{table*}[!ht]
\centering
\caption{Formation and destruction rates for the different species at different locations in the midplane of the disk for each model. The ice species are followed by a $\#$. The red and green shaded cells represent the reactions responsible for a relative overabundance (red) or underabundance (green).
The reaction rate is the reaction coefficient multiplied by the {number} densities of the reactants. The reaction type is given in the brackets under the reaction rate with the corresponding reaction equation number. The equations are given in appendix \ref{appendix:a}. The location was determined by choosing the point in the midplane where the differences between the column density of models Z01 and Z001 and the respective scaled-down column densities are the largest.}
\begin{adjustbox}{width=1\textwidth}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
\hline
species & location (x, z) {[}au{]} &
\multicolumn{3}{c|}{main formation reaction rate [cm$^{-3}$ s$^{-1}$]} & \multicolumn{3}{c}{main destruction reaction rate [cm$^{-3}$ s$^{-1}$]} \\ & &Z1 & Z01 & Z001 & Z1 & Z01 & Z001 \\ \hline
H2O\# & (2.70, 0)& \begin{tabular}[c]{@{}c@{}}4.35 x$10^{-14}$\\ (freeze out)\end{tabular} & \begin{tabular}[c]{@{}c@{}}5.19 x$10^{-14}$\\ (surface reaction)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}4.9 x$10^{-3}$\\ (surface reaction)\end{tabular} &
\cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}4.37 x$10^{-15}$\\ (thermal desorption)\end{tabular} & \cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}4.79 x$10^{-14}$\\ (thermal desorption)\end{tabular} &
\cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}4.92 x$10^{-3}$\\ (photodiss.,Eq.\ref{eq:d_H2O_Z01_Z001})\end{tabular}
\\ \hline
CH$_{4}$& (1.54, 0) & \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}3.44 x$10^{-13}$\\ (diss. recombination, Eq.\ref{eq:f_CH4_Z01})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}3.84 x$10^{-11}$\\ (diss. recombination, Eq.\ref{eq:f_CH4_Z01})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.21 x$10^{-7}$\\ (ion-neutral, Eq.\ref{eq:f_CH4_Z001})\end{tabular} & \begin{tabular}[c]{@{}c@{}}2.43 x$10^{-13}$\\ (ion-neutral)\end{tabular} & \begin{tabular}[c]{@{}c@{}}2.44 x$10^{-11}$\\ (ion-neutral)\end{tabular} & \begin{tabular}[c]{@{}c@{}}9.98 x$10^{-8}$\\ (photodiss.)\end{tabular} \\ \hline
CO & (6.4, 0) & \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}7.41 x$10^{-25}$\\ (diss. recombination, Eq.\ref{eq:f_CO_Z1})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}3.74 x$10^{-14}$\\ (neutral-neutral, Eq.\ref{eq:f_CO_Z01})\end{tabular}&
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.31 x$10^{-5}$\\ (photodiss, Eq.\ref{eq:f_CO_Z001})\end{tabular} & \begin{tabular}[c]{@{}c@{}}9.47 x$10^{-24}$\\ (ion-neutral)\end{tabular} & \begin{tabular}[c]{@{}c@{}}8.31 x$10^{-13}$\\ (ion-neutral)\end{tabular} & \begin{tabular}[c]{@{}c@{}}1.37 x$10^{-5}$\\ (neutral-neutral)\end{tabular} \\ \hline
CO\# & (27.6, 0)& \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}4.36 x$10^{-21}$\\ (surface reaction, Eq.\ref{eq:f_COi_Z1} )\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}9.56 x$10^{-9}$\\ (freeze out)\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}3.76 x$10^{-7}$\\ (freeze out)\end{tabular} & \begin{tabular}[c]{@{}c@{}}4.40 x$10^{-21}$\\ (surface reaction)\end{tabular} & \begin{tabular}[c]{@{}c@{}}1.82 x$10^{-8}$\\ (photodiss.)\end{tabular} & \begin{tabular}[c]{@{}c@{}}3.90 x$10^{-7}$\\ (photodiss.)\end{tabular} \\ \hline
CO2 & (9.55, 0)& \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}3.27 x$10^{-30}$\\ (thermal desorption)\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}2.74 x$10^{-14}$\\ (photon desorption)\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.32 x$10^{-6}$\\(neutral-neutral,Eq.\ref{eq:f_CO2_Z001})\end{tabular} &
\begin{tabular}[c]{@{}c@{}}2.72 x$10^{-30}$\\ (ion-neutral)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}2.77 x$10^{-14}$\\ (ion-neutral)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}1.36 x$10^{-6}$\\ (photodiss.)\end{tabular} \\ \hline
CO2\# & (9.55, 0) & \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}6.79 x$10^{-31}$\\ (freeze out)\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}7.20 x$10^{-16}$\\ (freeze out)\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}2.86 x$10^{-8}$\\ (freeze out)\end{tabular} & \begin{tabular}[c]{@{}c@{}}2.51 x$10^{-29}$\\ (photodiss.)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}2.74 x$10^{-14}$\\ (photodesorption)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}3.10 x$10^{-8}$\\ (photodesorption)\end{tabular} \\ \hline
HCN & (1.81, 0) & \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.89 x$10^{-12}$\\ (diss. recombination,Eq.\ref{eq:f_HCN_Z1_Z01})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}3.47 x$10^{-11}$\\ (diss. recombination,Eq.\ref{eq:f_HCN_Z1_Z01})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}2.49 x$10^{-4}$\\ (neutral-neutral,Eq.\ref{eq:f_HCN_Z001})\end{tabular} &
\begin{tabular}[c]{@{}c@{}}2.64 x$10^{-12}$\\ (ion-neutral)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}4.93 x$10^{-11}$\\ (ion-neutral)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}1.89 x$10^{-4}$\\ (photodiss.)\end{tabular} \\ \hline
HCN\# & (10.20, 0) & \begin{tabular}[c]{@{}c@{}}1.94 x$10^{-22}$\\ (surface reaction)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}6.77 x$10^{-12}$\\ (surface reaction)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}9.34 x$10^{-8}$\\ (surface reaction)\end{tabular} &
\cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}2.41 x$10^{-22}$\\ (thermal desorption)\end{tabular} & \cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}6.76 x$10^{-12}$\\ (photodiss., Eq.\ref{eq:d_HCN_Z001})\end{tabular} &
\cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}9.41 x$10^{-8}$\\ (photodiss., Eq.\ref{eq:d_HCN_Z001})\end{tabular} \\ \hline
CN & (2.53, 0)& \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}3.46 x$10^{-20}$\\ (diss. recombination, Eq.\ref{eq:f_CN_Z1_Z01})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.45 x$10^{-12}$\\ (diss. recombination, Eq.\ref{eq:f_CN_Z1_Z01})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.16 x$10^{-3}$\\ (photodiss., Eq.\ref{eq:f_CN_Z001})\end{tabular} & \begin{tabular}[c]{@{}c@{}}2.61 x$10^{-20}$\\ (neutral-neutral)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}1.10 x$10^{-12}$\\ (neutral-neutral)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}8.28 x$10^{-4}$\\ (neutral-neutraL)\end{tabular} \\ \hline
CN\# & (95.36, 0)& \cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}9.77 x$10^{-14}$\\ (photodiss., Eq.\ref{eq:d_HCN_Z001})\end{tabular} &
\cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}1.13 x$10^{-17}$\\ (photodiss., Eq.\ref{eq:d_HCN_Z001})\end{tabular} &
\cellcolor[rgb]{0.592,1,0.592} \begin{tabular}[c]{@{}c@{}}7.80 x$10^{-21}$\\ (photodiss., Eq.\ref{eq:d_HCN_Z001})\end{tabular} &
\begin{tabular}[c]{@{}c@{}}9.31 x$10^{-14}$\\ (surface reaction)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}7.48 x$10^{-18}$\\ (surface reaction)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}4.64 x$10^{-21}$\\ (surface reaction)\end{tabular} \\ \hline
HCO+ & (50.23, 0)& \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}5.10 x$10^{-19}$\\ (ion-neutral, Eq.\ref{eq:f_HCO+_Z1})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.70 x$10^{-8}$\\ (ion-neutral, Eq.\ref{eq:f_HCO+_Z01_Z001})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}5.32 x$10^{-8}$\\ (ion-neutral, Eq.\ref{eq:f_HCO+_Z01_Z001})\end{tabular} &
\begin{tabular}[c]{@{}c@{}}1.47 x$10^{-18}$\\ (diss. recombination)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}9.21 x$10^{-9}$\\ (diss. recombination)\end{tabular} &
\begin{tabular}[c]{@{}c@{}}4.66 x$10^{-8}$\\ (diss. recombination)\end{tabular} \\ \hline
N2H+ & (10.36, 0)& \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}5.21 x$10^{-47}$\\ (ion-neutral, Eq.\ref{eq:f_N2H+})\end{tabular} &
\cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}1.68 x$10^{-23}$\\ (ion-neutral, Eq.\ref{eq:f_N2H+})\end{tabular} & \cellcolor[rgb]{1,0.675,0.675} \begin{tabular}[c]{@{}c@{}}7.52 x$10^{-7}$\\ (ion-neutral, Eq.\ref{eq:f_N2H+})\end{tabular} &
\begin{tabular}[c]{@{}c@{}}4.85 x$10^{-47}$\\ (diss. recombination)\end{tabular}&
\begin{tabular}[c]{@{}c@{}}1.56 x$10^{-23}$\\ (diss. recombination)\end{tabular}&
\begin{tabular}[c]{@{}c@{}}6.93 x$10^{-7}$\\ (ion-neutral)\end{tabular} \\ \hline
\end{tabular}
\end{adjustbox}
\label{table:form_destr}
\end{table*}
\subsubsection{CN}\label{cd_cn}
In the top panel that displays the total column density for CN {in Fig. \ref{fig:column_CO_CO_CH4_HCN} we see that none of the models have a bump for a small radius. However, an oscillating behavior for the three models (Z1, Z01, and Z001) is noticeable.} This is because most CN at a small radius is located in a very thin layer of the disk (see Fig.~\ref{fig:abuncont}). The disk model is not able to resolve this thin layer properly and the result is the oscillating behavior of the total column density. We note two important aspects regarding this oscillatory behavior: First, the values for the CN column density are orders of magnitude less than the other molecules presented in this study. Second, we produced models with higher resolution to check how this would influence the oscillations. We found out that the produced column densities of CN are almost the same as in Fig. \ref{fig:column_CO_CO_CH4_HCN}, but with weaker oscillations.
Beyond 1 au CN behaves in a unique way compared to the other molecules. For a radius between 1 and 5 au the total column density increases by a couple of orders of magnitude as the metallicity decreases. This is the opposite impact that metallicity has on the total column density of the other molecules. It mirrors the clear increase of the CN abundance in the midplane (between a radius of 1 and 5 au) as metallicity decreases (see bottom row of the CN panels in Fig. \ref{fig:abuncont}). For the rest of the disk models, Z1 and Z01 have similar values, and model Z001 has lower values with a clear drop at 90 au.
Apart from the drop at 90 au for the Z001 model, both models (Z01 and Z001) exhibit an overabundance throughout the complete disk. The region around 2.5 au shows the greatest overabundance and the reaction rates in Table \ref{table:form_destr} show a clear increase as the metallicity decreases. Models Z1 and Z01 have dissociative recombination of HCNH$^+$ as the main formation reaction and for model Z001 the respective reaction is photodissociation of the HCN molecule (eq. \ref{eq:f_CN_Z001}). The clear dominant overabundance along the disk confirms the ratios for CN shown in Fig. \ref{fig:integ_ratio}.
\subsubsection{Bump at small radii for CO, CO$_2$, and HCN}\label{det_bump}
The already mentioned bump at small radii for CO has the following origin:
In the location of the bump, the rate of the most important formation reaction of CO for model Z1 is greater than the main formation reaction for the Z01 and Z001 models by more than three orders of magnitude. This is mainly the result of the differences in the abundances of gaseous H$_{2}$O between the different models. The main formation reaction of CO in that disk region is the ion-neutral reaction of H$_2$O and HCO$^+$ (Eq. \ref{eq:f_COb_Z1}). Gas-phase H$_{2}$O is the reactant of the formation reaction for CO and is much more abundant in the Z1 model in that particular region. Therefore, the formation of CO with the mentioned ion-neutral reaction is much more efficient for the Z1 model relative to models Z01 and Z001. Similarly, in the case of CO$_{2}$, the bump is the result of the greater CO abundance of the Z1 model. CO is the reactant for the main formation reaction of CO$_2$. Therefore a greater CO abundance leads to a greater CO$_2$ abundance.
For HCN all three models have the same main formation reaction. It is the ion-neutral reaction of NH$_{3}$ with HCNH$^{+}$ producing HCN and NH$_{4}^{+}$. Although all the models have the same main formation reaction, the bump is the result of the different reaction rates each model has.
\subsubsection{HCO$^{+}$ \emph{and} N$_{2}$H$^{+}$}\label{cd_hco+_n2h+}
\begin{figure}[!h]
\centering
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\textwidth]{plots/column_HCO+.pdf}
\end{subfigure
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\textwidth]{plots/column_N2H+.pdf}
\end{subfigure
\caption{Same as Fig. \ref{fig:column_H2O} but for HCO$^{+}$ and N$ _{2}$H$^+$.}
\label{fig:column_HCO+_N2H+}
\end{figure}
HCO$^{+}$ and N$_{2}$H$^{+}$ were put in this section together because they are electrically charged and are only present in the gas phase. The column density of the two ions N$_2$H$^+$ and HCO$^+$ is displayed in Fig.~\ref{fig:column_HCO+_N2H+}. The column density of N$_2$H$^+$ present an oscillating behavior similar to CN for a small radius. {As with CN, higher resolution models show that the oscillations become weaker as the resolution of the model increases.} This effect is also the result of a very thin layer in the inner parts of the disk which cannot be properly resolved in the model. In the top panel, HCO$^+$ shows similar values for the three models until a radius of 2 au is reached. Beyond that radius, models Z01 and Z001 have clearly higher total column densities than model Z1. This is expected because of the stronger radiation field that results in a more efficient ionization in the lower metallicity models. N$_{2}$H$^{+}$ shows a strong bump between 1 and 5 au and a following increase for models Z1 and Z01. Model Z001 does not have these strong features but it increases slightly as the radius increases. HCO$^{+}$ and N$_{2}$H$^{+}$ are more abundant than the scaled-down values for all radii. In fact, the overabundance is almost everywhere a couple of orders of magnitudes in both lower metallicity models (Z01 and Z001). Table \ref{table:form_destr} shows that for HCO$^{+}$ the respective ion-neutral formation reaction changes from H$^+$ + H$_2$CO for model Z1 to H$^+_3$ + CO for models Z01 and Z001. For N$_2$H$^+$ the ion-neutral formation reaction H$^+_3$ + N$_2$ does not change with metallicity but the rate of the reaction has a dramatic rise. The destruction reaction for the two ions is dissociative recombination (Table \ref{table:form_destr}). The electrons needed for this reaction usually come from the ionization of metals. However, as metallicity decreases the number of available metals is also reduced. The reduction of metals has a stronger impact on the destruction rate of HCO$^{+}$ and N$_{2}$H$^{+}$ than a stronger radiation field. {This is shown in Appendix \ref{appendix:b} where the reduction of the metals and the dust reduction are handled separately.} Eventually, this leads to an overabundance of the two ions. This general overabundance of both ions is displayed in a very clear way in the respective ratios shown in Fig. \ref{fig:integ_ratio}.
All the chemical reactions mentioned in this section are shown in detail in Table \ref{tab:react_eq} in Appendix \ref{appendix:a}.
\subsubsection{General trend for column densities and quantitative analysis}
In most cases, the decreased metallicity leads to a decrease in the column density in comparison to the Z1 model. However, the actual reduction does not follow the simple reduction of the abundances by the factors 10 and 100.
It is important to reiterate that the over- or underabundances of the species are not only the result of different rates of adsorption and desorption of each species. The different trends of the column density for the lower metallicity models are also the result of surface reactions on the dust grains, photodissociation in the gaseous and ice phase, and recombination processes. The different values for the formation and destruction rates of the species are triggered by the higher temperatures and the stronger radiation field as the metallicity decreases.
{In Table \ref{table:integ_ratios} we summarize the quantitative analysis of this comparison for each of the considered species. {For each species, the vertical column density $N_{sp}$ is defined as
\begin{equation}\label{vcoldens}
N_{sp}(r) =\int_{0}^{z_{max}(r)} n_{sp}(r,z) \,dz
\end{equation}
with $z$ as the disk height, $z_{max}$ as the maximal disk height, $r$ as the radius, and $n_{sp}$ as the number density of a chemical species at each point of the disk.}
For each species, we integrated the vertical column density $N_{sp}$ over the radius $r$ of models Z01 and Z001 and their scaled-down counterparts.
We used a simple integration from the inner radius $R_{in}$ to the outer radius $R_{out}$ of the disk for this step (Eq. \ref{integ}).}
{\begin{equation}\label{integ}
n_{tot}=2\pi \int_{R_{in}}^{R_{out}} N_{sp}r \,dr
\end{equation}
\begin{equation}\label{ratio}
\mathcal{R}= n_{tot}/n_{tot(sd)}
\end{equation}}
Next, we calculated the ratio of {the total molecular amount} of models Z01 and Z001 (n$_{tot}$) relative to the scaled-down results (n$_{tot(sd)}$). The ratio is calculated with Eq. \ref{ratio}. These results are also presented in Fig.~\ref{fig:integ_ratio}.
The ratio is higher than 1 in almost all gaseous species. It is particularly high for the ion-species HCO$^{+}$ and N$_2$H$^{+}$. The exception of gaseous CO$_2$ is due to the fact that the underabundance in the disk zone closest to the star is also the zone where most part of CO$_2$ is stored. However, all gaseous species show that the value for the ratio has a positive trend as the metallicity decreases.
\begin{figure}[!h]
\centering
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/integ_ratios.png}
\end{subfigure}
\caption{Ratio $\mathcal{R}$ (see Eq. \ref{ratio}) of the total molecular amount of the scaled-down Z1 and the low metallicity models Z01 and Z001 for each species. The red, blue, and black circles correspond to the ratios of the gaseous, ice, and total (gas + ice) species respectively. The dashed horizontal line shows the value for the ratio equal to 1.}
\label{fig:integ_ratio}
\end{figure}
\begin{table}[!h]
\begin{center}
\caption{Comparison of the total particle number of the considered gaseous species (first column) for the model abundances and scaled-down abundances. The second and third column is the ratio relative to the scaled-down of the gaseous species ($\mathcal{R}_{\rm{gas}}$) for the Z01 and the Z001 model respectively. The fourth and fifth columns correspond to the ice species ($\mathcal{R}_{\rm{ice(\#)}}$) and the sixth and seventh columns represent the total (gas + ice) values ($\mathcal{R}_{\rm{tot}}$).}
\label{table:integ_ratios}
\begin{tabularx}{0.49\textwidth}{c|*{2}{Y}|*{2}{Y}|*{2}{Y}}
\hline
species &
\multicolumn{2}{c|}{$\mathcal{R}$$_{\rm{gas}}$} &
\multicolumn{2}{c|}{$\mathcal{R}$$_{\rm{ice}(\#)}$} &
\multicolumn{2}{c}{$\mathcal{R}$$_{\rm{tot}}$}\\
& Z01 & Z001 & Z01 & Z001 & Z01 & Z001 \\
\hline
\hline
H$_2$O & 1.63 & 2.14 & 0.75 & 0.61 & 0.76 & 0.61\\
CH$_4$ & 3.67 & 10.42 & 0.77 & 0.56 & 0.77 & 0.56\\
CO & 4.78 & 16.92 & 3.05 & 1.96 & 4.34 & 13.06 \\
CO$_2$ & 0.27 & 0.34 & 3.25 & 0.62 & 3.24 & 0.62 \\
HCN & 2.61 & 20.97 & 1.71 & 1.28 & 1.71 & 1.30\\
CN & 8.20 & 32.57 & 1.94 & 0.45 & 4.32 & 12.65 \\
\hline
HCO$^{+}$ & 11.39 & 71.83 & & \\
N$_2$H$^{+}$ & 10.92 & 29.51 & & \\
\hline
\end{tabularx}
\end{center}
\end{table}
The ice species show a smaller than 1 ratio for H$_2$O$\#$ and CH$_4\#$ which decreases as the metallicity decreases. The rest of the ice species show a higher than 1 ratio for the Z01 model which decreases for the lowest metallicity model Z001. CO$_2\#$ and CN$\#$ even exhibit a smaller than 1 ratio for the Z001 model. All ice species show a negative trend as the metallicity decreases. This indicates that although there is still chemical production of certain ice species, the destruction due to photodesorption or thermal desorption of these species becomes more dominant with a decreasing metallicity. The total (gas + ice) values show a smaller than 1 ratio for H$_2$O and CH$_4$ with a negative trend with decreasing metallicity. CO and CN show a greater than 1 ratio with about three times higher ratios for the lowest metallicity Z001. CO$_2$ and HCN have greater than 1 ratios for the Z01 model but exhibit a negative trend as the metallicity decreases. CO$_2$ even shows a smaller than 1 ratio for the Z001 model. In summary, the strongest deviation from the scaled-down values for gaseous species is found in the ions HCO$^+$ and N$_2$H$^+$, followed by CN, HCN, CO, and CH$_4$. H$_2$O shows a slight deviation and CO$_2$ even shows smaller particle numbers than the scaled-down values. For the frozen species CN, CO, and CO$_2$ exhibit the strongest deviation from scaled-down values. Regarding the total (gas + ice) numbers, CO and CN are the molecules that have the strongest deviation.
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{0.8\textwidth}
\includegraphics[width=\textwidth]{plots/snowlines_ratio.png}
\end{subfigure}
\caption{Gas-to-ice ratio of the molecules. The left, middle and right columns represent the Z1, Z01, and Z001 models respectively. Red and blue regions represent the regions where gas and ice are dominant respectively. The black dashed line shows where the ice and gas abundances are equal (snowline). The locations of the snowlines at the midplane are summarized in Fig.\ref{fig:snowline_loc}.}
\label{fig:snowline}
\end{figure*}
\begin{figure}[!h]
\centering
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{plots/snowlines_loc.pdf}
\end{subfigure}
\caption{Snow regions at the midplane for the Z1(red), Z01(green), and Z001(blue) models for all neutral species. {The horizontal bars show the region in the midplane where the abundances of the ice species are larger than the abundances of the gaseous species. These regions are enclosed by the inner and outer snowline.}}
\label{fig:snowline_loc}
\end{figure}
\subsection{Snowline for each molecule}\label{ssec:Detection of the snowline}
Snowlines are the regions in protoplanetary disks where the temperature reaches the sublimation point for a particular volatile species. The position of the snowline is of great importance to the formation and basic architecture of a planetary system. It plays an important role in the formation and chemical composition of planetesimals, and therefore in the formation and composition of planets, e.g. \citep{2011ApJ...743L..16O,2016ApJ...821...82O,2015ApJ...815L..15B}
The temperature in a protoplanetary disk has a radial and a vertical gradient. Therefore the snowline is not a single radius but rather a two-dimensional surface. We show the location of this surface by displaying the regions in the disk where the chemical species are more abundant in the gas phase (red area) or the ice phase (blue area) (see Fig. \ref{fig:snowline}). The white regions represent a similar abundance of ice and gas and the black dashed line represents the location in the disk where gas and ice abundances are equal. Each row in the figure represents a different molecule and each column stands for a different metallicity model.
As a result of the radial and vertical temperature gradient in the disk, a snow region is enclosed spatially by the two-dimensional surface where ice and gas abundances are equal. We further define the intersection of this two-dimensional surface with the midplane as the snowline. The innermost intersection is defined as the inner snowline and the next intersection as the outer snowline. For most of the molecules studied here the outer snowline is not present. However, there are some cases where the outer snowline lies inside the disk and we speak of an ice ring inside the disk.
From Figure \ref{fig:snowline} it is clear that as the metallicity decreases, the snow region shrinks in its vertical and radial extension for all molecules, and in some cases, the snow region almost completely vanishes (e.g. CO$_2$ and CN). This is the result of the general increase in the disk's temperature caused by the stronger radiation field in the disk.
Fig.~\ref{fig:snowline_loc} summarizes the extension of the snow region and the location of the respective snowline at the midplane for each molecule for all models. The comparison between the models shows an influence of the metallicity on the location of the snowlines for all molecules. All the inner snowlines of each species are pushed outwards with decreasing metallicity. In some cases, the decrease in metallicity moves the outer snowline inside the disk so that an ice ring inside the disk is present.
Two of the species (CO$_{2}$ and CN) used for this section exhibit the presence of ice rings inside the disk. Figs.~\ref{fig:snowline} and ~\ref{fig:snowline_loc} show that these ice rings shrink with decreasing metallicity (CO$_2$ and CN in particular).
The effect of metallicity on the inner (outwards shift) and outer (inwards shift) snowlines is an indication that reducing the metallicity leads to a higher dust temperature (which is the temperature that determines if thermal desorption of frozen species from dust grains takes place) and an enhanced photo-desorption of ice species due to a stronger radiation field in the disk. Additionally, the fact that lower metallicity models have less amount of dust means that the gas phase molecules have less surface available to freeze out.
All this favors the depletion of ice species and, consequently, causes the ice rings to shrink as the metallicity decreases.
As a relevant example for planet formation, the snowline for H$_2$O (0.39, 0.45, and 0.59 au for models Z1, Z01, and Z001 respectively) does not change dramatically with metallicity. {Thus, the position of the H$_2$O snowline on planet formation would not be affected by metallicity changes.} Another snowline that does not exhibit a strong change with metallicity is the CO$_2$ snowline (1.22, 1.49, and 1.70 au). In the case of the CO snowline, the impact of metallicity is more noticeable (11.64, 14.21, and 85.45 au). In particular, for the Z001 model, the CO snowline is being pushed outwards to 85 au. The location of the snowlines for these three molecules has special significance in the formation and composition of planets, asteroids, and comets in a protoplanetary disk as these molecules are among the main carriers of C and O. Whether the atmosphere of a planet or a comet is carbon-rich or oxygen-rich, and whether it will be rich in more complex molecules, will depend on radial variations in disk midplane composition, and specifically the locations of the snowlines \citep{2016ApJ...823L..10W}.
As CO is considered to be the starting point for the formation of many complex organic molecules (COMs), the metallicity will certainly impact the location in the disk where COMs are more likely to be found and therefore affect the chemical composition of the planets, asteroids or comets that are formed at those radii.
\section{Limitations of the model and outlook for future studies}\label{Limitations of model}
In this section, we briefly discuss certain limitations of our model and their possible consequences.
In section \ref{ssec:gas disk structure} we introduce the physical structure of our model. We note that the column density power index \pmb{$\lambda$} of our models shown in Eq.(\ref{eq:sigma}) depends on the mass transport mechanisms operating in the disk, which in turn may depend on metallicity \citep{2013ApJ...775...68D}. This could lead to different timescales for accretion processes and the dissipation of the disk. In this study, we neglect these possible complications and assume that the disk structure is identical in the considered metallicity range.
We use a fixed stellar spectrum for all the models. In order to provide a more realistic model, we should take into account that the stellar spectrum changes with a different metallicity. This limitation is due to the fact that the model uses stellar spectra with solar metallicity by default. A consequence of this simplification is that the effective temperature of the star is underestimated as stars with lower metallicity tend to be hotter.
For this work, we focused on the behavior of certain molecules. We chose this set of molecules because of their relevance in dust growth, planetary atmospheric composition, and importance for observations. The inclusion of more complex organic molecules in the analysis will be addressed in future papers.
For this study, we assumed steady-state chemistry. Although this does not affect the consistency of our results, time-dependent chemistry becomes relevant when studying the impact of processes (e.g. accretion burst) that have a strong influence on the disk. {In a future study, we intend to include the impact of metallicity and accretion bursts in the chemistry of the disk.}
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{plots/co_h2o.pdf}
\end{subfigure}
\caption{CO/H$_2$O ratio for models with different metallicity}
\label{fig:co_h2o}
\end{figure}
We also note the particular case of CO. This molecule is very relevant because of its importance as a gas mass tracer (\cite{2017ApJ...849..130M}). Two very important conditions that allow the usage of CO as a gas mass tracer are that its emission has to be optically thin and it cannot be too strongly depleted in the gaseous state. {Our results suggest that changing the metallicity does not strongly affect the total abundance (gas + ice) of CO (relative to the response of other molecules and excluding the bump inside 1 au).} This would mean that CO-based observations would not be able to differentiate disks with different metallicity. This could be linked to the lower water abundance for lower metallicity disks and would suggest that low metallicity disks have a higher CO/H$_2$O ratio compared to "normal" disks (see Fig. \ref{fig:co_h2o}). To investigate the impact on observables requires proper modeling of line emission of the species in question, which is out of the scope of this paper but is planned in a future study.
In future studies, we will be looking more deeply at different aspects of the impact of metallicity on protoplanetary disks. In particular, the effects of time-dependent chemistry in the context of an accretion burst, the implementation of a more realistic stellar spectrum, and dust size distribution in the disk. The behavior of more complex organic species and the modeling of their observation will be studied as well.
\section{Summary and conclusions}\label{Summary and conclusions}
We started this study to look into the impact of lower metallicities on certain chemical species in the protoplanetary disk. We used the radiation thermo-chemical code for protoplanetary disks P{\tiny RO}D{\tiny I}M{\tiny O}\; to create a model with a reference metallicity (Z1) and two models with lower metallicities by a factor of 10 (Z01) and 100 (Z001). We simulated the lower metallicities by decreasing the dust-to-gas ratio in the disk by one and two orders of magnitude and by decreasing the initial element abundances by the same factor in the initial setup of the models respectively.
We analyzed the impact of metallicity on the chemical species H$_{2}$O, CO, CO$_{2}$, CH$_{4}$, CN, HCN, HCO+, and N$_{2}$H+ because they are often used in observations as useful tracers of different disk properties. The findings in this work could therefore give useful information for future observations in low metallicity environments.
One of the most important aspects of this study is the comparison of the lower metallicity model's vertical column density of chemical species with the reference metallicity vertical column density scaled-down by factors of 10 and 100.
We defined the notion of over-and underabundance of the models whenever the lower metallicity models show greater or smaller values for the vertical column density than the scaled-down vertical column density. We analyze the chemical formation and destruction reactions in the midplane wherever the over-or underabundance is strongest.
Our main findings are:
\begin{itemize}
\item The effects shown in the lower metallicity models cannot be explained by a simple scaled-down of the abundances of the chemical species. The obtained values for the abundances of the Z01 and Z001 models differ from the scaled-down values almost everywhere in the disk. This is because the lower metallicity models allow the radiation field to penetrate deeper regions of the disk. The temperature also increases in the regions where the radiation is stronger than in the Z1 model. This affects the formation and destruction reactions of the species.
\item The main effect of a stronger radiation field appears to be the enhancement of the different processes responsible for the destruction of ice species and the formation of gas species and the shifts in the type of destruction and formation reactions. Thus, the stronger radiation field causes a substantial increase in the formation efficiencies of gas-phase species. This is clearly shown in Fig. \ref{fig:integ_ratio} where the ratios between the scaled-down values and the lower metallicity values of all gaseous species (red dots) and the ice species (blue dots) are displayed. Furthermore, the impact of metallicity on the chemical reactions is different for each individual species. In some cases, the reaction type changes with metallicity, and in other cases the type of reaction does not change with metallicity but the rate of the reaction is clearly affected. This is shown in detail in Table \ref{table:form_destr} where the different main reactions and the respective rates of the three models are compared.
\item Our results show that CO shows a greater abundance than {the scaled-down values} for lower metallicities. This molecule shows a clear non-linear response to the change in metallicity. Beyond the bump at small radii, metallicity does not seem to have a strong influence on the column density of CO. This finding has very relevant implications for observations as CO would not be a useful indicator for the metallicity of the disk.
\item The impact of metallicity on the position of the snowlines is also present. With decreasing metallicity the snowline of the molecules analyzed in this study is "pushed" outwards. Additionally, decreasing the metallicity leads to the creation of ice rings for some species (e.g. CO$_2$) and to the shrinking of already existing ice rings for other species (e.g. CN).
\end{itemize}
The inclusion of other metallicity-dependent features will be performed in future studies (star spectrum and dust size distribution among others). The analysis of other scenarios (e.g. accretion bursts) will also help us gain more insight into the impact of metallicity in protoplanetary disks.
\begin{acknowledgements}
We would like to express our gratitude to the anonymous referee for the valuable comments that helped us improve the presentation of this work. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC). This work was supported by the Austrian Science Fund (FWF) under research grant P31635-N27. Ch. Rab is grateful for support from the Max Planck Society and acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 325594231.
\end{acknowledgements}
\bibliographystyle{aa}
|
1,314,259,996,754 | arxiv | \section*{References}%
\begin{quotation}\mbox{}\par}
\def\refer#1\par{{\setlength{\parindent}{-\leftmargin}\indent#1\par}}
\def\end{quotation}{\end{quotation}}
{\noindent\small{\bf Abstract:} The supersonic stellar and disk winds
possessed by massive young stellar objects will produce shocks when
they collide against the interior of a pre-existing bipolar cavity
(resulting from an earlier phase of jet activity). The shock heated
gas emits thermal X-rays which may be observable by spaceborne
observatories such as the Chandra X-ray Observatory. Hydrodynamical
models are used to explore the wind-cavity interaction. Radiative
transfer calculations are performed on the simulation output to
produce synthetic X-ray observations, allowing constraints to be
placed on model parameters through comparisons with
observations. The model reveals an intricate interplay between the
inflowing and outflowing material and is successful in reproducing
the observed X-ray count rates from massive young stellar objects.
}
\section{Introduction}
Observational and theoretical advances have provided increasing
evidence that massive star formation is not merely a scaled-up version
of lower-mass star formation. However, there are some
similarities. For instance, both involve outflows (Garay \& Lizano
1999; Reipurth \& Bally 2001; Banerjee \& Pudritz 2006, 2007) and
bipolar cavities are commonly observed around both high and low mass
young stellar objects (Garay \& Lizano 1999). The widths of IR
recombination line emission observed from massive young stellar
objects (MYSOs) indicates the presence of dense ionized outflows with
velocities ranging from 100 to $>340\;{\rm km\thinspace s}^{-1}$
(Drew, Bunn, \& Hoare 1993; Bunn, Hoare, \& Drew 1995). Further
confirmation of outflows has come from high angular resolution radio
observations (e.g. Hoare et al. 1994; Hoare 2006; Curiel et
al. 2006). One explanation would be a disk wind driven by radiation
pressure at the surface of the disk (e.g. Drew et al. 1998).
X-rays have been detected from deeply embedded MYSOs in star forming
regions by the \textit{Chandra X-ray observatory} (hereafter
\textit{Chandra})(e.g. Broos et al. 2007; Wang et al. 2007). Preibisch
et al. (2002) found that the X-ray emission from IRS3 A and C, with a
count rate of $0.166\pm 0.041\;{\rm ks^{-1}}$ for the former, could
not be explained by the standard scenario for massive stars (i.e. wind
embedded shocks produced by instabilities inherent in
radiatively-driven winds - see Owocki, Castor, \& Rybicki 1988), yet
the estimated stellar masses of these objects implies they will have
radiative outer envelopes which poses problems for the generation of
X-rays through magnetic star/disk interactions.
A potential source of X-rays which has not been considered before is
the collision between the stellar and disk winds and the infalling
envelope. We explore this scenerio using hydrodynamical models and
find that the interaction of the stellar and disk wind with the cavity
wall produces shock heated plasma which emits X-rays in agreement with
Chandra observations (Parkin et al. 2009).
The remainder of this work is structured as follows: in
\S~\ref{sec:model} we describe the winds-cavity model, in
\S~\ref{sec:results} we present the result of the hydrodynamic
simulations and X-ray calculations, and in \S~\ref{sec:conclusions} we
close with our conclusions.
\begin{figure}[h]
\centering
\includegraphics[width=12cm]{E_Parkin1_fig1.eps}
\caption{Density (top row) and temperature snapshots (bottom row) from
our fiducial model at a simulation time of $t=1000\;$(left), 1200
(middle), and 5000 yr (right). The grids extend to $x=5 \times
10^{16}\;$cm and $y= 8 \times 10^{16}\;$cm. The central star is
located in the bottom left corner. The reverse shock can be seen in
the lower left of the grid, where the enclosed preshock wind has a
temperature of $T=10^{4}\;$K. \label{fig:snapshots}}
\end{figure}
\section{The winds-cavity interaction model}
\label{sec:model}
The model consists of a MYSO situated at the centre of a previously
evacuated bipolar cavity which is surrounded by infalling molecular
cloud material. We include the stellar wind and a disk wind which
emanates from the surface of the circumstellar accretion disk; both
are assumed to be at terminal velocity. Due to the spatial scales
under consideration we do not attempt to model the radiative driving
of the winds as this requires high spatial resolution in the vicinity
of the star/disk (e.g. Proga et al. 1998). For simplicity we adopt an
angle dependent wind prescription based on the models of Proga et
al. (1998), Drew et al. (1998), and Sim et al. (2005), whereby the
stellar wind occupies a region from a polar angle $0-60^{\circ}$ and
the disk wind occupies the region from $60^{\circ}-85^{\circ}$. The
density and velocity distibutions for the infalling cloud material are
described by the equations of Terebey et al. (1984), and a simple
analytical prescription similar to that of Alvarez et al. (2004) is
used to determine the morphology of the pre-existing outflow
cavity. We model the winds-cavity interaction in 2D cylindrical
symmetric coordinate system using {\sc VH-1} (Blondin et
al. 1990). The code uses the piecewise-parabolic method of Colella \&
Woodward (1984) to solve the hydrodynamic equations on a fixed
grid. Further details of the model can be found in Parkin et
al. (2009).
We consider a 30$\;{\rm M_{\odot}}$~O8V star with a mass-loss rate of
$10^{-7}\;{\rm M_{\odot}~yr^{-1}}$ and terminal wind speed of
2000$\;{\rm km~s^{-1}}$. The disk wind has a mass-loss rate of
$10^{-6}\;{\rm M_{\odot}~yr^{-1}}$ and a terminal wind speed of
400$\;{\rm km~s^{-1}}$. The unshocked winds are assumed to be at a
temperature of $10^{4}\;$K. The mass infall rate for the cloud is
$2\times 10^{-4}\;{\rm M_{\odot}~yr^{-1}}$ and the cavity opening
angle is 30$^{\circ}$ . The winds-cavity interaction was followed for
a simulation time of $t = 5000\;$yrs.
To allow a comparison to be made between \textit{Chandra} X-ray
observations of MYSOs and the simulations we calculate attenuated
X-ray fluxes. The emissivity values used are for optically thin gas in
collisional ionization equilibrium obtained from look-up tables
calculated from the \textsc{MEKAL} plasma code (Liedahl, Osterheld, \&
Goldstein 1995 and references there-in). Ray-tracing calculations are
performed with an inclination angle (to the pole) of $60^{\circ}$.
In this work we only consider one set of parameters, and we refer the
reader to Parkin et al. (2009) for a detailed parameter space
exploration.
\section{Results}
\label{sec:results}
Fig.~\ref{fig:snapshots} shows the spatial distribution of material in
the simulation; the disk wind lines the cavity wall and separates the
stellar wind from the infalling molecular cloud material. The winds in
the simulation are supersonic, and their collision against the cavity
wall generates a reverse shock close to the star ($<
500\;$AU). Because the ram pressure of the inflow/outflow is angle
dependent and the base of the cavity is subject to instabilities the
position of the reverse shock oscillates, and its shape is often
non-spherical. This is due to small fluctuations in the shape and size
of the base of the cavity wall as inflowing material is ablated and
incorporated into the outflow, and as new inflowing material
replenishes it (see Fig.~\ref{fig:snapshots}). The shear layer between
the stellar and disk winds provides a site for the growth of $\sim
700\;$AU ($10^{16}\;$cm) amplitude Kelvin-Helmholtz instabilities on
timescales of $\sim$ a few years. By $t=1200\;$yrs an instability of
this proportion can be seen driving a clump of disk wind material into
the path of the stellar wind, which leads to mass-loading of the
latter.
\begin{figure}[h]
\centering
\includegraphics[width=14cm]{E_Parkin1_fig2.eps}
\caption{Ray-traced synthetic broadband X-ray images.\label{fig:bbimages}}
\end{figure}
At a simulation time of $t=1000\;$yrs, the intrinsic X-ray emission
comes mainly from disturbed cloud material ($L_{\rm int_{C}}\simeq 3.3
\times 10^{33}\;{\rm erg~s^{-1}}$), then the disk wind ($L_{\rm
int_{D}} \simeq 4.3 \times 10^{32}\;{\rm erg~s^{-1}}$), and finally
the stellar wind ($L_{\rm int_{S}}\simeq 5 \times 10^{30}\;{\rm
erg~s^{-1}}$). The shock driven into the cloud is too slow to heat
gas up to X-ray emitting temperatures, but large quantities of cloud
material are ablated by the outflowing disk wind and mixed into this
hotter flow\footnote{Note that some heating of the cloud material may
occur due to numerical heat conduction between the hot postshock
disk wind and the cooler cloud material (Parkin \& Pittard
2010).}. This process heats the cloud material to temperatures where
(soft) X-rays are emitted. However, the difference between the
\textit{attenuated} luminosities of the cloud and stellar wind
components is very small ($L_{\rm att_{C}} \simeq 8\times10^{28}\;{\rm
erg~s^{-1}}$ and $L_{\rm att_{S}} \simeq 7\times10^{28}\;{\rm
erg~s^{-1}}$, respectively). The explanation is that the stellar
wind emission is harder and extends to higher energies, and is less
affected by attenuation. In contrast, the cloud material, which is
heated to lower temperatures, emits prolifically at low energies and
has a much higher intrinsic luminosity, but its emission is subject to
considerable attenuation at $E < 1\;$keV. Although some variation in
X-ray emission occurs due to fluctuations in the position of the
reverse shock, these values are indicative of the mean luminosities
from the simulation.
The total attenuated luminosity from the model is $L_{\rm att_{tot}}
\simeq 2 \times 10^{29}\;{\rm erg~s^{-1}}$ which, when placed at a
distance of 1 kpc and convolved with the {\it Chandra} effective area,
equates to a count rate of $\simeq 0.1\;{\rm ks^{-1}}$. Interestingly,
this is in approximate agreement with X-ray count rates of $0.166\pm
0.041\;{\rm ks^{-1}}$ and $0.30\pm0.11\;{\rm ks^{-1}}$ inferred from
observations of Mon R2 IRS 3A by Preibisch et al. (2002) and for S106
IRS4 by Giardino, Favata, \& Micela (2004), respectively.
Examining synthetic broadband X-ray images calculated from the
simulation output one can see that the observable X-ray emission in
the 1-2, 2-4, and 4-10 keV bands originates from similar regions of
the cavity (Fig.~\ref{fig:bbimages}). The peak in the intensity in the
three bands have common origins near the reverse shock, and are mainly
generated by shocked stellar and disk wind material. Importantly, the
spatial extents of the detectable emission in all three energy bands
(1-2, 2-4, and 4-10 keV) are just below the resolution limit of
\textit{Chandra} ($\sim0.5''$) and so this model is consistent with
the unresolved nature of real MYSOs.
\section{Conclusions}
\label{sec:conclusions}
The wind-cavity interaction around an embedded MYSO has been studied
using hydrodynamic simulations and X-ray calculations. In summary, the
collision of the winds against the cavity wall generates a reverse
shock close to the star ($< 500\;$AU). The shock heated gas produces
X-ray emission with an integrated count rate ($\simeq 0.1\;{\rm
ks^{-1}}$) and spatial extent ($< 0.5''$) in agreement with
observations of MYSOs by \textit{Chandra}.
We close with a note that future X-ray satellites, such as the
International X-ray Observatory (IXO), may have the potential to
resolve the spatial extent of the X-ray emission from the winds-cavity
interaction. If this were the case, the winds-cavity model could be
used in concert with detailed analysis at infra-red wavelengths to
place unsurpassed constraints on the parameters of MYSO outflows.
\section*{Acknowledgements}
ERP thanks The University of Leeds and PRODEX for funding.
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|
1,314,259,996,755 | arxiv | \section{Introduction}
Mori dream spaces, introduced by Hu and Keel in \cite{HuKeelMDS}, are
varieties for which the minimal model program works particularly well.
In particular, the nef cone is polyhedral and generated by finitely
many semi-ample divisors, and the movable cone is the union of finitely
many polyhedral cones which are the nef cones of small $\mathbb Q$-factorial
modifications of the original variety. While this is a strong
condition, one consequence of \cite{BCHM} is that log Fano varieties
are Mori dream spaces.
In the aftermath of \cite{BCHM} understanding the birational geometry
of the moduli space of curves, following the Hassett-Keel program, has
become a major industry; see, for example, \cite{Hassett,
HassettHyeon, Alperetal}. A natural question is whether
$\overline{M}_{g,n}$ is a Mori dream space; the case $g=0$ was raised
in the original paper~\cite{HuKeelMDS}. The answer seems to be
increasingly no; $\overline{M}_{g,n}$ is not a Mori dream space for
large $g$ and $n$ \cite{ChenCoskun,Mullane}, and $\overline{M}_{0,n}$
is not a Mori dream space for $n \geq 10$
\cite{CastravetTevelevNotMDS, GonzalezKaru, HausenKeicherLaface}.
For smaller $n$, however, the situation is more tractable.
In \cite{Castravet} Castravet showed that the moduli space $\overline{M}_{0,6}$ is a
Mori dream space; since $\overline{M}_{0,6}$ is log Fano, this now also follows from \cite{BCHM}.
Castravet's work builds on earlier work of Keel, Vermeire~\cite{Vermeire}, and
Hassett and Tschinkel~\cite{HassettTschinkel} who showed that the
effective cone of $\overline{M}_{0,6}$ is generated by the $25$ boundary divisors,
and by $15$ extra divisors. She showed that the Cox ring of
$\overline{M}_{0,6}$ is generated by sections of these $40$ divisors. In this paper
we compute the relations between these generators to give a
presentation for the Cox ring.
The $25$ boundary divisors are indexed by subsets $I \subset
\{1,2,3,4,5,6\}$ with $2 \leq |I| \leq 3$ and $1 \in I$ if $|I|=3$.
We denote this set by $\mathcal I$, and a section of the divisor
$\delta_I$ indexed by $I$ by $x_I$. The $15$ extra Keel-Vermeire
divisors are indexed by permutations in the symmetric group $S_6$ of
the form $(ij)(kl)(mn)$. We denote the set of these permutations
by $\Pi$, and for $\pi \in \Pi$ write $y_{\pi}$ for a section of the
corresponding divisor.
The main result of this paper is:
\begin{theorem} \label{t:maintheorem}
Let $S = \mathbb C[x_I, y_{\pi} : I \in \mathcal I, \pi \in \Pi]$. The Cox ring
of $\overline{M}_{0,6}$ is $S/I_{\overline{M}_{0,6}}$, where $I_{\overline{M}_{0,6}}$ is an ideal with 225 generators,
which come in five symmetry classes:
\begin{enumerate}
\item $15$ of the form:
\begin{equation*}
x_{ij}x_{kl}x_{ijn}x_{kln}-x_{ik}x_{jl}x_{ikn}x_{jln}+x_{il}x_{jk}x_{iln}x_{jkn}
\end{equation*}
for $1\leq i<j<k<l\leq 6$, $m<n$ and $\{i,j,k,l,m,n\}=\{1,2,3,4,5,6\}$;
\item $60$ of the form:
\begin{equation*}
x_{1ik}y_{(1m)(ij)(kl)}+x_{1j}x_{mk}x_{il}x_{1mj}x_{1mk}x_{1ij}+ x_{1l}x_{mi}x_{jk}x_{1mi}x_{1ml}x_{1kl},
\end{equation*}
for $\{i,j,k,l,m\}=\{2,3,4,5,6\}$;
\item $45$ of the form:
\begin{equation*}
x_{ij}y_{(ij)(kl)(mn)} + x_{ik}x_{il}x_{jm}x_{jn}x_{ikl}^2 - x_{im}x_{in}x_{jl}x_{jk}x_{imn}^2,
\end{equation*}
for $\{i,j,k,l,m,n\}=\{1,2,3,4,5,6\}$ and $i<j$;
\item $45$ of the form:
\begin{equation*}
y_{(ij)(kl)(mn)}y_{(ij)(km)(ln)}-x_{il}x_{lj}x_{jm}x_{mi}x_{nk}^{2}x_{ijm}^{2}x_{ijl}^{2} +x_{in}x_{nj}x_{jk}x_{ki}x_{ml}^{2}x_{ijk}^{2}x_{ijn}^{2},
\end{equation*}
for $\{i,j,k,l,m,n\} = \{1,2,3,4,5,6\}$;
\item $60$ of the form:
\begin{multline*}
\qquad \quad y_{(1i)(jk)(lm)}y_{(1j)(il)(km)} - x_{1k}x_{1l}x_{ik}x_{im}x_{jl}x_{jm}x_{1ik}x_{1il}x_{1jk}x_{1jl}\\
+x_{1k}x_{1l}x_{ij}x_{im}x_{jm}x_{kl}x_{1ij}x_{1il}x_{1jk}x_{1kl} -x_{1k}x_{1m}x_{ij}x_{im}x_{jl}x_{kl}x_{1ij}x_{1im}x_{1jk}x_{1km}\\
-x_{1l}x_{1m}x_{ij}x_{ik} x_{jm} x_{kl} x_{1ij} x_{1il} x_{1jm} x_{1lm},
\end{multline*}
for $\{i,j,k,l,m\} = \{2,3,4,5,6\}$.
\end{enumerate}
Here we follow the convention that $x_{ij}\!=\!-x_{ji}$,
$x_{ijk}\!=\!-x_{jik}=\!-x_{ikj}$, $x_{ijk}\!=\!x_{lmn}$ for
\small$\{i,j,k,l,m,n \}= \{1,2,3,4,5,6\}$\normalsize, with $i<j<k$ and
$l<m<n$, and $y_{(ij)(kl)(mn)}=-y_{(kl)(ij)(mn)}=-y_{(ij)(mn)(kl)}$.
\end{theorem}
The proof exploits the embedding of $\overline{M}_{0,6}$ into a $24$-dimensional toric
variety determined by the generators of the Cox ring. It can be done
with various degrees of computer assistance. One contribution of this
paper is that we have taken care when choices were necessary so that
the $S_6$ action is transparent, and the representatives given here
for the generators have a simple form.
One application
of a complete finite presentation of the Cox ring of a variety $X$ is
that it allows the computation of all small birational models of $X$.
Using an earlier circulated version of this paper B\"ohm, Keicher, and
Ren have computed all the chambers of the Mori chamber decomposition
of $\overline{M}_{0,6}$ \cite{BKR}, finding $176, 512, 180$ maximal chambers that form $249,
605$ orbits under the $S_6$-action. While this suggests that the
problem of computing all small birational models is intractable, in
Section~\ref{s:smallbirationalmodels} we consider some simplifications
which imply that the chambers of the movable cone are simpler.
Identifying these models has already been considered in \cite{Moon}
for the two-dimensional slice of the Ner\'on-Severi group containing
$K_{\overline{M}_{0,6}}$ and $\sum_{i=1}^6 \psi_i$. This slice intersects two other
chambers, corresponding to the symmetric GIT quotient of $(\mathbb
P^1)^6$, and the symmetric Veronese quotient.
The structure of the paper is as follows.
In Section~\ref{s:background} we develop the necessary preliminaries
on the generators of $\Cox(\overline{M}_{0,6})$. In section~\ref{s:coxequations} we
explain how to compute the relations between these generators.
Theorem~\ref{t:maintheorem} is proved in Section~\ref{s:maintheorem}.
Finally, in Section~\ref{s:smallbirationalmodels} we consider the
problem of using Theorem~\ref{t:maintheorem} to compute the small
birational models of $\overline{M}_{0,6}$, and provide some computational
simplifications.
\section{The Cox ring of $\overline{M}_{0,6}$}
\label{s:background}
In this section we recall some background on the Cox ring of $\overline{M}_{0,6}$. The key
result of this section is Lemma~\ref{l:Qpiequation}, which will be
used to prove Theorem~\ref{t:maintheorem}.
The moduli space $\overline{M}_{0,6}$ of stable genus zero curves with $6$ marked
points is a smooth $3$-dimensional variety that compactifies
$M_{0,6} \cong (\mathbb C^* \setminus \{1\})^3 \setminus \bigcup_{1 \leq
i <j \leq 3}\{x_i =x_j \}$. The boundary $\overline{M}_{0,6} \setminus M_{0,6}$
parameterizes stable genus zero curves with $6$ marked points. The
boundary is a union of $25$ boundary divisors $\delta_I$, where $I
\subset \{1,\dots, 6\}$ with $|I|=2$ or $|I|=3$, and we identify
$\delta_I$ and $\delta_{\{1,\dots,6\} \setminus I}$ when $|I|=3$. The
boundary divisor $\delta_I$ is the closure of the locus in $\overline{M}_{0,6}$
consisting of those nodal curves with two irreducible components,
where the points with labels in $I$ lie on one component, and the
other points on the other.
The Picard group of $\overline{M}_{0,6}$ is isomorphic to $\mathbb Z^{16}$ and is
spanned by the boundary divisors.
Given a basis $\mathcal B = \{E_1,\dots,E_{16}\}$ for $\Pic(\overline{M}_{0,6})$, the Cox ring, first
introduced for any $\mathbb Q$-factorial variety $X$ with $\Pic(X)
\cong \mathbb Z^m$ in \cite{HuKeelMDS} is
$$\Cox(\overline{M}_{0,6}) = \oplus_{a \in \mathbb Z^{16}} H^0(\overline{M}_{0,6}, \mathcal
O(a_1E_1+\dots+a_{16}E_{16}) ).$$ This is naturally graded by $\Pic(\overline{M}_{0,6})$.
The pseudo-effective cone
$\Eff(\overline{M}_{0,6})$ is generated by the classes of the $25$ boundary divisors
and the classes of $15$ other effective divisors,
the Keel-Vermeire divisors~\cite{HassettTschinkel}, whose description
we recall below. In \cite{Castravet} Castravet showed that sections
of these divisors suffice to generate the Cox ring of $\overline{M}_{0,6}$.
\begin{theorem}[{\cite[Theorem 1.4]{Castravet}}] \label{t:Castravet}
The Cox ring of $\overline{M}_{0,6}$ is generated by sections of the
boundary divisors $\delta_I$ and the Keel-Vermeire divisors $Q_{\pi}$.
\end{theorem}
In this paper we extend Castravet's result to describe in addition the
relations on this ring. In the rest of this section we explicitly
describe the complement of these divisors in $\overline{M}_{0,6}$.
In \cite{KapranovChow} Kapranov shows that $\overline{M}_{0,n}$ is the Chow quotient of the
Grassmannian $G(2,n)$ by the action of the torus $(\mathbb C^*)^{n-1}
\cong (\mathbb C^*)^n/\mathbb C^*$ induced from the action of
$(\mathbb C^*)^{n-1}$ on $\mathbb P^{n-1}$. Under this identification
we have $M_{0,n} = G^0(2,n)/(\mathbb C^*)^{n-1}$, where $G^0(2,n)$ is
the locus in the Grassmannian $G(2,n)$ where all Pl\"ucker coordinates
are nonzero. This is naturally a subvariety of the quotient torus
\small$(\mathbb C^*)^{{n \choose 2}-1}/(\mathbb C^*)^{n-1}$\normalsize, whose
coordinate ring we denote by \small$\mathbb C[z_{24}^{\pm 1},
\dots,z_{n-1n}^{\pm 1}]$\normalsize. We denote by $\mathcal E$ the index set
\small$\{ \{2,4\},\dots,\{5,6\}\}$ \normalsize of the variables in this Laurent
polynomial ring.
The following lemma is the $n=6$ case of \cite[Proposition
6.4]{GibneyMaclaganEquations}.
\begin{lemma} \label{l:M06equations}
The moduli space $M_{0,6}$ is an affine variety with coordinate ring
$$\mathbb C[z_{24}^{\pm 1},z_{25}^{\pm 1},z_{26}^{\pm 1},z_{34}^{\pm 1},z_{35}^{\pm 1},z_{36}^{\pm 1}, z_{45}^{\pm 1},
z_{46}^{\pm 1},z_{56}^{\pm 1}]/ I_6 ,$$
where
\begin{align*} I_6 = \langle z_{34}-z_{24}+1, z_{35}-z_{25}+1,
z_{36}-z_{26}+1, z_{45}-z_{25}+z_{24}, z_{46}-z_{26}+z_{24}, z_{56}-z_{26}+z_{25} \rangle.
\end{align*}
\end{lemma}
This follows from the quotient construction, using the following identification of the variables $z_{ij}$. We denote by $\{ x_{ij} : 1 \leq i < j \leq n\}$ the
coordinates of $\mathbb P^{{n \choose 2}-1}$. Then
\begin{equation} \label{eqtn:zij}
z_{ij} = \begin{cases}
(x_{ij}x_{12}x_{13})/(x_{1i}x_{1j}x_{23}) & \text{ if } i,j \geq 4, \\
(x_{2j}x_{13})/(x_{1j}x_{23}) & \text{ if } i =2, \\
(x_{3j}x_{12}) / (x_{1j}x_{23}) & \text{ if } i =3.\\
\end{cases}
\end{equation}
For more details see \cite{GibneyMaclaganEquations}*{Section 6}.
Not all effective divisors on $\overline{M}_{0,6}$ lie in the cone spanned by the
boundary divisors, by work of Keel and Vermeire~\cite{Vermeire}. We
now recall their construction. The symmetric group $S_6$
acts naturally on $\overline{M}_{0,6}$ and $M_{0,6}$ by permuting the points.
Let $\pi=(ij)(kl)(mn) \in S_6$ be the product of three disjoint
transpositions. The divisor $Q_{\pi}$ is the closure in $\overline{M}_{0,6}$ of the
fixed locus of the action of $\pi$ on $M_{0,6}$.
The divisor $Q_{\pi}$ is effective by construction, but its class in
$\Pic(\overline{M}_{0,6})$ cannot be written as a nonnegative combination
of the classes of the boundary divisors.
The next lemma is the key result of this section, and is needed for the proof of Theorem~\ref{t:maintheorem}.
\begin{lemma} \label{l:Qpiequation}
Let $\pi=(12)(34)(56)$. The intersection of the divisor $Q_{\pi}$ with $M_{0,6}$ is the subvariety of $(\mathbb C^*)^9$ defined by the ideal
$$I_6 + \langle z_{24}-z_{25}z_{26} \rangle. $$
\end{lemma}
\begin{proof}
A point in $M_{0,6}$ corresponds to a configuration of six distinct
points in $\mathbb P^1$, which we may take to be $\{ \infty, 1, 0,
A,B,C \}$, or more formally $\{ [1:0], [1:1], [0:1], [A:1], [B:1],
[C:1] \}$, where $A,B,C \neq 0,1$. The transposition $\pi$ acts on
this configuration, taking it to $\{ [1:1], [1:0], [A:1], [0:1],
[C:1], [B:1] \}$. The automorphism of $\mathbb P^1$ given by the
matrix
$$\left( \begin{array}{rr} 1 & -A \\ 1 & -1 \\ \end{array} \right) \in
\PGL(2)$$ takes this configuration to the configuration $\{[1:0],
[1:1], [0:1], [A:1], [\frac{C-A}{C-1}:1], [\frac{B-A}{B-1}:1] \}$.
This point is thus fixed if and only if $\frac{C-A}{C-1} = B$ and
$\frac{B-A}{B-1}=C$. This means that $B+C-A-BC =0$. To translate
this into Pl\"ucker coordinates, we note that the Pl\"ucker
coordinates of the matrix
$$\left( \begin{array}{rrrrrr} 1 & 1 & 0 & A& B & C \\
0 & 1 & 1 & 1 & 1 & 1\\ \end{array} \right)$$
in the order $z_{24},z_{25},\dots,z_{56}$ are:
$$(- A + 1, - B + 1, - C + 1, -A, -B, -C, A - B, A - C, B - C),
$$
so $B+C-A-BC = z_{24}-z_{25}z_{26}$.
\end{proof}
Equations for the other Keel-Vermeire divisors can be obtained using
the action of the symmetric group $S_6$ on $\mathbb C[T^9]$, which we
now recall. The action comes from the description of $T^9$ as
$(\mathbb C^*)^{6 \choose 2}/(\mathbb C^*)^6$, or equivalently from
the formula for $z_{ij}$ given in \eqref{eqtn:zij}. For example, when
$i,j \geq 4$, we have \small $\textnormal{\small (12)} z_{ij} =
\textnormal{\small (12)} (x_{ij}x_{12}x_{13}/(x_{1i}x_{1j}x_{23})) =
-(x_{ij}x_{12}x_{23})/(x_{2i}x_{2j}x_{13}) =
-z_{ij}/z_{2i}z_{2j}$\normalsize. This is summarized in Table
\ref{t:S6action1}, where the entry in the row labeled by the adjacent
transposition \small $(i (i+1))$\normalsize \hspace{1pt} and column labeled by
$z_{kl}$ is \small $(i (i+1))z_{kl}$\normalsize .
\hspace{-5mm}
\small
\begin{table}[h]
{
\renewcommand{\arraystretch}{1.1}
\begin{center}
\begin{tabular}[h]{llllllllll}
& $z_{24}$ & $z_{25}$ & $z_{26}$ & $z_{34}$ & $z_{35}$ & $z_{36}$ & $z_{45}$ & $z_{46}$ & $z_{56}$ \\
$ (12) $& $ z_{24}^{\textnormal{-}1}$ & $ z_{25}^{\textnormal{-}1}$ & $ z_{26}^{\textnormal{-}1}$ &
-$ z_{34}/\!z_{24}$ & -$ z_{35}/\!z_{25}$& -$ z_{36}/\!z_{26}$ &
-$ z_{45}/\!z_{24}z_{25}$ & -$ z_{46}/\!z_{24}z_{26}$ &
-$ z_{56}/\!z_{25}z_{26}$ \\
$ (23) $& -$z_{34}$ & -$z_{35}$ & -$z_{36}$ & -$z_{24}$ & -$z_{25}$ &
-$z_{26}$ & -$z_{45}$ & -$z_{46}$ & -$z_{56}$\\
$(34) $& $ z_{24}^{\textnormal{-}1}$ & $ z_{25}/\!z_{24}$ & $ z_{26}/\!z_{24}$ &
-$ z_{34}/\!z_{24}$ & $ z_{45}/\!z_{24}$ & $ z_{46}/\!z_{24}$ &
$ z_{35}/\!z_{24}$ & $ z_{36}/\!z_{26}$ & $ z_{56}/\!z_{24}$ \\
$(45) $& $z_{25}$ & $z_{24}$ & $z_{26}$ & $z_{35}$ & $z_{34}$ & $z_{36}$
& -$z_{45}$ & $z_{56}$ & $z_{46}$ \\
$(56) $& $z_{24}$ & $z_{26}$ & $z_{25}$ & $z_{34}$ & $z_{36}$ & $z_{35}$
& $z_{46}$ & $z_{45}$ &-$z_{56}$ \\
&&&&&&&&&\\
\end{tabular}
\end{center}
\caption{The $S_6$ action on $\mathbb C[M_{0,6}]$}
\label{t:S6action1}
}
\end{table}
\normalsize
Applying the action to the equation $z_{24}-z_{25}z_{26}$ for
$Q_{(12)(34)(56)}$, we get the equations in Table~\ref{t:QpiEqns}.
\begin{table}[h!]
{
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{llll}
$\scriptstyle (12)(34)(56)$ & $z_{24}-z_{25}z_{26}$ &
$\scriptstyle (12)(35)(46)$ & $z_{25}-z_{24}z_{26}$\\
$\scriptstyle (12)(36)(45)$ & $z_{26}-z_{24}z_{25}$ &
$\scriptstyle (13)(24)(56)$ & $-z_{34}-z_{35}z_{36}$\\
$\scriptstyle (13)(25)(46)$ & $-z_{35}-z_{34}z_{36}$ &
$\scriptstyle (13)(26)(45)$ & $-z_{36}-z_{34}z_{35}$\\
$\scriptstyle (14)(23)(56)$ & $z_{24}z_{36}-z_{25}z_{46}$ &
$\scriptstyle (14)(25)(36)$ & $-z_{46}+z_{24}z_{56}$\\
$\scriptstyle (14)(26)(35)$ & $-z_{45}-z_{24}z_{56}$ &
$\scriptstyle (15)(23)(46)$ & $z_{25}z_{36}-z_{24}z_{56}$\\
$\scriptstyle (15)(24)(36)$ & $-z_{56}+z_{25}z_{46}$ &
$\scriptstyle (15)(26)(34)$ & $z_{45}-z_{25}z_{46}$\\
$\scriptstyle (16)(23)(45)$ & $z_{26}z_{35}+z_{24}z_{56}$ &
$\scriptstyle (16)(24)(35)$ & $z_{56}+z_{26}z_{45}$\\
$\scriptstyle (16)(25)(34)$ & $z_{46}-z_{26}z_{45}$ \\
\end{tabular}
\end{center}
\caption{Equations defining the KV-divisors}
\label{t:QpiEqns}
}
\end{table}
We denote the equation for $Q_\pi$ by $f_\pi$. The polynomial
$f_{\pi}$ depends on several choices. Firstly, adding any element of
the ideal $I_6$ gives the same element of $\mathbb C[M_{0,6}]$. More seriously,
there is a dependence on the choice of $\sigma \in S_6$ with $\pi =
\sigma \cdot \textnormal{\small (12)(34)(56)}$. For example, $\sigma
=\textnormal{\small (12)}$ gives $\pi = \textnormal{\small
(12)(34)(56)}$, but $\textnormal{\small (12)} (z_{24}- z_{25}z_{26})
= z_{24}^{-1} - z_{25}^{-1} z_{26}^{-1} =
-f_{(12)(34)(56)}/z_{24}z_{25}z_{26}$. This is a different element of
$\mathbb C[M_{0,6}]$. The corresponding divisor is still $Q_{(12)(34)(56)}$,
however.
For the explicit approach taken
here making a choice is necessary at this stage. The final result
given in Theorem~\ref{t:maintheorem} does not depend on this choice,
however.
\begin{remark}
We note that $Q_{\pi}$ is not the fixed locus of $\pi$ on all of $\overline{M}_{0,6}$.
This fixed locus contains the zero-stratum of $\overline{M}_{0,6}$ corresponding to
the curve with four components, three of which contain the pairs of
points $\{i,j\}$, $\{k,l\}$, and $\{m,n\}$, which is not in $Q_{\pi}$.
This can be seen, for example, by noting that $\trop(V(I_6+ \langle
f_{\pi} \rangle))$ does not intersect the cone of $\trop(M_{0,n})$
corresponding to this stratum.
\end{remark}
The set $Y =M_{0,6}\setminus \bigcup_{\pi\in S_6}(M_{0,6}\cap Q_\pi)$
is an open subvariety of $M_{0,6}$. An easy but important consequence
of Lemma~\ref{l:Qpiequation} is the following. Recall that $\mathcal
E =\{ \{2,4\},\dots, \{5,6\}\}$, and $\Pi$ is the set of products of
three disjoint transpositions in $S_6$.
\begin{lemma}\label{l:idealY}
The open set $Y=M_{0,6}\setminus\big(\bigcup_\pi {\mathbf
V}(f_\pi)\big)$ is an affine variety with coordinate ring
$\mathbb C[z_{ij}^{\pm 1},u_\pi^{\pm 1}: ij \in \mathcal E, \pi \in
\Pi]/J_{KV}$ where
\[
J_{KV}=I_6+\langle u_{\pi}-f_{\pi} \colon \pi \in \Pi \rangle
\subset \mathbb C[(\mathbb C^*)^{24}].
\]
\end{lemma}
\begin{proof}
For an affine variety $X$ with coordinate ring $K[X]$ and an
irreducible divisor $V(g) \subset X$ the complement $X \setminus V(g)$
has coordinate ring $K[X][u^{\pm 1}]/\langle u-f \rangle$. The lemma
follows from applying this inductively to the $f_{\pi}$.
\end{proof}
The $S_6$-action on the variables $z_{ij}$ induces an action on the
variables $u_{\pi}$ as well, by imposing the condition $\sigma \cdot
(u_{\pi}-f_{\pi}) = m(u_{\sigma \pi \sigma^{-1}}-f_{\sigma \pi
\sigma^{-1}})$ for some term $m$. For example, when $\pi =
(12)(34)(56)$ and $\sigma = (12)$, we have $(12) f_{\pi} =
-1/(z_{24}z_{25}z_{26}) f_{\pi}$, so $(12) u_{\pi} =
-1/(z_{24}z_{25}z_{26}) u_{\pi}$.
\section{Equations for Cox rings}
\label{s:coxequations}
In this section we outline an approach to computing relations for the
Cox ring of a Mori dream space when a set of generators is already
known, and begin to apply this for the case of $\overline{M}_{0,6}$.
We first recall notation for Cox rings. Throughout this section $X$
is a normal $\mathbb Q$-factorial projective variety defined over a field $K$ with
$\Pic(X)_{\mathbb Q} \cong \mathbb Z^r$.
Fix divisors $E_1,\dots,E_r$ whose classes form a basis for
$\Pic(X)_{\mathbb Q}$. The Cox ring of $X$ is
$$\Cox(X) = \oplus_{\mathbf a \in \mathbb Z^r} H^0(X,\mathcal
O(a_1E_1+\dots + a_r E_r).$$
The variety $X$ is a {\em Mori dream space}~\cite{HuKeelMDS} if
$\Cox(X)$ is finitely generated, so we can write $\Cox(X) = S/I$,
where $S = K[x_1,\dots,x_s]$ for sections $x_i \in
H^0(X,\mathcal O(D_i))$ of divisors $D_1,\dots,D_S$. This
description does involve choosing sections $x_i$, but in many cases,
including $\overline{M}_{0,6}$, the $x_i$ are determined up to scalar multiple. The
grading $\deg(x_i)=[D_i] \in \Pic(X)$ induces an action of the torus
$H = \Hom(\Pic(X),K^*)$ on $\mathbb A^s$ by $\phi \cdot x_i =
\phi([D_i]) x_i$, and this restricts to an action on $V(I)$. Fix an
ample class $\alpha \in \Pic(X)$, and let $X_{\Sigma}$ be the toric
variety $\mathbb A^{s} \ensuremath{\operatorname{/\!\!/}}_{\alpha} H$. This has torus $T=
(K^*)^s/H$. We assume that $\alpha$ is chosen sufficiently
generically so that the fan $\Sigma$ is simplicial. This is possible
as nonsimplicial $\Sigma$ only arise from $\alpha$ on a union
of hyperplanes in $\Pic(X) \otimes \mathbb R$. Then $X = V(I)
\ensuremath{\operatorname{/\!\!/}}_{\alpha} H$ embeds into $X_{\Sigma}$. Moreover, the restriction
$\Pic_\mathbb Q(X_\Sigma) \rightarrow \Pic_\mathbb Q(X)$ is an isomorphism and it
induces an isomorphism of the pseudo-effective cones
$\Eff(X_\Sigma)\rightarrow \Eff(X)$ \cite[Proposition
2.11]{HuKeelMDS}.
\begin{example} \label{e:M06embedding}
When $X=\overline{M}_{0,6}$, the Cox ring has $40$ generators as we now recall. Let
$\mathcal I = \{ I \subseteq [6] : 2 \leq |I| \leq 3 \text{ and } 1
\in I \text{ if } |I| =3 \}$ be the set of partitions indexing the
boundary divisors in $\overline{M}_{0,6}$ and let $\Pi = \{ (ij)(kl)(mn) \} \subseteq
S_6$ be the permutations indexing the Keel-Vermeire divisors.
Consider the polynomial ring $S = \mathbb C[x_I, y_{\pi} : I \in
\mathcal I, \pi \in \Pi]$ graded by $\deg(x_I)=\delta_{I}$ and
$\deg(y_\pi)=Q_\pi$. Then Theorem~\ref{t:Castravet} implies that there
is a graded surjective algebra homomorphism $S\rightarrow
\Cox(\overline{M}_{0,6})$. Let $I_{\overline{M}_{0,6}}$ be its kernel and $H=\Hom(\Pic(\overline{M}_{0,6}),\mathbb
C^*) \cong T^{16}$. Then $H$ acts on $\Spec(S)=\mathbb A^{40}$ and
given a sufficiently general ample class $\alpha \in \Pic(\overline{M}_{0,6})$ the GIT
quotient $X_\Sigma=\mathbb A^{40}\ensuremath{\operatorname{/\!\!/}}_\alpha H$ is a 24-dimensional
simplicial toric variety with $\overline{M}_{0,6} =V(I_{\overline{M}_{0,6}})\ensuremath{\operatorname{/\!\!/}}_\alpha H \subset
\mathbb A^{40} \ensuremath{\operatorname{/\!\!/}}_\alpha H$ \cite[Proposition 2.11]{HuKeelMDS}. The
intersection of $\overline{M}_{0,6}$ with the torus $T^{24}$ of the torus
$X_{\Sigma}:= \mathbb A^{40} \ensuremath{\operatorname{/\!\!/}}_\alpha H$ equals $M_{0,6} \setminus
\bigcup_{\pi \in \Pi} Q_{\pi}$.
\end{example}
The next proposition explains how to compute the ideal $I$ of
relations between the generators of the Cox ring. Recall that when
$X$ is a Mori Dream space the Mori chambers coincide with the GIT chambers
for $X = V(I) \ensuremath{\operatorname{/\!\!/}}_{\alpha} H$ \cite[Theorem 2.4]{HuKeelMDS}. Any
non-moving divisor on the toric variety $X_{\Sigma} =\mathbb A^s
\ensuremath{\operatorname{/\!\!/}}_{\alpha} H$ restricts to a non-moving divisor on $X$, so since
the moving cone of $X$ is full-dimensional, the same is true for
$X_{\Sigma}$. This implies that the polynomial ring $S$ is the Cox
ring of $X_\Sigma$. One consequence of this
is that the divisors on $X$ defined by generators of $\Cox(X)$ are in
bijection with the torus invariant divisors on $X_{\Sigma}$.
We write $S_{\mathbf{x}}$ for the localization of $S$ at the product
of the variables. By \cite{CoxToricTotalCoordinate}
$(S_{\mathbf{x}})_{\mathbf{0}}$ is the coordinate ring of the torus
$T$ of $X_{\Sigma}$, where the subscript denotes the degree zero part
in the $\Pic(X)$ grading.
\begin{proposition} \label{p:CoxRingRelations}
Let $X$ be a normal projective Mori Dream Space
and
let $ X \hookrightarrow X_{\Sigma}$ be a toric embedding given by
choosing $\alpha \in \Pic(X)$ ample.
Write $E_i$ for the effective divisor
on $X$ corresponding to the $i$th ray of $\Sigma$.
Let $Y= X \cap T$, where $T \cong (K^*)^n$ is the torus of
$X_{\Sigma}$. Let $I(Y) \subset K[z_1^{\pm 1},\dots,z_n^{\pm 1}]$ be
the ideal of $Y$. Then $\Cox(X) \cong S/I$, where
$$I = (\phi(I(Y))S_{\mathbf{x}})\cap S,$$
and $\phi \colon K[T]
\rightarrow S_{\mathbf{x}}$ is the homomorphism
$\phi \colon K[z_1^{\pm 1},\dots,z_n^{\pm 1}] \rightarrow S_{\mathbf{x}}$ given
by $$\phi(z_i) = \prod_{j=1}^s x_j^{\ord_{E_j}(z_i)}.$$
\end{proposition}
\begin{proof}
The Cox ring of a normal projective variety is a domain \cite[Proposition 1.4.1.5]{CoxRingBook}.
This means that $I$ is a prime ideal, so $I=IS_{\mathbf{x}} \cap S$.
Note that $IS_{\mathbf{x}}$ is generated by its degree zero part
$(IS_{\mathbf{x}})_0$, so it suffices to show that the map $\phi$
identifies $\phi(I(Y))$ with $(IS_{\mathbf{x}})_0$. By
\cite[Proposition 2.3]{GibneyMaclaganEquations} it in turn suffices to
show that $\phi$ induces an isomorphism $\Spec(K[z_1^{\pm
1},\dots,z_n^{\pm 1}]) \cong T \subset X_{\Sigma}$.
The coordinate function $z_i$ is a unit in $K[Y]$, and so a rational
function on $X$. Its image $\phi(z_i)$ is also a unit in $K[Y]$.
This means that the divisors $\divv(z_i)$ and $\divv(\phi(z_i))$ on $X$
are supported on the boundary $X \setminus Y = X \setminus T =
\cup_{i=1}^s E_i$. A rational function $f$ is determined up to scalar
multiplication by its divisor $\divv(f)$.
As the ideal $I$ is also defined only up to the multiplication of the
variables $x_i$ by scalars, it thus suffices to show that
$\ord_{E_i}(z_j) = \ord_{E_i}(\phi(z_j))$. This is the case by
construction.
\end{proof}
To apply Proposition~\ref{p:CoxRingRelations} to $\overline{M}_{0,6}$ we need to
compute $\ord_D(z_{ij})$ for $D \in \{ \delta_I, Q_{\pi} : I \in \mathcal I, \pi \in \Pi\}$ and $\{i,j\}
\in \mathcal E$. Our method relies on the calculation of the classes
of these divisors $D$ in an explicit basis for $\Pic(\overline{M}_{0,6})$. As we rely
heavily on the $S_6$-action, we choose a description in which this
action is transparent. Fix a copy of $\mathbb Z^{16}$ with basis
labelled by $\{\mathbf{e}_i : 1 \leq i \leq 6 \}$ and
$\{\mathbf{e}_{1ij} : 2 \leq i<j \leq 6 \}$. Set the boundary divisor
$\delta_{1ij}$ equal to $2 \mathbf{e}_{1ij}$, and $\delta_{ij}$ equal
to $\mathbf{e}_i+\mathbf{e}_j - \sum_{i,j \in \{1,k,l\} \text{ or }
i,j \not \in \{1,k,l\}} \mathbf{e}_{1kl}$.
In this basis, for
\small$\pi=(1m)(ij)(kl)$ \normalsize, we have $Q_{\pi} = \sum_{i\in [6]}\mathbf{e}_i-2\sum_{r\in\{i,j\}\text{ and }s\in \{k,l\}}\mathbf{e}_{1rs}$.
The $S_6$-action on
$\mathbb Z^{16}$ then just permutes the indices. To check that this is indeed a description of $\Pic(\overline{M}_{0,6})$ one only needs to check that the given
vectors obey the known relations between the boundary divisors.
Alternatively, we give the explicit change of basis from the Kapranov
basis for $\overline{M}_{0,6}$, coming from the description of $\overline{M}_{0,6}$ as the blow-up of
$\mathbb P^3$ at $5$ points and $10$ lines. This has basis the
exceptional divisors $E_i, E_{jk}$ for $1 \leq i \leq 5$, $1 \leq j<k
\leq 5$, and the pull-back $\psi_6 = H$ of the hyperplane class on
$\mathbb P^3$.
The change of basis between
these two bases is given by the formulas:
\begin{equation*}
\begin{split}
\mathbf{e}_i&=\frac{1}{2}\big(H+E_i-\sum_{j\in [5]\setminus\{i\}}E_j+\frac{1}{2}(\sum_{j\in [5]\setminus\{i\}}E_{ij}-\sum_{j,k\in [5]\setminus\{i\}}E_{jk}\quad)\big),\quad \text{if } i<6,\\
\mathbf{e}_6&=\frac{1}{2}\big(-H+\sum_{i\in[5]}E_i+\frac{1}{2}\sum_{i,j\in[5]}E_{ij}\big),
\end{split}
\end{equation*}
and for $\{i,j\} \subset \{2,\dots ,6\}$, and $i<j$,
\[
\mathbf{e}_{1ij} =
\begin{cases}
1/2 E_{kl} & \text{if } j<6\text{ and }\{1,i,j,k,l\}=[5] \\
1/2 E_{1i} & \text{if } j=6
\end{cases}.
\]
Let $A$ be the $16\times
40$-matrix with columns indexed by the boundary divisors $\delta_I$
and $Q_\pi$, in the order
\[
\{12\},\dots,\{56\},\{123\},\dots ,\{156\},(12)(34)(56),\dots ,(16)(25)(34),
\]
and such that $A_I=[\delta_I]$ and $A_\pi=[Q_\pi] \in \Pic(\overline{M}_{0,6})$ in this basis.
Explicitly, $A$ has the form
\[
A=(A_{\text{bnd}}|A_{KV})
\]
where $A_{bnd}$ is the matrix
\begin{equation}
\label{e:matrixGb}
A_{bnd}=\left(
\begin{smallmatrix}
1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
1&0&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&1&0&0&0&1&0&0&0&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&1&0&0&0&1&0&0&1&0&0&1&1&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&1&0&0&0&1&0&0&1&0&1&0&1&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&1&0&0&0&1&0&0&1&0&1&1&0&0&0&0&0&0&0&0&0&0\\
{-1}&{-1}&0&0&0&{-1}&0&0&0&0&0&0&{-1}&{-1}&{-1}&2&0&0&0&0&0&0&0&0&0\\
{-1}&0&{-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}&0&0&{-1}&0&2&0&0&0&0&0&0&0&0\\
{-1}&0&0&{-1}&0&0&0&{-1}&0&{-1}&0&{-1}&0&{-1}&0&0&0&2&0&0&0&0&0&0&0\\
{-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}&{-1}&0&{-1}&0&0&0&0&0&2&0&0&0&0&0&0\\
0&{-1}&{-1}&0&0&0&0&{-1}&{-1}&{-1}&0&0&0&0&{-1}&0&0&0&0&2&0&0&0&0&0\\
0&{-1}&0&{-1}&0&0&{-1}&0&{-1}&0&{-1}&0&0&{-1}&0&0&0&0&0&0&2&0&0&0&0\\
0&{-1}&0&0&{-1}&0&{-1}&{-1}&0&0&0&{-1}&{-1}&0&0&0&0&0&0&0&0&2&0&0&0\\
0&0&{-1}&{-1}&0&{-1}&0&0&{-1}&0&0&{-1}&{-1}&0&0&0&0&0&0&0&0&0&2&0&0\\
0&0&{-1}&0&{-1}&{-1}&0&{-1}&0&0&{-1}&0&0&{-1}&0&0&0&0&0&0&0&0&0&2&0\\
0&0&0&{-1}&{-1}&{-1}&{-1}&0&0&{-1}&0&0&0&0&{-1}&0&0&0&0&0&0&0&0&0&2
\end{smallmatrix}\right)
\end{equation}
and
\begin{equation}
\label{e:matrixKV}
A_{KV}=\left(
\begin{smallmatrix}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\
0&0&0&0&0&0&0&{-2}&{-2}&0&{-2}&{-2}&0&{-2}&{-2}\\
0&0&0&0&{-2}&{-2}&0&0&0&{-2}&0&{-2}&{-2}&0&{-2}\\
0&0&0&{-2}&0&{-2}&{-2}&0&{-2}&0&0&0&{-2}&{-2}&0\\
0&0&0&{-2}&{-2}&0&{-2}&{-2}&0&{-2}&{-2}&0&0&0&0\\
0&{-2}&{-2}&0&0&0&0&0&0&{-2}&{-2}&0&{-2}&{-2}&0\\
{-2}&0&{-2}&0&0&0&{-2}&{-2}&0&0&0&0&{-2}&0&{-2}\\
{-2}&{-2}&0&0&0&0&{-2}&0&{-2}&{-2}&0&{-2}&0&0&0\\
{-2}&{-2}&0&{-2}&{-2}&0&0&0&0&0&0&0&0&{-2}&{-2}\\
{-2}&0&{-2}&{-2}&0&{-2}&0&0&0&0&{-2}&{-2}&0&0&0\\
0&{-2}&{-2}&0&{-2}&{-2}&0&{-2}&{-2}&0&0&0&0&0&0
\end{smallmatrix}\right)
\end{equation}
We then get the following description of the matrix $R$ that induces
the morphism $\phi$ of Proposition~\ref{p:CoxRingRelations}.
\begin{lemma} \label{l:Rdefinition}
Let $R$ be the matrix with columns indexed by $\mathcal I$ and $\Pi$ ordered as before, and rows indexed by the generators $z_{ij},u_\pi$ of $\mathbb C[Y]$, and entries
\begin{equation}
\label{Rdefinition}
\begin{split}
R_{ij,I}&=\ord_{\delta_I}(z_{ij}),\\
R_{ij,\pi}&=\ord_{Q_\pi}(z_{ij}),\\
R_{\pi,I}&=\ord_{\delta_I}(u_\pi),\\
R_{\pi_1,\pi_2}&=\ord_{Q_{\pi_2}}(u_{\pi_1}).
\end{split}
\end{equation}
Then
\begin{equation}
\label{e:matrixR}
R=
\left(\begin{smallarray}{ccccccccccccccccccccccccc|ccccccccccccccc}
0& 1& -1& 0& 0& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
0& 1& 0& -1& 0& -1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& -1& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
0& 1& 0& 0& -1& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 0& -1& 0& 0& -1& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 0& 0& -1& 0& -1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& -1& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 0& 0& 0& -1& -1& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 1& 1& 0& 0& 0& 0& 0& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& 0& -1& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1& 0& 0& 1& 0& 0& 1& -1& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& 0& -1& -1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 1& 0& 0& 1& 0& -1& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& 0& -1& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 0& 0& 1& 0& 0& -1& -1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
\noalign{\vskip1pt\hrule\vskip1pt}
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& -1& -1& -2& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& -1& -2& -1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& -2& -1& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& -1& -1& -2& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& -1& -2& -1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& 0& -2& -1& -1& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 1& 1& -2& -2& -2& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1& 0& 1& 0& -2& -2& -1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& 1& -2& -2& -1& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 1& 0& 1& -2& -2& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1& 1& 0& 0& -2& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& 0& 1& -2& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 1& 1& 0& -2& -2& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 1& 0& 0& -1& -2& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\
1& 1& -1& -1& -1& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& 1& 0& -1& -2& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1
\end{smallarray}\right)
\end{equation}
\end{lemma}
\begin{proof}
Write $R$ for the matrix with entries given by orders of vanishing,
and $\tilde{R}$ for the matrix on the right hand side of \eqref{e:matrixR}. The $z_{ij}$ and
$u_{\pi}$ are rational functions on $\overline{M}_{0,6}$ whose divisors are supported
on the union of the boundary divisors $\delta_I$ and the $Q_{\pi}$. Note that $u_{\pi}=f_{\pi}$ in $\mathbb C[Y]$.
Since $\divv(z_{ij})$ and $\divv(f_{\pi})$ are principal divisors,
$\sum_{I} \ord_{\delta_I}(z_{ij}) \delta_I + \sum_{\pi} \ord_{Q_{\pi}}(z_{ij}) Q_{\pi}
= \sum_{I} \ord_{\delta_I}(f_{\pi})\delta_I + \sum_{\pi}
\ord_{Q_{\pi}}(f_{\pi})Q_{\pi} = 0 \in \Pic(\overline{M}_{0,6})$. The columns of the
matrix $A$ are representatives for the boundary divisors and the
Keel-Vermeire divisors in $\Pic(\overline{M}_{0,6})$, so this implies that $AR^T =0$. Since the rows
of $\tilde{R}$ are a basis for $\ker(A)$, every row of $R$ lies in the
rowspace of $\tilde{R}$.
Note that $\tilde{R}$ has the block form:
\[
\tilde{R}=\left(
\begin{array}{c|c}
R' &0 \\ \hline
C &I_{15}
\end{array}\right)
\]
where $I_{15}$ is the identity matrix of rank 15 and $R'$ is the $9
\times 25$ matrix appearing in Proposition 5.2 of
\cite{GibneyMaclaganEquations}. We can further decompose $R' = (R'_1
| I_9 | R'_2)$, where $R'_1$ is $9 \times 6$, and $R'_2$ is $9 \times
10$, and $C = (C_1 | \mathbf{0} | C_2)$, where $C_1$ is $15 \times 6$,
$\mathbf{0}$ is the $15 \times 9$ matrix with every entry $0$, and
$C_2$ is $15 \times 10$. An element of the rowspace of $\tilde{R}$
then has four blocks of coordinates, and is determined by its values
in the second and fourth blocks, which are indexed by the
$\delta_{ij}$ with $\{i,j\} \in \mathcal E$, and by the $Q_{\pi}$ with
$\pi \in \Pi$.
To show that $R = \tilde{R}$ it thus suffices to show the following four facts:
\begin{enumerate}
\item $\ord_{\delta_{ij}}(z_{kl}) = 1$ if $ij=kl$, and $0$ otherwise,
for $\{i,j\}, \{k,l\} \in \mathcal E$,
\item $\ord_{\delta_{ij}}(f_{\pi}) = 0$ for
$\{i,j\} \in \mathcal E$ and $\pi \in \Pi$,
\item $\ord_{Q_{\pi}}(z_{ij}) = 0$ for
$\{i,j\} \in \mathcal E$ and $\pi \in \Pi$, and
\item $\ord_{Q_{\pi}}(f_{\pi'}) = 1$ if $\pi = \pi'$, and $0$ otherwise,
for $\pi, \pi' \in \Pi$.
\end{enumerate}
Since $Q_{\pi}$ intersects $M_{0,6}$, the last two follow from the
fact that $f_{\pi}$ is an equation for $Q_{\pi}$, and no $z_{ij}$ with
$\{i,j\} \in \mathcal E$ or $f_{\pi'}$ with $\pi' \neq \pi$ lies in the ideal $I_6 + \langle f_{\pi} \rangle$.
To see the first two, we use the description of \cite[Theorem
5.7]{GibneyMaclaganEquations}, where it is shown that $\overline{M}_{0,6}$ is the
closure of $M_{0,6} \subset (\mathbb C^*)^9$ in a toric variety $X_{\Delta}$ given
by a fan $\Delta$ whose rays are spanned by the columns of $R'$. The
intersection of $\overline{M}_{0,6}$ with the torus-invariant divisor corresponding
to the ray indexed by $\delta_I$ is $\delta_I$. For $\{i,j \} \in
\mathcal E$ the ray indexed by $\delta_{ij}$ is the basis vector
$\mathbf{e}_{ij}$. Thus the variety of $\overline{I}_6 := I_6 \cap
\mathbb C[z_{ij}, z_{kl}^{\pm 1} : \{k,l \} \neq \{i,j\}]$ in $\mathbb A^1
\times T^8$ is the union of $M_{0,6}$ and an open part of
$\delta_{ij}$. This means that to show that $\ord_{\delta_{ij}}(g)=0$
for $g \in \mathbb C[z_{ij}, z_{kl}^{\pm 1}]/\overline{I}_6$ it suffices to
show that $g \not \in \overline{I}_6 + \langle z_{ij} \rangle$. This
is the case for $g=z_{kl}$ with $\{k,l \} \neq \{i,j\}$, and for
$g=f_{\pi}$. Since in addition we have $\ord_{\delta_{ij}}(z_{ij}) =
1$, this finishes the proof.
\end{proof}
We will make extensive computational use of the $S_6$-action on
$\Cox(\overline{M}_{0,6})$. We first review how this works for automorphisms of Mori
Dream Spaces in more generality.
Let $X$ be a Mori Dream Space with Cox ring $S/I$, where $S = \mathbb
C[x_1,\dots,x_s]$. Suppose that $G$ is a group acting on $S$ by
monomial maps: $g \cdot x_i = \alpha x^u$ for some $\alpha \in \mathbb
C$ and monomial $x^u$ with $\deg(x^u) = \deg(x_i)$. Write $\mathbf{x}
= \prod_{i=1}^s x_i$. The assumption that $S$ acts by monomial maps
means that the action of $G$ on $S$ extends to an action of $G$ on
$S_{\mathbf{x}}$, which then restricts to an action on the Laurent
polynomial ring $(S_{\mathbf{x}})_{\mathbf{0}}$. Write $X^0 =
\Spec(((S/I)_{\mathbf{x}})_{\mathbf{0}})$ for the complement in $X$ of
the divisors of the $x_i$ for $1 \leq i \leq s$. In the next lemma we
show that if the action restricts to a given action of $G$ on $X^0$,
then the $G$ action on $S$ fixes $I$.
\begin{lemma} \label{l:Gaction}
Let $X$ be a Mori dream space with Cox ring $\Cox(X) \cong S/I$ where
$S = \mathbb C[x_1,\dots,x_s]$. Suppose that a group $G$ acts by
degree-preserving monomial maps on $S$. Then the ideal $I$ is invariant under the action of
$G$ if the restriction of the action of $G$ to
$(S_{\mathbf{x}})_{\mathbf{0}}$ preserves
$I^0=(IS_{\mathbf{x}})_{\mathbf{0}}$.
\end{lemma}
\begin{proof}
It suffices to show that $g \cdot f \in I$ for all $f \in I$. That
shows that $g I \subset I$, and the reverse inclusion then follows
from applying $g^{-1}$. For $f \in I^0S_{\mathbf{x}} \cap S$,
we have $f = \sum f_i h_i/\mathbf{x}^{m_i}$ for some $h_i \in S$
and $f_i \in I^0$. Then $g \cdot f = \sum (g \cdot f_i) (g \cdot
h_i)/g \cdot \mathbf{x}^{m_i} = \sum (g\cdot f_i) (g \cdot h_i) /( g
\cdot \mathbf{x}^{m_i}) = \sum f'_i h'_i/\mathbf{x}^{n_i}$, where
$f'_i \in I_0$, by hypothesis, and $h'_i \in S$. Here we use that the
action of $G$ on $S$ is via monomial maps to know that $g \cdot
\mathbf{x}^{m_i}$ is again a scalar multiple of a monomial. Since $G$ acts on $S$, we have
$g \cdot f \in S$, so this shows that $g \cdot f \in
I^0S_{\mathbf{x}} \cap S$. As this equals $I$ by
Proposition~\ref{p:CoxRingRelations}, we have the desired result.
\end{proof}
Note that the induced map $\mathbb C[z_{ij}^{\pm 1}] \rightarrow
\mathbb C[x_I^{\pm 1}, y_{\pi}^{\pm 1} : I \in \mathcal I, \pi \in \Pi]$ given by the
matrix \eqref{e:matrixR} has the property that $\phi(z_{ij})$ is a
monomial in the variables $x_{ijk}$ times the formula given in
\eqref{eqtn:zij}.
We will apply Lemma~\ref{l:Gaction} to the $S_6$-action on $\overline{M}_{0,6}$. Let
$S = \mathbb C[x_I, y_{\pi} : I \in \mathcal I, \pi \in \Pi]$. We
have the isomorphism $(S_{\mathbf{x}})_{\mathbf{0}} \cong \mathbb
C[z_{24}^{\pm 1},\dots,z_{56}^{\pm 1}]$, where $S_6$ acts on $\mathbb
C[z_{24}^{\pm 1},\dots, z_{56}^{\pm 1}]$ as in
Table~\ref{t:S6action1}.
Consider the $S_6$-action on $S$ defined as follows. We set $\sigma
\cdot x_{ij} = x_{\sigma(i) \sigma(j)}$, with the convention that
$x_{ji} = -x_{ij}$. This can be regarded as induced from the natural
action of $S_6$ on $\wedge^2 \mathbb Z^6$. Similarly, the $S_6$-action on the variables $x_{ijk}$ is induced from the natural $S_6$-action on $\Sym^2(\wedge^3 \mathbb Z^6)$. Here the $\Sym^2$ arises
from the fact that $x_{123}=x_{456}$. Explicitly, this means that, for
example, $(12) x_{123} = -x_{123}$, $(12) x_{456} = -x_{456}$, and
$(12)x_{145}= x_{245}$. It is then a straightforward computation to
check that $\phi(\sigma \cdot z_{ij}) = \sigma \cdot \phi(z_{ij})$ for
all $\sigma \in S_6$. Since the action of $S_6$ on the variables
$z_{ij}$ is induced from the action on $x_{ij}$ using
\eqref{eqtn:zij}, it is only necessary for this to check that this
action is correct on the variables $x_{ijk}$.
The action of $S_6$ on the variables $y_{\pi}$ is then uniquely
determined by the requirement that $\sigma \cdot \phi(u_{\pi}) =
\phi(\sigma \cdot u_{\pi})$, as $\phi(u_{\pi}) = m y_{\pi}$ for a
monomial $m$ in the variables $x_I$. Explicitly, this action is
induced from the natural $S_6$-action on $\wedge^3 (\Sym^2 \mathbb
Z^6)$. For example, $(12) y_{(12)(34)(56)} = y_{(12)(34)(56)}$ since
$(12) ((\mathbf{e}_1+\mathbf{e}_2) \wedge (\mathbf{e}_3+\mathbf{e}_4)
\wedge (\mathbf{e}_5 + \mathbf{e}_6)) = (\mathbf{e}_1+\mathbf{e}_2)
\wedge (\mathbf{e}_3+\mathbf{e}_4) \wedge (\mathbf{e}_5 +
\mathbf{e}_6)$. Note that a careful choice of signs in
Table~\ref{t:QpiEqns} was required for the action to work in this
fashion.
\section{The relations}
\label{s:maintheorem}
In this section we apply the method of
Proposition~\ref{p:CoxRingRelations} to compute the relations in
$\Cox(\overline{M}_{0,6})$. By Castravet's theorem
(Theorem~\ref{t:Castravet}) this Cox ring is generated by sections
$x_I$ of the $25$ boundary divisors $\delta_{I}$, and by sections
$y_{\pi}$ of the $15$ Keel-Vermeire divisors $Q_{\pi}$. The main
theorem of this section describes the ideal of relations between these
$40$ generators.
{
\renewcommand{\thetheorem}{\ref{t:maintheorem}}
\begin{theorem}
Let $S = \mathbb C[x_I, y_{\pi} : I \in \mathcal I, \pi \in \Pi]$. The Cox ring
of $\overline{M}_{0,6}$ is $S/I_{\overline{M}_{0,6}}$, where $I_{\overline{M}_{0,6}}$ is an ideal with 225 generators,
which come in five symmetry classes:
\begin{enumerate}
\item \label{item:case1} $15$ of the form:
\begin{equation*}
x_{ij}x_{kl}x_{ijn}x_{kln}-x_{ik}x_{jl}x_{ikn}x_{jln}+x_{il}x_{jk}x_{iln}x_{jkn}
\end{equation*}
for $1\leq i<j<k<l\leq 6$, $m<n$ and $\{i,j,k,l,m,n\}=\{1,2,3,4,5,6\}$;
\item \label{item:case2} $60$ of the form:
\begin{equation*}
x_{1ik}y_{(1m)(ij)(kl)}+x_{1j}x_{mk}x_{il}x_{1mj}x_{1mk}x_{1ij}+ x_{1l}x_{mi}x_{jk}x_{1mi}x_{1ml}x_{1kl},
\end{equation*}
for $\{i,j,k,l,m\}=\{2,3,4,5,6\}$;
\item \label{item:case3} $45$ of the form:
\begin{equation*}
x_{ij}y_{(ij)(kl)(mn)} + x_{ik}x_{il}x_{jm}x_{jn}x_{ikl}^2 - x_{im}x_{in}x_{jl}x_{jk}x_{imn}^2,
\end{equation*}
for $\{i,j,k,l,m,n\}=\{1,2,3,4,5,6\}$ and $i<j$;
\item \label{item:case4} $45$ of the form:
\begin{equation*}
y_{(ij)(kl)(mn)}y_{(ij)(km)(ln)}-x_{il}x_{lj}x_{jm}x_{mi}x_{nk}^{2}x_{ijm}^{2}x_{ijl}^{2} +x_{in}x_{nj}x_{jk}x_{ki}x_{ml}^{2}x_{ijk}^{2}x_{ijn}^{2},
\end{equation*}
for $\{i,j,k,l,m,n\} = \{1,2,3,4,5,6\}$;
\item \label{item:case5} $60$ of the form:
\begin{multline*}
\qquad \quad y_{(1i)(jk)(lm)}y_{(1j)(il)(km)} - x_{1k}x_{1l}x_{ik}x_{im}x_{jl}x_{jm}x_{1ik}x_{1il}x_{1jk}x_{1jl}\\
+x_{1k}x_{1l}x_{ij}x_{im}x_{jm}x_{kl}x_{1ij}x_{1il}x_{1jk}x_{1kl} -x_{1k}x_{1m}x_{ij}x_{im}x_{jl}x_{kl}x_{1ij}x_{1im}x_{1jk}x_{1km}\\
-x_{1l}x_{1m}x_{ij}x_{ik} x_{jm} x_{kl} x_{1ij} x_{1il} x_{1jm} x_{1lm},
\end{multline*}
for $\{i,j,k,l,m\} = \{2,3,4,5,6\}$.
\end{enumerate}
Here we follow the convention that $x_{ij}\!=\!-x_{ji}$,
$x_{ijk}\!=\!-x_{jik}=\!-x_{ikj}$, $x_{ijk}\!=\!x_{lmn}$ for
\small$\{i,j,k,l,m,n \}= \{1,2,3,4,5,6\}$\normalsize, with $i<j<k$ and
$l<m<n$, and $y_{(ij)(kl)(mn)}=-y_{(kl)(ij)(mn)}=-y_{(ij)(mn)(kl)}$.
\end{theorem}
}
\begin{proof}
By Example~\ref{e:M06embedding} $\overline{M}_{0,6}$ has an embedding
$i:\overline{M}_{0,6} \hookrightarrow X_{\Sigma}$ into a $24$-dimensional toric
variety in such a way that the intersection of $\overline{M}_{0,6}$ with the
torus $T^{24}$ of $X_{\Sigma}$ is $Y=M_{0,6} \setminus \cup_{\pi \in \Pi} Q_{\pi}$.
By Lemma~\ref{l:idealY} we have
\[
\mathbb C[Y]=\mathbb C[z_{ij}^{\pm 1},u_\pi^{\pm 1}: 2\leq i<j \leq 6, ij \neq 23, \pi \in \
\Pi]/J_{KV},
\]
where $J_{KV}=I_6+\left< u_\pi-f_\pi: \pi \in \Pi \right> \subset
\mathbb C[T^{24}]$. Here $I_6$ and $f_{\pi}$ are as defined in
Lemma~\ref{l:M06equations} and Table~\ref{t:QpiEqns}.
We denote by $\mathbf{x}\mathbf{y}$ the product $\prod_{I \in \mathcal I}x_I
\prod_{\pi \in \Pi} y_{\pi}$. Let $\phi$ be given by
\begin{equation*}
\label{e:phi}
\begin{split}
\phi:\mathbb C[z_{ij}^{\pm 1},u_\pi^{\pm 1}] \rightarrow \big(S_{\mathbf {xy}}\big)_0\\
z_{ij} \mapsto \prod_{I}x_I^{R_{{ij},I}},\quad
\\
u_\pi \mapsto y_\pi\prod_{I}x_I^{R_{\pi,I}},
\end{split}
\end{equation*}
where $R$ is the matrix defined in (\ref{e:matrixR}) and $R_{{ij},I} $
(respectively $R_{{ij},\pi}$) denotes the entry with row labelled
${ij}$ and column labelled $I$ (respectively $\pi$).
Proposition~\ref{p:CoxRingRelations} then implies that the ideal
$I_{\overline{M}_{0,6}}$ equals
$$I_{\overline{M}_{0,6}} = (\phi(J_{KV})S_{\mathbf{x}\mathbf{y}}) \cap S.$$
In order to compute the ideal of relations we thus only need to
compute $\phi(I(Y))S_{\mathbf{xy}} \cap S$. We will exhibit the
different symmetry types as elements of $\phi(I(Y))S_{\mathbf{xy}}
\cap S$ and then argue that the ideal generated by these equations is
saturated with respect to the product of the variables, so equals
$I_{\overline{M}_{0,6}}$.
\begin{description}
\item[(1)]
For $j,k,l$ with $[6]=\{1,2,3,j,k,l\}$ we have
\[
\phi_R(z_{2j})=\frac{x_{2j}x_{13}x_{13k}x_{13l}}{x_{1j}x_{23}x_{1jk}x_{1jl}},
\]
\[
\phi_R(z_{3j})=\frac{x_{3j}x_{12}x_{12k}x_{12l}}{x_{1j}x_{23}x_{1jk}x_{1jl}}
\]
and
\[
\phi_R(z_{jk})=\frac{x_{jk}x_{12}x_{13}x_{123}x_{12l}x_{13l}}{x_{1j}x_{1k}x_{23}x_{1jk}x_{1jl}x_{1kl}}.
\]
Then,
\begin{align*}
\phi_R(z_{34}+1-z_{24})&=\frac{x_{12}x_{34}x_{125}x_{126}}{x_{14}x_{23}x_{145}x_{146}}-\frac{x_{13}x_{24}x_{135}x_{136}}{x_{14}x_{23}x_{145}x_{146}}+1\\
&=\frac{x_{12}x_{34}x_{125}x_{126}-x_{13}x_{24}x_{135}x_{136}+x_{14}x_{23}x_{145}x_{146}}{x_{14}x_{23}x_{145}x_{146}}.
\end{align*}
Thus the polynomial
\begin{equation} \label{eqtn:case1}
x_{12}x_{34}x_{125}x_{126}-x_{13}x_{24}x_{135}x_{136}+x_{14}x_{23}x_{145}x_{146}
\end{equation}
times a Laurent monomial in the variables $x_I$ is in $\phi_R(I(Y))$
and therefore it itself is in
$\big(\phi_R(I(Y))S_{\mathbf{xy}}\big)\cap S$. This is the case $i=1,
j=2, k=3, l=4, m=5, n=6$ of case \ref{item:case1}. The $S_6$-orbit of this
polynomial has size $15$; this can be checked directly by hand, or
using the accompanying {\tt Macaulay2} package available from the
authors' webpages \cite{M06M2package}.
\item[(2)]
Let $\pi=(12)(34)(56)$. Then $u_\pi -f_\pi=u_\pi-z_{24}+z_{25} z_{26}$ and
\begin{align*}
\qquad \quad h&=u_\pi+ z_{45} +z_{25} z_{36}\\
&=(u_{\pi}-f_{\pi}) +z_{25}(z_{36}+1-z_{26})+(z_{45}-z_{25}+z_{24})\quad \qquad
\end{align*}
is an element of $I(Y)$. Therefore
\begin{multline*}
\qquad \quad\phi_R(h)=\Big(\frac{x_{12}x_{13}x_{136}}{x_{14}x_{15}x_{16}x_{23}^{2}x_{145}x_{146}x_{156}^{2}}\Big)\cdot\\
(x_{135}y_{\pi}+x_{14}x_{25}x_{36}x_{124}x_{125}x_{134}+x_{16}x_{23}x_{45}x_{123}x_{126}x_{156})\qquad
\end{multline*}
and hence
\begin{equation} \label{eqtn:case2}
x_{135}y_{(12)(34)(56)}+x_{14}x_{25}x_{36}x_{124}x_{125}x_{134}+x_{16}x_{23}x_{45}x_{123}x_{126}x_{156}
\end{equation}
is an element of $\big(\phi_R(I(Y))S_{\mathbf{xy}}\big)\cap S$. This
the case $i=3, j=4, k=5, l=6, m=2$ of case~\ref{item:case2}. The
$S_6$-orbit has size $60$; again this can be checked by hand, or using the
accompanying package.
\item[(3)]
For $\pi=(12)(34)(56)$, we have that
\begin{multline*}
\qquad \quad\phi_R(u_\pi-f_\pi)=\Big(\frac{x_{13}x_{135}x_{136}}{x_{14}x_{15}x_{16}x_{23}^{2}x_{145}x_{146}x_{156}^{2}}\Big)\cdot\\
(x_{12}y_{\pi}+x_{13}x_{14}x_{25}x_{26}x_{134}^{2}-x_{15}x_{16}x_{23}x_{24}x_{156}^{2})\qquad \quad
\end{multline*}
and therefore
\begin{equation} \label{eqtn:case3}
x_{12}y_{\pi}+x_{13}x_{14}x_{25}x_{26}x_{134}^{2}-x_{15}x_{16}x_{23}x_{24}x_{156}^{2}
\end{equation}
is an element of $\big(\phi_R(I(Y))S_{\mathbf{xy}}\big)\cap S$. This
is the case $i=1, j=2, k=3, l=4, m=5, n=6$ of case~\ref{item:case3}.
The $S_6$-orbit has size $45$.
\item[(4)]
Let $\pi_1=(12)(34)(56)$ and $\pi_2=(12)(35)(46)$.
Let
\[
h=y_{\pi_1}y_{\pi_2}-x_{14}x_{15}x_{24}x_{25}x_{36}^{2}x_{124}^{2}x_{125}^{2}+x_{13}x_{16}x_{23}x_{26}x_{45}^{2}x_{123}^{2}x_{126}^{2}.
\]
Let $F_1$ be the polynomial \eqref{eqtn:case1},
let $F_2$ be the
polynomial~\eqref{eqtn:case2}, and let $F_3$ be the polynomial of \eqref{eqtn:case3}.
Set $g = F_3 ((45) \cdot F_3)$.
We claim that
$x_{12}^2h - g$ lies in the ideal
\[\langle (35) \cdot F_1, (46) \cdot F_1, (36)\cdot F_2, (45) \cdot F_2, (34)(56) \cdot F_2, F_2 \rangle \subseteq
\big(\phi_R(I(Y))S_{\mathbf{xy}}\big)\cap S.
\]
This can in principle be checked by hand, but is also confirmed in the accompanying {\tt Macaulay2} package. Since $F_3 \in
\big(\phi_R(I(Y))S_{\mathbf{xy}}\big)\cap S$, we also have $g \in
\big(\phi_R(I(Y))S_{\mathbf{xy}}\big)\cap S$. Thus
$x_{12}^2h$ is also in $\phi_R(Y)S_{\mathbf{xy}}$ and so $h \in
I_{\overline{M}_{0,6}}$. This is the case $i=1, j=2, k=3, l=4, m=5, n=6$ of case~\ref{item:case4}. The $S_6$-orbit has size $60$.
\item[(5)]
Finally, let
$\pi_1=(12)(34)(56)$ and $\pi_2=(13)(25)(46)$. Set
\begin{multline*}
\qquad \quad h=y_{\pi_1}y_{\pi_2}-x_{14}x_{15}x_{24}x_{26}x_{35}x_{36}x_{124}x_{125}x_{134}x_{135}\\
+x_{14}x_{15}x_{23}x_{26}x_{36}x_{45}x_{123}x_{125}x_{134}x_{145}\\\qquad-x_{14}x_{16}x_{23}x_{26}x_{35}x_{45}x_{123}x_{126}x_{134}x_{146}\\
-x_{15}x_{16}x_{23}x_{24} x_{36} x_{45} x_{123} x_{125} x_{136} x_{156}.
\end{multline*}
Set $g = F_3((23)(45) \cdot F_3) \in
\phi_R(I(Y))S_{\mathbf{xy}}$.
We claim that $x_{12}x_{13}h-g$ lies in the ideal
\begin{multline*}
\qquad \langle (25) \cdot F_1, (15) \cdot F_1, (25)(46) \cdot F_1, (35) \cdot F_1, (46) \cdot F_1, (36) \cdot F_1, \\ (236) \cdot F_2, (12) \cdot F_2, (154) \cdot F_3, (13)(24) \cdot F_3 \rangle.\\
\end{multline*}
Thus $x_{12}x_{13}h \in \phi_R(I(Y))S_{\mathbf{xy}}$ and therefore
$h\in I_{\overline{M}_{0,6}}$. This is the case $i=2, j=3, k=4, l=5, m=6$ of
case~\eqref{item:case5}. The $S_6$-orbit has size $60$.
\end{description}
This shows that one representative of each $S_6$-orbit from the
statement of the theorem lies in
$(\phi(J_{KV})S_{\mathbf{x}\mathbf{y}}) \cap S$.
Lemma~\ref{l:Gaction} and the explicit action of $S_6$ given at the
end of Section~\ref{s:coxequations} then imply that all $225$
polynomials given in the theorem statement are in $I_{\overline{M}_{0,6}}=
(\phi(J_{KV})S_{\mathbf{x}\mathbf{y}}) \cap S$. To show that these
polynomials in fact generate $I_{\overline{M}_{0,6}}$, we first note that the ideal
generated by these $225$ generators in $S_{\mathbf{x}\mathbf{y}}$
equals $\phi(J_{KV})S_{\mathbf{x}\mathbf{y}}$. This is the case
because the polynomial of case~\ref{item:case1} is a monomial multiple of the image under
$\phi$ of a representative of one $S_6$-orbit of generators of
$J_{KV}$: $z_{34}+1-z_{24}$. The polynomial of case~\ref{item:case2} is a monomial
multiple of the image under $\phi$ of the other $S_6$-orbit of
generators: $y_{\pi}-f_{\pi}$. It thus suffices to check that the
ideal generated by the given $225$ generators is saturated with
respect to the product of the variables. This can be done,
for example, using the accompanying {\tt Macaulay2}
package.
\end{proof}
\section{Small birational models}
\label{s:smallbirationalmodels}
A positive consequence of the fact that $\overline{M}_{0,6}$ is a Mori dream space
is that the movable cone of $\overline{M}_{0,6}$ decomposes into finitely many Mori
chambers, each of which is the pullback of the nef cone of a
birational model of $\overline{M}_{0,6}$, and each small $\mathbb Q$-factorial
modification of $\overline{M}_{0,6}$ occurs in this list. These chambers are
precisely the GIT chambers of the GIT description
\begin{equation} \label{eqtn:bigGIT}
\overline{M}_{0,6} = V(I_{\overline{M}_{0,6}}) \ensuremath{\operatorname{/\!\!/}}_{\alpha} H
\end{equation}
given in Example~\ref{e:M06embedding}.
The main result of this section is Proposition~\ref{p:Esuffices},
which gives a simpler GIT problem whose chambers also describe the
chamber decomposition of the movable cone. The new GIT problem is the
quotient of a variety in $\mathbb A^{25}$ instead of $\mathbb A^{40}$,
which makes the computations significantly simpler. An updated
version of \cite{BKR} (Remark 6.7) takes this decomposition into
account.
\begin{lemma} \label{l:Qpihyperplane}
Fix $\pi= (ij)(kl)(mn)$. The hyperplane $\Delta_{\pi}$ in $\Pic(\overline{M}_{0,6})$ spanned by
$\{ \delta_{ab} : \{a,b\} \not \in \{ \{i,j\}, \{k,l\},
\{m,n\} \} \cup \{ \delta_{abc} : i,j \in \{a,b,c\} \text{ or } k, l
\in \{a ,b ,c \} \text{ or } m,n \in \{a,b,c \} \}$ separates
$Q_{\pi}$ from all other boundary divisors and $Q_{\pi'}$.
\end{lemma}
\begin{proof}
Consider the element $\rho=\sum_{i=1}^6 \mathbf{e}_{i}^* +
\mathbf{e}^*_{ikm} + \mathbf{e}^*_{ikn} + \mathbf{e}^*_{ilm} +
\mathbf{e}^*_{iln} \in \Pic(\overline{M}_{0,6})^*$, where the $*$ denotes the dual
basis element in the symmetric basis for $\Pic(\overline{M}_{0,6})$. By examining the
matrices $A_{bnd}$ and $A_{KV}$ given in \eqref{e:matrixGb} and
\eqref{e:matrixKV}, we obtain $\rho(Q_{\pi}) = -2$,
$\rho(\delta_{ij})=\rho(\delta_{kl})=\rho(\delta_{mn}) = 2$,
$\rho(\delta_{ikm}) = \rho(\delta_{ikn}) = \rho(\delta_{ilm}) =
\rho(\delta_{iln}) = 2$, and all other $\rho(\delta_{I})=0$. In
addition $\rho(Q_{\pi'}) \in \{2,3\}$ for all $\pi' \in \Pi$ with $\pi'
\neq \pi$. Let $H$ be the hyperplane $\{ x \in \Pic(\overline{M}_{0,6}) \otimes
\mathbb R: \rho(x)=0 \}$. Then $H$ contains the generators of
$\Delta_{\pi}$, so it remains to check that $\spann(\Delta_{\pi})$ is
fifteen dimensional, which can be done by computing the rank of the
corresponding $16 \times 18$ matrix.
\end{proof}
Let $J = I_{\overline{M}_{0,6}} \cap \mathbb C [x_I : I \in \mathcal I]$. This is
generated by the $15$ polynomials of the first symmetry class of
Theorem~\ref{t:maintheorem}, and by the $S_6$-orbit, which also has
size $15$, of the polynomial $x_{12}x_{26}x_{34}x_{35}x_{45}x_{126}^2
- x_{13}x_{24}x_{25}x_{36}x_{45}x_{136}^2 +
x_{14}x_{23}x_{25}x_{35}x_{46}x_{146}^2 -
x_{15}x_{23}x_{24}x_{34}x_{56}x_{156}^2$. The Picard torus $H$ acts
on the affine space $\mathbb A^{25}$ with coordinates $\{x_I : I \in
\mathcal I\}$. Since $J$ is homogeneous with respect to the
induced grading, $H$ also acts on $V(J) \subset \mathbb A^{25}$. By
\cite{GibneyMaclaganEquations}*{Theorem 7.1} we have
\begin{equation} \label{eqtn:newGIT} \overline{M}_{0,6} = V(J) \ensuremath{\operatorname{/\!\!/}}_{\alpha} H
\end{equation} where $\alpha$ is any character of $H$
corresponding to an ample divisor on $\overline{M}_{0,6}$.
Let $E = \pos(\delta_I : I \in \mathcal I) \subseteq \Pic(\overline{M}_{0,6})_{\mathbb
R}$ be the cone generated by the boundary divisors.
\begin{proposition} \label{p:Esuffices}
The movable cone $\Mov(\overline{M}_{0,6})$ of $\overline{M}_{0,6}$ is contained in the cone $E$. The
GIT chambers of the GIT problem \eqref{eqtn:newGIT} that intersect
$\Mov(\overline{M}_{0,6})$ contain the Mori chambers for $\overline{M}_{0,6}$, and the corresponding
birational models equal the GIT quotients
$$ V(J) \ensuremath{\operatorname{/\!\!/}}_{\beta} H$$
as $\beta$ varies over the relative interior of the chambers.
\end{proposition}
\begin{proof}
The movable cone for $\overline{M}_{0,6}$ is the movable cone of the toric variety
$X_{\Sigma}$. For $\rho \in \mathcal I \cup \Pi$ we
write $D_{\rho}$ for $\delta_I$ if $\rho=I \in \mathcal I$ and for
$Q_{\pi}$ if $\rho = \pi \in \Pi$. The movable cone is then
$$\Mov(\overline{M}_{0,6}) = \cap_{\rho \in \mathcal I \cup \Pi } \pos(D_{\rho'} : \rho' \in
\mathcal I \cup \Pi, \rho' \neq \rho).$$
Write $S$ for the polynomial ring $\mathbb C[ x_I, y_{\pi} : I \in
\mathcal I, \pi \in \Pi]$, and $S'$ for the polynomial ring $\mathbb
C[ x_I : I \in \mathcal I]$. The inclusion $S'/J \rightarrow
S/I_{\overline{M}_{0,6}}$ induces an inclusion $$\phi_{\mathbf{v}} \colon \oplus_{\ell
\geq 0} (S'/J)_{\ell \mathbf{v}} \rightarrow \oplus_{\ell \geq 0}
(S/I_{\overline{M}_{0,6}})_{\ell \mathbf{v}}$$ for any $\mathbf{v} \in \Mov(\overline{M}_{0,6})$. It suffices to
show that $\phi_{\mathbf{v}}$ is surjective for any $\mathbf{v}
\in \Mov(\overline{M}_{0,6})$, and thus an isomorphism. This shows that the
birational models of $\overline{M}_{0,6}$ have the form $V(J) \ensuremath{\operatorname{/\!\!/}}_{\beta} H$ as
$\beta$ varies over the GIT chambers for \eqref{eqtn:newGIT}. Since
the target of $\phi_{\mathbf{v}}$ is nonzero for all $\mathbf{v} \in
\Mov(\overline{M}_{0,6})$, the source must be as well, which shows that $\mathbf{v}
\in E$, and thus $\Mov(\overline{M}_{0,6}) \subseteq E$.
Fix a monomial $m \in S$ of degree $l\mathbf{v}$ for some
$l>0$. We will show that there is a
polynomial $f \in S$ with no term divisible by any $y_{\pi}$ with $m-f \in
I_{\overline{M}_{0,6}}$. We proceed iteratively. Set $f_0=m$, so $m-f_0 \in I_{\overline{M}_{0,6}}$
holds trivially. Suppose that at some stage $i$ some monomial of
$f_i$ is divisible by $y_{\pi}y_{\pi'}$ with $\pi \neq \pi'$. Let the
maximum number of distinct $y_{\pi}$ dividing any monomial in $f_i$ be
$k_i$, and choose one such monomial $\tilde{m}$. We can then subtract
an appropriate multiple of one of the generators of $I_{\overline{M}_{0,6}}$ of the
form \eqref{item:case4} or \eqref{item:case5} to replace $\tilde{m}$.
This gives a polynomial $f_{i+1}$ with $m-f_{i+1} \in I_{\overline{M}_{0,6}}$. Note
that the number of monomials in $f_{i+1}$ divisible by $k_i$ distinct
$y_{\pi}$ has decreased under this operation, so after a finite number
of iterations $k_i$ must also decrease. As this cannot happen
indefinitely, after a finite number of iterations we have a polynomial
$f_k$ with $m-f_k \in I_{\overline{M}_{0,6}}$ and no term of $f_k$ divisible by more
than one distinct $y_{\pi}$.
Since $\mathbf{v} \in \Mov(\overline{M}_{0,6})$, by Lemma~\ref{l:Qpihyperplane} the
vector $\mathbf{v}$ is contained in the half-space on the opposite
side of the hyperplane $\Delta_{\pi}$ from $Q_{\pi}$. Thus every term
of $f_k$ divisible by some $y_{\pi}$ must also be divisible by some
$x_I$ with $\delta_I$ not on $\Delta_{\pi}$. We can then subtract an
appropriate multiple of one of the the generators of $I_{\overline{M}_{0,6}}$ of the
form \eqref{item:case2} or \eqref{item:case3} to obtain a polynomial
$f_{k+1}$ which has one fewer monomial divisible the maximal occuring power of any $y_{\pi}$.
After a finite number of iterations we thus obtain a polynomial $f$
with $m-f \in I_{\overline{M}_{0,6}}$ and no term of $f$ divisible by any $y_{\pi}$.
This shows that $m$ is in the image of $\psi_{\mathbf{v}}$. Since the
monomials of degree $l\mathbf{v}$ as $l$ varies span $\oplus_{l \geq
0} (S/I_{\overline{M}_{0,6}})_{l \mathbf{v}}$, this shows that $\psi_{\mathbf{v}}$
is surjective as required.
\end{proof}
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|
1,314,259,996,756 | arxiv | \section{Background and Preliminaries}
We consider the two-way wireless relaying scenario shown in Fig.\ref{relay_channel}, where bi-directional data transfer takes place between the nodes A and B with the help of the relay R. It is assumed that all the three nodes operate in half-duplex mode, i.e., they cannot transmit and receive simultaneously in the same frequency band. The relaying protocol consists of the following two phases: the \textit{multiple access} (MA) phase, during which A and B simultaneously transmit to R and the \textit{broadcast} (BC) phase during which R transmits to A and B. Network coding is employed at R in such a way that A (B) can decode the message of B (A), given that A (B) knows its own message.
\begin{figure}[htbp]
\centering
\subfigure[MA Phase]{
\includegraphics[totalheight=1in,width=2in]{2way_relay_MAC}
\label{fig:phase1}
}
\subfigure[BC Phase]{
\includegraphics[totalheight=1in,width=2in]{2way_relay_BC}
\label{fig:phase2}
}
\caption{The Two Way Relay Channel}
\label{relay_channel}
\end{figure}
\vspace{-0.1 cm}
The concept of physical layer network coding has attracted a lot of attention in recent times. The idea of physical layer network coding for the two way relay channel was first introduced in \cite{ZLL}, where the multiple access interference occurring at the relay was exploited so that the communication between the end nodes can be done using a two stage protocol. Information theoretic studies for the physical layer network coding scenario were reported in \cite{KMT},\cite{PoY}. The design principles governing the choice of modulation schemes to be used at the nodes for uncoded transmission were studied in \cite{APT1}. An extension for the case when the nodes use convolutional codes was done in \cite{APT2}. A multi-level coding scheme for the two-way relaying scenario was proposed in \cite{HeN}.
It was observed in \cite{APT1} that for uncoded transmission, the network coding map used at the relay needs to be changed adaptively according to the channel fade coefficient, in order to minimize the impact of the multiple access interference.
\subsection{Signal Model}
\subsubsection*{Multiple Access (MA) Phase}
Let $\mathcal{S}$ denote the symmetric $M$-PSK constellation used at A and B, where $M=2^\lambda$, $\lambda$ being a positive integer. Assume that A (B) wants to transmit an $\lambda$-bit binary tuple to B (A). Let $\mu: \mathbb{F}_{2^\lambda} \rightarrow \mathcal{S}$ denote the mapping from bits to complex symbols used at A and B. Let $x_A= \mu(s_A)$, $x_B=\mu(s_B)$ $\in \mathcal{S}$ denote the complex symbols transmitted by A and B respectively, where $s_A,s_B \in \mathbb{F}_{2^\lambda}$. The received signal at $R$ is given by,
\begin{align}
\nonumber
Y_R=H_{A} x_A + H_{B} x_B +Z_R,
\end{align}
where $H_A$ and $H_B$ are the fading coefficients associated with the A-R and B-R links respectively. The additive noise $Z_R$ is assumed to be $\mathcal{CN}(0,\sigma^2)$, where $\mathcal{CN}(0,\sigma^2)$ denotes the circularly symmetric complex Gaussian random variable with variance $\sigma ^2$. We assume a block fading scenario, with the ratio $ H_{B}/H_{A}$ denoted as $z=\gamma e^{j \theta}$, where $\gamma \in \mathbb{R}^+$ and $-\pi \leq \theta < \pi,$ is referred as the {\it fading state} and for simplicity, also denoted by $(\gamma, \theta).$ Also, it is assumed that $z$ is distributed according to a continuous probability distribution.
Let $\mathcal{S}_{R}(\gamma,\theta)$ denote the effective constellation at the relay during the MA Phase, i.e.,
\begin{align}
\nonumber
\mathcal{S}_{R}(\gamma,\theta)=\left\lbrace x_i+\gamma e^{j \theta} x_j \vert x_i,x_j \in \mathcal{S}\right \rbrace.
\end{align}
Let $d_{min}(\gamma e^{j\theta})$ denote the minimum distance between the points in the constellation $\mathcal{S}_{R}(\gamma,\theta)$, i.e.,
{\footnotesize
\begin{align}
\label{eqn_dmin}
d_{min}(\gamma e^{j\theta})=\hspace{-0.5 cm}\min_{\substack {{(x_A,x_B),(x'_A,x'_B)}\\{ \in \mathcal{S} \times \mathcal{S}} \\ {(x_A,x_B) \neq (x'_A,x'_B)}}}\hspace{-0.5 cm}\vert \left(x_A-x'_A\right)+\gamma e^{j \theta} \left(x_B-x'_B\right)\vert.
\end{align}
}
From \eqref{eqn_dmin}, it is clear that there exists values of $\gamma e^{j \theta}$ for which $d_{min}(\gamma e^{j\theta})=0$. Let $\mathcal{H}=\lbrace \gamma e^{j\theta} \in \mathbb{C} \vert d_{min}(\gamma,\theta)=0 \rbrace$. The elements of $\mathcal{H}$ are said to be the singular fade states. An alternative definition of singular fade state is as follows:
\begin{definition}
A fade state $\gamma e^{j \theta}$ is said to be a singular fade state, if the cardinality of the signal set $\mathcal{S}_{R}(\gamma, \theta)$ is less than $M^2$.
\end{definition}
For example, consider the case when symmetric 4-PSK signal set used at the nodes A and B, i.e., $\mathcal{S}=\lbrace (\pm 1 \pm j)/\sqrt{2} \rbrace$. For $\gamma e^{j \theta}=(1+j)/2$, $d_{min}(\gamma e^{j \theta})=0$, since,
\begin{align*}
\left\vert \left( \dfrac{1+j}{\sqrt{2}}-\dfrac{1-j}{\sqrt{2}} \right) + \dfrac{(1+j)}{2} \left( \dfrac{-1-j}{\sqrt{2}} - \dfrac{1+j}{\sqrt{2}} \right)\right\vert=0.
\end{align*}
\noindent
Alternatively, when $\gamma e^{j \theta}=(1+j)/2$, the constellation $\mathcal{S}_{R}(\gamma,\theta)$ has only 12 ($<$16) points.
Hence $\gamma e^{j \theta}=(1+j)/2$ is a singular fade state for the case when 4-PSK signal set is used at A and B.
Let $(\hat{x}_A,\hat{x}_B) \in \mathcal{S} \times \mathcal{S}$ denote the Maximum Likelihood (ML) estimate of $({x}_A,{x}_B)$ at R based on the received complex number $Y_{R}$, i.e.,
\begin{align}
(\hat{x}_A,\hat{x}_B)=\arg\min_{({x}'_A,{x}'_B) \in \mathcal{S} \times \mathcal{S}} \vert Y_R-H_{A}{x}'_A-H_{B}{x}'_B\vert.
\end{align}
\subsubsection*{Broadcast (BC) Phase}
Depending on the value of $\gamma e^{j \theta}$, R chooses a map $\mathcal{M}^{\gamma,\theta}:\mathcal{S} \times \mathcal{S} \rightarrow \mathcal{S}'$, where $\mathcal{S}'$ is the signal set (of size between $M$ and $M^2$) used by R during $BC$ phase. The elements in $\mathcal{S} \times \mathcal{S}$ which are mapped on to the same complex number in $\mathcal{S}'$ by the map $\mathcal{M}^{\gamma,\theta}$ are said to form a cluster. Let $\lbrace \mathcal{L}_1, \mathcal{L}_2,...,\mathcal{L}_l\rbrace$ denote the set of all such clusters. The formation of clusters is called clustering, denoted by $\mathcal{C}^{\gamma,\theta}$.
The received signals at A and B during the BC phase are respectively given by,
\begin{align}
Y_A=H'_{A} X_R + Z_A,\;Y_B=H'_{B} X_R + Z_B,
\end{align}
where $X_R=\mathcal{M}^{\gamma,\theta}(\hat{x}_A,\hat{x}_B) \in \mathcal{S'}$ is the complex number transmitted by R. The fading coefficients corresponding to the R-A and R-B links are denoted by $H'_{A}$ and $H'_{B}$ respectively and the additive noises $Z_A$ and $Z_B$ are $\mathcal{CN}(0,\sigma ^2$).
In order to ensure that A (B) is able to decode B's (A's) message, the clustering $\mathcal{C}^{\gamma,\theta}$ should satisfy the exclusive law \cite{APT1}, i.e.,
{\footnotesize
\begin{align}
\left.
\begin{array}{ll}
\nonumber
\mathcal{M}^{\gamma,\theta}(x_A,x_B) \neq \mathcal{M}^{\gamma,\theta}(x'_A,x_B), \; \mathrm{where} \;x_A \neq x'_A \; \mathrm{,} \;x_B \in \mathcal{S},\\
\nonumber
\mathcal{M}^{\gamma,\theta}(x_A,x_B) \neq \mathcal{M}^{\gamma,\theta}(x_A,x'_B), \; \mathrm{where} \;x_B \neq x'_B \; \mathrm{,} \;x_A \in \mathcal{S}.
\end {array}
\right\} \\
\label{ex_law}
\end{align}
\vspace{-.3 cm}
}
From an information theoretic perspective, the mapping $\mathcal{M}^{\gamma,\theta}$ needs to satisfy the exclusive law for the reason outlined below. Consider the ideal situation where the additive noises at the nodes are zero. It is assumed that the fading state $\gamma e^{j \theta} \notin \mathcal{H}$. The assumption is required since R can decode unambiguously to an element in $\mathcal{S} \times \mathcal{S}$ only if $\gamma e^{j \theta}$ is not a singular fade state and is justified since $\gamma e^{j \theta}$ takes values from a continuous probability distribution and the cardinality of $\mathcal{H}$ is finite. During the MA Phase, assume the relay jointly decodes correctly to the pair $(x_A,x_B)$ and transmits $X_R=\mathcal{M}^{\gamma,\theta}(x_A,x_B)$ during the BC Phase. The received complex symbols at A and B are respectively $Y_A=H'_A X_R$ and $Y_B=H'_B X_R$. At node A, the amount of uncertainty about $x_B$ which gets resolved after observing $Y_A$, $I(x_B;Y_A/x_A)=H(x_B\vert x_A)-H(x_B\vert Y_A,x_A)=H(x_B)-H(x_B \vert X_R, x_A)$ (since $x_B$ and $x_A$ are independent). Since $H(x_B \vert X_R, x_A)=0$ if and only if the mapping $\mathcal{M}^{\gamma,\theta}$ satisfies the exclusive law, the amount of uncertainty about $x_B$ ($x_A$) which gets resolved at A (B) is maximized if and only if the clustering satisfies the exclusive law.
\begin{definition}
The cluster distance between a pair of clusters $\mathcal{L}_i$ and $\mathcal{L}_j$ is the minimum among all the distances calculated between the points $x_A+\gamma e^{j\theta} x_B ,x'_A+\gamma e^{j\theta} x'_B \in \mathcal{S}_R(\gamma,\theta)$, where $(x_A,x_B) \in \mathcal{L}_i$ and $(x'_A,x'_B) \in \mathcal{L}_j$.
\end{definition}
\begin{definition}
The \textit{minimum cluster distance} of the clustering $\mathcal{C}^{\gamma,\theta}$ is the minimum among all the cluster distances, i.e.,
{\footnotesize
\begin{align}
\nonumber
d_{min}(\mathcal{C}^{\gamma, \theta})=\hspace{-0.8 cm}\min_{\substack {{(x_A,x_B),(x'_A,x'_B)}\\{ \in \mathcal{S}\times\mathcal{S},} \\ {\mathcal{M}^{\gamma,\theta}(x_A,x_B) \neq \mathcal{M}^{\gamma,\theta}(x'_A,x'_B)}}}\hspace{-0.8 cm}\vert \left( x_A-x'_A\right)+\gamma e^{j \theta} \left(x_B-x'_B\right)\vert.
\end{align}
}
\end{definition}
The minimum cluster distance determines the performance during the MA phase of relaying. The performance during the BC phase is determined by the minimum distance of the signal set $\mathcal{S}'$. Throughout, we restrict ourselves to optimizing the performance during the MA phase. For values of $\gamma e^{j \theta}$ in the neighborhood of the singular fade states, the value of $d_{min}(\gamma e^{j\theta})$ is greatly reduced, a phenomenon referred as {\it distance shortening}. To avoid distance shortening, for each singular fade state, a clustering needs to be chosen such that the minimum cluster distance at the singular fade state is non-zero and is also maximized.
A clustering $\mathcal{C}^{\lbrace h \rbrace}$ is said to remove a singular fade state $ h \in \mathcal{H}$, if the minimum cluster distance of the clustering $\mathcal{C}^{\lbrace h \rbrace}$ for $\gamma e^{j \theta}=h$ is greater than zero.
Let $\mathcal{C}_{\mathcal{H}}=\left\lbrace \mathcal{C}^{\lbrace h\rbrace} : h \in \mathcal{H} \right\rbrace$ denote the set of all such clusterings. Let $d_{min}({\mathcal{C}^{\lbrace h\rbrace}},\gamma,\theta)$ denote the minimum cluster distance of the clustering $\mathcal{C}^{\lbrace h\rbrace}$ evaluated at $\gamma e^{j\theta}$. For $\gamma e^{j \theta} \notin \mathcal{H}$, the clustering $\mathcal{C}^{\gamma,\theta}$ is chosen to be $\mathcal{C}^{\lbrace h\rbrace}$, which satisfies $d_{min}({\mathcal{C}^{\lbrace h\rbrace}},\gamma,\theta) \geq d_{min}({\mathcal{C}^{\lbrace h' \rbrace}},\gamma,\theta), \forall h \neq h' \in \mathcal{H}$.
\begin{note}
The clusterings which belong to the set $\mathcal{C}_{\mathcal{H}}$ need not be distinct, since a single clustering can remove more than one singular fade state.
\end{note}
\begin{example}
In the case of BPSK, if the fade state is $\gamma=1$ and $\theta=0$ the distance between the pairs $(0,1)(1,0)$ is zero as in Fig.\ref{fig:BPSK}(a).The following clustering remove this singular fade state:
$$\{\{(0,1)(1,0)\},\{(1,1)(0,0)\}\}$$
The minimum cluster distance is non zero for this clustering.
\end{example}
\begin{figure}[t]
\centering
\vspace{-.8 cm}
\includegraphics[totalheight=2.5in,width=2.5in]{bpsk1}
\caption{Effective Constellation at the relay for singular fade states, when the end nodes use BPSK constellation.}
\label{fig:BPSK}
\end{figure}
\subsection{Issues with Koike-Akino Popovski Tarokh's approach \cite{APT1}}
It is assumed that the channel state information is not available at the transmitting nodes A and B during the MA phase. A block fading scenario is assumed and the clustering used by the relay is indicated to A and B by using overhead bits.
The procedure suggested in \cite{APT1} to obtain the set of all clusterings, was using a computer search algorithm (closest neighbour clustering algorithm), which involved varying the fade state values over the entire complex plane, i.e., $0 \leq \gamma < \infty$, $0 \leq \theta < 2\pi$ and finding the clustering for each value of channel realization. The total number of network codes which would result is known only after the algorithm is run for all possible realizations $\gamma e^{j \theta}$ which is uncountably infinite and hence the number of overhead bits required is not known beforehand. Moreover, performing such an exhaustive search is extremely difficult in practice, especially when the cardinality of the signal set $M$ is large.
The implementation complexity of the scheme suggested in \cite{APT1} is extremely high. It appears that, for each realization of the singular fade state, the closest neighbour clustering algorithm \cite{APT1} needs to be run at R to find the clustering. In contrast, we provide a simple criterion (Section IV A, \cite{VNR}) based on which a clustering from the set $\mathcal{C}_{\mathcal{H}}$ is chosen depending on the value of $\gamma e^{j \theta}$. In this way, the set of all values of $\gamma e^{j\theta}$ (the complex plane) is quantized into different regions, with a clustering from the set $\mathcal{C}_{\mathcal{H}}$ used in a particular region.
In the closest neighbour clustering algorithm suggested in \cite{APT1}, the network coding map is obtained by considering the entire distance profile. The disadvantages of such an approach are two-fold.
\begin{itemize}
\item
Considering the entire distance profile, instead of the minimum cluster distance alone which contributes dominantly to the error probability, results in an extremely large number of network coding maps. For example, for 8-PSK signal set, the closest neighbour clustering algorithm results in more than 5000 maps.
\item
The closest neighbour clustering algorithm tries to optimize the entire distance profile, even after clustering signal points which contribute the minimum distance. As a result, for several channel conditions, the number of clusters in the clustering obtained is greater than the number of clusters in the clustering obtained by taking the minimum distance alone into consideration. This results in a degradation in performance during the BC Phase, since the relay uses a signal set with cardinality equal to the number of clusters.
\end{itemize}
In \cite{APT1}, to overcome the two problems mentioned above, another algorithm is proposed, in which for a given $\gamma e^{j \theta}$, an exhaustive search is performed among all the network coding maps obtained using the closest neighbour clustering algorithm and a map with minimum number of clusters is chosen. The difficulties associated with the implementation of the closest neighbour clustering algorithm carry over to the implementation of this algorithm as well.
To avoid all these problems we suggest a scheme, which is based on the removal of all the singular fade states. Since the number of singular fade states is finite (the exact number of singular fade states and their location in the complex plane are discussed in Section II), the total number of network coding maps used is upper bounded by the number of singular fade states. In fact, the total number of network coding maps required is shown to be lesser than the total number of singular fade states in Section VI. In other words, the total number of network coding maps required is known exactly, which determines the number of overhead bits required. It is shown in section III that the problem of obtaining clusterings which remove all the singular fade states reduces to completing a finite number of partially filled Latin Squares, which totally avoids the problem of performing exhaustive search for an uncountably infinite number of values.
In \cite{APT1}, an important observation was that for the case of QPSK modulation during the MA Phase, there exists several channel conditions under which the use of unconventional 5-ary signal constellation results in a larger throughput. It appears as if the mitigation of the multiple access interference, comes at the cost of degraded performance during the BC phase, because of the use of a signal set with a larger cardinality during the BC phase. By providing all the clusterings explicitly, it is shown in Section III B that the use of a signal set with larger cardinality is not required for 8-PSK signal set. In other words, the mitigation of multiple access interference does not come at the cost of degraded performance during the BC phase, for 8-PSK signal set.
The contributions and organization of the paper are as follows:
\begin{itemize}
\item It is shown that the requirement of satisfying the exclusive law is same as the clustering being represented by a Latin Square and can be used to get the clustering which removes singular fade states. In other words, it is shown that the problem of finding a clustering which removes a singular fade state reduces to filling a partially filled Latin Square.
\item Using the properties of the set of Latin Squares for a given set of parameters, the problem of finding the set of maps corresponding to all the singular fade states can be simplified to finding the same for only for a small subset of singular fade states. Specifically, it is shown that
\begin{enumerate}
\item For the set of all singular fade states lying on a circle, from a Latin Square corresponding to one singular fade state, Latin Squares for the other singular fade states can be obtained by appropriate permutation of the columns of the first Latin Square.
\item There is a one-to-one correspondence between a Latin Square corresponding to a singular fade state on a circle of radius $r$ and a Latin Square corresponding to a singular fade state on a circle of radius $\frac{1}{r}.$
\end{enumerate}
\item It is shown that the bit-wise XOR mapping can remove the singular fade state $(\gamma=1, \theta=0)$ for any $M$-PSK, (i.e., for $M$ any power of 2)
\item For any $M$-PSK signal set, all the clusterings which can remove the singularities can be obtained with the aid of Latin Squares along with their isotopes. As an example, this is shown explicitly for QPSK signal set and 8PSK signal set.
\item It was shown in \cite{APT1} that for the case when QPSK signal set is used during the MA phase, there exists certain values of fade state for which a clustering with cardinality 5 needs to be used to maximize the minimum cluster distance. While the use of clusterings with cardinality 5, reduces the impact of multiple access interference, it adversely impacts the performance during the BC phase. For 8-PSK signal set, clusterings which remove the singular fade states, all of which have a cardinality of 8 are explicitly given, which implies that there is no trade-off between the MAC Phase and the BC phase.
\item
Even though, the problem of completability using $M$ symbols of an $M \times M$ Latin Square in general is unsolved, it is shown that the structure of the partially filled Latin Square in the problem considered allows the construction of explicit Latin Squares, for some singular fade states. In other words, by providing some explicit constructions, for $M$-PSK signal set, it is shown that the partially filled Latin Squares corresponding to some singular fade states can be completed using $M$ symbols itself.
\item
It was observed in \cite{APT1} that for 4 PSK signal set there exists clusterings which remove more than one singular fade state. For any $M$-PSK signal set, certain special cases when multiple singular fade states are removed by the same Latin Square are identified. A construction algorithm is provided to obtain such Latin Squares. Each one of the Latin Squares obtained using the algorithm provided removes $M^2/8$ singular fade states.
\item
An explicit construction procedure is provided to show that certain Latin Squares for $M$-PSK signal set, are obtainable from the Latin Squares for $M/2$-PSK signal set.
\item
When the end nodes use constellations of different sizes $M(=2^\lambda)$ and $N(=2^\mu)$, to get the clusterings it is required to fill Latin Rectangles. These Latin rectangles are obtained by removing appropriate columns of the Latin Squares constructed for the case when both the end nodes use constellations of same size which is $\max(M,N)$.
\end{itemize}
\section{SINGULAR FADE STATES FOR M-PSK SIGNAL SET}
Throughout the paper the points in the symmetric $M$-PSK signal set are assumed to be of the form $e^{j (2k+1) \pi/M},0 \leq k \leq M-1$ and $M$ is of the form $2^\lambda$, where $\lambda$ is a positive integer.
Let $\Delta\mathcal{S}$ denote the difference constellation of the $M$-PSK signal set $\mathcal{S}$, i.e., $\Delta\mathcal{S}=\lbrace s_i-s'_i \vert s_i, s'_i \in \mathcal{S}\rbrace$.
For any M-PSK signal set, the set $\Delta\mathcal{S}$ is of the form,
{\footnotesize
\begin{align}
\nonumber
\Delta\mathcal{S}=&\left\lbrace 0\right\rbrace\cup \left\lbrace 2\sin(\pi n /M) e^{j k 2 \pi/M} \vert{n \; \textrm{odd} }\right\rbrace\\
\nonumber
&\hspace{2 cm}\cup\left\lbrace 2\sin(\pi n /M) e^{j (k 2 \pi/M + \pi/M)}\vert{n \; \textrm{even} }\right\rbrace,
\end{align}
}where $1 \leq n \leq M/2$ and $0 \leq k \leq M-1$.
In other words, the non-zero points in $\Delta\mathcal{S}$ lie on $M/2$ circles of radius $2\sin(\pi n/M), 1 \leq n \leq M/2$ with each circle containing $M$ points. The phase angles of the $M$ points on each circle is $2 k \pi/M$, if $n$ is odd and $2k \pi/M+\pi/M$ if $n$ is even, where $0 \leq k \leq M-1$. For example the difference constellation for 4-PSK and 8-PSK signal sets are shown in Fig. \ref{4psk_diff} and Fig. \ref{8psk_diff} respectively.
\begin{figure}[htbp]
\centering
\includegraphics[totalheight=3.5in,width=6in]{4psk_diff}
\caption{Difference constellation for 4-PSK signal set}
\label{4psk_diff}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[totalheight=3in,width=5in]{8psk_diff}
\caption{Difference constellation for 8-PSK signal set}
\label{8psk_diff}
\end{figure}
Let us define,
\begin{align}
\nonumber
x_{k,n}=\left\lbrace
\begin{array}{ll}
\nonumber
2\sin(\pi n /M) e^{j k 2 \pi/M} {\;\textrm{if}\; n \; \textrm{is} \; \textrm{odd}, }\\
2\sin(\pi n /M) e^{j (k 2 \pi/M + \pi/M)} {\;\textrm{if}\; n \; \textrm{is} \; \textrm{even}, }
\end{array}
\right.\\
\label{eqn_diff}
\end{align}
where $1 \leq n \leq M/2$ and $0 \leq k \leq M-1$.
From \eqref{eqn_dmin}, it follows that the singular fade states are of the form,
{\footnotesize
\vspace{-0.2 cm}
\begin{align*}
\gamma_{s} e^{j \theta_{s}}=-x_{k,n}/x_{k',n'},
\end{align*}
}for some $x_{k,n},x_{k',n'} \in \Delta\mathcal{S}$.
\begin{lemma}
\label{lemma_trig}
For integers $k_1$, $k_2$, $l_1$ and $l_2$, where $$1 \leq k_1,k_2,l_1,l_2 \leq \frac{M}{2},k_1 \neq k_2 \textrm{ and } l_1 \neq l_2,$$
{\footnotesize
\vspace{-0.4 cm}
\begin{align}
\nonumber
\dfrac{\sin(k_1 \pi/M)}{\sin(k_2 \pi/M)}=\dfrac{\sin(l_1 \pi/M)}{\sin(l_2 \pi/M)},
\end{align}
}if and only if $k_1 = l_1$ and $k_2 = l_2$.
\begin{proof}
See \cite{VNR}.
\end{proof}
\end{lemma}
The following lemma gives the location of the singular fade states in the complex plane.
\begin{lemma}
\label{lemma_singularity}
The singular fade states other than zero lie on $M^2/4-M/2+1$ circles with $M$ points on each circle, with the radii of the circles given by $\sin(k_1\pi/M)/\sin(k_2 \pi/M)$, where $1 \leq k_1,k_2 \leq M/2$. The phase angles of the $M$ points on each one of the circles are given by $k2\pi/M$, $0 \leq k \leq M-1$, if both $k_1$ and $k_2$ are odd or both are even and $k2\pi/M+\pi/M$, $0 \leq k \leq M-1$, if only one among $k_1$ and $k_2$ is odd.
\end{lemma}
From Lemma \ref{lemma_singularity}, it follows that if $\gamma e^{j\theta}$ is a singular fade state, $\frac{1}{\gamma} e^{-j \theta}$ is also a singular fade state.
\begin{example}
For the case when 4-PSK signal set is used during the MA Phase, the singular fade states lie on three circles as shown in Fig. \ref{4psk_sing}.
\end{example}
\begin{example}
For the case when 8-PSK signal set is used during the MA Phase, the singular fade states lie on thirteen circles as shown in Fig. \ref{8psk_sing}.
\end{example}
\begin{figure}[htbp]
\centering
\includegraphics[totalheight=3in,width=5.5in]{4psk_sing}
\caption{Singular fade states for 4-PSK signal set}
\label{4psk_sing}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[totalheight=2.5in,width=4.5in]{8psk_sing}
\caption{Singular fade states for 8-PSK signal set}
\label{8psk_sing}
\end{figure}
\section{The Exclusive Law and Latin Squares}
\textit{Definition 4:} \cite{Rod} A Latin Square L of order $M$ on the symbols from the set $\mathbb{Z}_t=\{0,1, \cdots ,t-1\}$ is an \textit{M} $\times$ \textit{M} array, in which each cell contains one symbol and each symbol occurs at most once in each row and column.
Let the nodes A and B use the same constellation of size $M.$ Consider an \textit{M} $\times$ \textit{M} array at the relay with the rows (columns) indexed by the constellation point used by node A (B), i.e., symbols from the set \{{0, 1, 2,}\dotso, {$M$-1}\}. The relay is allowed to use any constellation with size $\textit{t}\geq\text{M}$ (using $t > M$ may lead to some advantages, see \cite{APT1}). Our aim is to cluster the $M^{2}$ slots in the $\textit{M}\times\textit{M}$ array such that the exclusive law is satisfied. To do so, we will fill in the slots in the array with the elements of set $\mathbb{Z}_{t}$ in such a way that \eqref{ex_law} is satisfied, and the clusters are obtained by taking all the row-column pairs $(i,j),$ $i,j \in {0,1,\cdots,M-1},$ such that the entry in the $(i,j)-$th slot is the same symbol from $\mathbb{Z}_{t}.$ The specific symbols from $\mathbb{Z}_t$ are not important, but it is the set of clusters that are important.
Now, it is easy to see that if the exclusive law need to be satisfied, then the clustering should be such that an element in a row and also in a column cannot be repeated in the same row and column. Thus all the relay clusterings which satisfy the exclusive law form Latin Squares. Hence, we have the following:
\textit{ All the relay clusterings which satisfy the exclusive law forms Latin Squares, when the end nodes use constellations of same size.}
With this observation, the study of clustering which satisfies the exclusive law can be equivalently carried out as the study of Latin Squares with appropriate parameters.
\subsection{Removing Singular fade states and Constrained Latin Squares}
The relay can manage with constellations of size $M$ in BC phase, but it is observed that in some cases relay may not be able to remove the singular fade states with $t=M$ and results in severe performance degradation in the MA phase \cite{APT1}. Let $(k,l)(k^{\prime},l^{\prime})$ be the pairs which give same point in the effective constellation $\mathcal{S}_R$ at the relay for a singular fade state, where $k,k^{\prime},l,l^{\prime} \in \{0,1,....,M-1\}$. Let $k,k^{\prime}$ be the constellation points used by node A and $l,l^{\prime}$ be the constellation points used by node B. If they are not clustered together, the minimum cluster distance will be zero. To avoid this, those pairs should be in same cluster. This requirement is termed as {\it singularity-removal constraint}. So, we need to obtain Latin Squares which can remove singular fade states and with minimum value for $t.$ Therefore, initially we will fill the slots in the $\textit{M}\times\textit{M}$ array such that for the slots corresponding to a singularity-removal constraint the same element will be used to fill slots. This removes that particular singular fade state. Such a partially filled Latin Square is called a {\it Constrained Partial Latin Square}. After this, to make this a Latin Square, we will try to fill the other slots of the partially filled, Constrained Partial Latin Square with minimum number of symbols from the set $\mathbb{Z}_{t}.$
\begin{example}
Consider the case where both A and B uses BPSK as shown in Fig. \ref{fig:BPSK}, where the effect of noise is neglected. There are two singular fade states, one at $(\gamma =1, \theta=0)$ and the other at $(\gamma =1, \theta =\pi).$ We try to eliminate the singular fade states one by one. First to remove $(\gamma=1, \theta=0),$ the symbol in $0^{th}$ row $1^{st}$ column (henceforth the slot (0,1)) and symbol in (1,0) should be same. Otherwise, the minimum cluster distance will be zero. We are using symbol 1 (choice of this symbol will not alter the clustering) for this and we will get the Constrained Partial Latin Square as in Table. \ref{LL}. This uniquely completes to the Latin Square in Table. \ref{LLL}. Notice that this will remove the singular fade state $(\gamma = 1, \theta = \pi)$ also. This Latin Square corresponds to the bit-wise XOR mapping, but with higher order constellations the number of singular fade states increases and bit-wise XOR cannot remove (will be seen in the sequel) all the singular fade states.
\end{example}
\begin{table}
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|c|c|}
\hline & 1 \\
\hline 1 & \\
\hline
\end{tabular}
\caption{Partially filled Latin Square}
\label{LL}
}
\hfill
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|c|c|}
\hline 0 & 1 \\
\hline 1 & 0 \\
\hline
\end{tabular}
\caption{Completely Filled Latin Square}
\label{LLL}
}
\end{table}
Let the $M-$PSK points be $e^{j(2k+1) \pi/M}$, where $0 \leq k \leq M-1$. For simplicity, by the point $k$ we mean the point $e^{j(2k+1) \pi/M}.$ The following lemma shows that bit-wise XOR mapping removes the singular fade state $(\gamma=1,\theta=0)$, for any $M$-PSK signal set.
\begin{lemma}
When the user nodes use $2^{\lambda}$-PSK constellations, the singular fade state $(\gamma=1, \theta=0)$ is removed by bit-wise XOR mapping (denoted by $\oplus$), for all $\lambda.$
\end{lemma}
\begin{proof}
For $\lbrace (k,k^{\prime}), (l,l^{\prime})\rbrace$ to be a singularity constraint corresponding to the singular fade state $(\gamma=1, \theta=0)$,
{\footnotesize
\vspace{-.3 cm}
\begin{align}
\nonumber
&-\frac{e^{ \frac{jk\pi}{M}}-e^{\frac{jk^{\prime}\pi}{M}}}{e^{\frac{jl\pi}{M}}-e^{\frac{jl^{\prime}\pi}{M}}}=1,\textrm{ i.e.,}\\
\label{eqn_xor}
&\frac{\sin\left[\pi(k-k^{\prime})/2^{\lambda}\right]} {\sin \left[\pi(l^{\prime}-l)/2^{\lambda}\right]}e^{j \frac{\pi}{M}(k+k;-l-l')}=1.
\end{align}
}
Equating the amplitudes on both sides of \eqref{eqn_xor}, one gets $$ \sin\left[\pi(k-k^{\prime})/2^{\lambda}\right] = \sin \left[\pi(l^{\prime}-l)/2^{\lambda}\right], $$
which leads to the following two cases:
\noindent
{\it Case (i):} $k-k^{\prime}= l^{\prime}-l$ \\
{\it Cases (ii):}$\dfrac{\pi(k-k^{\prime})}{2^{\lambda}} = \pi-\dfrac{\pi(l^{\prime}-l)}{2^{\lambda}} \implies k-l = k^{\prime}-l^{\prime}+2^{\lambda}.$
Equating the phases on both the sides of \eqref{eqn_xor} leads to
\begin{equation}
\label{thetaresult}
\frac{\pi}{2^{\lambda}}(k+k^{\prime}-l-l^{\prime})=0 \implies k+k^{\prime} = l+l^{\prime}.
\end{equation}
Combining {\it Case (i)} above and \eqref{thetaresult} gives $(k^{\prime},l^{\prime}) = (l,k),$
i.e., the singularity-removal constraint is of the form $\{(k,l)(l,k)\}$. In other words, the clustering should satisfy this symmetry.
Combining {\it Case (ii)} above and \eqref{thetaresult} leads to $k= l+2^{\lambda-1}$ irrespective of $k^{\prime},l^{\prime}$ and $k^{\prime} = l^{\prime}+2^{\lambda-1}$ irrespective of $k,l.$ In other words, $\{(l+2^{\lambda-1},l)\}$, $l \in \{0,1,....,2^{\lambda}\}$ is the set of singularity-removal constraints.
From the above, one can conclude that a clustering which removes the singular fade state ($\gamma=1, \theta=0$) should have \\
(i) A symmetric Latin Square, meaning that the cells $(k,l)$ and $(l,k)$ should have the same symbol.\\
(ii) A Latin Square with the symbols in the cells $\{(l+2^{\lambda-1},l)\},$ and $l \in \{0,1,....,2^{\lambda}\}$ being the same.
The Latin Square produced by bit-wise XOR mapping is clearly symmetric. Moreover, the quantity $(l+2^{\lambda-1})\oplus l$ is always equal to $2^{\lambda-1}$ for all values of $l,$ i.e., the symbols in all the cells of the set $\{(l+2^{\lambda-1},l)\}, ~~ l \in \{0,1,....,2^{\lambda}\}$ are the same. Hence the XOR map removes the singular fade state ($\gamma=1, \theta=0$).
\end{proof}
\textit{Definition 5:} \cite{Sto} Two Latin Squares $L$ and $L$ $^{\prime}$ (using the same symbol set) are isotopic if there is a triple $(\textit{f,g,h}),$ where $f$ is a row permutation, $g$ is a column permutation and $h$ is a symbol permutation, such that applying these permutations on $L$ gives $L^{\prime}.$
\begin{lemma}
\label{lemma_col_perm}
Two Latin Squares $L$ and $L^\prime$ which remove the singular fade states $(\gamma, \theta)$ and $(\gamma, \theta^{\prime})$, respectively, (i.e., two singular fade states on the same circle), are Isotopic that are obtainable one from another by a column permutation alone. If $\theta'-\theta = k\frac{2\pi}{M}$, $L'$ can be obtained by cyclic shifting of the columns of $L$, $k$ times in the anticlockwise direction.
\end{lemma}
\begin{proof}
Let $L$ and $L^\prime$, respectively remove the singular fade states $(\gamma, \theta)$ and $(\gamma, \theta^\prime).$
The effect of rotation in the $z-$plane by an angle $\theta^\prime - \theta$ due to channel fade coefficients $H_A$ and $H_B$ can be viewed equivalently as a relative rotation of the constellation used by B by an angle $\theta^\prime - \theta$ with respect to the constellation used by A and no relative rotation between the channel fade coefficients $H_A$ and $H_B.$ Let $S$ and $S^{\prime}$ be the resulting rotated constellations after rotation in the constellation of node B corresponding to an angle $\theta^\prime - \theta.$
Since there are $M$ singular fade states for a specific $\gamma$, (shown in \cite{VNR}), and they are all spaced by same angular separation, $\theta^\prime - \theta$ is an integer multiple of $2\pi/M$ which is an angular separation of the $M$-PSK constellation points. That is, a rotation in the channel by an angle $\theta^\prime - \theta$ is equivalent to a rotation in the constellation points in the $M$-PSK constellation. So, we can obtain the Latin Square L$^{\prime}$ by column permutations in L, since the columns are indexed by constellation points used by node B. This means, if we obtain the Latin Square for a singular fade state $(\gamma, \theta),$ then by appropriately shifting the columns we obtain the Latin Squares that remove all the other singular fade states of the form $(\gamma, \theta^\prime).$ This completes the proof.
\end{proof}
From \ref{lemma_col_perm}, it follows that for each circle, it is enough if we obtain one Latin Square which removes a singular fade state on that circle. The Latin Squares which remove the other singular fade states can be obtained by column permutation. For example, all the Latin Squares which remove the singular fade states on the unit circle can be obtained from the bit-wise XOR map by column permutation.
\textit{Definition 6:} A Latin Square $L^T$ is said to be the Transpose of a Latin Square $L$, if $L^T(i,j)=L(j,i)$ for all $i,j \in \{0,1,2,..,M-1\}.$
Recall from Section II that if $\gamma e^{j\theta}$ is a singular fade state, then $\frac{1}{\gamma}e^{-j \theta}$ is also a singular fade state. The following Lemma shows that the transpose of the Latin Square which removes $\gamma e^{j\theta}$ removes the singular fade state $\frac{1}{\gamma}e^{-j \theta}$.
\begin{lemma}
\label{lemma_trans}
If the Latin Square $L$ removes the singular fade state $(\gamma, \theta),$ then the Latin Square $L^T$ removes the singular fade state $(\frac{1}{\gamma}, -\theta).$
\end{lemma}
\begin{proof}
Let $\{(k_1,l_1)(k_2,l_2)\}$ be a singularity-removal constraint for the singular fade state $(\gamma, \theta).$ Then,\\
$$\gamma=\dfrac{\sin\left(\frac{\pi(k_{1}-k_{2})}{M}\right)}{\sin\left(\frac{\pi(l_{2}-l_{1})}{M}\right)} \mbox{ and }\theta=\dfrac{\pi}{M}(k_{1}+k_{2}-l_{1}-l_{2}).$$
Taking transpose in the constraint we will obtain $\{(l_1,k_1)(l_2,k_2\}.$ Let this constraint correspond to the singular fade state $(\gamma^\prime, \theta^\prime).$ Then,
$$\gamma^\prime =\frac{\sin\left(\frac{\pi(l_{1}-l_{2})}{M}\right)}{\sin\left(\frac{\pi(k_{2}-k_{1})}{M}\right)} =\dfrac{\sin\left(\frac{\pi(l_{2}-l_{1})}{M}\right)}{\sin\left(\frac{\pi(k_{1}-k_{2})}{M}\right)}=1/\gamma.$$
\noindent
Similarly,
$$\theta^\prime=\frac{\pi}{M}(l_{1}+l_{2}-k_{1}-k_{2}) =-\frac{\pi}{M}(k_{1}+k_{2}-l_{1}-l_{2}) =-\theta.$$
This completes the proof.
\end{proof}
Lemma \ref{lemma_trans} implies that it is enough to obtain the Latin Squares which remove those singular fade states which lie on or inside the unit circle centered at the origin. The Latin Squares which remove the singular fade states which lie outside the unit circle can be obtained by taking transpose.
\section{Illustrations}
\subsection{End nodes use QPSK}
There are 12 singular fade states (shown in Fig.\ref{4psk_sing}), when both the end nodes A and B use QPSK as in Fig.\ref{fig:nodes}.
\begin{figure}[t]
\centering
\vspace{-2 cm}
\includegraphics[totalheight=4in,width=3in]{qpsk}
\vspace{-4cm}
\caption{QPSK Constellations used at the end nodes}
\label{fig:nodes}
\end{figure}
The singular fade states are
\begin{align}
\nonumber
\gamma=1 &;\hspace{10pt} \theta=0,+\pi/2,-\pi/2,\pi\\
\nonumber
\gamma=1/\sqrt{2} &;\hspace{10pt} \theta=+\pi/4,+3\pi/4,-\pi/4,-3\pi/4\\
\nonumber
\gamma=\sqrt{2}&;\hspace{10pt}\theta=+\pi/4,+3\pi/4,-\pi/4,-3\pi/4.
\end{align}
We remove singular fade states one by one. Consider first, the case $(\gamma=1, \theta=0).$
The singularity-removal constraints are
\begin{center}
$\{(0,1)(1,0)\};~~ \{(0,2)(1,3)(2,0)(3,1)\};~~ \{(0,3)(3,0)\};$ \\
$ \{(1,2)(2,1)\}; ~~\{(2,3)(3,2)\}.$
\end{center}
Satisfying these constraints, a Latin Square can be constructed with $t$=4, in three different ways, $L_{1}, L_{2}$ and $L_{3}$ as shown in figure \ref{L1}, figure \ref{L2} and figure \ref{L3}. All these three clusterings corresponding to each Latin Square give the same performance on the basis of first minimum cluster distance in the MA phase. But the advantage with the one shown in figure \ref{L1} is that it removes singular fade state at $(\gamma=1, \theta=\pi)$ also. This is easily verified, by seeing that after two cyclic shifts in the columns of $L_1$ the clustering that it results in is the same as the old one. This is explicitly shown in Fig \ref{fig:latinshift}.
The singularity-removal constraints for $(\gamma=1, \theta=\pi)$ are
\begin{center}
$\{(0,3)(1,2)\}; ~~ \{(0,0)(1,1)(2,2)(3,3)\}; ~~ \{(0,1)(3,2)\};$ \\
$\{(1,0)(2,3)\}; ~~\{(2,1)(3,0)\}.$
\end{center}
\noindent
The Latin squares to remove this singular fade state, $L_{1}, L_{4}$ and $L_{5}$ are shown in figure \ref{L1}, figure \ref{L4} and figure \ref{L5} respectively.
\begin{figure}
\centering
\vspace{0 cm}
\includegraphics[totalheight=1.4in,width=3in]{latinshift}
\vspace{0 cm}
\caption{Two rotations of $L_{1}$ gives same clustering as that of $L_{1}$}
\label{fig:latinshift}
\end{figure}
In order to reduce the total number of different clusterings we select the clustering corresponds to the Latin Square $L_{1}$ shown in figure \ref{L1} as the clustering to remove both these singular fade states. The corresponding clustering, $\mathcal{C}_0$ is
\begin{center}
\{(0,1)(1,0)(2,3)(3,2)\}, ~~ \{(0,2)(1,3)(2,0)(3,1)\}, \\
\{(0,3)(3,0)(1,2)(2,1)\}, ~~ \{(0,0)(1,1)(2,2)(3,3)\}.
\end{center}
Now consider the singular fade state $(\gamma=1, \theta=\pi/2).$ The singularity-removal constraints are
\begin{center}
$\{(0,0)(1,3)\};~~\{(0,1)(1,2)(2,3)(3,0)\}; ~~\{(0,2)(3,3)\}$\\
$\{(1,1)(2,0)\}; ~~\{(2,2)(3,1)\}$
\end{center}
In this case also Latin Square can be constructed with $t=4$, in three different ways as shown in figure \ref{L6}, figure \ref{L7} and figure \ref{L8}. But as in earlier case out of these three one, $L_{6}$ (shown in figure \ref{L6}) will remove singular fade state $(\gamma=1, \theta=-\pi/2)$. The singularity-removal constraints for $(\gamma=1, \theta=-\pi/2)$ are
\begin{center}
$\{(0,2)(1,1)\}; ~~\{(0,3)(1,0)(2,1)(3,2)\}; ~~\{(0,0)(3,1)\};$ \\
$\{(1,3)(2,2)\}; ~~ \{(2,0)(3,3)\}.$
\end{center}
All the Latin Squares which remove the singular fade state $(\gamma=1, \theta=-\pi/2)$ are shown in figure \ref{L6}, figure \ref{L9} and figure \ref{L10}. We will select that clustering which reduces total number of different clusterings, i.e, $L_{6}$ as was done before. The corresponding clustering, $\mathcal{C}_1$ is as follows:
\begin{center}
\{(0,0)(1,3)(2,2)(3,1)\}, ~~ \{(0,1)(1,2)(2,3)(3,0)\}, \\
\{(0,2)(3,3)(1,1)(2,0)\}, ~~ \{(0,3)(1,0)(2,1)(3,2)\}.
\end{center}
The interesting point here is that the Latin Squares $L_{1}$ and $L_{6}$ are Isotopic Latin Squares. That is, clustering corresponding to $L_{6}$ is obtained by, cyclically shifting the columns of $L_{1}$ (since columns are indexed by constellation points used by node B) as in Fig.\ref{fig:Obtaining L.6 from L.1 by column shifting}.
\begin{figure}[t]
\centering
\vspace{-1 cm}
\includegraphics[totalheight=2in,width=3in]{qpsklatin6}
\vspace{-.8 cm}
\caption{Obtaining $L_{6}$ from $L_{1}$ by column shifting}
\label{fig:Obtaining L.6 from L.1 by column shifting}
\end{figure}
Now we have removed four singular fade states till now all of them on the unit circle. Consider next, $(\gamma=1/\sqrt{2}, \theta=\pi/4).$ The singularity-removal constraints are
\begin{center}
$\{(0,1)(1,3)\};~~\{(0,2)(3,0)\};~~\{(1,2)(2,0)\};~~\{(2,3)(3,1)\}.$
\end{center}
The corresponding partially filled Latin Square is shown in figure {\ref{parfil1}.
It cannot be completed with $t$=4. That means that the relay has to use a constellation of size more than four. We can see that it can be completed with $t$=5. We get two clusterings as given in figure \ref{L11} and figure \ref{L12}. Both will remove this singular fade state and use constellation of size five.
The clustering corresponding to Latin Square $L_{11}$, denoted by $\mathcal{C}_2$ is
\begin{center}
\{(0,0)(2,3)(3,1)\},~~ \{(0,1)(1,3)(2,2)\},~~ \{(0,2)(1,1)(3,0)\},
\{(0,3)(1,0)(2,1)(3,2)\},~~ \{(1,2)(2,0)(3,3)\}.
\end{center}
The clustering corresponding to Latin Square $L_{12}$, denoted by $\mathcal{C}_3$ is
\begin{center}
\{(0,0)(1,1)(2,2)(3,3)\},~~ \{(0,1)(1,3)(3,2)\}, ~~ \{(0,2)(2,1)(3,0)\},
\{(0,3)(1,2)(2,0)\},~~ \{(1,0)(2,3)(3,1)\}\}.
\end{center}
Now considering the next singular fade state $(\gamma=\sqrt{2}, \theta=\pi/4),$ by the same procedure as before, the singularity-removal constraints are
\begin{center}
\{(0,1)(2,0)\},~ \{(0,2)(2,3)\},~\{(1,2)(3,1)\},~\{(1,3)(3,0)\}.
\end{center}
The partially filled Latin Square is shown in figure \ref{parfil2}. This cannot be completed with $t=4$, but by $t=5$ it can be completed in two ways as in $L_{13}$ and $L_{14}$ shown in figure \ref{L13} and figure \ref{L14}. The corresponding clusterings are shown in the Table \ref{table1}.
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 1 & 0 & 3 & 2\\
\hline 2 & 3 & 0 & 1\\
\hline 3 & 2 & 1 & 0\\
\hline
\end{tabular}}
\label{L1}
}
\subfigure[]{
{\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 1 & 3 & 0 & 2\\
\hline 2 & 0 & 3 & 1\\
\hline 3 & 2 & 1 & 0\\
\hline
\end{tabular}}
\label{L2}
}
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 1 & 0 & 3 & 2\\
\hline 2 & 3 & 1 & 0\\
\hline 3 & 2 & 0 & 1\\
\hline
\end{tabular}}
\label{L3}
}
\caption[]{Completely Filled Latin Squares for $\gamma=1, \theta=0$ }
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 2 & 0 & 3 & 1\\
\hline 1 & 3 & 0 & 2\\
\hline 3 & 2 & 1 & 0\\
\hline
\end{tabular}}
\label{L4}
}
\subfigure[]{
{\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 1 & 0 & 3 & 2\\
\hline 3 & 2 & 0 & 1\\
\hline 2 & 3 & 1 & 0\\
\hline
\end{tabular}}
\label{L5}
}
\caption[]{Completely Filled Latin Squares for $\gamma=1, \theta=\pi$ }
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 1 & 2 & 3 & 0\\
\hline 0 & 3 & 2 & 1\\
\hline 3 & 0 & 1 & 2\\
\hline 2 & 1 & 0 & 3\\
\hline
\end{tabular}}
\label{L6}
}
\subfigure[]{
{\begin{tabular}{|c|c|c|c|}
\hline 1 & 2 & 3 & 0\\
\hline 3 & 0 & 2 & 1\\
\hline 0 & 3 & 1 & 2\\
\hline 2 & 1 & 0 & 3\\
\hline
\end{tabular}}
\label{L7}
}
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 1 & 2 & 3 & 0\\
\hline 0 & 3 & 2 & 1\\
\hline 3 & 1 & 0 & 2\\
\hline 2 & 0 & 1 & 3\\
\hline
\end{tabular}}
\label{L8}
}
\caption[]{Completely Filled Latin Squares for $\gamma=1, \theta=\pi/2$ }
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 1 & 2 & 3 & 0\\
\hline 0 & 3 & 1 & 2\\
\hline 3 & 0 & 2 & 1\\
\hline 2 & 1 & 0 & 3\\
\hline
\end{tabular}}
\label{L9}
}
\subfigure[]{
{\begin{tabular}{|c|c|c|c|}
\hline 1 & 2 & 3 & 0\\
\hline 0 & 3 & 2 & 1\\
\hline 2 & 0 & 1 & 3\\
\hline 3 & 1 & 0 & 2\\
\hline
\end{tabular}}
\label{L10}
}
\caption[]{Completely Filled Latin Squares for $\gamma=1, \theta=-\pi/2$ }
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline & 0 & 1 & \\
\hline & & 2 & 0\\
\hline 2 & & & 3\\
\hline 1 & 3 & & \\
\hline
\end{tabular}}
\label{parfil1}
}
\subfigure[]{
{\begin{tabular}{|c|c|c|c|}
\hline & 0 & 1 & \\
\hline & & 2 & 3\\
\hline 0 & & & 1\\
\hline 3 & 2 & & \\
\hline
\end{tabular} }
\label{parfil2}
}
\caption[]{Partially Filled Latin Squares for $\gamma=1/\sqrt{2}, \theta=\pi/4$ and for $\gamma=\sqrt{2}, \theta=\pi/4$ }
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 3 & 0 & 1 & 4\\
\hline 4 & 1 & 2 & 0\\
\hline 2 & 4 & 0 & 3\\
\hline 1 & 3 & 4 & 2\\
\hline
\end{tabular}}
\label{L11}
}
\subfigure[]{
{\begin{tabular}{|c|c|c|c|}
\hline 4 & 0 & 1 & 2\\
\hline 3 & 4 & 2 & 0\\
\hline 2 & 1 & 4 & 3\\
\hline 1 & 3 & 0 & 4\\
\hline
\end{tabular} }
\label{L12}
}
\caption[]{$L_{11}$ for $\gamma=1/\sqrt{2}, \theta=\pi/4$ and $\gamma=\sqrt{2}, \theta=3\pi/4$ and $L_{12}$ for $\gamma=1/\sqrt{2}, \theta=\pi/4$ and $\gamma=\sqrt{2}, \theta=-\pi/4$}
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 4 & 0 & 1 & 2\\
\hline 1 & 4 & 2 & 3\\
\hline 0 & 3 & 4 & 1\\
\hline 3 & 2 & 0 & 4\\
\hline
\end{tabular}}
\label{L13}
}
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 2 & 0 & 1 & 4\\
\hline 4 & 1 & 2 & 3\\
\hline 0 & 4 & 3 & 1\\
\hline 3 & 2 & 4 & 0\\
\hline
\end{tabular}}
\label{L14}
}
\caption[]{$L_{13}$ for $\gamma=\sqrt{2}, \theta=\pi/4$ and $\gamma=1/\sqrt{2}, \theta=-\pi/4$ and $L_{14}$ for $\gamma=\sqrt{2}, \theta=\pi/4$ and $\gamma=1/\sqrt{2}, \theta=3\pi/4$}
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 1 & 4 & 0 & 2\\
\hline 2 & 0 & 3 & 4\\
\hline 3 & 2 & 4 & 1\\
\hline
\end{tabular} }
\label{L15}
}
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 1 & 3 & 4 & 2\\
\hline 2 & 4 & 1 & 0\\
\hline 3 & 2 & 0 & 4\\
\hline
\end{tabular}}
\label{L16}
}
\caption[]{$L_{15}$ for $\gamma=\sqrt{2}, \theta=-3\pi/4$ and $\gamma=1/\sqrt{2}, \theta=3\pi/4$ and $L_{16}$ for $\gamma=\sqrt{2}, \theta=3\pi/4$ and $\gamma=1/\sqrt{2}, \theta=-3\pi/4$}
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 4 & 3 & 1 & 0\\
\hline 3 & 2 & 4 & 1\\
\hline 1 & 4 & 0 & 2\\
\hline
\end{tabular} }
\label{L17}
}
\subfigure[]{
{
\begin{tabular}{|c|c|c|c|}
\hline 0 & 1 & 2 & 3\\
\hline 2 & 4 & 1 & 0\\
\hline 4 & 0 & 3 & 1\\
\hline 1 & 3 & 4 & 2\\
\hline
\end{tabular}}
\label{L18}
}
\caption[]{$L_{17}$ for $\gamma=\sqrt{2}, \theta=-\pi/4$ and $\gamma=1/\sqrt{2}, \theta=-3\pi/4$ and $L_{18}$ for $\gamma=\sqrt{2}, \theta=-3\pi/4$ and $\gamma=1/\sqrt{2}, \theta=-\pi/4$}
\end{figure}
\begin{table*}
\centering
\caption{Clusterings Obtained for different singular fade states when the end nodes use QPSK constellations}
\label{table1}
\begin{tabular}{|c|c|c|c|}
\hline Sl.No & Singular fade states & Clustering & Cluster\\
\hline 1& $\gamma=1,\theta=0$ & $\mathcal{C}_0$ &$\{$\{(0,1)(1,0)(2,3)(3,2)\}$,$\{(0,2)(1,3)(2,0)(3,1)\}$,$\{(0,3)(3,0)(1,2)(2,1)\}$,$\{(0,0)(1,1)(2,2)(3,3)\}$\}$\\
\hline 2&$\gamma=1,\theta=\pi/2$ & $\mathcal{C}_1$ &$\{$\{(0,0)(1,3)(2,2)(3,1)\}$,$\{(0,1)(1,2)(2,3)(3,0)\}$,$\{(0,2)(3,3)(1,1)(2,0)$\},$\{(0,3)(1,0)(2,1)(3,2)$\}\}$\\
\hline 3&$\gamma=1,\theta=\pi$ & $\mathcal{C}_0$ &$\{$\{(0,1)(1,0)(2,3)(3,2)\}$,$\{(0,2)(1,3)(2,0)(3,1)\}$,$\{(0,3)(3,0)(1,2)(2,1)\}$,$\{(0,0)(1,1)(2,2)(3,3)\}$\}$\\
\hline 4&$\gamma=1,\theta=-\pi/2$ & $\mathcal{C}_1$ &$\{$\{(0,0)(1,3)(2,2)(3,1)\}$,$\{(0,1)(1,2)(2,3)(3,0)\}$,$\{(0,2)(3,3)(1,1)(2,0)$\},$\{(0,3)(1,0)(2,1)(3,2)$\}\}$\\
\hline 5.a&$\gamma=1/\sqrt{2},\theta=\pi/4$ & $\mathcal{C}_2$ &$\{$\{(0,0)(2,3)(3,1)\}$,$\{(0,1)(1,3)(2,2)\}$,$\{(0,2)(1,1)(3,0)\}$,$\{(0,3)(1,0)(2,1)(3,2)$\},$\{(1,2)(2,0)(3,3)$\}\}$\\
5.b& & $\mathcal{C}_3$ &$\{$\{(0,0)(1,1)(2,2)(3,3)\}$,$\{(0,1)(1,3)(3,2)\}$,$\{(0,2)(2,1)(3,0)\}$,$\{(0,3)(1,2)(2,0)$\},$\{(1,0)(2,3)(3,1)$\}\}$\\
\hline 6.a&$\gamma=\sqrt{2},\theta=\pi/4$ & $\mathcal{C}_4$ &$\{$\{(0,0)(1,1)(2,2)(3,3)\}$,$\{(0,1)(2,0)(3,2)\}$,$\{(0,2)(1,0)(2,3)\}$,$\{(0,3)(1,2)(3,1)$\},$\{(1,3)(2,1)(3,0)$\}\}$\\
6.b& & $\mathcal{C}_5$ &$\{$\{(0,0)(1,2)(3,1)\}$,$\{(0,1)(2,0)(3,3)\}$,$\{(0,2)(1,1)(2,3)\}$,$\{(0,3)(1,0)(2,1)(3,2)$\},$\{(1,3)(2,2)(3,0)$\}\}$\\
\hline 7.a&$\gamma=1/\sqrt{2},\theta=3\pi/4$ & $\mathcal{C}_6$ &$\{$\{(0,0)(1,2)(2,1)\}$,$\{(0,1)(1,0)(3,3)\}$,$\{(0,2)(1,3)(2,0)(3,1)\}$,$\{(0,3)(2,2)(3,0)$\},$\{(1,1)(2,3)(3,2)$\}\}$\\
7.b& & $\mathcal{C}_5$ &$\{$\{(0,0)(1,2)(3,1)\}$,$\{(0,1)(2,0)(3,3)\}$,$\{(0,2)(1,1)(2,3)\}$,$\{(0,3)(1,0)(2,1)(3,2)$\},$\{(1,3)(2,2)(3,0)$\}\}$\\
\hline 8.a&$\gamma=\sqrt{2},\theta=3\pi/4$ & $\mathcal{C}_2$ &$\{$\{(0,0)(2,3)(3,1)\}$,$\{(0,1)(1,3)(2,2)\}$,$\{(0,2)(1,1)(3,0)\}$,$\{(0,3)(1,0)(2,1)(3,2)$\},$\{(1,2)(2,0)(3,3)$\}\}$\\
8.b& & $\mathcal{C}_7$ &$\{$\{(0,0)(2,3)(3,2)\}$,$\{(0,1)(1,0)(2,2)\}$,$\{(0,2)(1,3)(2,0)(3,1)\}$,$\{(0,3)(1,1)(3,0)$\},$\{(1,2)(2,1)(3,3)$\}\}$\\
\hline 9.a&$\gamma=1/\sqrt{2},\theta=-3\pi/4$ & $\mathcal{C}_7$ &$\{$\{(0,0)(2,3)(3,2)\}$,$\{(0,1)(1,0)(2,2)\}$,$\{(0,2)(1,3)(2,0)(3,1)\}$,$\{(0,3)(1,1)(3,0)$\},$\{(1,2)(2,1)(3,3)$\}\}$\\
9.b & & $\mathcal{C}_8$ &$\{$\{(0,0)(1,3)(3,2)\}$,$\{(0,1)(1,2)(2,3)(3,0)\}$,$\{(0,2)(2,1)(3,3)\}$,$\{(0,3)(1,1)(2,0)$\},$\{(1,0)(2,2)(3,1)$\}\}$\\
\hline 10.a&$\gamma=\sqrt{2},\theta=-3\pi/4$ & $\mathcal{C}_6$ &$\{$\{(0,0)(1,2)(2,1)\}$,$\{(0,1)(1,0)(3,3)\}$,$\{(0,2)(1,3)(2,0)(3,1)\}$,$\{(0,3)(2,2)(3,0)$\},$\{(1,1)(2,3)(3,2)$\}\}$\\
10.b& & $\mathcal{C}_9$ &$\{$\{(0,0)(1,3)(2,1)\}$,$\{(0,1)(1,2)(2,3)(3,0)\}$,$\{(0,2)(1,0)(3,3)\}$,$\{(0,3)(2,2)(3,1)$\},$\{(1,1)(2,0)(3,2)$\}\}$\\
\hline 11.a&$\gamma=1/\sqrt{2},\theta=-\pi/4$ & $\mathcal{C}_4$ &$\{$\{(0,0)(1,1)(2,2)(3,3)\}$,$\{(0,1)(2,0)(3,2)\}$,$\{(0,2)(1,0)(2,3)\}$,$\{(0,3)(1,2)(3,1)$\},$\{(1,3)(2,1)(3,0)$\}\}$\\
11.b& & $\mathcal{C}_9$ &$\{$\{(0,0)(1,3)(2,1)\}$,$\{(0,1)(1,2)(2,3)(3,0)\}$,$\{(0,2)(1,0)(3,3)\}$,$\{(0,3)(2,2)(3,1)$\},$\{(1,1)(2,0)(3,2)$\}\}$\\
\hline 12.a&$\gamma=\sqrt{2},\theta=-\pi/4$ & $\mathcal{C}_3$ &$\{$\{(0,0)(1,1)(2,2)(3,3)\}$,$\{(0,1)(1,3)(3,2)\}$,$\{(0,2)(2,1)(3,0)\}$,$\{(0,3)(1,2)(2,0)$\},$\{(1,0)(2,3)(3,1)$\}\}$\\
12.b& & $\mathcal{C}_8$ &$\{$\{(0,0)(1,3)(3,2)\}$,$\{(0,1)(1,2)(2,3)(3,0)\}$,$\{(0,2)(2,1)(3,3)\}$,$\{(0,3)(1,1)(2,0)$\},$\{(1,0)(2,2)(3,1)$\}\}$\\
\hline
\end{tabular}
\vspace{-.1 cm}
\end{table*}
Next, consider the singular fade state $(\gamma=1/\sqrt{2}, \theta=3\pi/4).$ The singularity-removal constraints are
\begin{center}
\{(0,0)(1,2)\},~\{(0,1)(3,3)\},~\{(1,1)(2,3)\},~\{(2,2)(3,0)\}.
\end{center}
The corresponding Latin Squares are shown in figure \ref{L14} and figure \ref{L15}.
For the singular fade state $(\gamma=\sqrt{2}, \theta=3\pi/4)$, the singularity-removal constraints are
\begin{center}
\{(0,0)(2,3)\},~\{(0,1)(2,2)\},~\{(1,1)(3,0)\},~\{(1,2)(3,3)\}
\end{center}
and the corresponding Latin Squares are shown in figure \ref{L11} and figure \ref{L16} .
Similarly, for the singular fade state $(\gamma=1/\sqrt{2}, \theta=-3\pi/4),$ the singularity-removal constraints are
\begin{center}
\{(0,0)(3,2)\},~\{(0,3)(1,1)\},~\{(1,0)(2,2)\},~\{(2,1)(3,3)\}
\end{center}
with the corresponding Latin Squares as shown in figure \ref{L16} and figure \ref{L17}.
The singularity-removal constraints for singular fade state $(\gamma=\sqrt{2}, \theta=-3\pi/4)$ are
\begin{center}
\{(0,0)(2,1)\},~\{(0,3)(2,2)\},~\{(1,0)(3,3)\},~\{(1,1)(3,2)\}.
\end{center}
and the Latin Squares are given in figure \ref{L15} and figure \ref{L18}. The singularity-removal constraints for singular fade state $(\gamma=1/\sqrt{2}, \theta=-\pi/4)$ are
\begin{center}
\{(0,2)(1,0)\},~\{(0,3)(3,1)\},~\{(1,3)(2,1)\},~\{(2,0)(3,2)\}.
\end{center}
The Latin Squares are given in figure.\ref{L13} and figure.\ref{L18}. The singularity-removal constraints for singular fade state $\gamma=\sqrt{2}$ and $\theta=-\pi/4$ are
\begin{center}
\{(0,2)(2,1)\},~\{(0,3)(2,0)\},~\{(1,0)(3,1)\},~\{(1,3)(3,2)\}.
\end{center}
The Latin Squares are given in figure \ref{L12} and figure \ref{L17} .
It is observed that to remove all other singular fade states not lying on unit circle the relay needs a constellation of size five. Table \ref{table1} shows the singular fade states and the corresponding clusterings. There are two clusterings to remove a singular fade state for all singular fade states except for those with $\gamma=1.$ We can select any one. Anyone from the two $\{\mathcal{C}_2,\mathcal{C}_3 \}$ can be selected to remove singular fade state $(\gamma=1/\sqrt{2}, \theta=\pi/4).$ After that, by column permutations we can remove the singular fade states with $(\gamma=1/\sqrt{2},$ and $\theta=+3\pi/4,-\pi/4,-3\pi/4$. By taking transpose of the Latin Square for $(\gamma=1/\sqrt{2}, \theta=\pi/4)$ we can remove singular fade state $(\gamma=\sqrt{2}, \theta=-\pi/4).$ After that by column permutations we can remove the singular fade states with $\gamma=\sqrt{2}$ and $\theta=+3\pi/4,+\pi/4,-3\pi/4$. If we select $\mathcal{C}_2$ to remove $(\gamma=1/\sqrt2, \theta=\pi/4),$ we will get the following set of clusterings $\{\mathcal{C}_0,\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_4,\mathcal{C}_6,\mathcal{C}_8\}$ to remove all the singular fade states. In the other case, when we select $\mathcal{C}_3$ to remove $(\gamma=1/\sqrt{2},\theta=\pi/4)$ we will get the following set of clusterings $\{\mathcal{C}_0,\mathcal{C}_1,\mathcal{C}_3,\mathcal{C}_5,\mathcal{C}_7,\mathcal{C}_9\}$ to remove all the singular fade states.
\subsection{End nodes use 8PSK}
Consider the scenario where the end nodes use 8-PSK constellation as shown in Fig.\ref{8psk}.
\begin{figure}[htbp]
\centering
\includegraphics[totalheight=1.5in,width=1.5in]{8psk}
\caption{8PSK constellation used by nodes A and B}
\label{8psk}
\end{figure}
For 8-PSK signal set, the 104 singular fade states are shown in Fig.\ref{8psk_sing}. These singular fade states lie in 13 circles, 8 in each circle separated by an angle $\pi/4$. It is sufficient to get Latin Squares for these 13 values, then by isotopic property of the Latin Squares all other singular fade states are removed. Further by taking transposes of the Latin Square for a singular fade state $\gamma, \theta$, another singular fade state $1/\gamma, -\theta$ is removed. As discussed XOR removes singular fade state $\gamma=1, \theta=0$. Finally it is seen that obtaining six Latin Squares for different singular fade state values for $\gamma < 1$ (or $\gamma > 1$) and the XOR is enough to remove all the 104 singular fade states. The 104 singular fade states are,\\
for $\theta=2m\pi/8$ , where $m=0,1,2\cdots7$\\\\
$\gamma=\dfrac{\sin\pi/8}{\sin\pi/8},\dfrac{\sin\pi/8}{\sin3\pi/8}, \dfrac{\sin3\pi/8}{\sin\pi/8}, \dfrac{\sin4\pi/8}{\sin2\pi/8}$ and $\dfrac{\sin2\pi/8}{\sin4\pi/8}$.\\\\
for $\theta=(2m+1)\pi/8$ , where $m=0,1,2\cdots7$\\\\
$\gamma=\dfrac{\sin\pi/8}{\sin4\pi/8},\dfrac{\sin\pi/8}{\sin2\pi/8}, \dfrac{\sin2\pi/8}{\sin3\pi/8}, \dfrac{\sin3\pi/8}{\sin4\pi/8},\dfrac{\sin4\pi/8}{\sin3\pi/8},$\\
\hspace*{.75cm}$ \dfrac{\sin3\pi/8}{\sin2\pi/8}, \dfrac{\sin2\pi/8}{\sin\pi/8}$ and $\dfrac{\sin4\pi/8}{\sin\pi/8}$.
We consider singular fade states one by one. The singularity constraints for $\gamma=1, \theta=0$ are
\begin{center}
$\{(0,1)(1,0)\}$, $\{(0,2)(2,0)\}$, $\{(0,3)(3,0)\}$, $\{(0,5)(5,0)\}$\\*
$\{(0,6)(6,0)\}$, $\{(0,7)(7,0)\}$, $\{(1,2)(2,1)\}$, $\{(1,3)(3,1)\}$\\*
$\{(1,4)(4,1)\}$, $\{(1,6)(6,1)\}$, $\{(1,7)(7,1)\}$, $\{(2,3)(3,2)\}$\\*
$\{(2,4)(4,2)\}$, $\{(2,6)(6,2)\}$, $\{(2,7)(7,2)\}$, $\{(3,4)(4,3)\}$\\*
$\{(3,5)(5,3)\}$, $\{(3,6)(6,3)\}$, $\{(4,5)(5,4)\}$, $\{(4,6)(6,4)\}$\\*
$\{(4,7)(7,4)\}$, $\{(5,6)(6,5)\}$, $\{(5,7)(7,5)\}$, $\{(6,7)(7,6)\}$\\*
$\{(0,4)(1,5)(2,6)(3,7)(4,0)(5,1)(6,2)(7,3)\}$\\*
\end{center}
XOR can remove this singular fade state. Further by appropriately column shifting all the singular fade states with $\gamma=1$ is removed. For example consider $\gamma=1, \theta=\pi/4$. The singularity constraints are
\begin{center}
$\{(0,0)(1,7)\}$, $\{(0,1)(2,7)\}$, $\{(0,2)(3,7)\}$, $\{(0,4)(5,7)\}$\\*
$\{(0,5)(6,7)\}$, $\{(0,6)(7,7)\}$, $\{(1,1)(2,0)\}$, $\{(1,2)(3,0)\}$\\*
$\{(1,3)(4,0)\}$, $\{(1,5)(6,0)\}$, $\{(1,6)(7,0)\}$, $\{(2,2)(3,1)\}$\\*
$\{(2,3)(4,1)\}$, $\{(2,5)(6,1)\}$, $\{(2,6)(7,1)\}$, $\{(3,3)(4,2)\}$\\*
$\{(3,4)(5,2)\}$, $\{(3,5)(6,2)\}$, $\{(4,4)(5,3)\}$, $\{(4,5)(6,3)\}$\\*
$\{(4,6)(7,3)\}$, $\{(5,5)(6,4)\}$, $\{(5,6)(7,4)\}$, $\{(6,6)(7,5)\}$\\*
$\{(0,3)(1,4)(2,5)(3,6)(4,7)(5,0)(6,1)(7,2)\}$\\*
\end{center}
Next consider $\gamma=\dfrac{\sin \pi/8}{\sin 3\pi/8} =0.414, \theta=0$. The singularity constraints are
\begin{center}
$\{(0,2)(1,7)\}$, $\{(0,3)(1,6)\}$, $\{(0,5)(7,2)\}$, $\{(0,6)(7,1)\}$\\*
$\{(1,3)(2,0)\}$, $\{(1,4)(2,7)\}$, $\{(2,4)(3,1)\}$, $\{(2,5)(3,0)\}$\\*
$\{(3,5)(4,2)\}$, $\{(3,6)(4,1)\}$, $\{(4,6)(5,3)\}$, $\{(4,7)(5,2)\}$\\*
$\{(5,0)(6,3)\}$, $\{(5,7)(6,4)\}$, $\{(6,0)(7,5)\}$, $\{(6,1)(7,4)\}$\\*
\end{center}
\begin{figure}[h]
\centering
\subfigure[$\gamma=1, \theta=0$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline 1 & 0 & 3 & 2 & 5 & 4 & 7 & 6 \\
\hline 2 & 3 & 0 & 1 & 6 & 7 & 4 & 5 \\
\hline 3 & 2 & 1 & 0 & 7 & 6 & 5 & 4 \\
\hline 4 & 5 & 6 & 7 & 0 & 1 & 2 & 3 \\
\hline 5 & 4 & 7 & 6 & 1 & 0 & 3 & 2 \\
\hline 6 & 7 & 4 & 5 & 2 & 3 & 0 & 1 \\
\hline 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\
\hline
\end{tabular}}
\label{8L1}
}
\subfigure[$\gamma=1, \theta=\pi/4$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 0 \\
\hline 0 & 3 & 2 & 5 & 4 & 7 & 6 & 1 \\
\hline 3 & 0 & 1 & 6 & 7 & 4 & 5 & 2 \\
\hline 2 & 1 & 0 & 7 & 6 & 5 & 4 & 3\\
\hline 5 & 6 & 7 & 0 & 1 & 2 & 3 & 4\\
\hline 4 & 7 & 6 & 1 & 0 & 3 & 2 & 5\\
\hline 7 & 4 & 5 & 2 & 3 & 0 & 1 & 6\\
\hline 6 & 5 & 4 & 3 & 2 & 1 & 0 & 7\\
\hline
\end{tabular}}
\label{8L2}
}
\subfigure[$\gamma=\dfrac{\sin \pi/8}{\sin 3\pi/8}, \theta=0$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline 5 & 4 & 7 & 6 & 1 & 0 & 3 & 2\\
\hline 6 & 3 & 0 & 5 & 2 & 7 & 4 & 1 \\
\hline 7 & 2 & 1 & 4 & 3 & 6 & 5 & 0 \\
\hline 4 & 5 & 6 & 7 & 0 & 1 & 2 & 3 \\
\hline 1 & 0 & 3 & 2 & 5 & 4 & 7 & 6 \\
\hline 2 & 7 & 4 & 1 & 6 & 3 & 0 & 5 \\
\hline 3 & 6 & 5 & 0 & 7 & 2 & 1 & 4 \\
\hline
\end{tabular}}
\label{8L3}
}
\subfigure[$\gamma=\dfrac{\sin \pi/8}{\sin 3\pi/8}, \theta=0$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 0\\
\hline 4 & 7 & 6 & 1 & 0 & 3 & 2 & 5\\
\hline 3 & 0 & 5 & 2 & 7 & 4 & 1 & 6\\
\hline 2 & 1 & 4 & 3 & 6 & 5 & 0 & 7\\
\hline 5 & 6 & 7 & 0 & 1 & 2 & 3 & 4\\
\hline 0 & 3 & 2 & 5 & 4 & 7 & 6 & 1\\
\hline 7 & 4 & 1 & 6 & 3 & 0 & 5 & 2\\
\hline 6 & 5 & 0 & 7 & 2 & 1 & 4 & 3\\
\hline
\end{tabular}}
\label{8L4}
}
\caption[]{Latin Squares Corresponding to Different singular fade states }
\end{figure}
\begin{figure}[h]
\centering
\subfigure[$\gamma=\dfrac{\sin 3\pi/8}{\sin \pi/8}, \theta=-\pi/4$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 1 & 4 & 3 & 2 & 5 & 0 & 7 & 6 \\
\hline 2 & 7 & 0 & 1 & 6 & 3 & 4 & 5\\
\hline 3 & 6 & 5 & 4 & 7 & 2 & 1 & 0 \\
\hline 4 & 1 & 2 & 3 & 0 & 5 & 6 & 7 \\
\hline 5 & 0 & 7 & 6 & 1 & 4 & 3 & 2 \\
\hline 6 & 3 & 4 & 5 & 2 & 7 & 0 & 1 \\
\hline 7 & 2 & 1 & 0 & 3 & 6 & 5 & 4 \\
\hline 0 & 5 & 6 & 7 & 4 & 1 & 2 & 3 \\
\hline
\end{tabular}}
\label{8L5}
}
\subfigure[$\gamma=\dfrac{\sin 2\pi/8}{\sin 4\pi/8}, \theta=0$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 4 & 0 & 6 & 3 & 1 & 5 & 7 & 2\\
\hline 2 & 7 & 0 & 1 & 3 & 6 & 5 & 4\\
\hline 0 & 4 & 7 & 2 & 5 & 1 & 6 & 3\\
\hline 3 & 6 & 4 & 0 & 2 & 7 & 1 & 5\\
\hline 5 & 1 & 2 & 6 & 0 & 4 & 3 & 7\\
\hline 7 & 2 & 1 & 5 & 6 & 3 & 4 & 0\\
\hline 1 & 5 & 3 & 7 & 4 & 0 & 2 & 6\\
\hline 6 & 3 & 5 & 4 & 7 & 2 & 0 & 1\\
\hline
\end{tabular} }
\label{8L6}
}
\subfigure[$\gamma=\dfrac{\sin 2\pi/8}{\sin4\pi/8}, \theta=\pi/2$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 6 & 3 & 1 & 5 & 7 & 2 & 4 & 0\\
\hline 0 & 1 & 3 & 6 & 5 & 4 & 2 & 7\\
\hline 7 & 2 & 5 & 1 & 6 & 3 & 0 & 4\\
\hline 4 & 0 & 2 & 7 & 1 & 5 & 3 & 6\\
\hline 2 & 6 & 0 & 4 & 3 & 7 & 5 & 1\\
\hline 1 & 5 & 6 & 3 & 4 & 0 & 7 & 2\\
\hline 3 & 7 & 4 & 0 & 2 & 6 & 1 & 5\\
\hline 5 & 4 & 7 & 2 & 0 & 1 & 6 & 3\\
\hline
\end{tabular}}
\label{8L7}
}
\subfigure[$\gamma=\dfrac{\sin 4\pi/8}{\sin 2\pi/8}, \theta=-\pi/2$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 6 & 0 & 7 & 4 & 2 & 1 & 3 & 5\\
\hline 3 & 1 & 2 & 0 & 6 & 5 & 7 & 4\\
\hline 1 & 3 & 5 & 2 & 0 & 6 & 4 & 7\\
\hline 5 & 6 & 1 & 7 & 4 & 3 & 0 & 2\\
\hline 7 & 5 & 6 & 1 & 3 & 4 & 2 & 0\\
\hline 2 & 4 & 3 & 5 & 7 & 0 & 6 & 1\\
\hline 4 & 2 & 0 & 3 & 5 & 7 & 1 & 6\\
\hline 0 & 7 & 4 & 6 & 1 & 2 & 5 & 3\\
\hline
\end{tabular} }
\label{8L8}
}
\caption[]{Latin Squares Corresponding to Different singular fade states }
\end{figure}
\begin{figure}[h]
\centering
\subfigure[$\gamma=\dfrac{\sin \pi/8}{\sin 4\pi/8}, \theta=\pi/8$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 4 & 3 & 0 & 6 & 2 & 1 & 7 & 5 \\
\hline 1 & 5 & 3 & 2 & 4 & 7 & 0 & 6 \\
\hline 5 & 6 & 7 & 1 & 3 & 0 & 4 & 2 \\
\hline 3 & 2 & 6 & 5 & 0 & 4 & 1 & 7 \\
\hline 0 & 4 & 2 & 7 & 6 & 3 & 5 & 1 \\
\hline 6 & 7 & 5 & 0 & 1 & 2 & 3 & 4 \\
\hline 7 & 0 & 1 & 4 & 5 & 6 & 2 & 3 \\
\hline 2 & 1 & 4 & 3 & 7 & 5 & 6 & 0 \\
\hline
\end{tabular}}
\label{8L9}
}
\subfigure[$\gamma=\dfrac{\sin \pi/8}{\sin 2\pi/8}, \theta=\pi/8$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 6 & 0 & 7 & 1 & 2 & 4 & 3 & 5\\
\hline 7 & 3 & 4 & 2 & 5 & 1 & 6 & 0\\
\hline 4 & 2 & 0 & 7 & 1 & 6 & 5 & 3\\
\hline 0 & 7 & 1 & 4 & 3 & 5 & 2 & 6\\
\hline 2 & 4 & 3 & 5 & 6 & 0 & 7 & 1\\
\hline 5 & 1 & 6 & 0 & 7 & 3 & 4 & 2\\
\hline 1 & 6 & 5 & 3 & 4 & 2 & 0 & 7\\
\hline 3 & 5 & 2 & 6 & 0 & 7 & 1 & 4\\
\hline
\end{tabular} }
\label{8L10}
}
\subfigure[$\gamma=\dfrac{\sin 2\pi/8}{\sin 3\pi/8}, \theta=\pi/8$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 7 & 5 & 0 & 1 & 2 & 3 & 4 & 6 \\
\hline 0 & 1 & 3 & 4 & 5 & 6 & 7 & 2 \\
\hline 6 & 4 & 5 & 7 & 3 & 2 & 1 & 0 \\
\hline 4 & 0 & 2 & 3 & 1 & 7 & 6 & 5 \\
\hline 2 & 3 & 4 & 6 & 7 & 5 & 0 & 1 \\
\hline 5 & 6 & 7 & 2 & 0 & 1 & 3 & 4 \\
\hline 3 & 2 & 1 & 0 & 6 & 4 & 5 & 7 \\
\hline 1 & 7 & 6 & 5 & 4 & 0 & 2 & 3 \\
\hline
\end{tabular} }
\label{8L11}
}
\subfigure[$\gamma=\dfrac{\sin 3\pi/8}{\sin 4\pi/8}, \theta=\pi/8$]{
{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 0 & 1 & 2 & 3 & 6 & 5 & 4 & 7\\
\hline 6 & 5 & 0 & 7 & 3 & 4 & 2 & 1\\
\hline 5 & 6 & 7 & 0 & 4 & 3 & 1 & 2\\
\hline 1 & 7 & 3 & 6 & 0 & 2 & 5 & 4\\
\hline 3 & 0 & 1 & 4 & 2 & 7 & 6 & 5\\
\hline 4 & 3 & 6 & 2 & 5 & 1 & 7 & 0\\
\hline 2 & 4 & 5 & 1 & 7 & 0 & 3 & 6\\
\hline 7 & 2 & 4 & 5 & 1 & 6 & 0 & 3\\
\hline
\end{tabular} }
\label{8L12}
}
\caption[]{Latin Squares Corresponding to Different singular fade states }
\end{figure}
\begin{table*}
\centering
\caption{Clusterings Obtained for different singular fade states on each circle (for $\gamma \leq 1$) when the end nodes use 8-PSK constellations}
\label{table2}
\begin{tabular}{|c|c|c|}
\hline Sl.No & Singular fade states & Cluster\\
\hline & &$\{\{(0,1)(1,0)(2,3)(3,2)(5,4)(4,5)(6,7)(7,6)\},\{(0,2)(1,3)(2,0)(3,1)(4,6)(6,4)(5,7)(7,5)\}$ \\ 1&$\gamma=1,\theta=0$ & $\{(0,3)(3,0)(1,2)(2,1)(4,7)(5,6)(6,5)(7,4)\},\{(0,4)(4,0)(1,5)(5,1)(3,7)(2,6)(6,2)(7,3)\}$ \\ & & $\{(0,5)(1,4)(2,7)(3,6)(4,1)(5,0)(6,3)(7,2)\},\{(0,6)(6,0)(1,7)(7,1)(4,2)(5,3)(3,5)(2,4)\}$ \\ & & $\{(0,7)(7,0)(1,6)(6,1)(4,3)(5,2)(2,5)(3,4)\},\{(0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6),(7,7)\}\}$\\
\hline & &$\{\{(0,0)(1,5)(2,2)(3,7)(4,4)(5,1)(6,6)(7,3)\},\{(0,1)(1,4)(2,7)(3,2)(4,5)(5,0)(6,6)(7,3)\}$ \\ & &$\{(0,2)(3,1)(1,7)(2,4)(4,6)(5,3)(6,0)(7,5)\},\{(0,3)(1,6)(2,1)(3,4)(4,7)(5,2)(6,5)(7,0)\}$ \\ 2&$\gamma=\dfrac{\sin\pi/8}{\sin3\pi/8},\theta=0$ &$\{(0,4)(1,1)(2,6)(3,3)(4,0)(5,5)(6,2)(7,7)\},\{(0,5)(3,6)(1,0)(2,3)(4,1)(5,4)(6,7)(7,2)\}$ \\ & &$\{(0,6)(3,5)(1,3)(2,0)(4,2)(5,3)(6,4)(7,1)$\},$\{(0,7)(3,0)(1,2)(2,5)(4,3)(5,6)(6,1)(7,4)\}\}$\\
\hline & &$\{\{(0,0)(1,7)(2,1)(3,2)(4,5)(5,6)(6,4)(7,3)\},\{(0,1)(1,2)(2,0)(3,3)(4,4)(5,7)(6,5)(7,6)\}$ \\ & &$\{(0,2)(3,1)(1,5)(2,6)(4,3)(5,4)(6,7)(7,0)\},\{(0,3)(1,4)(2,7)(3,0)(4,6)(5,5)(6,2)(7,1)\}$ \\3&$\gamma=\dfrac{\sin2\pi/8}{\sin4\pi/8},\theta=0$ &$\{(0,4)(1,3)(2,5)(3,6)(4,1)(5,2)(6,0)(7,7)\},\{(0,5)(1,6)(3,7)(2,4)(4,0)(5,3)(6,1)(7,2)\}$ \\ & &$\{(0,6)(3,5)(1,1)(2,2)(4,7)(5,0)(6,3)(7,4)$\},$\{(0,7)(1,0)(2,3)(3,4)(4,2)(5,1)(6,6)(7,5)\}\}$\\
\hline & &$\{\{(0,0)(1,4)(2,6)(3,5)(4,1)(5,7)(6,3)(7,2)\},\{(0,1)(1,2)(2,4)(3,0)(4,5)(5,6)(6,7)(7,3)\}$\\ & &$\{(0,2)(1,6)(2,5)(3,4)(4,0)(5,3)(6,1)(7,7)\},\{(0,3)(1,7)(2,1)(3,2)(4,4)(5,0)(6,5)(7,6)\}$ \\ 4&$\gamma=\dfrac{\sin\pi/8}{\sin4\pi/8},\theta=\pi/8$ &$\{(0,4)(1,3)(2,7)(3,1)(4,2)(5,5)(6,6)(7,0)\},\{(0,5)(1,0)(2,3)(3,6)(4,7)(5,4)(6,2)(7,1)\}$ \\ & &$\{(0,6)(1,5)(2,2)(3,7)(4,3)(5,1)(6,0)(7,4)\},\{(0,7)(1,1)(2,0)(3,3)(4,6)(5,2)(6,4)(7,5)\}\}$\\
\hline & &$\{\{(0,0)(1,6)(2,5)(3,7)(4,4)(5,2)(6,1)(7,3)\},\{(0,1)(1,7)(2,2)(3,0)(4,5)(5,3)(6,6)(7,4)\}$ \\& &$\{(0,2)(1,0)(2,3)(3,1)(4,6)(5,4)(6,7)(7,5)\},\{(0,3)(1,5)(2,4)(3,2)(4,7)(5,1)(6,0)(7,6)\}$ \\ 5&$\gamma=\dfrac{\sin\pi/8}{\sin2\pi/8},\theta=\pi/8$ &$\{(0,4)(1,3)(2,1)(3,6)(4,0)(5,7)(6,5)(7,2)\},\{(0,5)(1,2)(2,0)(3,3)(4,1)(5,6)(6,4)(7,7)\}$ \\ & &$\{(0,6)(1,1)(2,7)(3,4)(4,2)(5,5)(6,3)(7,0)\},\{(0,7)(1,4)(2,6)(3,5)(4,3)(5,0)(6,2)(7,1)\}\}$\\
\hline & &$\{\{(0,0)(1,6)(2,3)(3,5)(4,4)(5,2)(6,7)(7,1)\},\{(0,1)(1,4)(2,2)(3,7)(4,5)(5,0)(6,6)(7,3)\}$ \\ & &$\{(0,2)(1,0)(2,7)(3,1)(4,6)(5,4)(6,3)(7,5)\},\{(0,3)(1,1)(2,6)(3,4)(4,7)(5,5)(6,2)(7,0)\}$ \\ 6&$\gamma=\dfrac{\sin2\pi/8}{\sin3\pi/8},\theta=\pi/8$ &$\{(0,4)(1,7)(2,5)(3,2)(4,0)(5,3)(6,1)(7,6)\},\{(0,5)(1,2)(2,4)(3,3)(4,1)(5,6)(6,0)(7,7)\}$ \\ & &$\{(0,6)(1,3)(2,1)(3,0)(4,2)(5,7)(6,5)(7,4)\},\{(0,7)(1,5)(2,0)(3,6)(4,3)(5,1)(6,4)(7,2)\}\}$\\
\hline & &$\{\{(0,0)(1,2)(2,3)(3,4)(4,1)(5,7)(6,5)(7,6)\},\{(0,1)(1,7)(2,6)(3,0)(4,2)(5,5)(6,3)(7,4)\}$\\ & &$\{(0,2)(1,6)(2,7)(3,5)(4,4)(5,3)(6,0)(7,1)\},\{(0,3)(1,4)(2,5)(3,2)(4,0)(5,1)(6,6)(7,7)\}$ \\ 7&$\gamma=\dfrac{\sin3\pi/8}{\sin4\pi/8},\theta=\pi/8$ &$\{(0,4)(1,0)(2,1)(3,3)(4,6)(5,2)(6,7)(7,5)\},\{(0,5)(1,1)(2,0)(3,6)(4,7)(5,4)(6,2)(7,3)\}$ \\ & &$\{(0,6)(1,5)(2,4)(3,7)(4,3)(5,0)(6,7)(7,2)\},\{(0,7)(1,3)(2,2)(3,1)(4,5)(5,6)(6,4)(7,0)\}\}$\\
\hline
\end{tabular}
\vspace{-.1 cm}
\end{table*}
The Latin Square shown in figure.\ref{8L3} removes this singular fade state. Further by column shifting all the singular fade states with $\gamma=0.414$ are removed. Consider $\gamma=\dfrac{\sin \pi/8}{\sin 3\pi/8} =0.414, \theta=\pi/4$. The singularity constraints are
\begin{center}
$\{(0,1)(1,6)\}$, $\{(0,2)(1,5)\}$, $\{(0,4)(7,1)\}$, $\{(0,5)(7,0)\}$\\*
$\{(1,2)(2,7)\}$, $\{(1,3)(2,6)\}$, $\{(2,3)(3,0)\}$, $\{(2,4)(3,7)\}$\\*
$\{(3,4)(4,1)\}$, $\{(3,5)(4,0)\}$, $\{(4,5)(5,2)\}$, $\{(4,6)(5,1)\}$\\*
$\{(5,7)(6,2)\}$, $\{(5,6)(6,3)\}$, $\{(6,7)(7,4)\}$, $\{(6,0)(7,3)\}$\\*
\end{center}
The Latin Square which removes the above singular fade state is shown in figure.\ref{8L4}. Latin Squares which removes the singular fade states with $\gamma=\dfrac{\sin 3\pi/8}{\sin \pi/8}$ are obtained by taking transposes of the Latin Squares which removes the singular fade states with $\gamma=\dfrac{\sin \pi/8}{\sin 3\pi/8}$. For example, consider $\gamma=\dfrac{\sin 3\pi/8}{\sin \pi/8}, \theta=-\pi/4$. The singularity constraints are
\begin{center}
$\{(1,0)(6,1)\}$, $\{(2,0)(5,1)\}$, $\{(4,0)(1,7)\}$, $\{(5,0)(0,7)\}$\\*
$\{(2,1)(7,2)\}$, $\{(3,1)(6,2)\}$, $\{(3,2)(0,3)\}$, $\{(4,2)(7,3)\}$\\*
$\{(4,3)(1,4)\}$, $\{(5,3)(0,4)\}$, $\{(5,4)(2,5)\}$, $\{(6,4)(1,5)\}$\\*
$\{(7,5)(2,6)\}$, $\{(6,5)(3,6)\}$, $\{(7,6)(4,7)\}$, $\{(0,6)(3,7)\}$\\*
\end{center}
The Latin Square to remove above singular fading state (figure.\ref{8L5}) is obtained by taking transpose of the Latin Square shown in figure.\ref{8L4}. Then by column shifting all the singular fading states with $\gamma=\dfrac{\sin 3\pi/8}{\sin \pi/8}$ is removed.
Next Consider $\gamma=\sin \pi/4, \theta=0$. The singularity constraints are
\begin{center}
$\{(0,3)(2,7)\}$, $\{(0,5)(6,1)\}$, $\{(1,4)(3,0)\}$, $\{(1,6)(7,2)\}$\\*
$\{(2,5)(4,1)\}$, $\{(3,6)(5,2)\}$, $\{(4,7)(6,3)\}$, $\{(5,0)(7,4)\}$\\*
\end{center}
The Latin Square shown in figure.\ref{8L6} removes this singular fade state. Then by shifting columns all the singular fade states with $\gamma=\sin \pi/4$ is removed. By taking transposes all the singular fade states with $\gamma=\dfrac{1}{\sin \pi/4}$ is removed.
For example, the Latin Square for singular fade state $\gamma=\sin \pi/4, \theta=\pi/2$ is obtained by two column shifting from Latin Square for $\gamma=\sin \pi/4, \theta=0$. The singularity constraints for $\gamma=\sin \pi/4, \theta=\pi/2$ are,
\begin{center}
$\{(0,1)(2,5)\}$, $\{(0,3)(6,7)\}$, $\{(1,2)(3,6)\}$, $\{(1,4)(7,0)\}$\\*
$\{(2,3)(4,7)\}$, $\{(3,4)(5,0)\}$, $\{(4,5)(6,1)\}$, $\{(5,6)(7,2)\}$\\*
\end{center}
The Latin Square to remove singular fade state $\gamma=\sin \pi/4, \theta=\pi/2$ is shown in Table.\ref{8L7}.
Now consider the singular fade state $\gamma=\dfrac{1}{\sin \pi/4}, \theta=-\pi/2$. The singularity constraints for this singular fade state are
\begin{center}
$\{(1,0)(5,2)\}$, $\{(3,0)(7,6)\}$, $\{(2,1)(6,3)\}$, $\{(4,1)(0,7)\}$\\*
$\{(3,2)(7,4)\}$, $\{(4,3)(0,5)\}$, $\{(5,4)(1,6)\}$, $\{(6,5)(2,7)\}$\\*
\end{center}
From the Latin Square to remove singular fade state $\gamma=\sin \pi/4, \theta=\pi/2$ the Latin Square to remove singular fade state $\gamma=\dfrac{1}{\sin \pi/4}, \theta=-\pi/2$ (figure.\ref{8L8}) is obtained by taking transpose.
Till now we have removed 40 singular fade states lying on the angles $\theta=2m\pi/8$, where $m=0,1,2\cdots7$. Now $\theta=(2m+1)\pi/8$, where $m=0,1,2\cdots7$. is considered.
The singularity constraints for the singular fade state $\gamma=\dfrac{\sin\pi/8}{\sin4\pi/8}$ and $\theta=\pi/8$ are
\begin{center}
$\{(0,2)(1,6)\}, \{(0,5)(7,1)\}, \{(1,3)(2,7)\}, \{(2,4)(3,0)\}$\\*
$\{(3,5)(4,1)\}, \{(4,6)(5,2)\}, \{(5,7)(6,3)\}, \{(6,0)(7,4)\}$\\*
\end{center}
The Latin Square to remove this singular fade state is given in figure.\ref{8L9}. By shifting the columns of this Latin Square all other Latin Squares to remove singular fade states with $\gamma=\dfrac{\sin\pi/8}{\sin4\pi/8}$ is obtained. The transposes of these Latin Squares remove singular fade states with $\gamma=\dfrac{\sin4\pi/8}{\sin\pi/8}$.
The singularity constraints for the singular fade state $\gamma=\dfrac{\sin\pi/8}{\sin2\pi/8}$ and $\theta=\pi/8$ are
\begin{center}
$\{(0,1)(1,7)\}, \{(0,3)(1,5)\}, \{(0,4)(7,2)\}, \{(0,6)(7,0)\}$\\*
$\{(1,2)(2,0)\}, \{(1,4)(2,6)\}, \{(2,3)(3,1)\}, \{(2,5)(3,7)\}$\\*
$\{(3,4)(4,2)\}, \{(3,6)(4,0)\}, \{(4,5)(5,3)\}, \{(4,7)(5,1)\}$\\*
$\{(5,0)(6,2)\}, \{(5,6)(6,4)\}, \{(6,1)(7,3)\}, \{(6,7)(7,5)\}$\\*
\end{center}
The Latin Square to remove this singular fade state is given in figure.\ref{8L10}. By shifting the columns of this Latin Square all other Latin Squares to remove singular fade states with $\gamma=\dfrac{\sin\pi/8}{\sin2\pi/8}$ is obtained. The transposes of these Latin Squares remove singular fade states with $\gamma=\dfrac{\sin2\pi/8}{\sin\pi/8}$.
The singularity constraints for the singular fade state $\gamma=\dfrac{\sin2\pi/8}{\sin3\pi/8}$ and $\theta=\pi/8$ are
\begin{center}
$\{(0,2)(2,7)\}, \{(0,3)(2,6)\}, \{(0,4)(6,1)\}, \{(0,5)(6,0)\}$\\*
$\{(1,3)(3,0)\}, \{(1,4)(3,7)\}, \{(1,5)(7,2)\}, \{(1,6)(7,1)\}$\\*
$\{(2,4)(4,1)\}, \{(2,5)(4,0)\}, \{(3,5)(5,2)\}, \{(3,6)(5,1)\}$\\*
$\{(4,6)(6,3)\}, \{(4,7)(6,2)\}, \{(5,0)(7,3)\}, \{(5,7)(7,4)\}$\\*
\end{center}
The Latin Square to remove this singular fade state is given in figure.\ref{8L11}. By shifting the columns of this Latin Square all other Latin Squares to remove singular fade states with $\gamma=\dfrac{\sin2\pi/8}{\sin3\pi/8}$ is obtained. The transposes of these Latin Squares remove singular fade states with $\gamma=\dfrac{\sin3\pi/8}{\sin2\pi/8}$.
The singularity constraints for the singular fade state $\gamma=\dfrac{\sin3\pi/8}{\sin4\pi/8}$ and $\theta=\pi/8$ are
\begin{center}
$\{(0,3)(3,7)\}, \{(0,4)(5,0)\}, \{(1,4)(4,0)\}, \{(1,5)(6,1)\}$\\*
$\{(2,5)(5,1)\}, \{(2,6)(7,2)\}, \{(3,6)(6,2)\}, \{(4,7)(7,3)\}$\\*
\end{center}
The Latin Square to remove this singular fade state is given in figure.\ref{8L12}. By shifting the columns of this Latin Square all other Latin Squares to remove singular fade states with $\gamma=\dfrac{\sin3\pi/8}{\sin4\pi/8}$ is obtained. The transposes of these Latin Squares remove singular fade states with $\gamma=\dfrac{\sin4\pi/8}{\sin3\pi/8}$.
From above discussions it is clear that by obtaining one Latin Square for each value of $\gamma \leq 1$ the other Latin Squares can be obtained by column permutation or by taking transpose. The clustering for the seven $\gamma$ values when end nodes use 8-PSK is explicitly given in \ref{table2}. All the clusterings are having 8 clusters only. Thus all the singular fade states are removed with 8 symbols only. That is, relay can use 8-PSK constellation.
\section{Singularity Removal Constraints FOR $M$-PSK SIGNAL SET}
It was shown in Section III A that the singularity removal constraints can be equivalently viewed as a Constrained Partial Latin Square (CPLS). In the following subsection, the structure of the CPLS for $M$-PSK signal set is obtained.
\subsection{The Structure of Constrained Partial Latin Square for $M$-PSK Signal Set}
In this subsection, the structure of the CPLS is shown to be the one referred to as $t$-plex which is defined as follows:
\begin{definition}
A $n \times n$ Partially Filled Latin Square (PFLS) is said to be a $t$-plex of order $n$ with $q$ symbols, if exactly $t$ cells are filled in each row and in each column, with each one of the $q$ symbols filling exactly $nt/q$ cells in total. A $n \times n$ PFLS which is a $t$-plex with $n$ symbols is simply referred to as a $t$-plex of order $n$.
\end{definition}
\begin{figure}
\centering
\includegraphics[totalheight=1in,width=1in]{k_plex_ex2}
\caption{Example showing a 2-plex of order 4 with 8 symbols}
\label{fig:k_plex_ex2}
\end{figure}
\begin{figure}
\centering
\includegraphics[totalheight=1in,width=1in]{kplex_ex1}
\caption{Example showing a 2-plex of order 4}
\label{fig:k_plex_ex1}
\end{figure}
For example, Fig. \ref{fig:k_plex_ex2} shows a 2-plex of order 4 with 8 symbols and Fig. \ref{fig:k_plex_ex1} shows a 2-plex of order 4. Note that a $t$-plex of order $n$ need not be always completable to a filled Latin Square with $n$ symbols. For example, the 2-plex of order 4 shown in Fig. \ref{fig:k_plex_ex1} cannot be completed using 4 symbols, since the cell corresponding to the zeroth row and third column cannot be filled with any of the four symbols $0,1,2,3$.
The CPLS for the singular fade states which lie on the unit circle and the corresponding Latin Squares which remove them, were obtained in Section III A. The following lemma gives the structure of the CPLS for those singular fade states which lie on circles $\frac{\sin(\frac{\pi k}{M})}{\sin(\frac{\pi l}{M})}, k \neq l$, for the case when both $k$ and $l$ are odd or both $k$ and $l$ are even.
\begin{lemma}
\label{lemma_sing_const1}
For a singular fade state $\frac{\sin\left(\frac{\pi k}{M}\right)}{\sin\left(\frac{\pi l}{M}\right)}e^{j 2\pi m/M}$, where both $k$ and $l$ are odd or both are even, the singularity removal constraints are given by,
{\footnotesize
\begin{align}
\nonumber
&\left\lbrace\left(i,i-m-\frac{M}{2}-\frac{\left(k-l\right)}{2}\right),\left(i-k,i-m+\frac{M}{2}-\frac{\left(k+l\right)}{2}\right)\right\rbrace,\\
\nonumber
&\left\lbrace\left(i,i-m-\frac{\left(k+l\right)}{2}\right),\left(i-k,i-m-\frac{\left(k-l\right)}{2}\right)\right\rbrace,
\end{align}
}where $0 \leq i \leq M-1$.
\begin{proof}
For $\lbrace (k_1,l_1), (k_2,l_2)\rbrace$ to be a singularity constraint corresponding to the singular fade state $\frac{\sin\left(\frac{\pi k}{M}\right)}{\sin\left(\frac{\pi l}{M}\right)}$,
{\footnotesize
\begin{align}
\nonumber
-\frac{e^{ \frac{jk_1\pi}{M}}-e^{\frac{jk_2\pi}{M}}}{e^{\frac{jl_1\pi}{M}}-e^{\frac{jl_2\pi}{M}}}=\frac{\sin\left(\frac{\pi k}{M}\right)}{\sin\left(\frac{\pi l}{M}\right)}e^{j 2\pi m/M},\textrm{ i.e.,}
\end{align}
}
{\footnotesize
\begin{align}
\label{eqn_equate}
\frac{\sin(\dfrac{\pi(k_1-k_2)}{M})}{\sin(\dfrac{\pi(l_2-l_1)}{M})}e^{\frac{j(k_1+k_2-l_1-l_2)\pi}{M}}=\frac{\sin\left(\frac{\pi k}{M}\right)}{\sin\left(\frac{\pi l}{M}\right)}e^{j 2\pi m/M}.
\end{align}
}
Equating the amplitude and phase terms in \eqref{eqn_equate}, we get, $$k_1+k_2=l_1+l_2,k_1-k_2=k,l_2-l_1=l$$ or, $$k_1+k_2=l_1+l_2+2m,k_1-k_2=k,l_2-l_1=M-l.$$
Note that by Lemma \ref{lemma_trig}, separately equating the numerator and denominator of the amplitude on both the sides of \eqref{eqn_equate} is allowed.
Assuming $k_1=i$, the two solutions given in the statement of the lemma are obtained.
\end{proof}
\end{lemma}
\begin{figure}
\centering
\includegraphics[totalheight=2in,width=2in]{PFLS_3_1}
\caption{CPLS corresponding to the singular fade state $\frac{\sin(\frac{3 \pi}{8})}{\sin(\frac{\pi}{8})}$}
\label{fig:PFLS_3_1}
\vspace{-.9 cm}
\end{figure}
For example, the CPLS corresponding to the singular fade state $\frac{\sin(\frac{3 \pi}{8})}{\sin(\frac{\pi}{8})}$, is shown in Fig. \ref{fig:PFLS_3_1}.
The following lemma gives the structure of the CPLS for those singular fade states which lie on circles $\frac{\sin(\frac{\pi k}{M})}{\sin(\frac{\pi k}{M})}, k \neq l$, for the case when only one among $k$ and $l$ is odd.
\begin{lemma}
\label{lemma_sing_const2}
For a singular fade state $\frac{\sin\left(\frac{\pi k}{M}\right)}{\sin\left(\frac{\pi l}{M}\right)}e^{j \frac{(2m+1)\pi}{M}}$, where only one among $k$ and $l$ is odd, the singularity removal constraints are given by,
{\scriptsize
\begin{align*}
&\left\lbrace\left(i,i-m-\frac{M}{2}-\frac{\left(k+1-l\right)}{2}\right),\left(i-k,i-m+\frac{M}{2}-\frac{\left(k+1+l\right)}{2}\right)\right\rbrace,\\
&\left\lbrace\left(i,i-m-\frac{\left(k+1+l\right)}{2}\right),\left(i-k,i-m-\frac{\left(k+1-l\right)}{2}\right)\right\rbrace,
\end{align*}
}where $0 \leq i \leq M-1$.
\begin{proof}
The proof is similar to the proof of Lemma \ref{lemma_sing_const1} and is omitted.
\end{proof}
\end{lemma}
The structure of the CPLS for $M$-PSK signal set, is shown to be a 4-plex of order $M$ with $2M$ symbols or a 2-plex of order $M$, as stated in the following lemma.
\begin{lemma}
The CPLS obtained after filling in the singularity removal constraints corresponding to the singular fade state which lies on the circle with radius $\frac{\sin(\frac{\pi k}{M})}{\sin(\frac{\pi l}{M})}$, $k \neq l$, forms a 4-plex of order $M$ with $2M$ symbols if $k \neq M/2,l \neq M/2$ and forms a 2-plex of order $M$ otherwise.
\begin{proof}
\noindent
{\it Case (i) $k \neq M/2$, $l \neq M/2$:}
From Lemma \ref{lemma_sing_const1} and Lemma \ref{lemma_sing_const2}, each value of $i$, $0 \leq i \leq M-1$, gives rise to two distinct constraints, which fills two cells in the $i$-th row. Similarly, the two constraints corresponding to $i'=i+k$ also fills two cells in the $i$-th row. Hence, in each row, 4 cells are filled. By a similar argument, 4 cells are filled in each column. Since the same symbol fills in two of the cells, a total of $4M/2=2M$ symbols are used in the partial filling of Latin Square. As a result, a $4$-plex with $2M$ symbols is obtained.
\noindent
{\it Case (ii) $k = M/2$ or $l = M/2$:}
When $k = M/2$ or $l = M/2$, from Lemma \ref{lemma_sing_const1} and Lemma \ref{lemma_sing_const2}, for each value of $i$, $0 \leq i \leq M-1$, it can be verified that the two singularity constraints are not distinct. Only 2 cells are filled in each row and each column. As a result, a $2$-plex is obtained.
\end{proof}
\end{lemma}
For example, the PFLS corresponding to the singular fade state $\frac{\sin(\frac{3 \pi}{8})}{\sin(\frac{\pi}{8})}$, shown in Fig. \ref{fig:PFLS_3_1}, is a 4-plex of order 8 with 16 symbols.
If the same symbol is filled in those cells which belong to two different singularity constraints, the singularity removal constraints are said to be combined. It is shown in the following lemma that by combining the singularity constraints, the CPLS which is a $4$-plex of order $M$ with $2M$ symbols can be transformed into a $4$-plex of order $M$.
\begin{lemma}
\label{lemma_PFLS}
For $k\neq M/2,l \neq M/2$, the CPLS corresponding to the singular fade state which lies on the circle $\frac{\sin(\frac{\pi k}{M})}{\sin(\frac{\pi k}{M})}, k \neq l$, obtained after filling in the singularity constraints, can be transformed in to a $4$-plex by appropriately combining the singularity constraints.
\begin{proof}
Since the elements of both the sets $\lbrace i,i-k,i+M/2,i-k+M/2 \rbrace$ and $\lbrace i-M/2-(k-l)/2,i+M/2-(k+l)/2,i-(k-l)/2,i-(k+l)/2 \rbrace$ are distinct, the following constraints in Lemma \ref{lemma_sing_const1}, can be combined:
{\footnotesize
\begin{align*}
&\left\lbrace\left(i,i-\frac{M}{2}-\frac{\left(k-l\right)}{2}\right),\left(i-k,i+\frac{M}{2}-\frac{\left(k+l\right)}{2}\right)\right\rbrace,\\
&\left\lbrace\left(i+\frac{M}{2},i-\frac{\left(k-l\right)}{2}\right),\left(i-k+\frac{M}{2},i-\frac{\left(k+l\right)}{2}\right)\right\rbrace.
\end{align*}
}
Similarly, the following constraints can also be combined:
{\footnotesize
\begin{align*}
&\left\lbrace\left(i,i-\frac{\left(k+l\right)}{2}\right),\left(i-k,i-\frac{\left(k-l\right)}{2}\right)\right\rbrace,\\
&\left\lbrace\left(i+\frac{M}{2},i+\frac{M}{2}-\frac{\left(k+l\right)}{2}\right),\left(i-k+\frac{M}{2},i+\frac{M}{2}-\frac{\left(k-l\right)}{2}\right)\right\rbrace.
\end{align*}
}
In this way, a 4-plex is obtained from the PFLS for the case when $k,l \neq \frac{M}{2}$ and both $k$ and $l$ are even or both are odd. Following, exactly the same procedure, the lemma can be proved for the case when one among $k$ and $l$ is odd and the other is even.
\end{proof}
\end{lemma}
\begin{figure}
\centering
\includegraphics[totalheight=2in,width=2in]{PFLS_3_1_combined}
\caption{4-plex of order 8 corresponding to the singular fade state $\frac{\sin(\frac{3 \pi}{8})}{\sin(\frac{\pi}{8})}$}
\label{fig:PFLS_3_1_combined}
\vspace{-.8 cm}
\end{figure}
For example, the CPLS corresponding to the singular fade state $\frac{\sin(\frac{3 \pi}{8})}{\sin(\frac{\pi}{8})}$, shown in Fig. \ref{fig:PFLS_3_1}, can be transformed into a 4-plex of order 8, as shown in Fig. \ref{fig:PFLS_3_1_combined}.
The completability of a $t$-plex of order $M$ with $M$ symbols is discussed in the following subsection.
\subsection{Completability of a $t$-plex of order $M$ with $M$ symbols}
The completion of the 2-plexes and the 4-plexes of order $M$, obtained from the singularity removal constraints, using exactly $M$ symbols is advantageous for the reason outlined as follows: It was shown in \cite{APT1} that for the case when QPSK signal set is used during the MA phase, there exists certain values of fade state for which a clustering with cardinality 5 needs to be used to maximize the minimum cluster distance. While the use of clusterings with cardinality 5, reduces the impact of multiple access interference, it adversely impacts the performance during the BC phase. The 2-plexes and 4-plexes can be completed using $M$ symbols, means that the optimizing the performance during the MA phase does not come at the cost of a degraded performance during the BC phase.
Whether a partially filled $M \times M$ Latin Square is completable using $M$ symbols or not is an open problem. Burton \cite{Bu} conjectured the following regarding the completability of the partially filled Latin Square which forms a $t$-plex of order $M$.
\begin{conjecture}
\label{Thm_comp}
A $t$-plex of a $M \times M$ Latin Square is completable using $M$ symbols, for $t \leq M/4$.
\end{conjecture}
By explicitly providing all the Latin Squares, for 8-PSK signal set, it was shown in the previous section that the CPLS corresponding to all the singular fade states are completable using 8 symbols. If Conjecture \ref{Thm_comp} were indeed true, the 4-plexes and the 2-plexes obtained from the CPLS corresponding to the different singular fade states, can be completed using $M$ symbols, for $M$ any power of 2 greater than or equal to 16. Even though, the problem of completability of a $t$-plex in general remains unsolved, in the next section, it is shown that the structure of the singularity constraints allows the construction of explicit Latin Squares, for some singular fade states. In other words, by providing some explicit constructions, it is shown that the $4$-plexes of order $M$ corresponding to some of the singular fade states are always completable using $M$ symbols.
\section{SOME SPECIAL CONSTRUCTIONS OF LATIN SQUARES}
As discussed in section II, the singular fade states lie on circles with radii $\frac{\sin\left(\frac{\pi k}{M}\right)} {\sin\left(\frac{\pi l}{M}\right)}, 1 \leq k,l \leq M/2$. It was shown in Section II that all the Latin Squares which remove the singular fade states on the unit circle (i.e., the case when $k=l$) can be obtained from the bit-wise XOR map, by appropriate column permutation. In Subsection A of this section, an explicit construction algorithm is provided to obtain Latin Squares for the case when both $k$ and $l$ are odd. In Subsection B of this section, a construction procedure to obtain Latin Squares for the case when both $k$ and $l$ are even is provided. Note that the cells in the Latin Squares obtained are filled using exactly $M$ symbols.
It was observed in \cite{APT1} that for 4 PSK signal set there exists clusterings which remove more than one singular fade state. The Latin Squares obtained from the construction algorithm given in Subsection A remove multiple singular fade states.
The construction procedure given in Subsection B involves obtaining Latin squares for $M$-PSK signal set from the ones already obtained for $M/2$-PSK signal set.
\subsection{Construction of Latin Squares which remove Multiple Singular Fade States}
Let $\mathcal{L}_{k,l}^e$ denote the Latin Square which removes multiple singular fade states on the circle $\frac{\sin\left(\frac{\pi k}{M}\right)} {\sin\left(\frac{\pi l}{M}\right)}, 1 \leq k,l \leq M/2$, both $k$ and $l$ are odd, which is obtained as follows:
\begin{algorithmic}[1]
\STATE Start with an empty Latin Square.
\STATE The first row of the $M \times M$ Latin Square is filled with numbers $0$ to $M-1$ in the same order.
\FOR{$0 \leq s \leq M-1$}
\FOR{$1 \leq t \leq M$}
\IF{$s$ is even}
\STATE The entry $(kt,s+lt)$ is filled in the Latin Square is filled to be $s$.
\ELSE
\STATE The entry $(kt,s-lt)$ is filled in the Latin Square is filled to be $s$ (Note that all the arithmetic operations are modulo $M$.)
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic}
Since both $k$ and $l$ are odd, the algorithm given above is always guaranteed to result in a completely filled Latin Square.
Let $\mathcal{L}_{k,l}^o$ denote the Latin Square obtained by replacing the word \textit{even} by the word \textit{odd} in Step 5 of the algorithm.
\begin{figure}
\centering
\subfigure[]{
\includegraphics[totalheight=2in,width=2in]{latin_8psk_1}
\label{fig:latin_8psk_1}
}
\subfigure[]{
\includegraphics[totalheight=2in,width=2in]{latin_8psk_2}
\label{fig:latin_8psk_2}
}
\subfigure[]{
\includegraphics[totalheight=2in,width=2in]{latin_8psk_3}
\label{fig:latin_8psk_3}
}
\subfigure[]{
\includegraphics[totalheight=2in,width=2in]{latin_8psk_4}
\label{fig:latin_8psk_4}
}
\caption{Construction of the Latin Square $\mathcal{L}_{3,1}^e$}
\label{fig:latin_8psk_even}
\end{figure}
\begin{figure}
\centering
\includegraphics[totalheight=2in,width=2in]{latin_8psk_odd}
\caption{The Latin Square $\mathcal{L}_{3,1}^o$}
\label{fig:latin_8psk_odd}
\end{figure}
For example consider the case when $k=3,l=1$. We start with only the first row filled as shown in Fig. \ref{fig:latin_8psk_1}. The cells in which $0$'s appear are filled next, by traversing the Latin Square, $k=3$ steps forward and $l=1$ step to the right, as depicted in Fig. \ref{fig:latin_8psk_2}. The cells in which $1$'s appear are filled next, by traversing the Latin Square, $k=3$ steps forward and $l=1$ step to the left, as depicted in Fig. \ref{fig:latin_8psk_3}. In a similar way, the remaining cells in the Latin Square can also be filled to obtain the Latin Square $\mathcal{L}_{3,1}^e$ shown in Fig. \ref{fig:latin_8psk_4}. Similarly, the Latin Square $\mathcal{L}_{3,1}^o$ can be obtained as shown in Fig. \ref{fig:latin_8psk_odd}.
The following Theorem states that multiple singular fade states on multiple circles are removed by the Latin Squares $\mathcal{L}_{k,l}^e$ and $\mathcal{L}_{k,l}^o$.
\begin{theorem}
\label{thm_can_LS}
One of two the Latin Squares $\mathcal{L}_{k,l}^e$ and $\mathcal{L}_{k,l}^o$ (both $k$ and $l$ odd), removes any singular fade state of the form $\frac{\sin{\frac{k_1 \pi}{M}}} {\sin{\frac{l_1 \pi}{M}}} e^{\frac{\j m 2 \pi}{M}}$, where $0 \leq m \leq M-1$ and $(k_1,l_1)$ belongs to the set, $\left\lbrace (nk,nl) \vert 1 \leq n \leq \frac{M}{2}-1, n \textrm{ odd}\right\rbrace$.
\begin{proof}
The singularity constraints for the singular fade state $\frac{\sin{\frac{k \pi}{M}}} {\sin{\frac{l \pi}{M}}} e^{\frac{\j m 2 \pi}{M}}$ are given by Lemma \ref{lemma_sing_const1}.
Assuming $i'=i-k$, The singularity constraints given in Lemma \ref{lemma_sing_const1} can be rewritten as,
{\footnotesize
\begin{align}
\label{eqn_sing_const3}
\left\lbrace\left(i',i'+m+\frac{M}{2}+\frac{\left(k-l\right)}{2}\right),\left(i'+k,i'+m-\frac{M}{2}+\frac{\left(k+l\right)}{2}\right)\right\rbrace\\
\label{eqn_sing_const4}
\left\lbrace\left(i',i'+m+\frac{\left(k+l\right)}{2}\right),\left(i'+k,i'+m+\frac{\left(k-l\right)}{2}\right)\right\rbrace,
\end{align}
}where $0 \leq i' \leq M-1$.
From \eqref{eqn_sing_const3}, the second entry in the the singularity constraint can be obtained from the first entry by traversing $k$ rows downwards and $l$ columns to the left. Similarly, from \eqref{eqn_sing_const4}, the second entry in the the singularity constraint can be obtained from the first entry by traversing $k$ rows downwards and $l$ columns to the right. Furthermore, for a given $i'$, the column indices of the first cells given in the singularity constraint given in \eqref{eqn_sing_const3} and \eqref{eqn_sing_const4} differ by an odd number. Hence, one of the two Latin squares $\mathcal{L}_{k,l}^e$ and $\mathcal{L}_{k,l}^o$ removes the singular fade state $\frac{\sin{\frac{k \pi}{M}}} {\sin{\frac{l \pi}{M}}} e^{\frac{\j m 2 \pi}{M}}$, depending on whether $m+\frac{\left(k+l\right)}{2}$ is odd or even. From, the construction procedure described in the beginning of the sub-section, it can be verified easily that $\mathcal{L}_{k,l}^e$ is the same as $\mathcal{L}_{nk,nl}^e$ and $\mathcal{L}_{k,l}^o$ is the same as $\mathcal{L}_{nk,nl}^o$, for all odd $n$. This completes the proof.
\end{proof}
\end{theorem}
It follows from Theorem \ref{thm_can_LS} that a single Latin Square $\mathcal{L}_{k,l}^e$ removes half of the singular fade states which lie on $M/4$ circles, i.e., a total of $\frac{M^2}{8}$ singular fade states. Two Latin Squares $\mathcal{L}_{k,l}^e$ and $\mathcal{L}_{k,l}^o$ remove all the fade states which lie on $M/4$ circles. Also, the Latin square $\mathcal{L}_{k,l}^e$ ($\mathcal{L}_{k,l}^o$) is the same as the Latin Squares $\mathcal{L}_{nk,nl}^e$ ($\mathcal{L}_{nk,nl}^o$), where $3\leq n \leq M/2-1$, $n$ odd.
For example, the Latin Squares $\mathcal{L}_{3,1}^e$ and $\mathcal{L}_{3,1}^o$ , shown respectively in Fig. \ref{fig:latin_8psk_4} and Fig. \ref{fig:latin_8psk_odd}, remove all the 16 singular fade states which lie on the 2 circles with radii $\frac{\sin(\frac{k_1 \pi}{M})}{{\sin(\frac{l_1 \pi}{M})}}$, where $(k_1,l_1) \in \lbrace(1,3),(3,1)\rbrace$.
\begin{note}
The Latin Square $\mathcal{L}_{k,l}^o$ is the same as the one obtained by a single column permutation of $\mathcal{L}_{k,l}^e$, with the symbols alone permuted. Since the clustering is unchanged by symbol permutation, it is enough if we specify either $\mathcal{L}_{k,l}^e$ or $\mathcal{L}_{k,l}^o$.
\end{note}
Out of the $\frac{M^2}{4}-\frac{M}{2}+1$ circles on which the singular fade states lie, $\frac{M^2}{16}-\frac{M}{4}$ circles have radii of the form $\frac{\sin(\frac{k \pi}{M})}{{\sin(\frac{l \pi}{M})}}$, where both $k$ and $l$ are odd. From Theorem 1, the Latin Squares which remove all the singular fade states which lie on such circles can be obtained from $\frac{\frac{M^2}{16}-\frac{M}{4}}{\frac{M}{4}}=\frac{M}{4}-1$ Latin Squares, which are constructed using the algorithm given in the beginning of this subsection.
\subsection{Construction of Latin Squares for $M$ PSK signal set from that of $M/2$ PSK signal set}
In the previous subsection, an explicit construction of Latin Squares which remove the singular fade states on the circle $\frac{\sin{\frac{\pi k }{M}}}{\sin{\frac{\pi l }{M}}}$, both $k$ and $l$ odd, was provided. In this subsection, it is shown that for $M$ PSK signal set, Latin Squares which remove the singular fade states on the circle $\frac{\sin{\frac{\pi k }{M}}}{\sin{\frac{\pi l }{M}}}$, both $k$ and $l$ even, $k \neq l, k \neq M/2$ and $l \neq M/2$, can be constructed from the Latin Squares obtained for $M/2$ PSK signal set.
Let $\mathcal{L}(M,k,l,\phi)$ denote an $M \times M$ Latin Square which removes the singular fade state $\frac{\sin(\frac{k\pi}{M})} {\sin(\frac{l\pi}{M})} e^{j \phi}$.
For the $M \times M$ Latin Square $\mathcal{L}(M,k,l,\phi)$, let $\mathcal{L}^{oe}(M,k,l,\phi)$ denote the $M/2 \times M/2$ Latin Square obtained by taking only the odd numbered rows and even numbered columns of $\mathcal{L}(M,k,l,\phi)$. In a similar way, the the $M/2 \times M/2$ Latin Squares $\mathcal{L}^{ee}(M,k,l,\phi),\mathcal{L}^{oo}(M,k,l,\phi),\mathcal{L}^{eo}(M,k,l,\phi)$ can be defined.
\begin{definition}
The \textit{quadruplicate} of an $M \times M$ Latin Square $\mathcal{L}(M,k,l,\phi)$ is the ordered set consisting of the four $M/2 \times M/2$ Latin Squares
{\small
$\lbrace \mathcal{L}^{ee}(M,k,l,\phi),\mathcal{L}^{oo}(M,k,l,\phi),\mathcal{L}^{eo}(M,k,l,\phi),\mathcal{L}^{oe}(M,k,l,\phi) \rbrace.$
}
\end{definition}
The Latin Square is $\mathcal{L}(M,k,l,\phi)$ uniquely determined by its quadruplicate.
For the Latin Square $\mathcal{L}(M,k,l,\phi)$, let $\mathcal{L}_{c}(M,k,l,\phi)$ denote the Latin Square obtained by adding integer $c$ to all the cells of the Latin Square.
The following theorem gives the construction of the Latin Square $\mathcal{L}(M,k,l,\phi)$, for the case when both $k$ and $l$ are even, $k \neq l, k \neq M/2$, $l \neq M/2$ and $\mod\left(\frac{k}{2}+\frac{l}{2},2\right)=0$.
\begin{theorem}
\label{theorem_even_odd1}
The elements of the quadruplicate of $\mathcal{L}(M,k,l,\phi)$, if $\mod\left(\frac{k}{2}+\frac{l}{2},2\right)=0$, are given by,
{\footnotesize
\begin{align*}
\mathcal{L}^{ee}(M,k,l,\phi)=\mathcal{L}^{oo}(M,k,l,\phi)&=\mathcal{L}(M/2,k/2,l/2,\phi),\\
\mathcal{L}^{eo}(M,k,l,\phi)=\mathcal{L}^{oe}(M,k,l,\phi)&=\mathcal{L}_{M/2}(M/2,k/2,l/2,\phi).
\end{align*} }
\begin{proof}
From Lemma \ref{lemma_sing_const1}, the singularity constraints are given by,
{\footnotesize
\begin{align}
\label{eqn_sing_const5}
\left\lbrace\left(i,i-\frac{M}{2}-\frac{\left(k-l\right)}{2}\right),\left(i-k,i+\frac{M}{2}-\frac{\left(k+l\right)}{2}\right)\right\rbrace\\
\label{eqn_sing_const6}
\left\lbrace\left(i,i-\frac{\left(k+l\right)}{2}\right),\left(i-k,i-\frac{\left(k-l\right)}{2}\right)\right\rbrace,
\end{align}
}where $0 \leq i \leq M-1$.
The singularity constraints given above can be further split into cases depending on whether $i$ is even or odd.
\noindent{Case 1: $i$ is even}
\noindent
Assume $i'=i/2$, the singularity constraints in \eqref{eqn_sing_const5} and \eqref{eqn_sing_const6} can be rewritten as,
{\footnotesize
\begin{align}
\nonumber
&\left\lbrace\left(2i',2i'-2\frac{M/2}{2}-\frac{2\left(k/2-l/2\right)}{2}\right)\right.,\\
\label{eqn_sing_const7}
&\hspace{2 cm}\left.\left(2i'-2\frac{k}{2},2i'+2\frac{M/2}{2}-2\frac{\left(k/2+l/2\right)}{2}\right)\right\rbrace\\
\nonumber
&\left\lbrace\left(2i',2i'-2\frac{\left(k/2+l/2\right)}{2}\right),\right.\\
\label{eqn_sing_const8}
&\hspace{3 cm}\left.\left(2i'-2\frac{k}{2},2i'-2\frac{\left(k/2-l/2\right)}{2}\right)\right\rbrace,
\end{align}
}where $0 \leq i' \leq M/2-1$.
The singularity constraints given in \eqref{eqn_sing_const7} and \eqref{eqn_sing_const8}, are the same as that of the singular fade state $\frac{\sin(\frac{\pi k/2}{M/2})}{\sin(\frac{\pi l/2}{M/2})}$ with the row and column indices doubled. Hence, $\mathcal{L}(M,2k,2l,\phi)^{ee}=\mathcal{L}(M/2,k,l,\phi)$.
\noindent{Case 2: $i$ is odd}
\noindent
Using a similar argument as that of case 1, it can be shown that the singularity constraints for this case are the same as that of the singular fade state $\frac{\sin(\frac{\pi k/2}{M/2})}{\sin(\frac{\pi l/2}{M/2})}$ with the row and column indices doubled and shifted by one place. Hence, $\mathcal{L}^{oo}(M,k,l,\phi)=\mathcal{L}(M/2,k/2,l/2,\phi)$. Note that by taking both $\mathcal{L}^{ee}(M,k,l,\phi)$ and $\mathcal{L}^{oo}(M,k,l,\phi)$ to be $\mathcal{L}(M/2,k/2,l/2,\phi)$, the conditions required for $\mathcal{L}(M,k,l,\phi)$ to be a Latin Square are not violated.
Since {\footnotesize $\mathcal{L}^{ee}(M,k,l,\phi)=\mathcal{L}^{oo}(M,k,l,\phi)=\mathcal{L}(M/2,k/2,l/2,\phi)$}, half of the cells in the Latin Square $\mathcal{L}(M,k,l,\phi)$ are filled with numbers from $0$ to $M/2-1$. Hence, $\mathcal{L}^{oe}(M,k,l,\phi)$ and $\mathcal{L}^{eo}(M,k,l,\phi)$ can be taken to be $\mathcal{L}_{M/2}(M/2,k/2,l/2,\phi)$.
\end{proof}
\end{theorem}
\begin{figure}
\centering
\subfigure[]{
\includegraphics[totalheight=2in,width=2in]{latin_16psk_2}
\label{fig:latin_16psk_2}
}
\subfigure[]{
\includegraphics[totalheight=2in,width=2in]{latin_16psk_3}
\label{fig:latin_16psk_3}
}
\caption{Construction of the Latin Square $\mathcal{L}(16,6,2,0)$}
\label{fig:latin_16psk_even}
\end{figure}
For example, the Latin Square $\mathcal{L}(16,6,2,0)$ can be obtained from $\mathcal{L}(8,3,1,0)$ (which is shown in Fig. \ref{fig:latin_8psk_odd}) as follows: $\mathcal{L}^{ee}(16,6,2,0)$ and $\mathcal{LS}^{oo}(16,6,2,0)$ are chosen to be $\mathcal{L}(8,3,1,0)$ as shown in Fig. \ref{fig:latin_16psk_2}. The Latin Squares $\mathcal{L}^{oe}(16,6,2,0)$ and $\mathcal{L}^{eo}(16,6,2,0)$ are chosen to be $\mathcal{L}_8(8,3,1,0)$ to obtain the Latin Square $\mathcal{L}(16,6,2,0)$ shown in Fig. \ref{fig:latin_16psk_3}.
The following theorem gives the construction of the Latin Square $\mathcal{L}(M,k,l,\phi)$, for the case when both $k$ and $l$ are even, $k \neq l, k \neq M/2$, $l \neq M/2$ and $\mod\left(\frac{k}{2}+\frac{l}{2},2\right)=1$.
\begin{theorem}
The elements of the quadruplicate of $\mathcal{L}(M,k,l,\phi)$, if $\mod\left(\frac{k}{2}+\frac{l}{2},2\right)=1$, are given by,
{\footnotesize
\begin{align*}
&\mathcal{L}^{oe}(M,k,l,\phi)=\mathcal{L}(M/2,k/2,l/2,\phi),\\
&\mathcal{L}^{eo}(M,k,l,\phi)=\mathcal{L}(M/2,k/2,l/2,\phi-2\pi/M),\\
&\mathcal{L}^{oo}(M,k,l,\phi)=\mathcal{L}^{ee}(M,k,l,\phi)=\mathcal{L}_{M/2}(M/2,k/2,l/2,\phi).
\end{align*} }
\begin{proof}
The proof is similar to Theorem \ref{theorem_even_odd1} and is omitted.
\end{proof}
\end{theorem}
In this subsection, construction procedure was provided to obtain the Latin Squares which remove the singular fade states which lie on circles with radii $\frac{\sin(\frac{k \pi}{M})}{\sin(\frac{l \pi}{M})}$, where both $k$ and $l$ are even, $k \neq l, k \neq M/2$ and $l \neq M/2$. The total number of such circles is $\left(\frac{M}{4}-1\right)^2-\left(\frac{M}{4}-1\right)=\frac{M^2}{16}-\frac{3M}{4}+2$. It is enough if we obtain one Latin Square per circle for those circles which lie inside the unit circle, since the Latin Square corresponding to all other singular fade states can be obtained by appropriately performing column permutation and taking transpose. Hence, to remove all the singular fade states on circles with radii $\frac{\sin(\frac{k \pi}{M})}{\sin(\frac{l \pi}{M})}$, where both $k$ and $l$ are even, $k \neq l, k \neq M/2$ and $l \neq M/2$, it is enough if we obtain $\frac{\left(\frac{M^2}{16}-\frac{3M}{4}+2\right)}{2}$ Latin Squares from the construction procedure described in this subsection.
The Latin Squares constructed so far (the bit-wise XOR map and the ones obtained in Subsection A and B of this section), which are
{\footnotesize
$$1+\left(\frac{M}{4}-1\right)+\frac{\left(\frac{M^2}{16}-\frac{3M}{4}+2\right)}{2}=\frac{M^2}{32}-\frac{M}{8}+1$$
}in number, are sufficient to obtain all the Latin Squares which remove the singular fade states which lie on
{\footnotesize
$$1+\left(\frac{M^2}{16}-\frac{M}{4}\right)+\left(\frac{M^2}{16}-\frac{3M}{4}+2\right)=\frac{M^2}{8}-M+3$$
}circles. The Latin Squares which remove the singular fade states which lie on the other circles inside the unit circle can be found using computer search. It can be verified that it is sufficient to obtain $\frac{3M^2}{32}+\frac{M}{8}$ Latin Squares in total, from which all the other Latin Squares can be obtained, for $M \geq 8$.
\section{Bidirectional Relaying With Unequal Transmission Rates and Latin Rectangles}
In this section we consider the scenario in which both the end nodes use PSK constellations of different sizes. This results in different transmission rates for both the users. Assume node A and node B use PSK constellations $\mathcal{S}_A$ and $\mathcal{S}_B$ of size $M (=2^\lambda)$ and $N (=2^\mu)$ respectively. The relay should use a constellation of size at least $\max(M,N)$. The exclusive law for this case is given by,
{\footnotesize
\begin{align}
\left.
\begin{array}{ll}
\nonumber
\mathcal{M}^{\gamma,\theta}(x_A,x_B) \neq \mathcal{M}^{\gamma,\theta}(x'_A,x_B), \; \mathrm{where} \;x_A \neq x'_A \; \mathrm{,} \;x_B \in \mathcal{S}_B,\\
\nonumber
\mathcal{M}^{\gamma,\theta}(x_A,x_B) \neq \mathcal{M}^{\gamma,\theta}(x_A,x'_B), \; \mathrm{where} \;x_B \neq x'_B \; \mathrm{,} \;x_A \in \mathcal{S}_A.
\end {array}
\right\} \\
\label{ex_law}
\end{align}
}
\begin{definition} A Latin rectangle L on the symbols from the set $\mathbb{Z}_t=\{0,1, \cdots ,t-1\}$ is an \textit{M} $\times$ \textit{N} array, in which each cell contains one symbol and each symbol occurs at most once in each row and column.
\end{definition}
The relay clusterings which satisfy the exclusive law form Latin rectangles with $M$ rows and $N$ columns. As before, the study of clusterings which satisfy the exclusive law can be equivalently carried out as the study of Latin rectangles with appropriate parameters. Without loss of generality, it is assumed that $M> N$. It is shown that the set of singular fade states for this case is a subset of the set of singular fade states for the case when both A and B use $M$-PSK signal set. Also, it is shown that the $M \times N$ Latin Rectangle for this case can be obtained from $M \times M$ Latin Square obtained for the case when both A and B use $M$-PSK signal set.
\begin{lemma}
The set of singular fade states for the case when $M$-PSK signal set is used at A and $N$-PSK signal set is used at B, both $M$ and $N$ powers of 2 and $M >N$, is a subset of the set of singular fade states for the case when both A and B use $M$-PSK signal set.
\end{lemma}
\begin{proof}
Let $\mathcal{S}_M$ and $\mathcal{S}_N$ denote the symmetric $M$-PSK and $N$-PSK signal sets used at A and B respectively.
Assume $M=2^{\lambda}$ and $N=2^{\mu}$, $\lambda > \mu$.
Note that the signal set $\mathcal{S}_N$ is a subset of the signal set $\mathcal{S}_M$.
The singular fade states for this case are of the form,
{\footnotesize
\begin{align}
\label{eqn_sing_rectan}
\gamma e^{j \theta}=-\frac{x(k)-x(k')}{y(l)-y(l')}, x(k)\neq x(k') \in \mathcal{S}_M, y(l)\neq y(l') \in \mathcal{S}_N
\end{align}
}
The singular fade states given by \eqref{eqn_sing_rectan}, are also singular fade states for the case when both A and B use $M$-PSK signal set, since $y(l)$ and $y(l')$ are points in the $M$-PSK signal set as well. This completes the proof.
\end{proof}
The $M \times N$ system can be interpreted as a $M \times M$ system with node B not transmitting some symbols. The corresponding Latin Rectangle is obtained from the Latin Square for $M \times M$ system by removing the columns corresponding to the unused symbols.
The following example illustrates this.
\begin{example}
Consider the case when 8-PSK signal set is used at A and 4-PSK signal set is used at B. For $\gamma=\sin(\pi/4)$ and $\theta=0$, by removing the columns 1, 3, 5 and 7 (equivalently viewed as B transmitting only the 4 symbols 0, 2, 4 and 6 in the 8-PSK signal set) from the Latin Square given in Fig. \ref{8L6}, the Latin Rectangle shown in Fig. \ref{latin_rectang} is obtained.
\end{example}
\begin{figure}[htbp]
\centering
{
\begin{tabular}{|c|c|c|c|}
\hline 4 & 6 & 1 & 7 \\
\hline 2 & 0 & 3 & 5 \\
\hline 0 & 7 & 5 & 6 \\
\hline 3 & 4 & 2 & 1 \\
\hline 5 & 2 & 0 & 3 \\
\hline 7 & 1 & 6 & 4 \\
\hline 1 & 3 & 4 & 2 \\
\hline 6 & 5 & 7 & 0 \\
\hline
\end{tabular}}
\caption{Latin Rectangle for $\gamma=\sin(\pi/4), \theta=0$}
\label{latin_rectang}
\end{figure}
\vspace{-0.1 cm}
\section{DISCUSSION}
In this paper, for the design of modulation schemes for the physical layer network-coded two way relaying scenario with the protocol which employs two phases: Multiple access (MA) Phase and Broadcast (BC) phase, we identified a relation between the required exclusive laws satisfying clusterings and Latin Squares. This relation is used to get all the maps to be used at the relay efficiently. Further we illustrated the results presented for the case, where both the end nodes use QPSK constellation as well as 8-PSK constellations. Here we concentrated only on singular fade states and the clusterings to remove that with only the minimum cluster distance under consideration. We are not considering the entire distance profile as done in \cite{APT1}. Our work eliminate the singular fade states effectively and these clusterings can be used in other regions in the complex plane of $(\gamma, \theta)$, as shown in \cite{VNR}.
\section*{Acknowledgement}
This work was supported partly by the DRDO-IISc program on Advanced Research in Mathematical Engineering through a research grant as well as the INAE Chair Professorship grant to B.~S.~Rajan.
|
1,314,259,996,757 | arxiv | \section{Introduction}
Unconditional security, or information theoretical security, is always the end goal of cryptography. It is a challenging task for classical cryptography, which is based on mathematical complexity. Luckily, quantum cryptography, based on the basic principle of quantum mechanics, provides a way to reach such terminal goal. Taking advantage of conventional techniques from optical communication and cutting-edge technologies from quantum optics, some quantum cryptography primitives, such as quantum key distribution (QKD)~\cite{bennett1984}, quantum coin tossing (QCT)~\cite{bennett1984}, quantum digital signature (QDS)~\cite{gottesman2001} and quantum secret sharing (QSS)~\cite{cleve1999}, have been implemented~\cite{schmitt-manderbach2007,rubenok2013,pappa2014,yin2017a,collins2016,bogdanski2008}. Of these, QKD implementations are relatively mature, have been demonstrated in a field environment~\cite{stucki2011, sasaki2011, tang2014a,yin2016} and commercialized~\cite{comqkdsystems}.
However, along with QKD maturation, deviation of the behaviour in implementation from the theoretical assumptions has been explored. Such deviation might open backdoors and be exploited by an eavesdropper (Eve). Once any security assumptions are broken by Eve, she could compromise theory-proved security and spy on the secret key in practice. The feasibility and capability of Eve's attacks have been shown in various cases~\cite{xu2010,lydersen2010a,gerhardt2011,sun2011,jain2011,tang2013,bugge2014,sajeed2015a,huang2016,makarov2016}. Similar vulnerability may occur in the other quantum cryptography implementations~\cite{yin2017a, collins2016, bogdanski2008}, since they employ similar optical components and devices.
In practical quantum cryptography systems, a weak coherent source (WCS) is widely used to replace the single photon source. An inherent imperfection in WCS is the emission of multiphoton pulses, which gives Eve more than one copy of Alice's quantum states. Then Eve could perform the photon-number-splitting (PNS) attack~\cite{huttner1995,brassard2000}, in which she blocks all single-photon pulses, and keeps one photon from the multiphoton state. Then she could get the entire final key after Alice and Bob announce their basis choices. Note that a modified PNS attack based on a beam splitter has been demonstrated~\cite{liu2011a}. Thus, the danger of PNS attack is not only theoretical, but also practical. Fortunately, decoy state protocols~\cite{hwang2003, wang2005a, lo2005} were proposed to beat such attack, and have been implemented in many QKD systems~\cite{schmitt-manderbach2007, rosenberg2007, liu2010, tang2014a, yin2016}. They have also been employed in other quantum cryptography systems~\cite{bogdanski2008, fu2015, yin2017a, yin2017} to guarantee their security.
Generally speaking, in the decoy state protocol, signal and decoy states only have different mean photon numbers. Decoy states are used to estimate the detection gain and error rate of single-photon pulses in signal states. If Eve could not distinguish the signal and decoy states, she would change the photon number in both signal and decoy pulses during the PNS attack~\cite{huttner1995,brassard2000}. Thus, she would disturb the yield and error rate of decoy states, which affects the estimation of single-photon detection gain and error rate in signal states. It results in a decrease of the secure key rate~\cite{lo2005}.
However, the essential assumption, indistinguishability of the signal and decoy states, may not be guaranteed in practice. In fact, Eve might exploit practical imperfections to find a side-channel which allows her to distinguish the signal and decoy states. Then she could perform different hacking strategies to keep the normal statistic distributions, meanwhile spy some secret information silently without being discovered. Several types of source imperfections and corresponding attacks have been shown in different QKD systems~\cite{wang2009, nauerth2009, jiang2012, tang2013, tamaki2016}. Importantly, the first quantum satellite also employs one of such imperfect sources~\cite{liao2017}. Recently, the security of decoy-state QKD with a leaky source was proved by K.~Tamaki {\it et al.}\ in Ref.~\onlinecite{tamaki2016}. According to their analysis, the trace distance can be used to characterize the leaked information due to the imperfection of source, which may come from Alice's imperfect state preparation or Eve's attack (such as the Trojan-horse attack~\cite{vakhitov2001,gisin2006,jain2014}). Then the key rate can be bounded with the trace distance, in which the yield and error rate of single-photon pulses are estimated by solving an optimization problem. In this Article, based on the work of Ref.~\onlinecite{tamaki2016}, we obtain two analytical formulas to estimate the yield and error rate of the single photon pulses, which produce the secure key rate given that the signal and decoy states are distinguishable. A further analysis shows that the key rate can be improved by calibrating the transmittance of Bob's optical devices (the calibrated method has also been used to protect a single photon detector of Bob from the blinding attack~\cite{maroy2010,maroy2017}).
In this Article, we first recap the basic theory of decoy state protocol in~\cref{basic}. In~\cref{measurement}, we test experimentally two intensity modulation methods: laser diode gain switching by modulating its pump current, and external intensity modulator connected after the laser diode. The test of pump-current modulation shows a side-channel in the time domain, which allows Eve to distinguish the signal state from the decoy state. We then model a PNS attack that bypasses the decoy state protocol in~\cref{attack}. By optimizing the attack's time windows, Eve should be able to eavesdrop the secret key successfully. On the other hand, from Alice and Bob's point of view, we give a method to analyze the security of the decoy state protocol with imperfect source in~\cref{key rate}. Here the imperfect source means that the signal state and decoy state are partially distinguishable in any degrees of freedom. Two analytic formulas are given to estimate the key rate under such an imperfect source. To improve the secure key rate, an advanced method with calibrated Bob is provided as well. Our method is based on the decoy state protocol with three intensities (a signal state, a weak decoy state, and a vacuum state), which is the most widely used setting. In~\cref{discussion}, our method is applied to three QKD systems to calculate their secure key rate versus distance. We conclude in~\cref{conclusion}.
\section{Decoy state protocol}
\label{basic}
As a fundamental theory of our research, we recap the decoy state protocol first in this section. Here we take the weak~+~vacuum decoy state protocol~\cite{ma2005} as an example to explain the basic idea of the decoy state protocol. This simple one weak~+~vacuum decoy state protocol is commonly used in Bennett-Brassard~1984 (BB84) QKD system~\cite{bennett1984}, as it provides the optimal key rate in the case of only two decoy states~\cite{ma2005}. The security analysis in~\cref{attack,key rate,discussion} also follows this decoy state model.
According to the analysis of Gottesman-Lo-L\"utkenhaus-Preskill (GLLP)~\cite{gottesman2004}, the key rate of QKD with the WCS can be written as
\begin{equation}\label{GLLP}
R \geq q\{-Q_\mu H_2(E_\mu) f(E_\mu) +P_1^\mu Y_1^\mu[1-H_2(e_1^\mu)]\}.
\end{equation}
Here $q=1/2$ for BB84 protocol (if one uses the efficient BB84 protocol~\cite{lo2005a}, $q\approx 1$), the subscript $\mu$ means the intensity of a signal state, $Q_\mu$ ($E_{\mu}$) is the total gain (error rate) of the signal state, $Y_1^\mu$ and $e_1^\mu$ are the yield and error rate of single-photon pulses, $P_1^\mu$ is the probability of single-photon pulses, $f(x)$ is the bidirectional error correction efficiency, normally $f(x)\geq 1$ with Shannon limit $f(x)=1$, and $H_2(x)=-x \log2(x) -(1-x)\log2(1-x)$ is the binary Shannon information entropy.
In~\cref{GLLP}, $Q_\mu$ and $E_\mu$ are directly obtained in an experiment, and $P_1^\mu$ is known for a given source. Thus, the major task of the decoy state is to tightly estimate the lower bound of $Y_1^\mu$ and upper bound of $e_1^\mu$. Note the fact that, if the phase of the WCS is randomized from 0 to $2\pi$ (the phase randomization assumption), the density matrix of the WCS can be written as
\begin{equation}\label{Rho}
\rho_{\omega}=\sum_{n=0}^{\infty} P_n^\omega |n\rangle\langle n|,
\end{equation}
where $\omega=\{\mu,\nu,0\}$ represents the average intensity of pulse signal state $\mu$, decoy state $\nu$, and vacuum state that is always 0. $P_n^\omega$ is the probability distribution of $n$-photon number from the source with the intensity $\omega$. For the WCS, $P_n^\omega=e^{-\omega}\omega^n/n!$ . Without loss of generality, we assume $\mu>\nu$. Thus, the total gain and error rate can be written as
\begin{equation}\label{QE_calcu}
\begin{split}
Q_\omega&=\sum_{n=0}^\infty P_n^\omega Y_n^\omega,\\
Q_\omega E_\omega&=\sum_{n=0}^\infty P_n^\omega Y_n^\omega e_n^\omega.
\end{split}
\end{equation}
Here $Y_n^\omega$ (or $e_n^\omega$) is the yield (or error rate) given that Alice sends a $n$-photon pulse from the source with intensity $\omega$. Obviously, if Eve does not have any prior information about the intensity of Alice's pulse, we can assume that
\begin{equation}\label{ASSump}
\begin{split}
Y_n^\mu =Y_n^\nu =Y_n,\\
e_n^\mu =e_n^\nu =e_n.
\end{split}
\end{equation}
Then the lower bound of $Y_1$ and upper bound of $e_1$ can be estimated by solving the linear Eqs.~\labelcref{QE_calcu} with weak~+~vacuum decoy states~\cite{ma2005}.
\section{Intensity modulation test}
\label{measurement}
To evaluate the realization of the weak~+~decoy state protocol, we test two intensity modulation methods. The implementation of each has been obtained from a third party, and is tested as supplied, without any tampering or making adjustments.
The first method under testing is the pump-current modulation, similar to Refs.~\onlinecite{yin2017a,liao2017a}. For the signal and weak decoy states, different intensities are produced by applying different pulses of pump current to a laser diode. Thus, the laser diode directly emits optical pulses with different intensities. The vacuum state is generated by turning off the pump current. An optical attenuator then applies a fixed attenuation to all the optical pulses, to reach single-photon level. The second method under testing is an external intensity modulator, similar to Refs.~\onlinecite{rosenberg2007,yuan2007a,dixon2008}. Optical pulses could be produced with a constant intensity from a laser diode first, and then the different intensities of signal and decoy states are modulated by an intensity modulator (IM). Similarly to the former method, a fixed attenuator provides attenuation to the single-photon level.
Our intensity measurement of the optical pulses is taken before the fixed attenuation is applied. The optical pulses are measured by a photodetector ($40~\giga\hertz$ bandwidth) and an oscilloscope ($33~\giga\hertz$ bandwidth), averaging over $\gtrsim 5000$ pulses. We obtain the normalized probability distribution of emitting photons over time which is shown in Figs.~\ref{fig:current}(a) and~\ref{fig:IM}. Although we measure the intensity of classical optical pulses, the probability of emitting single photon should follow the same distribution, because constant attenuation is applied.
\begin{figure}
\scalebox{1}{\includegraphics[width=\columnwidth]{current_drive.pdf}}
\caption{\label{fig:current}(Color online) Pump-current modulation for the laser diode generating $\ket{H}$ polarization state. (a) Normalized intensity distribution of the signal state and the decoy state measured in the time domain. For ease of comparison, the pulses are normalized to have the same area. The original signal-to-decoy intensity ratio is 3:1 ($\mu=0.6, \nu=0.2$). $W_s$ and $W_d$ indicate the typical time windows for Eve to conduct PNS attack, as detailed in~\cref{attack}. (b) Laser-diode's pump current. The current is calculated from the measured voltage across the laser diode module. The lasing threshold, $14~\milli\ampere$, is shown as a line. The relative time alignment between (a) and (b) is a guess.}
\end{figure}
For the case of pump-current modulation, we measure the intensities of signal and decoy states for the polarization state $\ket{H}$. \Cref{fig:current}(a) clearly shows that the probability distributions of emitting the signal state and the decoy state do not totally overlap. The main peaks of these two distributions are mismatched. The signal state emits earlier than the decoy state with high probability, and has a secondary peak from $662$ to $937~\pico\second$. Over the same time interval, the probability distribution of the decoy state drops to low values. The timing mismatch of the signal state and the decoy state clearly violates the basic assumption of indistinguishability in the decoy state protocol. As we show numerically in~\cref{attack}, this can be exploited by Eve to bypass the protection of the decoy state protocol. However, the measured result of external intensity modulation in \cref{fig:IM} does not show a measurable timing mismatch between signal and decoy states. This is expected, because the pulse generation and intensity modulation in this type of source are physically decoupled and performed by separate devices. As long as there is no electrical crosstalk between the laser diode driver and intensity modulator driver, no correlation is expected. This is the case, as~\cref{fig:IM} shows.
\begin{figure}
\scalebox{1}{\includegraphics[width=\columnwidth]{IM.pdf}}
\caption{\label{fig:IM}(Color online) External intensity modulation. Normalized intensity distribution of the signal state and the decoy state measured in the time domain. For ease of comparison, the pulses are normalized to have the same area.}
\end{figure}
To investigate the reason for timing mismatch in the case of pump-current modulation, we measure the current flowing through the laser diode (Agilecom WSLS-940010C4123). A differential probe with $30~\giga\hertz$ bandwidth (Agilent N5445A) is used to measure the differential voltage $V$ across the laser diode and its built-in serial resistor $R_s = 20~\ohm$. Since the laser diode forward voltage $V_d = 1.23~\volt$ is known from its test sheet, we can calculate the pump current $I=(V-V_d)/R_s$. This calculated current is shown in~\cref{fig:current}(b).
If the laser diode were pumped by a constant current, any current above the lasing threshold $I_\text{th} = 14~\milli\ampere$ (shown in the figure) would result in continuous-wave (c.w.)\ laser emission. However, when the current is initially zero then rapidly increased above $I_\text{th}$, the diode does not begin to lase instantly~\cite{agrawal1993}. First, a certain number of carriers has to be injected into the p-n junction before the diode reaches population inversion, and that takes time (the higher the current, the less time). Once the population inversion is reached and the diode attains light amplification condition, the few spontaneously emitted photons present in the optical cavity need time to amplify into a strong coherent light. This results in a fraction-of-nanosecond delay between the application of current and the start of strong light emission. In this process, the population inversion and emitted light power briefly overshoot the steady-state. They then undergo a few oscillations with $\sim 100~\pico\second$ period and eventually settle at the steady-state c.w.\ level if the pump current continues~\cite{agrawal1993}. However, if the pump current is interrupted, as is the case with our device under test, the lasing stops. As can be seen in~\cref{fig:current}, the signal state is produced by a higher peak current pulse, the laser begins emitting light earlier and has time to emit two light pulses (i.e.,\ light power oscillations) before the current stops. The decoy state is produced by a lower peak current pulse, light emission begins later and the laser only has time to emit one light pulse. This physics of laser diode operation is well-known to the manufacturers of pulsed laser diodes (e.g.,\ PicoQuant). However, the engineers who selected this modulation method for the QKD system under our test did not initially realise that it created a security loophole.
While the mismatch might be reduced by adjusting the timing and shape of the pump current between the states, it is unlikely to be fully eliminated. The difference in light pulse shapes is due to the physics of laser diode operation. Further testing is needed to find how well the mismatch can be controlled in a practical device.
\section{PNS attack}
\label{attack}
In the case of pump-current modulation, because the signal state and the decoy state are partially distinguishable in the time domain, the PNS attack becomes possible again. Here we consider a special PNS attack summarized in~\cref{tbl:strategy}. Eve selects time windows $W_s$ and $W_d$ to observe states sent by Alice. By properly setting the intervals of $W_s$ and $W_d$, Eve treats all states observed in $W_s$ ($W_d$) as the signal state (the decoy state). Then she performs the PNS attack. For single-photon states, Eve blocks or forwards those that are in the observation windows, while blocks all of those that are out of the observation windows. Once the states contain two or more photons, Eve keeps one photon and either blocks or forwards the rest of the photons to Bob in the observation windows, but forwards all the photons to Bob when the states are out of the windows. If Eve obtains photons in both $W_s$ and $W_d$, she randomly keeps photons in only one window, and forwards the rest of photons to Bob.
By following the criteria of a successful attack proposed in Ref.~\onlinecite{tang2013}, the success ability of the above attack could be analyzed. A successful attack lets Eve know partial information about the final secret key. In other words, Alice and Bob's key remains partially insecure after post-processing. To show this, a lower bound of the key rate under Alice and Bob's estimation, $R^l$, and an upper bound of the key rate under Eve's attack, $R^u$, are compared. If
\begin{equation}\label{cond}
R^l > R^u,
\end{equation}
the shared final key must be partially insecure and Eve knows some amount of information. This is the result Eve would like to reach in her attack.
\renewcommand{\arraystretch}{1.9}
\begin{table}
\caption{Hacking strategy and corresponding yields.}
\label{tbl:strategy}
\begin{tabular}{c|cc}
PNS attack &In the time windows & \makecell{Outside the \\ time windows}\\
\hline
Single-photon states & Forward or block\footnote{\label{a}Depends on the optimized yield $Z_n^{\omega}$ in simulation.} & Block\\
Multiphoton states & \makecell{Keep one photon and \\forward or block others\footref{a}} & Forward \\
Yield $Y_n^{\omega_{\textrm{Eve}}}$ & $Z_n^{\omega}$ & \makecell{0 ($n = 1$) \\ $Y_n^{\omega}$ ($n\geq2$)} \\
\end{tabular}
\end{table}
The lower bound of the key rate is the one used in the decoy state protocol~\cite{ma2005}:
\begin{equation}\label{lowerbound}
R^l = -Q_{\mu}H_2(E_{\mu}) f(E_\mu) + Y_1^{\mu}\mu e^{-\mu}[1-H_2(e_1^{\mu})],
\end{equation}
which is consistent with~\cref{GLLP} when we consider the efficient BB84 protocol~\cite{lo2005a}, $q = 1$. Here $Y_1^{\mu}$ and $e_1^{\mu}$ are single-photon yield and error rate in the normal decoy state protocol. It is the secure key rate from Alice and Bob's point of view under the attack. Since Alice and Bob do not know about Eve's attack, the estimation of the lower bound of $Y_1^{\mu}$ and the upper bound of $e_1^{\mu}$ still follows the weak~+~vacuum decoy protocol~\cite{ma2005} with the assumption of indistinguishability. The actual upper bound of the key rate under the PNS attack~\cite{tang2013} is
\begin{equation}\label{upperbound}
R^u = Y_1^{\mu_{\textrm{Eve}}}\mu e^{-\mu},
\end{equation}
where $Y_1^{\mu_{\textrm{Eve}}}$ is the real overall yield of single photon states under Eve's attack. The goal of our attack is to minimize the upper bound in~\cref{upperbound} to satisfy inequality~\labelcref{cond}, while matching the value of $Q_{\omega}$ and even reaching lower QBER than $Q_{{\omega}}E_{\omega}$. Then the attack will remain unnoticed.
Based on the measurement result in~\cref{fig:current}(a), Eve can only partially distinguish the signal state and the decoy state. In a certain observation window, we define the following guessing probability. The conditional probability $P(i\big\vert j)$ is defined as Eve guesses the state is $i$ given Alice actually sending the $j$ state. Here $i, j \in [s, d]$, which means $i$ or $j$ is either the signal state, $s$, or the decoy state, $d$. Thus, $P(s\big\vert s)$ and $P(d\big\vert d)$ are the probabilities of correct guess in $W_s$ and $W_d$ respectively, while $P(s\big\vert d)$ and $P(d\big\vert s)$ are the probabilities of wrong guess in the same windows.
As mentioned in the hacking strategy, once Eve observes multiphoton states in $W_s$ or $W_d$, she keeps a single photon and might forward or block the remaining photons to Bob. In order to maintain the statistics of $Q_{\omega}$ and $Q_{{\omega}}E_{\omega}$, Eve has to manipulate detection yield in the observation windows from $Y_n^{\omega}$ to $Z_n^{\omega}$ as shown in~\cref{tbl:strategy}. In the time window $W_s$~($W_d$), the yield is denoted as $Z_n^{\mu}$~($Z_n^{\nu}$). Please note that Eve is allowed to use a lower-loss, or even lossless, channel, which means $Z_n^{\omega}$ could be greater than $Y_n^{\omega}$. At the phase of decoy announcement in QKD protocol, Bob classifies detection slots according to Alice's signal and decoy information. Thus, under Eve's attack, the yields $Y_n^{\omega_{\textrm{Eve}}}$ actually should be recalculated as the following. For the single-photon states, Eve fully controls the yields, since the single-photon states outside the time windows are blocked. Thus, $Y_1^{\omega_{\textrm{Eve}}} $ are given by
\begin{equation}\label{newY1}
\begin{split}
Y_1^{\mu_{\textrm{Eve}}} = P(s\big\vert s)Z_1^{\mu} + P(d\big\vert s)Z_1^{\nu},\\
Y_1^{\nu_{\textrm{Eve}}} = P(s\big\vert d)Z_1^{\mu} + P(d\big\vert d)Z_1^{\nu}.
\end{split}
\end{equation}
For multiphoton states ($n\geq2$), Eve forwards the states to Bob when these states are outside the observation windows, so $Y_n^{\omega_{\textrm{Eve}}} $ are given by
\begin{equation}\label{newY}
\begin{split}
Y_n^{\mu_{\textrm{Eve}}} = P(s\big\vert s)Z_n^{\mu} + P(d\big\vert s)Z_n^{\nu} + [1 \!-\! P(s\big\vert s) \!-\! P(d\big\vert s)]Y_n^{\mu},\\
Y_n^{\nu_{\textrm{Eve}}} = P(s\big\vert d)Z_n^{\mu} + P(d\big\vert d)Z_n^{\nu} + [1 \!-\! P(s\big\vert d) \!-\! P(d\big\vert d)]Y_n^{\nu}.
\end{split}
\end{equation}
Correspondingly, the overall gains of the signal state and the decoy state are
\begin{equation}\label{newQ}
\begin{split}
Q_{\mu_{\textrm{Eve}}} = Y_0^{\mu_{\textrm{Eve}}}e^{-\mu} + \sum_{n=1}^{\infty} Y_n^{\mu_{\textrm{Eve}}} e^{-\mu}\frac{\mu^{n}}{n!},\\
Q_{\nu_{\textrm{Eve}}} = Y_0^{\nu_{\textrm{Eve}}}e^{-\nu} + \sum_{n=1}^{\infty} Y_n^{\nu_{\textrm{Eve}}} e^{-\nu}\frac{\nu^{n}}{n!},
\end{split}
\end{equation}
where $Y_0^{\omega_{\textrm{Eve}}}$ are dark count rates under the attack. The overall QBERs are given by
\begin{equation}\label{newQBER}
\begin{split}
E_{\mu_{\textrm{Eve}}}Q_{\mu_{\textrm{Eve}}} = \frac{1}{2}Y_0^{\mu_{\textrm{Eve}}}e^{-\mu} + \sum_{n=1}^{\infty} \frac{1}{2}P(d\big\vert s)Z_n^{\nu} e^{-\mu}\frac{\mu^{n}}{n!},\\
E_{\nu_{\textrm{Eve}}}Q_{\nu_{\textrm{Eve}}} = \frac{1}{2}Y_0^{\nu_{\textrm{Eve}}}e^{-\nu} + \sum_{n=1}^{\infty} \frac{1}{2}P(s\big\vert d)Z_n^{\mu} e^{-\nu}\frac{\nu^{n}}{n!}.
\end{split}
\end{equation}
Here we consider an extreme case. A dark count introduces error half the time. There is no error if signal and decoy states are correctly distinguished by Eve or states are outside the windows $W_s$ and $W_d$. However, wrong guess in $W_s$ and $W_d$ results in random clicks, which introduces error half the time.
According to the standard decoy state protocol~\cite{ma2005}, the normal overall gains should be
\begin{equation}\label{Q}
\begin{split}
Q_{\omega} = Y_0 + 1- e^{-\eta\omega},
\end{split}
\end{equation}
where $Y_0$ is the dark count rate, and $\eta$ is the total transmittance of the QKD system. $\eta$ is given by
\begin{equation}\label{eta}
\eta= \eta_{\textrm{Bob}} 10^{-\alpha L/10},
\end{equation}
where $\eta_{\textrm{Bob}}$ is the transmittance of Bob's optical device, including detector efficiency, and $\alpha$ is the transmittance of channel between Alice and Bob. Typically, $\alpha=0.21~\deci\bel/\kilo\metre$ for the commercial fibre at $1550~\nano\metre$. $L$ is the length of channel. The normal overall QBER should be
\begin{equation}\label{QBER}
\begin{split}
E_{\omega}Q_{\omega} =\frac{1}{2} Y_0 +e_{\textrm{detector}}( 1- e^{-\eta\omega}),
\end{split}
\end{equation}
where $e_{\textrm{detector}}$ is the probability that a photon goes to erroneous detector, characterizing the alignment and stability of a QKD system.
To achieve a successful attack, the upper bound of the key rate $R^u$ should be minimized, which is equivalent to minimizing $Y_1^{\mu_{\textrm{Eve}}}$ in~\cref{newY1}. Meanwhile, to achieve a traceless attack, the attack has to follow the same detection statistics by optimizing $Z_n^{\mu}$, $Z_n^{\nu}$, $P(s\big\vert s)$, $P(d\big\vert s)$, $P(s\vert d)$ and $P(d\vert d)$ for every distance value. Therefore, it becomes an optimization problem under certain constraints:
\begin{equation}\label{optimization}
\begin{split}
\min_{Z_n^{\mu}, Z_n^{\nu}; P(s\vert s), P(d\vert s)} Y_1^{\mu_{\textrm{Eve}}}
\end{split}
\end{equation}
subject to
\begin{equation}\label{constraint}
\begin{split}
Q_{\mu} = Q_{\mu_{\textrm{Eve}}},\\
Q_{\nu} = Q_{\nu_{\textrm{Eve}}},\\
E_{\mu}Q_{\mu} \geqslant E_{\mu_{\textrm{Eve}}}Q_{\mu_{\textrm{Eve}}},\\
E_{\nu}Q_{\nu} \geqslant E_{\nu_{\textrm{Eve}}}Q_{\nu_{\textrm{Eve}}},\\
Z_n^{\mu}, Z_n^{\nu} \in [0,1],\\
P(s\vert s), P(d\vert s), P(s\vert d), P(d\vert d) \in [0,1].
\end{split}
\end{equation}
Ideally, the detection efficiency could be $100\%$, so the yield $Z_n^{\omega}$ could reach 1. We also remark that the probabilities $P(i\big\vert j)$ are taken from the measured probability distribution of the states sent by Alice in~\cref{fig:current}(a). $P(s\big\vert s)$ and $P(s\big\vert d)$ should be taken from the time window $W_s$; $P(d\big\vert s)$ and $P(d\big\vert d)$ should be taken from the time window $W_d$. Importantly, since every time window could contain several timing intervals, any observation probabilities mentioned above should be a sum over the time window.
\begin{figure}
\scalebox{1}{\includegraphics[width=\columnwidth]{Optimized-interval-keyrate-GYS-Y1-01.pdf}}
\caption{\label{fig:optimization}(Color online) The lower bound $R^l$ and optimized upper bound $R^u$ of key rate under our simulated attack. The detection parameters are taken from GYS experiment~\cite{gobby2004}: the dark count rate $Y_0= 1.7\times 10^{-6}$, the transmission in Bob's apparatus $\eta_{\textrm{Bob}}= 4.5 \%$, the misalignment error rate $e_{\textrm{detector}} = 3.3 \%$ and the error correction efficiency $f(E_\mu) = 1.22$.}
\end{figure}
The simulation result is shown in~\cref{fig:optimization}. To follow the initial analysis of weak~+~vacuum decoy state protocol in Ref.~\onlinecite{ma2005}, we also use the detection parameters from Gobby-Yuan-Shields (GYS) experiment~\cite{gobby2004} in our attack simulation. However, we assume the source has characteristics as in~\cref{fig:current}(a); this source actually comes from a different QKD system with the mean photon number $\mu$ = 0.6 for the signal state and $\nu = 0.2$ for the weak decoy state. According to inequality~\labelcref{cond}, once the optimized upper bound starts becoming smaller than the lower bound, Eve can successfully execute the PNS attack. \Cref{fig:optimization} shows that Eve is able to successfully hack it and eavesdrop some of the secret key when the distance between Alice and Bob is longer than $45~\kilo\metre$. The attack windows $W_s$ and $W_d$ are optimized for every distance point to get the lowest $R^u$ at this distance. For example, when the distance between Alice and Bob is $49~\kilo\metre$, the optimized $W_s$ and $W_d$ are shown as grey zones in~\cref{fig:current}(a).
\section{Secure key rate with imperfect source}
\label{key rate}
The previous section shows the effect of partial distinguishability between signal and decoy states in the time domain. However, the side-channel that partially distinguishes signal and decoy states could be more general. For example, generating signal and decoy states by individual laser diodes is widely employed in QKD systems~\cite{peng2007,yin2008,liu2010}, even in the first quantum satellite~\cite{liao2017}. Unfortunately, it has been shown that this type of state preparation might leak the modulation information in the time and frequency (spectral) domains~\cite{nauerth2009}. For another preparation method of one laser diode with an IM in a plug-and-play system, Eve can shift the arriving time of pulses to the rising edge of intensity modulation, obtaining a side-channel in the frequency domain in the plug-and-play system~\cite{jiang2012}. Moreover, modulation information of IM might be read out by an active Trojan-horse attack~\cite{vakhitov2001,tamaki2016}. Even if the intensity modulation is perfect, the laser pulses with non-random phases give Eve a chance to distinguish signal and decoy states~\cite{tang2013}. Therefore, it is important to build a general security model that tolerates such side-channels. In this section, we modify the model of the decoy state protocol to consider such imperfect sources, and derive two analytic formulas that estimate the contribution of single-photon pulses.
\subsection{Model}
\label{sec:keyrate-model}
We analyze the weak~+~vacuum decoy state protocol with intensities $\omega=\{\mu,\nu,\nu_1\}$. Without loss of generality, we assume $\mu>\nu>\nu_1$ and $\mu>\nu+\nu_1$. Here $\nu_1$ represents the intensity of a vacuum state.
If the imperfection of source is taken into account, the density matrix of Alice's states [\cref{Rho}] can be rewritten as
\begin{equation}\label{Rho_2}
\rho'_{\omega}=\rho_\omega\otimes \rho_\omega(\lambda)=\sum_{n=0}^{\infty} \int_\lambda d\lambda P_n^\omega f_\omega (\lambda) |n,\lambda\rangle\langle n,\lambda| .
\end{equation}
Here $\rho_\omega(\lambda)$ is the quantum state used by Eve to distinguish the signal state and the decoy state for each pulse. Note that $\rho_\omega(\lambda)$ can be an actual quantum state, or any additional dimension of Alice's pulses. $\lambda$ represents the parameter (for example, the time, frequency and so on) measured by Eve, which is used to distinguish the signal state or the decoy state. $f_\omega(\lambda)$ is the normalized probability distribution of $\lambda$ ($\int_\lambda f_\omega(\lambda) d\lambda=1$), which depends on the intensities of Alice's pulse $\omega$. Note that if $\lambda$ is a discrete variable, $\int d\lambda$ should be changed to $\sum_\lambda$ in~\cref{Rho_2}. We assume that the measured parameter of Eve, $\lambda$, is independent of the photon number of Alice's state, $n$. I.e., $|n,\lambda\rangle=|n\rangle|\lambda\rangle$. Obviously, if $\rho_\omega(\lambda)$ is independent on the intensities of Alice's pulse, which means $\rho_\mu(\lambda)=\rho_\nu(\lambda)=\rho_{\nu_1}(\lambda)\equiv \rho(\lambda)$, then \cref{Rho_2} becomes~\cref{Rho}. Thus, the general decoy state method can be used to estimate the bound of yield and error rate for the single photon pulses~\cite{ma2005}.
By combining Eqs.~\labelcref{QE_calcu} with~\cref{Rho_2}, the total gain and error rate of Alice's states should be rewritten as
\begin{equation}\label{QE_calcu2}
\begin{split}
Q_\omega&=\sum_{n=0}^\infty P_n^\omega Y_n^\omega =\sum_{n=0}^\infty P_n^\omega \sum_{\lambda} f_{\omega}(\lambda)Y_n(\lambda),\\
Q_\omega E_\omega&=\sum_{n=0}^\infty P_n^\omega Y_n^\omega e_n^\omega =\sum_{n=0}^\infty P_n^\omega \sum_{\lambda} f_{\omega}(\lambda)Y_n(\lambda)e_n(\lambda),
\end{split}
\end{equation}
where $Y_n(\lambda)$ and $e_n(\lambda)$ are the yield and error rate given that Alice sends an $n$-photon pulse and Eve obtains $\lambda$ in her measurement. Thus $Y_n(\lambda)$ and $e_n(\lambda)$ depend on the parameter $\lambda$, but are independent on the intensities of Alice's pulses $\omega$.
As Ref.~\onlinecite{tamaki2016} mentioned, the imperfection of source can be characterized by the distance between $\rho_\omega(\lambda)$ and $\rho_{\omega'}(\lambda)$, which is given by
\begin{equation}\label{Duv_rho}
D_{\omega\omega'}=\frac{1}{2}tr|\rho_\omega-\rho_{\omega'}|.
\end{equation}
Here $tr{|x|}$ is the trace distance of quantum state.
\bigskip
\subsection{Lower bound of $\bm{Y_1^\mu}$}
\label{sec:keyrate-lowerbound}
Now we derive the lower bound of $Y_1^\mu$ according to the model given above. From~\cref{Duv_rho}, it is easy to obtain inequalities
\begin{equation}\label{Yen_bound}
\begin{split}
|Y_n^{\omega}-Y_n^{\omega'}|\leq 2D_{\omega \omega'},\\
|Y_n^{\omega}e_n^{\omega}-Y_n^{\omega'}e_n^{\omega'}|\leq 2D_{\omega\omega'},
\end{split}
\end{equation}
where $\omega, \omega' = \mu, \nu, \nu_1$ and $1>\mu>\nu>\nu_1>0$. Note that by following the proof of Ref.~\onlinecite{tamaki2016}, the factor 2 in~\cref{Yen_bound} may be removed to improve the key rate.
To estimate the lower bound of $Y_1^\mu$, we follow the procedure in Ref.~\onlinecite{ma2005}. We assume that $Y_0^{\mu} =Y_0^{\nu}=Y_0^{\nu_1}=Y_0$, since there is no difference for the vacuum pulse and Eve cannot get any information from such pulse. The lower bound of $Y_1^\mu$ is given by (see \cref{A_Y1} for derivation)
\begin{equation}\label{Y12}
\begin{split}
Y_1^\mu \geq\ &\frac{\mu[e^{\nu}Q_{\nu}\!-\!e^{\nu_1}Q_{\nu_1}\!-\!\frac{\nu^2\!-\!\nu_1^2}{\mu^2}(e^{\mu}Q_{\mu\!}-\!Y_0^L)]}{\mu(\nu-\nu_1)-(\nu^2-\nu_1^2)} -g(\mu,\nu,\nu_1)\\
\equiv\ &G(\mu,\nu,\nu_1) - g(\mu,\nu,\nu_1).
\end{split}
\end{equation}
Here,
\begin{equation}
g(\mu,\nu,\nu_1)\equiv \frac{2\mu[D_{\mu\nu} (e^{\nu}-1) + D_{\mu\nu_1}(e^{\nu_1}-1)]}{\mu(\nu-\nu_1)-(\nu^2-\nu_1^2)}.
\end{equation}
It is easy to check that $G(\mu,\nu,\nu_1)$ is the same as the lower bound of $Y_1^\mu$ in the standard decoy-state protocol (Eq.~(21) in Ref.~\onlinecite{ma2005}), which represents the yield of the single photon pulse with a perfect source. Thus, $g(\mu,\nu,\nu_1)$ represents the leaked information due to the imperfection of source.
In general decoy state experiments, weak~+~vacuum state protocol is used, which means $\nu_1=0$. Then~\cref{Y12} can be written as
\begin{equation}\label{Y1}
Y_1^\mu \! \geq\! \frac{\mu}{\mu\nu\!-\!\nu^2} [e^{\nu}Q_{\nu}\!-\!\frac{\nu^2}{\mu^2}e^{\mu}Q_{\mu}\!-\!\frac{\mu^2\!-\!\nu^2}{\mu^2}Q_\text{vac}\!-\!2 D_{\mu\nu}(e^{\nu}\!-\!1)].
\end{equation}
\subsection{Upper bound of $e_1^\mu$}
\label{sec:keyrate-upperbound}
The upper bound of $e_1^\mu$ can be estimated by (see \cref{A_e1} for derivation)
\begin{equation}\label{e12}
e_1^\mu \leq \min\{K^\mu, K^\nu, K^{\nu_1}, K^{\mu\nu}, K^{\mu\nu_1}\},
\end{equation}
where
\begin{equation}
\begin{split}
K^\mu = \ & \frac{e^{\mu}Q_\mu E_\mu-e_0 Y_0^L}{\mu Y_1^\mu},\\
K^\nu = \ & \frac{e^\nu Q_\nu E_\nu -e_0 Y_0^L +2\nu D_{\mu\nu}}{\nu Y_1^\mu},\\
K^{\nu_1} = \ & \frac{e^{\nu_1} Q_{\nu_1} E_{\nu_1} -e_0 Y_0^L +2\nu_1 D_{\mu\nu_1}}{\nu_1 Y_1^\mu},\\
K^{\mu\nu} = \ & \frac{e^\mu Q_\mu E_\mu -e^\nu Q_\nu E_\nu +2 D_{\mu \nu}(e^\nu-1)}{(\mu-\nu)Y_1^\mu},\\
K^{\mu\nu_1} = \ & \frac{e^\mu Q_\mu E_\mu -e^{\nu_1} Q_{\nu_1} E_{\nu_1} +2 D_{\mu \nu_1}(e^{\nu_1} -1)}{(\mu-\nu_1)Y_1^\mu}. \nonumber
\end{split}
\end{equation}
When $D_{\mu \nu_1} = 0$, \cref{e12} becomes as the same as the upper bound of $e_1^\nu$ in the standard decoy-state protocol (Eq.~(25) in Ref.~\onlinecite{ma2005}).
For weak~+~vacuum decoy state protocol, $\nu_1=0$ and $Y_0 = Q_\text{vac}$. Then, the upper bound of $e_1^\mu$ can be rewritten as
\begin{equation}\label{e1}
e_1^\mu \leq \min\{K^\mu, K^\nu, K^{\mu\nu}\}.
\end{equation}
\subsection{Numerical simulation}
\label{sec:keyrate-numerics}
\begin{figure}
\scalebox{1}{\includegraphics[width=\columnwidth]{keyrate.pdf}}
\caption{\label{fig:keyrate}(Color online) Estimated system parameters with imperfect source. (a) yield and (b) error rate of the signal state for the single photon pulse, and (c) key rate are shown for different amounts of imperfection $D_{\mu\nu}$. The detection parameters used in the simulations are the same as those in~\cref{fig:optimization}. The intensities of the signal state and the decoy state are optimized with step 0.01 from $\mu \in [0.01,0.5]$, $\nu\in [0.01,0.2]$. We only show the estimated $Y_1^\mu$ and $e_1^\mu$ where the final key rate is positive. No secure key can be generated for $D_{\mu\nu}=10^{-1}$ and $10^{-2}$.}
\end{figure}
We simulate weak~+~vacuum decoy state protocol. When Eve is absent, the total gain and error rate of the signal state and the decoy state are given by~\cref{Q,QBER}. The rest of parameters follow the definition in standard decoy state protocol~\cite{ma2005}. Submitting~\cref{Q,QBER} into~\cref{Y1,e1}, we can obtain the lower bound of yield and the upper bound of error rate for the single photon pulse as shown in~\cref{fig:keyrate}(a) and (b). Then the estimated key rate is given in~\cref{fig:keyrate}(c). This clearly shows that the imperfection of source will reduce the key rate between Alice and Bob rapidly. For example, when the source is perfect ($D_{\mu\nu} = 0$), the maximum distance is about $141~\kilo\metre$. The secure key rate estimated by our method matches that estimated by the original decoy-state protocol in Ref.~\onlinecite{ma2005}, which confirms the estimation is tight. However, the maximum distances are reduced to $124, 92,48~\kilo\metre$ for $D_{\mu\nu}=10^{-5}, 10^{-4}, 10^{-3}$. No positive key rate is possible at any distance for $D_{\mu\nu}=10^{-2}, 10^{-1}$.
\subsection{Theory improvement}
\label{sec:keyrate-improved}
From the modified security proof in the last section, even for the relatively small imperfection $D_{\mu\nu}=10^{-3}$, the maximum secure distance drops quickly from $141~\kilo\metre$ to $48~\kilo\metre$. Here, we propose an advanced security proof to improve the final key rate with the imperfect source by setting a reasonable assumption. Then we could loosen the security constraint when estimating $Y_1^\mu$ and $e_1^\mu$, theoretically improving the secure key rate and the maximum secure distance.
\begin{figure}
\scalebox{1}{\includegraphics[width=\columnwidth]{keyrate2.pdf}}
\caption{\label{fig:keyrate2}(Color online) Estimated key rate assuming calibrated transmittance in Bob's optical devices. The detection parameters used here are the same as those in~\cref{fig:optimization}.}
\end{figure}
In practical QKD systems based on prepare-and-measure protocol, Bob's devices are located within his protected zone. Thus, it is possible for Bob to calibrate the optical transmittance of his optical devices. Note that here we do not mean that Eve could not change the parameters of Bob's system (for example, change the SPD from Geiger mode to linear mode by performing the blinding attack), but mean that Bob could actively calibrate the transmittance of his partial or all devices. In fact, this assumption has been used to secure the single photon detector of Bob~\cite{maroy2010,maroy2017}. Thus, we think this assumption is reasonable and practical. We can then obtain
\begin{equation}
Y_n(\lambda) \leq 1-(1-\eta_{\textrm{Bob}}^{\textrm{cal}})^n.
\end{equation}
Please note that $\eta_{\textrm{Bob}}^{\textrm{cal}}$ is the calibrated transmittance in Bob, which should be equal with or lower than the total transmittance of Bob $\eta_{\textrm{Bob}}$. In the simulation, we could assume that Bob can calibrate the whole transmittance in his system, thus we could have $\eta_{\textrm{Bob}}^{\textrm{cal}} =\eta_{\textrm{Bob}}$. Then \cref{Yen_bound} can be rewritten as
\begin{equation}\label{Yen_bound2}
\begin{split}
|Y_n^{\omega}-Y_n^{\omega'}|\leq 2D_{\omega \omega'}[1-(1-\eta_{\textrm{Bob}})^n],\\
|Y_n^{\omega}e_n^{\omega}-Y_n^{\omega'}e_n^{\omega'}|\leq 2D_{\omega\omega'}[1-(1-\eta_{\textrm{Bob}})^n].
\end{split}
\end{equation}
Then in weak~+~vacuum decoy state protocol, it is easy to check that the lower bound of $Y_1^\mu$ [\cref{Y1}] and the upper bound of $e_1^\mu$ [\cref{e1}] can be rewritten as
\begin{equation}
\label{modified}
\begin{split}
Y_1^\mu \geq\ & \frac{\mu}{\mu\nu-\nu^2} [e^{\nu}Q_{\nu}-\frac{\nu^2}{\mu^2}e^{\mu}Q_{\mu}-\frac{\mu^2-\nu^2}{\mu^2}Y_0\\
&-2 D_{\mu\nu}(e^{\nu}-e^{\nu(1-\eta_{\textrm{Bob}})})],\\
e_1^\mu \leq\ &\min\{K^\mu,K^\nu,K^{\mu \nu}\},\textrm{where}\\
K^\mu =\ &\frac{e^{\mu}Q_\mu E_\mu-e_0 Y_0}{\mu Y_1^\mu},\\
K^\nu =\ &\frac{e^\nu Q_\nu E_\nu -e_0 Y_0 +2\nu D_{\mu\nu}\eta_{\textrm{Bob}}}{\nu Y_1^\mu},\\
K^{\mu \nu} =\ &\frac{e^\mu Q_\mu E_\mu -e^\nu Q_\nu E_\nu +2 D_{\mu \nu}(e^\nu-e^{\nu(1-\eta_{\textrm{Bob}})})}{(\mu-\nu)Y_1^\mu}.
\end{split}
\end{equation}
Then we could estimate the final key rate with the same method given above. The estimation result in~\cref{fig:keyrate2} clearly shows that when the transmittance of Bob's optical devices ($\eta_{\textrm{Bob}}=4.5 \%$) is taken into account, the final key rate and the maximum distance are improved. For example, in the case that $D_{\mu\nu}=10^{-3}$, the maximum distance increases to $105~\kilo\metre$ from $48~\kilo\metre$. Also note that for $D_{\mu\nu}=10^{-2}$ and $10^{-1}$, the improved proof provides positive key rate up to $64$ and $18~\kilo\metre$. We remark that the assumption of calibrated transmission loss for Bob's devices is not applicable to measurement-device-independent QKD~(MDI QKD), in which the detection part is not in the protected zone and can be fully controlled by Eve.
\section{Discussion and application examples}
\label{discussion}
In~\cref{key rate}, the method used to estimate $Y_1^\mu$ and $e_1^\mu$ in a decoy state protocol considers a type of imperfect source which could partially distinguish signal and decoy states in any degrees of freedom. Once these imperfections are experimentally measured, this method could provide a standard way to calculate the final key rate under such imperfections. Note that this method focuses on the imperfect modulation of signal and decoy states, but does not handle the distinguishability among different BB84 states. We currently assume the identical mismatch of signal and decoy states for each BB84 state. Removing this theoretical limitation could be future work.
Another limitation lies in our experiment. We have measured the distinguishability between signal and decoy states only in the time domain. However, the two modulation methods we have tested might also introduce time-dependent spectral mismatch, which we have not measured. For the gain-switched semiconductor laser, a short pulse usually has a so-called chirp, a fast-changing wavelength modulation~\cite{koch1984,linke1985}. The spectral and intensity modulation contribute simultaneously to the distinguishability, resulting in a joint distribution of $D_{\mu\nu}$ as explained later in this section. The external intensity modulator may also affect the spectrum of pulses~\cite{koyama1988,kawanishi2001}. However, the requisite time-resolved spectroscopy is a more complex measurement~\cite{linke1985,saunders1994,niemi2002}, which could be investigated in the future. For the two devices tested, we henceforth assume distinguishability in the time domain only.
We now apply our security proof to the measurement results of the two sources tested in~\cref{measurement}, and to one more published source measurement in Ref.~\onlinecite{nauerth2009}. Both the initial proof in~\cref{sec:keyrate-numerics} and the advanced proof in~\cref{sec:keyrate-improved} are applied in each case. The purpose of this application is quantifying the imperfection of signal and decoy states preparation, and showing its effect on the secure key rate. In order to compare the three source implementations, we arbitrarily assume that these different sources are used in the same fibre-based QKD system with GYS parameters at the detection side. The resulting secure key rates are shown in~\cref{fig:real key}.
\begin{figure}
\scalebox{1}{\includegraphics[width=\columnwidth]{real_key_rate.pdf}}
\caption{\label{fig:real key} (Color online) Estimated key rate for different experimental distinguishability of signal and decoy states. The detection parameters used here are the same as in~\cref{fig:optimization}. The key rate is estimated by the initial proof with~\cref{Y1,e1} and also the improved proof with~\cref{modified}. Note that in the case of $D_{\mu\nu}=0.1400$, no secure key can be generated with the initial proof, but the positive key rate is possible with our improved proof. $D_{\mu\nu}=0.4005$ cannot generate positive key rate in either proof.}
\end{figure}
For the first case shown in~\cref{fig:current}(a), the corresponding value of $D_{\mu\nu}$ given by~\cref{Duv_rho} is 0.4005. Then both our security proofs are applied. For the improved proof, we assume $\eta_{\textrm{Bob}}=4.5 \%$. The secure key rate is \textit{zero} under either key rate estimation. This shows that the modulation imperfection can make the system insecure.
On the contrary, the value of $D_{\mu\nu}$ for the case in~\cref{fig:IM} is only $3.6\times10^{-3}$. This non-zero value probably stems from the noise in our characterization apparatus. Nevertheless, we have to conservatively treat all mismatch as belonging to the source under test. This non-zero value of $D_{\mu\nu}$ still indicates a certain degree of mismatch. As shown in~\cref{fig:real key}, under the initial proof, the maximum distance drops to $22~\kilo\metre$, while under the advanced proof with $\eta_{\textrm{Bob}}=4.5 \%$, it improves to $83~\kilo\metre$. I.e.,\ owing to the much lower mismatch in this case, the positive key rate could be generated. However, the maximum transmission distance is sensitive even to such low mismatch values.
\begin{figure}
\scalebox{1}{\includegraphics[width=\columnwidth]{Reprint-mismatch-from-Germany.pdf}}
\caption{\label{fig:reprint} (Color online) Initial mismatch of signal and decoy states for the vertical-polarization pair of laser diodes in (a) time domain and (b) frequency domain. The timing mismatch can be reduced by tuning electrical delay lines in the laser diodes' driver. Data reprinted from Ref.~\onlinecite{nauerth2009}.}
\end{figure}
Another case of imperfect preparation for signal and decoy states is published in Ref.~\onlinecite{nauerth2009}. In that study, the signal and decoy states are generated by individual laser diodes, which is a common technique~\cite{peng2007,yin2008,liu2010,liao2017}. It shows that mismatches between signal and decoy states are both in the time domain and frequency domain for each individual BB84 state~\cite{nauerth2009}. Because our proof cannot handle the BB84 states individually, we have chosen a typical mismatch between the signal and decoy states in vertical polarization as reprinted in~\cref{fig:reprint}, and assumed arbitrarily that the other three BB84 polarization states have the mismatch identical to that. Even though Ref.~\onlinecite{nauerth2009} studies an imperfect source in a free-space QKD system, we remark that it is reasonable to expect mismatch for any QKD implementations that generate signal and decoy states by individual laser diodes~\cite{peng2007,yin2008,liu2010,liao2017}. Please note that~\cref{fig:reprint} illustrates the initial mismatches between two independent lasers. The timing mismatch can be reduced by adjusting the delay between the laser diodes~\cite{nauerth2009,rau2015}.
The security proof in~\cref{key rate} is able to handle mismatch in arbitrary degrees of freedom, because we do not specify the dimensions of the probability $f_\omega(\lambda)$. $f_\omega(\lambda)$ can be a joint probability. For example, the joint probability distribution of $\omega$ state in the time and frequency domains can be $f_\omega(t, f)$, where $t$ represents the time domain and $f$ represents the frequency domain. Thus, $D_{\mu\nu}$ can be defined as
\begin{equation}\label{Duv_mul}
D_{\mu\nu}=\frac{1}{2}\sum_t \sum_f |f_\mu(t,f)-f_{\nu}(t,f)|.
\end{equation}
Similarly, the calculation of $D_{\mu\nu}$ can be expanded to more than two dimensions. In the specific case shown in~\cref{fig:reprint}, time-resolved spectroscopy necessary to measure the joint probability was not performed. It has been arbitrarily assumed instead that the probability distributions in the time and frequency domains are independent, with a remark that this will need to be verified experimentally~\cite{nauerth2009}. Then, $f_\omega(t, f) = f_\omega(t) f_\omega(f)$ can be calculated from the available experimental data. The corresponding $D_{\mu\nu}$ is 0.1400. With such value of $D_{\mu\nu}$, the initial security proof cannot generate positive key rate for any distance, while the improved proof with $\eta_{\textrm{Bob}}=4.5 \%$ could generate the secure key up to only $10~\kilo\metre$, as shown in~\cref{fig:real key}.
According to the above analysis and comparison of the three cases, the external intensity modulator shows the smallest mismatch between signal and decoy states, resulting in the highest key rate and longest transmission distance. However, Trojan-horse attack could read out the modulation information from the intensity modulator~\cite{vakhitov2001,tamaki2016}. Thus, countermeasures against Trojan-horse attack are also necessary~\cite{lucamarini2015,sajeed2015}. The use of intensity modulator may also result in non-zero vacuum state, which can be handled by our security proof. The secure key rate can then be estimated by applying~\cref{Y12,e12}.
All the measurement results shown above contain a measurement error. The error may come from the thermal noise of electronic devices, the nonlinearity of optical-to-electrical converters and digital-to-analog converters in the oscilloscope. We have simply treated the measured $D_{\mu\nu}$ as the real mismatch. Thus, the key rates shown in~\cref{fig:real key} are conservative estimates. It is an open question how to extract the real parameter from the noisy test results.
\section{Conclusion}
\label{conclusion}
In this Article, we have investigated the imperfect sources in QKD systems that implement the decoy state protocol. By testing two intensity modulation methods, we have found that the basic assumption about indistinguishability of signal and decoy states does not hold in practice, especially in the case of laser diode pump current modulation. This pump-current modulation shows timing mismatch between the signal and decoy states. We have modeled a PNS attack based on the timing mismatch that breaks the security of the QKD system. To make the system robust against this loophole, we have considered the distinguishability of signal and decoy states in the security proof, and obtained two analytical formulas to estimate the yield and error rate of the single photon pulses. The result shows that the distinguishability would reduce the secure key rate. Fortunately, the key rate under such imperfection can be improved by calibrating the transmittance of Bob's unit. We have applied this method to three implementations of the decoy state protocol to estimate their secure key rate, which in some cases has become reduced (also limiting the transmission distance), and in some just zero.
The estimation of $Y_1^\mu$ and $e_1^\mu$ with the distinguishable decoy state provides a method to guarantee the security of practical quantum cryptography systems. This method could be employed as a standard tool to estimate the secure key rate, once the distinguishability in the decoy state protocol is quantified in all degrees of freedom. A conceptually similar evaluation method has been proposed for the Trojan-horse attack~\cite{lucamarini2015}. Please note that the key rate shown in this work is based on prepare-and-measure QKD. For MDI QKD and the other quantum cryptography systems, the secure key rate and the other security properties under such imperfect source model should be derived separately.
\acknowledgments
We thank Y.~Zhou for helpful discussions, S.~Nauerth and H.~Weinfurter for providing the raw data from Ref.~\onlinecite{nauerth2009}. This work was funded by the National Natural Science Foundation of China (grant numbers 11674397 and 61601476) and NSERC of Canada (programs Discovery and CryptoWorks21).
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\begin{document}
\begin{center}
\large{\textbf{The Abel, Fourier and Radon transforms on
symmetric spaces}}
\end{center}
\begin{center}
{\footnotesize SIGURDUR HELGASON}
\end{center}
\begin{center}
\textbf{To Gerrit van Dijk on the occasion of his 65\st{th} birthday}
\end{center}
\normalsize
\section{Introduction}
\label{sec:1}
In this paper we prove some recent results on the three
transforms in the title and discuss their relationships to older
results. The spaces we deal with are symmetric spaces $X=G/K$ of
the noncompact type, $G$ being a connected noncompact semisimple
Lie group with finite center and $K$ a maximal compact subgroup.
For the two natural Radon transforms on $X$ we prove a new
inversion formula and a sharpening of an old support theorem; for
the Abel transform we prove some new identities with some
applications and for the Fourier transform a result for
integrable functions which has a strong analog of the
Riemann--Lebesgue lemma. These latter results are from a
collaboration with Rawat, Sengupta and Sitaram.
\noindent\emph{Notation.} Following Schwartz we use the
notation $\D (X)$ for $\Cinf_c(X)$, $\E (X)$ for $\Cinf (X)$
and $\SS (\mathbf{R}^n)$ for the space of rapidly decreasing functions
on $\mathbf{R}^n$.
\section{Different Radon transforms on the symmetric space~$X$}
\label{sec:2}
Radon's paper \cite{R} suggested the general problem of
determining a function on a manifold on the basis of its
integrals over certain submanifolds. A natural case of this
problem is the inversion of the X-ray transform on a Riemannian
manifold. It is the transform $f \to \widehat{f}$ defined by the
arc-length integral
\begin{equation}
\label{eq:2.1}
\widehat{f} (\gamma ) = \int_{\gamma} f(x) \, dm (x) \, ,
\end{equation}
$f$ being an ``arbitrary'' function on the Riemannian
manifold~$X$ and $\gamma$ any complete geodesic in $X$.
In general this injectivity problem seems to be unresolved. For
a Cartan symmetric space $X \neq \SS^n$ the injectivity, however,
holds. For a symmetric~$X$ of the noncompact the injectivity
holds in the stronger form of the
\vspace{2ex}
\begin{theorem*} {\textup{\cite{He10}}}
\vspace{1ex}
If $\widehat{f} (\gamma) =0$ for all geodesics~$\gamma$ disjoint from
a ball $B \subset X$ then $f(x) =0$ for $x \not\in B$.
\end{theorem*}
This last result requires stronger decay assumption
at~$\infty$ than the injectivity result does.
Here we shall prove an explicit inversion formula for the X-ray
transform for
rank~$X>1$. See \S\ref{sec:5} for the contact with Rouvi\`ere's
different solution.
Funk \cite{F} and Radon \cite{R} inverted this transform for the
sphere $\SS^2$ and $\mathbf{R}^2$. Denoting the set of geodesics by
$\Xi$ we have the coset space representations
\begin{eqnarray*}
\SS^2 = \mathbf{O} (3) /\mathbf{O} (2) \, , & \Xi = \mathbf{O} (3) / \mathbf{O} (2) \mathbf{Z}^2\\
\mathbf{R}^2 = \mathbf{M} (2) /\mathbf{O} (2) \, , & \Xi = \mathbf{M} (2) /\mathbf{M} (1) \mathbf{Z}_2
\end{eqnarray*}
$\mathbf{M} (n) $ denoting the isometry group of $\mathbf{R}^n$.
This suggests the following generalization. Let $X=G/K$ and
$\Xi =G/H$ be coset spaces of the same locally compact group
$G$, $K$ and $H$ being closed subgroups. Here it will be
convenient to assume all these groups as well as $L=K \cap H$
to be unimodular. We do not assume the elements $\xi \in \Xi$
to be subsets of $X$ but instead use Chern's concept of incidence:
\begin{displaymath}
x= gK \hbox{ is incident to } \xi =\gamma H
\end{displaymath}
if $gK \cap \gamma H \neq \emptyset$ as subsets of $G$. Given
$x \in X$, $\xi \in \Xi$ define
\begin{eqnarray*}
\bigcheck{x} &=&\{ \xi = \Xi : x\, ,\,
\xi\hbox{ incident }\}\, ,\\
\widehat{\xi} &=& \{ x \in X : x,\xi
\hbox{ incident } \} \, .
\end{eqnarray*}
These are orbits of certain subgroups of $G$ and have natural
measures $dm$, $d\mu$ (up to factors) and we define the abstract
Radon transform $f \to \widehat{f}$ and its dual $\varphi \to
\checkvarphi$ by
\begin{equation}
\label{eq:2.2}
\widehat{f} (\xi) = \int_{\widehat{\xi}} f(x) \, dm (x) \, , \quad
\checkvarphi (x) =\int_{\bigcheck{x}} \varphi (\xi) \,
d \mu (\xi) \, .
\end{equation}
The normalizations of $dm$ and $d\mu$ are unified by taking $x_0
=eK$, $\xi_0 =eH$ and
\begin{equation}
\label{eq:2.3}
\widehat{f} (\gamma H) =\int_{H/L} f (\gamma h \cdot x_0)\, dh_L \,
, \quad
\checkvarphi(gK) =\int_{K/L} \varphi (gk \cdot \xi_0)
\, dk_L
\end{equation}
the invariant measures $dh_L$, $dk_L$ being fixed by Haar
measures of $H$, $K$ and $L$.
\subsection*{Main Problems:}
\romanparenlist
\begin{enumerate}
\item
Injectivity of $f \to \widehat{f}$, $\varphi \to \checkvarphi$.
\item
Inversion formulas.
\item
Range and kernel question for these transforms.
\item
Applications elsewhere.
\end{enumerate}
An easy general result relevant to problem~(iii) is the
following. For a suitable normalization of the measures $dx=
dg_K$, $d\xi = d\gamma_H$ we have
\begin{equation}
\label{eq:2.4}
\int_X f(x) \checkvarphi (x) \, dx =
\int_{\Xi} \widehat{f} (\xi) \varphi (\xi) \, d\xi \, ,
\end{equation}
a result which suggests the extension of (\ref{eq:2.3}) to
distributions.
\section{$d$-planes in $\mathbf{R}^n$}
\label{sec:3}
Here we consider the space $X=\mathbf{R}^n$ and $\Xi = \mathbf{G} (d,n)$ the
set of $d$-dimensional planes in $\mathbf{R}^n$. These are both
homogeneous under the group $G=\mathbf{M} (n)$. Fix $x_0 \in \mathbf{R}^n$,
$\xi_0 \in \mathbf{G} (d,n)$ at distance $d(x_0 \, , \, \xi_0)=p$. Then
we have
\begin{equation}
\label{eq:3.1}
X=\mathbf{R}^n =\mathbf{M} (n) / K_p \, , \quad
\Xi = \mathbf{G} (d,n) = \mathbf{M} (n) / H_p\, ,
\end{equation}
where $K_p$ and $H_p$, respectively, are the stability groups of
$x_0$ and $\xi_0$. Since various $p$ will be considered the
transforms (\ref{eq:2.2}) will be denoted by $\widehat{f}_p$ and
$\sucheck{\varphi}_p$. Since the action of $\mathbf{M} (n) $ on $X$ and
$\Xi$ is quite rich it turns out that for the coset space
representation~(\ref{eq:3.1})
\begin{displaymath}
z \in X \hbox{ is incident to } \eta \in \Xi \Leftrightarrow
d(z,\eta) =p \, .
\end{displaymath}
Thus the transform $\widehat{f}_p$ and $\checkvarphi_p$ can be
written
\begin{equation}
\label{eq:3.2}
\widehat{f}_p (\xi) = \int_{d(x,\xi)=p} f(x) \, dm (x) \, , \,
\checkvarphi_p (x) =\int_{d(x,\xi)=p}
\varphi (\xi) \, d\mu (\xi)\, .
\end{equation}
In particular, $\widehat{f}_0$ is the usual $d$-plane transform
$\widehat{f}$, but in order to invert it we need $\checkvarphi_p$ for
variable $p$. One of several versions of the inversion formula
is the following (\cite{He9}, \cite{He11}):
\begin{equation}
\label{eq:3.3}
f(x) = c(d) \left[ \left( \frac{d}{d(r^2)} \right)^d
\int^{\infty}_r p(p^2-r^2)^{\frac{d}{2}-1}
(\widehat{f})^{\vee}_p (x) \, dp
\right]_{r=0}
\end{equation}
with $c(d)$ and constant. Note that $(\widehat{f})^{\vee}_p (x)$ is
the average of the integrals of $f$ over all $d$-planes at
distance $p$ from $x$.
For $d=1$ this formula reduces to
\begin{equation}
\label{eq:3.4}
f(x) = - \frac{1}{\pi} \int^{\infty}_0 \,
\frac{d}{dp} ((\widehat{f})^{\vee}_p (x)) \,
\frac{dp}{p} \, ,
\end{equation}
which for $n=2$ coincides with Radon's original formula. Radon's
proof is very elegant and is based on an exhaustion of the
exterior $|x| >r$ by lines. As far as I know this proof has not
been extended to higher dimensions~$n$. Formula~(\ref{eq:3.4})
for $n>2$ is crucial for the inversion of (\ref{eq:2.1}) given in
Theorem~5.1.
\section{$d$-dimensional totally geodesic submanifolds in
hyperbolic space $\mathbf{H}^n$}
\label{sec:4}
A similar method works here and the analog of (\ref{eq:3.3}) is
the formula
\begin{equation}
\label{eq:4.1}
f(x) = C (d) \left[\left( \frac{d}{d(r^2)} \right)^d
\int^{\infty}_r (t^2-r^2)^{\frac{d}{2}-1} t^d
(\widehat{f})^{\vee}_{s(t)} (x) \, dt \right]_{r=1}\, ,
\end{equation}
where $C (d)$ is a constant and $s(p) = \cosh^{-1} (p)$
(\cite{He9}, \cite{He11}). Other versions of the inversion exist
(e.g,~\cite{He12} and \cite{BC}). For $d=1$ this reduces to
\begin{equation}
\label{eq:4.2}
f(x) = - \frac{1}{\pi} \int^{\infty}_0 \,
\frac{d}{dp} ((\widehat{f})^{\vee}_p (x))\,
\frac{dp}{\sinh p}
\end{equation}
a formula which for $n=2$ is stated without proof in Radon \cite{R}.
\section{X-ray inversion on the symmetric space $X=G/K$}
\label{sec:5}
In communication from 2003, Rouvi\`ere proved an extension of
formula~(\ref{eq:4.2}) to symmetric spaces~$X$ of rank $\ell
=1$. Inspired by his methods, I proved the inversion formula (\ref{eq:5.2}) for
the X-ray transform for $X$ of rank~$\ell >1$. Then Rouvi\`ere
\cite{Ro2} extended his formula to $X$ of arbitrary rank~$\ell$.
Actually he has several such formulas but they are all different
from the formula~(\ref{eq:5.2}) below.
Fix a flat totally geodesic submanifold $E$ of $X$ with $\dim E
=\ell >1$\break ($\ell$ the rank of $X$) passing through the origin
$o=eK$ of $X$. Let $p>0$ and $S=S_p(o)$ be the sphere in $E$
with radius~$p$ and center~$o$. The geodesics $\gamma$ in $E$
tangent to $S$ are permuted transitively by the orthogonal group
$\mathbf{O} (E)$. Let $du$ and $dk$ denote the normalized Haar measures
on $U$ and $K$. The spaces $k \cdot E$ as $k$ runs through $K$
constitute all flat totally geodesic subspaces of $X$ through $o$
of dimension~$\ell$. Thus the images $k \cdot \gamma$ ($k \in
K$, $\gamma$ tangent to $S$) constitute the set $\Gamma_p$ of all
geodesics $\gamma$ in $X$ lying in some flat $\ell$-dimensional
totally geodesic submanifold of $X$ through $o$ and $d(o,\gamma)=p$. The set
$\Gamma_p$ has a natural measure $\omega_p$ given by the
functional
\vspace{-.5ex}
\begin{equation}
\label{eq:5.1}
\omega_p =\varphi \to \int_K \biggl( \int_{\mathbf{O} (E)}
\varphi (k(u \cdot \gamma)) \, du \biggr) \, dk \, .
\end{equation}
\begin{theorem}
\label{th:5.1}
The X-ray transform (2.1) on a symmetric space
$X=G/K$ of rank $\ell >1$ is inverted by the formula
\begin{equation}
\label{eq:5.2}
f(o) =- \frac{1}{\pi} \int^{\infty}_0 \biggl( \frac{d}{dp}
\int_{\Gamma_p} \widehat{f} (\gamma) \, d\omega_p (\gamma)\biggr)
\frac{dp}{p}\, , \quad f \in \D (X) \, .
\end{equation}
\end{theorem}
Since $\Gamma_p$ and $d\omega_p$ are $K$-invariant the formula holds
at each point $x$ by replacing $f$ by $f \circ g$ where $g \in G$
is such that $g \cdot o=x$.
\begin{proof}
First assume $f$ to be $K$-invariant and consider the
restriction $f|E$. Fix an orthonormal frame $H_0,H \in E_0$,
the tangent space to $E$ at $o$, consider the one parameter
subgroups $\exp tH_0$, $\exp tH$ and the geodesic $\gamma_0
(t) = \exp tH_0 \cdot o$. Then the geodesic $\gamma (t) =
\exp pH \cdot \gamma_o (t)$ lies in $E$ and is tangent to
$S_p (o)$. Because of (\ref{eq:3.4}) we have
\begin{equation}
\label{eq:5.3}
f(o) =- \frac{1}{\pi} \int^{\infty}_0 \frac{d}{dp}\,
(\widehat{f})^E_p (o) \frac{dp}{p}\, ,
\end{equation}
where the superscript $E$ stands for the dual transform on
geodesics in the space~$E$. Thus
\begin{equation}
\label{eq:5.4}
(\widehat{f})^E_p (o) = \int\limits_{\substack{\gamma \subset E \\ d(o,\gamma)=p}}
(\widehat{f})(\gamma)\, d\nu (\gamma) =
\int_{\mathbf{O} (E)} (\widehat{f}) (u \cdot \gamma)\, du \, ,
\end{equation}
where $\nu$ stands for the average over the set of
geodesics tangent to $S_p(o)$.
For $f \in \D (X)$ arbitrary we use (\ref{eq:5.2}) on the
function
\begin{displaymath}
f^{\sharp} (x) = \int_K f(k \cdot x)\, dk \, .
\end{displaymath}
Taking into account the definition (\ref{eq:5.1}) the inversion
formula (\ref{eq:5.2}) follows immediately.
\end{proof}
\begin{remark}
Note that the measure $\omega_p$ in (\ref{eq:5.1}) is a kind of
convolution of the Haar measures $dk$ and $du$. However it is
not a strict convolution since the product $ku$ is not defined.
\end{remark}
\section{The horocycle transform in $X=G/K$}
\label{sec:6}
Consider the usual Iwasawa decomposition of $G$, $G= NAK$ where
$N$ and $A$ are nilpotent and abelian, respectively. A
\emph{horocycle} is by definition (\cite{GG}) an orbit in $X$ of a
conjugate $gNg^{-1}$ of $N$. The group $G$ permutes the
horocycles transitively and the space $\Xi$ of horocycles can
be written $\Xi = G/MN$ where $M$ is the centralizer of $A$ in
$K$. In the double fibration
\begin{center}
\begin{picture}(90,50)
\put(-35,0){$X=G/K$}
\put(45,0){$G/MN =\Xi$}
\put(5,40){$G/M$}
\put(-10,15){\line(2,3){15}}
\put(25,35){\line(2,-3){15}
\end{picture}
\end{center}
it turns out that $x=gK$ is incident to $\xi = \gamma MN$ if and
only if $x \in \xi$. The transforms (\ref{eq:2.2}) become
\begin{equation}
\label{eq:6.1}
\widehat{f} (\gamma MN) =\int_N f(\gamma n \cdot o)\, dn \, , \quad
\checkvarphi (gK) = \int_K \varphi (gk \cdot \xi_0)\, dk\, ,
\end{equation}
where $\xi_o = N \cdot o$. While the map $\varphi \to
\checkvarphi$ has a big kernel, the horocycle transform $f \to
\widehat{f}$ is injective (\cite{He1} or \cite{GK}). The following
result (\cite{He5}) is considerably stronger.
\begin{theorem}{(Support theorem.)}
\label{th:6.1}
Let $B$ be a closed ball in $X$. Then
\begin{eqnarray*}
\begin{array}{ll}
\widehat{f} (\xi) =0 &\hbox{for } \xi \cap B =\emptyset
\hbox{ implies}\\
f(x) =0 & \hbox{for } x \not\in B \, .
\end{array}
\end{eqnarray*}
\end{theorem}
Here one requires stronger decay conditions on $f$ than for the
injectivity. A different proof was given in \cite{GQ}. We have also the following inversion and Plancherel
formula for the Radon transform (\cite{He2,He3}). The
pseudodifferential operator $\Lambda$ and the differential
operator~$\square$ below are constructed by means of the Harish--Chandra
$c$-function, and $w$ denotes the order of the Weyl group. For
$G$ complex a result similar to (\ref{eq:6.2}) appears in \cite{GG}.
\begin{theorem}
\label{th:6.2}
\begin{trivlist}{}{}
\item
For $f \in \D (X)$ or sufficiently rapidly decreasing we have the
inversion formula
\begin{equation}
\label{eq:6.2}
f=\frac{1}{w} (\Lambda \bar{\Lambda}\widehat{f})^{\vee} \, .
\end{equation}
\item
If all Cartan subgroups of $G$ are conjugate the formula has the
improved version
\begin{displaymath}
f=\frac{1}{w} \square ((\widehat{f})^{\vee})\, .
\end{displaymath}
\item
For $G$ arbitrary
\begin{equation}
\label{eq:6.3}
w \int_X | f(x) |^2 \, dx = \int_{\Xi} |\Lambda \widehat{f}|^2
(\xi)\, d\xi\, ,
\end{equation}
with a suitable normalization of the invariant measures $dx$ and
$d\xi$.
\end{trivlist}
\end{theorem}
The range question~(iii) for $f \to \widehat{f}$ is more
complicated. Consider first the hyperbolic plane $\mathbf{H}^2$ in the
Poincar\'e unit disk model $D$. Here the horocycles are the
circles in the disk tangential to the boundary $\{ e^{i\theta}:
\theta \in \mathbf{R} \}$. Let $\xi_{t,\theta}$ denote the horocycle
through $e^{i\theta}$ with distance $t$ (with sign) from the
origin. Then we have the following result (\cite{He7}).
\begin{theorem}
\label{th:6.3}
The range $\D (D)^{\widehat{}}$ consists of the functions $\psi \in \D
(\Xi)$
\begin{displaymath}
\psi (\xi_{t,\theta}) =\sum_n \psi_n (t) e^{in\theta}
\end{displaymath}
where
\begin{equation}
\label{eq:6.4}
\psi_n (t) = e^{-t} \biggl(\frac{d}{dt}-1\biggr) \cdots
\biggl(\frac{d}{dt}-2|n| +1\biggr)\varphi_n (t)
\end{equation}
where $\varphi_n \in \D (D)$ is even.
\end{theorem}
This implies a relationship between $\psi (\xi_{t,\theta})$ and
$\psi (\xi_{-t,\theta})$. More specifically, if $f'(t) = f(-t)$,
$\Psi_n = e^t \psi_n$ then $*$ denoting convolution on $\mathbf{R}$
\begin{displaymath}
\Psi'_n = S_n * \Psi_n
\end{displaymath}
where the distribution $S_n$ on $\mathbf{R}$ has Fourier transform
\begin{displaymath}
\widehat{S}_n = \frac{(i\lambda +1)\ldots (i\lambda +2|n|-1)}
{(i\lambda -1)\ldots (i\lambda -2|n|+1)}\, ,
\quad \lambda \in \mathbf{R}\, .
\end{displaymath}
This relationship between $\psi (\xi_{-t,\theta})$ and $\psi
(\xi_{t,\theta})$ implies that in (\ref{eq:6.3}) $f \to \Lambda
\widehat{f}$ does not map $L^2 (X)$ onto $L^2 (\Xi)$.
For the generalization of (\ref{eq:6.4}) to $X=G/K$ we need some
additional notation. Let $\widehat{K}$ be the unitary dual of $K$
and $d(\delta)$ the degree of a $\delta \in \widehat{K}$. Given
$\delta$ acting on $V_{\delta}$ let
\begin{displaymath}
V^M_{\delta} = \{ v \in V_{\delta} : \delta (m) v =v
\hbox{ for } m \in M \}
\end{displaymath}
and put $\ell (\delta)=\dim V^M_{\delta}$. Let
\begin{displaymath}
\widehat{K}_M = \{ \delta\in \widehat{K} : \ell (\delta) >0 \} \, .
\end{displaymath}
In the following theorem \cite{He9} the expansion (\ref{eq:6.5})
is a generalization of (\ref{eq:6.4}). Put $\rho (H)
=\frac{1}{2}\mathop{\rm Trace\,}\nolimits (\mathop{\rm ad\,}\nolimits H |{\mathfrak n} )$.
\begin{theorem}
\label{th:6.4}
The range $\D (X)\sphat$ consists of the functions $\psi \in \D
(\Xi)$
\begin{equation}
\label{eq:6.5}
\psi (ka \cdot \xi_0) = \sum_{\delta \in \widehat{K}_M} d(\delta)
\mathop{\rm Tr\,}\nolimits (\delta (k) \Psi_{\delta}(a)) \qquad (\mathop{\rm Tr\,}\nolimits = \mathop{\rm Trace\,}\nolimits)
\end{equation}
where $\Psi_{\delta}$ is a function on $A$ with values in $\mathop{\rm Hom\,}\nolimits
(V_{\delta}, V^M_{\delta})$, i.e.,~$\Psi_{\delta} \in \D (A,\mathop{\rm Hom\,}\nolimits
(V_{\delta},V^M_{\delta}))$, given by
\begin{equation}
\label{eq:6.6}
\Psi_{\delta} (a) = e^{-\rho (\log a)} Q^{\delta}(D)_a
(\Phi_{\delta}(a)) \qquad a \in A
\end{equation}
where
\begin{equation}
\label{eq:6.7}
\Phi_{\delta} \in \D (A,\mathop{\rm Hom\,}\nolimits (V_{\delta},V^M_{\delta}))
\end{equation}
is $W$-invariant and $Q^{\delta}(D)$ is a certain $\ell (\delta )
\times \ell (\delta)$ matrix of constant coefficient differential
operators on $A$.
\end{theorem}
From this result we can derive the following (unpublished)
refinement of the support theorem above. Let $A^+$ be the Weyl
chamber corresponding to the choice of the group~$N$.
\begin{theorem}
\label{th:6.5}
Suppose $f \in \D (X)$ satisfies
\begin{displaymath}
\widehat{f} (ka \cdot \xi_0)=0 \hbox{ for }
k \in K \, ,\, a \in A^+\, , \, |\log a |>R \, .
\end{displaymath}
Then
\begin{displaymath}
\widehat{f} (ka \cdot\xi_0) =0 \hbox{ for } k \in K\, ,\,
|\log a |>R \, ,\, a \in A
\end{displaymath}
so by Theorem~\ref{th:6.1}
\begin{displaymath}
f(x) =0 \hbox{ for } d (0,x)>R\, .
\end{displaymath}
\end{theorem}
\begin{proof}
Let $Q_c (D)$ be the matrix of cofactors of $Q^{\delta} (D)$ so
that
\begin{equation}
\label{eq:6.8}
Q_c (D) Q^{\delta}(D) = \det Q^{\delta}(D) I \, .
\end{equation}
Then (\ref{eq:6.6}) implies
\begin{equation}
\label{eq:6.9}
Q_c (D) (e^{\rho} \Psi_{\delta}) =\det Q^{\delta}(D)
\Phi_{\delta}\, .
\end{equation}
Now it is known (\cite{K}, \cite{He9} pp.~267, 348) that $\det
Q^{\delta}(D)$ is a product of linear factors $\delta (H_i)+c$
where $H_i \in {\mathfrak a}$ and $\partial (H_i)$ the corresponding
directional derivative.
Suppose the function $\psi = \widehat{f}$ satisfies
\begin{displaymath}
\psi (ka \cdot \xi_0)=0 \hbox{ for } k\in K \, , \,
a \in A^+ \, , \, |\log a| >R \, .
\end{displaymath}
Since
\begin{displaymath}
\Psi_{\delta}(a) = \int_K \psi (ka \cdot \xi_0)
\delta (k^{-1}) \, dk
\end{displaymath}
we deduce from (\ref{eq:6.8}) and (\ref{eq:6.9}) that
\begin{displaymath}
\det Q^{\delta} (D) \Phi_{\delta} (a) =0 \hbox{ for }
a \in A^+ \, , \, |\log a |>R \, .
\end{displaymath}
Consider this equation on a ray in $A^+$ starting at $e$.
Because of the mentioned factorization of $\det Q^{\delta}(D)$ we
deduce that on this ray $\Phi_{\delta}$ satisfies an ordinary
differential equation on the interval $(R,\infty)$. Having
compact support we deduce that $\Phi_{\delta}(a) =0$ for $a \in
A^+$, $|\log a|>R$. By its Weyl group invariance it vanishes for
all $a \in A$, $|\log a |>R$ which by (\ref{eq:6.5}) proves the theorem.
\end{proof}
Consider the case rank $X=1$. Let $B_R (o)$ be a ball in $X$
with radius $R$ and center $0$. Fix a unit vector $H$ in the
Lie algebra of $A$ such that $\exp H \in A^+$. Put $a_t=\exp
tH$. The \emph{interior} of the horocycle $k Na_t \cdot o$ is
the union $\bigcup_{\tau >t} k N a_{\tau} \cdot o$. A horocycle
$\xi$ is said to be \emph{external} to $B_R(o)$ if its interior
is disjoint from $B_R(o)$; $\xi$~is said to
\emph{enclose} $B_R (o)$ if its interior contains $B_R (o)$.
\begin{corollary}
\label{cor:6.6}
Let $X$ have rank one and $B_R (o)$ as above. Let $f \in \D
(X)$. Then the following are equivalent:
\begin{list}{}{}
\item (i)~~$\widehat{f} (\xi)=0$ whenever $\xi$ is external to
$B_R(o)$.
\item (ii)~~$\widehat{f} (\xi) = 0$ whenever $\xi$ encloses $B_R
(o)$.
\item (iii)~$f \equiv 0$ outside $B_R (o)$.
\end{list}
\end{corollary}
For hyperbolic space this is clear from Theorem~\ref{th:6.3} and
was proved in a different way by Lax--Phillips \cite{LP}.
Problem (iii) for the dual transform $\varphi \to \checkvarphi$
has a satisfactory answer (see \cite{He9} IV \S\S 2 and 4). The
kernel can be described in the spirit of Theorem~\ref{th:6.5} and
for the range one has the surjectivity
\begin{equation}
\label{eq:6.10}
\E (\Xi)^{\vee} = \E (X)\, .
\end{equation}
\section{The Abel transform}
\label{sec:7}
Let $\D_K (X)$ denote the space of $K$-invariant functions in
$\D(X)$. The \emph{Abel transform} $f \to \A f$ is defined by
\begin{equation}
\label{eq:7.1}
(\A f) (a) = e^{\rho (\log a)} \int_N f(an \cdot o)\, dn\, ,
\quad a \in A \, , f \in \D_K (X) \, .
\end{equation}
Except for the factor $e^{\rho}$ it is the restriction of the
Radon transform to $K$-invariant functions
\begin{equation}
\label{eq:7.2}
\A f = e^{\rho} \widehat{f} \, .
\end{equation}
Some of its properties are best analyzed by means of the
spherical functions
\begin{equation}
\label{eq:7.3}
\varphi_{\lambda}(g) =\int_K e^{(i\lambda -\rho)(H(gk))}
\, dk \, , \quad g \in G \, , \, \lambda \in {\mathfrak a}^*_c\, ,
\end{equation}
where $H(g) \in {\mathfrak a}$ is determined by $g \in k \exp H(g) N$ and
${\mathfrak a}^*_c$ is the complex dual of ${\mathfrak a}$. The \emph{spherical
transform}
\begin{equation}
\label{eq:7.4}
\widetilde{f} (\lambda)=\int_X f(x) \varphi_{-\lambda}(x)\, dx
\quad f \in \D_K (X)
\end{equation}
(where $\varphi_{\lambda}(g \cdot o) = \varphi_{\lambda}(g)$) is
a homomorphism relative to convolution $\times$ on $X$:
\begin{equation}
\label{eq:7.5}
(f_1 \times f_2)^{\sim}(\lambda) = \widetilde{f}_1 (\lambda )
\widetilde{f}_2 (\lambda )\, .
\end{equation}
As proved in \cite{H}, $\A$ intertwines the spherical transform
and the Euclidean Fourier transform $F \to F^*$ on $A$ so
\begin{equation}
\label{eq:7.6}
\int_A (\A f) (a) e^{-i\lambda (\log a)}\, da =
\int_X \varphi_{-\lambda}(x)f(x)\, dx \, ,
\quad (\A f)^* =\widetilde{f} \, .
\end{equation}
Thus $\A f$ is $W$-invariant and by (\ref{eq:7.5})
\begin{equation}
\label{eq:7.7}
\A (f_1 \times f_2) = \A f_1 * \A f_2 \, ,
\end{equation}
where $*$ is convolution on $A$. Let $\mathbf{D} (X)$ denote the
algebra of $G$-invariant differential operators on $X$ and
$\Gamma : \mathbf{D} (X) \to \mathbf{D}_W (A)$ the isomorphism onto the
$W$-invariant constant coefficient differential operators on~$A$.
The Abel transform is a simultaneous transmution operator between
$\mathbf{D} (X)$ and $\mathbf{D}_W(A)$, i.e.,
\begin{equation}
\label{eq:7.8}
\A Df = \Gamma (D) \A f \, , \quad D \in \mathbf{D} (X) \, , \,
f \in \D_K (X)
\end{equation}
(\cite{He13}) which for example can be used to prove that each
$D$ has a fundamental solution. By the Paley--Wiener theorem for
(\ref{eq:7.3}) one has that $\A :\D_K(X) \to \D_W (A)$ is a
bijective homeomorphism. (Here the subscript $W$ means
$W$-invariance.) Hence we have a bijection
\begin{equation}
\label{eq:7.9}
\A^* : \D'_W (A) \to \D'_K (X)
\end{equation}
between the corresponding distribution spaces. Also if $\varphi
\in \E_W (A)$ we have easily (\cite{Be} or \cite{He9} IV, \S4)
\begin{equation}
\label{eq:7.10}
(\A^* \varphi) (gK) = \int_{K/M} \varphi (\exp H(gk))
e^{-\rho (H(gk))} \, dk \, .
\end{equation}
The Radon transform has the advantage over $\A$ that it commutes
with the action of $G$. Thus we can deduce from (\ref{eq:6.10})
and (\ref{eq:7.2}) that (\cite{He9})
\begin{equation*}
\A^* \E_W (A) = \E_K (X) \, .
\end{equation*}
We now add a few new results about $\A$ and $\A^*$ which will be
useful later. Some are closely related to rank-one results in
Bagchi and Sitaram in \cite{BS}.
Because of the convolution property (\ref{eq:7.7}) one can ask
how $\A^*$ behaves relative to convolution. Let $L$ be the
operator on $\s (A)$ given by
\begin{equation}
\label{eq:7.11}
(L \varphi)^* (\lambda) = |c(\lambda)|^{-2} \varphi^*
(\lambda), \quad \lambda \in {\mathfrak a}^* \, ,
\end{equation}
where $c(\lambda)$ is Harish--Chandra's $c$-function.
\begin{theorem}
\label{th:7.1}
Let $\varphi \in \D_W (A)$, $\psi \in \E_W (A)$. Then
\begin{displaymath}
\A^* (L\varphi) =w \,\, \A^{-1}(\varphi) \qquad
(w = \hbox{ order of }W)
\end{displaymath}
and
\begin{equation}
\label{eq:7.12}
\A^* (\varphi * \psi) = \frac{1}{w}\A^* (L\varphi)
\times \A^* \psi \, .
\end{equation}
\end{theorem}
\begin{proof}
Using the inversion formula for the spherical transform we have
\begin{eqnarray*}
\A^* (L\varphi)(gK) &=& \int_K (L\varphi)
(\exp H(gk))^{-\rho (H(gk))}\, dk\\
&=& \int_K \biggl(\int_{{\mathfrak a}^*} (L\varphi)^* (\lambda)
e^{i\lambda (H(gk))}\, d\lambda \biggr)
e^{-\rho (H(gk))}\, dk\\
&=& \int_{{\mathfrak a}^*} |c(\lambda)|^{-2} \varphi^*_{\lambda}
(g) \, d\lambda =F(gK)\, , \\
\end{eqnarray*}
where $\widetilde{F} (\lambda) = \varphi^* (\lambda) w$. But
$\widetilde{F} = (\A F)^* =\varphi^* w$ so
\begin{displaymath}
\varphi = \frac{1}{w}\A F \, , \quad F=w \A^{-1}(\varphi)\, .
\end{displaymath}
Thus $\A^* (L\varphi) = w \A^{-1}(\varphi)$. Consider now the average
\begin{displaymath}
\psi^{\lambda}(a) = \frac{1}{w}\sum_{s \in W}
e^{is\lambda (\log a)}\, .
\end{displaymath}
Then $ \A^* \psi^{\lambda}=\varphi_{\lambda}$ and
$\varphi * \psi^{\lambda}=\varphi^* (\lambda)\psi^{\lambda}$. But $\A^{-1}\varphi = \frac{1}{w} F \in \D_K(X)$ and
\begin{displaymath}
\A^{-1} \varphi \times \varphi_{\lambda} =\frac{1}{w}
\widetilde{F} (\lambda) \varphi_{\lambda} =\varphi^*
(\lambda) \varphi_{\lambda}\, .
\end{displaymath}
Combining these formulas we have
\begin{equation}
\label{eq:7.13}
\A^* (\varphi * \psi^{\lambda}) = \A^{-1}\varphi
\times \A^* (\psi^{\lambda})\, .
\end{equation}
Now $\psi \in \D_W (A)$ is a superposition
\begin{displaymath}
\psi (a) =\int_{{\mathfrak a}^*}\psi^* (\lambda)\psi^{\lambda}
(a) \, d\lambda
\end{displaymath}
so the identity (\ref{eq:7.12}) follows from (\ref{eq:7.13}) for
such $\psi$. For $\psi \in \E_W(A)$ the identity follows by an
approximation because $\varphi$ and $\A^* (L\varphi)$ have
compact support and $\A^*$ is continuous on $\E_W(A)$.
\end{proof}
Theorem~\ref{th:7.1} implies the following inversion formula
which in reality is a special case of (\ref{eq:6.2}). It
appears also in \cite{Be}.
\begin{corollary}
\label{cor:7.2}
The transform $f \to \A f$ has inversion
\begin{displaymath}
f=\frac{1}{w} \A^* (L\A f)\, \qquad f \in \D_K(X)\, .
\end{displaymath}
\end{corollary}
The above results suggest various ways of defining $\A$ on the
space $\E'_K(X)$ of $K$-invariant compactly supported
distributions on $X$ although formula (\ref{eq:7.1}) does not
work.
\begin{trivlist}{}{}
\item
\emph{Spherical transform method.}\quad If $T \in \E'_K(X)$, the
spherical transform
\begin{displaymath}
\widetilde{T} (\lambda) = \int_X \varphi_{-\lambda} (x)
\, dT (x)
\end{displaymath}
is a $W$-invariant entire function of exponential type on
${\mathfrak a}^*_c$ and of polynomial growth. (See \cite{EHO} or
\cite{He5}, Theorem~8.5.) By the Euclidean
Paley--Wiener theorem there exists an $S \in \E'_W (A)$ such that
$\widetilde{T} =S^*$. Thus in accordance with (\ref{eq:7.6}) we put
\begin{equation}
\label{eq:7.14}
\A T=S \, .
\end{equation}
\item
\emph{Radon transform method.}\quad Because of (\ref{eq:2.4}) the
Radon transform of a distribution $T \in \E'(X)$ is defined by
\begin{displaymath}
\widehat{T} (\varphi) = T(\checkvarphi)\qquad \varphi \in \E
(\Xi)\, .
\end{displaymath}
If $T$ is $K$-invariant then so is $\widehat{T}$ and since $\Xi
=K/M \times A$ under the bijection $(kM,a) \to ka \cdot
\xi_o$ we see that $\widehat{T}$ has the form $\widehat{T} = 1
\otimes \sigma$ where $\sigma \in \E'(A)$. Because of (\ref{eq:7.2}) we put
\begin{equation}
\label{eq:7.15}
\A T = e^{\rho} \sigma \, .
\end{equation}
\item
\emph{Functional analysis method.}\quad As remarked $\A^*$ is a
bijection of $\D'_W (A)$ onto $\D'_K(X)$. The restriction of
$\A^*$ to $\E_W(A)$ is a continuous bijection onto $\E_K(X)$ and
in fact a homeomorphism since both spaces are Fr\'echet. Thus we
have $(\A^*)^* : \E'_K(X) \to \E'_W (A)$ bijectively so we can
define
\begin{equation}
\label{eq:7.16}
\A T = (\A^*)^* (T) \, .
\end{equation}
\end{trivlist}
\begin{proposition}
\label{prop:7.3}
All the definitions (\ref{eq:7.14})--(\ref{eq:7.16}) coincide.
\end{proposition}
The convolution property in Theorem~\ref{th:7.1} extends readily
to distributions so
\begin{eqnarray*}
\A^* (\E'_W (A)*\psi) &=& \A^{-1}(\E'_W (A))
\times \A^* \psi\\
&=& \E'_K(X) \times \A^* \psi \, .
\end{eqnarray*}
Thus putting
\begin{eqnarray*}
V_{\psi} = \E'_W(A) * \varphi \, , \quad
\psi \in \E_W (A)\, ; \quad
W_f = \E'_K(X) \times f \, , \quad
f \in \E_K (X)
\end{eqnarray*}
we conclude that
\begin{equation}
\label{eq:7.17}
\A^* (V_{\psi}) = W_{\A^* \psi} \, .
\end{equation}
\begin{theorem}[(Bagchi--Sitaram)]
\label{th:7.4}
If $X$ has rank one and $f \in \E_K(X)$ then the closure of the
space $W_f =\E'_K(X) \times f$ contains a spherical function.
\end{theorem}
The authors use (\ref{eq:7.17}) to reduce the question to the
analogous one for the one-dimensional space $A \sim \mathbf{R}$ where by
Schwartz's theorem stated in \S9 below some exponentials $e^{i\mu}$ and $e^{-i\mu}$
belong to the closure and $\A^* (e^{i\mu} + e^{-i\mu}) = 2\varphi_{\mu}$.
\section{The Fourier transform on $X=G/K$}
\label{sec:8}
We now go to the notation of \S6 with the Iwasawa decomposition
$G= NAK$, and ${\mathfrak g} ={\mathfrak n} +{\mathfrak a} +{\mathfrak k}$ for the corresponding Lie
algebras. For $g \in G$ let $A(g) \in {\mathfrak n}$ be determined by $g=n
\exp A(g)k$ $(n \in N , k \in K)$. Given $x=gK$ in $X$, $b=kM$
in $B=K/M$ we put
\begin{displaymath}
A(x,b) = A (k^{-1}g)
\end{displaymath}
and as usual we put $\rho (H) =\frac{1}{2}\mathop{\rm Trace\,}\nolimits (\mathop{\rm ad\,}\nolimits H|{\mathfrak n})$.
Let ${\mathfrak a}^*_c$ denote the space of complex-valued linear forms on
${\mathfrak a}$.
Given a function $f$ on $X$ we define its \emph{Fourier
transform} by
\begin{equation}
\label{eq:8.1}
\widetilde{f} (\lambda ,b) = \int_X f(x)
e^{(-i\lambda +\rho) (A (x,b))}\, dx
\end{equation}
for those $(\lambda ,b) \in {\mathfrak a}^*_c \times B$ for which the
integral is defined. Many of the principal theorems for Fourier
transforms on $\mathbf{R}^n$ have analogs for $X=G/K$.
\begin{trivlist}{}{}
\item \emph{Inversion Formula}(\cite{He3}).\quad For $f \in
\D(X)$ we have
\begin{displaymath}
f(x) =\frac{1}{w}\int_{{\mathfrak a}^*} \!\! \int_{B}\widetilde{f}
(\lambda,b)e^{(i\lambda +\rho) (A(x,b))}
|c(\lambda)|^{-2}\, d\lambda \, db
\end{displaymath}
where $c(\lambda)$ is Harish--Chandra's $c$-function.
\item \emph{Plancherel Formula}(\cite{He4}).\quad The map $f \to
\widetilde{f}$ extends to an isometry of $L^2(X)$ onto $L^2
({\mathfrak a}^*_+ \times B)$:
\begin{equation}
\label{eq:8.2}
\int_X |f(x)|^2 \, dx = \int_{{\mathfrak a}^*_+ \times B}
|\widetilde{f} (\lambda ,b|^2)|c(\lambda)|^{-2}\,
d\lambda \, db \, .
\end{equation}
\item \emph{Paley-Wiener Theorem}(\cite{He5}).\quad The map $f
\to \widetilde{f}$ maps the space $\D(X)$ onto the space of smooth
$\varphi (\lambda ,b)$ on ${\mathfrak a}^*_c \times B$ which are
holomorphic on ${\mathfrak a}^*_c$ of exponential type (uniformly in $B$)
satisfying the invariance condition
\begin{equation}
\label{eq:8.3}
\int_B \varphi (\lambda ,b)
e^{(i\lambda +\rho)(A(x,b))}\, db \,\,
\hbox{ is $W$-invariant in } \lambda \, .
\end{equation}
\end{trivlist}
For the next result we refer to Eguchi's paper for full
explanations of notation.
\begin{trivlist}{}{}
\item \emph{The Schwartz Theorem}(\cite{Eg}).\quad Let $0<p\leq2$
and $\s^p(X) \subset L^p(X)$ the corresponding Schwartz
space. Let $\epsilon = \frac{2}{p}-1$ and $\s ({\mathfrak a}^*_{\epsilon}
\times B)$ the space of functions which are holomorphic in the
``tube'' ${\mathfrak a}^*_{\epsilon} \times B$, are rapidly decreasing
and satisfy (\ref{eq:8.3}). Then $f \to \widetilde{f}$ is a
bijection of $\s^p(X)$ onto $\s({\mathfrak a}^*_{\epsilon} \times B)$.
\end{trivlist}
These results leave out the space $L^1 (X)$ and one should think
that a self-respecting Fourier transform should be defined here.
We shall now show (modifying a bit the proof of \cite{HRSS}) that this
can be done and that a strong analog of the classical
Riemann--Lebesgue lemma holds for $\widetilde{f}$ in (\ref{eq:8.1}).
Let $C(\rho)$ denote the convex hull in ${\mathfrak a}^*$ of the set $\{ s
\rho : s \in W \}$ of Weyl group transforms of $\rho$.
\begin{theorem}
\label{th:8.1}
Let $f \in L^1 (B)$. Then there exists a subset $B' \subset B$
with $B-B'$ of measure $0$ such that for each $b \in B'$
\begin{trivlist}{}{}
\item
(i)~~$\widetilde{f} (\lambda ,b)$ is defined for $\lambda$ in the tube
${\mathfrak a}^* +i C(\rho)$ and holomorphic in its interior.
\item
(ii)~~$\lim_{\xi \to \infty}\, \widetilde{f} (\xi + i \eta ,b)=0$
uniformly for $\eta \in C(\rho)$.
\end{trivlist}
\end{theorem}
\begin{proof}
Let $\lambda =\xi +i\eta$ where $\xi \in {\mathfrak a}^* \, , \, \eta \in C
(\rho)$. Then
\begin{equation}
\label{eq:8.4}
\int_B |\widetilde{f} (\lambda ,b)| \, db \leq \int_X |f(x)|
\int_B e^{(\eta + \rho)(A(x,b))}\, db \, dx \, .
\end{equation}
The integral over $B$ is the spherical function
$\varphi_{-i\eta}$ which is bounded by~$1$ (\cite{HJ}). Thus
\begin{displaymath}
\| \widetilde{f} (\lambda , \cdot) \|_1 \leq \| f \|_1
\end{displaymath}
and for each $\lambda \in {\mathfrak a}^* +i C(\rho)$,
$\widetilde{f} (\lambda ,b)$ exists for all $b$ in a subset
$B_{\lambda} \subset B$ of full invariant measure. Let
\begin{displaymath}
B' = B' (f) =\cap_{s \in W} B_{is\rho}\, .
\end{displaymath}
For the statements~(i) and (ii) we may assume $f \geq 0$ in
(\ref{eq:8.1}). Since $b \in B_{is\rho}$ for each $s \in W$ we
have
\begin{equation}
\label{eq:8.5}
\int_X f(x) e^{(s\rho +\rho)(A(x,b))} \, dx <\infty \, .
\end{equation}
Fix $b \in B'$, $\eta \in C(\rho)$. Then
\begin{equation}
\label{eq:8.6}
\int_X f(x) e^{(\rho + \eta) (A(x,b))} \, dx =
\sum_{\sigma \in W} \int_{X_{\sigma}} f(x)
e^{(\rho + \eta)(A(x,b))} \, dx \,
\end{equation}
where
\begin{displaymath}
X_{\sigma}=\{ x \in X : \sigma ( A(x,b)) \in \overline{{\mathfrak a}^+}\,
.
\end{displaymath}
Replace $\eta (A(x,b))$ by $(\sigma \eta) $ $(\sigma(A(x,b)))$ and
let $(\sigma \eta)^+$ be the element in $\overline{{\mathfrak a}^*_+}$,
which is $W$-conjugate to $\sigma \eta$. Then since $(\sigma
\eta)^+-\sigma \eta \geq 0$ on ${\mathfrak a}^+$ we have
\begin{displaymath}
\int_{X_{\sigma}} f(x) e^{(\rho +\eta)(A(x,b))} \, dx \leq
\int_{X_{\sigma}} f(x)
e^{(\rho + (\sigma \eta)^+)(\sigma (A(x,b)))}\, dx \, .
\end{displaymath}
Now by Lemma~8.3, Ch.~IV in \cite{He8}
\begin{displaymath}
\overline{{\mathfrak a}^*_+} \cap C(\rho) =
\overline{{\mathfrak a}^*_+} \cap (\rho + {}^- {\mathfrak a}^*) \, ,
\end{displaymath}
where
\begin{displaymath}
{}^-{\mathfrak a}^* = \{ \lambda \in {\mathfrak a}^* | \langle \lambda ,\mu \rangle
\leq 0 \hbox{ for } \mu \in {\mathfrak a}^*_+ \} \, .
\end{displaymath}
Thus
\begin{displaymath}
(\sigma \eta)^+ \in \overline{{\mathfrak a}^*_+} \cap
(\rho + {}^-{\mathfrak a}^*)\,,
\end{displaymath}
whence
\begin{equation}
\label{eq:8.7}
(\sigma \eta)^+ -\rho \leq 0 \hbox{ on } {\mathfrak a}^+ \, .
\end{equation}
Thus the last integral is bounded by
\begin{displaymath}
\int_{X_{\sigma}} f(x)
e^{(\rho +\sigma^{-1}\rho)(A(x,b))} \, dx <\infty
\end{displaymath}
by (\ref{eq:8.5}). This shows by (\ref{eq:8.6}) that if $b \in
B'$ and $\lambda \in {\mathfrak a}^* +i C (\rho)$ the integral
(\ref{eq:8.1}) is absolutely convergent. The holomorphy
statement follows by Morera's theorem. This proves~(i).
For part (ii) we use the Radon transform (\ref{eq:6.1}). Since
$f \in L^1(X)$, $\widehat{f} (\xi)$ exist for almost all $\xi \in
\Xi$ (\cite{He4}). Since $(kM,a) \to ka \cdot \xi_0$ is a
diffeomorphism of $K/M$ onto $\Xi$ we write $\widehat{f}(kM,a)$ for
$\widehat{f} (ka \cdot \xi_0)$. Enlarging $B'$ to another subset of
$B$ of full invariant measure we may assume $\widehat{f} (b,a)$
exists for $b \in B'$ and almost all $a$. Now we have
\begin{equation}
\label{eq:8.8}
\int_X f(x) \, dx = \int_{AN} f(anK)\, da \, dn
\end{equation}
for suitable Haar measures on $A$ and $N$. Applying this to the
function $x \to f(k \cdot x)$ with $kM =b \in B'$ we get
\begin{equation}
\label{eq:8.9}
\int_X f(x) \, dx =\int_A \widehat{f} (kM,a)\, da
\end{equation}
so since $A (an \cdot o)= \log a$,
\begin{eqnarray}
\label{eq:8.10}
\widetilde{f} (\lambda ,kM) &=& \int_A \widehat{f} (kM,a)
e^{(\rho +\eta) (\log a)} e^{-i\xi (\log a)}\, da\\
\nonumber
&=& \sum_{s \in W} \int_{s^{-1}A^+} \widehat{f}
(kM,a) e^{(\rho +\eta)(\log a)}
e^{-i\xi(\log a)}\, da\, .
\end{eqnarray}
Now $a \in s^{-1} A^+$ implies $sa \in A^+$ and
\begin{displaymath}
\eta (\log a) = (s\eta) (s\log a) \leq (s\eta)^+
(s\log a) \leq \rho (s \log a)
\end{displaymath}
by (\ref{eq:8.7}). Thus on $s^{-1} A^+$,
\begin{equation}
\label{eq:8.11}
\widehat{f} (kM,a) e^{(\rho + \eta)(\log a)} \leq
\widehat{f} (kM,a) e^{(\rho + s^{-1}\rho)(\log a)}\, .
\end{equation}
For $b=kM \in B'$ the integral in (\ref{eq:8.1}) is absolutely
convergent so by (\ref{eq:8.9}) the function
\begin{equation}
\label{eq:8.12}
a \to \widehat{f} (kM,a) e^{(\rho + \eta)(\log a)}
\end{equation}
belongs to $L^1 (A)$. The first part of (\ref{eq:8.10}) combined
with the Riemann--Lebesgue lemma for the Fourier transform on $A$
shows that for each $\eta \in C (\rho)$
\begin{displaymath}
\lim_{\xi \to \infty} \widetilde{f} (\lambda ,b)=0\, .
\end{displaymath}
For the uniform convergence in (ii) we use the second part of
(\ref{eq:8.10}). Let $f_n$ be positive in $\D (X)$ such that $f_n
\to f$ a.e. and $f_n (x) \leq f(x)$. In (\ref{eq:8.10}) and
(\ref{eq:8.11}) we replace $f$ by the function $g_n =f-f_n$.
Then
\begin{eqnarray*}
|\widetilde{g}_n (\lambda ,kM) | & \leqq & \sum_{s \in W}
\int_{s^{-1}A^+} \widehat{g}_n (kM,a)
e^{(\rho +\eta)(\log a)}\, da \\
&\leq & \sum_{s \in W} \int_{s^{-1}A^+}
\widehat{g}_n (kM,a) e^{(\rho +s^{-1}\rho)(\log a)}\,
da \, ,
\end{eqnarray*}
which tends to $0$ as $n \to \infty$ by (\ref{eq:8.9}),
(\ref{eq:8.12}) and the dominated convergence theorem. Thus
given $\epsilon >0$ we can fix $N$ such that $|\widetilde{g}_N
(\lambda ,kM)| <\epsilon$ for all $\lambda \in {\mathfrak a}^* +iC_{\rho}$. By
the Paley--Wiener theorem for $\D (X)$ there is an $L$ such that
$|\widetilde{f}_N (\xi +i\eta ,kM)|\leq \epsilon$ for $|\xi | >L$ and
$\eta \in C (\rho)$. Since $\widetilde{g}_N = \widetilde{f}-\widetilde{f}_N$
this proves (ii).
\end{proof}
\begin{remark}
Another version of (ii) involving the $L^1$ norm over $B$ is
given in \cite{SS}.
\end{remark}
\section{Spectral analysis on $X$}
\label{sec:9}
A theorem of Schwartz \cite{Sc} states that if $f$ is a function
in $\E (\mathbf{R})$ $(f \not\equiv 0)$ the closed subspace of $\E(X)$ (in
its usual Fr\'echet space topology) generated by all the
translates of $f$ contains an exponential $e^{\mu x}$ for some
$\mu \in \mathbf{C}$.
We shall now give the proof from \cite{HS} of the following
analog of Schwartz's theorem.
\begin{theorem}
\label{th:9.1}
Let $X =G/K$ have rank one and $f\neq 0$ a function in $\E(X)$.
Then the closed subspace $V_f$ of $\E(X)$ generated by the
$G$-translates of $f$ contains a function
\begin{displaymath}
x \to e_{\mu ,b} (x) = e^{\mu (A(x,b))}
\end{displaymath}
for some $\mu \in {\mathfrak a}^*_c$.
\end{theorem}
For this we consider for $\lambda \in {\mathfrak a}^*_c$ the \emph{Poisson
transform}
\begin{displaymath}
\P_{\lambda}: F(b) \to f(x)\, , \, \qquad F \in L^1(B)\, ,
\end{displaymath}
where
\begin{equation}
\label{eq:9.1}
f(x) =\int_B e^{(i\lambda +\rho)(A(x,b))}F(b) \, db \, .
\end{equation}
The element $\lambda$ is said to be \emph{simple} if $\P_{\lambda}$
is injective. The simplicity criterion for $\lambda$ \cite{He6}
implies that for each $\lambda \in {\mathfrak a}^*_c$ one of the transforms
$s\lambda$\,\,\, $(s \in W)$ is simple. Consider now the spherical
function $\varphi_{\lambda}$ (\ref{eq:7.3}) which can also be
written
\begin{displaymath}
\varphi_{\lambda} (x) =\int_B
e^{(i\lambda +\rho)(A(x,b))}\, db \, .
\end{displaymath}
We know from \cite{He9}, III, Lemma~2.3 that if $-\lambda$ is
simple then the closed space $\E_{(\lambda)}(X) \subset \E (X)$
generated by the $G$-translates of $\varphi_{\lambda}$ contains
the space $\P_{\lambda} (L^2 (B))$.
Coming to the proof of the theorem we conclude from the
Bagchi--Sitaram result (\ref{eq:7.4}) that the space
$V^K_f$ of $K$-invariants in $V_f$ contains a spherical
function $\varphi_{\lambda}$. By the simplicity result quoted,
either $\lambda$ or $-\lambda$ is simple so we can take
$-\lambda$ simple. Thus by the conclusion above, $V_f$ contains
the space $\P_{\lambda} (L^2 (B))$. Now by \cite{He9}, III,
Exercise~B1, pp.~371 and 570,
\begin{equation}
\label{eq:9.2}
e^{(i\lambda +\rho) A(x,eM)} =\sum_{\delta \in \widehat{K}_M}
\, d(\delta) \varphi_{\lambda,\delta}(x) \, ,
\end{equation}
with $\delta$ and $\widehat{K}_M$ as in (\ref{eq:6.5}) and
\begin{displaymath}
\varphi_{\lambda ,\delta} (x) = \int_{K}
e^{(i\lambda +\rho) (A (x,k))}\langle \delta (k) v,v \rangle
\, dk \, .
\end{displaymath}
Thus $\varphi_{\lambda ,\delta} \in V_f$ so since (\ref{eq:9.2})
converges in the topology of $ \E (X)$ the theorem follows.
\begin{remark}
Since Schwartz's theorem fails for $\mathbf{R}^n$ $(n>1)$ the proof
above via the Bagchi--Sitaram theorem is limited to the case of
rank~$X=1$. However, this does not rule out the possibility
that Theorem~\ref{th:9.1} might remain valid for $X$ of higher rank.
\end{remark}
\section{Further results on the Fourier transform}
\label{sec:10}
A result of Hardy's \cite{Ha} shows limitations on how fast a
function on $\mathbf{R}^n$ and its Fourier transform can decay at
$\infty$. Precisely, if
\begin{displaymath}
|f(x)|\leq A e^{-\alpha |x|^2}\, , \,
|\widetilde{f} (u) |\leq B e^{-\beta |u|^2}\quad
\alpha ,\beta >0
\end{displaymath}
and if $\alpha \beta >\frac{1}{4}$ then $f=0$. Sitaram and
Sundari \cite{SiSu} proved an analog for a class of spaces $X$
and Sengupta \cite{Se} extended this to all~$X$. Many other
variations of the result have been proved by Ray and Sarkar,
Cowling, Sitaram and Sundari, Narayanan and Ray, Shimeno,
Thangavelu. (See References.)
The following classical result is closely related to Wiener's
Tauberian theorem. \emph{Let $f \in L^1 (\mathbf{R}^n)$ such that
$\widetilde{f} (u) \neq 0$ for all $u \in \mathbf{R}^n$. Then the
translates of $f$ span a dense subspace of $\mathbf{R}^n$}. Many papers
deal with analogies of this result for semisimple Lie groups and
symmetric spaces. See \cite{EM}, \cite{Sa}, \cite{Si1},
\cite{Si2}, \cite{SS}, \cite{MRSS} for a sample.
The polar coordinate representation $(kM,a)\to kaK$ of $X$
identifies $X$ with $K/M \times A^+$ up to a null set. Thus one
might interpret the Plancherel formula (\ref{eq:8.2}) as
identifying $X$ with its ``dual''. But in contrast to $\mathbf{R}^n$
where the Fourier transform is essentially equal to its inverse,
the Fourier transform
\begin{eqnarray}
\label{eq:10.1}
\widetilde{f} (\lambda ,b) &=& \int_{X} f(x)
e^{(-i\lambda +\rho)(A(x,b))}\, dx \, ,\\
\noalign{\nonumber\hbox{and the inverse}}\\
\label{eq:10.2}
(\F^{-1}\varphi) (x) &=& \int_{{\mathfrak a}^* \times B}
\varphi (\lambda ,b) e^{(i\lambda +\rho)(A(x,b))}
|c(\lambda)|^2 \, d \lambda \, db
\end{eqnarray}
are quite different. Hence it is a natural problem to prove the
analog of the Paley--Wiener theorem for $\F^{-1}$.
This was done by A.~Pasquale \cite{Pa} for the spherical
transform for $X$ of rank one or the case of $G$ complex, and by
N.~Andersen \cite{AN} in general. Let $L$ denote the Laplacian
on $X$.
\begin{theorem}
\label{th:10.1}
The image of $\F^{-1} (\D ({\mathfrak a}^* \times B))$ consists of the
functions $f$ on $X$ satisfying
\begin{displaymath}
(1+ \, d(o,x))^m L^n f \in L^2 (X) \quad
\hbox{for all } m,n \in \mathbf{Z}^+
\end{displaymath}
and
\begin{displaymath}
\lim_{n \to \infty} \| (L+\langle\rho ,\rho\rangle )^n
\|^{1/2n}_2 < \infty \, .
\end{displaymath}
\end{theorem}
Another characterization was given by Pesenson \cite{P}, namely
\begin{displaymath}
\| L^{\sigma}f\|_2 \leq (\omega^2 +|\rho |^2)^{\sigma}
\| f \|_2 \quad \hbox{for all } \sigma >0 \, .
\end{displaymath}
|
1,314,259,996,759 | arxiv | \section{Introduction and the main result}
\label{s: intro}
The main aim of this paper is to prove the following product
formul{\ae} for short-time Schr\"odinger unitary groups and
orthogonal projections:
\begin{theorem} \label{th:main}
Let $H$
be a nonnegative self-adjoint operator acting on a separable Hilbert space $\mathcal{H}$ and $P$ an orthogonal projection onto a closed subspace of $\mathcal{H}$. Suppose that $H^{1/2}P$ is densely defined, so that $H_P := (H^{1/2}P)^*(H^{1/2}P)$ is a self-adjoint operator. Then for any $f\in \mathcal{H}$ and $\varepsilon = \pm 1$ the following relations hold,
\begin{align}
& \lim_{n\rightarrow \infty} (P\,\mathrm{e}^{-\varepsilon itH/n}
P)^nf
= \mathrm{e}^{-\varepsilon itH_P}Pf\,, \label{Pboth} \\
& \lim_{n\rightarrow \infty} (\mathrm{e}^{-\varepsilon itH/n} P)^nf
= \mathrm{e}^{-\varepsilon itH_P}Pf\,, \label{Pright} \\
& \lim_{n\rightarrow \infty} (P\,\mathrm{e}^{-\varepsilon itH/n})^nf
= \mathrm{e}^{-\varepsilon itH_P}Pf \label{Pleft}
\end{align}
in the Hilbert space norm, and moreover, the convergence is uniform on every
bounded $t$-interval in $\mathbb{R}$.
\end{theorem}
\noindent Needless to say, the claim is nontrivial only if $H$ and $P$ do not
commute.
The main motivation to study such product formul{\ae} comes from the
behavior of quantum systems exposed to frequent measurements. Turing
was the first to notice \cite{Ho04} that if we ascertain repeatedly
whether a quantum system is in a given state, then in the limit
of infinite measurement frequency it becomes impossible to leave
this state. The idea was rediscovered in the context of unstable
system decays -- see, e.g., \cite{BN67, Fr72} -- but it attracted a
wide attention only after Misra and Sudarshan \cite{MS77} invented a
catchy name calling such a behavior \emph{quantum Zeno effect} in an
allusion to the classical Zeno aporia about a flying arrow. More
about the early history can be found in \cite{Ex85}.
The second breaking moment came in 1990 when Itano et al.
\cite{IHBW90} demonstrated the existence of the effect
experimentally. Since then it became object of an extensive
examination, both from the theoretical and experimental points of
view, and it led even to various practical applications; a partial
summary can be found in the review paper \cite{FP08}.
If the projection $P$ describing the measurement has a dimension
larger than one, the question about the time evolution in the
subspace to which the permanent observation confines the state of
the system becomes nontrivial. It is natural to expect that such a
`Zeno dynamics' will be governed by the part of the original
Hamiltonian $H$ acting in the subspace $P\mathcal{H}$, and it was
shown in \cite[Sect.~2.4]{Ex85} that the generator is indeed
associated with appropriate quadratic form constructed from the
operator $H$ and the projection $P$. It was not easy, however, to
establish the existence of the Zeno dynamics beyond the situations
when the dimension of $P$ is finite or the operator $H$ is bounded;
needless to say that this is often not the case with actual physical
systems. A prime example is a permanent ascertaining whether a free
quantum particle dwells within a prescribed region $\Omega$ of the
configuration space discussed in \cite{FPSS01}, see also
\cite{FP08}, with the conclusion that the Zeno generator is (the
multiple of) the corresponding Dirichlet Laplacian. The argument
made use of the stationary phase method but the existence of the
limit was not actually established by the authors.
Motivated by the said paper we addressed the question of the Zeno
dynamics existence in \cite{EI05}, where we have managed to
establish the existence of the limits of the expressions appearing
on the left-hand sides of \eqref{Pboth}--\eqref{Pleft} in the
topology of a larger space, namely, the Fr\'echet space
$L^2_{\text{\rm loc}}({\mathbb R}; {\mathcal H}) = L^2_{\text{\rm
loc}}({\mathbb R})\otimes {\mathcal H}$, provided that $H$ is
semibounded and the operator $H_P$ is densely defined; the validity
of the formul{\ae} is preserved if the exponential in
\eqref{Pboth}--\eqref{Pleft} is replaced by functions of a wider
class, in particular, by the resolvent $(I+itH)^{-1}\,$
\cite{EINZ07, Ich15}. We argued in \cite{EI05} that such a result
can be regarded as sufficient from the viewpoint of physics due to
the fact that every measurement, in particular, that of time is
burdened with errors, and any actual experiment typically involves
averaging over a large number of system copies.
It is desirable, though, to answer the question without such a
underpinning by demonstrating the result with the convergence in a
stronger sense, namely that of the strong operator topology. This is
the aim of the present paper. In addition to the described physical
motivation, the obtained relation are of independent mathematical
interest belonging to the genre of the product formul{\ae} of
Trotter and Trotter-Kato \cite{Ka78, Tr59}, see also \cite{RS80}. In
fact, we are going to prove a slightly more general claim with a fixed
$P$ replaced by a projection-valued function of~$t$ satisfying
certain regularity assumptions.
\begin{theorem} \label{th:main'}
Let $\mathcal{H}$ be a separable Hilbert space, and $H,\,P$, and thus also $H_P$, be the same as in Theorem~\ref{th:main}. Let further $P(\cdot)$ be a strongly continuous function the values of which are orthogonal projections in $\mathcal{H}$ defined in a right neighbourhood of zero and satisfying $P(0)=P$. Moreover, suppose that
\begin{equation} \label{hypoth-to-H1/2P}
\lim_{\tau\to 0+} [\tau^{-1}(I-\mathrm{e}^{-it\tau H})]^{1/2}P(\tau) v
= \mathrm{e}^{\pi i/4}(tH)^{1/2}Pv\,,
\end{equation}
for every $v \in D[H^{1/2}P]$. Then for any $f \in \mathcal{H}$ and $\varepsilon = \pm 1$ we have
\begin{align}
& \lim_{n\rightarrow \infty} (P(1/n)\,\mathrm{e}^{-\varepsilon
itH/n} P(1/n))^nf
= \mathrm{e}^{-\varepsilon itH_P}Pf\,, \label{P(t)both} \\
& \lim_{n\rightarrow \infty} (\mathrm{e}^{-\varepsilon itH/n}
P(1/n))^nf
= \mathrm{e}^{-\varepsilon itH_P}Pf\,, \label{P(t)right} \\
& \lim_{n\rightarrow \infty} (P(1/n)\,\mathrm{e}^{-\varepsilon
itH/n})^nf
= \mathrm{e}^{-\varepsilon itH_P}Pf\,, \label{P(t)left}
\end{align}
in the Hilbert space norm, where the convergence is uniform on every bounded $t$-interval in $\mathbb{R}$.
\end{theorem}
\begin{remark} \label{}
Note that the hypothesis made in Theorem~\ref{th:main'} about the convergence of $[\tau^{-1}(I- \mathrm{e}^{-\tau H})]^{1/2}P(\tau)v$ is slightly weaker in comparison with \cite[Theorem~2.1]{EI05} where we assumed that $D[H^{1/2}P(\tau)] \supset D[H^{1/2}P]$ and $\lim_{\tau \rightarrow 0+} \|H^{1/2}P(\tau)v\| = \|H^{1/2}Pv\|$ holds for every $v \in D[H^{1/2}P]$. This, in fact, was not fully necessary there as a footnote in \cite[p.~206]{EI05} briefly mentioned.
\end{remark}
Note that the assumption of positivity of $H$ is made for
convenience only, it is obvious that the result remains to be valid
if $H$ is replaced by $H+cI$ with a fixed $c\in\mathbb{R}$, i.e. for
any self-adjoint operator bounded from below. On the other hand, the
density hypothesis is crucial; in \cite[Rem.~2.7]{EI05} we cited an
example showing that in its absence the expressions
$(\mathrm{e}^{-itH/n} P)^n$ may not converge in any sense. It may
also happen that they converge but not strongly. Examples were found
by Matolcsi and Svidkoy \cite{MS03}, however, they do not contradict
Theorem~\ref{th:main'} because in one of them the analogue of $H_P$
is not densely defined and in the other the operator $H$ is not
semibounded.
Let us now describe briefly our strategy to prove Theorem~\ref{th:main'}. The main tool is Chernoff's theorem \cite{Ch74}, see also \cite{Ch68}, which for reader's convenience we reproduce in Sect.~\ref{s: proof} below. It will yield the sought result if we show that the $\tau$-family $\{[I+ \tau^{-1}(I-P(\tau)\,\mathrm{e}^{-\varepsilon it\tau H} P(\tau))]^{-1}\}_{\tau>0}$ converges to $(I+\varepsilon itH_P)^{-1}$ as $\tau\to 0+$ in the strong operator topology. The proof might basically follow the argument used by Kato in \cite{Ka78} to establish his celebrated {\it self-adjoint} Trotter-Kato product formula for the form sum of two nonnegative self-adjoint operators, since these two problems appear to have notable similarities. However, a straightforward analogy of Kato's argument is not sufficient due to a difficulty one encounters, to be specified in Sect.~\ref{s: proof}, Step IV of the proof. The point is that the argument \emph{can} be applied to a certain class of admissible functions $\phi(x)$ which contains beside real exponentials, e.g., $(1+\varepsilon ix)^{-1}$, as shown in the mentioned papers \cite{EINZ07, Ich15}, but unfortunately \emph{this class fails to include} $\mathrm{e}^{-\varepsilon ix}$ corresponding to our unitary group $\mathrm{e}^{-itH}$.
Our way to overcome this obstacle is to start from the weaker result mentioned above. What we did in \cite{EI05} was to complement the modified Kato's argument by the Vitali theorem from complex function theory; in that way we proved that for each $f \in {\mathcal H}$ the $\tau$-family $\{[I+ \tau^{-1}(I-P(\tau)\mathrm{e}^{-\varepsilon it\tau H}P(\tau))]^{-1}f\}_{\tau>0}$ converges in the the Fr\'echet space $L^2_{\text{\rm loc}}({\mathbb R}; {\mathcal H})$. This conclusion serves as a departing point here, although it only implies that for some set $M_f \subset [0,\infty)$ of Lebesgue measure zero, the $\tau$-family $\{[I+ \tau^{-1}(I-P(\tau)\mathrm{e}^{-\varepsilon it\tau H}P(\tau))]^{-1}f\}_{\tau>0}$ converges in the Hilbert space norm for every $t \in [0,\infty) \setminus M_f$, not excluding the possibility that the convergence \emph{does} hold at some points of $M_f$. Furthermore, using the separability hypothesis made about the Hilbert space ${\mathcal H}$, we can choose a countable dense subset ${\mathcal D} := \{f_l\}_{l=1}^{\infty}$ in ${\mathcal H}$. Putting then $M = M_{\mathcal D} := \cup_{l=1}^{\infty} M_{f_l}$, which is also a set of Lebesgue measure zero, we may say that the above indicated $\tau$-family converges for all $t \in {\mathbb R}\setminus M$ and for every $f \in {\mathcal D}$, and therefore, in view of the density, also for every $f \in {\mathcal H}$.
To pass from the `almost all $t$' to the `all $t$' stage, one has to demonstrate that the exceptional set $M$ is in fact empty. This task may seem a small step, but in reality it proved to be a deep and highly nontrivial question. Our way to deal with it is to establish the equicontinuity of the above $\tau$-family, cf.~Lemma~\ref{l:equiconti}, which would allow us to achieve our goal by means of the Ascoli-Arzel\`{a} theorem.
A detailed description of the argument we have sketched here, given
in Sect.~\ref{s: proof} and Sect.~\ref{s: proof-lemma}, is a core part
of the paper. As a preliminary, we characterize in the next section the limit
self-adjoint operator $H_P$ appearing in the theorems. Finally, the
paper will be concluded with a short section in which we return
briefly to example of the permanent position measurement considered
in \cite{FPSS01}.
\section{Concerning the limit self-adjoint operator~$H_P$}
\label{s: limitop}
Throughout the paper $H$ will be a nonnegative self-adjoint operator in a
separable Hilbert space ${\mathcal H}$, and $P$ will be an orthogonal
projection. As we have indicated above, the nonnegativity assumption
is made for convenience; our main result extends easily to any
self-adjoint operator $H$ bounded from below as well as, by sign
change, to one bounded from above, i.e. to each semi-bounded
self-adjoint operator in ${\mathcal H}$.
Our aim here is to elucidate what is the Zeno generator $H_P$ appearing in Theorems~\ref{th:main} and \ref{th:main'} and to indicate some of its properties which we will need in the sequel. Consider the quadratic form $u \mapsto \|H^{1/2}Pu\|^2$ with \emph{form domain} $D[H^{1/2}P]$, being the domain of the \emph{operator} $H^{1/2}P$. While the domains of $HP$ and $H^{1/2}P$ are nontrivial only as subspaces of $P{\mathcal H}$, we consider these operators always as acting in the whole Hilbert space ${\mathcal H}$ writing, if needed, $HP = HP\!\!\restriction\!_{P{\mathcal H}}\oplus\, 0\!\restriction\!_{(I-P){\mathcal H}}$ with the domain
\begin{equation} \label{HP}
D[HP] = D[HP\!\!\restriction\!_{P{\mathcal H}}]\oplus (I-P){\mathcal H}, \quad D[HP\!\!\restriction\!_{P{\mathcal H}}] = D[H]\cap P{\mathcal H},
\end{equation}
and similarly for $H^{1/2}P$. Note that the latter is a \emph{closed} linear operator, because $H^{1/2}$ is closed; from the same reason the operator $HP$ is also closed. We have $(H^{1/2}P)^* \supset PH^{1/2}$ in general. On the other hand, the operator $PH^{1/2}$ may not be closed. This would be the case if and only if there were $c_0,\,c_1 > 0$
such that $\|H^{1/2}u\| \leq c_0\|PH^{1/2}u\| + c_1\|u\|$ holds for all $u$ in the domain $D[H^{1/2}]$ of $PH^{1/2}$, however, the necessary part of this condition may not be satisfied. The same is true for $PH$.
By $H_P$ we denote the unique self-adjoint operator associated with the above mentioned quadratic form. Its (operator) domain $D[H_P]$ is a subspace of $D[H^{1/2}P]$ consisting of all $u \in D[H^{1/2}P]$ which satisfy $|\langle H^{1/2}Pu, H^{1/2}Pv \rangle| \leq C\|v\|$ for all $v \in D[H^{1/2}P]$ with a constant $C \geq 0$, cf.~\cite[Sect.~VI.2.1]{Ka76}, so that
\begin{equation} \label{H_P}
H_P = (H^{1/2}P)^*(H^{1/2}P).
\end{equation}
Needless to say, the form domain of $H_P$ is the (operator) domain of $H_P^{1/2}$, for which we have by polar decomposition, cf.~\cite[Sect.~VI.2.7]{Ka76},
\begin{align}
&H_P^{1/2} = |H^{1/2}P| = [(H^{1/2}P)^*(H^{1/2}P)]^{1/2}\,,
\nonumber\\
&D[H_P^{1/2}] = D[H^{1/2}P]
= (D[H^{1/2}] \cap P{\mathcal H}) \oplus (I-P){\mathcal H}.
\label{domain-H1/2P}
\end{align}
Then it is not difficult to check the following claim:
\begin{proposition} \label{p:H_P}
The operator $H_P$ in \eqref{H_P} decomposes as
\begin{equation} \label{decomp-HP}
H_P = H_P\!\!\restriction\!_{P{\mathcal H}}
\oplus\, 0\!\restriction\!_{(I-P){\mathcal H}}
\end{equation}
having the dense domain
$$
D[H_P] := D[H_P\!\!\restriction\!_{P{\mathcal H}}] \oplus (I-P){\mathcal H}
\,\subset\, P{\mathcal H} \oplus (I-P){\mathcal H} = {\mathcal H}\,.
$$
Here $H_P\!\!\restriction\!_{P{\mathcal H}}$ is the $P{\mathcal H}$-component of $H_P$, which is the self-adjoint operator in the subspace $P{\mathcal H}\subset\mathcal{H}$ associated with the quadratic form
in $P{\mathcal H}$ with the form domain $D[{H_P}^{1/2}]\cap P{\mathcal H}$,
\begin{equation} \label{quad-H1/2P}
D[{H_P}^{1/2}]\cap P{\mathcal H} = D[H^{1/2}P]\cap P{\mathcal H} \ni w
\mapsto \|H^{1/2}Pw\|^2 = \|H_P^{1/2}w\|^2\,,
\end{equation}
and the zero operator $0\!\!\restriction\!\!_{(I-P){\mathcal H}}$ is its $(I\!-\!P){\mathcal H}$-component, trivially bounded and self-adjoint on the subspace $(I\!-\!P){\mathcal H}$ orthogonal to $P\mathcal{H}$.
\end{proposition}
From Proposition~\ref{p:H_P}, we can see that $H_P$ is in general not a restriction of $H$, and furthermore, the inclusion $D[H_P] \subset D[H]$ does not hold either. This is obvious, since $H$ is the unique self-adjoint operator in ${\mathcal H}$ associated with the quadratic form
\begin{equation} \label{quad-H1/2}
D[H^{1/2}] \ni w \mapsto \|H^{1/2}w\|_{\mathcal H}^2\,,
\end{equation}
so that $(I-P){\mathcal H}$ is not a subset of $D[H]$ as long as $H$ is \emph{unbounded}.
\smallskip
We can make, however, a weaker claim described in the following proposition; we note in passing that it is well illustrated by the inclusion
$D[(-\Delta)_{\Omega}] \subset D[-\Delta]$ from the example treated in Sect.~\ref{s: example} below.
\begin{proposition} \label{p:DHP<DH}
Let $H$ be a nonnegative unbounded self-adjoint operator acting on a Hilbert space ${\mathcal H}$ and $P$ an orthogonal projection which may not commute with $H$. Assume that $H^{1/2}P$ is densely defined, then the $P{\mathcal H}$-component $H_P\!\!\restriction\!_{P{\mathcal H}}$ of $H_P$ in \eqref{decomp-HP} satisfies $D[H_P\!\!\restriction\!_{P{\mathcal H}}] \subset D[H]$.
\end{proposition}
\noindent {\it Proof:} The form \eqref{quad-H1/2} restricted to $D[H^{1/2}P]$ becomes \eqref{quad-H1/2P}, so that $\|H^{1/2}w\|^2 = \|H^{1/2}Pw\|^2 = \|H_P^{1/2}w\|^2$ holds if $w \in D[H_P^{1/2}]$ proving thus the claim. \hfill
\qed
\begin{remark}
One can modify $H_P$ restricting it to the self-adjoint operator $H_P^\mathrm{mod}$ such that $H_P^\mathrm{mod}\!\!\restriction\!_{P{\mathcal H}} = H_P\!\!\restriction\!_{P{\mathcal H}}$ {\it and, at the same time,} $\,D[H_P^\mathrm{mod}] \subset D[H]$. To this end, it is enough to replace the zero operator in \eqref{decomp-HP} on $(I-P){\mathcal H}$ by its restriction to $(I-P)D[H]$, so that
\begin{align} \label{HPmd}
& H_P^\mathrm{mod}
:= H_P\!\!\restriction\!_{P{\mathcal H}}
\oplus 0\!\!\restriction\!_{(I-P)D[H]}\,,\\
& D[H_P^\mathrm{mod}]
:= D[H_P\!\!\restriction\!_{P{\mathcal H}}] \oplus (I-P)D[H]\,.\nonumber
\end{align}
Let us stress that in general $H$ is not an extension of $H_P^\mathrm{mod}$ either.
\end{remark}
Unless $H$ is bounded, the operator $H_P$ is generally different
from $PHP$. When $HP$ is densely defined, the symmetric operator
$PHP$ is not necessarily essentially self-adjoint on $D[HP]$, and
consequently, $H_P$ may not be the closure of $PHP$ either.
On the other hand, the quadratic form $u \mapsto \|H^{1/2}Pu\|^2$
defined on $D[H^{1/2}P]$ is a closed extension of the form $u
\mapsto \langle Pu,HPu \rangle$ defined on $D[HP]$, but in general
the former is not the closure of the latter, because $D[HP]$ is not
necessarily dense in $D[H^{1/2}P]$. Indeed, if $H$ is unbounded,
$D[H]$ is a proper subspace of $D[H^{1/2}]$. Take $u_0 \in D
[H^{1/2}]\backslash D[H]$ such that the vector $H^{1/2}u_0$ is
nonzero, and set $P$ to be the orthogonal projection onto the
one-dimensional subspace spanned by $u_0$. Taking into account that
$D[HP] = \{u \in {\mathcal H};\,\, Pu \in D[H]\}$ which $u_0 =Pu_0$
does not belong to, we find $HPu = 0$ for $u \in D[HP]$, while
$H^{1/2}Pu_0 = H^{1/2}u_0 \not= 0$ by assumption.
\section{Proof of Theorem~\ref{th:main'}}
\label{s: proof}
Most parts of the argument work for a general Hilbert space, however, there is a place where we have to assume ${\mathcal H}$ to be separable. It is clearly sufficient to prove formula \eqref{P(t)both} in Theorem~\ref{th:main'} because \eqref{P(t)right} and \eqref{P(t)left} easily follow from it, and furthermore, it is enough to consider $\varepsilon = 1$ and $t \geq 0$. For the sake of definiteness we use here and in the following the physicist convention about the inner product $\langle \cdot, \cdot \rangle$ on ${\mathcal H}$ supposing that it is antilinear in the \emph{first} argument.
\smallskip
The nonnegative self-adjoint operator $H$ we deal with can be conventionally represented through its spectral family $\{E(\lambda)\}_{\lambda\geq 0}$,
\begin{equation} \label{H-specrep}
H = \int_{0-}^{\infty} \lambda E(\mathrm{d}\lambda)
\end{equation}
Given $\kappa>0$, we introduce
\begin{align}
K(\kappa) :&=\frac1{\kappa}[I-\mathrm{e}^{-i\kappa H}]
= \frac{I-\cos \kappa H}{\kappa} + i\frac{\sin \kappa H}{\kappa}
\nonumber \\[-.5em]\nonumber\\[-.5em]
&=: G(\kappa) + iH(\kappa)\,, \label{KGH}
\end{align}
where $G(\kappa)$ and $H(\kappa)$ are bounded self-adjoint
operators, and $G(\kappa)$ is nonnegative. Let $H^{\pm}(\kappa)$ be
nonnegative bounded self-adjoint operators which are the
nonnegative/negative parts of $H(\kappa)$, respectively. Then we
have
\begin{align}
|K(\kappa)| &= [G(\kappa)^2+H(\kappa)^2]^{1/2} = \tfrac{|\sin \frac12
\kappa H|}{\frac12\kappa}\,, \nonumber \\[.2em]
|I+ K(\kappa)| &=
\Big(I + (1+\kappa) \big(\tfrac{\sin \frac12\kappa H}{\frac12\kappa}\big)^2\Big)^{1/2}\,,
\label{KGH2} \\[.5em]
H(\kappa) &= H^+(\kappa) - H^-(\kappa)\,,
\quad |H(\kappa)| = H^+(\kappa) + H^-(\kappa)\,. \nonumber
\end{align}
Using the spectral family $\{E(\lambda)\}_{\lambda\geq 0}$ from
\eqref{H-specrep}, we denote by $E_H^+(\kappa)$ the orthogonal projection
\begin{equation}
E_H^+(\kappa) : \mathcal{H} \rightarrow \bigoplus_{m=0}^{\infty}
E\big([\tfrac{2m\pi}{\kappa}, \tfrac{(2m+1)\pi}{\kappa})\big)
\mathcal{H}
\equiv E\Big(\bigcup_{m=0}^{\infty}
[\tfrac{2m\pi}{\kappa},
\tfrac{(2m+1)\pi}{\kappa})\Big)\mathcal{H}\,, \nonumber
\end{equation}
and put $E_H^-(\kappa) := I- E_H^+(\kappa)$. Note that
$E_H^+(\kappa) \overset s \to I$ and $E_H^-(\kappa) \overset s \to
0$ holds as $\kappa \rightarrow 0+$. Of course, both the
$E_H^{\pm}(\kappa)$ commute with $H(\kappa)$. As $E_H^+(\kappa)$ and
$E_H^-(\kappa)$ are nothing but the projections onto the closed
subspaces of $\mathcal{H}$ where $H(\kappa)$ becomes respectively a
nonnegative and negative self-adjoint operator, we have
\begin{equation} \label{Hpm}
H^+(\kappa) = H(\kappa)E_H^+(\kappa)\,,
\quad H^-(\kappa) = -H(\kappa)E_H^-(\kappa)\,.
\end{equation}
We put $\kappa = |t|\tau$ with $\tau >0$, so that
$\kappa = \pm\, t\tau$ holds for $\pm\, t \geq 0$.
We will use the same notation as in \cite{EI05}, $F(\zeta;\tau) := P(\tau)\, \mathrm{e}^{-\zeta\tau H}P(\tau)$ and $S(\zeta;\tau) := \tau^{-1}[I-F(\zeta;\tau)]$ with $\text{\rm Im}\, \zeta \geq0$. These operator families are uniformly bounded and holomorphic for $\text{\rm Im}\, \zeta > 0$ which are the properties we used there. In this paper, however, we need only $\zeta= it$, in other words
\begin{align}
F(it;\tau) &= P(\tau)\, \mathrm{e}^{-it\tau H}P(\tau)\,, \label{Ftau}\\
S(it;\tau) &= \tau^{-1}[I-F(it;\tau)]
=\tau^{-1}[I- P(\tau)\,\mathrm{e}^{-it\tau H}P(\tau)]\,. \label{Stau}
\end{align}
It is easy to see that $F(it;\tau)$ in \eqref{Ftau} is a contraction and
$S(it;\tau)$ in \eqref{Stau} satisfies
\begin{align}
\mathrm{Re}\,(f,S(it;\tau)f)
& =\tau^{-1} \big[(f,f) -(f, P(\tau)\,\mathrm{e}^{-it\tau H}P(\tau)f) \big] \nonumber \\[.2em]
& \geq \tau^{-1} \big[\|f\|^2 -\|f\|\,\|P(\tau)\,\mathrm{e}^{-it\tau H}P(\tau)f\| \big] \nonumber \\[.2em]
& \geq \tau^{-1} \big[\|f\|^2- \|f\|^2 \big] =0 \nonumber
\end{align}
for all $f \in \mathcal{H}$, and consequently, $S(it;\tau)$ is an
$m$-accretive operator \cite{Ka76}. This means that $I+S(it;\tau)$ has a
bounded inverse and that $(I+ S(it;\tau))^{-1}$ is also a contraction.
The crucial observation is that to prove Theorem~\ref{th:main'} it is
sufficient to refer to Chernoff's result cited below,
and to verify that
\begin{equation} \label{Chern}
(I+S(it;\tau))^{-1} \overset s \longrightarrow (I+itH_P)^{-1}P
\quad \mathrm{as}\;\; \tau \rightarrow 0+
\end{equation}
holds for $t \in {\mathbb R}$.
\smallskip\noindent
\textbf{Chernoff's Theorem} (cf.~\cite[Theorem~1.1, pp.~4--6]{Ch74}, see also \cite{Ch68}):
\textsl{For a $t$-family $\{F(t)\}_{t\geq 0}$ of linear contractions on a Banach space and the generator $A$ of a strongly continuous contraction semigroup, the following two conditions are equivalent: }
\textrm{(a)} \textsl{For some $\lambda_0>0$, the family $\{\lambda_0 I + \tfrac{I-F(\varepsilon)}{\varepsilon}\}_{\varepsilon >0}$ converges strongly to $(\lambda_0 I+A)^{-1}$ as $\varepsilon \to 0+$. }
\textrm{(b)} \textsl{As $n\to \infty$, $\{F(\tfrac{t}{n})\}_{n=1}^{\infty}$ converges strongly to $\mathrm{e}^{tA}$, uniformly on bounded $t$-intervals.}
\medskip
Let us stress that our proof requires to demonstrate the convergence in \eqref{Chern} \emph{pointwise for any fixed $t$}; this will be sufficient to establish the convergence in the product formul{\ae} \eqref{P(t)both}--\eqref{P(t)left} as \emph{locally uniform} in $t \in {\mathbb R}$; in fact, we have only to deal with \eqref{P(t)both} as mentioned at the beginning of this section.
\begin{remark}
Since the function $F(it;\tau)$ in \eqref{Ftau} differs slightly from $F(t)$ appearing in condition (a) of Chernoff's Theorem, let us explain in detail how the product formula \eqref{P(t)both} follows from \eqref{Chern}, modifying to that purpose Chernoff's proof of the implication (a)$\,\Rightarrow\,$(b).
Consider the first the nontrivial part referring to the subspace $P{\mathcal H}$. Note that $S(it;\tau)$ generates a strongly continuous contraction semigroup $\{\mathrm{e}^{-\theta S(it;\tau)}\}_{\theta\geq 0}$ on ${\mathcal H}$, and the resolvent convergence \eqref{Chern} is equivalent to the convergence of the corresponding semigroups \cite[Theorem~IX.2.16]{Ka76}, hence for any $f\in P{\mathcal H}$ we have $\mathrm{e}^{-\theta S(it;\tau)}f \overset s \longrightarrow \mathrm{e}^{-i\theta tH_P}f$ as $\tau\to 0+$, uniformly on bounded intervals of the variable $\theta\geq 0$. In particular, choosing $\theta=1$ we get
\begin{equation} \label{remv-theta}
\mathrm{e}^{-S(it;\tau)}f \longrightarrow \mathrm{e}^{-itH_P}f\quad\text{\rm as}\;\; \tau\to 0+\,
\end{equation}
for a fixed $t\geq 0$, and using the same equivalence in the opposite direction we infer that
$$
(I+\lambda S(it;\tau))^{-1}f \longrightarrow
(I+i\lambda tH_P)^{-1}Pf \quad\text{\rm as}\;\;\tau\to 0+
$$
holds for any $\lambda\ge 0$ and $t\ge 0$. In particular, using the diagonal trick in the last relation with $\tau=1/n$ and $\lambda = 1/\sqrt{n}$, we obtain
\begin{equation} \label{diag-trick}
(I +\tfrac1{\sqrt{n}}S(it; 1/n))^{-1}f \longrightarrow Pf
\quad\text{\rm as}\;\; n \to \infty
\end{equation}
for every $t\geq 0$. Next we refer to \cite[Lemma 2]{Ch68} by which we have
$$
\|[F(it; 1/n)^ng -\mathrm{e}^{-n(I- F(it; 1/n))}g\|
\leq \sqrt{n}\,\|(I-F(it; 1/n))g\|
$$
for any $g \in {\mathcal H}$. Choosing $g= (I + \tfrac1{\sqrt{n}}S(it; 1/n))^{-1}f$ and having in mind that $f=Pf$, we conclude from here that
\begin{align*}
&\big\|[F(it; 1/n)^n - \mathrm{e}^{-S(it; 1/n)}]
(I + \tfrac1{\sqrt{n}}S(it; 1/n))^{-1}f\big\| \\
&\qquad \leq \big\|(I + \tfrac1{\sqrt{n}}S(it; 1/n))^{-1}f - Pf\|\,,
\end{align*}
where by \eqref{diag-trick} the right-hand side tends to zero uniformly on bounded $t$-intervals as $n \to \infty$. Using the diagonal trick once again we get
$
\lim_{n\to\infty} \|F(it; 1/n)^nf - \mathrm{e}^{-S(it; 1/n)}f\| = 0\,
$
uniformly on bounded $t$-intervals. Then the sought conclusion \eqref{P(t)both} follows immediately from relations \eqref{remv-theta} and \eqref{diag-trick}, since by \eqref{Ftau} we have $F(it; 1/n)^n = [P(1/n) \mathrm{e}^{-itH/n} P(1/n)]^n$. Having dealt with the subspace $P\mathcal{H}$, the remaining case, $f \in (P{\mathcal H})^{\perp}$, is trivial: we have
$$
[P(1/n) \mathrm{e}^{-itH/n} P(1/n)]^n f \longrightarrow 0 \quad\text{\rm as}\;\; n \to \infty
$$
for each $t \geq 0$, since $P(1/n)f = P(1/n)(I-P)f$ tends by assumption to $P(I-P)f =0$ and $\mathrm{e}^{-itH_P}Pf =0$ holds at the same time.
\end{remark}
\smallskip
Let us add a remark on the conventions: here and in the following the convergence of operator families in the strong operator topology is denoted by $\overset s \longrightarrow$, and in the weak operator topology by $\overset w \longrightarrow$. A simple arrow is reserved for the convergence with respect to the norm of the Hilbert space $\mathcal{H}$, sometimes also dubbed `strong' -- we will occasionally use this term too -- while for the weak convergence in $\mathcal{H}$ we will again employ the symbol $\overset w \longrightarrow$. Later, in Proposition~\ref{p:sigmaweak-bdd}, we will use still another topology for convergence of families of Hilbert space vectors.
To proceed with the argument, we rewrite relation \eqref{Stau} as follows
\begin{align*}
S(it;\tau) &= \tau^{-1} \big[I- P(\tau)(\cos t\tau H- i\sin t\tau H)P(\tau)\big] \\[.2em]
&= \tfrac{I-P(\tau)}{\tau}
+ P(\tau)\,\tfrac{I-\cos t\tau H}{\tau}\,P(\tau)
+ i\,P(\tau)\,\tfrac{\sin t\tau H}{\tau}\,P(\tau) \\[.2em]
&= \tfrac{I-P(\tau)}{\tau} +P(\tau)tG(t\tau)P(\tau)
+ i\,P(\tau)tH(t\tau)P(\tau)\,,
\end{align*}
and consequently,
\begin{align} \label{IplusS}
I &+ S(it;\tau) \\[.2em]
&= I + \tau^{-1}(I-P(\tau))
+P(\tau)tG(t\tau)P(\tau) +iP(\tau)tH(t\tau)P(\tau) \nonumber \\[.2em]
&= (1 +\tau^{-1})(I- P(\tau)) \oplus P(\tau)(I+ tG(t\tau)
+ itH(t\tau))P(\tau)\,, \nonumber
\end{align}
where $\oplus$ denotes the direct sum corresponding to the
decomposition of the Hilbert space into $P(\tau)\mathcal{H}$ and its
orthogonal complement.
To prove the sought relation \eqref{Chern} we need first a pair of lemmata.
\begin{lemma} \label{l:inverse}
The inverse of $I+S(it;\tau)$ in \eqref{IplusS} is given by
\begin{align} \label{inverse-of-IplusS}
(I &+ S(it;\tau))^{-1} \\
&= (1+\tau^{-1})^{-1}(I- P(\tau))
\oplus \big[P(\tau)(I+ tG(t\tau) + itH(t\tau))P(\tau)\big]^{-1}. \nonumber
\end{align}
\end{lemma}
\vspace{-.3em}
\begin{proof}
The above expression and \eqref{IplusS} clearly multiply to
identity.
\end{proof}
\begin{lemma} \label{l:conv}
Let $u \in D[H^{1/2}]$, then in the limit $\kappa \rightarrow 0+$ we have
\begin{align*}
\mathrm{(i)}\,\, &G(\kappa)^{1/2} u \longrightarrow 0\,, \\[.2em]
\mathrm{(ii)}\,\,
&H^+(\kappa)^{1/2} u \longrightarrow H^{1/2}u\,, \;
H^-(\kappa)^{1/2} u \longrightarrow 0\,, \; \mathrm{and}\;\;
|H(\kappa)|^{1/2} u \longrightarrow H^{1/2}u
\end{align*}
\end{lemma}
\begin{proof}
Using spectral theorem together with dominated convergence theorem we infer
that as $\kappa \rightarrow 0+$,
\begin{align*}
G(\kappa)^{1/2} u
&= \int_{0-}^{\infty}
\Big|\tfrac{\sin\tfrac{\kappa \lambda}2}{\tfrac{\kappa\lambda}2}\Big|^{1/2}
|\sin\tfrac{\kappa \lambda}2|^{1/2}\,\lambda^{1/2}
E(\mathrm{d}\lambda)u \,\longrightarrow 0\,,
\end{align*}
\vspace{.5em}
$$
|H(\kappa)|^{1/2} u = \int_{0-}^{\infty} \Big|\tfrac{\sin \kappa
\lambda}{\kappa}\Big|^{1/2}
E(\mathrm{d}\lambda)u \longrightarrow \int_{0-}^{\infty}
\lambda^{1/2} E(\mathrm{d}\lambda) u = H^{1/2}u
$$
and
\begin{align*}
H^+(\kappa)^{1/2} u
&= |H(\kappa)|^{1/2}E_H^+(\kappa) u\\[.2em]
&= \int_{0-}^{\infty} \Big|\tfrac{\sin \kappa
\lambda}{\kappa}\Big|^{1/2}
\chi_{\bigcup_{m=0}^{\infty}
[\tfrac{2m\pi}{\kappa}, \tfrac{(2m+1)\pi}{\kappa})}(\lambda)\,
E(\mathrm{d}\lambda) u \\
&\qquad\qquad
\longrightarrow \int_{0-}^{\infty} \lambda^{1/2}
E(\mathrm{d}\lambda) u = H^{1/2}u
\end{align*}
This implies at the same time $H^-(\kappa)^{1/2} u \longrightarrow 0\,$ by \eqref{KGH2}.
\end{proof}
After these preliminaries, we are going to start the proof of
\eqref{Chern}. As we said in the introduction, the core of our
reasoning is the argument used by Kato \cite{Ka78}, see also
\cite[Supplements to Sect.~VIII.8]{RS80}.
In the same vein, with \eqref{Stau} in mind, we put for
$\tau>0$ and $t \in {\mathbb R}$
\begin{equation} \label{def-of-utau}
u_{\tau}(t) := (I+S(it;\tau))^{-1}f
\end{equation}
for an arbitrary but fixed $f \in \mathcal{H}$. We see from the expression
\eqref{inverse-of-IplusS} of $(I+S(it;\tau))^{-1}$ that
$u_{\tau}(t)$ is strongly continuous in $t$. At the same time, the
$\tau$-family $\{u_{\tau}(t)\} \subset {\mathcal H}$ with
$u_{\tau}(t)$ defined by
\eqref{def-of-utau} and $t \in {\mathbb R}$ fixed is uniformly bounded
by $\|f\|$, because $(I+S(it;\tau))^{-1}$ is a contraction.
Our next aim is to show that for each fixed $t \in {\mathbb R}$, the
family $\{u_{\tau}(t)\}$ converges in the Hilbert space norm to some
$u(t) \in {\mathcal H}$ as $\tau\to 0+$ and that $u(t) =
(I+itH_P)^{-1}Pf$. To achieve this goal we will have to analyze, in
particular, a new uniform property of the $\tau$-family
$\{u_{\tau}(t)\}$ at the final stage of the argument.
Using the identity \eqref{IplusS} we can invert relation
\eqref{def-of-utau} explicitly as
\begin{align} \label{fIplusS}
f&= (I+ S(it;\tau))u_{\tau}(t) \nonumber \\[.2em]
&= u _{\tau}(t) + \tau^{-1}(I-P(\tau))u _{\tau}(t)
+P(\tau)tG(t\tau)P(\tau)u_{\tau}(t) \nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad
+iP(\tau)tH(t\tau) P(\tau)u _{\tau}(t) \nonumber \\[.2em]
&= (1+ \tau^{-1})(I-P(\tau))u_{\tau}(t)
\oplus P(\tau)[I+ tG(t\tau) +itH(t\tau)] P(\tau)u _{\tau}(t)\,.
\end{align}
Taking the inner product of $f$ with $(I-P(\tau))u _{\tau}(t)$ and
$P(\tau)u _{\tau}$, we get
\begin{equation} \label{innerIminusP}
\langle (I-P(\tau))u _{\tau}(t),f \rangle
= (1+\tau^{-1})\|(I-P(\tau))u _{\tau}(t)\|^2
\end{equation}
and
\begin{align} \label{innerP}
\langle P(\tau)u_{\tau}(t), f \rangle
&= \langle P(\tau)u_{\tau}(t), u_{\tau}(t) \rangle
+ \langle P(\tau)u_{\tau}(t), tG(t\tau)P(\tau)u_{\tau}(t)\rangle
\nonumber\\
& \quad +i\langle P(\tau)u _{\tau}(t),
tH(t\tau) P(\tau)_{\tau}(t)\rangle \nonumber\\[.2em]
&= \|P(\tau)u _{\tau}(t)\|^2
+ \|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2 \nonumber \\
&\quad +i\big(\|(|t|H^+(t\tau))^{1/2} P(\tau)u_{\tau}(t) \|^2 \nonumber \\
&\quad\quad -\|(|t|H^-(t\tau))^{1/2} P(\tau)u_{\tau}(t) \|^2\big)
\end{align}
for $\tau >0\,$; recall that $H(\kappa) = H^+(\kappa) - H^-(\kappa)$.
From the real part of \eqref{innerP} with \eqref{innerIminusP}, we infer,
using Schwarz inequality, that the $\tau$-families $\{P(\tau)u
_{\tau}(t)\}$ and $\{(I-P(\tau))u _{\tau}(t)\}$, as well as
$\{\tau^{-1}(I-P(\tau))u _{\tau}(t)\}$, are uniformly bounded by
$\|f\|$; the last claim naturally extends to
$\{\tau^{-1/2}(I-P(\tau))u _{\tau}(t)\}$ as long as $\tau\le 1$.
Moreover, using the real part of \eqref{innerP} again we can
conclude that the same is true for $\{(|t|G(\tau))^{1/2}
P(\tau)u_{\tau}(t)\}$. Let $f \in {\mathcal H}$ and $t \in {\mathbb
R}$ be arbitrary but fixed. It follows from the obtained uniform
boundedness that for each $t \in {\mathbb R}$, there exist a
(sub)sequence $\{\tau'\}_{0<\tau'\leq 1}$ of the family
$\{\tau\}_{0<\tau\le 1}$ with $\tau' \rightarrow 0+$ and vectors
$u(t),\, u_0(t)$ and $g(t)$ in ${\mathcal H}$ such that the
sequences $\{u_{\tau'}(t)\}$,
$\{(\tau')^{-1/2}(I-P(\tau'))u_{\tau'}(t)\}$ and $\{t^{1/2}
G(|t|\tau')^{1/2}u_{\tau'}(t)\}$ converge weakly to $u(t)$, $u_0(t)$
and $g(t)$, respectively, in ${\mathcal H}$ as $\tau' \to 0+$;
thus $\{P(\tau')u_{\tau'}(t)\}$ converges weakly to $Pu(t)$. We keep in mind
that, with the knowledge we have at the present stage, the limit
$u(t)$ may depend on the chosen sequence
$\{u_{\tau'}(t)\}_{0<\tau'\le 1}$.
\begin{lemma}\label{l:weaklimit}
One has
\begin{equation}\label{weaklimit}
u(t) = Pu(t), \quad u_0(t)=0, \quad g(t)=0,
\end{equation}
and therefore, as $\tau'\to 0+$,
\begin{align} \label{weakconv}
&u_{\tau'}(t) \overset w \longrightarrow u(t) \,,\;\;
({\tau'})^{-1/2}(I-P(\tau'))u_{\tau'}(t)\overset w \longrightarrow 0\,,\;\; \\
& P(\tau')u _{\tau'}(t) \overset w \longrightarrow Pu(t)\,,\;\;\nonumber
(|t|G(t\tau'))^{1/2} P(\tau')u_{\tau'}(t) \overset w \longrightarrow 0\,.
\end{align}
\end{lemma}
\noindent\emph{Proof.}
To begin with, the second limit in \eqref{weakconv} implies, in
particular, that $(I-P(\tau'))u _{\tau'}(t) \to 0$, hence we have
$(I-P)u(t) =0$ or $u(t) =Pu(t) $. Indeed, the uniform boundedness of
$\|(\tau')^{-1/2}(I-P(\tau'))u_{\tau'}(t)\|$ means that
$\|(I-P(\tau'))u_{\tau'}(t)\|\to 0$ which together with the weak convergence
implies $u(t) = Pu(t)$. Moreover, the same
uniform boundedness following from \eqref{innerIminusP} means that
$\tau^{-1}(I-P(\tau)) u_{\tau}(t)$ also converges weakly along
$\{\tau'\}$ which implies $u_0(t)=0$.
Secondly, for any fixed $w \in D[H^{1/2}]$ and $t \in {\mathbb R}$
we have the relation
\begin{align*}
\langle w, g(t) \rangle
& = \lim_{\tau'\to 0+}\, \langle w, (|t|G(t\tau'))^{1/2} P(\tau')u_{\tau'}(t)\rangle \\
& = \lim_{\tau'\to 0+}\, \langle (|t|G(t\tau'))^{1/2}w, P(\tau')u_{\tau'}(t)\rangle \\
& = \langle 0,Pu(t) \rangle =0\,,
\end{align*}
because by Lemma~\ref{l:conv} we have $(|t|G(t\tau'))^{1/2}w \rightarrow 0$
as $\tau'\rightarrow 0+$. This means that $g(t)=0$, because $D[H^{1/2}]$
is dense by assumption.
\qe
\medskip
Next we want to find out what one can deduce from the imaginary part of \eqref{innerP} about the properties of the operators $H^\pm(t\tau)$ introduced in \eqref{KGH2} and \eqref{Hpm}. To this aim we have to employ a topology different from those used up to now. Given a dense subspace $\mathcal{K}$ of $\mathcal{H}$, the symbol $\sigma(\mathcal{H}, \mathcal{K})$ denotes the \emph{weak topology on $\mathcal{H}$ defined by the dual pairing $\langle \mathcal{H}, \mathcal{K} \rangle$} or the \emph{$\mathcal{K}$-weak topology on $\mathcal{H}$} -- see, e.g., \cite[Sect.~IV.20.2]{Ko69} or \cite[Sect.~IV.5]{RS80}. In general, it is weaker (coarser) than the usual weak topology on $\mathcal{H}$, the latter being nothing else than
$\sigma(\mathcal{H}, \mathcal{H})$ in the just introduced notation
We see from the imaginary part of \eqref{innerP} that the difference between the respective elements of the $\tau$-families $\{\|(|t|H^+(t\tau))^{1/2} P(\tau)u_{\tau}(t) \|^2\}$ and $\{\|(|t|H^-(t\tau))^{1/2} P(\tau)u_{\tau}(t) \|^2\}$ is uniformly bounded, because by Schwarz inequality the modulus of the left-hand side in
\begin{align*}
&\mathrm{Im}\,\langle Pu(t),f \rangle \\
& = \lim_{\tau\to 0+}\,\big[\|(|t|H^+(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2
-\|(|t|H^-(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2\big]
\end{align*}
does not exceed $\|f\|^2$. This fact, unfortunately, does not tell us whether the two $\tau$-families of vectors, $\{(|t|H^{\pm}(t\tau))^{1/2} P(\tau)u_{\tau}(t)\}$, are separately
(uniformly) bounded, that is, whether each of them is (uniformly) weakly bounded. We have, however, at least the following result.
\begin{proposition} \label{p:sigmaweak-bdd}
Let $\{u _{\tau'}(t)\}$ be the subsequence appearing in Lemma~\ref{l:weaklimit}, weakly convergent to $u(t)$. Then the $\tau'$-families $\{(|t|H^{\pm}(t\tau'))^{1/2} P(\tau')u_{\tau'}(t)\}$ are \emph{Cauchy} sequences in the $\sigma(\mathcal{H}, D[H^{1/2}])$-weak topology, and as a result they are $\sigma(\mathcal{H}, D[H^{1/2}])$-weakly bounded. Furthermore, the family $\{(|t|H^{-}(t\tau'))^{1/2} P(\tau')u_{\tau'}(t)\}$ converges to zero in this topology as $\tau' \rightarrow 0+$.
\end{proposition}
\begin{proof}
Take an arbitrary $\phi \in D[H^{1/2}]$. We use Lemma~\ref{l:conv} which states, in particular, that $(|t|H^+(t\tau))^{1/2} \phi \rightarrow (|t|H)^{1/2}\phi$ holds when $\tau \rightarrow 0+$. In combination with \eqref{weakconv}, this yields
\begin{align*}
\langle \phi, (|t|H^+(t\tau'))^{1/2} P(\tau')u_{\tau'}(t)\rangle
&= \langle (|t|H^+(t\tau'))^{1/2} \phi, P(\tau') u_{\tau'}(t)\rangle
\nonumber\\
&\longrightarrow \langle (|t|H)^{1/2}\phi, Pu(t)\rangle
\end{align*}
as $\tau' \rightarrow 0+$. For the minus sign, on the other hand, Lemma~\ref{l:conv} says that $(|t|H^-(t\tau))^{1/2}\phi \rightarrow 0$, and this implies
$
\langle \phi, (|t|H^-(t\tau'))^{1/2}P(\tau')u_{\tau'}(t)\rangle
\longrightarrow 0, \quad \tau' \rightarrow 0+,
$
because the self-adjointness of $(|t|H^-(t\tau'))^{1/2}$ in combination with Schwarz inequality gives
\begin{align*}
|\langle (|t|H^-(t\tau'))^{1/2}\phi, P(\tau')u_{\tau'}(t) \rangle|
&\leq \|(|t|H^-(t\tau'))^{1/2}\phi\|\,\|P(\tau')u_{\tau'}(t) \| \\
&\leq \|(|t|H^-(t\tau'))^{1/2}\phi\|\,\|f\|\
\longrightarrow 0\,.
\end{align*}
This yields the stated assertion, including the fact that the family $\{(|t|H^-(t\tau'))^{1/2}P(\tau')u_{\tau'}(t)\}$ is $\sigma(\mathcal{H}, D[H^{1/2}])$-weakly convergent to zero.
\end{proof}
\medskip
On the other hand, it is not clear whether $Pu(t)$ belongs to $D[H^{1/2}]$, that is, whether the plus-sign family $\{(|t|H^{+}(t\tau))^{1/2} P(\tau)u_{\tau}(t)\}$ converges to $H^{1/2}Pu(t)$ in the $\sigma(\mathcal{H}, D[H^{1/2}])$ topology. What is important, the information we were able to deduce in this way about the convergence of the families $\{(|t|H^\pm(t\tau'))^{1/2} P(\tau')u_{\tau'}(t)\}$ is too limited; it does not seem possible to apply the same procedure as we used in Lemma~\ref{l:weaklimit} for $\{(|t|G(t\tau'))^{1/2} P(\tau')u_{\tau'}(t)\}$.
This forces us to seek a different strategy for the proof of Theorem~\ref{th:main'} that would allow us to identify the vector $u(t)$ with $(I +itH_P)^{-1}Pf$, in other words, to demonstrate relation \eqref{Chern} claiming that for every fixed $t\in {\mathbb R}$, the family $\{(I+S(it;\tau))^{-1}f\}$, or otherwise $\{u_{\tau}(t)\}$ in accordance with \eqref{def-of-utau}, converges to $(I+itH_P)^{-1}Pf$ in the Hilbert space norm as $\tau \to 0+$.
The argument is somewhat subtle and relies on our previous work \cite{EI05}, see also \cite{EINZ07}, about the Zeno product formul{\ae} related to
Theorems~\ref{th:main} and \ref{th:main'}. In those papers we demonstrated that the family $\{u_{\tau}(t)\}$ has a unique limit, namely
$(I+itH_P)^{-1}Pf$, as $\tau \to 0+$. As we mentioned in the introduction, however, the obtained convergence referred neither to the norm of ${\mathcal H}$ nor even to the weak topology. Precisely speaking, it is shown in \cite{EI05} that \eqref{Pboth}--\eqref{Pleft} and \eqref{P(t)both}--\eqref{P(t)left} hold in the topology of the Fr\'echet space $L^2_{\text{\rm loc}}({\mathbb R}; {\mathcal H}) = L^2_{\text{\rm loc}}({\mathbb R})\otimes {\mathcal H}$ of the ${\mathcal H}$-valued strongly measurable functions $v(\cdot)$ on ${\mathbb R}$ such that the $\|v(\cdot)\|$ are locally square integrable there, equipped with the topology induced by the family of semi-norms $v \mapsto \big(\int_a^b \|v(t)\|^2\, \mathrm{d}t\big)^{1/2}$ for any bounded interval $(a,b)$ with $a<b$. This follows from \cite[Lemma 3.1, p.~200]{EI05} which says that for every bounded closed interval $[a,b] \subset {\mathbb R}$ one has
\begin{equation} \label{L2locCov}
\int_a^b \|u_{\tau}(t) - (I+ itH_P)^{-1}Pf\|^2\, \mathrm{d}t
\; \longrightarrow \, 0 \quad\mathrm{as} \;\: \tau \to 0+\,;
\end{equation}
the reader may compare this result with \eqref{Chern}.
Note also that \eqref{L2locCov} implies that for every
$f \in {\mathcal H}$, there exist a set
$M_f \subset {\mathbb R}$ of Lebesgue measure zero, possibly dependent
on $f$, and a (sub)sequence $\{\tau'\}_{0<\tau' \leq 1}$ of
$\{\tau\}_{0<\tau \leq 1}$ along which it holds that for all $s \in
{\mathbb R}\setminus M_f$,
\begin{equation}\label{a.e.Cov}
u_{\tau}(s) \longrightarrow (I+ isH_P)^{-1}Pf
\quad\text{\rm in\;the\;norm\;of}\;\: {\mathcal H},
\end{equation}
in other words, $u_{\tau'}(s) \longrightarrow (I+ isH_P)^{-1}Pf$ as $\tau'\to 0$. With the coming argument in mind, it is useful to note that the set $\mathbb {R} \setminus M_f$ at which the convergence takes place is dense in ${\mathbb R}$. Furthermore, since ${\mathcal H}$ is separable by assumption, we can choose a countable dense subset ${\mathcal D} := \{f_l\}_{l=1}^{\infty}$ in ${\mathcal H}$. Putting $M = M_{\mathcal D} := \cup_{l=1}^{\infty} M_{f_l}$, which is also a set of Lebesgue measure zero, we may then say that \eqref{a.e.Cov} holds for all $s \in {\mathbb R}\setminus M$ and for every $f \in {\mathcal D}$, and hence, in view of the density, also for every $f \in {\mathcal H}$.
Moreover, we note that $s=0$ does not belong to $M_f$ for any $f \in {\mathcal H}$, and therefore it neither belongs to $M$. Indeed, using Lemma~\ref{l:inverse} and \eqref{inverse-of-IplusS} in combination with the continuity of $\tau \mapsto P(\tau)$ it is easy to see that
\begin{equation}\label{Cov-at-0}
u_{\tau}(0) = (I+S(0;\tau))^{-1}f
= (I+\tau^{-1})^{-1}(I-P(\tau)) \oplus P(\tau)f
\longrightarrow Pf,
\end{equation}
which means $0 \notin M_f$.
\medskip
Now let us first briefly outline our plan of \emph{how to complete the proof of Theorem~\ref{th:main'}}; we recall that one has to verify relation \eqref{Chern} showing that for every $t\in {\mathbb R}$, $\{u_{\tau}(t)=(I+S(it;\tau))^{-1}f\}$ converges to $(I+itH_P)^{-1}Pf$ in the Hilbert space norm for all $t$ as $\tau \to 0+$. Let us recall here that, as already mentioned in the text following \eqref{Chern}, the use of Chernoff's theorem only requires to establish the convergence in \eqref{Chern} pointwise for all $t \in {\mathbb R}$. Here and in the following we keep in mind that $\{u_{\tau}(t)\}_{0<\tau\leq 1}$ is uniformly bounded in both $t$ and $0<\tau\leq 1$, i.e.
\begin{equation} \label{utau:bdd-tau-t}
\sup_{0<\tau\leq 1}\sup_{t\in {\mathbb R}} \|u_{\tau}(t)\| \leq \|f\|,
\end{equation}
since $(I+S(it;\tau))^{-1}$ is a contraction. We need not strive to show its local uniformity, however, the following reasoning will establish this property with respect to $t \in {\mathbb R} \setminus \{0\} $. We will proceed in four steps demonstrating validity of the following claims:
\smallskip
I. The $\tau$-family $\{u_{\tau}(t)\}_{0<\tau \leq 1}$ of vectors
$u_{\tau}: {\mathbb R} \ni t \mapsto u_{\tau}(t) \in {\mathcal H}$ is
equicontinuous in $t \in {\mathbb R} \setminus \{0\}$ with respect to
the strong topology (i.e., Hilbert space norm) of ${\mathcal H}$.
\smallskip
II. The family $\{u_{\tau}(t)\}_{0<\tau\leq 1}$ converges as $\tau\to 0+$ for each fixed $t \in {\mathbb R}$ to some $u(t) \in{\mathcal H}$ in the weak topology of ${\mathcal H}$, and furthermore, the convergence is \emph{even} locally uniform with respect to $t \in {\mathbb R} \setminus \{0\}$. The limit function $u(t)$ turns out to be continuous in $t \in {\mathbb R}$ in the weak topology of ${\mathcal H}$.
\smallskip
III. The limit satisfies
\begin{equation} \label{u(t)=}
u(t) = (I+ itH_P)^{-1}Pf \quad \text{for\,\, all}\,\, t.
\end{equation}
\smallskip
IV. Finally, the family $\{u_{\tau}(t)\}_{0<\tau\leq 1}$ converges as $\tau\to 0+$ for any fixed $t \in {\mathbb R}$ to $u(t) \equiv (I+ itH_P)^{-1}Pf$ in the strong topology of ${\mathcal H}$, and furthermore, the convergence is \emph{even} locally uniform with respect to $t \in {\mathbb R} \setminus \{0\}$.
\medskip
\emph{Step I.} The claim is expressed in the following lemma about the
local equicontinuity of the $\tau$-family $\{u_{\tau}(t)\}$
in a neighbourhood of $t=s$ in ${\mathbb R}\setminus \{0\}$. It plays
a crucial role in concluding the proof of Theorem~\ref{th:main'}.
\begin{lemma} \label{l:equiconti}
Let $f \in {\mathcal H}$.
Then the $\tau$-family $\{u_{\tau}(t)\}_{0<\tau \leq 1}$ of vectors
$u_{\tau}: {\mathbb R}\setminus \{0\} \ni t \mapsto u_{\tau}(t) \in {\mathcal H}$
is equicontinuous locally in $t$ with respect to the strong topology
on ${\mathcal H}$. More explicitly, for every $\varepsilon>0$ and for
every $s \in {\mathbb R}\setminus \{0\}$ there exists an
$s$-dependent constant $\delta=\delta(f;\varepsilon;s)>0$ such that if
$t,s > 0$ or $\:t,s< 0$ with $|t-s|<\delta$, then
$\|u_{\tau}(t)-u_{\tau}(s)\| < \varepsilon$ holds for all $0<\tau \leq 1$.
\end{lemma}
\smallskip
We postpone for the moment the proof of Lemma~\ref{l:equiconti}, returning
to it in Sect.~\ref{s: proof-lemma}, and accept its claim, to finish
first the proof of Theorem~\ref{th:main'}.
\medskip
\emph{Step II.}
Without loss of generality we may suppose $f \not= 0$. If $t=0$, the family $\{u_{\tau}(0)\}_{0<\tau\leq 1}$ converges strongly to $Pf$ as $\tau\to 0+\,$ as mentioned above, cf. \eqref{Cov-at-0}. Consider thus a nonzero $t \in {\mathbb R}\setminus \{0\}$ and take \emph{arbitrary} (sub)se\-quen\-ce $\{\tau'\}_{0<\tau' \leq 1}$ of $\{\tau\}_{0<\tau \leq 1}$ with $\tau'\to 0+$. By Lemma~\ref{l:equiconti}, to be yet proven, we see the $\tau'$-family $\{u_{\tau'}(t)\}_{0<\tau' \leq 1}$ is equicontinuous in the strong topology and therefore in the weak topology, because the `full' $\tau$-family $\{u_{\tau}(t)\}_{0<\tau\leq 1}$ is. We observed in \eqref{utau:bdd-tau-t} that the vectors $u_{\tau}(t)$ with any $t \in {\mathbb R}\setminus \{0\}$ and $\tau\in(0,1]$ lie in the closed ball $\bar{B}(0;\|f\|) \subset {\mathcal H}$ with the center at the origin and radius $\|f\|$, which is weakly compact. To proceed, note that $\bar{B}(0;\|f\|)$ is {\it metrizable in the weak topology}, since ${\mathcal H}$ is separable by assumption, see e.g. \cite[Problem/Solution 18, p.~12 and 181]{Ha67}. Thus the equicontinuity holds with respect to the metric on the space $\bar{B}(0;\|f\|)$ equivalent to the weak topology on it. Then, by virtue of the Ascoli--Arzel\`a theorem, see e.g. \cite[p.~81]{KeNa76} or \cite[Thm.~1.5.3]{Si15}, there exists a (sub)sequence $\{\tau^{\prime\prime}\}_{0<\tau^{\prime\prime}\leq 1}$ of $\{\tau'\}_{0<\tau' \leq 1}$ with $\tau^{\prime\prime} \to 0+$, along which $\{u_{\tau}(t)\}$ converges \emph{weakly} to some limit $u(t)$, and moreover, the convergence is \emph{(locally) uniform} in $t \in {\mathbb R}\setminus \{0\}$. This means that the numerical family $\{\langle \psi, u_{\tau^{\prime\prime}}(t) \rangle\}$ converges for every fixed $\psi \in {\mathcal H}$ as $\tau^{\prime\prime} \to 0+$, (locally) uniformly in $t \in {\mathbb R}\setminus \{0\}$. Putting this together with the case $t=0$ mentioned above, we see that the limit $\langle \psi, u(t)\rangle$ is $t$-continuous everywhere in ${\mathbb R}$, in other words, that $u(t)$ is $t$-continuous everywhere in ${\mathbb R}$ with respect to the weak topology of ${\mathcal H}$.
\medskip
\emph{Step III.} We have established above two ways of convergence of the original $\tau$-family $\{u_{\tau}(t)\}_{0<\tau \leq 1}$ of ${\mathcal H}$-valued $t$-continuous functions as $\tau\to 0+$, or more specifically, convergence with respect to two different topologies. One is the convergence to $u(t)$ in the weak topology of ${\mathcal H}$, (locally) uniformly for $t \in {\mathbb R}\setminus \{0\}$, and the other is the convergence to $(I+itH_P)^{-1}Pf$ in $L^2_{\text{\rm loc}}({\mathbb R}; {\mathcal H})$ in the strong, and therefore also weak sense. This allows us to conclude that these two limit vectors coincide for all $t$, in other words, to establish the equality \eqref{u(t)=}. Indeed, for any $\psi \in {\mathcal H}$ and an arbitrary bounded interval $(a,b)$ with $a<b$ we have the following elementary estimate,
\begin{align*}
&\int_a^b |\langle \psi, u(t)- (I+itH_P)^{-1}Pf\rangle|\, \mathrm{d}t \\
&\leq \int_a^b |\langle \psi, u(t)- u_{\tau}(t)\rangle|\, \mathrm{d}t
+\int_a^b |\langle \psi, u_{\tau}(t)- (I+itH_P)^{-1}Pf\rangle|\, \mathrm{d}t\,,
\end{align*}
the last term of which tends to zero as $\tau\to 0+$ by \eqref{L2locCov}. At the same time, the first term on the right-hand side also tends to zero in view of the weak convergence and the dominated convergence theorem. Consequently the integrated expression on the left-hand side vanishes identically,
$$
\langle \psi, u(t)- (I+itH_P)^{-1}Pf\rangle = 0,
$$
for all $t$ and for all $\psi \in {\mathcal H}$, which yields the desired claim \eqref{u(t)=}.
In this way, we have obtained the relations
\begin{align}
&u(t) =Pu(t) \in D[H_P] = D[(H^{1/2} P)^*(H^{1/2} P)]\,,\nonumber\\[-.6em]
\label{limits2}\\[-.6em]
&Pf= P(I+ itH_P)u(t) = P[I+it(H^{1/2} P)^*(H^{1/2} P)]u(t)\,, \nonumber
\end{align}
which show that $\{u_{\tau}(t)\}$ converges to $u(t)= (I+ itH_P)^{-1}Pf$ in the weak topology of ${\mathcal H}$
along $\tau \to 0+$, and therefore also along $\tau\to 0+$.
\medskip
\emph{Step IV.} It remains to demonstrate that $\{u_{\tau}(t)\}$ converges to $u(t)= (I+ itH_P)^{-1}Pf$ also \emph{in the Hilbert space norm} for all $t \in {\mathbb R}$. For the sake of completeness we shall show that the same claim can also be made about the $\tau$-family $\{(tG(\tau))^{1/2}P(\tau)u_{\tau}(t)\}$ in \eqref{innerP}; note that the analogous question concerning $\{(|t|H^{\pm}(t\tau))^{1/2}P(\tau) u_{\tau}(t)\}$ also appearing in \eqref{innerP} is more complicated and we are able to provide only a partial answer to it, cf. Proposition~\ref{p:sigmaweak-bdd}.
Since we have already established the weak convergence of the $\tau$-family
$\{u_{\tau}(t)\}$ for all $t$, we need only to show that the $\tau$-families
of the norms of these vectors converge. To this end, we observe again the real
part of \eqref{innerP}, however, replacing now $f$ by $P(\tau)f$ in the inner
product; we write also the imaginary part for purpose of a further discussion.
Writing the left-hand side of \eqref{innerP} as
$\langle u_{\tau}(t),P(\tau)f \rangle$, we obtain for its real and imaginary
part
\begin{align}
\mathrm{Re}\,\langle u_{\tau}(t),P(\tau)f \rangle
&= \|P(\tau)u_{\tau}(t)\|^2 +\|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2, \label{real} \\[.5em]
\mathrm{Im}\,\langle u_{\tau}(t),P(\tau)f \rangle
&= \|(|t|H^+(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2 \nonumber\\
&\qquad\qquad\qquad
- \|(|t|H^-(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2.
\label{imaginary}
\end{align}
In view of \eqref{limits2}, the continuity of the projection family $\{P(\tau)\}$, and the weak convergence of $\{u_{\tau}(t)\}$ as $\tau\rightarrow 0+$, which we have already established, the left-hand sides of the last two relations converge, leading to the following limits
\begin{align}
\text{\rm Re}\,\langle u_{\tau}(t),P(\tau)f \rangle
&\longrightarrow \langle u(t),Pu(t) \rangle = \|Pu(t)\|^2.
\label{real2} \\[.5em]
\text{\rm Im}\,\langle u_{\tau}(t),P(\tau)f \rangle
&\longrightarrow \langle u(t),tH_Pu(t) \rangle
= \|(|t|H)^{1/2}Pu(t)\|^2\label{imaginary2}
\end{align}
valid for \emph{all} $t$. Comparing now the right-hand sides of relations \eqref{real} and \eqref{real2} we get
\begin{align*}
\|Pu(t)\|^2 &= \lim_{\tau \to 0+}\, \big[\|P(\tau)u_{\tau}(t)\|^2
+\|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2 \big]\\
&= \liminf_{\tau \to 0+}\, \big[\|P(\tau)u_{\tau}(t)\|^2
+ \|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2 \big] \\
&\geq \liminf_{\tau \to 0+}\, \|P(\tau)u_{\tau}(t)\|^2
+\liminf_{\tau \to 0+}\, \|(|t|G(\tau))^{1/2}P(\tau)u_{\tau}(t)\|^2 \\
&\geq \|Pu(t)\|^2 + \|0\|^2 = \|Pu(t)\|^2,
\end{align*}
which means that the two terms on the right-hand side of \eqref{real} converge
to $\|Pu(t)\|^2$ and to zero, respectively, yielding thus the sought
convergence in the Hilbert space norm along the sequence $\tau \to 0+$,
\begin{equation} \label{strongconv}
P(\tau)u_{\tau}(t) \longrightarrow Pu(t)\,, \quad
(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t) \longrightarrow 0\,.
\end{equation}
The first convergence in \eqref{strongconv} also implies
$u_{\tau}(t) \rightarrow u(t) = Pu(t)$, i.e. convergence of
$\tau$-family$\{u_{\tau}(t)\}$ to $u(t)=Pu(t)$ in the Hilbert space
norm, because we know already that $(I-P(\tau'))u _{\tau'}(t) \to
0$.
This shows nothing but our sought statement of the convergence \eqref{Chern} for the family $\{u_{\tau}(t)\}$ which, with the reference to Chernoff's criterion \cite{Ch68, Ch74}, concludes the proof of Theorem~\ref{th:main'}.
\qed
\medskip
The proof of Lemma~\ref{l:equiconti} will be given in the next section.
\section{Proof of Lemma~\ref{l:equiconti}}
\label{s: proof-lemma}
Recall first that the Hilbert space ${\mathcal H}$ we are handling is assumed to be separable which is the property we needed to construct the exceptional set $M$, cf. the text following \eqref{a.e.Cov}. Recall also
the notation we introduced in \eqref{def-of-utau} for the vector obtained by application of the operator $(I+ S(it;\tau))^{-1}$ to an arbitrary $f \in {\mathcal H}$,
\begin{align}
u_{\tau}&(t)
= (I+ S(it;\tau))^{-1}f \nonumber\\
&= \big\{(1+\tau^{-1})^{-1}(I-P(\tau))
\oplus
[P(\tau)(I+ tG(t\tau) + itH(t\tau))P(\tau)]^{-1}\big\}f \nonumber\\
&=\big\{(1+\tau^{-1})^{-1}(I-P(\tau))\oplus T^P(t;\tau)\big\}f\,,
\end{align}
where the operator
\begin{equation} \label{def-TP(t;tau)}
T^P(t;\tau) := [P(\tau)(I+ tG(t\tau) +itH(t\tau))P(\tau)]^{-1}
\end{equation}
may be also considered on the whole Hilbert space ${\mathcal H}$,
although in the proper sense it is an operator with the domain and range
included in the closed subspace $P(\tau){\mathcal H}$; one may regard it
as vanishing on the orthogonal complement $(I-P(\tau)){\mathcal H}$.
Next, for fixed $t,s >0$ or $t,s <0$, we put
\begin{align} \label{D(t,s;tau)}
D(t,s;\tau)f
:&= u_{\tau}(t) - u_{\tau}(s) \nonumber\\
&= [(I+ S(it;\tau))^{-1}- (I+ S(is;\tau))^{-1}]f \nonumber\\
&= (I+ \tau^{-1})^{-1}(I-P(\tau))
\oplus \big[T^P(t;\tau) -T^P(s;\tau)\big]f\,.
\end{align}
With the above direct sum decomposition in mind, one may for a fixed $\tau$ deal with the operator on the subspace $P(\tau){\mathcal H}$,
\begin{equation} \label{TP}
T^P(t,s;\tau) := D(t,s;\tau)\restriction_{P(\tau){\mathcal H}}
= T^P(t;\tau) -T^P(s;\tau)\,,
\end{equation}
however, since $\tau$ is varying, we have to consider it on the whole space ${\mathcal H}$. Note that the $\tau$-family $\{D(t,s;\tau)\}_{0<\tau \leq 1}$ is
strongly continuous in $0<\tau \leq 1$, and uniformly bounded, i.e. $\|D(t,s;\tau)\| \leq 2$, because both $(I+ S(it;\tau))^{-1}$ and $(I+ S(is;\tau))^{-1}$ are contractions. Note also that we have $ D(t,s;\tau) = P(\tau)D(t,s;\tau)$.
\medskip
To verify the assertion of Lemma~\ref{l:equiconti}, we need to show that for any fixed $f \in {\mathcal H}$, the difference $D(t,s;\tau)f$ in \eqref{D(t,s;tau)} with $t,s \in {\mathbb R} \setminus \{0\}$ converges to zero in the Hilbert space norm as $|t-s| \to 0$, and that the convergence is uniform with respect to $\tau\in(0,1]$.
We use a small trick showing first that it is sufficient to establish the claim of Lemma~\ref{l:equiconti} \emph{under the additional assumption} that one of the $t$ and $s$, say the latter, belongs to ${\mathbb R} \setminus (M\cup \{0\})$. Indeed, if this is the case, i.e. if for any fixed $s\in{\mathbb R} \setminus (M\cup \{0\})$ and arbitrary $f \in {\mathcal H},\;\varepsilon>0$, there is a
$\delta =\delta(f;\varepsilon;s)>0$ such that
$$
\|u_{\tau}(t) -u_{\tau}(s)\| < \tfrac{\varepsilon}2 \;\;
\text{\rm holds for all}\;\; t \in (s-\delta,s+\delta)\;\; \text{\rm and}\;\; \tau\in (0,1],
$$
the lemma is valid in the general case as well. To see that, we take any two points $t_1,\,t_2 \in {\mathbb R} \setminus \{0\}$ in the vicinity of the chosen $s$ satisfying $|t_i-s|<\tfrac{\delta}2$ for $i=1,2$, then
\begin{align*}
&|t_1-t_2| \leq |t_1-s|+|s-t_2| < \tfrac{\delta}2+ \tfrac{\delta}2=\delta, \\
&\|u_{\tau}(t_1) -u_{\tau}(t_2)\|
\leq \|u_{\tau}(t_1) -u_{\tau}(s)\| +\|u_{\tau}(s) -u_{\tau}(t_2)\|
< \tfrac{\varepsilon}2 +\tfrac{\varepsilon}2 = \varepsilon
\end{align*}
holds independently of $\tau$. This yields the `full' claim of Lemma~\ref{l:equiconti} in view of the fact that the set ${\mathbb R} \setminus (M\cup \{0\})$ from which the number $s$ is chosen is dense in ${\mathbb R} \setminus \{0\}$.
\smallskip
Let us thus turn to the nontrivial part of the proof which consists of establishing the assertion of Lemma~\ref{l:equiconti} for $s \in {\mathbb R} \setminus (M\cup \{0\})$ and $t \in {\mathbb R} \setminus \{0\}$; the aim is to complete the proof of our main result in the forthcoming Lemma~\ref{l:estimate-T}(ii). We will work out the argument assuming that $t,s > 0$, the case with the opposite signs is completely analogous.
To begin with, we use functional calculus with \eqref{KGH}, \eqref{KGH2}
to rewrite the differences containing $H(t\tau)$ and $G(t\tau)$
in \eqref{D(t,s;tau)} as
\begin{align} %
\hspace{-1em}tH(t\tau) - sH(s\tau)
&= \tfrac{\sin t\tau H - \sin s\tau H}{\tau}
= \tfrac{2}{\tau} \cos(\tfrac{t+s}2\tau H)\, \sin(\tfrac{t-s}2\tau H),
\label{tH(t)-sH(s)}\\
tG(t\tau) - sG(s\tau)
&= \tfrac{-\cos t\tau H + \cos s\tau H}{\tau}
= \tfrac{2}{\tau} \sin(\tfrac{t+s}2\tau H)\, \sin(\tfrac{t-s}2\tau H).
\label{tG(t)-sG(s)}
\end{align}
To simplify expressions in the discussion to follow, we recall the quantity
$K(\kappa)$ introduced in \eqref{KGH} which for $\kappa = t\tau$ allows us
to write
\begin{equation}
I + tK(t\tau) = I + tG(t\tau) +itH(s\tau)
= I + \tfrac{I-\cos t\tau H}{\tau} + i\tfrac{\sin t\tau H}{\tau}\,,
\label{KGH3}
\end{equation}
and similarly for $k=s\tau$. Furthermore, for a given $\tau\in (0,1]$ we
intro\-duce the self-adjoint operator
\begin{equation} \label{Htau}
H_{\tau} := H(I +\tau H)^{-1}
\end{equation}
which is positive and bounded, and note that $I+|s|H_\tau$ has a bounded
inverse for any $s\in\mathbb{R}$. The difference $D(t,s;\tau)$ in
\eqref{D(t,s;tau)} which can be identified with its nontrivial part
$T^P(t,s;\tau)$ in \eqref{TP} can be then rewritten as
\begin{align}
D(t,s;\tau) &= T^P(t,s;\tau) \nonumber\\
&= [P(\tau)(I+ tK(t\tau))P(\tau)]^{-1}
- [P(\tau)(I+ sK(s\tau))P(\tau)]^{-1} \nonumber\\
&= [P(\tau)(I+ tK(t\tau))P(\tau)]^{-1} \nonumber\\
&\qquad \cdot \big\{P(\tau)(I+ sK(s\tau))P(\tau)
-P(\tau)(I+ tK(t\tau))P(\tau)\big\} \nonumber\\
&\quad\qquad \cdot [P(\tau)(I+ sK(s\tau))P(\tau)]^{-1} \nonumber\\
&= [P(\tau)(I+ tK(t\tau))P(\tau)]^{-1}
[P(\tau)(sK(s\tau) -tK(t\tau))P(\tau)] \nonumber\\
&\qquad \cdot [P(\tau)(I+ sK(s\tau))P(\tau)]^{-1} \nonumber\\
&= [P(\tau)(I+ tK(t\tau))P(\tau)]^{-1} \nonumber\\
&\qquad \cdot [(sK(s\tau) -tK(t\tau))(I+ |s|H_{\tau})^{-1}] \nonumber\\
&\quad\qquad \cdot (I+ |s|H_{\tau}) P(\tau)[P(\tau)(I+ sK(s\tau))P(\tau)]^{-1} \nonumber\\
&=: T_1(t;\tau)\, T_2(t,s;\tau)\, T_3(s;\tau),
\label{split-D(t,s;tau)to123}
\end{align}
where we have introduced
\begin{align}
&T_1(t;\tau) = T^P(t;\tau) = [P(\tau)(I+ tK(t\tau))P(\tau)]^{-1},
\label{T1tau} \\
&T_2(t,s;\tau) = (sK(s\tau) -tK(t\tau))(I+ |s|H_{\tau})^{-1},\label{T2tau} \\
&T_3(s;\tau)
=(I+ |s|H_{\tau})P(\tau)[P(\tau)(I+ sK(s,\tau))P(\tau)]^{-1},\label{T3tau}
\end{align}
which are all bounded operators on $P(\tau){\mathcal H}$ as well as on ${\mathcal H}$; in \eqref{T2tau} we are able to drop the projections $P(\tau)$ appearing in the last formula since the operator \eqref{def-TP(t;tau)} maps the subspace $P(\tau)\mathcal{H}$ onto itself.
Next we consider the following two operator-valued functions, which may be thought of as the $\tau$-limits of the families $\{T_1(t;\tau)\}_{0<\tau\leq 1}\,$ in \eqref{T1tau}, and $\{T_3(s;\tau)\}_{0<\tau\leq 1}\,$ in \eqref{T3tau}, namely
\begin{align}
T_1(t) :&= [(I+ itH_P)^{-1}P], \label{T1} \\
T_3(s) :&= (I+ |s|H)P[(I+ isH_P)^{-1}P]
= (I+ |s|H)[(I+ isH_P)^{-1}P], \label{T3}
\end{align}
where we can remove again one $P$ from the second expression of \eqref{T3} since $(I+ isH_P)^{-1}$ maps $P{\mathcal H}$ to itself. We already know from \eqref{strongconv} that $T_1(t)$ is the strong limit as $\tau\to 0+$ of the family $\{T_1(t;\tau)\}$ of contractions, for the moment at least as long as $t \in {\mathbb R}\setminus (M \cup \{0\})$.
Next we are going to show that $T_3(s)$ can be extended to a bounded operator on ${\mathcal H}$. We begin with a crucial observation.
\begin{lemma} \label{l:T0}
Let $H$ be our nonnegative self-adjoint operator acting in ${\mathcal H}$ and $H_P$ the self-adjoint operator introduced in Sect. 2 referring to the orthogonal projection $P$. Consider the operator
\begin{equation} \label{def-T0}
T_0 := (I+H)[(I+ H_P)^{-1}P]
\end{equation}
in ${\mathcal H}$ for which we have: \text{\rm (i)} the domain and range of $T_0$ are
\begin{align}
D[T_0] &= P(I+H_P) D[HP]
= P(I+H_P) (D[H]\cap P{\mathcal H}) \oplus (P{\mathcal H})^{\perp}, \nonumber \\
R[T_0] &= (I+H) D[HP]
= (I+H) (D[H]\cap P{\mathcal H}),
\end{align}
as $T_0$ is the direct sum, $T_0=T_0\!\restriction_{P{\mathcal H}} \oplus\, 0\,$, in accordance with \eqref{HP}, \\
\text{\rm (ii)} and, in addition,
\begin{equation} \label{T0-eq}
T_0\, g =g\,, \quad g \in D[T_0\!\restriction_{P{\mathcal H}}],
\end{equation}
thus $T_0$ can be extended to a bounded operator $\tilde{T}_0$ on ${\mathcal H}$ such that
\begin{align} \label{T0}
&\text{\it the closure of \,$T_0\!\restriction_{P{\mathcal H}}$\,
is the identity operator\,$I_{P{\mathcal H}}$\,\,on\,} P{\mathcal H}, \nonumber\\
&\text{\it and}\,\,\tilde{T_0} = I_{P{\mathcal H}} \oplus 0 \,\,\,
\text{\rm on }\,\, {\mathcal H}
= P{\mathcal H} \oplus (P{\mathcal H})^{\perp}. \end{align}
In particular, $\tilde{T_0}$ is a contraction.
\end{lemma}
\begin{proof}
Product of linear operators $A$ and $B$ in ${\mathcal H}$ has the domain $D[AB] = B^{(-1)}D[A] := \{\,g \in D[B]:\, Bg \in D[A]\,\}$. Since $P$ commutes with $H_P$, we can rewrite \eqref{def-T0} as $T_0 := [(I+H)P][(I+ H_P)^{-1}P]$ with the domain
\begin{align*}
D[T_0] &= [P(I+H_P)^{-1}]^{(-1)}D[(I+H)P] \\
&= \{\,g \in D[(I+ H_P)^{-1}P]:\,(I+ H_P)^{-1}Pg \in D[(I+H)P]\} \\
&= \{\,g \in D[H_P]:\,(I+ H_P)^{-1}Pg \in D[HP]\} \\
&= P(I+H_P)D[HP]= P(I+H_P) (D[H]\cap P{\mathcal H}) \oplus (P{\mathcal H})^{\perp}
\end{align*}
and the range
\begin{align*}
R[T_0] &= T_0D[T_0] \\
&= [(I+H)P][(I+ H_P)^{-1}P]P(I+H_P)D[HP] \\
&= [(I+H)P]D[HP]
= (I+H)(D[H]\cap P{\mathcal H}).
\end{align*}
Let us now turn to the claim (ii). Just as $H_P$ is the self-adjoint operator in ${\mathcal H}$ associated with the quadratic form $u \mapsto \|H^{1/2}Pu\|^2$ defined on $D[H^{1/2}P] = (D[H^{1/2}]\cap P{\mathcal H}) \oplus (P{\mathcal H})^{\perp}$, the self-adjoint operator $P(I+H_P)$ is associated with the form $u \mapsto \|(I+H)^{1/2}Pu\|^2$ defined in view of the inequalities
$$
\frac{1}{\sqrt{2}}(I+H^{1/2}) \le (I+H)^{1/2} \le I+H^{1/2}.
$$
on the same domain $D[(I+H)^{1/2}P] = D[H^{1/2}P]$. Consequently,
\begin{equation} \label{I+H_P-1}
P(I+H_P) = P(I+H)_P = ((I+H)^{1/2}P)^* (I+H)^{1/2}P\,.
\end{equation}
We are going to use it to show \eqref{T0-eq}. The adjoint $((I+H)^{1/2}P)^*$ to $(I+H)^{1/2}P$ is a closed extension of the closable (in general, non-closed) operator $P(I+H)^{1/2}$, in other words, $((I+H)^{1/2}P)^*\supset P(I+H)^{1/2}$, which yields
$$
((I+H)^{1/2}P)^*(I+H)^{1/2}P \supset
P(I+H)^{1/2}(I+H)^{1/2}P = P(I+H)P,
$$
i.e. the operator on the left-hand side is an extension of the operator on the right. As both sides are invertible, the analogous inclusion holds for their inverses,
$$
[(I+H_P)^{-1}P]
= [((I+H)^{1/2}P)^*(I+H)^{1/2}P]^{-1} \supset [P(I+H)P]^{-1}.
$$
It follows that for any $g \in D[T_0]$ specified in \eqref{T0} we have
$$
T_0\, g = (I+H)[(I+H_P)^{-1}P]g = (I+H)[P(I+H)P]^{-1}g = Pg,
$$
which yields the desired claim \eqref{T0-eq}, since $T_0$ is reduced by the projection $P$ and $D[T_0\!\restriction_{P{\mathcal H}}] \subseteq P{\mathcal H}$. Thus we see that the closure of $T_0\!\restriction_{P{\mathcal H}}$ is the identity operator $I_{P{\mathcal H}}$ on $P{\mathcal H}$, and $\tilde{T_0}$ as the closed extension of $T_0$ to the whole ${\mathcal H}$ has the norm not exceeding one.
\end{proof}
\begin{lemma} \label{l:estimate-T3}
The operator $T_3(s)$ in \eqref{T3}, acting in ${\mathcal H}$ with the domain
$P(I+isH_P)D[HP]
= (I+isH_P)(D[H]\cap P{\mathcal H})$, can be extended to a bounded operator
on the whole space ${\mathcal H}$ with the norm satisfying
$\|T_3(s)\|\leq \sqrt{2}$.
\end{lemma}
\begin{proof}
Note first that the claim of Lemma~\ref{l:T0} remains valid when we replace $H_P,\,H$ by $|s|H_P,\,|s|H$, respectively, for any $s \not= 0$.
We rewrite the operator $T_3(s)$ in \eqref{T3} as
\begin{align*}
T_3(s)
&= \big\{(I+ |s|H)[(I+ |s|H_P)]^{-1}P]\big\} \\
&\qquad\qquad \cdot \big\{[P(I+ |s|H_P)][(I+ isH_P)^{-1}P]\big\}\,.
\end{align*}
By spectral theorem the norms of the first and second factors on the right-hand side are one and $\sqrt{2}$, respectively, independently of $s$, and Lemma~\ref{l:T0} (ii) in combination with the above observation implies that the first factor is a contraction, which gives $\|T_3(s)\| \leq \sqrt{2}$.
\end{proof}
\smallskip
In the next step, we turn to investigation of the second and third factors, \eqref{T2tau} and \eqref{T3tau}, of the operator $T^P(t,s;\tau)$ in \eqref{split-D(t,s;tau)to123}. We begin with proving a crucial property of the $\tau$-family $\{T_3(s;\tau)\}_{0<\tau\leq 1}$ with $s \in {\mathbb R} \setminus (M \cup \{0\})$, where the set $M \subseteq {\mathbb R} $ of Lebesgue measure zero was introduced in the text following \eqref{a.e.Cov}; recall that we adopted the separability assumption. After doing that, we will focus on the $\tau$-family $\{T_2(t,s;\tau)\}_{0<\tau\leq 1}$ with $t,\,s \in {\mathbb R} \setminus \{0\}$.
\begin{lemma} \label{l:unif-bound-T3}
Let the Hilbert space ${\mathcal H}$ be separable and consider a number
$s \in {\mathbb R} \setminus (M \cup \{0\})$, so that for every vector
$f \in {\mathcal H}$ the $\tau$-family
$\{[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}f\}$ converges in the Hilbert
space norm to $[(I+isH_P)^{-1}P]f$ as $\tau\to 0+$.
Then the following claims are valid:
\text{\rm (i)} For fixed $s$, the operator family $\{T_3(s;\tau)\}_{0<\tau\leq 1}$ defined by \eqref{T3tau} is uniformly bounded on ${\mathcal H}$, and converges strongly to $T_3(s)$ in \eqref{T3} as $\tau\to 0+$.
\text{\rm (ii)} To be specific, for a fixed $s \in {\mathbb R} \setminus (M \cup \{0\})$ there is a constant $C_{T_3}(s)\ge\sqrt{2}$ such that
\begin{equation}
\|T_3(s;\tau)\| \leq C_{T_3}(s) \quad \text{for all}\;\: \tau \in (0,1]\,.
\label{unif-bound-T3}
\end{equation}
\end{lemma}
\medskip
\begin{proof}
(i) Our aim is to verify that $T_3(s;\tau)g$ converges for $g \in {\mathcal H}$ to $T_3(s)g$ as $\tau\to 0+$ in the Hilbert space norm and to use this fact to establish the uniform boundedness of $\{T_3(s;\tau)g\}_{0<\tau\leq 1}$.
To this end, let us recall the notion of convergence in the strong graph sense: given a sequence $\{A_n\}_{n=1}^{\infty}$ of operators in a Hilbert space ${\mathcal H}$ we say that $(\psi,\varphi) \in {\mathcal H}\times {\mathcal H}$ belongs to the strong graph limit if one can find $\psi_n \in D[A_n],\, n= 1,2, \dots,$ such that
$$
\psi_n \longrightarrow \psi,\quad
A_n \psi_n \longrightarrow \varphi
\quad\;\text{\rm as}\;\; n\to\infty\,.
$$
We denote the set of such pairs $(\psi,\varphi)$ by $\Gamma_{\infty}^s$. If $\Gamma_{\infty}^s$ is the graph of an operator $A$, i.e. $\Gamma_{\infty}^s = \{ (\psi, A \psi) \in {\mathcal H}\times {\mathcal H};\, \psi \in D[A] \}$, $\:A$ is also said to be the \emph{strong graph limit} of $\{A_n\}$. We have the following result \cite[Thm~4]{GJ69}, see also \cite[Thm~VIII.26]{RS80}:
\begin{proposition} \label{p:graphlim}
Let $\{A_n\}_{n=1}^{\infty}$ and $A$ be self-adjoint operators in ${\mathcal H}$. Then $\{A_n\}$ converges to $A$ in strong resolvent sense if and only if $A$ is the strong graph limit of $\{A_n\}$.
\end{proposition}
To make use of this result, let us consider the family $\{A_{\tau}\}_{0< \tau \leq 1}$ of the following bounded and self-adjoint operators,
$$
A_{\tau} : = I+ |s|H_{\tau}\,,
\quad\; 0< \tau \leq 1.
$$
By spectral theorem its elements converge in the strong operator topology to the self-adjoint operator $I+|s|H$ as $\tau\to 0+$, and that by Proposition~\ref{p:graphlim} means that $I+|s|H$ is the strong graph limit of $\{A_{\tau}\}$. Since $s \in {\mathbb R} \setminus (M\cup \{0\})$ holds by assumption, we have
\begin{align*}
&\quad \psi_{\tau} := [P(\tau)(I+ sK(s\tau))P(\tau)]^{-1}g
\longrightarrow \psi := [(I+isH_P)^{-1}P]g\,, \\
\intertext{as $\tau\to 0+$ for $g\in {\mathcal H}$, with}
&\quad [(I+isH_P)^{-1}P]g \in D[H_P]
\subset D[(I+ |s|H)P] \subset D[I+ |s|H]\,.
\end{align*}
In this way, the pair $(\psi, (I+ |s|H)\psi)$ belongs to the set $\Gamma_{\infty}^s$ resulting from the strong graph limit of the family $\{A_{\tau}\}$, and consequently, $A_{\tau}\psi_{\tau} \longrightarrow (I+ |s|H)\psi$, i.e.
\begin{align*}
&A_{\tau}[P(\tau)(I+ sK(s\tau))P(\tau)]^{-1}g
\longrightarrow (I+ |s|H)[(I+isH_P)^{-1}P]g\,,
\end{align*}
in other words, the sought convergence, $T_3(s;\tau)g \longrightarrow\, T_3(s)g$. This implies, by the uniform boundedness principle (or Banach-Steinhaus theorem), the family $\{T_3(s;\tau)\}_{0\le\tau< 1}$ is uniformly bounded in the operator norm.
(ii) Consider again an arbitrary $g \in {\mathcal H}$. In view of the convergence established above in combination with the bound on the norm of $T_3(s)$ from Lemma~\ref{l:estimate-T3}(ii), we see that for a fixed $s \in {\mathbb R}\setminus (M\cup\{0\})$ there is a positive, $g$-dependent number $\tau_s = \tau_s(g) \leq 1$ such that
\begin{align*}
\|T_3(s;\tau)g\| \leq \|T_3(s)g\| + \tfrac12
\leq \|T_3(s)\|\|g\| + \tfrac12
\leq \sqrt{2}\|g\| + \tfrac12\,,
\;\: \tau \in (0,\tau_s]\,.
\end{align*}
This helps us to complement our knowledge of the uniform boundedness using
the estimate
$$
\sup_{0<\tau\leq 1} \|T_3(s;\tau)g\|
\leq \max\big\{\sqrt{2}\|g\| + \tfrac12,\,
\sup_{\tau_s<\tau\leq 1}\|T_3(s;\tau)g\|\,\big\}\,.
$$
We note that the function $\tau\mapsto\|T_3(s;\tau)g\|$ is continuous on the compact set $[\tau_s,1] \subset {\mathbb R}$, hence the right-hand side is bounded for any $g$ and, applying again the uniform boundedness principle, we conclude that the $\tau$-family $\{T_3(s;\tau)\}_{0<\tau\leq 1}$ is uniformly bounded by a constant $C_{T_3}(s) \geq \sqrt{2}$ depending on $s$. This completes the proof of Lemma~\ref{l:unif-bound-T3}.
\end{proof}
\medskip
It remains to deal with the middle factor $T_2(t,s;\tau)$ in \eqref{T2tau}. We rewrite it as
\begin{align}
T_2(t,s;\tau)
&= i(sH(s\tau) -tH(t\tau))(I+ |s|H_{\tau})^{-1} \nonumber\\
&\quad +(sG(s\tau) -tG(t\tau))I+ |s|H_{\tau})^{-1} \nonumber\\
&=: iT_{2,H}(t,s;\tau) + T_{2,G}(t,s;\tau). \label{T2HGtau}
\end{align}
For a fixed $\tau\in(0,1]$, the operators $T_{2,H}(t,s;\tau)$, $T_{2,G}(t,s;\tau)$, as well as their linear combination, are bounded operators which are in the functional calculus sense obtained from $H$ as its appropriate bounded and continuous functions; in view of \eqref{tH(t)-sH(s)} and \eqref{tG(t)-sG(s)} we have
\begin{align*}
T_{2,H}(t,s;\tau)
&= \phi_H(t,s;\tau,H)
:= (tH(t\tau)-sH(s\tau))(I+ |s|H_{\tau})^{-1} \\
&= \frac{\tfrac{2}{\tau} \sin(\tfrac{t-s}2\tau\lambda)
\cos(\tfrac{s+t}2\tau\lambda)}{I+ |s|\lambda(I+\tau\lambda)^{-1}}\,,
\\
T_{2,G}(t,s;\tau)
&= \phi_G(t,s;\tau,H)
:= (tG(t\tau)- sG(s\tau))(I+ |s|H_{\tau})^{-1} \nonumber\\
&= \frac{\tfrac{2}{\tau} \sin(\tfrac{t-s}2\tau\lambda)\,
\sin(\tfrac{s+t}2\tau\lambda)}{I+ |s|\lambda(I+\tau\lambda)^{-1}}\,,
\end{align*}
and
$$
T_2(t,s;\tau)
= \phi(t,s;\tau,H)
:= i\phi_H(t,s;\tau,H) +\phi_G(t,s;\tau,H)
$$
It is straightforward to check that
\begin{equation}
|\phi(t,s;\tau,\lambda)|^2 = \Big|\frac{\tfrac{2}{\tau} \sin(\tfrac{t-s}2\tau\lambda)}
{I+ |s|\lambda(I+\tau\lambda)^{-1}}\Big|^2, \label{bound-phi}
\end{equation}
which means that for any $g \in {\mathcal H}$ we have
\begin{align}
\|T_2(t,s;\tau)g\|^2 &= \|T_{2,H}(t,s;\tau)g\|^2 + \|T_{2,G}(t,s;\tau)g\|^2 \nonumber\\
&= \int_{0-}^{\infty} \Big|\frac{\tfrac{2}{\tau} \sin(\tfrac{t-s}2\tau\lambda)}
{I+ |s|\lambda(I+\tau\lambda)^{-1}}\Big|^2
\|E(\mathrm{d}\lambda)g\|^2\,. \label{norm-T2}
\end{align}
The following lemma represents the second crucial element in establishing Lemma~\ref{l:equiconti}, and in this way, proving our main theorems.
\begin{lemma} \label{l:estimate-T}
Consider $t,s \in {\mathbb R} \setminus\{0\}$ such that either $t,s >0$ or $t,s < 0$. Then for $T_2(t,s;\tau)$ given by \eqref{T2tau} and for $D(t,s;\tau)$ given by \eqref{D(t,s;tau)} or \eqref{split-D(t,s;tau)to123} the following holds:
\smallskip
\noindent \text{\rm (i)} The $\tau$-family $\{T_2(t,s;\tau\}$ of bounded operators in \eqref{T2HGtau} is uniformly bounded locally uniformly for $t,s \in {\mathbb R} \setminus\{0\}$. Further, for every $f \in {\mathcal H}$ and any $\varepsilon>0$, there is an $s$-dependent number $\delta = \delta(f;\varepsilon;s) >0$ such that
\begin{equation}
t\in {\mathbb R}\setminus \{0\}\;\; \text{\it with}\;\, |t-s|< \delta \;\;
\Longrightarrow\;\; \|T_{2}(t,s;\tau)f\| < \varepsilon, \label{equicont-T2}
\end{equation}
uniformly with respect to $\tau\in(0,1]$.
\smallskip
\noindent \text{\rm (ii)} Let further $s \in {\mathbb R}\setminus (M\cup\{0\})$, then for every $f \in {\mathcal H}$ and any $\varepsilon>0$,
there is a number \mbox{$\delta = \delta(f;\varepsilon;s) >0$} such that
\begin{equation
t\in {\mathbb R}\setminus \{0\}\;\;
\text{\it with}\;\; |t-s|< \delta
\;\;\Longrightarrow\;\;
\|D(t,s;\tau)f\|< \, \varepsilon, \label{equicont-D}
\end{equation
uniformly with respect to $\tau\in(0,1]$.
\end{lemma}
\begin{proof}
(i) The $\tau$-families of bounded operators appearing in \eqref{T2HGtau} converge in the strong operator topology for any fixed pair $t,s \in {\mathbb R} \setminus\{0\}$; by functional calculus we find easily that
\begin{align*}
&T_{2,H}(t,s;\tau) \overset s \to T_{2,H}(t,s) :=\tfrac{(t-s)H}{I+|s|H},
\quad T_{2,G}(t,s;\tau) \overset s \to T_{2,G}(t,s) := 0, \\
&T_2(t,s;\tau) \overset s \longrightarrow
T_2(t,s) := iT_{2,H}(t,s) + T_{2,G}(t,s)
\end{align*}
holds as $\tau\to 0+$, and therefore
$$
|T_2(t,s;\tau)|^2 \overset s \longrightarrow
|T_2(t,s)|^2 = T_{2,H}(t,s)^2 + T_{2,G}(t,s)^2
= \big(\tfrac{(t-s)H}{I+|s|H}\big)^2.
$$
Our way to prove the claim is to apply the uniform boundedness principle to these operator families. From \eqref{norm-T2} we obtain
$$
\|T_2(t,s;\tau)\| = \sup_{\|g\|=1}\|T_2(t,s;\tau)g\| = \sup_{\lambda\geq 0}|\phi(t,s;\tau,\lambda)|\,,
$$
and for $T_2(t,s) = \phi(t,s;H)$ with $\phi(t,s;\lambda):= \tfrac{(t-s)\lambda}{1+|s|\lambda}$ we have similarly
\begin{equation}
\|T_2(t,s)\| = \sup_{\lambda\geq 0}|\phi(t,s;\lambda)| = \tfrac{|t-s|}{|s|}\,. \label{T2ts-norm}
\end{equation}
We note that $\phi(t,s;\tau,\lambda)$ is continuous with respect to all the variables, $\tau \in (0,1]$, $\lambda \in [0,\infty)$, and $t,s \in {\mathbb R} \setminus\{0\}$. Our intention is now to regard the family $\{T_2(t,s;\tau)\}$ as depending on a broader set of parameters: we fix a nonzero $s$ and consider $\tau$ in $(0,1]$ and $t$ in a neighborhood of $s$, more specifically,
\begin{equation} \label{larger-parameter-set}
\tau \in (0,1], \quad
t \in D(s) := \{t \in {\mathbb R}\setminus \{0\};\, |t-s|
\leq \tfrac{|s|}2\}.
\end{equation}
For each fixed $f \in {\mathcal H}$ and $s \in {\mathbb R} \setminus\{0\}$ we put
\begin{equation}
\Phi_{f,s}(t;\tau) := \left\{
\begin{array}{lcc}
\|T_2(t,s;\tau)f\| & \quad \dots\;\; & 0<\tau\leq 1, \\[.3em]
\|T_2(t,s)f\| & \quad \dots\;\; & \tau=0,
\end{array}\right.
\end{equation}
which is a bounded and continuous function on the compact set $D(s) \times [0,1] \subset {\mathbb R}^2$. Then
$
C_{f,s} := \max_{t,\tau \in D(s) \times [0,1]}\Phi_{f,s}(t;\tau)
$
exists, depending on $f$ and $s$, but being independent of $t \in D(s)$ and $\tau \in(0,1]$, and from the strong convergence of $\{T_2(t,s;\tau)\}$ to $T_2(t,s)$ noted above we infer that
$$
\|T_2(t,s;\tau)f\| \leq C_{f,s}
$$
holds for given $f \in {\mathcal H}$, $\:s\ne 0$, and all $(t,\tau) \in D(s) \times (0,1]$. Applying now the uniform boundedness principle to this operator family with the extended set of parameters \eqref{larger-parameter-set}, we establish the existence of a constant $C_{T_2}(s)>0$, depending on $s$ but independent of $t \in D(s)$ and $\tau \in (0,1]$, such that
\begin{equation} \label{T2-bound}
\|T_2(t,s;\tau)\| \leq C_{T_2}(s).
\end{equation}
In view of the strong convergence, it cannot be smaller than the norm of $T_2(t,s)$ in \eqref{T2ts-norm}, that is, $C_{T_2}(s) \geq \tfrac{|t-s|}{|s|}$.
Since $E([\lambda,\infty)) \overset s \to 0$ holds as $\lambda\to\infty$, one can find for a fixed $s$, any $f \in {\mathcal H}$, and each $\varepsilon>0$ a number $\lambda_0 = \lambda_0(f;\varepsilon;s) >0$ such that
\begin{equation} \label{phi-tail}
\|E([\lambda_0,\infty))f\|^2
< \tfrac1{C_{T_2}(s)^2}\cdot\tfrac{\varepsilon^2}{2}\,.
\end{equation}
In this case using \eqref{bound-phi} and \eqref{phi-tail}, we get
\begin{align*}
\|T_2&(t,s;\tau)f\|^2
= \|\phi(t,s;\tau,H)f\|^2 \\
&= \Big(\int_{0-}^{\lambda_0}\,\, +\,\, \int_{\lambda_0}^{\infty}\Big)
|\phi(t,s;\tau,\lambda)|^2 \|E(\mathrm{d}\lambda)f\|^2 \\
&\leq \int_{0-}^{\lambda_0}
\bigg|\tfrac{\tfrac{2}{\tau}\sin(\tfrac{t-s}2\tau\lambda)}
{1+ (s\lambda)(I+\tau\lambda)^{-1}}\bigg|^2
\|E(\mathrm{d}\lambda)f\|^2
+C_{T_2}(s)^2\|E([\lambda_0,\infty))f\|^2 \\
&\leq (|t-s|\lambda_0)^2 \|E([0,\lambda_0))f\|^2
+ \tfrac{\varepsilon^2}2\,,
\end{align*}
where the last inequality comes from the estimate of the numerator of the squared modulus factor in the above integral,
$$
\big|\tfrac{2}{\tau}\sin(\tfrac{t-s}2\tau\lambda)\big|
\leq |t-s|\lambda_0, \qquad 0\leq \lambda \leq \lambda_0.
$$
Since the set $\{(\tau,\lambda); \, 0<\tau\leq 1,\, 0\leq \lambda \leq \lambda_0\}$ is bounded, and therefore its closure is compact, we infer that for a fixed $s\in {\mathbb R}\setminus \{0\}$ there is a number $\delta = \delta(f;\varepsilon;s) >0$ such that
$$
t \in {\mathbb R}\setminus \{0\}\,\, \text{\rm with}\,\, |t-s|<\delta \,\,
\Longrightarrow
(|t-s|\lambda_0)^2 (1+\|f\|^2) < \tfrac{\varepsilon^2}{2}
$$
uniformly for $0<\tau\leq 1$. In fact, we can find it explicitly; it is enough to choose
$$
\delta(f;\varepsilon;s)
:= \min\big\{\tfrac1{\sqrt{2}\lambda_0(1+\|f\|^2)^{1/2}}\,\varepsilon,
\tfrac{|s|}2\big\}.
$$
Thus we have
$
\|T_2(t,s;\tau)f\|^2
\leq (1+\|f\|^2)^{-1}
\tfrac{\varepsilon^2}{2} \|E([0,\lambda_0))f\|^2
+ \tfrac{\varepsilon^2}2
< \varepsilon^2,
$
in other words,
$$
\|T_2(t,s;\tau,H)f\| < \varepsilon,
$$
which yields the implication \eqref{equicont-T2}.
\smallskip\noindent
(ii) To prove the remaining claim of the lemma, we first note that $T_1(t;\tau) \equiv T^P(t;\tau)$ is a contraction, $\|T_1(t;\tau)\| \leq 1$, and therefore
\begin{align*}
\|D(t,s;\tau)f\|
&= \|T^P(t,s;\tau)f\| = \|T_1(t;\tau) T_2(t,s;\tau) T_3(s;\tau)f\|\\
&\leq \|T_2(t,s;\tau) T_3(s;\tau)f\|.
\end{align*}
To deal with the last norm we repeat the argument from the part (i) replacing $f$ by $T_3(s;\tau)f$, obtaining for any $\lambda_0>0$ the estimate
\begin{align}
&\|T_2(t,s;\tau) T_3(s;\tau)f\|^2 \nonumber \\
&= \Big(\int_{0-}^{\lambda_0} +\int_{\lambda_0}^{\infty} \Big)
|\phi(t,s;\tau,\lambda)|^2\|E(\mathrm{d}\lambda) T_3(s;\tau)f\|^2
\nonumber\\
&\leq \Big[\int_{0-}^{\lambda_0}
\Big|\tfrac{\tfrac{2}{\tau} \sin\tfrac{(t-s)}2\tau\lambda}
{1+ (s\lambda)(I+\tau\lambda)^{-1}}\Big|^2
\|E(\mathrm{d}\lambda) T_3(s;\tau)f\|^2 \nonumber \\
&\qquad\qquad\qquad\qquad\qquad
+ C_{T_2}(s)^2\|E([\lambda_0,\infty))T_3(s;\tau)f\|^2\Big] \label{T_2T_3} \\
&\leq (|t-s|\lambda_0)^2 \|E([0,\lambda_0))T_3(s;\tau)f\|^2
+ C_{T_2}(s)^2\|E([\lambda_0,\infty)) T_3(s;\tau)f\|^2 \nonumber
\end{align}
with the constant $C_{T_2}(s)$ from \eqref{T2-bound}. The second term of the last expression converges to zero as $\lambda_0 \to \infty$ for any \emph{fixed} $\tau\in(0,1]$.
To make use of the estimate, however, we need to find a $\lambda_0$ independent of $\tau$ in order to obtain convergence \emph{uniform} with respect to $0 <\tau \leq 1$. This can be achieved, however, because for fixed
$f \in {\mathcal H}$ and $s \in {\mathbb R} \setminus (M\cup \{0\})$, the set
$$
V_f := \{ T_3(s;\tau)f ;\,\, 0 <\tau \leq 1 \} \subset {\mathcal H}
$$
is bounded by \eqref{unif-bound-T3}, and has a compact closure, as a consequence of the strong continuity of $T_3(s;\tau)f$ with respect to the variable $\tau>0$. Then for every $f \in {\mathcal H}$ and every $\varepsilon>0$ one can find an $s$-dependent number $\lambda_{00} = \lambda_{00}(f;\varepsilon;s) >0$ such that
\begin{equation} \label{uniform-for-tau}
\sup_{0<\tau \leq 1}\|E([\lambda_{00},\infty))T_3(s;\tau)f\|^2
< \tfrac1{C_{T_2}(s)^2}\cdot\tfrac{\varepsilon^2}{2}
\end{equation}
with the constant $C_{T_2}(s)$ in \eqref{T2-bound}. Let us be more explicit about the last claim. Since $V_f$ is totally bounded, there is a finite family of vectors $\{y_j\}_{j=1}^N \subset V_f$, $y_j = T_3(s;\tau_j)f$ for some
$\tau_j\in(0,1]$, and an open ball $B(0; \tfrac{\varepsilon}{\sqrt{8}C_{T_2}(s)})$ with the center at the origin and radius $\tfrac{\varepsilon}{\sqrt{8}C_{T_2}(s)}$ for which we have
$
V_f\,\, \subset\,\, \cup_{j=1}^N \big(y_j
+ B(0; \tfrac{\varepsilon}{\sqrt{8}C_{T_2}(s)})\big).
$
Using again the fact that $E([\lambda,\infty)) \overset s \to 0$ holds as $\lambda\to\infty$ we infer that there is a family $\{\lambda_j\}_{j=1}^N$ of large positive numbers such that $C_{T_2}(s)\|E([\lambda_j,\infty)) y_j\| \leq \tfrac{\varepsilon}{\sqrt{8}}$ holds for $j=1,2,\dots, N$ and we put $\lambda_{00} := \max_{j=1,2,\dots, N}{\lambda_j}$. The finite union coverage means that any $\phi \in V_f$ satisfies $\phi \in y_{j_{\phi}} + B(0; \tfrac{\varepsilon}{\sqrt{8}C_{T_2}(s)})$ for some $1 \leq j_{\phi} \leq N$ with $0< \tau_{j_{\phi}} \leq 1$. Noting that $\|E([\lambda_{00},\infty))\|\le 1$, we have
\begin{align*}
&C_{T_2}(s)^2\|E([\lambda_{00},\infty))\phi\|^2 \\
&\leq C_{T_2}(s)^2\big(\max_{j=1,2,\dots, N}\|E([\lambda_{00},\infty))y_j\|
+ \tfrac{\varepsilon}{\sqrt{8}C_{T_2}(s)}\big)^2 \\
&\leq 2C_{T_2}(s)^2\big[\max_{j=1,2,\dots, N}\|E([\lambda_{00},\infty))y_j\|^2
+ \big( \tfrac{\varepsilon}{\sqrt{8}C_{T_2}(s)})^2\big] \\
&< \tfrac{\varepsilon^2}4 + \tfrac{\varepsilon^2}4
= \tfrac{\varepsilon^2}2.
\end{align*}
\smallskip
\noindent Returning to the first term on the right-hand side of \eqref{T_2T_3} where take $\lambda_0=\lambda_{00}$, and noting that $\|T_3(s;\tau)f\| \leq C_{T_3}(s)\|f\|$ holds in view of \eqref{unif-bound-T3}, we see that there exists a number $\delta = \delta(f;\varepsilon;s) >0$ such that for a fixed $s$ we have
$$
t \in {\mathbb R}\setminus \{0\}\; \text{\rm with}\;
|t-s|<\delta \;
\Longrightarrow
(|t-s|\lambda_{00})^2 (C_{T_3}(s)\|f\|)^2 < \tfrac{\varepsilon^2}{2},
$$
uniformly for all $\tau\in(0,1]$; one can be again explicit and choose
$$
\delta(f;\varepsilon;s) :=
\min\{\tfrac{1}{\sqrt{2}\lambda_{00}(1+(C_{T_3}(s)\|f\|)^2)^{1/2}}\,
\varepsilon,\, \tfrac{|s|}2\}\,.
$$
The estimate \eqref{T_2T_3} then gives
$$
\|T_2(t,s;\tau) T_3(s;\tau)f\|^2
< \tfrac{\varepsilon^2}2 + \tfrac{\varepsilon^2}2
= \varepsilon^2,
$$
or in other words
$$
\|D(t,s;\tau)f\| \leq \|T_2(t,s;\tau) T_3(s;\tau)f\|
< \varepsilon,
$$
which yields \eqref{equicont-D}. This concludes the proof of the present lemma,
and in this way also of Lemma~\ref{l:equiconti} as we have outlined in the opening of this section.
\end{proof}
\section{An example}
\label{s: example}
Let us finally mention briefly a typical situation in which Zeno
dynamics occurs as indicated, for instance, in the
paper~\cite{FPSS01}, namely the perpetual position ascertaining. We
consider an open domain $\Omega\subset\mathbb{R}^d $ with a smooth
boundary, thought of as the detector volume, and associate with it
the orthogonal projection $P$ on $L^2(\mathbb{R}^d)$ acting as the
multiplication operator by the indicator function $\chi_\Omega$ of
the set $\Omega$. Suppose that the dynamics of the particle
undisturbed by the measurement is free, that is, described by the
Hamiltonian $H = -\Delta$, i.e. the Laplacian in $\mathbb{R}^d$
which is a nonnegative self-adjoint operator in $L^2(\mathbb{R}^d)$.
The assumption of density of the domain of $H^{1/2}P =
(-\Delta)^{1/2}\chi_{\Omega}$ is satisfied, since it contains
$C_0^{\infty}(\Omega) \cup C_0^{\infty}({\mathbb R}^d \setminus
\overline{\Omega})$, where $\overline{\Omega}$ is the closure of
$\Omega$, and this family of functions is dense in $L^2({\mathbb
R}^d)$.
Consider further the Dirichlet Laplacian $-\Delta_\Omega$ in
$L^2(\Omega)$ defined in the usual way \cite[Thm~XIII.15]{RS78} as the
Friedrichs extension of the appropriate quadratic form.
It can be checked \cite[Prop.~6.1]{EI05} that
$$
(-\Delta)_P := ((-\Delta)^{1/2}P)^* (-\Delta)^{1/2}P
$$
is densely defined and its restriction to the subspace $L^2(\Omega)$ is nothing
but the Dirichlet Laplacian $-\Delta_\Omega$ with the domain
$D[-\Delta_\Omega] = W_0^1(\Omega) \cap W^2(\Omega)$, and
$$
-\Delta_\Omega = (-\Delta)_P\restriction_{PL^2({\mathbb R}^d)}
= (-\Delta)_P\restriction_{L^2(\Omega)}\,.
$$
Then Theorem~\ref{th:main} says that
$$
\text{s\,-}\!\!\lim_{n\rightarrow \infty} (P\,\mathrm{e}^{-it(-\Delta/n)}P)^n
= \mathrm{e}^{-it(-\Delta_\Omega)}P
$$
holds in the strong operator topology of $\mathcal{B}(L^2(\mathbb{R}^d))$, the Banach space of the bounded linear operators on $L^2(\mathbb{R}^d)$, in other words, that the perpetual reduction of the wave function forces the particle to move within the region $\Omega$ as if its boundary was Dirichlet, i.e. hard wall. This is, of course, the expected conclusion indicated, e.g. in \cite{FP08, FPSS01}, however, only Theorem~\ref{th:main} allows one to state such a claim with the proper rigor.
\subsection*{Acknowledgments}
The research was supported by the Czech Science Foundation within the project 17-01706S and in part by Grant-in Aid for Scientific Research 16K05230, Japan Society for the Promotion of Science, and by the EU project CZ.02.1.01/0.0/0.0/16\textunderscore 019/0000778. The authors are grateful to Tsuyoshi Ando for valuable discussions, to Hiroshi Tamura, Valentin Zagrebnov, and late Hagen Neidhardt for a number of useful comments, and to Hideo Tamura for his unceasing encouragement.
|
1,314,259,996,760 | arxiv | \section*{Appendix \thesection\protect\indent \parbox[t]{11.15cm}{#1}}
\addcontentsline{toc}{section}{Appendix \thesection\ \ \ #1}}
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\def\hat D'{\hat D'}
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\def {LF} {{LF}}
\def {LG} {{LG}}
\def {LP} {{LP}}
\def {BG} {{BG}}
\def {BP} {{BP}}
\def {BF} {{BF}}
\def G_{-+{\bar{\alpha}}} {G_{-+p}}
\def G_{\bar{\alpha} \beta}{}^\beta {G_{-1p}}
\def \epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} G_{\beta_1 \beta_2 \beta_3} {G_{- \bar{1} p}}
\def \epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} F_{-+\beta_1 \beta_2 \beta_3} {G_{+1p}}
\def F_{-+{\bar{\alpha}} \beta}{}^\beta {G_{+ \bar{1} p}}
\def G_{1 \bar{1} p} {G_{1 \bar{1} p}}
\def G_{pq}{}^q {G_{pq}{}^q}
\def G_{- {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {G_{- {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def G_{+ {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {G_{+ {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def G_{1 {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {G_{1 {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def G_{\bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {G_{\bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{\beta,}{}^\beta{}_{\bar{\alpha}} {F_{-+1 \bar{1} p}}
\def \epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} \Omega_{\beta_1, \beta_2 \beta_3} {F_{-+pq}{}^q}
\def \Omega_{-,+\bar{\alpha}} {F_{-1pq}{}^q}
\def \Omega_{\bar{\alpha},-+} {F_{- \bar{1} pq}{}^q}
\def \Omega_{\bar{\alpha}, \beta}{}^\beta {F_{+1pq}{}^q}
\def F_{+ \bar{1} pq}{}^q {F_{+ \bar{1} pq}{}^q}
\def F_{-+1 {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {F_{-+1 {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def F_{-+\bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {F_{-+\bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def F_{-1 \bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {F_{-1 \bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def F_{+ 1 \bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {F_{+ 1 \bar{1} {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{p,q}{}^q {\Omega_{p,q}{}^q}
\def \Omega_{p, 1 \bar{1}} {\Omega_{p, 1 \bar{1}}}
\def \Omega_{p,-+} {\Omega_{p,-+}}
\def \Omega_{p,1-} {\Omega_{p,1-}}
\def \Omega_{p,\bar{1}-} {\Omega_{p,\bar{1}-}}
\def \Omega_{p,+\bar{1}} {\Omega_{p,+\bar{1}}}
\def \Omega_{p,+1} {\Omega_{p,+1}}
\def \Omega_{1, {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{1, {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{1,p-} {\Omega_{1,p-}}
\def \Omega_{1,p+} {\Omega_{1,p+}}
\def \Omega_{1,p \bar{1}} {\Omega_{1,p \bar{1}}}
\def \Omega_{1,1p} {\Omega_{1,1p}}
\def \Omega_{{\bar{q}}_1 , {\bar{q}}_2 \bar{1}} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{{\bar{q}}_1 , {\bar{q}}_2 \bar{1}} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{\bar{q},}{}^{\bar{q}}{}_p {\Omega_{\bar{q},}{}^{\bar{q}}{}_p}
\def \Omega_{{\bar{q}}_1 , {\bar{q}}_2 -} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{{\bar{q}}_1 , {\bar{q}}_2 -} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{{\bar{q}}_1 , {\bar{q}}_2 +} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{{\bar{q}}_1 , {\bar{q}}_2 +} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{{\bar{q}}_1 , {\bar{q}}_2 1} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{{\bar{q}}_1 , {\bar{q}}_2 1} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{\bar{1} , {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{\bar{1} , {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{\bar{1},1p} {\Omega_{\bar{1},1p}}
\def \Omega_{\bar{1},p-} {\Omega_{\bar{1},p-}}
\def \Omega_{\bar{1},p+} {\Omega_{\bar{1},p+}}
\def \Omega_{\bar{1},p\bar{1}} {\Omega_{\bar{1},p\bar{1}}}
\def \Omega_{-, {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{-, {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{-,1p} {\Omega_{-,1p}}
\def \Omega_{-,p-} {\Omega_{-,p-}}
\def \Omega_{-,+p} {\Omega_{-,+p}}
\def \Omega_{-,p\bar{1}} {\Omega_{-,p\bar{1}}}
\def \Omega_{+, {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p {\Omega_{+, {\bar{q}}_1 {\bar{q}}_2} \epsilon^{{\bar{q}}_1 {\bar{q}}_2}{}_p}
\def \Omega_{+,1p} {\Omega_{+,1p}}
\def \Omega_{+,p-} {\Omega_{+,p-}}
\def \Omega_{+,+p} {\Omega_{+,+p}}
\def \Omega_{+,\bar{1}p} {\Omega_{+,\bar{1}p}}
\def D_p f {D_p f}
\def D_p g {D_p g}
\def G_{-+\bar{p}} {G_{-+\bar{p}}}
\def G_{-1{\bar{p}}} {G_{-1{\bar{p}}}}
\def G_{- \bar{1} {\bar{p}}} {G_{- \bar{1} {\bar{p}}}}
\def G_{+1{\bar{p}}} {G_{+1{\bar{p}}}}
\def G_{+ \bar{1} {\bar{p}}} {G_{+ \bar{1} {\bar{p}}}}
\def G_{1 \bar{1} {\bar{p}}} {G_{1 \bar{1} {\bar{p}}}}
\def G_{{\bar{p}}q}{}^q {G_{{\bar{p}}q}{}^q}
\def G_{- q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {G_{- q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def G_{+ q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {G_{+ q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def G_{1 q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {G_{1 q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def G_{\bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {G_{\bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def F_{-+1 \bar{1} {\bar{p}}} {F_{-+1 \bar{1} {\bar{p}}}}
\def F_{-+{\bar{p}}q}{}^q {F_{-+{\bar{p}}q}{}^q}
\def F_{-1{\bar{p}}q}{}^q {F_{-1{\bar{p}}q}{}^q}
\def F_{- \bar{1} {\bar{p}}q}{}^q {F_{- \bar{1} {\bar{p}}q}{}^q}
\def F_{+1{\bar{p}}q}{}^q {F_{+1{\bar{p}}q}{}^q}
\def F_{+ \bar{1} {\bar{p}}q}{}^q {F_{+ \bar{1} {\bar{p}}q}{}^q}
\def F_{-+1 q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {F_{-+1 q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def F_{-+\bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {F_{-+\bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def F_{-1 \bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {F_{-1 \bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def F_{+ 1 \bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {F_{+ 1 \bar{1} q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{q_1 , q_2 \bar{1}} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{q_1 , q_2 \bar{1}} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{q,}{}^q{}_{\bar{p}} {\Omega_{q,}{}^q{}_{\bar{p}}}
\def \Omega_{q_1 , q_2 -} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{q_1 , q_2 -} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{q_1 , q_2 +} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{q_1 , q_2 +} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{q_1 , q_2 1} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{q_1 , q_2 1} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{1 , q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{1 , q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{1,\bar{1} {\bar{p}}} {\Omega_{1,\bar{1} {\bar{p}}}}
\def \Omega_{1,{\bar{p}}-} {\Omega_{1,{\bar{p}}-}}
\def \Omega_{1,{\bar{p}}+} {\Omega_{1,{\bar{p}}+}}
\def \Omega_{1,{\bar{p}}1} {\Omega_{1,{\bar{p}}1}}
\def \Omega_{{\bar{p}},q}{}^q {\Omega_{{\bar{p}},q}{}^q}
\def \Omega_{{\bar{p}}, 1 \bar{1}} {\Omega_{{\bar{p}}, 1 \bar{1}}}
\def \Omega_{{\bar{p}},-+} {\Omega_{{\bar{p}},-+}}
\def \Omega_{{\bar{p}},\bar{1}-} {\Omega_{{\bar{p}},\bar{1}-}}
\def \Omega_{{\bar{p}},1-} {\Omega_{{\bar{p}},1-}}
\def \Omega_{{\bar{p}},+1} {\Omega_{{\bar{p}},+1}}
\def \Omega_{{\bar{p}},+\bar{1}} {\Omega_{{\bar{p}},+\bar{1}}}
\def \Omega_{\bar{1}, q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{\bar{1}, q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{\bar{1},{\bar{p}}-} {\Omega_{\bar{1},{\bar{p}}-}}
\def \Omega_{\bar{1},{\bar{p}}+} {\Omega_{\bar{1},{\bar{p}}+}}
\def \Omega_{\bar{1},{\bar{p}} 1} {\Omega_{\bar{1},{\bar{p}} 1}}
\def \Omega_{\bar{1},\bar{1}{\bar{p}}} {\Omega_{\bar{1},\bar{1}{\bar{p}}}}
\def \Omega_{-, q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{-, q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{-,\bar{1} {\bar{p}}} {\Omega_{-,\bar{1} {\bar{p}}}}
\def \Omega_{-,{\bar{p}}-} {\Omega_{-,{\bar{p}}-}}
\def \Omega_{-,+{\bar{p}}} {\Omega_{-,+{\bar{p}}}}
\def \Omega_{-,{\bar{p}}1} {\Omega_{-,{\bar{p}}1}}
\def \Omega_{+, q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}} {\Omega_{+, q_1 q_2} \epsilon^{q_1 q_2}{}_{\bar{p}}}
\def \Omega_{+,\bar{1}{\bar{p}}} {\Omega_{+,\bar{1}{\bar{p}}}}
\def \Omega_{+,{\bar{p}}-} {\Omega_{+,{\bar{p}}-}}
\def \Omega_{+,+{\bar{p}}} {\Omega_{+,+{\bar{p}}}}
\def \Omega_{+,1{\bar{p}}} {\Omega_{+,1{\bar{p}}}}
\def D_{\bar{p}} f {D_{\bar{p}} f}
\def D_{\bar{p}} g {D_{\bar{p}} g}
\def G_{-+1} {G_{-+1}}
\def G_{-+ \bar{1}} {G_{-+ \bar{1}}}
\def G_{1p}{}^p {G_{1p}{}^p}
\def G_{\bar{1} p}{}^p {G_{\bar{1} p}{}^p}
\def G_{- 1 \bar{1}} {G_{- 1 \bar{1}}}
\def G_{+ 1 \bar{1}} {G_{+ 1 \bar{1}}}
\def G_{234} {G_{234}}
\def G_{\bar{2} \bar{3} \bar{4}} {G_{\bar{2} \bar{3} \bar{4}}}
\def G_{+p}{}^p {G_{+p}{}^p}
\def G_{-p}{}^p {G_{-p}{}^p}
\def F_{-+1p}{}^p {F_{-+1p}{}^p}
\def F_{-+\bar{1}p}{}^p {F_{-+\bar{1}p}{}^p}
\def F_{-1\bar{1}p}{}^p {F_{-1\bar{1}p}{}^p}
\def F_{+1 \bar{1}p}{}^p {F_{+1 \bar{1}p}{}^p}
\def F_{-1234} {F_{-1234}}
\def F_{-\bar{1} \bar{2} \bar{3} \bar{4}} {F_{-\bar{1} \bar{2} \bar{3} \bar{4}}}
\def F_{-+234} {F_{-+234}}
\def F_{-+ \bar{2} \bar{3} \bar{4}} {F_{-+ \bar{2} \bar{3} \bar{4}}}
\def F_{+1 \bar{2} \bar{3} \bar{4}} {F_{+1 \bar{2} \bar{3} \bar{4}}}
\def F_{+ \bar{1} 234} {F_{+ \bar{1} 234}}
\def \Omega_{p,}{}^p{}_{\bar{1}} {\Omega_{p,}{}^p{}_{\bar{1}}}
\def \Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_1 {\Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_1}
\def \Omega_{p,}{}^p{}_+ {\Omega_{p,}{}^p{}_+}
\def \Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_+ {\Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_+}
\def \Omega_{p}{}^p{}_- {\Omega_{p}{}^p{}_-}
\def \Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_- {\Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_-}
\def \Omega_{p_1 , p_2 p_3} \epsilon^{p_1 p_2 p_3} {\Omega_{p_1 , p_2 p_3} \epsilon^{p_1 p_2 p_3}}
\def \Omega_{{\bar{p}}_1 , {\bar{p}}_2 {\bar{p}}_3 {\Omega_{{\bar{p}}_1 , {\bar{p}}_2 {\bar{p}}_3}
\epsilon^{{\bar{p}}_1 {\bar{p}}_2 {\bar{p}}_3}}
\def \Omega_{p,}{}^p{}_{1} {\Omega_{p,}{}^p{}_{1}}
\def \Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_{\bar{1}} {\Omega_{{\bar{p}},}{}^{{\bar{p}}}{}_{\bar{1}}}
\def \Omega_{1,p}{}^p {\Omega_{1,p}{}^p}
\def \Omega_{\bar{1},p}{}^p {\Omega_{\bar{1},p}{}^p}
\def \Omega_{1,1 \bar{1}} {\Omega_{1,1 \bar{1}}}
\def \Omega_{\bar{1},1 \bar{1}} {\Omega_{\bar{1},1 \bar{1}}}
\def \Omega_{1,1-} {\Omega_{1,1-}}
\def \Omega_{\bar{1}, \bar{1}-} {\Omega_{\bar{1}, \bar{1}-}}
\def \Omega_{1,-+} {\Omega_{1,-+}}
\def \Omega_{\bar{1},-+} {\Omega_{\bar{1},-+}}
\def \Omega_{1, \bar{1}-} {\Omega_{1, \bar{1}-}}
\def \Omega_{\bar{1},1-} {\Omega_{\bar{1},1-}}
\def \Omega_{1,1+} {\Omega_{1,1+}}
\def \Omega_{\bar{1}, \bar{1}+} {\Omega_{\bar{1}, \bar{1}+}}
\def \Omega_{1, \bar{1}+} {\Omega_{1, \bar{1}+}}
\def \Omega_{\bar{1},1+} {\Omega_{\bar{1},1+}}
\def \Omega_{-,p}{}^p {\Omega_{-,p}{}^p}
\def \Omega_{-,1 \bar{1}} {\Omega_{-,1 \bar{1}}}
\def \Omega_{-,-+} {\Omega_{-,-+}}
\def \Omega_{-,1-} {\Omega_{-,1-}}
\def \Omega_{-,\bar{1}-} {\Omega_{-,\bar{1}-}}
\def \Omega_{-,+1} {\Omega_{-,+1}}
\def \Omega_{-,+\bar{1}} {\Omega_{-,+\bar{1}}}
\def \Omega_{+,p}{}^p {\Omega_{+,p}{}^p}
\def \Omega_{+,1\bar{1}} {\Omega_{+,1\bar{1}}}
\def \Omega_{+,-+} {\Omega_{+,-+}}
\def \Omega_{+,1-} {\Omega_{+,1-}}
\def \Omega_{+,\bar{1}-} {\Omega_{+,\bar{1}-}}
\def \Omega_{+,+1} {\Omega_{+,+1}}
\def \Omega_{+,+\bar{1}} {\Omega_{+,+\bar{1}}}
\def D_+ f {D_+ f}
\def D_- f {D_- f}
\def D_1 f {D_1 f}
\def D_{\bar{1}} f {D_{\bar{1}} f}
\def D_+ g {D_+ g}
\def D_- g {D_- g}
\def D_1 g {D_1 g}
\def D_{\bar{1}} g {D_{\bar{1}} g}
\def P_+ {P_+}
\def P_- {P_-}
\def P_1 {P_1}
\def P_{\bar{1}} {P_{\bar{1}}}
\def G_{-+{\bar{\alpha}}} {G_{-+{\bar{\alpha}}}}
\def G_{\bar{\alpha} \beta}{}^\beta {G_{\bar{\alpha} \beta}{}^\beta}
\def \epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} G_{\beta_1 \beta_2 \beta_3} {\epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} G_{\beta_1 \beta_2 \beta_3}}
\def \epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} F_{-+\beta_1 \beta_2 \beta_3} {\epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} F_{-+\beta_1 \beta_2 \beta_3}}
\def F_{-+{\bar{\alpha}} \beta}{}^\beta {F_{-+{\bar{\alpha}} \beta}{}^\beta}
\def \Omega_{\beta,}{}^\beta{}_{\bar{\alpha}} {\Omega_{\beta,}{}^\beta{}_{\bar{\alpha}}}
\def \epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} \Omega_{\beta_1, \beta_2 \beta_3} {\epsilon_{\bar{\alpha}}{}^{\beta_1 \beta_2 \beta_3} \Omega_{\beta_1, \beta_2 \beta_3}}
\def \Omega_{-,+\bar{\alpha}} {\Omega_{-,+\bar{\alpha}}}
\def \Omega_{\bar{\alpha},-+} {\Omega_{\bar{\alpha},-+}}
\def \Omega_{\bar{\alpha}, \beta}{}^\beta {\Omega_{\bar{\alpha}, \beta}{}^\beta}
\def G_{-+\alpha} {G_{-+\alpha}}
\def G_{\alpha \beta}{}^\beta {G_{\alpha \beta}{}^\beta}
\def \epsilon_\alpha{}^{{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3 {\epsilon_\alpha{}^{{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3}
G_{{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3}}
\def \epsilon_\alpha{}^{{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3 {\epsilon_\alpha{}^{{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3}
F_{-+{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3}}
\def F_{-+\alpha \beta}{}^\beta {F_{-+\alpha \beta}{}^\beta}
\def \Omega_{{\bar{\beta}},}{}^{{\bar{\beta}}}{}_\alpha {\Omega_{{\bar{\beta}},}{}^{{\bar{\beta}}}{}_\alpha}
\def \epsilon_\alpha{}^{{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3 {\epsilon_\alpha{}^{{\bar{\beta}}_1 {\bar{\beta}}_2 {\bar{\beta}}_3}
\Omega_{{\bar{\beta}}_1, {\bar{\beta}}_2 {\bar{\beta}}_3}}
\def \Omega_{-,+\alpha} {\Omega_{-,+\alpha}}
\def \Omega_{\alpha,-+} {\Omega_{\alpha,-+}}
\def \Omega_{\alpha, \beta}{}^\beta {\Omega_{\alpha, \beta}{}^\beta}
\def\begin{equation}{\begin{equation}}
\def\end{equation}{\end{equation}}
\def \cite {\cite}
\def \bibitem {\bibitem}
\def \label{\label}
\def {\rm f} {{\rm f}}
\def \footnote {\footnote}
\def u {u }
\def v {v}
\def \tau {\tau}
\def {\cal A} { {\cal A} }
\def {\rm w} {{\rm w}}
\def \Sigma {\Sigma}
\def {\rm p} {{\rm p}}
\def \nonumber {\nonumber}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{issue}{Issue}
\newtheorem{question}{Question}
\newtheorem{conjecture}{Conjecture}
\newcommand{\noindent{\it Proof: }}{\noindent{\it Proof: }}
\newcommand{\ q.~e.~d. \vspace{0.2in}}{\ q.~e.~d. \vspace{0.2in}}
\begin{document}
\date{November 2002}
\begin{titlepage}
\begin{center}
\hfill hep-th/0606049 \\
\hfill KUL-TF-06/20 \\
\vspace{3.0cm} {\Large \bf N=31 is not IIB}
\\[.2cm]
\vspace{1.5cm}
{\large U. Gran$^1$, J. Gutowski$^2$, G. Papadopoulos$^3$ and D. Roest$^3$}
\vspace{0.5cm}
${}^1$ Institute for Theoretical Physics, K.U. Leuven\\
Celestijnenlaan 200D\\
B-3001 Leuven, Belgium\\
\vspace{0.5cm}
${}^2$ DAMTP, Centre for Mathematical Sciences\\
University of Cambridge\\
Wilberforce Road, Cambridge, CB3 0WA, UK
\vspace{0.5cm}
${}^3$ Department of Mathematics\\
King's College London\\
Strand\\
London WC2R 2LS, UK\\
\end{center}
\vskip 1.5 cm
\begin{abstract}
We adapt the spinorial geometry method to investigate supergravity backgrounds
with near maximal number of supersymmetries. We then apply the formalism to
show that the IIB supergravity backgrounds with 31 supersymmetries preserve an additional supersymmetry and
so they are maximally supersymmetric. This rules out the existence of IIB supergravity preons.
\end{abstract}
\end{titlepage}
\newpage
\setcounter{page}{1}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\setcounter{section}{0}
\setcounter{subsection}{0}
It has been known for some time that a priori in type II and
eleven-dimensional supergravities there may exist backgrounds with
any number of supersymmetries. This is because the holonomy of the
supercovariant connection of these theories is a subgroup of
$SL(32,\bb{R})$ and so any $N<32$ spinors have a non-trivial stability
subgroup in the holonomy group. For a more detailed explanation see
\cite{hull, duffl, gpta} for the M-theory and \cite{gptb} for IIB.
Furthermore, it was argued in \cite{ugjggpdr} that the Killing
spinor bundle ${\cal K}$ can be any subbundle of the Spin bundle and
the spacetime geometry depends on the trivialization of ${\cal K}$.
This is unlike what happens in the case of Riemannian and Lorentzian
geometries \cite{berger, figueroab} and heterotic and type I
supergravities\footnote{This is provided the parallel spinors are
Killing.} \cite{gpgran}, where there are restrictions both on the
number of Killing spinors and the Killing spinor bundle.
In this paper, we shall show that IIB backgrounds with 31
supersymmetries are maximally supersymmetric. Backgrounds with 31
supersymmetries have been considered before in the context of
M-theory \cite{bandos} and have been termed as preons. To our
knowledge this is the first example which demonstrates that there
are restrictions on the number of supersymmetries of type II
backgrounds. To do this, we shall adapt the spinorial method
\cite{jguggp} of solving Killing spinor equations to backgrounds
that admit near maximal number of supersymmetries. We shall mostly
focus on IIB and eleven-dimensional supergravity but most of
the analysis extends to all supergravity theories.
To adapt the spinorial method to backgrounds with near maximal number of supersymmetries, we
introduce a ``normal'' ${\cal K}^\perp$ to the Killing spinor bundle ${\cal K}$ of a supersymmetric background.
The spinors of IIB supergravity
are complex positive chirality Weyl spinors, so the Spin bundle is
${\cal S}^c_+={\cal S}_+\otimes\bb{C}$, where ${\cal S}_+$ is the rank
sixteen bundle of positive chirality Majorana-Weyl spinors. ${\cal
S}^c_+$ may also be thought of as an associated bundle of a
principal bundle with fibre $SL(32, \bb{R})$, the holonomy group of the
supercovariant connection, acting with the fundamental
representation on $\bb{R}^{32}$. If a background admits $N$ Killing
spinors which span the fibre of the Killing spinor bundle ${\cal
K}$, then one has the sequence \begin{eqnarray} 0\rightarrow {\cal K}\rightarrow
{\cal S}_+^c\rightarrow {\cal S}_+^c/{\cal K}\rightarrow 0~.
\label{exseq} \end{eqnarray} The inclusion $i:\,{\cal K}\rightarrow {\cal
S}_+^c$ can be locally described as \begin{eqnarray} \epsilon_r=\sum_{i=1}^{32}
f^i{}_r \eta_i~,~~~r=1,\dots, N \,,\end{eqnarray} where $\eta_p$,
$p=1,\dots,16$, is a basis in the space of positive chirality
Majorana-Weyl spinors, $\eta_{16+p}= i\eta_p$ and the coefficients
$f$ are real spacetime functions. For our notation and spinor
conventions see \cite{ugjggpdr}. Any $N$ Killing spinors related by
a local $Spin(9,1)$ transformation give rise to the same spacetime
geometry. This is because the Killing spinor equations and the field
equations of IIB supergravity are Lorentz invariant. Therefore any
bundles of Killing spinors and any choice of sections related by a
$Spin(9,1)$ gauge transformation\footnote{IIB supergravity has a
$Spin(9,1)\times U(1)$ gauge symmetry but the restriction to
$Spin(9,1)$ will suffice.}
should be identified.
To construct ${\cal K}^\perp$, first consider the dual ${}^\star{\cal S}_+^c$ of ${\cal S}_+^c$ and
introduce a basis $\eta^i$, $\eta^i(\eta_j)=\delta^i{}_j$, i.e. $\eta^{16+p}=-i\eta^p$.
Next consider the sections $\alpha $ of ${}^\star{\cal S}_+^c$ that annihilate the Killing spinors $\epsilon_r$, i.e
$\alpha (\epsilon)=0$, or equivalently
\begin{eqnarray}
f^i{}_r u_i=0~,~~~~\alpha =u_i \eta^i~,
\end{eqnarray}
where $u_i$ are real spacetime functions. Since the matrix $f=(f^i{}_r)$ has rank $N$, there are $32-N$ solutions
to this equation. These solutions span the sections of the co-kernel, ${\rm coker}\, i\subset {}^\star{\cal S}^c_+$
of the inclusion map
$i: {\cal K}\rightarrow {\cal S}_+^c$. It is well-known that $Spin(9,1)$ has an invariant
inner product $B: {\cal S}_+\otimes {\cal S}_-\rightarrow \bb{R}$
\begin{eqnarray}
B(\epsilon, \zeta)=-B(\zeta, \epsilon)=<B(\epsilon^*), \zeta>~,
\end{eqnarray}
which extends to $B: {\cal S}^c_+ \otimes {\cal S}^c_-\rightarrow
\bb{C}$ as a bi-linear in both entries. Next consider
\begin{eqnarray}
{\cal B}(\epsilon, \zeta)={\rm Re}\, B(\epsilon, \zeta)~,
\end{eqnarray}
which defines a non-degenerate pairing ${\cal B}: {\cal S}^c_+
\otimes {\cal S}^c_-\rightarrow \bb{R}$. This in turn induces a
isomorphism $j: {}^\star{\cal S}^c_+\rightarrow {\cal S}^c_-$ as
${\cal B}(j(\alpha ), \epsilon)=\alpha (\epsilon)$. We identify the image of
$j$, $j({\rm coker}\, i)\subset {\cal S}^c_- $, as the ``normal''
bundle ${\cal K}^\perp$ of ${\cal K}$, i.e. $j({\rm coker}\,
i)={\cal K}^\perp$. Clearly if $\alpha \in {\rm coker}\, i$ and
$\epsilon\in {\cal K}$, then $\alpha (\epsilon)=0$, and so one gets the
``orthogonality'' condition,
\begin{eqnarray}
{\cal B}(j(\alpha ), \epsilon)=0~.
\label{normcol}
\end{eqnarray}
Observe that ${\cal S}_+^c/{\cal K}={}^\star{\cal K}^\perp$.
To write this orthogonality condition in components, introduce a basis in ${\cal S}^c_-$, say $\theta_{i'}=-\Gamma_0\eta_i$. Then
write $j(\alpha )=\nu=n^{i'} \theta_{i'}$ and the condition (\ref{normcol}) can be written as
\begin{eqnarray}
n^{i'} {\cal B}_{i' j} f^j{}_r=0~,
\label{normcolcom}
\end{eqnarray}
where ${\cal B}_{i'j}={\cal B}(\theta_{i'}, \eta_j)$.
The condition (\ref{normcol}), or equivalently (\ref{normcolcom}),
leads to a correspondence
between the $N$ Killing spinors and the $32-N$ normal directions, i.e.
\begin{eqnarray}
N\longleftrightarrow 32-N~.
\end{eqnarray}
This is because instead of specifying
the Killing spinors, one can determine the normal spinors. Substituting the normal
spinors into these equations, one can then solve for
the Killing spinors.
In addition, the construction of ${\cal K}^\perp$ and (\ref{normcol}) or (\ref{normcolcom}) are
$Spin(9,1)$ covariant. Because of this, the $Spin(9,1)$ gauge symmetry can be used
to bring the normal spinors instead of the Killing spinors into a canonical form. In turn, this leads to a simplification
in the expression for the Killing spinors which can be used to solve
the Killing spinor equations for backgrounds with near maximal number of supersymmetries. We shall demonstrate this
for IIB backgrounds with 31 supersymmetries.
Furthermore, one may
consider cases such that the sections of ${\cal K}^\perp$ are invariant under some non-trivial stability
subgroup of $Spin(9,1)$. It is clear these cases are related to (e.g.~maximal and half-maximal) $G$-backgrounds
\cite{ugjggpdr, gmaxsusy}, where the invariance condition was imposed on the Killing spinors.
The spinorial geometry techniques that we use to investigate backgrounds with $N$
supersymmetries can be adapted to examine backgrounds with $32-N$ supersymmetries and vice-versa.
One can easily extend the construction described above to M-theory. In particular, one again has
\begin{eqnarray}
0\rightarrow {\cal K}\rightarrow {\cal S}\rightarrow {\cal S}/{\cal
K}\rightarrow 0 \,,
\end{eqnarray}
where ${\cal S}$ is the spin bundle associated with the Majorana representation of $Spin(10,1)$. The inclusion map
$i:\, {\cal K}\rightarrow {\cal S}$ can be written locally as $\epsilon_r= \sum_{i=1}^{32} f^i{}_r \eta_i$, where $f^i{}_r$ are real
spacetime functions and $(\eta_i, i=1, \dots, 32)$ is a basis of Majorana spinors. As in the IIB case, we consider the
the co-kernel of the inclusion map $i:\, {\cal K}\rightarrow {\cal S}$, ${\rm coker}\, i\subset {}^\star {\cal S}$.
It is well known that ${\cal S}$ admits a $Spin(10,1)$ invariant
inner product $B$ which gives rise to an isomorphism $j:\, {}^\star {\cal S}\rightarrow {\cal S}$. As in the IIB
case, we define the normal bundle of the Killing spinor bundle as ${\cal K}^\perp= j({\rm coker}\, i)$.
In this case, ${\cal K}^\perp$ is a subbundle of ${\cal S}$ and ${\cal S}/{\cal K}={\cal K}^\perp$.
Taking a section $\nu= n^i\eta_i$ of ${\cal K}^\perp$,
the orthogonality condition analogous to (\ref{normcol}) and (\ref{normcolcom}) is
\begin{eqnarray}
n^i B_{ij} f^j{}_r=0~,
\label{elcon}
\end{eqnarray}
where $B_{ij}=B(\eta_i, \eta_j)$. The condition (\ref{elcon}) is $Spin(10,1)$ covariant.
As an example consider IIB backgrounds that admit 31 supersymmetries. According to the correspondence $N\leftrightarrow 32-N$,
these are related to backgrounds
with one supersymmetry investigated in \cite{ugjggpa, ugjggpdr}. To carry out the computation,
we need to find the canonical form of spinors in ${\cal S}^c_-$ up to $Spin(9,1)$ transformations.
It is easy to deduce using an argument similar to \cite{ugjggpa} that there are three kinds of orbits of $Spin(9,1)$
in the negative chirality Weyl spinors with stability subgroups $Spin(7)\ltimes\bb{R}^8$, $SU(4)\ltimes \bb{R}^8$ and
$G_2$. A canonical form of these spinors is
\begin{eqnarray}
&&\nu_1=(n+im) (e_5+e_{12345})~,~~~\nu_2= (n-\ell+im) e_5+ (n+\ell+im) e_{12345}~,
\cr
&&\nu_3=n(e_5+e_{12345})+i m (e_1+e_{234})~,
\end{eqnarray}
respectively.
Using the $Spin(9,1)$ gauge symmetry, we choose ${\cal K}^\perp$ to lie along the directions of one of the
above spinors. Consider first the $\nu_1$ case. Write the
Killing spinors as
\begin{eqnarray}
\epsilon_r= f^1{}_r (1+e_{1234})+ f^{17}{}_r i (1+e_{1234})+ f^k{}_r
\eta_k \,,
\end{eqnarray}
where $\eta_k$ are remaining basis elements complementary to $1+e_{1234}$ and $i(1+e_{1234})$.
In what follows, we use the basis constructed from the five types of spinors in \cite{ugjggpdr}.
Substituting $\epsilon_r$ into (\ref{normcol}), we
get
\begin{eqnarray}
f^1{}_r n- f^{17}{}_r m=0~.
\end{eqnarray}
Without loss of generality, we take $n\not=0$. Using this, we solve for $f^1{}_r$ and substitute back
into the Killing spinors to find
\begin{eqnarray}
\epsilon_r={f^{17}{}_r\over n} (m+in) (1+e_{1234})+ f^k{}_r \eta_k~.
\end{eqnarray}
Similarly for the normal spinors $\nu_2$ and $\nu_3$, we find that
\begin{eqnarray}
&&\epsilon_r={f^{17}{}_r\over n}[
(m+in)(1+e_{1234})]+{f^{18}{}_r\over n} [\ell (1+e_{1234})-n
(1-e_{1234})]+ f^{k}{}_r \eta_{k} \,,
\cr
&&\epsilon_r={f^{19}{}_r\over n} [m (1+e_{1234})+i n(e_{15}+e_{2345})]+ f^k{}_r \eta_k~,
\end{eqnarray}
correspondingly, where $\eta_k$ are the remaining basis elements in each case.
Substituting these spinors into the algebraic Killing spinor equation and using that the rank of
the matrix $(f^i{}_r)$ is 31, for the $Spin(7)\ltimes \bb{R}$ case one finds that
\begin{eqnarray}
&&P_M\Gamma^M C*[(m+in) (1+e_{1234})]+{1\over24} G_{M_1M_2M_3}\Gamma^{M_1M_2M_3} (m+in) (1+e_{1234})=0~,
\cr
&&P_M\Gamma^M\eta_p=0~,~~~G_{M_1M_2M_3}\Gamma^{M_1M_2M_3}\eta_p=0~,~~~p=2,\dots, 16~,
\label{cona}
\end{eqnarray}
and similarly
\begin{eqnarray}
&&P_M\Gamma^M C*[(m+in) (1+e_{1234})]+{1\over24} G_{M_1M_2M_3}\Gamma^{M_1M_2M_3} (m+in) (1+e_{1234})=0~,
\cr
&&P_M\Gamma^M C*[\ell (1+e_{1234})-n (1-e_{1234})]
\cr
&&~~~~~~~~~~+{1\over24} G_{M_1M_2M_3}\Gamma^{M_1M_2M_3} [\ell (1+e_{1234})-n (1-e_{1234})]=0~,
\cr
&&P_M\Gamma^M C*[i (1-e_{1234})]+{1\over24} G_{M_1M_2M_3}\Gamma^{M_1M_2M_3} [i (1-e_{1234})]=0~,
\cr
&&P_M\Gamma^M\eta_{p}=0~,~~~G_{M_1M_2M_3}\Gamma^{M_1M_2M_3}\eta_{p}=0~,~~~p=3,\dots,16~,
\label{conb}
\end{eqnarray}
and
\begin{eqnarray}
&&P_M\Gamma^M C*[m (1+e_{1234})+in(e_{15}+e_{2345})]
\cr
&&~~~~~~~~~~~~~~+{1\over24} G_{M_1M_2M_3}\Gamma^{M_1M_2M_3} [m (1+e_{1234})+in(e_{15}+e_{2345})]=0~,
\cr
&&P_M\Gamma^M C*(i (1+e_{1234})+{1\over24} G_{M_1M_2M_3}\Gamma^{M_1M_2M_3} (i (1+e_{1234})=0~,
\cr
&&P_M\Gamma^M C* (e_{15}+e_{2345})+{1\over24} G_{M_1M_2M_3}\Gamma^{M_1M_2M_3} (e_{15}+e_{2345})=0~,
\cr
&&P_M\Gamma^M\eta_p=0~,~~~G_{M_1M_2M_3}\Gamma^{M_1M_2M_3}\eta_p=0~,~~~p=2,4,\dots,16~,
\label{conc}
\end{eqnarray}
for the other two cases. The factorization of $P$ and $G$ flux terms on $\eta_p$ occurs because some of the
remaining basis elements $\eta_k$
come in complex conjugate pairs $(\eta_p, i\eta_p)$, where $\eta_p$ are Majorana-Weyl spinors. Since
the $P$ flux term in the Killing spinor equations contains the charge conjugation matrix, $C*\eta_p=\eta_p$ and
$C*(i \eta_p)=-i\eta_p$, there is a relative sign between the $P$ and $G$ flux terms when the algebraic Killing spinor
equation is evaluated on
$\eta_p$ and $i\eta_p$. It now remains to solve these equations.
First, focus on the equation $P_M \Gamma^M\eta_p=0$. Observe
that in all cases, the remaining spinors $\eta_p$ contain spinors which are annihilated by either
$\Gamma^-$ or $\Gamma^+$. In the former case, the condition $P_M\Gamma^M\eta_p=0$ implies that only the $P_-$
component is non-vanishing
while in the latter case implies that only the component $P_+$ is non-vanishing. Since spinors of both types occur, $P=0$.
Next consider the conditions on the $G$ flux. It turns out that (\ref{cona}), (\ref{conb}) or (\ref{conc}) imply
that $G_{M_1M_2M_3}\Gamma^{M_1M_2M_3}\epsilon=0$ for all spinors
$\epsilon$ and so $G=0$. To see this consider the $Spin(7)\ltimes\bb{R}^8$ case. Setting $P=0$ in the first condition
in (\ref{cona}), we deduce that
$G_{M_1M_2M_3}\Gamma^{M_1M_2M_3}(1+e_{1234})=0$. Since the algebraic Killing spinor equations with $P=0$
are linear over the complex numbers, we also have that
$G_{M_1M_2M_3}\Gamma^{M_1M_2M_3}i(1+e_{1234})=0$. This together with the remaining conditions in (\ref{cona}) imply
that $G_{M_1M_2M_3}\Gamma^{M_1M_2M_3}\eta_i=0$ for all the basis elements $\eta_i$. A similar argument applies to the rest of the
cases. Thus we have found that the algebraic Killing spinor equations imply that $P=G=0$. We have also verified this by
an explicit computation.
Finally, if the $P$ and $G$ fluxes vanish, then the gravitino Killing spinor equation
of IIB supergravity becomes linear over the complex numbers. This means that backgrounds with vanishing
$P$ and $G$ fluxes always preserve an even number of supersymmetries. Thus backgrounds with 31
supersymmetries preserve an additional supersymmetry and so they
are maximally supersymmetric. In particular, they are
locally isometric \cite{jfgpa} to Minkowski spacetime, $AdS_5\times S^5$ \cite{schwarz} and the maximally supersymmetric plane wave
\cite{bfhp}. As a corollary, we have shown that IIB supergravity preons do not exist.
Our proof has relied on the algebraic Killing spinor equation of IIB supergravity and so does not straightforwardly generalize
to eleven-dimensional supergravity. Nevertheless, as we have seen the normal Killing spinor bundle construction generalizes to M-theory.
In addition,
one can show that the 31 Killing spinors of M-theory preon backgrounds take a simple form and it may
be possible to solve the Killing spinor equations. We hope to report on the existence
of M-theory preons in the future.
\vskip 0.5cm
\noindent{\bf Acknowledgements} \vskip 0.1cm The research of D.R.~is
funded by the PPARC grant PPA/G/O/2002/00475 and U.G.~has a
postdoctoral fellowship funded by the Research Foundation
K.U.~Leuven.
\vskip 0.5cm
|
1,314,259,996,761 | arxiv | \section{Introduction}
The $K$-matrix formalism for reactions between
particles with an internal structure is widely used in many domains of physics, including
molecular collisions~\cite{lin19}, mesoscopic physics~\cite{fyo96,alh00},
hadronic spectra~\cite{lon86}, nuclear reactions~\cite{kaw15}, and statistical
reaction theory in general~\cite{mit10}.
Its advantage over the competing $R$-matrix theory~\cite{wig47,lane58, desc10} is a
simplified connection between internal states of the compound
system and the channel wave functions of the incoming or outgoing
particles.\footnote{We use the term `particles' both for the elementary
constituents and the (possibly composite) reactants
in the initial or final states of the reaction.}
In particular, in the $K$-matrix approach, the Hamiltonian dynamics within the internal
states can be treated by well-known configuration-interaction (CI)
methods~\cite{low55}. However, the derivation of equations relating the $K$ matrix
to the Hamiltonian can be rather obscure
in the literature. The derivations often start from the Lippmann-Schwinger
equation and its associated $T$ matrix, which is already several steps removed from the Hamiltonian
equation expressed in a computationally transparent basis~\cite{dal61,mah69,chu95,tay06,mit10}. Here
we carry out the derivations starting from a representation of the Hamiltonian
$H$ in a discrete basis.
As a benefit, we find an expression for the dispersive couplings
of the internal states to the continuum that is computationally quite
simple. In contrast, many derivations in the literature suppress
these terms in the final formulas.
A simple version of our formalism has been applied in
nuclear reaction theory~\cite{fan18,ber18}. In the Mazama code
introduced in Ref.~\cite{fan18}, the diagonal $S$-matrix element is computed for
one specific channel, providing the elastic cross section in that channel and the total reaction cross section.
Here we consider a general scattering problem of any number of two-particle channels.
\section{Discrete-basis formulation of the scattering problem}
\subsection{Discretized two-particle Hilbert space}
The Hilbert space of the two-particle scattering system consists of two subspaces.
The first contains configurations, labeled by $\lambda$, that are used to
construct internal wave functions of the compound system;
the scattering wave function amplitude for each internal configuration $\lambda$ will be denoted $\psi_\lambda$.
The second subspace contains all
the scattering channels. Each channel $c$ is defined by
the set of configurations having the same internal structures for the two
particles and differing only in the relative coordinate between the particles'
centers of mass.
We introduce a discretized mesh of separation distances $r_n = R_0 + n \Delta r$ ($n=0,1,\ldots$)
with finite spacing $\Delta r$.
The channel wave function in channel $c$ then consists of the
set of amplitudes $\varphi_c(n)$ of the configurations on the mesh points,
\be
\label{phi_c}
\vec \varphi_c = \{ \varphi_c(0),\varphi_c(1),...\} \;.
\ee
$R_0$ is assumed to be sufficiently large such that
potential interactions between the reactants at larger
distances $r >R_0$ can be ignored. The first configuration $\varphi_c(1)$
is connecting to the internal states, either
directly or through some extension of the chain into the interacting
region. A less restrictive definition of the channel wave
function that allows for a potential interaction $V_c(r)$ in each channel $c$ is given in Appendix A.
\subsection{Hamiltonian matrix elements}
\subsubsection{Channel Hamiltonian}
The Hamiltonian in the channel space is taken to be
the kinetic energy operator of the relative motion of the two particles. It is approximated by the second-order
difference formula on neighboring mesh points.
Following nomenclature from condensed-matter physics, we denote the
Hamiltonian matrix element between adjacent states in the
channel by $t_c$ (here $t_c=\hbar^2/2 M_c(\Delta r)^2$ where $M_c$ is the reduced mass of the two fragments).
Then the Hamiltonian matrix $H^c$ describing the relative motion of the fragments in channel $c$ has the matrix elements
\be
\label{H^c}
H^c_{n,n'} = -t_c \delta_{n,n'+1} + (2 t_c +E_c ) \delta_{n,n'}
-t_c \delta_{n,n^\prime-1} \;,
\ee
where $E_c$ is the summed energy of the two reactants at rest.
In the region $r_n > R_0$ the Hamiltonian is invariant under translations,
so its eigenfunctions at energy $E$ can be expressed as a superposition of an incoming wave and an outgoing wave with
wave number $k_c$ and amplitudes $a_c^{(-)}$ and $a_c^{(-)}$, respectively
\be
\label{a_c}
\varphi_c(n) = a_c^{(-)} e^{-ik_c r_n} - a_c^{(+)} e^{i k_c r_n } \;.
\ee
Using $[H^c \vec \varphi_c](n) = E \varphi_c(n)$ for $n >0$ together with (\ref{a_c}), the energy-momentum dispersion is given by
\be\label{k_c}
E-E_c =2t_c(1-\cos \kappa_c) \;.
\ee
where
$\kappa_c=k_c \Delta r$. In the continuum limit, Eq.~(\ref{k_c}) reduces to the usual quadratic dispersion
$E-E_c= (\hbar^2/ 2 M_c) k_c^2$.
\subsubsection{Interaction with internal states}
The Hamiltonian matrix elements involving states in the interaction
region $r \leq R_0$ are of
two kinds: those strictly between internal states and those
that connect with the channel wave functions at the $n=1$ site.
We denote the latter matrix elements connecting the internal state $\lambda$ with channel $c$ by $v_{\lambda,c}$.
Fig.~\ref{Hconnectivity} demonstrates the states and the Hamiltonian matrix elements
that connect them.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.8\columnwidth]{fig1.pdf}
\caption{Connectivity of the discretized Hamiltonian. The
internal states are enclosed in the large dashed circle.
Small open circles represent states
of the internal Hamiltonian and the solid lines indicate off-diagonal
matrix elements of the internal Hamiltonian.
Solid circles represent
the discretized channel configurations. They are coupled to each other
through the dotted lines to generate the channel Hamiltonian.
The dashed lines denote matrix elements $v_{\lambda,c}$ connecting the channels to the internal states.
\label{Hconnectivity}
}
\end{center}
\end{figure}
We consider $N_i$ internal states and $N_c$ channels, and assume that the internal state Hamiltonian is diagonal
with energies $E_\lambda$. For each channel $c$, the wave function is regular at $n=0$, i.e., $\varphi_c(0)=0$.
At radial site $n = 1$ the scattering wave function satisfies the Hamiltonian equation
\be
\label{H1}
-t_c \varphi_c(2) + ( 2 t_c+E_c) \varphi_c(1) + \sum_{\lambda = 1}^{N_{i}} v_{\lambda,c} \psi_\lambda = E \varphi_c(1) \,, \qquad c = 1,...,N_c \;,
\ee
while the corresponding equations for the internal-state amplitudes are
\be
\label{Hr}
\sum_{c=1}^{N_c} v_{\lambda,c}\varphi_c(1) + E_\lambda \psi_\lambda = E \psi_\lambda \,,\qquad \lambda = 1,...,N_i \;.
\ee
Eliminating the internal state amplitudes $\psi_\lambda$ from Eqs.~(\ref{Hr}) and substituting in Eqs.~(\ref{H1}), we find
\be\label{Hcc'}
-t_c\varphi_c(2) + (2t_c \cos\kappa_c) \varphi_c (1) + \sum_{\lambda, c'} \frac{v_{\lambda c} v_{\lambda c'}}{E-E_\lambda} \varphi_{c'}(1) =0\;.
\ee
\section{$D$ matrix}
Substituting the channel wave function form (\ref{a_c}) in Eqs.~(\ref{Hcc'}) yields a set of coupled linear equations relating the
vector of outgoing amplitudes $\vec a^{(+)}= (a_1^{(+)},a_2^{(+)},...,a_{N_c}^{(+)})$
to the vector of incoming amplitudes
$\vec a^{(-)}= (a_1^{(-)},a_2^{(-)},...,a_{N_c}^{(-)})$
\be\label{a-a+}
t_c a_c^{(-)} + \sum_{\lambda c'} \frac{v_{\lambda c} v_{\lambda c'}}{E-E_\lambda} e^{-i\kappa_{c'} }a_{c'}^{(-)} =
t_c a_c^{(+)} + \sum_{\lambda c'} \frac{v_{\lambda c} v_{\lambda c'}}{E-E_\lambda} e^{i\kappa_{c'} }a_{c'}^{(+)} \;,
\ee
where we have absorbed a factor of $e^{-ik_c R_0}$ in $ a_c^{(-)}$ and a factor of $e^{ik_c R_0}$ in $ a_c^{(+)}$.
In principle, we could define an $N_c \times N_c$ matrix
that transforms $\vec a^{(-)}$ to $\vec a ^{(+)}$ but this is
not the $S$-matrix. The $S$-matrix preserves the total
probability flux and requires the amplitudes
$a_c^{(\pm)}$ to be normalized to the unit flux. To change to
flux-normalized variables, we note that, for a tridiagonal channel Hamiltonian, the probability current $J_c(n \to n+1)$ from a
site $n$ to the neighboring site $n+1$ is given by
\be\label{current}
J_c(n \to n+1) = i H^c_{n,n+1} [\varphi_c(n)\varphi_c^*(n+1) - \varphi^*_c(n)\varphi_c(n+1))]\;
\ee
up to a channel-independent constant.
Applying (\ref{current}) to the wave functions $a_c^{(\pm)} e^{\pm i k_c r_n}$ for the Hamiltonian (\ref{H^c}), we find for the current
$J_c$ in channel $c$
\be\label{model_curr}
J_c = \pm 2 t_c \sin \kappa_c \, |a_c^{(\pm)}|^2 \;,
\ee
which is independent of $n$.
The flux-normalized amplitudes are thus
\be\label{bc_def}
b_c^{(\pm)} = a_c^{(\pm)}/d_c \;,
\ee
where
\be\label{dcdef}
d_c = \left(2 t_c \sin \kappa_c\right)^{-1/2} \;.
\ee
Eqs.~(\ref{a-a+}) can be rewritten for the flux-normalized amplitudes $b_c^{(\pm)}$
\be\label{b-b+}
d_c t_c b_c^{(-)} + \sum_{\lambda c'} \frac{v_{\lambda c} v_{\lambda c'}}{E-E_\lambda} d_{c'}e^{-i\kappa_{c'} }b_{c'}^{(-)} =
d_c t_c b_c^{(+)} + \sum_{\lambda c'} \frac{v_{\lambda c} v_{\lambda c'}}{E-E_\lambda} d_{c'} e^{i\kappa_{c'} }b_{c'}^{(+)} \;.
\ee
Dividing both sides of the equation by $d_c t_c$, we obtain
\be\label{DbDb}
D \vec b^{(-)} = D^* \vec b^{(+)} \;,
\ee
where the matrix $D$ is defined by\footnote{The definitions include factors
of $2\pi$ and $(2\pi)^{-1/2}$ following the convention in the
literature~\cite{mit10}}.
\be\label{Ddef}
D_{c,c^\prime} = \delta_{c,c^\prime} + 2\pi \sum_{\lambda=1}^{N_i} \frac{W_{\lambda c} W_{\lambda,c^\prime}}{E - E_\lambda} d^2_{c^\prime} t_{c'} e^{-i\kappa_{c'}}
\ee
with
\be\label{Wdef}
W_{\lambda c} =\frac{1}{\sqrt{2\pi}}\frac{v_{\lambda c} } {d_c t_c} \;.
\ee
$D^*$ is obtained from $D$ by simply replacing $e^{-i\kappa_{c'}} \to e^{i \kappa_{c'}}$.
The $S$ matrix is defined by $ \vec b^{(+)} = S \vec b^{(-)}$ and, using Eq.~(\ref{DbDb}), is given by
\be
\label{SDD}
S={D^*}^{-1} D \;.
\ee
\section{$K$ matrix}
The $K$ matrix is defined from the $S$ matrix by the implicit relation
\be
\label{S-K}
S = \frac{1+iK}{1-iK} \;.
\ee
Substituting Eq.~(\ref{SDD}) in Eq.~(\ref{S-K}), and solving for $K$, we express $K$ in terms of $D$ and $D^*$
\be\label{KfromD}
K =- i (D+D^*)^{-1} (D-D^*)\;.
\ee
In the following we derive an explicit expression for the matrix elements of $K$. Using Eq.~(\ref{Ddef}), we have
\be\label{DpmD*}
\frac{D + D^*}{2} = 1+ W^T (E - H)^{-1} V \;,\;\;\; \frac{D - D^*}{2} = -i \pi W^T (E - H)^{-1} W \;,
\ee
where
\be
H=\sum_\lambda | \lambda\rangle E_\lambda \langle \lambda|
\ee
is the internal state Hamiltonian of the compound system. The matrix $V$ is defined by
\be\label{V}
V_{\lambda c} = \pi W_{\lambda c} \cot \kappa_c \;,
\ee
where we have used $d_c^2 t_c =(2 \sin \kappa_c)^{-1}$.
Substituting Eq.~(\ref{DpmD*}) in (\ref{KfromD}), we find
\be\label{K-X}
K = -(1+X)^{-1} X \tan \kappa = -[1-(1+X)^{-1}] \tan \kappa \;,
\ee
where the matrix $X$ is defined by
\be
X= W^T (E - H)^{-1} V\;,
\ee
and $\tan \kappa$ is a diagonal matrix with elements $\tan \kappa_c$ along its diagonal.
To invert $1+X$ we use the operator identity $B^{-1}(B-A)A^{-1} = A^{-1}-B^{-1}$ with $A=E- H$ and $B=E-H +V W^T$ to find
\be
(E-H +V W^T)^{-1} V W^T (E-H)^{-1} = (E-H)^{-1} - (E-H + V W^T)^{-1} \;.
\ee
Multiplying by $W^T$ on the left and by $V$ on the right, we obtain
\be\label{X-Y}
YX=X-Y \;,
\ee
where
\be
Y= W^T (E-H+ VW^T )^{-1} V \;.
\ee
Solving (\ref{X-Y}), we find $(1+X)^{-1}=1-Y$. Substituting in (\ref{K-X}), we find
\be
K=- Y \tan\kappa = -W^T (E-H+ VW^T)^{-1} V \tan\kappa = -\pi W^T (E-H+ VW^T)^{-1} W \;,
\ee
where we have used Eq.~(\ref{V}).
The final expression for $K$ is thus
\be\label{K-matrix}
K= \pi W^T (H+\Delta -E)^{-1} W \;,
\ee
where $W$ is given in Eq.~(\ref{Wdef}) and describes the coupling matrix of the channels to the internal states, while $\Delta = - VW^T$ is the real shift matrix
\be\label{Delta_def}
\Delta_{\lambda \lambda'} = -\pi \sum_c W_{\lambda c} W_{\lambda' c} \cot \kappa_c \;.
\ee
The above expression for $K$ has the same form as the usual $K$ matrix, c.f.~Eq.~(18) of Ref.~\cite{alh00}.
However, our term includes the real shift
matrix $\Delta$ that is usually ignored in expressions for the $K$ matrix.
In other derivations of the $S$ matrix, this shift arises from off-shell
couplings to the channels; see, e.g., Eqs.~(28-30) of Ref.~\cite{mit10}. In
our approach, this shift arises naturally from the matrix algebra.
The $K$ matrix in (\ref{K-matrix}) is real symmetric, which guarantees that the $S$ matrix in (\ref{S-K}) is symmetric and unitary.
\section{$S$ matrix}
To find an explicit expression for the $S$ matrix, we use again the operator identity ${B^{-1}(B-A)A^{-1} = A^{-1}-B^{-1}}$ but now for $A=E- (H+\Delta)$ and
$B=E- (H+\Delta -i\pi WW^T)$. We obtain
\begin{eqnarray}
i \pi [E- (H+\Delta -i\pi WW^T)]^{-1} W W^T [E-(H+\Delta)]^{-1} \nonumber \\
= [E- (H+\Delta)]^{-1} - [E- (H+\Delta -i\pi WW^T)]^{-1} \;.
\end{eqnarray}
Multiplying by $\pi W^T$ on the left and by $W$ on the right, we find
\be \label{K-Z1}
i\pi ZK = K+\pi Z \;,
\ee
where
\be\label{Z}
Z = W^T [E- (H+\Delta -i\pi WW^T)]^{-1} W \;,
\ee
and we have used the expression (\ref{K-matrix}) for the $K$ matrix. Relation (\ref{K-Z1}) can be rewritten in the form
\be\label{K-Z2}
\frac{K}{1-iK} = -\pi Z \;.
\ee
Using the relation (\ref{S-K}) between the $S$ matrix and the $K$ matrix, we find
\be
S=1+2i \frac{K}{1-iK} = 1 - 2\pi i Z
\ee
where we have used (\ref{K-Z2}) to obtain second equality. We thus find an explicit expression for $S$
\be
S= 1 - 2\pi i W^T [E- (H+\Delta -i\pi WW^T)]^{-1} W \;,
\ee
which includes both a real shift $\Delta$ and an imaginary shift $-i\pi WW^T$ to the Hamiltonian $H$. This expression coincides formally
with Eqs.~(28-30) in Ref.~\cite{mit10} for the $S$ matrix in the absence of background scattering.
\section{Concluding remarks}
We have described an alternative derivation of the $K$ matrix of
scattering theory, as shown for the Hamiltonian specified
by Eqs.~(\ref{H^c}), (\ref{H1}) and (\ref{Hr}). Using this derivation, we are able to avoid imposing
formal structures such as the continuum Green's functions of
Lippmann-Schwinger reaction theory. It practice, it is well-suited to
many-body Hamiltonians of equal-mass particles, in which case it may be difficult to
identify a relative coordinate. This includes nuclei and atomic
condensates where common practice follows the
Hartree-Fock or Hartree-Fock-Bogoliubov approximations and their extensions
in the CI framework. For large systems, this approach needs much less
computational effort than other reaction formalisms, which rely on explicit
antisymmetrization and/or the use of a Jacobi coordinate representation
to separate out a channel wave function $\varphi^r_c(r)$ in the relative
coordinate $r$ of the two particles.
In the above derivation, we left unspecified the exact relationship of the
usual channel wave function $\varphi^r_c$ to the discrete-basis wave function
$\vec \varphi_c$. These quantities have different
dimensions: the components of $\vec \varphi_c$ are dimensionless amplitudes in the CI formalism
while $\varphi^r_c$ has dimension $[{\rm length}]^{-1/2}$, the same as ordinary
coordinate-space wave functions. The formal connection between the two
is not obvious, since it is difficult to separate out a relative
coordinate wave function unless it is already defined in the CI basis.
Our approach only involves the role of the relative coordinate at large
separations, where the absence of interactions leads to
the simplified Hamiltonian approximation in Eq.~(\ref{H^c}).
The present formalism might be applicable
to problems in nuclear reaction theory such as fission~\cite{ber19}. It should also
simplify the treatment of the interaction between droplets
of atomic condensates, such as the fusion reaction described in Ref.~\cite{shin04}.
\section*{Acknowledgements}
We thank J.J.~Rehr for discussion on possible applications to molecular
reactions. The work of Y.A. and P.F. was supported in part by the U.S. DOE grant No.~DE-SC0019521, and by the U.S. DOE NNSA Stewardship
Science Graduate Fellowship under cooperative agreement No.~DE-NA0003864.
|
1,314,259,996,762 | arxiv |
\section{Concluding remarks}
\label{sec:conclusion}
Having sharply characterized the worst-case excess risk of \problemName{Seq\smallDash{}LS}{} and \problemName{LDS\smallDash{}LS}{}, we see more
precisely the trade-offs---or arguably the lack thereof---presented by resetting %
a system, or by simply observing parallel runs from one, where possible.
After sufficient resets, one learns roughly as though examples were independent altogether
(as reflected in the \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} and \problemName{Ind\smallDash{}LDS\smallDash{}LS}{} baselines).
In addition to the theoretical upshot that it presents,
this phenomenon seems encouraging insofar as the setup may describe reality:
one does not learn to ride a bicycle by witnessing thousands of unrelated pedal strokes,
nor by watching one cyclist endure the entire Tour de France,
but rather by seeing and attempting many moderate rides and maneuvers.
We see a number of future directions for research,
primarily in further
charting out the reach of the iid{}-like phenomenon in learning from multiple sequences.
Our work offers the trajectory small-ball criterion (\Cref{def:trajectory_small_ball})
as a vehicle for proving that this phenomenon occurs,
or otherwise for bounding the minimax rate from above.
What other notable sequential processes,
outside of those covered in~\Cref{sec:results:upper:small_ball_examples},
can we capture as trajectory small-ball instances?
One might look to covariate sequences generated by, say,
input-to-state stable (ISS) non-linear systems,
stochastic polynomial difference equations,
or various Markov decision processes.
On the flip side, when must we necessarily pay a price for dependent data?
One answer from our work is that a necessary gap between
independent and sequentially dependent learning
appears when there are insufficiently many trajectories ($m \lesssim n$).
As outlined in~\Cref{sec:proof_ideas:beyond_diag} and~\Cref{sec:beyond-diag-results}, we conjecture that this gap can be
made much wider, namely by considering non-diagonalizable linear dynamical systems.
That said, other pertinent problems may exhibit gaps as well.
Finding them would help inform where the limits of learning from sequential data lie.
On the regression side, one might look to move beyond a well-specified linear regression model,
extend to other loss functions,
analyze regularized least-squares estimators in place of OLS,
or consider a more adversarial analysis (e.g.\ measuring regret rather than risk, in an online setting).
\section{Beyond diagonalizability: a conjecture for the general case}
\label{sec:beyond-diag-results}
Recall that various results in
\Cref{sec:upper_bounds:lds} required a diagaonalizability
assumption (\Cref{assume:diagonalizability}) on the dynamics matrix $A$,
specifically in the many trajectories regime when ${T'} > T$
(\Cref{stmt:upper_bound_many_trajectories}),
or in the few trajectories regime (\Cref{sec:upper_bounds:lds:few_trajectories}).
In this section, we conjecture how removing
the diagonalizability assumption would affect the results.
For simplicity, we focus on the few trajectories regime, and
further assume that ${T'} = T$.
Building on potential extensions of this paper's analysis, and numerical evidence
detailed in \Cref{sec:proof_ideas}, we conjecture the following
extensions of
\Cref{thm:sublinear_trajectories_bound} and \Cref{stmt:lower_bound_main}:
\begin{myconjecture}[Risk for \DLR{} with few trajectories under non-diagonalizable systems]
\label{conj:general-few-trajectories}
There are universal positive constants $c_0$, $c_1$, $c_2$, $c_3$,
and a universal mapping $\varphi : \mathbb{N}_{+} \rightarrow \ensuremath{\mathbb{R}}_{>0}$
such that the following holds for any instance of \DLR{} satisfying \Cref{assume:marginal_stability} (marginal stability)
and \Cref{assume:one_step_controllability} (one-step controllability).
Let $A = S J S^{-1}$ denote the Jordan normal form of the dynamics matrix $A$.
Define
$\gamma := \frac{\lambda_{\max}(S^{-1} BB^\mathsf{T} S^{-*})}{\lambda_{\min}(S^{-1} BB^\mathsf{T} S^{-*})}$,
and let $r$ be the size of the largest Jordan block in $J$.
If $n \geq c_0$, $m \leq c_1 n$, and $mT \geq c_2 n$, then:
\begin{align}
\mathbb{E}[L(\hat{W}_{m,T}; {T'}, \PxA{A,B})]
\leq c_3 \sigma_\xi^2 \varphi(r) \gamma
\cdot \frac{pn^{2r}} {m^{2r} T}.
\label{eq:conjecture_upper_bound}
\end{align}
Additionally,
there exist universal positive constants $c'_0$, $c'_1$, $c'_2$, $c'_3$, and $c'_4$ such that the following is true.
Suppose $\mathcal{A} \subseteq \ensuremath{\mathbb{R}}^{n \times n}$ is any set
containing all $n \times n$ matrices with Jordan blocks of size at most $r$.
Let $T \geq c'_0$, $n \geq c'_1$, $mT \geq c'_2 n$, and $m \leq c'_3 n$.
Then:
\begin{align}
\mathsf{R}(m, T, T; \{ \PxA{A} \mid A \in \mathcal{A} \})
\geq c'_4 \sigma_\xi^2 \varphi(r) \gamma
\cdot \frac{pn^{2r}}{m^{2r} T}.
\label{eq:conjecture_lower_bound}
\end{align}
\end{myconjecture}
\Cref{stmt:upper_bound_general}
provides a viable path towards proving the upper bound
\eqref{eq:conjecture_upper_bound} from \Cref{conj:general-few-trajectories}
up to logarithmic factors
in the regime of constant Jordan block size $r$,
by reducing the problem to understanding
the scaling of $\underline{\lambda}(k,t;A,B) = \underline{\lambda}(\Gamma_k(A, B), \Gamma_t(A, B))$ when $k \leq t$.
Our analysis uses diagonalizability (\Cref{assume:diagonalizability})
of the dynamics matrices to show that $\underline{\lambda}(k,t;A,B) \gtrsim \gamma^{-1} \cdot k/t$.
Without such an assumption, analyzing $\underline{\lambda}(k,t;A,B)$ is
substantially more involved.
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\linewidth]{figures/ulam_T_k_plot.pdf}
\caption{
A plot of the ratio $\alpha$ versus $1/\underline{\lambda}(k,t)$ with $k$ fixed to $5$ and $t$ fixed to $k \alpha$.
Here, $\underline{\lambda}(k,t) := \underline{\lambda}(k,t;J_r,I_r)$.
The slope of the line
(in log-log space) computed via linear regression is reported.
We conjecture that in general, $\underline{\lambda}(k,t;A,B) \gtrsim c_{r} \gamma^{-1} \cdot (k/t)^{2r-1}$.}
\label{fig:ulam_T_k}
\end{figure}
A numerical simulation (\Cref{fig:ulam_T_k})
suggests that $\underline{\lambda}(k,t;A,B) \gtrsim c_{r} \gamma^{-1} \cdot (k/t)^{2r-1}$ is
the general rate for dynamics matrices $A$ with Jordan
blocks at most size $r$, where $c_{r}$ is a constant depending only on $r$.
Assuming this scaling to be correct and plugging the rate
into \Cref{stmt:upper_bound_general} yields
\eqref{eq:conjecture_upper_bound} up to logarithmic factors.
Partial progress towards analyzing $\underline{\lambda}(k,t;A,B)$
was made in \cite[Proposition~7.6]{sarkar2019sysid}, where
it is shown that $\underline{\lambda}(k,t;A,B) \gtrsim c_{r} \gamma^{-1} \cdot (k/t)^{r^2}$,
with $1/c_{r}$ depending exponentially on $r$.
We do not conjecture a form for the
mapping $\varphi(r)$;
$\underline{\lambda}(k,t;A,B)$ becomes numerically ill-conditioned when
$r$ is large, hindering simulation with large blocks.
On the other hand, the analytic arguments in \Cref{sec:lower_bound_proof_sketch:few_trajectories}
combined with the numerical evidence in \Cref{fig:risk_lower_bound}
suggest that the bound \eqref{eq:conjecture_lower_bound} holds
(up to the condition number factor $\gamma$).
The one caveat is that, even if we were to analytically
characterize the eigenvalues of $\Theta_{r,T,T}$ for all $r$, our proof strategy
would most likely not be able to give a sharp characterization
of the leading constant $\varphi(r)$ in the lower bound.
This is because our
proof inherently exploits the independence between
decoupled subsystems, and does not tackle the harder challenge
of understanding the coupling effects within a Jordan block.
We conclude this section by noting that \Cref{conj:general-few-trajectories}
does not include any logarithmic factors in
the upper bound rate \eqref{eq:conjecture_upper_bound},
and includes the condition number factor $\gamma$
in the lower bound \eqref{eq:conjecture_lower_bound}.
In other words, \Cref{conj:general-few-trajectories}
applied to the special case of $r=1$ conjectures
that \Cref{thm:sublinear_trajectories_bound} is loose by $\log^2(\gamma n/m)$,
and that \Cref{stmt:lower_bound_main} is loose by a factor of $\gamma$.
\section{Numerical simulation}
\label{sec:experiments}
We conduct a simple numerical simulation illustrating the benefits of
multiple trajectories on learning. We construct a family of
\problemName{LDS\smallDash{}LS}\ problem instances, parameterized by a scalar $\rho \in (0, \infty)$ as follows.
The covariate distribution $\mathsf{P}_x$ is
the linear dynamical system $x_{t+1} = A x_t + w_t$ with:
\begin{align*}
A = U \diag( \underbrace{\rho, \dots, \rho}_{\floor{n/2} \textrm{ times}}, -\rho, \dots, -\rho ) U^\mathsf{T}, \quad U \sim \mathrm{Unif}(O(n)), \quad w_t \sim N(0, I/4).
\end{align*}
Here, $O(n)$ denotes the set of $n \times n$ orthonormal matrices.
By construction, $\rho$ is the spectral radius of $A$.
The labels $y_t$ are set as $y_{t} = x_{t+1}$, so that
the ground truth $W_\star \in \ensuremath{\mathbb{R}}^{n \times n}$ is equal to $A$.
We compare the risk of the OLS estimator \eqref{eq:ols_definition}
on the \problemName{LDS\smallDash{}LS}\ problem instance, compared with its risk on the corresponding
\problemName{Ind\smallDash{}LDS\smallDash{}LS}\ baseline. Specifically,
we plot the ratio between OLS excess risks $\mathbb{E}[L(\cdot; T, \mathsf{P}_x)]$ on the two problem instances ($\mathsf{P}_x$),
respectively.
We fix the covariate dimension $n=5$ and
the trajectory horizon length $T=10n$, and vary the number of trajectories
$m \in \{1, \dots, 10\}$. \Cref{fig:multitraj} shows the result of this experiment, where we also vary $\rho \in \{0.98, 0.99, 1.0, 1.01, 1.02\}$.
The error bars are plotted over $1000$ trials. All computations are implemented
using \texttt{jax}~\cite{jax2018github}, and run with \texttt{float64} precision on a single machine.
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\linewidth]{figures/multitraj.pdf}
\caption{
Plot of the ratio of excess risk for
\problemName{LDS\smallDash{}LS}\ problem instances over its corresponding
\problemName{Ind\smallDash{}LDS\smallDash{}LS}\ baseline instance, as a function of the number of
trajectories $m$, holding both covariate dimension $n$ and
horizon length $T$ fixed.
The vertical blue line marks the transition between few trajectories ($m \leq n$) and many
trajectories ($m \geq n$).}
\label{fig:multitraj}
\end{figure}
In \Cref{fig:multitraj}, we see that for the few trajectories regime ($m \leq n$) appearing to the left of the
vertical blue line, the instability of
the covariate process plays an outstanding role in determining the
value of the ratio. On the other hand, for the
many trajectories regime ($m \geq n$) appearing to the right of the blue line, the ratios quickly converge to a constant no greater than two (at $m=10$). This behavior is consistent with
\Cref{stmt:upper_bound_many_trajectories}.
Finally, \Cref{stmt:lower_bound_main} suggests that the scaling
behavior of the $\rho=1$ curve with respect to $m$ is on the order of
$1/m$.
\subsection{High probability upper bounds}
\label{sec:app:high_prob_upper_bounds}
\subsubsection{Weak trajectory small ball}
We first present a modified definition of trajectory small-ball
(cf.~\Cref{def:trajectory_small_ball}) which we will use to establish
high probability bounds.
\begin{mydef}[Weak trajectory small-ball (wTrajSB{})]
\label{def:weak_trajectory_small_ball}
Fix a trajectory length $T \in \ensuremath{\mathbb{N}}_+$,
a parameter $k \in \{1, \dots, T\}$,
positive definite matrices $\{\Psi_j\}_{j=1}^{\floor{T/k}} \subset \mathsf{Sym}^{n}_{> 0}$,
and constants $\alpha, \beta \in (0, 1)$.
The distribution $\mathsf{P}_x$ satisfies the
$(T, k, \{\Psi_j\}_{j=1}^{\floor{T/k}}, \alpha, \beta)$-\emph{weak-trajectory-small-ball (wTrajSB{})}
condition if:
\begin{enumerate}
\item $\frac{1}{\floor{T/k}} \sum_{j=1}^{\floor{T/k}} \Psi_j \preccurlyeq \Gamma_T(\mathsf{P}_x)$,
\item $\{x_t\}_{t \geq 1}$ is adapted to a filtration $\{\mathcal{F}_t\}_{t \geq 1}$, and
\item for all $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$, $j \in \{1, \dots, \floor{T/k}\}$:
\begin{align}
\Pr_{\{x_t\} \sim \mathsf{P}_x}\left\{
\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \ip{v}{x_t}^2 \leq \alpha \cdot v^\mathsf{T} \Psi_j v
~\Bigg|~ \mathcal{F}_{(j-1)k} \right\} \leq \beta \:\:
\textrm{a.s.}\label{eq:weak_trajectory_small_ball}
\end{align}
\end{enumerate}
\end{mydef}
The main difference between \Cref{def:weak_trajectory_small_ball}
vs.\ \Cref{def:trajectory_small_ball} is the third condition
\eqref{eq:weak_trajectory_small_ball}, which only needs to hold for a
\emph{fixed} resolution $\alpha$ and failure probability $\beta$.
By contrast, in \Cref{def:trajectory_small_ball}, the condition
must hold of for \emph{all} resolutions---there denoted by
$\varepsilon$---with failure probabilities that tend to zero as
the resolution $\varepsilon \to 0$
(cf.~\eqref{eq:trajectory_small_ball}).
\subsubsection{Ordinary least squares bounds}
\begin{mylemma}[Minimum eigenvalue bound via weak trajectory small-ball]
\label{stmt:weak_small_ball_to_min_eval}
Suppose that $\mathsf{P}_x$ satisfies
the $(T,k,\{\Psi_j\}_{j=1}^{\floor{T/k}},\alpha,\beta)$-wTrajSB{} condition
(\Cref{def:weak_trajectory_small_ball}).
Put $S := \floor{T/k}$ and $\Gamma_t := \Gamma_t(\mathsf{P}_x)$
for $t \in \ensuremath{\mathbb{N}}_{+}$.
Fix any $\underline{\Gamma} \in \mathsf{Sym}^{n}_{> 0}$ satisfying
$\frac{1}{S} \sum_{j=1}^{S} \Psi_j \preccurlyeq \underline{\Gamma} \preccurlyeq \Gamma_T$,
and define the constants:
\begin{align}
C_S := \frac{\frac{1}{S}\sum_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma})^2}{\left(\frac{1}{S}\sum_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma})\right)^2}, \quad \bar\mu := \frac{1}{S} \sum_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma}). \label{eq:weak_small_ball_growth_conditions}
\end{align}
(Note that $1 \leq C_S \leq S$ always).
Fix $\delta \in (0, 1)$, and
suppose that:
\begin{align*}
n \geq 2, \quad \frac{mT}{kn} \geq \frac{64C_S}{1-\beta}\log\left( \frac{1280 C_S}{\alpha (1-\beta) \underline{\lambda}(\underline{\Gamma}, \Gamma_T) \bar\mu \delta}\right).
\end{align*}
With probability at least $1-\delta$,
the following events simulatenously hold:
\begin{align}
\lambda_{\min}\left( \underline{\Gamma}^{-1/2}\sum_{i=1}^{m}\sum_{t=1}^{T} x_t^{(i)} (x_t^{(i)})^\mathsf{T} \underline{\Gamma}^{-1/2}\right) &\geq \frac{\alpha (1-\beta) mT \bar\mu}{8}, \label{eq:high_prob_min_eval_bound} \\
\Tr\left( \underline{\Gamma}^{-1/2}\sum_{i=1}^{m}\sum_{t=1}^{T} x_t^{(i)} (x_t^{(i)})^\mathsf{T} \underline{\Gamma}^{-1/2} \right) &\leq \frac{2mTn }{\underline{\lambda}(\underline{\Gamma}, \Gamma_T) \cdot \delta}. \nonumber
\end{align}
\end{mylemma}
\begin{proof}
The proof proceeds quite similarly to the proof of
\Cref{stmt:small_ball_to_min_eval}.
Thus, we focus mostly on the parts that differ.
For notational brevity, let:
\begin{align*}
\beta' := 1 - \beta, \quad \underline{\lambda} := \underline{\lambda}(\underline{\Gamma}, \Gamma_T), \quad \underline{\lambda}_j := \underline{\lambda}(\Psi_j, \underline{\Gamma}).
\end{align*}
Since $\underline{\Gamma} \preccurlyeq \Gamma_T$ by assumption, we have
$\underline{\lambda} \in (0, 1]$.
The first step, in preparation for applying the PAC-Bayes deviation
inequality, is to construct a family of random variables with moment generating function upper bounded by one.
To do this, we utilize the weak trajectory small-ball condition \eqref{eq:weak_trajectory_small_ball}, which
implies for any $v \in \mathbb{S}^{n-1}$ and $j \in \{1, \dots, S\}$:
\begin{align*}
\Pr\left\{ \frac{1}{k}\sum_{t=(j-1)k+1}^{jk} \ip{v}{\underline{\Gamma}^{-1/2} x_t}^2 \leq \alpha \underline{\lambda}(\Psi_j, \underline{\Gamma}) \,\Bigg|\, \mathcal{F}_{(j-1)k} \right\} \leq \beta.
\end{align*}
Let $\tilde{x}_t := \underline{\Gamma}^{-1/2} x_t$ be the whitened vector.
Define the random indicator variables
for $i=1, \dots, m$ and $j=1, \dots, S$:
\begin{align*}
B_j^{(i)} &:= \mathbf{1}\left\{ \frac{1}{k}\sum_{t=(j-1)k+1}^{jk} \ip{v}{\tilde{x}_t^{(i)}}^2 \geq \alpha \underline{\lambda}(\Psi_j, \underline{\Gamma}) \right\}.
\end{align*}
By Markov's inequality:
\begin{align*}
\sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2
\geq k\alpha \sum_{j=1}^{S} \underline{\lambda}_j \mathbf{1}\{ B_j^{(i)} = 1 \}.
\end{align*}
Hence for any $\eta > 0$ and $v \in \mathbb{S}^{n-1}$:
\begin{align*}
\mathbb{E}\exp\left( -\eta \sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2 \right) \leq \mathbb{E} \exp\left( -\eta k \alpha\sum_{j=1}^{S} \underline{\lambda}_j \mathbf{1}\{ B_j^{(i)} = 1 \} \right).
\end{align*}
Now observe:
\begin{align*}
\mathbb{E}[\exp(-\eta k \alpha \underline{\lambda}_j \mathbf{1}\{B_j^{(i)}=1\}) \mid \mathcal{F}_{(j-1)k}] &= e^{-\eta k \alpha \underline{\lambda}_j} \Pr(B_j^{(i)}=1 \mid \mathcal{F}_{(j-1)k}) + \Pr(B_j^{(i)}=0 \mid \mathcal{F}_{(j-1)k}) \\
&= (e^{-\eta k \alpha \underline{\lambda}_j} - 1) \Pr(B_j^{(i)}=1 \mid \mathcal{F}_{(j-1)k}) + 1 \\
&\leq (e^{-\eta k \alpha \underline{\lambda}_j} - 1) \beta' + 1 \\
&\stackrel{(a)}{\leq} 1 + \left( -\eta k \alpha \underline{\lambda}_j + \frac{1}{2} \eta^2 k^2 \alpha^2 \underline{\lambda}_j^2 \right)\beta' \\
&\stackrel{(b)}{\leq} \exp\left( \left( -\eta k \alpha \underline{\lambda}_j + \frac{1}{2} \eta^2 k^2 \alpha^2 \underline{\lambda}_j^2 \right)\beta' \right).
\end{align*}
Above, we used the facts
(a) for $x > 0$, we have
$e^{-x} - 1 \leq -x + \frac{x^2}{2}$,
and (b) for $x \in \ensuremath{\mathbb{R}}$, we have $1 + x \leq e^x$.
Hence by the tower property:
\begin{align*}
\mathbb{E}\exp\left( -\eta\sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2\right)
&\leq \exp\left( \sum_{j=1}^{S} \left( -\eta k \alpha \underline{\lambda}_j + \frac{1}{2} \eta^2 k^2 \alpha^2 \underline{\lambda}_j^2 \right)\beta' \right) \\
&\leq \exp\left( - \eta k \alpha \beta' \sum_{j=1}^{S} \underline{\lambda}_j + \frac{1}{2}\eta^2 k^2 \alpha^2 \beta' \sum_{j=1}^{S} \underline{\lambda}_j^2 \right) \\
&= \exp\left( -\eta k \alpha \beta' \left( \sum_{j=1}^{S} \underline{\lambda}_j\right) \left( 1 - \frac{\eta k \alpha}{2} \frac{\sum_{j=1}^{S} \underline{\lambda}_j^2}{\sum_{j=1}^{S} \underline{\lambda}_j} \right)\right).
\end{align*}
Now, let us set
\begin{align*}
\eta = \frac{1}{k\alpha} \frac{\sum_{j=1}^{S} \underline{\lambda}_j}{\sum_{j=1}^{S} \underline{\lambda}_j^2} = \frac{1}{k\alpha} \cdot \frac{1}{\bar\mu} \cdot \frac{1}{C_S},
\end{align*}
from which we conclude:
\begin{align*}
\mathbb{E}\exp\left( -\eta\sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2\right)
&\leq \exp\left( -\frac{\beta'}{2} \frac{\left(\sum_{j=1}^{S} \underline{\lambda}_j\right)^2}{\sum_{j=1}^{S} \underline{\lambda}_j^2} \right) = \exp\left(-\frac{S \beta'}{2} \frac{\left(\frac{1}{S}\sum_{j=1}^{S} \underline{\lambda}_j\right)^2}{\frac{1}{S}\sum_{j=1}^{S} \underline{\lambda}_j^2}\right) \\
&= \exp\left(-\frac{S\beta'}{2C_S}\right).
\end{align*}
By independence across the $m$ trajectories:
\begin{align*}
\mathbb{E} \exp\left( -\eta \sum_{i=1}^{m} \sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2 + \frac{mS \beta'}{2C_S} \right) \leq 1.
\end{align*}
As desired, we have constructed a family of random variables indexed by
$v \in \mathbb{S}^{n-1}$, with MGF bounded by one.
Using the PAC-Bayes arguments from
\Cref{stmt:small_ball_to_min_eval}
followed by Markov's inequality,
with probability at least $1-2e^{-t}$, for all
$v \in \mathbb{S}^{n-1}$ and $\gamma \in [0, 1/2]$:
\begin{align}
\sum_{i,t} \ip{\tilde{x}_t^{(i)}}{v}^2 \geq \frac{1}{\eta}\left[ \frac{mS\beta'}{2C_S} - n\log\left(\frac{5}{4\gamma^2}\right) - t\right] -
\frac{4\gamma^2 e^t mT}{\underline{\lambda}}. \label{eq:high_prob_min_eval_ineq_1}
\end{align}
Choosing $\gamma^2 = \frac{n \underline{\lambda}}{4 \eta mT e^{t}}$, we have that:
\begin{align}
\frac{n}{\eta} \log\left(\frac{5}{4\gamma^2}\right) + \frac{4 mT e^{t}}{\underline{\lambda}} \gamma^2 = \frac{n}{\eta} \left[ 1 + \log\left( \frac{5 m T e^{t} \eta }{ n \underline{\lambda} } \right)\right]. \label{eq:high_prob_min_eval_ineq_2}
\end{align}
Note that this choice of $\gamma$ satisfies $\gamma \in [0, 1/2]$, since:
\begin{align*}
\frac{n \underline{\lambda}}{4 \eta mT e^{t}} \leq \frac{1}{4} &\Longleftarrow \frac{n}{\eta mT} \leq 1 &&\text{since } t \geq 0 \text{ and } \underline{\lambda} \leq 1.
\end{align*}
The RHS above is ensured by:
\begin{align*}
\frac{mT}{kn} \geq \alpha \frac{\sum_{j=1}^{S} \underline{\lambda}_j^2}{\sum_{j=1}^{S} \underline{\lambda}_j} = \alpha C_S \cdot \bar\mu \Longleftarrow \frac{mT}{kn} \geq C_S &&\text{since } \alpha, \bar\mu \leq 1.
\end{align*}
If we further enforce that:
\begin{align}
\frac{mS\beta'}{4C_S} \geq (n+1)\left[t + \log\left(\frac{5mT\eta}{n\underline{\lambda}}\right)\right], \label{eq:high_prob_min_eval_data_req}
\end{align}
then combining \eqref{eq:high_prob_min_eval_ineq_1}
with \eqref{eq:high_prob_min_eval_ineq_2}:
\begin{align*}
\sum_{i,t} \ip{\tilde{x}_t^{(i)}}{v}^2 &\geq \frac{1}{\eta}\left[ \frac{mS\beta'}{2C_S} - t - n - n\log\left(\frac{5mT e^t \eta}{n\underline{\lambda}}\right) \right] \\
&= \frac{1}{\eta}\left[ \frac{mS\beta'}{2C_S} - (n+1)t - n - n \log\left( \frac{5mT\eta}{n\underline{\lambda}} \right) \right] \\
&\geq \frac{1}{\eta}\left[ \frac{mS\beta'}{2C_S} - (n+1)t - (n+1) \log\left(\frac{5mT\eta}{n\underline{\lambda}}\right) \right] \\
&\geq \frac{mS\beta'}{4\eta C_S} = \frac{\alpha \beta' mT \bar\mu}{8}.
\end{align*}
For \eqref{eq:high_prob_min_eval_data_req}, it suffices that:
\begin{align*}
\frac{mT}{kn} \geq \frac{32 C_S t}{\beta'}, \quad \frac{mT}{kn} \geq \frac{16 C_S}{\beta'} \log\left( \frac{5}{\underline{\lambda} \alpha} \frac{\sum_{j=1}^{S} \underline{\lambda}_j}{\sum_{j=1}^{S} \underline{\lambda}_j^2} \frac{mT}{kn} \right) = \frac{16 C_S}{\beta'} \log\left(\frac{5}{\underline{\lambda} \alpha \bar\mu C_S}\right) + \frac{16 C_S}{\beta'} \log\left(\frac{mT}{kn}\right).
\end{align*}
For the RHS inequality, by \Cref{prop:invert_log_t_over_t}
it suffices that:
\begin{align*}
\frac{mT}{kn} \geq \frac{32 C_S}{\beta'} \max\left\{\log\left( \frac{5}{\underline{\lambda} \alpha \bar\mu C_S} \right), 0\right\}, \quad \frac{mT}{kn} \geq \frac{64 C_S}{\beta'} \log\left( \frac{128 C_S}{\beta'} \right).
\end{align*}
Note that $x \log(1/x) \leq 1/e$ for all $x > 0$, and therefore:
\begin{align*}
C_S\log\left( \frac{5}{\underline{\lambda} \alpha \bar\mu C_S} \right) &= C_S\log\left(\frac{5}{\alpha \underline{\lambda} \bar\mu}\right) + C_S\log\left(\frac{1}{C_S} \right) \leq C_S\log\left(\frac{5}{\alpha \underline{\lambda} \bar\mu}\right) + 1
\leq 2 C_S \log\left(\frac{5}{\alpha \underline{\lambda} \bar\mu}\right).
\end{align*}
Hence it suffices that:
\begin{align*}
\frac{mT}{kn} \geq \frac{64C_S}{\beta'} \max\left\{\log\left(\frac{5}{\alpha \underline{\lambda} \bar\mu} \right), \log\left( \frac{128 C_S}{\beta'} \right)\right\}.
\end{align*}
The claim now follows by simplifying all the required inequalities
for the quantity $mT/kn$.
\end{proof}
To contrast the effects of the wTrajSB{} assumption from those
of the TrajSB{} assumption,
let us compare \Cref{stmt:weak_small_ball_to_min_eval}
to its counterpart \Cref{stmt:small_ball_to_min_eval}.
The minimum eigenvalue
bound \eqref{eq:high_prob_min_eval_bound} from \Cref{stmt:weak_small_ball_to_min_eval}
differs from the corresponding TrajSB{}
bound \eqref{eq:traj_sb_min_eval_bound} in the role of the
eigenvalues of the matrices $\{ \Psi_j\}_{j=1}^{\floor{T/k}}$
from the small-ball definition.
However, due to the differing requirements on the amount of data
$mT$, neither result is necessarily sharper than the other, as
detailed in the following remark:
\begin{myremark}
\normalfont
When the matrices $\{ \Psi_j\}_{j=1}^{\floor{T/k}}$
from the trajectory small-ball definition
vary across $j$,
both \Cref{stmt:small_ball_to_min_eval}
and \Cref{stmt:weak_small_ball_to_min_eval}
yield different dependencies
on the eigenvalues $\{\underline{\lambda}(\Psi_j, \underline{\Gamma})\}_{j=1}^{\floor{T/k}}$.
In particular,
\Cref{stmt:small_ball_to_min_eval}
yields a minimum eigenvalue bound scaling as
$mT \underline{\mu}$, where $\underline{\mu}$ is the
\emph{geometric mean} of the eigenvalues
$\{\underline{\lambda}(\Psi_j, \underline{\Gamma})\}$, whereas
\Cref{stmt:weak_small_ball_to_min_eval}
yields a bound scaling as $mT \bar\mu$,
where $\bar\mu$ is the \emph{arithmetic
mean} of the eigenvalues.
By the AM-GM inequality, we have that
$\bar\mu \geq \underline{\mu}$, so the latter bound
is stronger than the former.
However, \Cref{stmt:weak_small_ball_to_min_eval}
has a stronger requirement on
the amount of data, requiring that
$mT \gtrsim kn C_S$, where $C_S \in [1, S]$
is defined in \eqref{eq:weak_small_ball_growth_conditions}, whereas \Cref{stmt:small_ball_to_min_eval}
has the weaker requirement that $mT \gtrsim kn$.
In the worst case when $C_S \asymp S$,
then the $mT \gtrsim kn C_S$ requirement
simplifies to the many trajectories
assumption $m \gtrsim n$.
Thus, the qualitative behavior of these two bounds
are not necessarily comparable.
\end{myremark}
Meanwhile, although neither bound is strictly sharper than
the other, if we assume polynomial growth of the $\{\Psi_j\}$
matrices, then the two bounds are roughly on par:
\begin{myremark}
\normalfont
When the matrices $\{\Psi_j\}$ exhibit low degree polynomial
growth, both \Cref{stmt:small_ball_to_min_eval}
and \Cref{stmt:weak_small_ball_to_min_eval}
yield similar qualitative behavior.
Concretely, let us suppose that
$k=1$, $\Psi_j = j^p \cdot I$ for $j \in [T]$,
and $\underline{\Gamma} = \frac{1}{T} \sum_{j=1}^{T} \Psi_j$.
Then, $\bar\mu = 1$, whereas
$\underline{\mu} \geq \frac{1+p}{e^p}$.
Thus, if we consider $p$ as constant, then $\bar\mu \asymp \underline{\mu}$.
\end{myremark}
We now state our general OLS upper bound under
the weak trajectory small-ball condition.
\begin{mylemma}[General OLS upper bound, high probability]
\label{stmt:upper_bound_main_high_prob}
There are universal positive constants $c_0$ and $c_1$ such that the following holds.
Suppose that $\mathsf{P}_x$ satisfies
the $(T,k,\{\Psi_j\}_{j=1}^{\floor{T/k}},\alpha,\beta)$-wTrajSB{} condition
(\Cref{def:weak_trajectory_small_ball}).
Put $S := \floor{T/k}$ and $\Gamma_t := \Gamma_t(\mathsf{P}_x)$
for $t \in \ensuremath{\mathbb{N}}_{+}$.
Fix any $\underline{\Gamma} \in \mathsf{Sym}^{n}_{> 0}$ satisfying
$\frac{1}{S} \sum_{j=1}^{S} \Psi_j \preccurlyeq \underline{\Gamma} \preccurlyeq \Gamma_T$,
and the constants:
\begin{align*}
C_S := \frac{\frac{1}{S}\sum_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma})^2}{\left(\frac{1}{S}\sum_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma})\right)^2}, \quad \bar\mu := \frac{1}{S} \sum_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma}).
\end{align*}
Fix $\delta \in (0, 1/e)$.
Suppose that:
\begin{align*}
n \geq 2, \quad \frac{mT}{kn} \geq \frac{c_0 C_S}{1-\beta}\log\left( \frac{C_S}{\alpha (1-\beta) \underline{\lambda}(\underline{\Gamma}, \Gamma_T) \bar\mu \delta}\right).
\end{align*}
Then, for any $\Gamma' \in \mathsf{Sym}^n_{> 0}$,
with probability at least $1-\delta$:
\begin{align*}
\norm{\hat{W}_{m,T} - W_\star}^2_{\Gamma'} \leq c_1 \sigma_\xi^2 \left[ \frac{pn \log\left(\frac{1}{\alpha(1-\beta) \underline{\lambda}(\underline{\Gamma}, \Gamma_T) \bar\mu \delta}\right)}{ \underline{\lambda}(\underline{\Gamma}, \Gamma') \alpha(1-\beta) mT \bar\mu }\right].
\end{align*}
\end{mylemma}
\begin{proof}
Put $\beta' := 1-\beta$ and $\tilde{X}_{m,T} := X_{m,T} \underline{\Gamma}^{-1/2}$.
By the arguments in the proof of \Cref{stmt:upper_bound_general},
\begin{align*}
\norm{ \hat{W}_{m,T} - W_\star }_{\Gamma'}^2
&\leq \min\{ n, p \} \frac{ \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 }{\underline{\lambda}(\underline{\Gamma}, \Gamma') \cdot \lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} )}.
\end{align*}
Put $M := (\alpha \beta' mT \bar\mu/8) \cdot I := \zeta \cdot I$.
By \Cref{prop:yasin_vector_easier},
with probability at least $1-\delta/2$:
\begin{align*}
&~~~~\mathbf{1}\{\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \succcurlyeq M\} \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 \\
&\leq 16 \sigma_\xi^2 \left[ p \log{5} + \frac{1}{2}\log\det\left(I_n + \zeta^{-1} \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}\right) + \log(2/\delta) \right] \\
&\leq 32 \sigma_\xi^2 \left[ p + \log\det\left(I_n + \zeta^{-1} \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}\right) + \log(2/\delta) \right] \\
&\leq 32 \sigma_\xi^2 \left[ p + n \log(1 + \zeta^{-1} \Tr(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})/n) + \log(2/\delta) \right].
\end{align*}
Now, by \Cref{stmt:weak_small_ball_to_min_eval}, with
probability at least $1-\delta/2$, we also have:
\begin{align*}
\lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) \geq \zeta, \quad \Tr(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) \leq \frac{4mTn}{\underline{\lambda} \delta}.
\end{align*}
On both events:
\begin{align*}
\opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 &\leq 32\sigma_\xi^2 \left[ p + n \log\left(1 + \frac{32}{\alpha\beta' \underline{\lambda} \bar\mu \delta}\right) + \log(2/\delta) \right] \\
&\leq 64 \sigma_\xi^2 \left[ p + n \log\left(\frac{33}{\alpha\beta'\underline{\lambda}\bar\mu\delta}\right)\right].
\end{align*}
Combining the inequalities:
\begin{align*}
\norm{\hat{W}_{m,T} - W_\star}^2_{\Gamma'} \leq 512 \sigma_\xi^2 \min\{n,p\} \left[\frac{ p + n \log\left(\frac{33}{\alpha\beta'\underline{\lambda}\bar\mu\delta}\right) }{ \underline{\lambda}(\underline{\Gamma}, \Gamma') \alpha\beta' mT \bar\mu }\right] \leq 1024 \sigma_\xi^2 \left[\frac{pn \log\left(\frac{33}{\alpha\beta' \underline{\lambda} \bar\mu \delta}\right)}{ \underline{\lambda}(\underline{\Gamma}, \Gamma') \alpha\beta' mT \bar\mu }\right].
\end{align*}
By a union bound, both events hold with probability
at least $1-\delta$, which concludes the proof.
\end{proof}
\subsubsection{Mixing implies weak trajectory small-ball}
One advantage of \Cref{def:weak_trajectory_small_ball} is that
it is implied by the standard notions of $\phi$-mixing in the literature (see e.g.~\cite{mohri2008rademachermixing,duchi2012ergodicmd,kuznetsov2017mixing}).
In this section, we prove this reduction.
First, we state the definition of $\phi$-mixing.
\begin{mydef}[$\phi$-mixing covariate sequence]
\label{def:phi_mixing}
Let $\{x_t\}_{t \geq 1}$ be a covariate sequence which is adapted
to a filtration $\{\mathcal{F}_t\}_{t \geq 1}$.
Define the function $\phi(k)$ as:
\begin{align}
\phi(k) := \sup_{t \in \ensuremath{\mathbb{N}}_+} \sup_{B \in \mathcal{F}_t} \tvnorm{ \Pr_{x_{t+k}}(\cdot \mid B) - \Pr_{x_{t+k}} }. \label{eq:phi_mixing}
\end{align}
The process $\{x_t\}_{t \geq 1}$ is called
\emph{$\phi$-mixing} if $\lim_{k \to \infty} \phi(k) = 0$.
We also let $\bar{\phi}(k)$ denote the \emph{upper envelope} of $\phi(k)$, i.e., $\bar{\phi}(k) := \sup_{k' \geq k} \phi(k)$.
\end{mydef}
The following result shows that a $\phi$-mixing covariate sequence where
each marginal distribution is weakly small-ball satisfies the
weak trajectory small-ball condition.
\begin{myprop}
\label{stmt:phi_mixing_implies_weak_small_ball}
Fix $\alpha \in (0, 1)$ and $\beta \in (0, 1/4)$.
Suppose that
the covariate sequence $\{x_t\}_{t \geq 1}$ is $\phi$-mixing,
and that
for every $t \in \ensuremath{\mathbb{N}}_+$ and $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$ we have:
\begin{align}
\Pr_{x_t}\{ \ip{v}{x_t}^2 \leq \alpha v^\mathsf{T} \Sigma_t v \} \leq \beta, \quad \Sigma_t := \mathbb{E}[x_tx_t^\mathsf{T}]. \label{eq:weak_small_ball_marginal}
\end{align}
Let $k_{\mathrm{mix}} := \inf\{ k \in \ensuremath{\mathbb{N}}_+ \mid \bar{\phi}(k) \leq \beta \}$
and assume that $T \geq 2k_{\mathrm{mix}}$.
Put $S := \floor{T/(2k_{\mathrm{mix}})}$ and suppose that $\{\Psi_j\}_{j=1}^{S}$ satisfies:
\begin{align*}
\Psi_j \preccurlyeq \frac{1}{4} \Sigma_t \quad\forall j \in [S], \, t \in [k_{\mathrm{mix}}(2j - 1) + 1, 2j k_{\mathrm{mix}}].
\end{align*}
Then, $\mathsf{P}_x$ satisfies the
$\left(T, 2k_{\mathrm{mix}}, \{\Psi_j\}_{j=1}^{T/(2k_{\mathrm{mix}})}, \alpha, \frac{4}{3}\left(\frac{1}{2}+\beta\right)\right)$-wTrajSB\ condition (cf.~\Cref{def:weak_trajectory_small_ball}).
\end{myprop}
\begin{proof}
Fix $j \in [S]$.
Since $\Psi_j \preccurlyeq \frac{1}{4}\Sigma_t$
for all $t \in [k_{\mathrm{mix}}(2j - 1) + 1, 2j k_{\mathrm{mix}}]$, we have:
\begin{align*}
\Psi_j \preccurlyeq \frac{1}{4k_{\mathrm{mix}}}\sum_{t=k_{\mathrm{mix}}(2j-1)+1}^{2jk_{\mathrm{mix}}} \Sigma_t.
\end{align*}
Hence,
\begin{align*}
\frac{1}{S} \sum_{j=1}^{S} \Psi_j \preccurlyeq \frac{1}{4S k_{\mathrm{mix}}} \sum_{j=1}^{S}\sum_{t=k_{\mathrm{mix}}(2j-1)+1}^{2jk_{\mathrm{mix}}} \Sigma_t \preccurlyeq \frac{1}{4S k_{\mathrm{mix}}} \sum_{t=1}^{T} \Sigma_t \preccurlyeq \Gamma_T.
\end{align*}
Above, the last inequality holds since $\floor{T/(2k_{\mathrm{mix}})} \geq T/(4k_{\mathrm{mix}})$.
By definition of $\phi$-mixing (cf.~\Cref{def:phi_mixing})
and the upper envelope $\bar\phi$,
for any $j \in [S]$ and $t \geq k_{\mathrm{mix}}(2j-1) + 1$:
\begin{align*}
\Pr_{x_t}\left\{ \ip{v}{x_t}^2 \leq 4\alpha \cdot v^\mathsf{T} \Psi_j v ~\Big|~ \mathcal{F}_{(j-1)2k_{\mathrm{mix}}} \right\} &\leq \Pr_{x_t}\left\{ \ip{v}{x_t}^2 \leq 4\alpha \cdot v^\mathsf{T} \Psi_j v \right\} + \beta \\
&\leq \Pr_{x_t}\left\{ \ip{v}{x_t}^2 \leq \alpha \cdot v^\mathsf{T} \Sigma_t v \right\} + \beta \\
&\leq 2\beta.
\end{align*}
Therefore:
\begin{align*}
&\frac{1}{2k_{\mathrm{mix}}} \sum_{t=(j-1)2k_{\mathrm{mix}}+1}^{2jk_{\mathrm{mix}}} \Pr_{x_t}\left\{ \ip{v}{x_t}^2 \leq 4\alpha \cdot v^\mathsf{T} \Psi_j v ~\Big|~ \mathcal{F}_{(j-1)2k_{\mathrm{mix}}} \right\} \\
&\leq \frac{1}{2k_{\mathrm{mix}}} [ k_{\mathrm{mix}} + 2\beta k_{\mathrm{mix}} ] = \frac{1}{2} + \beta.
\end{align*}
The claim now follows from \Cref{stmt:avg_small_ball_implies_block}.
\end{proof}
We conclude by noting that $\phi$-mixing is a stronger notion of mixing
than $\beta$-mixing, where \eqref{eq:phi_mixing} is only required to hold in
expectation. We leave to future work an analysis that only relies on
the weaker $\beta$-mixing.
\section{Introduction}
\label{sec:intro}
Statistical learning theory aims to characterize the worst-case efficiency
of learning from example data. Its most common setup assumes
that examples are independently and identically distributed (\emph{iid{}})
draws from an underlying data distribution, but
various branches of theory---not to mention deployed applications
of machine learning---consume non-independent data as well.
An especially fruitful setting, and the focus of this paper, is
in learning from sequential data,
where examples are generated by some ordered stochastic process
that renders them possibly correlated.
Naturally, sequential processes describe application domains
spanning engineering and the sciences, such as
robotics~\citep{nguyentuong2011modellearning},
data center cooling~(e.g.\ \cite{lazic2018cooling}),
language (e.g.\ \cite{sutskever2014seq2seq,belanger2015ldstext}),
neuroscience (e.g.\ \cite{linderman2017basketball,glaser2020recurrent}),
and economic forecasting~\citep{mcdonald2017timeseries}.
Learning over sequential data can also capture some
formulations of imitation learning~\citep{osa2018imitation} and reinforcement learning~\citep{chen2021decisiontransformer,janner2021rlseq}.
In supervised learning, one learns to predict output \emph{labels} from input \emph{covariates},
given example pairings of the two.
Formal treatments of learning from sequential data
typically concern a \emph{single} inter-dependent chain of covariates.
Where these treatments vary is in their assumptions
about the underlying process that generates the covariate chain.
For instance, some assume that the process is auto-regressive (e.g.\
\cite{lai1983autoregressive,goldenshluger2001autoregressive,gonzalez2020autoregressive})
or ergodic (e.g.\ \cite{yu1994mixing,duchi2012ergodicmd}).
Others assume that it is a linear dynamical system
(e.g.\ \cite{simchowitz18learning,faradonbeh2018unstable,sarkar2019sysid}).
In this paper, we examine what happens when we learn from \emph{many}
independent chains rather than from one, as one does anyway in many applications
(e.g.\ \cite{pomerleau1989alvinn,khansari2011lfd,brants2007language,jozefowicz2016exploring}).
\Cref{fig:dependence-schematic} depicts the data dependence structure of our setup in
comparison with its two natural counterparts.
Learning from a dataset of many short (constant length) chains
ought to be similar to independent learning, even if each chain is highly intra-dependent.
On the other hand, for any non-trivial chain length,
intuition suggests that the error can degrade relative to the total sample size in the worst case,
since a greater proportion of the data may contain correlations.
Lower bounds even show that, when one sees only a single chain,
this degradation is outright necessary in the worst case \citep{bresler2020leastsquaresmarkov}.
Do we see any such effect with many chains?
We study this question by sharply characterizing worst-case error
rates of a fundamental task---linear regression---imposed over
a general sequential data model.
Our findings reveal a remarkable phenomenon: %
after seeing sufficiently many chains ($m$) relative to the example dimension $n$,
no matter the chain length $T$,
\emph{the error rate matches that of learning from the same total number $mT$ of independent examples},
drawn from their respective marginal distributions.
In our data model, each chain, called a \emph{trajectory},
comprises a sequence of covariates $\{x_t\}$ generated from a stochastic process.
Each covariate is accompanied by a noisy linear response $y_t$ as its label.
A training set $\{(x_t^{(i)}, y_t^{(i)})\}_{i=1,t=1}^{m,T}$ comprises $m$ independent chains,
each of length $T$.
From such a training set, an estimator produces a hypothesis that predicts the label of any covariate.
The resulting hypothesis is evaluated according to its mean-squared prediction error
over a fresh chain of length ${T'}$, possibly unequal to $T$---a notion of
risk defined naturally over a trajectory.
All of our risk upper bounds are guarantees for the ordinary least-squares estimator in particular.
A concrete, recurring example in this paper takes the covariate-generating process
to be a linear dynamical system (LDS).
Specifically, fixing matrices $A \in \ensuremath{\mathbb{R}}^{n \times n}$,
$B \in \ensuremath{\mathbb{R}}^{n \times d}$, and $W_\star \in \ensuremath{\mathbb{R}}^{p \times n}$,
a single trajectory $\{(x_t, y_t)\}_{t \ge 1}$ is generated as follows.
Let $x_0 = 0$, and for $t \ge 1$:
\begin{align*}
x_t &= A x_{t-1} + Bw_t, &&\text{\textcolor{gray}{(linear dynamics)}}\\
y_t &= W_\star x_t + \xi_t, &&\text{\textcolor{gray}{(linear regression)}}
\end{align*}
where the $\{w_t\}_{t \ge 1}$ are iid{} centered isotropic Gaussian draws
and $\{\xi_t\}_{t \ge 1}$ is a sub-Gaussian martingale difference sequence
(with respect to past covariates $\{x_k\}_{k=1}^t$ and noise variables $\{\xi_k\}_{k=1}^{t-1}$).
Incidentally, combining linear dynamical systems with linear regression
captures the basic problem of linear system identification (as in \cite{simchowitz18learning})
as a special case.
In other instantiations of learning from trajectories, the covariates $\{x_t\}$
may be generated by a different process;
what remains common is the superimposed regression task
set up by the ground truth $W_\star$ and the noise $\{\xi_t\}$.
The key condition that we will introduce, which renders a
covariate process amenable to regression,
is that it satisfies a \emph{trajectory small-ball} criterion (\Cref{def:trajectory_small_ball}).
\Cref{sec:results:upper:small_ball_examples} shows that LDS-generated data conforms to
the trajectory small-ball condition in particular, as do many other distributions.
\input{intro-figure}
Our main results (\Cref{sec:results:upper,sec:results:lower_bounds})
sharply characterize worst-case rates
of learning from trajectory data as a function of
the training trajectory count $m$,
the training trajectory length $T$,
the evaluation length ${T'}$,
the covariate and response dimensions $n$ and $p$,
and scale parameters of noise in the data model
(such as the variance of the noise $\{\xi_t\}$).
Restricting only to terms of covariate dimension $n$,
training set size $m$ and $T$,
and evaluation length ${T'}$, our bounds imply the following summary statement:
\begin{thm}[\informal{error rate with many small-ball trajectories, ${T'} \le T$}]
\label{thm:simple-rate-many-traj-in-window}
If $m \gtrsim n$, ${T'} \le T$, %
and covariate trajectories are drawn from a \emph{trajectory small-ball} distribution,
then the worst-case excess prediction risk (over evaluation horizon ${T'}$)
for linear regression from $m$ many
trajectories of $n$-dimensional covariates, each of length $T$, is $\Theta(n / (m T))$.
\end{thm}
In drawing comparisons to learning from independent examples,
it makes sense to consider training and evaluations lengths $T$ and ${T'}$
equal (cf.~\Cref{sec:problem}),
rendering \Cref{thm:simple-rate-many-traj-in-window} applicable.
The theorem thus echoes our main point above:
the same rate of $\Theta(n/(mT))$ describes regression on $mT$ independent examples
(details on this point are expanded in \Cref{sec:prob:separations}).
Further structural assumptions are needed (cf.~\Cref{sec:prob:separations}) in order
to cover the remaining range of problem dimensions,
namely few trajectories ($m \lesssim n$) or extended evaluations (${T'} > T$),
and to that end we return to linear dynamical systems as a focus.
Our remaining risk upper bounds, targeting learning under linear dynamics,
require that
the dynamics matrix $A$ be \emph{marginally unstable}
(meaning that its spectral radius $\rho(A)$ is at most one) and diagonalizable.
When trajectories are longer at test time than during training (i.e., ${T'} > T$),
marginal instability is practically necessary, otherwise the risk
can scale exponentially in ${T'} - T$.
The assumption otherwise still allows for unstable---and therefore non-ergodic---systems at $\rho(A) = 1$.
For simplicity, we also require that the control matrix $B$ have full row rank.
Our bounds then imply the following summary statement
about regression when the number of trajectories is limited:
\begin{thm}[\informal{error rate with few LDS trajectories}]
\label{thm:simple-rate-few-traj}
If $m \lesssim n$, $mT \gtrsim n$,
and covariate trajectories are drawn from a linear dynamical system whose
dynamics $A$ are marginally unstable and diagonalizable,
then the worst-case excess prediction risk (over evaluation horizon ${T'}$)
for linear regression from $m$ many
trajectories of $n$-dimensional covariates, each of length $T$,
is $\tilde\Theta(n / (m T) \cdot \max\{n{T'}/(mT), 1\})$.
\end{thm}
If the evaluation horizon ${T'}$ is a constant, the rate in~\Cref{thm:simple-rate-few-traj}
recovers that of~\Cref{thm:simple-rate-many-traj-in-window}, up to log factors and extra assumptions.
To draw further comparison, suppose that the training and evaluation horizons are equal,
i.e.,\ that ${T'} = T$.
On the face of it, the rate in~\Cref{thm:simple-rate-few-traj}
is evidently weaker than that of~\Cref{thm:simple-rate-many-traj-in-window},
by up to a factor of the covariate dimension $n$.
But the varying premises---of many vs.\ few trajectories---necessarily constrain
the risk definitions to differ. %
Under a fixed data budget $N := mT = m{T'}$, fewer trajectories $m$ imply a
longer horizon ${T'}$ over which the risk is evaluated.
Intuitively, a longer evaluation horizon makes for a different problem,
and renders the rate comparison invalid.
A more sound comparison
across regimes is possible by first normalizing the notion of performance within
a problem instance.
To this end,
we can consider the worst-case risk of learning from trajectories \emph{relative} to that of learning
from independent examples \emph{in the same regime}.
Constructing the latter baseline is somewhat subtle (cf.~\Cref{sec:prob:general-problems}).
To decorrelate the problem of learning from trajectories while maintaining
its temporal structure otherwise, we can imagine drawing from its marginal distributions
independently at each time step.
The resulting dataset is independent, but not identically distributed.
Although the rates for the sequential and decorrelated regression problems are---as
already highlighted---remarkably the same under many trajectories, %
the few-trajectory rate in~\Cref{thm:simple-rate-few-traj} is indeed weaker than
the $\Theta(n/(mT))$
rate that we prove for its decorrelated baseline (cf.~\Cref{stmt:upper_bound_ind_lds_ls}).
Since the more general~\Cref{thm:simple-rate-many-traj-in-window}
already describes what happens under many trajectories ($m \gtrsim n$)
and a strict evaluation horizon (${T'} \le T$),
what remains is a somewhat niche regime: many trajectories and an extended evaluation horizon
${T'} > T$.
For completeness, our bounds supply the following summary statement:
\begin{thm}[\informal{error rate with many LDS trajectories}]
\label{thm:simple-rate-many-traj}
If $m \gtrsim n$
and covariate trajectories are drawn from a linear dynamical system whose
dynamics $A$ are marginally unstable and diagonalizable,
then the worst-case excess prediction risk (over evaluation horizon ${T'}$)
for linear regression from $m$ many
trajectories of $n$-dimensional covariates, each of length $T$,
is $\Theta(n / (m T) \cdot \max\{{T'}/T, 1\})$.
\end{thm}
Using the tools of our analysis,
we also develop upper bounds for parameter error instead of prediction risk,
which inform recovery of the ground truth $W_\star$
and (by reduction) of the dynamics matrix $A$ in LDS.
The latter captures the linear system identification problem.
Our upper bounds improve on its worst-case guarantees by a factor of $1/T$ where applicable,
and extend the parameter ranges in which guarantees hold at all.
\subsection{Lower bounds}
\label{sec:lower_bound_proof_sketch}
\subsubsection{Observation noise behind \Cref{thm:trace_inv_lower_bounds_minimax_risk}}
Our definition of minimax risk
$\mathsf{R}(m, T, {T'}; \mathcal{P}_x)$ in \eqref{eq:minimax_risk_def}
involves a supremum over the worst case
$\sigma_\xi$-sub-Gaussian MDS distribution that
models the observation noise.
The proof of \Cref{thm:trace_inv_lower_bounds_minimax_risk} bounds this supremum from below
by considering a noise model that
decouples the observation noise $\{\xi_t\}_{t \geq 1}$ from the randomness
that drives the trajectory $\{x_t\}_{t \geq 1}$:
\begin{mydef}[Gaussian observation noise]
\label{def:gaussian_observation_noise}
The \emph{Gaussian observation noise model} holds when
$\xi_t \sim N(0, \sigma_\xi^2 I_p)$, $\xi_t \perp \xi_{t'}$ if $t \neq t'$, and the process $\{\xi_t\}_{t \geq 1}$ is independent from the process $\{x_t\}_{t \geq 1}$.
\end{mydef}
Decoupling the noise processes %
orthogonalizes the two problems simultaneously present in \problemName{Seq\smallDash{}LS}{}:
learning the dynamics of covariates and learning the responses from covariates.
\Cref{def:gaussian_observation_noise} draws attention to the latter.
It will unfortunately exclude us from addressing linear system identification
specifically with our lower bound,
but it allows a sharp and simple characterization of the minimax risk in general.
The proof of \Cref{thm:trace_inv_lower_bounds_minimax_risk}
is given in \Cref{sec:appendix:lower_bounds:trace_inv_proof}.
\subsubsection{An analysis of non-isotropic gramian matrices}
\label{sec:proof_ideas:gramian}
A key technical challenge for our analysis lies in constructing a sharp lower bound
on the expected trace inverse of a gramian matrix formed by random non-isotropic Gaussian
random vectors. Specifically, for integers $q, n \in \ensuremath{\mathbb{N}}_{+}$ with $q \geq n$,
and for a fixed positive definite matrix $\Sigma \in \mathsf{Sym}^q_{> 0}$, we are interested
in a lower bound on the quantity $\mathbb{E} \Tr((W^\mathsf{T} \Sigma W)^{-1})$,
where $W \in \ensuremath{\mathbb{R}}^{q \times n}$ has iid\ $N(0, 1)$ entries.
The matrix $W^\mathsf{T} \Sigma W$ is equal in distribution to the gramian matrix $Y \in \ensuremath{\mathbb{R}}^{n \times n}$ of the vectors
$g_1, \dots, g_n \in \ensuremath{\mathbb{R}}^q$, which are drawn iid\ from $N(0, \Sigma)$, i.e.,
$Y_{ij} = \ip{g_i}{g_j}$.
The main tool we use to analyze $\mathbb{E} \Tr((W^\mathsf{T} \Sigma W)^{-1})$ is
the convex Gaussian min-max theorem (CGMT) from \cite{thrampoulidis14gmt},
which allows us to bound from below the expected trace inverse by studying
a two dimensional min-max game that is more amenable to analysis.
The key idea is to cast the expected trace inverse as a least-norm optimization problem, and apply CGMT to the value of the optimization problem.
We believe the following result to be of independent
interest.
\begin{restatable}{mylemma}{traceinvlowerbound}\label{lemma:trace_inv_lower_bound}
Let $q, n$ be positive integers with $q \geq n$ and $n \geq 2$.
Let $W \in \ensuremath{\mathbb{R}}^{q \times n}$ have iid\ $N(0, 1)$ entries, and let $\Sigma \in \ensuremath{\mathbb{R}}^{q \times q}$ be positive definite.
Let $g \sim N(0, I_q)$ and $h \sim N(0, I_{n-1})$, with $g$ and $h$ independent.
Also, let $\{e_i\}_{i=1}^{q}$ be the standard basis vectors in $\ensuremath{\mathbb{R}}^q$.
We have:
\begin{align}
\mathbb{E} \Tr((W^\mathsf{T} \Sigma W)^{-1}) \geq \frac{n}{\sum_{i=1}^{q} \mathbb{E}\min_{\beta \geq 0} \max_{\tau \geq 0}\left[ -\frac{\beta \norm{h}_2}{\tau} + \norm{\beta g - e_i}^2_{(\Sigma^{-1} + \beta \norm{h}_2 \tau I_q)^{-1}} \right]}. \label{eq:inverse_gram_matrix_min_max}
\end{align}
\end{restatable}
The proof of \Cref{lemma:trace_inv_lower_bound} appears
in \Cref{sec:appendix:lower_bounds:gramians}.
We now discuss how to analyze the two-dimensional min-max game appearing
in \Cref{lemma:trace_inv_lower_bound}.
We first start by heuristically replacing it with a \emph{stylized problem},
where the random quantities which appear in \eqref{eq:inverse_gram_matrix_min_max}
are replaced by their expected scaling:
\begin{align}
\mathsf{SP}(\Sigma, n) :=
\sum_{i=1}^{q} \min_{\beta \geq 0} \max_{\tau \geq 0}
\underbrace{\left[
-\frac{\beta \sqrt{n}}{\tau} +
\beta^2 \Tr((\Sigma^{-1} + \beta\sqrt{n} \tau I_q)^{-1}) +
(\Sigma^{-1} + \beta\sqrt{n}\tau I_q)^{-1}_{ii}
\right]}_{=:\: \ell_i(\beta,\tau)}. \label{eq:stylized_problem_general}
\end{align}
While \eqref{eq:stylized_problem_general} is not a valid upper bound on the value of the min-max game appearing in \eqref{eq:inverse_gram_matrix_min_max}, analyzing
\eqref{eq:stylized_problem_general} is simpler and gives the correct intuition;
we give a rigorous upper bound in \Cref{stmt:Z_i_helper}.
We start by observing that if $\beta = 0$, then regardless of the choice
of $\tau$, $\ell_i(0, \tau) = \Sigma_{ii}$,
and therefore $\sum_{i=1}^{q} \ell_i(0, \tau_i) = \Tr(\Sigma)$ for any $\{\tau_i\}_{i=1}^{q} \subset \ensuremath{\mathbb{R}}^q_{\geq 0}$.
On the other hand, if $\beta > 0$, then
$\ell_i(\beta, \tau)$
tends to $-\infty$ as $\tau \rightarrow 0^+$
and to $0$ as $\tau \rightarrow \infty$.
Therefore, if we can show that
there exists a $v \in (0, \Tr(\Sigma))$, such that
every
set of points
$\{(\beta_i,\tau_i)\}_{i=1}^{q} \subset \ensuremath{\mathbb{R}}_{> 0}^{2}$
satisfying:\footnote{The conditions given in \eqref{eq:first_order_conditions} are \emph{not}
in general necessary first-order optimality conditions for a nonconvex/nonconcave game~\citep[see e.g.][Proposition 21]{jin2020localoptimality}.
However,
since for every $\beta > 0$, the function $\tau \mapsto \ell_i(\beta, \tau)$ has only strictly concave stationary points (\Cref{stmt:local_strict_concave_max}), these conditions are necessary for this
particular problem.}
\begin{align}
\frac{\partial \ell_i}{\partial \beta}(\beta_i, \tau_i) = \frac{\partial \ell_i}{\partial \tau}(\beta_i, \tau_i) = 0, \quad i=1, \dots, q, \label{eq:first_order_conditions}
\end{align}
also satisfies
$v = \sum_{i=1}^{q} \ell_i(\beta_i, \tau_i)$,
then $\mathsf{SP}(\Sigma, n) = v$.
To uncover the critical points,
we define the functions $f$ and $q_i$, for $i=1, \dots, q$, as:
\begin{align*}
f(x) := -x\sqrt{n} + x^2 \Tr((\Sigma^{-1} + x \sqrt{n} I_q)^{-1}), \quad
q_i(x) := (\Sigma^{-1} + x \sqrt{n} I_q)^{-1}_{ii}.
\end{align*}
With these definitions, we can write:
\begin{align*}
\ell_i(\beta,\tau) = \frac{1}{\tau^2} f(\beta\tau) + q_i(\beta \tau).
\end{align*}
Calculating
$\frac{\partial \ell_i}{\partial \beta}(\beta,\tau) = \frac{\partial \ell_i}{\partial \tau}(\beta,\tau) = 0$ yields,
for $\tau \neq 0$:
\begin{align}
0 &= \frac{\partial \ell_i}{\partial \tau}(\beta, \tau) = \tau^{-2} f'(\beta \tau) \beta - 2 \tau^{-3} f(\beta\tau) + q_i'(\beta \tau) \beta, \label{eq:critical_point_one} \\
0 &= \frac{\partial \ell_i}{\partial \beta}(\beta, \tau) = \tau^{-2} f'(\beta\tau)\tau + q_i'(\beta\tau) \tau. \label{eq:critical_point_two}
\end{align}
The second condition \eqref{eq:critical_point_two}
implies that $q_i'(\beta\tau) = - \tau^{-2}f'(\beta\tau)$.
Plugging this condition into \eqref{eq:critical_point_one} implies that
$f(\beta\tau) = 0$, and hence $\ell_i(\beta, \tau) = q_i(\beta\tau)$ for the critical point $(\beta, \tau)$.
We now study the positive roots of the equation $f(x) = 0$, or equivalently:
\begin{align*}
x \sqrt{n} = x^2 \Tr((\Sigma^{-1} + x\sqrt{n} I_q)^{-1}).
\end{align*}
Using the variable substitution $y := x \sqrt{n}$, we have, when $y > 0$, the equivalent problem:
\begin{align*}
\psi(y; \Sigma) := y \Tr((\Sigma^{-1} + y I_q)^{-1}) = n.
\end{align*}
Observe that $\psi(0; \Sigma) = 0$ and $\lim_{y\rightarrow\infty} \psi(y; \Sigma) = q$.
Furthermore, $\psi(y; \Sigma)$ is continuous and monotonically increasing with $y$.
Therefore, as long as $q > n$, there is exactly one $\bar{y} \in (0, \infty)$ such that $\psi(\bar{y}; \Sigma) = n$,
or equivalently there is exactly one $\bar{x} \in (0, \infty)$ such that
$\psi(\bar{x} \sqrt{n}; \Sigma) = n$.
Such a quantity $\bar{x}$ supplies the curve of critical points
$\mathsf{Crit}(\bar{x}) := \{(\beta,\tau) \in \ensuremath{\mathbb{R}}_{> 0}^2 \mid \beta\tau = \bar{x} \}$.
Note that $\mathsf{Crit}(\bar{x})$ is the set of critical points for
every $\ell_i(\beta, \tau)$, $i=1, \dots, q$.
Furthermore,
for any $(\beta_\star, \tau_\star) \in \mathsf{Crit}(\bar{x})$
and $i \in \{1, \dots, q\}$, we have that
$\ell_i(\beta_\star, \tau_\star) = q_i(\beta_\star \tau_\star) = (\Sigma^{-1} + \bar{x}\sqrt{n} I_q)^{-1}_{ii}$.
Therefore:
\begin{align*}
\{(\beta_i, \tau_i)\}_{i=1}^{T} \subset \mathsf{Crit}(\bar{x}) \Longrightarrow \sum_{i=1}^{q} \ell_i(\beta_i, \tau_i) = \Tr((\Sigma^{-1} + \bar{x}\sqrt{n} I_q)^{-1}) \in (0, \Tr(\Sigma)),
\end{align*}
and thus:
\begin{align}
\mathsf{SP}(\Sigma, n) = \frac{\sqrt{n}}{\bar{x}}, \:\: \textrm{with $\bar{x}$ the solution to} \:\: \psi(\bar{x}\sqrt{n}; \Sigma) = n. \label{eq:SP_solution}
\end{align}
In light of \eqref{eq:SP_solution},
\Cref{lemma:trace_inv_lower_bound} then suggests that:
\begin{align}
\mathbb{E} \Tr((W^\mathsf{T} \Sigma W)^{-1}) \gtrapprox \frac{n}{\mathsf{SP}(\Sigma, n)} = \bar{x} \sqrt{n}, \label{eq:heuristic_approx_v1}
\end{align}
where the $\gtrapprox$ notation indicates the heuristic nature
of replacing the expected min-max game appearing in the bound
\eqref{eq:inverse_gram_matrix_min_max} with the approximation
\eqref{eq:stylized_problem_general}.
If we briefly check \eqref{eq:heuristic_approx_v1} in the simple case when $\Sigma = I_q$,
we see that:
\begin{align*}
n = \psi(\bar{x}\sqrt{n};I_q) = \bar{x}\sqrt{n} \frac{q}{1 + \bar{x}\sqrt{n}} \Longrightarrow \bar{x}\sqrt{n} = \frac{n}{q} (1 + \bar{x}\sqrt{n}) \geq \frac{n}{q}.
\end{align*}
Hence, \eqref{eq:heuristic_approx_v1} yields that
$\mathbb{E} \Tr((W^\mathsf{T} W)^{-1}) \gtrapprox n/q$,
which is the correct scaling; the exact result is
$\mathbb{E} \Tr((W^\mathsf{T} W)^{-1}) = n/(q-n-1)$ for $q \geq n + 2$.
\subsubsection{Ideas behind \Cref{stmt:ind_seq_ls_lower_bound}}
We let $X_{m,T}$ denote the data matrix associated with $m$ iid\ copies of $\{x_t\}_{t=1}^{T}$, with $x_t \sim N(0, 2^t \cdot I_n)$ and
$x_{t} \perp x_{t'}$ for $t \neq t'$.
We also define
$\Gamma_T := \frac{1}{T} \sum_{t=1}^{T} 2^t \cdot I_n = \frac{2}{T} (2^T - 1) \cdot I_n$,
and observe that $\Gamma_T \succcurlyeq \frac{2^T}{T} \cdot I_n$.
By \Cref{lemma:trace_inv_lower_bound}, it suffices to lower bound the quantity
$\mathbb{E} \Tr(\Gamma_T^{1/2} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_T^{-1/2})$.
Since each column of $X_{m,T}$ is independent,
the matrix
$X_{m,T}2^{-T/2}$ has the same distribution as
$\mathsf{BDiag}(\Theta^{1/2}, m)W$,
where $\Theta \in \ensuremath{\mathbb{R}}^{T \times T}$ is diagonal,
$\Theta_{ii} = 2^{i-T}$ for $i \in \{1, \dots, T\}$, and
$W \in \ensuremath{\mathbb{R}}^{mT \times n}$ has iid\ $N(0, 1)$ entries.
In other words, we have:
\begin{align*}
\mathbb{E} \Tr(\Gamma_T^{1/2} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_T^{-1/2}) \geq \frac{1}{T} \mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta, m) W)^{-1}).
\end{align*}
By the arguments in \Cref{sec:proof_ideas:gramian}, we have:
\begin{align*}
\mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta, m) W)^{-1}) \gtrapprox \frac{n}{\mathsf{SP}(\mathsf{BDiag}(\Theta, m), n)},
\end{align*}
where the notation $\gtrapprox$ indicates the heuristic nature of the inequality as explained previously.
From \eqref{eq:SP_solution}, we want to find $\bar{x}$ such that:
\begin{align*}
n = \psi(\bar{x}\sqrt{n}; \mathsf{BDiag}(\Theta, m)) = \bar{x}\sqrt{n} \cdot m \sum_{j=0}^{T-1} \frac{1}{2^j + \bar{x}\sqrt{n}}.
\end{align*}
While solving this equation exactly for $\bar{x}\sqrt{n}$ is not tractable, we can estimate a lower bound on $\bar{x}\sqrt{n}$ quite easily.
For any integer $T_c \in \{0, \dots, T\}$, we have the following estimate:
\begin{align*}
\frac{n}{m} = \bar{x}\sqrt{n}\sum_{j=0}^{T-1} \frac{1}{2^j + \bar{x}\sqrt{n}} \leq T_c + 2 \bar{x}\sqrt{n} \cdot 2^{-T_c}.
\end{align*}
Let us first assume that $\bar{x}\sqrt{n} \in [1, 2^{T-1}]$, so that $\ceil{\log_2(\bar{x}\sqrt{n})} \in \{0, \dots, T\}$.
Setting $T_c = \ceil{\log_2(\bar{x}\sqrt{n})}$ then yields
the lower bound $\bar{x}\sqrt{n} \geq 2^{n/m-3}$.
On the other hand, if $\bar{x}\sqrt{n} > 2^{T-1}$, then since we assume $mT \geq c_1 n$,
we also have $\bar{x}\sqrt{n} > 2^{c_1n/m - 1}$.
Finally, if $\bar{x}\sqrt{n} < 1$, we have:
\begin{align*}
\frac{n}{m} = \bar{x}\sqrt{n} \sum_{j=0}^{T-1} \frac{1}{2^j + \bar{x}\sqrt{n}} < \sum_{j=0}^{T-1} \frac{1}{2^j + \bar{x}\sqrt{n}} \leq 2 \Longrightarrow m \geq n/2.
\end{align*}
This yields a contradiction, since by assumption $m \leq c_2 n$, if $c_2 < 1/2$,
so we must have $\bar{x}\sqrt{n} \geq 2^{c' n/m - 3}$ with $c' = \min\{1, c_1\}$.
Now by \eqref{eq:SP_solution} and \eqref{eq:heuristic_approx_v1}:
\begin{align*}
\mathsf{SP}(\mathsf{BDiag}(\Theta, m), n) = \frac{n}{\bar{x}\sqrt{n}} \leq n 2^{-c'n/m+3} \Longrightarrow \mathbb{E} \Tr(\Gamma_T^{1/2} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_T^{-1/2}) \gtrapprox \frac{2^{c' n/m}}{T}.
\end{align*}
We make this argument rigorous in \Cref{sec:appendix:ind_seq_ls_lower_bound}.
\subsubsection{Ideas behind \Cref{stmt:lower_bound_main}}
\label{sec:lower_bound_proof_sketch:few_trajectories}
We focus here on the hard instance when $A=I_n$ and $m \lesssim n$, since the cases when $A=0_{n\times n}$
or $A=I_n$ and $m \gtrsim n$
are straightforward applications of Jensen's inequality
and some basic manipulations (see \Cref{stmt:minimax_jensen_rate}).
The proof used by \Cref{stmt:lower_bound_main} when $A=I_n$ and $m \lesssim n$ is
actually a special case of a general proof indexed by
the largest Jordan block size of the hard instance.
For a maximum Jordan block size $r$, the hard instances are
$A = \mathsf{BDiag}(J_r, n/r)$,
where we assume for simplicity that $r$ divides $n$;
this reduces to $A=I_n$ when $r=1$.
We associate two important matrices with these hard instances.
To define them, let $\mathcal{I}_r := \{1, 1 + r, \dots, 1 + (T-1)r\}$,
and let $E_{\mathcal{I}_r} \in \ensuremath{\mathbb{R}}^{T \times Tr}$ denote
the linear operator that extracts the coordinates in $\mathcal{I}_r$.
The following matrices then play a key role in our analysis:
\begin{align}
\Psi_{r,T,{T'}} := \mathsf{BDiag}(\Gamma_{{T'}}^{-1/2}(J_r), T) \mathsf{BToep}(J_r, T), \quad \Theta_{r,T,{T'}} := E_{\mathcal{I}_r} \Psi_{r,T,{T'}} \Psi_{r,T,{T'}}^\mathsf{T} E_{\mathcal{I}_r}^\mathsf{T}. \label{eq:Theta_T_r_def}
\end{align}
The next step is to use a simple decoupling argument
(see \Cref{lemma:decoupling_blocks}) to argue that,
for $A = \mathsf{BDiag}(J_r, d)$:
\begin{align*}
\mathbb{E} \Tr(\Gamma_{{T'}}^{1/2} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}^{1/2}) \geq \mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta_{r,T,{T'}}, m) W)^{-1}),
\end{align*}
where $W \in \ensuremath{\mathbb{R}}^{mT \times d}$ has iid\ $N(0, 1)$ entries.
This positions us to use the arguments in \Cref{sec:proof_ideas:gramian} again.
We first focus on the $r=1$ case.
We reduce the problem to assuming ${T'} = T$,
by observing that since $\Gamma_t(I_n) = \frac{t+1}{2} \cdot I_n$ for any $t \in \ensuremath{\mathbb{N}}_{+}$, then
$\Theta_{1,T,{T'}} = \frac{T+1}{{T'}+1} \cdot \Theta_{1,T,T}$.
Therefore,
\begin{align}
\mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta_{1,T,{T'}}, m) W)^{-1}) &= \frac{{T'} + 1}{T+1} \cdot \mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta_{1,T,T}, m) W)^{-1}) \label{eq:Tnew_equals_T_wlog_r_equals_one_lower_bound} \\
&\gtrapprox \frac{{T'} + 1}{T+1} \cdot \frac{n}{\mathsf{SP}(\mathsf{BDiag}(\Theta_{1,T,T}, m), n)} \nonumber ,
\end{align}
where again the $\gtrapprox$ notation highlights the heuristic nature of the bound, used to build intuition.
To proceed, let $L_T \in \ensuremath{\mathbb{R}}^{T \times T}$ be the lower triangular matrix of all ones
and define $S_T := (L_TL_T^\mathsf{T})^{-1}$.
A computation yields that $\Theta_{1,T,T}^{-1} = \frac{T+1}{2} S_T$.
Note that we can write $S_T$ as a rank-one
perturbation to a tri-diagonal matrix.
Specifically, $S_T = \Tri{2}{-1}{T} - e_Te_T^\mathsf{T}$,
where $\Tri{a}{b}{T}$ denotes the symmetric $T \times T$ tri-diagonal matrix
with $a$ on the diagonal and $b$ on the lower and upper off-diagonals.
By the standard formula for the eigenvalues of a tri-diagonal matrix, we have that
$\lambda_{T-k+1}(\Tri{2}{-1}{T}) = 2\left(1 - \cos\left(\frac{k\pi}{T+1}\right)\right) \asymp k^2/T^2$.
In \Cref{sec:appendix:lower_bounds:eigenvalues},
we apply the work of \cite{kulkarni99tridiagonal}
to show that the rank-one perturbation is negligible:
$\lambda_{T-k+1}(S_T) \asymp k^2/T^2$ as well.
Therefore $\lambda_{T-k+1}(\Theta_{1,T,T}^{-1}) \asymp k^2/T$.
With this bound, we have:
\begin{align*}
n &= \psi(\bar{x}\sqrt{n}; \mathsf{BDiag}(\Theta_{1,T,T}, m)) = \bar{x}\sqrt{n} \cdot m \sum_{i=1}^{T} \frac{1}{\lambda_i(\Theta_{1,T,T}^{-1}) + \bar{x}\sqrt{n}} \\
&\lesssim \bar{x}\sqrt{n} \cdot m\sum_{i=1}^{T} \frac{1}{ i^2/T + \bar{x}\sqrt{n}}
\leq \bar{x}\sqrt{n} \cdot m\int_0^T \frac{1}{x^2/T + \bar{x}\sqrt{n}} \,\mathrm{d} x
\lesssim \sqrt{\bar{x}\sqrt{n}} \cdot m \sqrt{T}.
\end{align*}
This implies that $\bar{x}\sqrt{n} \gtrsim n^2/(m^2T)$, and therefore by \eqref{eq:SP_solution} and \eqref{eq:heuristic_approx_v1}:
\begin{align*}
\mathsf{SP}(\mathsf{BDiag}(\Theta_{1,T,T}, m), n) = \frac{n}{\bar{x}\sqrt{n}} \lesssim \frac{m^2 T}{n} \Longrightarrow \mathbb{E} \Tr(\Gamma_{{T'}}^{1/2} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}^{1/2}) \gtrapprox \frac{{T'}}{T} \cdot \frac{n^2}{m^2 T}.
\end{align*}
We make this argument rigorous in \Cref{sec:appendix:lower_bounds:r_equals_one}.
\subsubsection{Beyond diagonalizability}
\label{sec:proof_ideas:beyond_diag}
\begin{figure}[t]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/fix_m_vary_n_plot.pdf}
\subcaption{Plot of $\frac{Td}{2m} (\mathsf{SP})^{-1}$ versus $n$.}
\label{fig:fix_m_vary_n}
\end{minipage}%
\hfill
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{figures/fix_n_vary_m_plot.pdf}
\subcaption{Plot of $\frac{Td}{2m} (\mathsf{SP})^{-1}$ versus $m$.}
\label{fig:fix_n_vary_m}
\end{minipage}
\caption{Plot of $\frac{Td}{2m} (\mathsf{SP})^{-1}$ versus $n$ in (\subref{fig:fix_m_vary_n}) and versus $m$ in (\subref{fig:fix_n_vary_m}), both on a log-log scale.
For (\subref{fig:fix_m_vary_n}), $m$ and $p$ are fixed to one,
$d$ is fixed to $n/r$, and $T$ is fixed to $2n$.
For (\subref{fig:fix_n_vary_m}), $n$ is fixed to $150$, $p$ is fixed to one, and $T$ is fixed to $2n$.
In the legends, the slope of the line (in log-log space) computed via
linear regression is shown.
Based on these plots, we conjecture that $\frac{Td}{2m}(\mathsf{SP})^{-1} \gtrsim c_r (d/m)^{2r}$, where $c_r$ depends only on $r$.}
\label{fig:risk_lower_bound}
\end{figure}
When $r \geq 2$, the analytic complexity
of characterizing the solution to $n = \psi(\bar{x}\sqrt{n}; \mathsf{BDiag}(\Theta_{r,T,{T'}}, m))$
increases significantly.
Nevertheless, we can still solve for $\bar{x}\sqrt{n}$ by numerical root finding,
to look at the scaling patterns
for small values of $r$ and ${T'}=T$.
This computation (\Cref{fig:risk_lower_bound}) leads
us to conjecture
a general bound of
$\mathsf{R}(m,T,T;\{\PxA{\mathsf{BDiag}(J_r,n/r)}\}) \gtrapprox c_rn^{2r}/(m^{2r} T)$
when $m \lesssim n$,
where $c_r$ is a constant depending only on $r$.
A complete and precise statement is given in \Cref{sec:beyond-diag-results}.
\section{Analysis for lower bounds}
\label{sec:appendix:lower_bounds}
\subsection{Preliminaries}
Here, we collect the necessary auxiliary results we
will use to prove the lower bound.
The first result is an instance of the well-known fact that the conditional mean
is the estimator which minimizes the mean squared error.
\begin{myprop}
\label{prop:mean_minimizes_least_squares}
Let $T \in \ensuremath{\mathbb{N}}_{+}$ and $\{\Pxt{t}\}_{t=1}^{T}$ be a sequence of distributions over $\ensuremath{\mathbb{R}}^n$ with finite second moments $\Sigma_t := \mathbb{E}_{x_t \sim \Pxt{t}}[x_t x_t^\mathsf{T}]$.
Let $P_W$ be any arbitrary distribution on $\ensuremath{\mathbb{R}}^{p \times n}$.
Put $\Gamma_T := \frac{1}{T} \sum_{t=1}^{T} \Sigma_t$.
We have:
\begin{align*}
\inf_{\hat{W}} \mathbb{E}_{W \sim P_W}\left[ \frac{1}{T}\sum_{t=1}^{T} \mathbb{E}_{x_t \sim \Pxt{t}} \norm{\hat{W}(x_t) - Wx_t}_2^2 \right] = \mathbb{E}_{W \sim P_W} \norm{\mathbb{E}_{W' \sim P_W}[W'] - W}_{\Gamma_T}^2,
\end{align*}
where the infimum ranges over measurable functions $\hat{W} : \ensuremath{\mathbb{R}}^n \rightarrow \ensuremath{\mathbb{R}}^{p}$.
\end{myprop}
\begin{proof}
Let $\mu_T := \frac{1}{T} \sum_{t=1}^{T} \Pxt{t}$ denote the uniform mixture distribution,
so that
\begin{align*}
\frac{1}{T}\sum_{t=1}^{T} \mathbb{E}_{x_t \sim \Pxt{t}} \norm{\hat{W}(x_t) - Wx_t}_2^2 = \mathbb{E}_{\bar{x} \sim \mu_T} \norm{\hat{W}(\bar{x}) - W \bar{x}}_2^2.
\end{align*}
By repeated applications of Fubini's theorem,
\begin{align*}
\inf_{\hat{W}} \mathbb{E}_{W \sim P_W} \mathbb{E}_{\bar{x} \sim \mu_T} \norm{\hat{W}(\bar{x}) - W\bar{x}}_2^2 &= \inf_{\hat{W}} \mathbb{E}_{\bar{x} \sim \mu_T} \mathbb{E}_{W \sim P_W} \norm{\hat{W}(\bar{x}) - W\bar{x}}_2^2 \\
&= \mathbb{E}_{\bar{x} \sim \mu_T}\left[ \inf_{\hat{y} \in \ensuremath{\mathbb{R}}^p} \mathbb{E}_{W \sim P_W} \norm{\hat{y} - W\bar{x}}_2^2 \right] \\
&= \mathbb{E}_{\bar{x} \sim \mu_T} \mathbb{E}_{W \sim P_W} \norm{ \mathbb{E}_{W'\sim P_W}[W'] \bar{x} - W \bar{x} }_2^2 \\
&= \mathbb{E}_{W \sim P_W} \mathbb{E}_{\bar{x} \sim \mu_T} \norm{ \mathbb{E}_{W'\sim P_W}[W'] \bar{x} - W \bar{x} }_2^2 \\
&= \mathbb{E}_{W \sim P_W} \norm{\mathbb{E}_{W'\sim P_W}[W'] - W}_{\Gamma_T}^2.
\end{align*}
\end{proof}
The next result is a simple fact which states that
if a function is strictly increasing and
concave on an interval, then any root of the
function is lower bounded by the root of the
linear approximation at any point in the interval.
\begin{myprop}
\label{prop:linear_approx_root_lower_bound}
Let $f : I \rightarrow \ensuremath{\mathbb{R}}$ be a $C^1(I)$ function
that is strictly increasing and concave on an interval $I \subseteq \ensuremath{\mathbb{R}}$.
Suppose that $f$ has a (unique) root $x_0 \in I$.
For any $x \in I$, we have that:
\begin{align*}
x - \frac{f(x)}{f'(x)} \leq x_0.
\end{align*}
\end{myprop}
\begin{proof}
Because $f$ is concave on $I$, we have that:
\begin{align*}
0 = f(x_0) \leq f(x) + f'(x)(x_0 - x).
\end{align*}
Next, because $f$ is strictly increasing on $I$, we have that $f'(x) > 0$.
The claim now follows by re-arranging the previous inequality.
\end{proof}
The next result states that
the trace inverse of any positive definite matrix
is lower bounded by the trace inverse of any
priciple submatrix.
The claim is immediate from Cauchy's eigenvalue interlacing theorem, but we give a
more direct proof.
\begin{restatable}{myprop}{traceinvselector}\label{prop:trace_inv_selector_lower_bound}
Let $M \in \ensuremath{\mathbb{R}}^{q \times n}$ have full column rank.
Let $I \subseteq \{1, \dots, n\}$ be any index set, and let
$E_I : \ensuremath{\mathbb{R}}^n \rightarrow \ensuremath{\mathbb{R}}^{\abs{I}}$ denote any linear map which
extracts the coordinates associated to $I$.
We have:
\begin{align*}
\Tr((M^\mathsf{T} M)^{-1}) \geq \Tr((E_I M^\mathsf{T} M E_I^\mathsf{T})^{-1}).
\end{align*}
\end{restatable}
\begin{proof}
Fix a $z \in \ensuremath{\mathbb{R}}^n$.
Since $M$ has full column rank, we have
that $(M^\mathsf{T})^{\dag} = M (M^\mathsf{T} M)^{-1}$.
Therefore,
\begin{align*}
\min_{c \in \ensuremath{\mathbb{R}}^q : M^\mathsf{T} c = z} \norm{c}_2^2 = \norm{(M^\mathsf{T})^{\dag} z}_2^2 = z^\mathsf{T} (M^\mathsf{T} M)^{-1} z.
\end{align*}
Taking expectation with $z \sim N(0, I_n)$,
\begin{align*}
\Tr((M^\mathsf{T} M)^{-1}) = \mathbb{E}_{z \sim N(0, I_n)}\left[ \min_{c \in \ensuremath{\mathbb{R}}^q : M^\mathsf{T} c = z} \norm{c}_2^2 \right].
\end{align*}
On the other hand, we have that:
\begin{align*}
\min_{c \in \ensuremath{\mathbb{R}}^q : M^\mathsf{T} c = z} \norm{c}_2^2 \geq \min_{c \in \ensuremath{\mathbb{R}}^q : E_I M^\mathsf{T} c = E_I z} \norm{c}_2^2.
\end{align*}
This is clear because
for any $c \in \ensuremath{\mathbb{R}}^q$ satisfying $M^\mathsf{T} c = z$,
the equality $E_I M^\mathsf{T} c = E_I z$ trivially holds.
This means we have the following set inclusion:
\begin{align*}
\{ c \in \ensuremath{\mathbb{R}}^q \mid M^\mathsf{T} c = z \} \subseteq \{ c \in \ensuremath{\mathbb{R}}^q \mid E_I M^\mathsf{T} c = E_I z \}.
\end{align*}
Therefore, minimizing any function over the
first set will be lower bounded by minimizing
the same function over the second set.
From this inclusion, we conclude for any index set $I$:
\begin{align*}
\Tr((M^\mathsf{T} M)^{-1}) &= \mathbb{E}_{z \sim N(0, I_n)}\left[ \min_{c \in \ensuremath{\mathbb{R}}^q : M^\mathsf{T} c = z} \norm{c}_2^2 \right] \geq \mathbb{E}_{z \sim N(0, I_n)}\left[ \min_{c \in \ensuremath{\mathbb{R}}^q : E_I M^\mathsf{T} c = E_I z} \norm{c}_2^2 \right] \\
&= \mathbb{E}_{z \sim N(0, I_{\abs{I}})}\left[ \min_{c \in \ensuremath{\mathbb{R}}^q : E_I M^\mathsf{T} c = z} \norm{c}_2^2 \right] = \Tr((E_I M^\mathsf{T} M E_I^\mathsf{T} )^{-1}).
\end{align*}
\end{proof}
Next, we state well-known upper and lower tail bounds
for chi-squared random variables.
\begin{mylemma}[{\cite[Lemma~1]{laurent00adaptive}}]
\label{lemma:chi_squared_tail_bounds}
Let $g_1, \dots, g_D$ be iid\ $N(0, 1)$ random variables, and let $a_1, \dots, a_D$ be non-negative
scalars.
For any $t > 0$, we have:
\begin{align*}
\Pr\left\{ \sum_{i=1}^{D} a_i(g_i^2 - 1) \geq 2 \sqrt{t} \sqrt{\sum_{i=1}^{D} a_i^2} + 2 t \max_{i=1, \dots, D} a_i \right\} &\leq e^{-t}, \\
\Pr\left\{ \sum_{i=1}^{D} a_i(g_i^2 - 1) \leq - 2 \sqrt{t} \sqrt{\sum_{i=1}^{D} a_i^2} \right\} &\leq e^{-t}.
\end{align*}
\end{mylemma}
Finally, we conclude with a convex extension of
Gordon's min-max theorem.
\begin{mythm}[{\cite[Theorem~II.1]{thrampoulidis14gmt}}]
\label{thm:gaussian_min_max}
Let $A \in \ensuremath{\mathbb{R}}^{m \times n}$, $g \in \ensuremath{\mathbb{R}}^m$, and $h \in \ensuremath{\mathbb{R}}^n$ have iid\ $N(0, 1)$ entires and be independent of each other.
Suppose that $S_1 \subset \ensuremath{\mathbb{R}}^n$ and $S_2 \subset \ensuremath{\mathbb{R}}^m$ are non-empty compact convex sets,
and let $\psi : S_1 \times S_2 \rightarrow \ensuremath{\mathbb{R}}$ be a continuous, convex-concave function.
For every $t \in \ensuremath{\mathbb{R}}$, we have:
\begin{align*}
\Pr\left\{ \min_{x \in S_1} \max_{y \in S_2} \left[ y^\mathsf{T} A x + \psi(x, y) \right] \geq t \right\} \leq 2 \Pr\left\{ \min_{x \in S_1} \max_{y \in S_2} \left[\norm{x}_2 g^\mathsf{T} y + \norm{y}_2 h^\mathsf{T} x + \psi(x, y)\right] \geq t \right\}.
\end{align*}
\end{mythm}
\subsection{Proof of \Cref{thm:trace_inv_lower_bounds_minimax_risk}}
\label{sec:appendix:lower_bounds:trace_inv_proof}
We first prove the following intermediate result, which holds
under the Gaussian observation noise model (\Cref{def:gaussian_observation_noise}).
\begin{mylemma}
\label{stmt:minimax_risk_trace_inverse_one_dist}
Let $T \in \ensuremath{\mathbb{N}}_{+}$,
$\{\Pxt{t}\}_{t=1}^{T}$ be a sequence of distributions over $\ensuremath{\mathbb{R}}^n$ with finite second moments $\Sigma_t := \mathbb{E}_{x_t \sim \Pxt{t}}[x_t x_t^\mathsf{T}]$,
and $\sigma_{\xi} > 0$.
Let $P_X$ be a distribution on $\ensuremath{\mathbb{R}}^{q \times n}$ with $q \geq n$
such that for $X \sim P_X$,
$X^\mathsf{T} X$ is invertible almost surely.
For $W \in \ensuremath{\mathbb{R}}^{p \times n}$, let $P_W$ be the distribution over $\ensuremath{\mathbb{R}}^{q \times n} \times \ensuremath{\mathbb{R}}^{q \times p}$
with $(X, Y) \sim P_W$ satisfying $X \sim P_X$ and $Y \mid X = X W^\mathsf{T} + \Xi$,
where $\Xi \in \ensuremath{\mathbb{R}}^{q \times p}$ has iid\ $N(0, \sigma_{\xi}^2)$ entries (and is independent of everything else).
Put $\Gamma_T := \frac{1}{T} \sum_{t=1}^{T} \Sigma_t$.
We have that:
\begin{align*}
&\inf_{\hat{W}} \sup_{W \in \ensuremath{\mathbb{R}}^{p \times n}} \mathbb{E}_{(X,Y) \sim P_W} \left[ \frac{1}{T} \sum_{t=1}^{T} \mathbb{E}_{x_t \sim \Pxt{t}} \norm{ \hat{W}(X, Y, x_t) - Wx_t}_2^2 \right] \\
&\qquad\geq \sigma_{\xi}^2 p \cdot \mathbb{E}_{X \sim P_X} \Tr(\Gamma_T^{1/2} (X^\mathsf{T} X)^{-1} \Gamma_T^{1/2}),
\end{align*}
where the infimum ranges over all measurable functions
$\hat{W} : \ensuremath{\mathbb{R}}^{q \times n} \times \ensuremath{\mathbb{R}}^{q \times p} \times \ensuremath{\mathbb{R}}^{n} \rightarrow \ensuremath{\mathbb{R}}^{p}$.
\end{mylemma}
\begin{proof}
The proof extends the Bayesian argument from \cite[Theorem~1]{mourtada19exactminimax}.
Let $p_\lambda$ be any prior distribution over $\ensuremath{\mathbb{R}}^{p \times n}$.
Let $\mu_T := \frac{1}{T} \sum_{t=1}^{T} \Pxt{t}$ denote the uniform mixture.
Bounding the minimax risk from below by the Bayes risk:
\begin{align*}
&~~~\,\inf_{\hat{W}} \sup_{W \in \ensuremath{\mathbb{R}}^{p \times n}} \mathbb{E}_{(X,Y) \sim P_W} \mathbb{E}_{\bar{x} \sim \mu_T} \norm{ \hat{W}(X, Y, \bar{x}) - W\bar{x}}_2^2 \\
&\geq \inf_{\hat{W}} \mathbb{E}_{W_\lambda \sim p_\lambda} \mathbb{E}_{(X,Y) \sim P_{W_\lambda}} \mathbb{E}_{\bar{x} \sim \mu_T} \norm{ \hat{W}(X, Y, \bar{x}) - W_\lambda \bar{x}}_2^2 \\
&= \inf_{\hat{W}} \mathbb{E}_{(X,Y)} \mathbb{E}_{W_\lambda \mid (X, Y)} \mathbb{E}_{\bar{x} \sim \mu_T} \norm{ \hat{W}(X, Y, \bar{x}) - W_\lambda \bar{x}}_2^2 &&\textrm{using Fubini's theorem} \\
&= \mathbb{E}_{(X,Y)} \inf_{\hat{W}_{X,Y}} \mathbb{E}_{W_\lambda \mid (X, Y)} \mathbb{E}_{\bar{x} \sim \mu_T} \norm{ \hat{W}_{X,Y}(\bar{x}) - W_\lambda \bar{x}}_2^2 &&\textrm{where }\hat{W}_{X,Y} \text{ maps } \ensuremath{\mathbb{R}}^n \rightarrow \ensuremath{\mathbb{R}}^p \\
&= \mathbb{E}_{(X,Y)} \mathbb{E}_{W_\lambda \mid (X,Y)} \norm{ \mathbb{E}[W_\lambda \mid X,Y] - W_\lambda }_{\Gamma_T}^2 &&\textrm{using \Cref{prop:mean_minimizes_least_squares}}.
\end{align*}
Now let $W_\lambda \sim p_\lambda$ have iid\ $N(0, 1/\lambda)$ entries for $\lambda > 0$.
Noting that
\begin{align*}
\vec(Y) = (I_p \otimes X) \vec(W_\lambda^\mathsf{T}) + \vec(\Xi),
\end{align*}
we see that the vector $\cvectwo{\vec(W_\lambda^\mathsf{T})}{\vec(Y)}$ is jointly Gaussian conditioned on $X$:
\begin{align*}
\cvectwo{\vec(W_\lambda^\mathsf{T})}{\vec(Y)} \mid X \sim N\left( 0, \bmattwo{ \frac{1}{\lambda} I_{pn} }{ \frac{1}{\lambda}(I_p \otimes X^\mathsf{T}) }{*}{ \frac{1}{\lambda}(I_p \otimes XX^\mathsf{T}) + \sigma_{\xi}^2 I_{qp} } \right).
\end{align*}
Therefore, the distribution of $\vec(W_\lambda^\mathsf{T}) \mid X, Y$ is:
\begin{align*}
\vec(W_\lambda^\mathsf{T}) \mid X,Y &~\sim N(\mu_\lambda, \Sigma_\lambda), \\
\mu_\lambda &:= \frac{1}{\lambda}(I_p \otimes X^\mathsf{T})\left[ \frac{1}{\lambda}(I_p \otimes XX^\mathsf{T}) + \sigma_{\xi}^2 I_{qp} \right]^{-1} \vec(Y), \\
\Sigma_\lambda &:= \frac{1}{\lambda} I_{pn} - \frac{1}{\lambda^2} (I_p \otimes X^\mathsf{T}) \left[ \frac{1}{\lambda}(I_p \otimes XX^\mathsf{T}) + \sigma_{\xi}^2 I_{qp} \right]^{-1} (I_p \otimes X) .
\end{align*}
A generalization of the identity
$X^\mathsf{T}( \frac{1}{\lambda} XX^\mathsf{T} + \sigma_{\xi}^2 I_q)^{-1} = (\frac{1}{\lambda} X^\mathsf{T} X + \sigma_{\xi}^2 I_n)^{-1} X^\mathsf{T}$ yields:
\begin{align*}
(I_p \otimes X^\mathsf{T}) \left[ \frac{1}{\lambda}(I_p \otimes XX^\mathsf{T}) + \sigma_{\xi}^2 I_{qp} \right]^{-1} &= \left[ \frac{1}{\lambda}(I_p \otimes X^\mathsf{T} X) + \sigma_{\xi}^2 I_{np} \right]^{-1} (I_p \otimes X^\mathsf{T}).
\end{align*}
Therefore,
\begin{align*}
&~~~~\mathbb{E}[ \vec(W_\lambda^\mathsf{T}) \mid X, Y] - \vec(W_\lambda^\mathsf{T}) \\
&= \mu_\lambda - \vec(W_\lambda^\mathsf{T}) \\
&= \left[ (I_p \otimes X^\mathsf{T} X) + \sigma_{\xi}^2 \lambda I_{np} \right]^{-1} (I_p \otimes X^\mathsf{T}) \vec(Y) - \vec(W_\lambda^\mathsf{T}) \\
&= \left[ \left[ (I_p \otimes X^\mathsf{T} X) + \sigma_{\xi}^2 \lambda I_{np}\right]^{-1} (I_p \otimes X^\mathsf{T} X) - I_{np} \right] \vec(W_\lambda^\mathsf{T}) \\
&\qquad+ \left[ (I_p \otimes X^\mathsf{T} X) + \sigma_{\xi}^2 \lambda I_{np} \right]^{-1} (I_p \otimes X^\mathsf{T}) \vec(\Xi).
\end{align*}
Observing that
\begin{align*}
\norm{\mathbb{E}[W_\lambda \mid X,Y] - W_\lambda}_{\Gamma_T}^2 = \norm{\mathbb{E}[\vec(W_\lambda^\mathsf{T}) \mid X, Y] - \vec(W_\lambda^\mathsf{T})}_{I_p \otimes \Gamma_T}^2,
\end{align*}
and defining $M_X(\lambda) := (I_p \otimes X^\mathsf{T} X) + \sigma_{\xi}^2 \lambda I_{np}$,
we have the following bias-variance decomposition:
\begin{align*}
&~~~~\mathbb{E}_{X,\Xi,W_\lambda} \norm{\mathbb{E}[W_\lambda \mid X,Y] - W_\lambda}_{\Gamma_T}^2 \\
&= \mathbb{E}_{X,\Xi,W_\lambda} \bignorm{\left[ M_X^{-1}(\lambda) (I_p \otimes X^\mathsf{T} X) - I_{np} \right] \vec(W_\lambda^\mathsf{T}) }_{I_p \otimes \Gamma_T}^2 \\
&\qquad + \sigma_{\xi}^2 \mathbb{E}_{X} \Tr\left( (I_p \otimes \Gamma_T^{1/2}) M_X^{-1}(\lambda) (I_p \otimes X^\mathsf{T} X) M_X^{-1}(\lambda) (I_p \otimes \Gamma_T^{1/2}) \right) \\
&\geq \sigma_{\xi}^2 \mathbb{E}_{X} \Tr\left( (I_p \otimes \Gamma_T^{1/2}) M_X^{-1}(\lambda) (I_p \otimes X^\mathsf{T} X) M_X^{-1}(\lambda) (I_p \otimes \Gamma_T^{1/2}) \right) .
\end{align*}
Since $\lambda \mapsto \Tr\left( (I_p \otimes \Gamma_T^{1/2}) M_X^{-1}(\lambda) (I_p \otimes X^\mathsf{T} X) M_X^{-1}(\lambda) (I_p \otimes \Gamma_T^{1/2}) \right)$
is non-negative and decreasing in $\lambda$ for $\lambda > 0$,
by the monotone convergence theorem:
\begin{align*}
&~~~~\lim_{\lambda \rightarrow 0^+} \mathbb{E}_{X,\Xi,W_\lambda} \norm{\mathbb{E}[W_\lambda \mid X,Y] - W_\lambda}_{\Gamma_T}^2 \\
&\geq \sigma_\xi^2 \lim_{\lambda \rightarrow 0^+} \mathbb{E}_{X} \Tr\left( (I_p \otimes \Gamma_T^{1/2}) M_X^{-1}(\lambda) (I_p \otimes X^\mathsf{T} X) M_X^{-1}(\lambda) (I_p \otimes \Gamma_T^{1/2}) \right) \\
&=\sigma_\xi^2 \mathbb{E}_{X} \Tr\left( (I_p \otimes \Gamma_T^{1/2}) M_X^{-1}(0) (I_p \otimes X^\mathsf{T} X) M_X^{-1}(0) (I_p \otimes \Gamma_T^{1/2}) \right) \\
&= \sigma_\xi^2 \mathbb{E}_{X} \Tr( (I_p \otimes \Gamma_T^{1/2} (X^\mathsf{T} X)^{-1} \Gamma_T^{1/2} ) ) \\
&= \sigma_\xi^2 p \cdot \mathbb{E}_{X} \Tr( \Gamma_T^{1/2} (X^\mathsf{T} X)^{-1} \Gamma_T^{1/2} ).
\end{align*}
Since the first expression above lower bounds the minimax risk,
this concludes the proof.
\end{proof}
We now restate and prove \Cref{thm:trace_inv_lower_bounds_minimax_risk}.
\traceinvlowerboundsminimaxrisk*
\begin{proof}
Fix a $\mathsf{P}_x \in \mathcal{P}_x$, and let $\{\Pxt{t}\}_{t=1}^{{T'}}$ denote its marginal
distributions up to time ${T'}$.
Let $\mathsf{P}_{\xi}^{\mathsf{g}}$ denote the $\sigma_\xi$-MDS corresponding to the
Gaussian observation noise model (\Cref{def:gaussian_observation_noise}).
Note that for any hypothesis $f : \ensuremath{\mathbb{R}}^n \rightarrow \ensuremath{\mathbb{R}}^p$, we have from \eqref{eq:risk_def}:
\begin{align*}
L(\hat{f}; {T'}, \mathsf{P}_x) =
\mathbb{E}_{\mathsf{P}_x} \left[
\frac{1}{{T'}} \sum_{t=1}^{{T'}} \norm{ \hat{f}(x_t) - W_\star x_t }^2_2 \right] = \frac{1}{{T'}} \sum_{t=1}^{{T'}} \mathbb{E}_{x_t \sim \Pxt{t}} \norm{\hat{f}(x_t) - W_\star x_t}_2^2.
\end{align*}
By the definition of $\mathsf{R}(m,T,{T'};\mathcal{P}_x)$ from \eqref{eq:minimax_risk_def}
and \Cref{stmt:minimax_risk_trace_inverse_one_dist}:
\begin{align*}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) &\geq
\inf_{\mathsf{Alg}}
\sup_{W_\star}
\mathbb{E}_{\otimes_{i=1}^{m} \mathsf{P}_{x,y}^{W_\star}[\mathsf{P}_x,\mathsf{P}_{\xi}^{\mathsf{g}}]}
\left[ L \left(
\mathsf{Alg}(\{ (x_t^{(i)}, y_t^{(i)} ) \}_{i=1,t=1}^{m,T}); {T'}, \mathsf{P}_x \right) \right] \\
&\geq \sigma_\xi^2 p \cdot \mathbb{E}_{\otimes_{i=1}^{m} \mathsf{P}_x}\left[ \Tr\left(\Gamma_{{T'}}^{1/2}(\mathsf{P}_x)(X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}^{1/2}(\mathsf{P}_x) \right) \right].
\end{align*}
Since the bound above holds for any $\mathsf{P}_x \in \mathcal{P}_x$, we can take the
supremum over $\mathsf{P}_x \in \mathcal{P}_x$, from the which the claim follows.
\end{proof}
\subsection{A general risk lower bound}
We now state a lower bound which applies
with an arbitrary number of trajectories.
\begin{restatable}{mylemma}{minimaxjensenrate}\label{stmt:minimax_jensen_rate}
Suppose that $\mathcal{P}_x$ is any set containing $\PxA{0_{n \times n}}$ and
$\PxA{I_n}$. Let $mT \geq n$. Then:
\begin{align*}
\mathsf{R}(m,T,{T'};\mathcal{P}_x)
\geq \frac{\sigma_\xi^2 }{2}
\cdot \frac{pn}{mT}
\cdot \max \left\{ \frac{{T'}}{T}, 1 \right\}.
\end{align*}
\end{restatable}
\begin{proof}
Define $\zeta(A) := \mathbb{E}_{\otimes_{i=1}^{m} \PxA{A}} \left[\Tr\left(\Gamma_{{T'}}^{1/2}(A) (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}^{1/2}(A) \right)\right]$.
By \Cref{thm:trace_inv_lower_bounds_minimax_risk}:
\begin{align}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) \geq \sigma_\xi^2 p \cdot
\max\{\zeta(0_{n \times n}), \zeta(I_n)\}. \label{eq:lower_bound_risk_zero_and_eye}
\end{align}
Next, for any $M \in \mathsf{Sym}^n_{\geq 0}$,
the function $X \mapsto \Tr(M^{1/2} X^{-1} M^{1/2})$ is convex
on the domain $\mathsf{Sym}^n_{> 0}$.
To see this, we define $f(X; v) := v^\mathsf{T} X^{-1} v$ for $X \in \mathsf{Sym}^n_{> 0}$.
We can write
$f(X;v)$ as $f(X;v) = \sup\left\{ -z^\mathsf{T} X z + 2 v^\mathsf{T} z \mid z \in \ensuremath{\mathbb{R}}^n \right\}$;
therefore $X \mapsto f(X;v)$ is convex on $\mathsf{Sym}^n_{> 0}$, since it is the pointwise supremum of an affine function in $X$.
Now we see that $X \mapsto \Tr(M^{1/2} X^{-1} M^{1/2})$
is convex, since
$\Tr(M^{1/2} X^{-1} M^{1/2}) = \sum_{i=1}^{n} f(X; M^{1/2} e_i)$,
which is the sum of convex functions.
Therefore by Jensen's inequality,
whenever $X_{m,T}^\mathsf{T} X_{m,T}$ is invertible almost surely,
\begin{align*}
\zeta(A) &= \mathbb{E}_{\otimes_{i=1}^{m} \PxA{A}} \left[\Tr\left( \Gamma_{{T'}}^{1/2}(A) (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}^{1/2}(A) \right)\right] \\
&\geq \Tr\left( \Gamma_{{T'}}^{1/2}(A) (\mathbb{E}_{\otimes_{i=1}^{m} \PxA{A}}[X_{m,T}^\mathsf{T} X_{m,T}])^{-1} \Gamma_{{T'}}^{1/2}(A) \right) \\
&= \Tr\left( \Gamma_{{T'}}^{1/2}(A) (mT \cdot \Gamma_T(A))^{-1} \Gamma_{{T'}}^{1/2}(A) \right) \\
&= \frac{\Tr(\Gamma_{{T'}}(A) \Gamma_T^{-1}(A))}{mT}.
\end{align*}
We first consider the case when $A = 0_{n \times n}$.
Under these dynamics,
it is a standard fact that when $mT \geq n$, then
$X_{m,T}^\mathsf{T} X_{m,T}$ is invertible almost surely.
Furthermore, $\Gamma_t(0_{n \times n}) = I_n$ for all $t$, Hence,
$\zeta(0_{n \times n}) \geq \frac{n}{mT}$.
Next, we consider the case when $A = I_n$.
We first argue that as long as $mT \geq n$,
the matrix $X_{m,T}^\mathsf{T} X_{m,T}$ is invertible almost surely.
We write $x_t^{(i)} = \sum_{k=1}^{t} w_k^{(i)}$,
where $\{w_t^{(i)}\}_{i=1,t=1}^{m,T}$ are all iid\ $N(0, I_n)$ vectors.
Let $p : \ensuremath{\mathbb{R}}^{mTn} \rightarrow \ensuremath{\mathbb{R}}$ be the polynomial
$p(\{w_t^{(i)}\}) = \det(X_{m,T}^\mathsf{T} X_{m,T})$.
The zero-set of $p$ is either all of $\ensuremath{\mathbb{R}}^{mTn}$, or
Lebesgue measure zero.
We will select $\{w_t^{(i)}\}$ so that $p(\{w_t^{(i)}\}) \neq 0$,
which shows that the zeros of this polynomial are not all of $\ensuremath{\mathbb{R}}^{mTn}$, and hence Lebesgue measure zero.
Since the Gaussian measure on $\ensuremath{\mathbb{R}}^{mTn}$ is absolutely
continuous w.r.t.\ the Lebesgue measure on $\ensuremath{\mathbb{R}}^{mTn}$,
this implies that $\det(X_{m,T}^\mathsf{T} X_{m,T}) \neq 0$ almost surely.
To select $\{w_t^{(i)}\}$, we introduce some notation.
Let $e_i \in \ensuremath{\mathbb{R}}^n$ denote the $i$-th standard basis vector.
For any positive integer $k$,
let $U(k) \in \ensuremath{\mathbb{R}}^{k \times k}$
be the upper triangular matrix with ones for all
its non-zero entries. Let $S(k) = U(k) U(k)^\mathsf{T}$.
By construction, $S(k)$ is invertible since $U(k)$ is invertible.
We put $w_t^{(i)} = e_{(i-1)T + t} \cdot \mathbf{1}\{ (i-1)T + t \leq n \}$. We now claim that with this choice of $\{w_t^{(i)}\}$,
the matrix $X_{m,T}^\mathsf{T} X_{m,T}$ is invertible.
Suppose first that $T \geq n$.
Then we have that
$X_{m,T}^\mathsf{T} X_{m,T} = S(n)$, and therefore $\det(X_{m,T}^\mathsf{T} X_{m,T}) \neq 0$.
On the other hand, suppose that $T < n$.
Because $mT \geq n$, then we have that:
\begin{align*}
X_{m,T}^\mathsf{T} X_{m,T} = \mathsf{BDiag}(\underbrace{S(T), \dots, S(T)}_{\floor{n/T} \textrm{ times}}, S(n - T \floor{n/T})),
\end{align*}
where $\mathsf{BDiag}(M_1, \dots, M_k)$
denotes the block diagonal matrices with block diagonals
$M_1$, \dots, $M_k$.
Since $S(T)$ and $S(n-T\floor{n/T})$ are both invertible, so is
$X_{m,T}^\mathsf{T} X_{m,T}$ and therefore
$\det(X_{m,T}^\mathsf{T} X_{m,T}) \neq 0$.
Thus, $X_{m,T}^\mathsf{T} X_{m,T}$ is invertible almost surely.
Next, we note that
$\Sigma_t(I_n) = t \cdot I_n$ and
$\Gamma_t(I_n) = \left(\frac{1}{t} \sum_{k=1}^{t} k\right) \cdot I_n = \frac{t+1}{2} \cdot I_n$.
Hence we have
$ \Gamma_{{T'}}(I_n) \Gamma_T^{-1}(I_n) = \frac{{T'} + 1}{T + 1} \cdot I_n \succcurlyeq \frac{{T'}}{2T} \cdot I_n$,
and therefore $\zeta(I_n) \geq \frac{n}{2mT} \frac{{T'}}{T}$.
Combining our bounds on $\zeta(0_{n \times n})$ and
$\zeta(I_n)$, we have the desired claim:
\begin{align*}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) \geq \sigma_\xi^2 p \cdot \max\left\{ \frac{n}{mT}, \frac{n}{2mT} \frac{{T'}}{T} \right\} \geq \frac{\sigma_\xi^2}{2} \cdot \frac{pn}{mT} \cdot \max\left\{\frac{{T'}}{T}, 1\right\}.
\end{align*}
\end{proof}
\subsection{Non-isotropic random gramian matrices}
\label{sec:appendix:lower_bounds:gramians}
The goal of this subsection is to prove \Cref{lemma:trace_inv_lower_bound}, which gives a bound on the expected
trace inverse of a non-isotropic random gramian matrix.
We first prove an auxiliary lemma, which will be used
as a building block in the proof.
\begin{mylemma}
\label{lemma:min_max_upper_bound}
Fix any $x \in \ensuremath{\mathbb{R}}^q$.
Let $g \in \ensuremath{\mathbb{R}}^q$ and $h \in \ensuremath{\mathbb{R}}^n$ be random vectors with iid\ $N(0, 1)$ entries,
and let $W \in \ensuremath{\mathbb{R}}^{q \times n}$ be a random matrix with iid\ $N(0, 1)$ entries.
Let $\Sigma \in \ensuremath{\mathbb{R}}^{q \times q}$ be positive definite.
We have that:
\begin{align*}
\mathbb{E} \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \norm{ \Sigma^{1/2} W \alpha - x }_2^2 \leq \mathbb{E} \min_{\beta \geq 0} \max_{\tau \geq 0} \left[ -\frac{\beta\norm{h}_2}{\tau} + \norm{\beta g - \Sigma^{-1/2} x}_{(\Sigma^{-1} + \beta \norm{h}_2 \tau I_q)^{-1}}^2 \right] .
\end{align*}
\end{mylemma}
\begin{proof}
The proof invokes the convex Gaussian min-max lemma (\Cref{thm:gaussian_min_max})
via a limiting argument.
In what follows,
let $\{\alpha_k\}_{k \geq 1}$
and $\{v_k\}_{k \geq 1}$ be any two positive, increasing sequences of scalars tending to $+\infty$.
It is clear that for every $W$,
\begin{align*}
\lim_{k \rightarrow \infty} \min_{\norm{\alpha}_2 \leq \alpha_k} \norm{\Sigma^{1/2} W \alpha - x}_2^2 = \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \norm{\Sigma^{1/2} W \alpha - x}_2^2.
\end{align*}
Since $\alpha = 0$ is always a feasible solution to
$\min_{\norm{\alpha}_2 \leq \alpha_k} \norm{\Sigma^{1/2} W \alpha - x}_2^2$,
we have for every $k \geq 1$:
\begin{align*}
0 \leq \min_{\norm{\alpha}_2 \leq \alpha_k} \norm{\Sigma^{1/2} W \alpha - x}_2^2 \leq \norm{x}_2^2.
\end{align*}
Therefore, by the dominated convergence theorem,
\begin{align}
\mathbb{E} \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \norm{ \Sigma^{1/2} W \alpha - x }_2^2 = \mathbb{E} \lim_{k \rightarrow \infty} \min_{\norm{\alpha}_2 \leq \alpha_k} \norm{\Sigma^{1/2} W \alpha - x}_2^2 = \lim_{k \rightarrow \infty} \mathbb{E} \min_{\norm{\alpha}_2 \leq \alpha_k} \norm{ \Sigma^{1/2} W \alpha - x }_2^2. \label{eq:first_dominated_convergence}
\end{align}
We next state two variational forms which we will use:
\begin{align}
\frac{1}{2}\norm{x}_2^2 &= \max_{v \in \ensuremath{\mathbb{R}}^q} \left\{ v^\mathsf{T} x - \frac{\norm{v}_2^2}{2} \right\}, \label{eq:x_sq_variation} \\
\norm{x}_2 &= \min_{\tau \geq 0} \left\{ \frac{\norm{x}_2^2 \tau}{2} + \frac{1}{2\tau} \right\}. \label{eq:x_variation}
\end{align}
Using the first variational form \eqref{eq:x_sq_variation}, we have for every $W$ and $k_1 \geq 1$,
\begin{align*}
\min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \frac{1}{2}\norm{\Sigma^{1/2} W \alpha - x}_2^2 &= \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[ v^\mathsf{T} (\Sigma^{1/2} W \alpha - x) - \frac{\norm{v}_2^2}{2} \right] \\
&= \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&= \min_{\norm{\alpha}_2 \leq \alpha_{k_1}}\max_{\norm{v}_2 \leq \opnorm{\Sigma W} \alpha_{k_1} + \norm{\Sigma^{1/2} x}_2} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&= \lim_{k_2 \rightarrow \infty} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] .
\end{align*}
Observe that for every $k_2 \geq 1$,
\begin{align*}
0 &\leq \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&\leq \max_{\norm{v}_2 \leq v_{k_2}} \left[ - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&\leq \max_{v \in \ensuremath{\mathbb{R}}^q} \left[ - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right]
= \frac{1}{2}\norm{x}_2^2.
\end{align*}
Therefore, by \eqref{eq:first_dominated_convergence}
and another application of the dominated convergence theorem:
\begin{align}
\mathbb{E} \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \norm{ \Sigma^{1/2} W \alpha - x }_2^2 &=
\lim_{k_1 \rightarrow \infty} \mathbb{E} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \norm{\Sigma^{1/2} W \alpha - x}_2^2 \nonumber \\
&= \lim_{k_1 \rightarrow \infty} \mathbb{E} \lim_{k_2 \rightarrow \infty} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \nonumber \\
&= \lim_{k_1 \rightarrow \infty} \lim_{k_2 \rightarrow \infty} \mathbb{E} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right]. \label{eq:second_dominated_convergence}
\end{align}
We now apply
\Cref{thm:gaussian_min_max}
to the expectation on the RHS of \eqref{eq:second_dominated_convergence}:
\begin{align}
&~~~~\mathbb{E} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \nonumber \\
&= \int_0^\infty \Pr\left\{ \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[ v^\mathsf{T} W \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \geq t \right\} \mathrm{d} t \nonumber \\
&\stackrel{(a)}{\leq} 2 \int_0^\infty \Pr\left\{ \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \geq t \right\} \mathrm{d} t \nonumber \\
&= 2 \mathbb{E} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{\norm{v}_2 \leq v_{k_2}} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \nonumber \\
&\leq 2 \mathbb{E} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] . \label{eq:gaussian_min_max_inequality}
\end{align}
Above, inequality (a) is
an application of \Cref{thm:gaussian_min_max}.
Now for every $k_1$, $g$, and $h$,
define
\begin{align*}
\psi_{k_1}(g, h) := \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right].
\end{align*}
For every $k_1$, $g$, and $h$, we have
\begin{align*}
0 \leq \psi_{k_1}(g,h)
\leq \max_{v \in \ensuremath{\mathbb{R}}^q} \left[ - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right]
= \frac{\norm{x}^2_2}{2}.
\end{align*}
Furthermore, since $\{\alpha_k\}$ is an increasing sequence,
the sequence $ \{ \psi_{k}(g, h) \}_{k \geq 1}$ is
montonically decreasing.
Therefore, by the monotone convergence theorem,
\begin{align*}
\lim_{k \rightarrow \infty} \psi_{k}(g,h) &= \inf\{ \psi_{k}(g,h) \mid k \in \ensuremath{\mathbb{N}}_{+} \} \\
&= \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right].
\end{align*}
Therefore by another application of the dominated convergence theorem, we have that:
\begin{align}
&\lim_{k_1 \rightarrow \infty} \mathbb{E} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \nonumber \\
&= \mathbb{E} \lim_{k_1 \rightarrow \infty} \min_{\norm{\alpha}_2 \leq \alpha_{k_1}} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \nonumber \\
&= \mathbb{E} \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right]. \label{eq:third_dominated_convergence}
\end{align}
Chaining together inequalities \eqref{eq:second_dominated_convergence},
\eqref{eq:gaussian_min_max_inequality}, \eqref{eq:third_dominated_convergence}, we have:
\begin{align}
\mathbb{E} \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \norm{ \Sigma^{1/2} W \alpha - x }_2^2 \leq 2 \mathbb{E} \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[\norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right]. \label{eq:min_max_upper_bound}
\end{align}
We now proceed to study the RHS of \eqref{eq:min_max_upper_bound}, which we denote by $(\mathsf{AO})$ (the \emph{auxiliary optimization} problem):
\begin{align*}
(\mathsf{AO}) &:= \min_{\alpha \in \ensuremath{\mathbb{R}}^n} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[ \norm{\alpha}_2 g^\mathsf{T} v + \norm{v}_2 h^\mathsf{T} \alpha - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&= \min_{\beta \geq 0} \min_{\theta \in [-1,1]} \max_{v \in \ensuremath{\mathbb{R}}^q}\left[ \beta g^\mathsf{T} v + \beta \norm{v}_2 \norm{h}_2 \theta - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&\stackrel{(a)}{=} \min_{\beta \geq 0} \max_{v \in \ensuremath{\mathbb{R}}^q} \left[ \beta g^\mathsf{T} v - \beta \norm{v}_2 \norm{h}_2 - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&\stackrel{(b)}{=} \min_{\beta \geq 0} \max_{v \in \ensuremath{\mathbb{R}}^q} \max_{\tau \geq 0} \left[ \beta g^\mathsf{T} v - \beta\norm{h}_2 \norm{v}_2^2 \frac{\tau}{2} - \frac{\beta \norm{h}_2}{2 \tau} - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&= \min_{\beta \geq 0} \max_{\tau \geq 0} \max_{v \in \ensuremath{\mathbb{R}}^q}\left[ \beta g^\mathsf{T} v - \beta\norm{h}_2 \norm{v}_2^2 \frac{\tau}{2} - \frac{\beta \norm{h}_2}{2 \tau} - v^\mathsf{T} \Sigma^{-1/2} x - \frac{v^\mathsf{T} \Sigma^{-1} v}{2} \right] \\
&= \min_{\beta \geq 0} \max_{\tau \geq 0} \left[ -\frac{\beta\norm{h}_2}{2\tau} + \frac{1}{2} \norm{\beta g - \Sigma^{-1/2} x}_{(\Sigma^{-1} + \beta \norm{h}_2 \tau I_q)^{-1}}^2 \right] .
\end{align*}
Above, (b) holds by the variational form \eqref{eq:x_variation}.
The proof is now finished after justifying (a).
First, let $h_\beta(\theta, v)$ denote the term in the bracket,
so that
\begin{align*}
(\mathsf{AO}) = \min_{\beta \geq 0} \min_{\theta \in [-1, 1]} \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(\theta, v).
\end{align*}
Fix a $\beta \geq 0$.
By weak duality,
\begin{align*}
\min_{\theta \in [-1, 1]} \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(\theta, v) \geq \max_{v \in \ensuremath{\mathbb{R}}^q} \min_{\theta \in [-1, 1]} h_\beta(\theta, v) = \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(-1, v).
\end{align*}
On the other hand,
\begin{align*}
\min_{\theta \in [-1, 1]} \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(\theta, v) \leq \min_{\theta \in [-1, 0]} \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(\theta, v) = \max_{v \in \ensuremath{\mathbb{R}}^q} \min_{\theta \in [-1, 0]} h_\beta(\theta, v) = \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(-1, v).
\end{align*}
The first equality above is Sion's minimax theorem, since the function $\theta \mapsto h_\beta(\theta, v)$ is
affine for every $v$ and the function $v \mapsto h_\beta(\theta, v)$ is
concave for $\theta \in [-1, 0]$.
Therefore,
\begin{align*}
\min_{\theta \in [-1, 1]} \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(\theta, v) = \max_{v \in \ensuremath{\mathbb{R}}^q} h_\beta(-1, v).
\end{align*}
\end{proof}
With \Cref{lemma:min_max_upper_bound} in hand,
we can now restate and prove \Cref{lemma:trace_inv_lower_bound}.
\traceinvlowerbound*
\begin{proof}
We rewrite $\mathbb{E} \Tr((W^\mathsf{T} \Sigma W)^{-1})$ in a way that
is amenable to \Cref{lemma:min_max_upper_bound}.
Let $w_1 \in \ensuremath{\mathbb{R}}^{q}$ denote the first column of $W$, so that $W = \begin{bmatrix} w_1 & W_2 \end{bmatrix}$ with $W_2 \in \ensuremath{\mathbb{R}}^{q \times (n-1)}$.
We write:
\begin{align*}
W^\mathsf{T} \Sigma W = \begin{bmatrix} \norm{w_1}^2_{\Sigma} & w_1^\mathsf{T} \Sigma W_2 \\
W_2^\mathsf{T} \Sigma w_1 & W_2^\mathsf{T} \Sigma W_2 \end{bmatrix}.
\end{align*}
Using the block matrix inversion formula to compute
the $(1, 1)$ entry of
$(W^\mathsf{T} \Sigma W)^{-1}$:
\begin{align*}
((W^\mathsf{T} \Sigma W)^{-1})_{11} &= ( w_1^\mathsf{T} (\Sigma - \Sigma W_2^\mathsf{T} (W_2^\mathsf{T} \Sigma W_2)^{-1} W_2^\mathsf{T} \Sigma ) w_1)^{-1} \\
&= (w_1^\mathsf{T} \Sigma^{1/2}( I - P_{\Sigma^{1/2} W_2}) \Sigma^{1/2} w_1)^{-1} \\
&= (w_1^\mathsf{T} \Sigma^{1/2} P^\perp_{\Sigma^{1/2} W_2} \Sigma^{1/2} w_1)^{-1}.
\end{align*}
Since the columns of $W$ are all independent and identically distributed, this calculation shows that
the law of $((W^\mathsf{T} \Sigma W)^{-1})_{ii}$ is the
same as the law of $((W^\mathsf{T} \Sigma W)^{-1})_{11}$ for all $i=1, \dots, n$.
Therefore:
\begin{align*}
\mathbb{E} \Tr((W^\mathsf{T} \Sigma W)^{-1}) &= \sum_{i=1}^{n} \mathbb{E} ((W^\mathsf{T} \Sigma W)^{-1})_{ii} = n \cdot \mathbb{E} (w_1^\mathsf{T} \Sigma^{1/2} P^\perp_{\Sigma^{1/2} W_2} \Sigma^{1/2} w_1)^{-1} \\
&\geq \frac{n}{\mathbb{E} \Tr(\Sigma^{1/2} P^\perp_{\Sigma^{1/2} W_2} \Sigma^{1/2})}.
\end{align*}
The last inequality follows from Jensen's inequality combined with the independence of $w_1$ and $W_2$.
By decomposing
$\Tr( \Sigma^{1/2} P^{\perp}_{\Sigma^{1/2} W_2} \Sigma^{1/2} ) = \sum_{i=1}^{q} \norm{ P^\perp_{\Sigma^{1/2} W_2} \Sigma^{1/2}e_i}_2^2$
and observing that
\begin{align*}
\norm{P^\perp_{\Sigma^{1/2} W_2} x}_2^2 = \min_{\alpha \in \ensuremath{\mathbb{R}}^{n-1}} \norm{ \Sigma^{1/2} W_2 \alpha - x }_2^2 \quad \forall x \in \ensuremath{\mathbb{R}}^q,
\end{align*}
we have the following identity:
\begin{align*}
\mathbb{E}\Tr( \Sigma^{1/2} P^{\perp}_{\Sigma^{1/2} W_2} \Sigma^{1/2} ) = \sum_{i=1}^{q} \mathbb{E}\min_{\alpha_i \in \ensuremath{\mathbb{R}}^{n-1}} \norm{\Sigma^{1/2} W_2 \alpha_i - \Sigma^{1/2} e_i}_2^2.
\end{align*}
Invoking \Cref{lemma:min_max_upper_bound} with $x = \Sigma^{1/2} e_i$ for $i = 1, \dots, q$ yields
\begin{align*}
\mathbb{E}\Tr( \Sigma^{1/2} P^{\perp}_{\Sigma^{1/2} W_2} \Sigma^{1/2} ) \leq \sum_{i=1}^{q} \mathbb{E} \min_{\beta \geq 0} \max_{\tau \geq 0} \left[ -\frac{\beta\norm{h}_2}{\tau} + \norm{\beta g - e_i}_{(\Sigma^{-1} + \beta \norm{h}_2 \tau I_q)^{-1}}^2 \right],
\end{align*}
where $g \sim N(0, I_q)$ and $h \sim N(0, I_{n-1})$.
The claim now follows.
\end{proof}
We conclude this section with the following technical result
which we will use in the sequel.
\begin{mylemma}
\label{stmt:Z_i_helper}
Let $q, n \in \ensuremath{\mathbb{N}}_{+}$ with $q \geq n$ and $n \geq 6$, and let $\Sigma \in \mathsf{Sym}^{q}_{>0}$.
Let $g \sim N(0, I_q)$ and $h \sim N(0, I_{n-1})$ with $g$ and $h$ independent.
Define the random variables $Z_i$ for $i \in \{1, \dots, q\}$ as:
\begin{align}
Z_i := \min_{\beta \geq 0} \max_{\tau \geq 0} \left[ -\frac{\beta\norm{h}_2}{2\tau} + \beta^2 \norm{g}^2_{(\Sigma^{-1} + \beta \norm{h}_2 \tau I_q)^{-1}} + (\Sigma^{-1} + \beta \norm{h}_2 \tau I_q)^{-1}_{ii} \right].
\end{align}
Let $\{\lambda_i\}_{i=1}^{q}$ denote the eigenvalues of $\Sigma^{-1}$ listed
in decreasing order. Define $n_1$ and the random function $p(y)$ as:
\begin{align}
n_1 := \frac{n}{64}, \quad p(y) := \sum_{i=1}^{q} \frac{y}{\lambda_i + y} g_{i}^2 - \frac{n_1}{2}. \label{eq:n_1_and_p_y_def}
\end{align}
There exists an event $\mathcal{E}$ (over the probability of $g$ and $h$)
such that the following statements hold:
\begin{enumerate}[label=(\alph*)]
\item $\Pr(\mathcal{E}^c) \leq e^{-n/128} + e^{-q/16}$.
\item On $\mathcal{E}$, there exists a unique root $y^* \in (0, \infty)$ such that $p(y^*) = 0$.
\item The following bounds hold for $i \in \{1, \dots, q\}$:
\begin{align}
Z_i \leq \Sigma_{ii}, \quad \mathbf{1}\{\mathcal{E}\} Z_i \leq \mathbf{1}\{\mathcal{E}\} (\Sigma^{-1} + y^* I_q)^{-1}_{ii}. \label{eq:Z_i_inequalities}
\end{align}
\end{enumerate}
\end{mylemma}
\begin{proof}
First, we observe that we can trivially upper bound the value of $Z_i$
by setting $\beta = 0$ and obtaining the bound $Z_i \leq \Sigma_{ii}$.
Furthermore, by the rotational invariance of $g$ and the fact that $g$ and $h$ are independent, we have that $Z_i$ is equal in distribution to:
\begin{align*}
Z_i = \min_{\beta \geq 0} \max_{\tau \geq 0} \left[ -\frac{\beta\norm{h}_2}{2\tau} + \beta^2 \sum_{i=1}^{q} \frac{g_i^2}{\lambda_i + \beta\norm{h}_2 \tau} + (\Sigma^{-1} + \beta \norm{h}_2 \tau I_q)^{-1}_{ii} \right].
\end{align*}
Define the following events:
\begin{align*}
\mathcal{E}_h := \left\{ \norm{h}_2 \geq \sqrt{n}/8 \right\}, \quad \mathcal{E}_g := \left\{ \sum_{i=1}^{q} g_{i}^2 \geq q/2 \right\},
\end{align*}
and put $\mathcal{E} := \mathcal{E}_h \cap \mathcal{E}_g$.
Since $n \geq 6$, by a standard computation
we have that $\mathbb{E}\norm{h}_2 \geq \sqrt{n}/4$.
Therefore, by Gaussian concentration of Lipschitz functions
\citep[cf.][Chapter~2]{wainwright2019book},
$\Pr(\mathcal{E}_h^c) \leq e^{-n/128}$.
Furthermore, \Cref{lemma:chi_squared_tail_bounds}
yields that $\Pr(\mathcal{E}_g^c) \leq e^{-q/16}$.
By a union bound,
$\Pr(\mathcal{E}^c) \leq e^{-n/128} + e^{-q/16}$.
We now focus on upper bounding the quantity:
\begin{align*}
\mathbf{1}\{\mathcal{E}\} Z_i \leq \mathbf{1}\{\mathcal{E}\} \min_{\beta \geq 0} \max_{\tau \geq 0} \underbrace{\left[ -\frac{\beta\sqrt{n}}{16\tau} + \beta^2 \sum_{i=1}^{q} \frac{g_i^2}{\lambda_i + \beta\sqrt{n} \tau/8} + (\Sigma^{-1} + \beta\sqrt{n} \tau/8 I_q)^{-1}_{ii} \right]}_{=: \ell_i(\beta, \tau)}.
\end{align*}
Let us bracket the value
of the game $\min_{\beta \geq 0} \max_{\tau \geq 0} \ell_i(\beta, \tau)$.
We previously noted that
$\ell_i(0, \tau) = \Sigma_{ii}$
for all $\tau \in [0, \infty)$.
Next, for any $\beta > 0$,
$\lim_{\tau \rightarrow \infty} \ell_i(\beta, \tau) = 0$.
Hence,
\begin{align*}
\min_{\beta \geq 0} \max_{\tau \geq 0} \ell_i(\beta, \tau) \in [0, \Sigma_{ii}].
\end{align*}
Recalling from \eqref{eq:n_1_and_p_y_def} that $n_1 = n/64$ (so that $\sqrt{n_1} = \sqrt{n}/8$) and defining $f, q_i$ as:
\begin{align*}
f(x) &:= -\frac{x\sqrt{n_1}}{2} + x^2 \sum_{i=1}^{q} \frac{g_{i}^2}{\lambda_i + x \sqrt{n_1}}, \\
q_i(x) &:= (\Sigma^{-1} + x \sqrt{n_1})^{-1}_{ii},
\end{align*}
we have that $\ell_i(\beta, \tau) = \frac{1}{\tau^2} f(\beta \tau) + q_i(\beta \tau)$.
In order to sharpen our estimate for the value of the game,
we will study the positive critical points $(\beta, \tau) \in \ensuremath{\mathbb{R}}^2_{> 0}$
of the game
$\min_{\beta} \max_{\tau} \ell_i(\beta, \tau)$, i.e.,
the points $(\beta, \tau) \in \ensuremath{\mathbb{R}}^2_{> 0}$ satisfying
$\frac{\partial \ell_i}{\partial \beta}(\beta, \tau) = 0$
and $\frac{\partial \ell_i}{\partial \tau}(\beta, \tau) = 0$.
Note that in general for a nonconvex/nonconcave game,
this is \emph{not} a necessary first order optimality condition for the global min/max value~\citep[see e.g.][Proposition 21]{jin2020localoptimality}.
However, for every fixed $\beta > 0$, stationary points of the function $\tau \mapsto \ell_i(\beta, \tau)$ on $\ensuremath{\mathbb{R}}_{> 0}$ are strictly concave by \Cref{stmt:local_strict_concave_max}. Hence, by the implicit
function theorem (or alternatively \cite[Theorem 23]{jin2020localoptimality}), the first order stationarity conditions $\frac{\partial \ell_i}{\partial \beta}(\beta, \tau) = 0$
and $\frac{\partial \ell_i}{\partial \tau}(\beta, \tau) = 0$
are necessary for global min/max optimality.
For $\tau \neq 0$, this yields:
\begin{align*}
0 &= \tau^{-2} f'(\beta \tau) \beta - 2 \tau^{-3} f(\beta \tau) + q_i'(\beta \tau) \beta, \\
0 &= \tau^{-2} f'(\beta \tau) \tau + q_i'(\beta \tau) \tau.
\end{align*}
Together, these conditions imply that $f(\beta\tau) = 0$,
and that the value of the game at such a
critical point is $q_i(\beta \tau)$.
Thus, we are interested in the positive roots of $f(x) = 0$.
To proceed, recall the definition of $p(y)$ from \eqref{eq:n_1_and_p_y_def}:
\begin{align*}
p(y) = \sum_{i=1}^{q} \frac{y}{\lambda_i + y} g_{i}^2 - \frac{n_1}{2}.
\end{align*}
Note that $y^*$ is a positive root of $p$ iff $y^*/\sqrt{n_1}$ is a positive root of $f$.
Since $q \geq n$ by assumption,
observe that on $\mathcal{E}$:
\begin{align*}
\lim_{y \rightarrow \infty} p(y) = \sum_{i=1}^{q} g_{i}^2 - n_1/2 \geq q/2 - n_1/2 \geq n/2 - n/64 > 0.
\end{align*}
On the other hand, $p(0) = -n_1/2 < 0$.
Since $p(y)$ is continuous and strictly increasing,
on $\mathcal{E}$ there exists a unique $y^* \in (0, \infty)$ such that
$p(y^*) = 0$.
Thus,
\begin{align*}
\mathbf{1}\{ \mathcal{E} \} Z_i \leq \mathbf{1}\{ \mathcal{E} \} (\Sigma^{-1} + y^* I_q)^{-1}_{ii}.
\end{align*}
\end{proof}
\begin{myprop}
\label{stmt:local_strict_concave_max}
Let $M, A$ be $n \times n$ positive definite matrices, and
let $\alpha, \beta$ be positive numbers.
Consider the function:
\begin{align*}
f(\tau) := -\frac{\alpha}{\tau} + \ip{ (A + \beta \tau I)^{-1}}{M}.
\end{align*}
Suppose there exists a $\tau \in (0, \infty)$ satisfying
$f'(\tau) = 0$.
Then, $f''(\tau) < 0$.
\end{myprop}
\begin{proof}
A straightforward computation yields the following expressions for $f'(\tau)$ and $f''(\tau)$:
\begin{align*}
f'(\tau) &= \alpha \tau^{-2} - \beta \ip{(A + \beta \tau I)^{-2}}{M}, \\
f''(\tau) &= -2\alpha\tau^{-3} + 2\beta^2\ip{(A+\beta \tau I)^{-3}}{M}.
\end{align*}
The assumed condition $f'(\tau) = 0$ implies that:
\begin{align*}
\alpha \tau^{-2} = \beta \ip{(A+\beta\tau I)^{-2}}{M} \Longrightarrow -2\alpha \tau^{-3} = -2\beta\tau^{-1} \ip{(A+\beta \tau I)^{-2}}{M}.
\end{align*}
Next, let $A = Q \Lambda Q^\mathsf{T}$ be the eigendecomposition of $A$, with $\Lambda = \diag(\{\lambda_i\}_{i=1}^{n})$.
For any integer $k$:
\begin{align*}
\ip{(A+\beta \tau I)^{-k}}{M} = \Tr(M Q (\Lambda + \beta\tau I)^{-k} Q^\mathsf{T}) = \ip{ QMQ^\mathsf{T}}{ (\Lambda + \beta \tau I)^{-k}} = \sum_{i=1}^{n} \frac{(QMQ^\mathsf{T})_{ii}}{(\lambda_i + \beta \tau)^k}.
\end{align*}
Now, since $M$ is positive definite, $(QMQ^\mathsf{T})_{ii} > 0$ for all $i \in [n]$.
Furthermore, since $A$ is positive definite, $\lambda_i > 0$ for all $i \in [n]$.
Hence plugging these expressions into the expression of $f''(\tau)$:
\begin{align*}
f''(\tau) &= -2\beta^2 \sum_{i=1}^{n} \frac{(QMQ^\mathsf{T})_{ii}}{(\lambda_i + \beta \tau)^2 \beta \tau} + 2\beta^2 \sum_{i=1}^{n}\frac{(QMQ^\mathsf{T})_{ii}}{(\lambda_i + \beta\tau)^2 (\lambda_i + \beta\tau)} \\
&< -2\beta^2 \sum_{i=1}^{n} \frac{(QMQ^\mathsf{T})_{ii}}{(\lambda_i + \beta \tau)^2 \beta \tau} + 2\beta^2 \sum_{i=1}^{n}\frac{(QMQ^\mathsf{T})_{ii}}{(\lambda_i + \beta\tau)^2 \beta \tau} \\
&= 0.
\end{align*}
\end{proof}
\subsection{Proof of \Cref{stmt:ind_seq_ls_lower_bound}}
\label{sec:appendix:ind_seq_ls_lower_bound}
\indseqlslowerbound*
\begin{proof}
Let $\Gamma_T := \Gamma_T(\mathsf{P}_x)$.
We have that $\Gamma_T = \frac{2}{T} (2^T-1) I_n \succcurlyeq \frac{2^T}{T} I_n$.
By \Cref{thm:trace_inv_lower_bounds_minimax_risk}:
\begin{align*}
\mathsf{R}(m,T,T;\{\mathsf{P}_x\}) \geq \sigma_\xi^2 p \cdot \mathbb{E} \Tr(\Gamma_T^{1/2} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_T^{1/2}) \geq \frac{\sigma_\xi^2 p}{T} \cdot \mathbb{E} \Tr((2^{-T/2} X_{m,T}^\mathsf{T} X_{m,T} 2^{-T/2})^{-1}).
\end{align*}
Since each column of $X_{m,T}$ is independent,
the matrix
$X_{m,T}2^{-T/2}$ has the same distribution as
$\mathsf{BDiag}(\Theta^{1/2}, m)W$,
where $\Theta \in \ensuremath{\mathbb{R}}^{T \times T}$ is diagonal,
$\Theta_{ii} = 2^{i-T}$ for $i \in \{1, \dots, T\}$, and
$W \in \ensuremath{\mathbb{R}}^{mT \times n}$ has iid\ $N(0, 1)$ entries.
Let $\lambda_t = 2^{T-t}$ for $t \in \{1, \dots, T\}$.
With this notation:
\begin{align*}
\mathbb{E} \Tr((2^{-T/2} X_{m,T}^\mathsf{T} X_{m,T} 2^{-T/2})^{-1}) = \mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta, m) W)^{-1}).
\end{align*}
Let $\{g_{j}\}_{j=1}^{m}$ be independent
isotropic Gaussian random vectors in $\ensuremath{\mathbb{R}}^{T}$,
and let $h \sim N(0, I_{n-1})$ be independent from $\{g_j\}$.
Define the random variables $\{Z_i\}_{i=1}^{T}$ as:
\begin{align}
Z_i := \min_{\beta \geq 0} \max_{\tau \geq 0} \left[ -\frac{\beta \norm{h}_2}{2\tau} + \beta^2\sum_{j=1}^{m} \sum_{t=1}^{T} \frac{g_{j,t}^2}{\lambda_{t} + \beta \norm{h}_2 \tau} + \frac{1}{\lambda_i + \beta \norm{h}_2 \tau} \right].
\end{align}
By \Cref{lemma:trace_inv_lower_bound},
\begin{align*}
\mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta, m) W)^{-1}) \geq \frac{n}{2m} \left[ \sum_{i=1}^{T} \mathbb{E}[Z_i] \right]^{-1}.
\end{align*}
Next, define
\begin{align*}
n_1 := \frac{n}{64}, \quad
p(y) := \sum_{j=1}^{m} \sum_{t=1}^{T} \frac{y}{\lambda_{t} + y} g_{j,t}^2 - \frac{n_1}{2}.
\end{align*}
Since $n \geq 6$ and $mT \geq n$, we
can invoke \Cref{stmt:Z_i_helper} to conclude there exists
an event $\mathcal{E}_1$ (over the probability of $\{g_j\}$ and $h$)
such that:
\begin{enumerate}[label=(\alph*)]
\item on $\mathcal{E}_1$,
there exists a unique root $y^* \in (0, \infty)$
such that $p(y^*) = 0$,
\item the following inequalities holds:
\begin{align}
Z_i \leq \frac{1}{\lambda_i}, \quad \mathbf{1}\{\mathcal{E}_1\} Z_i \leq\mathbf{1}\{\mathcal{E}_1\} \frac{1}{\lambda_i + y^*}, \label{eq:bound_on_Z_i_good}
\end{align}
\item the following estimate holds:
\begin{align*}
\Pr(\mathcal{E}_1^c) \leq e^{-n/128} + e^{-mT/16}.
\end{align*}
\end{enumerate}
Now, let $c = 1/20$, and assume that
$c n_1 / m \geq 4$.
We can check easily that $\ceil{c n_1/m} \leq T$.
Fix a $\delta \in (0, e^{-2}]$ to be chosen later.
Define the integer $T_c := \ceil{c n_1/m} \in \{4, \dots, T\}$,
and the events (over the probability of $\{g_j\}$ and $h$):
\begin{align*}
\mathcal{E}_2^{g,T_c} := \left\{ \sum_{j=1}^{m} \sum_{t=1}^{T_c} g_{j,t}^2 \leq 5 mT_c \right\}, \quad \mathcal{E}_2^{g,+} := \left\{ \max_{t=1, \dots, T} \sum_{j=1}^{m} g_{j,t}^2 \leq 2 m + 4 \log\left( \frac{t^2 \pi^2}{6\delta} \right) \right\}.
\end{align*}
By \Cref{lemma:chi_squared_tail_bounds},
$\Pr((\mathcal{E}_2^{g,T_c})^c) \leq e^{-mT_c}$.
Next, Gaussian concentration for Lipschitz functions \citep[cf][Chapter~2]{wainwright2019book} yields,
for any $\eta \in (0, 1)$:
\begin{align*}
\max_{t=1, \dots, T} \Pr\left\{ \sqrt{\sum_{j=1}^{m} g_{j,t}^2} \geq \sqrt{m} + \sqrt{2\log(1/\eta)} \right\} \leq \eta.
\end{align*}
Hence by a union bound, and
the fact that
$6\delta/\pi^2 \sum_{t=1}^{T} t^{-2} \leq 6\delta/\pi^2 \sum_{t=1}^{\infty} t^{-2} = \delta$,
we have that $\Pr((\mathcal{E}_2^{g,+})^c) \leq \delta$.
Putting $\mathcal{E} := \mathcal{E}_1 \cup \mathcal{E}_2^{g,T_c} \cup \mathcal{E}_2^{g,+}$, we have:
\begin{align}
\Pr(\mathcal{E}^c) &\leq e^{-n/128} + e^{-mT/16} + e^{-mT_c} + \delta \nonumber \\
&\leq e^{-n/128} + e^{-mT/16} + e^{-c n_1} + \delta. \label{eq:prob_bad_event}
\end{align}
Next, noting that $t/2 \geq \log_2\log((t+1)^2\pi^2/6)$ for all $t \geq 4$:
\begin{align}
&~~~~\sum_{t=T_c}^{T-1} 2^{-t} \log((t+1)^2 \pi^2/(6\delta)) \nonumber \\
&= \sum_{t=T_c}^{T-1} 2^{-t + \log_2\log((t+1)^2 \pi^2/6)} + \log(1/\delta) \sum_{t=T_c}^{T-1} 2^{-t} \nonumber \\
&\leq \sum_{t=T_c}^{T-1} 2^{-t/2} + \log(1/\delta) \sum_{t=T_c}^{T-1} 2^{-t} && \text{since } T_c \geq 4 \nonumber \\
&= \sqrt{2}/(\sqrt{2}-1) (2^{-T_c/2} - 2^{-T/2}) + 2 \log(1/\delta) (2^{-T_c} - 2^{-T}) \nonumber \\
&\leq (4 + 2 \log(1/\delta)) 2^{-T_c/2} \nonumber \\
&\leq 4 \log(1/\delta) 2^{-T_c/2} &&\text{since } \delta \in (0, e^{-2}). \label{eq:geometric_series_with_log_bound}
\end{align}
Now, on $\mathcal{E}$:
\begin{align*}
\frac{n_1}{2} &= \sum_{j=1}^{m} \sum_{t=1}^{T} \frac{y^*}{\lambda_{t} + y^*} g_{j,t}^2 &&\text{since } p(y^*) = 0 \\
&\leq \sum_{j=1}^{m} \sum_{t=1}^{T_c} g_{j,t}^2 + y^* \sum_{j=1}^{m} \sum_{t=T_c}^{T-1} 2^{-t} g_{j,t+1}^2 \\
&= \sum_{j=1}^{m} \sum_{t=1}^{T_c} g_{j,t}^2 + y^* \sum_{t=T_c}^{T-1} 2^{-t} \left[\sum_{j=1}^{m} g_{j,t+1}^2\right] \\
&\leq 5mT_c + y^* \sum_{t=T_c}^{T-1} 2^{-t} \left[ 2m + 4 \log((t+1)^2 \pi^2/(6 \delta)) \right] &&\text{using } \mathcal{E} \\
&\leq 5mT_c + 4m y^* 2^{-T_c} + 16 y^* \log(1/\delta) 2^{-T_c/2} &&\text{using \eqref{eq:geometric_series_with_log_bound}} \\
&\leq 5mT_c + 18 m y^* \log(1/\delta) 2^{-T_c/2} &&\text{since } \delta \in (0, e^{-2}).
\end{align*}
This inequality implies the following
lower bound on $y^*$:
\begin{align*}
y^* &\geq \frac{2^{cn_1/(2m)}}{18 \log(1/\delta)} \left[ \frac{n_1}{2m} - 5 c \frac{n_1}{m} - 5 \right] \\
&= \frac{2^{cn_1/(2m)}}{18 \log(1/\delta)} \left[ \frac{n_1}{4m} - 5 \right] &&\text{since } c = 1/20 \\
&\geq \frac{2^{cn_1/(2m)}}{144 \log(1/\delta)} \frac{n_1}{m} &&\text{since } cn_1/m \geq 4 \Longrightarrow n_1/(8m) \geq 5 \\
&=: \underline{y}^*.
\end{align*}
We now bound,
\begin{align*}
\sum_{i=1}^{T} \mathbb{E}[Z_i] &= \sum_{i=1}^{T} \left[\mathbb{E}[\mathbf{1}\{\mathcal{E}\} Z_i] + \mathbb{E}[\mathbf{1}\{\mathcal{E}^c\} Z_i] \right] \\
&\leq \sum_{i=1}^{T} \left(\mathbb{E}\left[ \mathbf{1}\{\mathcal{E}\} \frac{1}{\lambda_i + y^*}\right] + \Pr(\mathcal{E}^c) \frac{1}{\lambda_i} \right) &&\text{using \eqref{eq:bound_on_Z_i_good}} \\
&\leq \sum_{i=1}^{T} \frac{1}{\lambda_i + \underline{y}^*} + \Pr(\mathcal{E}^c) \sum_{t=1}^{T} \frac{1}{\lambda_i} &&\text{since } y^* \geq \underline{y}^* \text{ on } \mathcal{E} \\
&\leq \sum_{t=0}^{T-1} \frac{1}{2^t + \underline{y}^*} + 2\left(e^{-n/128} + e^{-mT/16} + e^{-c n_1} + \delta \right) &&\text{using } \eqref{eq:prob_bad_event} \\
&\leq \frac{T_c}{y^*} + 2 \cdot 2^{-T_c} + 2\left(e^{-n/128} + e^{-mT/16} + e^{-c n_1} + \delta \right) \\
&\leq 288 c \log(1/\delta) 2^{-c n_1/(2m)} + 2 \cdot 2^{-cn_1/m} \\
&\qquad+ 2\left( e^{-n/128} + e^{-mT/16} + e^{-c n_1} + \delta\right) .
\end{align*}
Since $c n_1/m \geq 4$,
we can choose $\delta = e^{-c n_1/(2m)} \in (0, e^{-2}]$ and
obtain:
\begin{align*}
\sum_{i=1}^{T} \mathbb{E}[Z_i] &\leq 144c^2 \frac{n_1}{m} 2^{-c n_1/(2m)} + 2 \cdot 2^{-c n_1/m} + 2\left(e^{-n/128} + e^{-mT/16} + e^{-c n_1} + e^{-c n_1/(2m)} \right).
\end{align*}
Since $1 \leq c n_1/(4m)$,
$mT \geq n$, and $m \geq 1$,
this inequality implies there
exists universal positive constants
$c_1, c_2$ such that:
\begin{align*}
\sum_{i=1}^{T} \mathbb{E}[Z_i] \leq \frac{c_1n}{m} 2^{-c_2 n/m}.
\end{align*}
Hence:
\begin{align*}
\mathsf{R}(m,T,T;\{\mathsf{P}_x\}) \geq \frac{\sigma_\xi^2 p}{T} \frac{n}{2m} \left[ \sum_{i=1}^{T} \mathbb{E}[Z_i] \right]^{-1} \geq \frac{\sigma_\xi^2 p}{T} \frac{n}{2m} \frac{m}{c_1n} 2^{c_2 n/m} = \frac{\sigma^2_\xi p 2^{c_2 n/m}}{2c_1 T}.
\end{align*}
\end{proof}
\subsection{Block decoupling}
\label{sec:appendix:block_decoupling}
We now use a block decoupling argument to study lower bounds
on the risk.
The first step is the following result,
which bounds the risk from below by a
particular random gramian matrix.
\begin{mylemma}
\label{lemma:decoupling_blocks}
Let $n = dr$ with both $d,r$ positive integers.
Define $\mathcal{I}_r := \{1, 1 + r, \dots, 1 + (T-1)r\}$,
and let $E_{\mathcal{I}_r} \in \ensuremath{\mathbb{R}}^{T \times Tr}$ denote
the linear operator which extracts the coordinates in $\mathcal{I}_r$,
so that $(E_{\mathcal{I}_r} x)_i = x_{1 + (i-1)r}$ for $i=1, \dots, T$.
Recall the following definitions from Equation~\eqref{eq:Theta_T_r_def}:
\begin{align*}
\Psi_{r,T,{T'}} &= \mathsf{BDiag}(\Gamma_{{T'}}^{-1/2}(J_r), T) \mathsf{BToep}(J_r, T) \in \ensuremath{\mathbb{R}}^{Tr \times Tr}, \\
\Theta_{r,T,{T'}} &= E_{\mathcal{I}_r} \Psi_{r,T,{T'}} \Psi_{r,T,{T'}}^\mathsf{T} E_{\mathcal{I}_r}^\mathsf{T} \in \ensuremath{\mathbb{R}}^{T \times T}.
\end{align*}
Then, for $A = \mathsf{BDiag}(J_r, d)$ we have:
\begin{align*}
\mathbb{E}_{\otimes_{i=1}^{m} \PxA{A}} \left[\Tr\left(\Gamma_{{T'}}^{1/2}(A) (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}^{1/2}(A)\right)\right] \geq \mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta_{r,T,{T'}}, m) W)^{-1}),
\end{align*}
where $W \in \ensuremath{\mathbb{R}}^{mT \times d}$ is a matrix with independent $N(0, 1)$ entries.
\end{mylemma}
\begin{proof}
We apply \Cref{prop:trace_inv_selector_lower_bound}
with:
\begin{align*}
M = X_{m,T} \Gamma_{{T'}}^{-1/2}, \quad I = \{1, 1 + r, 1 + 2r, \dots, 1 + (d-1)r\}, \quad \abs{I} = d.
\end{align*}
Note that the block diagonal structure of $A$ yields
the same block diagonal structure on $\Gamma_{{T'}}$ and its inverse square root, specifically
$\Gamma_{{T'}}(A) = \mathsf{BDiag}(\Gamma_{{T'}}(J_r), d)$
and $\Gamma_{{T'}}^{-1/2}(A) = \mathsf{BDiag}(\Gamma_{{T'}}^{-1/2}(J_r), d)$.
Hence, it is not hard to see that the columns
of $M E_I^\mathsf{T}$ are not only independent, but also
identically distributed. Furthermore, the distribution
of each column obeys a multivariate Gaussian in $\ensuremath{\mathbb{R}}^{mT}$.
Hence, $ME_I^\mathsf{T}$ is equal in distribution to $Q_{m,T}^{1/2} W$,
where $W \in \ensuremath{\mathbb{R}}^{mT \times d}$ is a matrix of iid\ Gaussians and
$Q_{m,T} \in \mathsf{Sym}^{mT}_{> 0}$ is a positive definite covariance matrix to be determined.
Furthermore, because $ME_I^\mathsf{T}$ contains the vertical
concatenation of $m$ independent trajectories,
$Q_{m,T}$ itself is block diagonal:
\begin{align*}
Q_{m,T} = \mathsf{BDiag}(Q_T, m), \quad Q_T \in \mathsf{Sym}^T_{> 0}.
\end{align*}
Let us now compute an expression for $Q_{T}$.
Consider the dynamics:
\begin{align*}
x_{t+1}^r = J_r x_t^r + w_t^r, \quad x_0^r = 0, \quad w_t^r \sim N(0, \sigma_w^2 I_r).
\end{align*}
It is not hard to see that,
with $w_{0:T-1}^r = (w_0, \dots, w_{T-1}) \in \ensuremath{\mathbb{R}}^{rT}$,
\begin{align*}
\begin{bmatrix} \Gamma_{{T'}}^{-1/2}(J_r) x_1^r \\ \vdots \\ \Gamma_{{T'}}^{-1/2}(J_r) x_T^r \end{bmatrix} = \Psi_{r,T,{T'}} w_{0:T-1}^r.
\end{align*}
From this, we see that every column of $ME_I^\mathsf{T}$
is equal in distribution to $E_{\mathcal{I}_r} \Psi_{r,T,{T'}} w^r_{0:T-1}$, and therefore
has distribution
$N(0, E_{\mathcal{I}_r} \Psi_{r,T,{T'}} \Psi_{r,T,{T'}}^\mathsf{T} E_{\mathcal{I}_r}^\mathsf{T})$.
Therefore:
\begin{align*}
Q_T = E_{\mathcal{I}_r} \Psi_{r,T,{T'}} \Psi_{r,T,{T'}}^\mathsf{T} E_{\mathcal{I}_r}^\mathsf{T} = \Theta_{r,T,{T'}}.
\end{align*}
The claim now follows.
\end{proof}
\subsection{Eigenvalue analysis of a tridiagonal matrix}
\label{sec:appendix:lower_bounds:eigenvalues}
For any $T \in \ensuremath{\mathbb{N}}_{+}$, recall that
$L_T$ denotes the $T \times T$ lower triangle matrix
with ones in the lower triangle,
and $\Tri{a}{b}{T}$ denotes the symmetric $T \times T$ tri-diagonal matrix with
$a$ on the digonal and $b$ on the lower and upper off-diagonals.
In this section, we study the eigenvalues of $(L_TL_T^\mathsf{T})^{-1}$,
which we denote by $S_T$:
\begin{align}
S_T = (L_TL_T^\mathsf{T})^{-1} = \Tri{2}{-1}{T} - e_Te_T^\mathsf{T}. \label{eq:S_T_defn}
\end{align}
Understanding the eigenvalues of this matrix will be necessary
in the proof of \Cref{thm:lower_bound_r_equals_1}.
The following result sharply characterizes the spectrum of
$S_T$ up to constant factors.
\begin{mylemma}
\label{cor:SST_lower_upper_bounds}
Suppose $T \geq 8$. For all $k=1, \dots, T$, we have that:
\begin{align*}
0.02 \frac{k^2}{T^2} \leq \lambda_{T-k+1}(S_T) \leq \pi^2\frac{k^2}{T^2}.
\end{align*}
\end{mylemma}
\begin{proof}
We prove the upper bound in \Cref{prop:SST_upper_bounds},
and the lower bound in \Cref{prop:SST_lower_bounds}.
\end{proof}
The next result gives the necessary upper bounds on the eigenvalues
of $S_T$.
\begin{myprop}
\label{prop:SST_upper_bounds}
We have that:
\begin{align*}
\lambda_{T-k+1}(S_T) \leq \pi^2 \frac{k^2}{T^2}, \quad k=1, \dots, T.
\end{align*}
\end{myprop}
\begin{proof}
By \eqref{eq:S_T_defn}, we immediately produce a semidefinite upper bound on $S_T$:
\begin{align*}
S_T = \Tri{2}{-1}{T} - e_Te_T^\mathsf{T} \preccurlyeq \Tri{2}{-1}{T}.
\end{align*}
Therefore by the Courant min-max theorem, followed by the closed-form expression for the eigenvalues of $\Tri{2}{-1}{T}$, we have:
\begin{align*}
\lambda_{T-k+1}(S_T) \leq \lambda_{T-k+1}(\Tri{2}{-1}{T}) = 2\left(1 - \cos\left(\frac{k\pi}{T+1}\right)\right), \quad k=1, \dots, T.
\end{align*}
Next, we have the following elementary lower bounds for $\cos(x)$
on $x \in [0, \pi]$:
\begin{align*}
\cos(x) \geq \begin{cases}
1 - x^2/2 &\text{if } x \in [0, 2\pi/3], \\
(x-\pi)^2/4 - 1 &\text{if } x \in [2\pi/3, \pi].
\end{cases}
\end{align*}
Therefore, when $ k \in \left\{1, \dots, \bigfloor{\frac{2(T+1)}{3}} \right\}$, we immediately have that:
\begin{align*}
\lambda_{T-k+1}(S_T) \leq
\pi^2 \frac{k^2}{(T+1)^2}.
\end{align*}
For the case when $k \in \{ \floor{\frac{2(T+1)}{3}} + 1, \dots, T\}$, we use the cosine lower bounds to bound:
\begin{align*}
\lambda_{T-k+1}(S_T) &\leq 4 - \frac{\pi^2}{2} \left( 1 - \frac{k}{T+1}\right)^2 \\
&\leq 4\left[ 1 - \left( 1 - \frac{k}{T+1} \right)^2 \right] \\
&= 4 \left( \frac{k}{T+1} \right) \left( 2 - \frac{k}{T+1} \right) \\
&= 4 \left( \frac{k}{T+1} \right) \left( \frac{2(T+1)-k}{T+1} \right) \\
&\leq 4 \left( \frac{k}{T+1} \right) \left( \frac{ 3k - k }{T+1} \right) &&\text{since } k \geq 2(T+1)/3 \\
&= 8 \frac{k^2}{(T+1)^2}.
\end{align*}
The claim now follows by taking the maximum over the upper bounds.
\end{proof}
We now move to the lower bound on $\lambda_{T-k+1}(S_T)$.
At this point, it would be tempting to use Weyl's inequalities,
which imply that
$\lambda_i(S_T) \geq \lambda_i(\Tri{2}{-1}{T}) - 1$.
However, this bound becomes vacuous, since
$\lambda_{T}(\Tri{2}{-1}{T}) \lesssim 1/T^2$.
To get finer grained control, we need to use the
eigenvalue interlacing result of~\cite{kulkarni99tridiagonal}.
This is done in the following result:
\begin{myprop}
\label{prop:SST_lower_bounds}
Suppose that $T \geq 8$.
We have that
\begin{align*}
\lambda_{T-k+1}(S_T) \geq 0.02 \frac{k^2}{T^2}, \quad k=1, \dots, T.
\end{align*}
\end{myprop}
\begin{proof}
The proof relies on the interlacing result
from~\cite[Theorem~4.1]{kulkarni99tridiagonal}.
However, the interlacing result does not cover the
minimum eigenvalue of $S_T$, so we first explicitly derive a lower bound for $\lambda_{\min}(S_T)$. To do this, we note that:
\begin{align*}
\lambda_{\min}(S_T) = \lambda_{\min}((L_TL_T^\mathsf{T})^{-1}) = \frac{1}{\opnorm{L_T}^2}.
\end{align*}
Letting $l_i \in \ensuremath{\mathbb{R}}^T$ denote the $i$-th column of $L_T$,
by the variational form of the operator norm followed by Cauchy-Schwarz,
\begin{align*}
\opnorm{L_T} = \max_{\norm{v}_2 = 1} \norm{L_T v}_2 \leq \max_{\norm{v}_2 = 1} \sum_{i=1}^{T} \norm{l_{i}}_2 \abs{v_i} \leq \sqrt{ \sum_{i=1}^{T} \norm{l_{i}}_2^2 }
= \sqrt{\sum_{i=1}^{T} i} = \sqrt{T (T+1)/2}.
\end{align*}
Hence:
\begin{align*}
\lambda_{\min}(S_T) \geq \frac{2}{T(T+1)} \geq \frac{1}{T^2}.
\end{align*}
Now we may proceed with the remaining eigenvalues.
We can write $S_T$ as the following block matrix, with $e_{T-1} \in \ensuremath{\mathbb{R}}^{T-1}$
denoting the $(T-1)$-th standard basis vector:
\begin{align*}
S_T = \bmattwo{\Tri{2}{-1}{T-1}}{-e_{T-1}}{-e_{T-1}^\mathsf{T}}{1}.
\end{align*}
This matrix is of the form studied in \cite[Theorem~4.1]{kulkarni99tridiagonal};
for what follows we will borrow their notation.
Let $U_T(x)$ denote the $T$-th degree Chebyshev
polynomial of the 2nd kind.
We know that the eigenvalues of $S_T$ are
given by $\lambda = 2(1-x)$,
where $x$ are the roots of the polynomial $p_T(x)$ defined as:
\begin{align}
p_T(x) := (1+2x) U_{T-1}(x) - U_{T-2}(x). \label{eq:char_poly_SST}
\end{align}
Therefore, letting $\psi_1 \leq \dots \leq \psi_T$
denote the roots of \eqref{eq:char_poly_SST}
listed in increasing order, we have:
\begin{align*}
\lambda_{i}(S_T) = 2(1-\psi_i), \quad i=1, \dots, T.
\end{align*}
Let $\eta_1 < \dots < \eta_{T-2}$ denote the $T-2$ roots of $U_{T-2}(x)$
listed in increasing order. Put $\eta_0 := -\infty$ and $\eta_{T-1} := +\infty$.
Because the roots of
$U_{T-2}(x)$ are given by $x = \cos(\frac{k\pi}{T-1})$, $k=1, \dots, T-2$, we have that:
\begin{align*}
\eta_{i} = \cos\left( \frac{(T-1-i)\pi}{T-1} \right), \quad i = 1, \dots, T-2.
\end{align*}
\cite[Theorem~4.1]{kulkarni99tridiagonal} states that
there is exactly one root of $p_T(x)$
in each of the intervals
$(\eta_{j}, \eta_{j+1})$ for $j \in \{0, \dots, T-2\} \setminus \{i_\star\}$,
with $i_\star$ satisfying:
\begin{align*}
i_\star \in \begin{cases}
\{ \floor{\frac{2(T-1)}{3}} \} &\text{if } 2(T-1) \mod 3 \neq 0, \\
\{ \frac{2(T-1)}{3}, \frac{2(T-1)}{3} + 1 \} &\text{otherwise},
\end{cases}
\end{align*}
and furthermore $(\eta_{i_\star}, \eta_{i_\star+1})$ contains exactly two roots of $p_T(x)$.
Therefore, for $i \in \{i_\star+3, \dots, T-1\}$:
\begin{align*}
\psi_i \leq \eta_{i-1} \Longrightarrow \lambda_i(S_T) \geq 2(1-\eta_{i-1}) = 2\left(1 - \cos\left( \frac{( T - i)\pi}{T-1} \right)\right).
\end{align*}
For $i \in \{i_\star+3, \dots, T-1\}$, we have:
\begin{align*}
\frac{T-i}{T-1} \leq \frac{T-i_\star-3}{T-1} = \frac{ T - (\frac{2(T-1)}{3}-1) - 3}{T-1} = \frac{1}{3} - \frac{1}{T-1} \leq \frac{1}{3}.
\end{align*}
It is elementary to check that:
\begin{align*}
2 (1-\cos(x)) \geq \frac{x^2}{2} \quad \forall x \in [0, \pi/3].
\end{align*}
Therefore for $i \in \{i_\star+3, \dots, T-1\}$,
\begin{align*}
\lambda_i(S_T) \geq \frac{\pi^2}{2} \left( \frac{T-i}{T-1} \right)^2.
\end{align*}
Furthermore,
for $i \in \{1, \dots, i_\star + 2\}$,
\begin{align*}
\psi_i \leq \eta_{i_\star+1} \Longrightarrow \lambda_i(S_T) \geq 2(1 - \eta_{i_\star+1}) = 2 \left( 1 - \cos\left( \frac{(T-1-i_\star-1)\pi }{T-1 }\right) \right) \geq 2(1-\cos(\pi/21)).
\end{align*}
The last inequality holds by:
\begin{align*}
\cos\left( \frac{(T-1-i_\star-1)\pi }{T-1 }\right) &\leq \cos\left( \frac{(T-1)-(2(T-1)/3+2)}{T-1} \pi \right) &&\text{since } i_\star \leq \frac{2(T-1)}{3} + 1 \\
&= \cos\left( \left(\frac{1}{3} - \frac{2}{T-1}\right)\pi\right) \\
&\leq \cos(\pi/21) &&\text{since } T \geq 8.
\end{align*}
Summarizing, we have shown that:
\begin{align*}
\lambda_{T-k+1}(S_T) \geq \begin{cases}
\frac{1}{T^2} &\text{if } k = 1, \\
\frac{\pi^2}{2} \left( \frac{k-1}{T-1} \right)^2 &\text{if } k \in \{2, \dots, T-i_\star-2\}, \\
2(1-\cos(\pi/21)) &\text{if } k \in \{T-i_\star-1, \dots, T\}.
\end{cases}
\end{align*}
Since $\frac{k-1}{T-1} \geq \frac{k}{2T}$ when $k \geq 2$,
and since $2(1-\cos(\pi/21)) \geq 2(1-\cos(\pi/21)) \frac{k^2}{T^2}$ trivially, we have shown the desired conclusion:
\begin{align*}
\lambda_{T-k+1}(S_T) \geq \min\left\{ 1, \frac{\pi^2}{8}, 2(1-\cos(\pi/21)) \right\} \frac{k^2}{T^2} \geq 0.02 \frac{k^2}{T^2}, \quad k = 1, \dots, T.
\end{align*}
\end{proof}
\subsection{A risk lower bound in the few trajectories regime}
\label{sec:appendix:lower_bounds:r_equals_one}
\begin{mylemma}
\label{thm:lower_bound_r_equals_1}
There exist universal positive constants $c_0$, $c_1$, $c_2$, and $c_3$ such that the following is true.
Suppose $\mathcal{A} \subseteq \ensuremath{\mathbb{R}}^{n \times n}$ is any set containing
$I_{n}$.
Let $T \geq c_0$, $n \geq c_1$, $mT \geq n$, and $m \leq c_2 n$.
We have that:
\begin{align*}
\mathsf{R}(m, T, {T'}; \{ \PxA{A} \mid A \in \mathcal{A}\}) \geq c_3 \sigma_\xi^2 p \cdot \frac{n^2}{m^2 T} \cdot \frac{{T'}}{T}.
\end{align*}
\end{mylemma}
\begin{proof}
Let $\{g_j\}_{j=1}^{m}$ be independent $N(0, I_T)$ random vectors,
and let $h \sim N(0, I_{n-1})$ be independent from $\{g_j\}$.
Let $\{\lambda_t\}_{t=1}^{T}$ denote the eigenvalues of $\Theta_{1,T,T}^{-1}$ listed in decreasing order.
Define the random variables $\{Z_i\}_{i=1}^{T}$ as:
\begin{align}
Z_i := \min_{\beta \geq 0} \max_{\tau \geq 0} \left[ -\frac{\beta \norm{h}_2}{2\tau} + \beta^2\sum_{j=1}^{m} \sum_{t=1}^{T} \frac{g_{j,t}^2}{\lambda_t + \beta \norm{h}_2 \tau} + (\Theta_{1,T,T}^{-1} + \beta \norm{h}_2 \tau I_T)^{-1}_{ii} \right].
\end{align}
We now lower bound the minimax risk as follows:
\begin{align}
&~~~~\mathsf{R}(m,T,{T'};\{\PxA{I_n}\}) \nonumber \\
&\geq \sigma_\xi^2 p \cdot \mathbb{E}_{\otimes_{i=1}^{m} \PxA{I_n}}\left[ \Tr\left( \Gamma_{{T'}}(I_n)^{1/2} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}(I_n)^{1/2} \right) \right] && \text{by \Cref{thm:trace_inv_lower_bounds_minimax_risk}} \nonumber \\
&\geq \sigma_\xi^2 p \cdot \mathbb{E} \Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta_{1,T,{T'}}, m) W)^{-1}) && \text{by \Cref{lemma:decoupling_blocks}} \nonumber \\
&= \sigma_\xi^2 p \cdot \frac{{T'} + 1}{T+1} \cdot \mathbb{E}\Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta_{1,T,T}, m) W)^{-1}) &&\text{using \eqref{eq:Tnew_equals_T_wlog_r_equals_one_lower_bound}} \nonumber \\
&\geq \sigma_\xi^2 p \cdot \frac{{T'}}{2T} \cdot \mathbb{E}\Tr((W^\mathsf{T} \mathsf{BDiag}(\Theta_{1,T,T}, m) W)^{-1}) \nonumber \\
&\geq \sigma_\xi^2 p \cdot \frac{{T'}}{2T} \cdot \frac{n}{2m} \cdot \left[ \sum_{i=1}^{T} \mathbb{E}[Z_i] \right]^{-1} &&\text{by \Cref{lemma:trace_inv_lower_bound}}. \label{eq:lower_bound_with_sum_Zis_inverse}
\end{align}
Next, define:
\begin{align*}
n_1 := \frac{n}{64}, \quad
p(y) := \sum_{j=1}^{m} \sum_{t=1}^{T} \frac{y}{\lambda_{t} + y} g_{j,t}^2 - \frac{n_1}{2}.
\end{align*}
Assuming that $c_1 \geq 6$ so that $n \geq 6$ and $mT \geq n$, we
can invoke \Cref{stmt:Z_i_helper} to conclude there exists
an event $\mathcal{E}_1$ (over the probability of $\{g_j\}$ and $h$)
such that:
\begin{enumerate}[label=(\alph*)]
\item on $\mathcal{E}_1$,
there exists a unique root $y^* \in (0, \infty)$
such that $p(y^*) = 0$,
\item the following inequalities holds:
\begin{align}
Z_i \leq (\Theta_{1,T,T})_{ii}, \quad \mathbf{1}\{\mathcal{E}_1\} Z_i \leq\mathbf{1}\{\mathcal{E}_1\} \frac{1}{\lambda_i + y^*}, \label{eq:bound_on_Z_i_good_v2}
\end{align}
\item the following estimate holds:
\begin{align*}
\Pr(\mathcal{E}_1^c) \leq e^{-n/128} + e^{-mT/16}.
\end{align*}
\end{enumerate}
The remainder of the proof is to estimate a lower bound on
$y^*$.
Towards this goal, we define an auxiliary function:
\begin{align*}
\tilde{p}(y) := \mathbb{E}[p_1(y)] = m\sum_{t=1}^{T} \frac{y}{\lambda_t + y} - \frac{n_1}{2}.
\end{align*}
Let $\bar{y}^*$ be the unique solution
to $\tilde{p}(y) = 0$.
A unique root exists because $\tilde{p}(0) < 0$,
$\lim_{y \rightarrow \infty}\tilde{p}(y) = mT - n_1/2 \geq n - n/64 > 0$,
and $\tilde{p}$ is continuous and strictly increasing.
We derive a lower bound on $y^*$ through a
lower bound on $\bar{y}^*$.
For any fixed $\alpha > 0$, the function
$x \mapsto \frac{x}{\alpha + x}$ is monotonically
increasing and concave on $\ensuremath{\mathbb{R}}_{> 0}$.
Therefore, the function $p(y)$ is monotonically
increasing and concave on $\ensuremath{\mathbb{R}}_{> 0}$.
By \Cref{prop:linear_approx_root_lower_bound}, the root of the linear approximation to
$p(y)$ at $\bar{y}^*$ is a lower bound
to $y^*$:
\begin{align}
\mathbf{1}\{ \mathcal{E}_1 \} y^* \geq \mathbf{1}\{ \mathcal{E}_1 \} \left[\bar{y}^* - \frac{p(\bar{y}^*)}{p'(\bar{y}^*)}\right]. \label{eq:y_i_lower_bound}
\end{align}
Equation~\eqref{eq:y_i_lower_bound} is a crucial step for the proof, because
it turns analyzing $y^*$, which is the root of a random function, into analyzing the pointwise evaluation
of a random function on a deterministic quantity.
To lower bound the RHS,
we need a upper bound on $p(\bar{y}^*)$
and lower bounds on both $\bar{y}^*$ and $p'(\bar{y}^*)$.
\paragraph{Upper and lower bounds on $\bar{y}^*$.}
We first derive a crude upper bound by Jensen's inequality.
Observe that $\tilde{p}(\bar{y}^*) = 0$
implies that:
\begin{align*}
mT - \frac{n_1}{2} = m \sum_{t=1}^{T} \frac{\lambda_t}{\lambda_t + \bar{y}^*}.
\end{align*}
The function $x \mapsto x / (x + \bar{y}^*)$
is concave on $\ensuremath{\mathbb{R}}_{> 0}$.
Let $\bar{\lambda} := \frac{1}{T} \sum_{t=1}^{T} \lambda_t$.
Jensen's inequality states that
$T \frac{\bar{\lambda}}{\bar{\lambda} + \bar{y}^*} \geq \sum_{t=1}^{T} \frac{\lambda_t}{\lambda_t + \bar{y}^*}$.
Therefore:
\begin{align*}
1 - \frac{n_1}{2mT} \leq \frac{\bar{\lambda}}{\bar{\lambda} + \bar{y}^*} \Longrightarrow \bar{y}^* \leq \bar{\lambda} \frac{n_1}{2mT} \frac{1}{1-n_1/(2mT)}.
\end{align*}
Recalling the definition of $S_T$ from \eqref{eq:S_T_defn},
we can immediately bound
\begin{align*}
\bar{\lambda} = \frac{1}{T}\sum_{t=1}^{T} \lambda_t = \frac{1}{T} \Tr(\Theta_{1,T,T}^{-1}) = \frac{1}{T} \Tr\left( \frac{T+1}{2} S_T\right) \leq \Tr(S_T) \leq 2T.
\end{align*}
Therefore, since $mT \geq n$,
\begin{align*}
\bar{y}^* \leq \frac{n_1}{m} \frac{1}{1-n_1/(2mT)} \leq \frac{2 n_1}{m}.
\end{align*}
Now for the lower bound on $\bar{y}^*$.
Noting that $\lambda_{T-k+1} = \lambda_{T-k+1}(\Theta_{1,T,T}^{-1}) = \frac{T+1}{2} \lambda_{T-k+1}(S_T)$,
Corollary~\ref{cor:SST_lower_upper_bounds}
implies (assuming that $c_0 \geq 8$ so $T \geq 8$) that
\begin{align}
0.01 \frac{k^2}{T} \leq \lambda_{T-k+1} \leq \pi^2 \frac{k^2}{T}, \quad k=1, \dots, T. \label{eq:lambda_upper_lower_bounds}
\end{align}
Therefore, $\tilde{p}(\bar{y}^*) = 0$ implies that:
\begin{align*}
\frac{1}{\bar{y}^*} &= \frac{2m}{n_1} \sum_{t=1}^{T} \frac{1}{\lambda_t + \bar{y}^*} \leq \frac{2m}{n_1} \sum_{t=1}^{T} \frac{1}{0.01 t^2/T + \bar{y}^*} \\
&\leq \frac{2m}{n_1} \int_0^{T} \frac{1}{0.01 x^2/T + \bar{y}^*} \,\mathrm{d} x
= \frac{20 m\sqrt{T}}{n_1 \sqrt{\bar{y}^*}} \tan^{-1}\left(\frac{\sqrt{T}}{10\sqrt{\bar{y}^*}}\right) \leq \frac{10 \pi m\sqrt{T}}{n_1 \sqrt{\bar{y}^*}}.
\end{align*}
Solving for $\bar{y}^*$ yields:
\begin{align*}
\bar{y}^* \geq \frac{1}{100\pi^2} \frac{n_1^2}{m^2T}.
\end{align*}
Next, we use this lower bound on $\bar{y}^*$ to
bootstrap our upper bound $\bar{y}^* \leq 2 n_1/m$ into something stronger.
Using the upper bounds on $\lambda_t$ from \eqref{eq:lambda_upper_lower_bounds},
\begin{align*}
\frac{1}{\bar{y}^*} &= \frac{2m}{n_1} \sum_{t=1}^{T} \frac{1}{\lambda_t + \bar{y}^*} \geq \frac{2m}{n_1} \sum_{t=1}^{T} \frac{1}{ \pi^2 t^2/T + \bar{y}^*} \geq \frac{2m}{n_1} \int_1^{T+1} \frac{1}{\pi^2 x^2/T + \bar{y}^*} \,\mathrm{d} x \\
&= \frac{2 m \sqrt{T}}{\pi n_1\sqrt{\bar{y}^*}} \left[ \tan^{-1}\left( \frac{(T+1)\pi}{\sqrt{T \bar{y}^*}}\right) - \tan^{-1}\left(\frac{\pi}{\sqrt{T\bar{y}^*}}\right) \right].
\end{align*}
The function $\tan^{-1}(x)$ is increasing.
Using the $\bar{y}^* \leq 2 n_1/m$ upper bound
and the assumption that $mT \geq n$,
\begin{align*}
\frac{(T+1)\pi}{\sqrt{T\bar{y}^*}} \geq \pi \sqrt{\frac{mT}{2n_1}} \geq \pi \sqrt{32} \Longrightarrow \tan^{-1}\left(\frac{(T+1)\pi}{\sqrt{T\bar{y}^*}}\right) \geq \tan^{-1}( \pi\sqrt{32} ).
\end{align*}
On the other handing, using the bound $\bar{y}^* \geq \frac{1}{100\pi^2} \frac{n_1^2}{m^2T}$
and the assumption that $m \leq \sqrt{2}n/320$,
\begin{align*}
\frac{\pi}{\sqrt{T\bar{y}^*}} \leq 10\pi \frac{m}{n_1} \leq \pi\sqrt{32}/2 \Longrightarrow \tan^{-1}\left(\frac{\pi}{\sqrt{T\bar{y}^*}}\right) \leq \tan^{-1}( \pi\sqrt{32}/2 ).
\end{align*}
Combining these inequalities:
\begin{align*}
\frac{1}{\bar{y}^*} \geq \frac{2m \sqrt{T}}{\pi n_1\sqrt{\bar{y}^*}}\left[ \tan^{-1}(\pi\sqrt{32}) - \tan^{-1}(\pi\sqrt{32}/2) \right] \geq \frac{2 \cdot 0.05}{\pi} \frac{m\sqrt{T}}{n_1\sqrt{\bar{y}^*}} \Longrightarrow \bar{y}^* \leq 791\pi^2 \frac{n_1^2}{m^2 T}.
\end{align*}
Therefore we have the following upper and lower bounds on $\bar{y}^*$:
\begin{align}
\frac{1}{100\pi^2} \frac{n_1^2}{m^2 T} \leq \bar{y}^* \leq \min\left\{ 791\pi^2 \frac{n_1^2}{m^2 T}, 2 \frac{n_1}{m} \right\}.
\label{eq:ybar_lower_upper_bounds}
\end{align}
For the remainder of the proof,
in order to avoid precisely tracking constants,
we let $c_0, c_1, c_2, c_3$
be any positive universal constants
such that:
\begin{align}
c_0 \frac{k^2}{T} &\leq \lambda_{T-k+1} \leq c_1 \frac{k^2}{T}, \quad k=1, \dots, T, \label{eq:lambda_bounds_c_0_c_1} \\
c_2 \frac{n_1^2}{m^2 T} &\leq \bar{y}^* \leq c_3 \frac{n_1^2}{m^2 T}. \label{eq:bar_y_star_bounds_c_2_c3}
\end{align}
Equations \eqref{eq:lambda_upper_lower_bounds} and \eqref{eq:ybar_lower_upper_bounds} give one valid setting
of these constants.
\paragraph{Upper bound on $p(\bar{y}^*)$.}
To upper bound $p(\bar{y}^*)$,
we note that:
\begin{align*}
p(\bar{y}^*) &= \sum_{j=1}^{m}\sum_{t=1}^{T} \frac{\bar{y}^*}{\lambda_t + \bar{y}^*} g_{j,t}^2 - \frac{n_1}{2} \\
&= \sum_{j=1}^{m}\sum_{t=1}^{T} \frac{\bar{y}^*}{\lambda_t + \bar{y}^*} (g_{i,j}^2-1) + \sum_{j=1}^{m}\sum_{t=1}^{T} \frac{\bar{y}^*}{\lambda_t + \bar{y}^*} - \frac{n_1}{2} \\
&= \sum_{j=1}^{m}\sum_{t=1}^{T} \frac{\bar{y}^*}{\lambda_t + \bar{y}^*} (g_{i,j}^2-1) &&\text{since } \tilde{p}(\bar{y}^*) = 0.
\end{align*}
Therefore, by \Cref{lemma:chi_squared_tail_bounds},
\begin{align}
\Pr\left( p(\bar{y}^*) > 2 \sqrt{t} \sqrt{m\sum_{t=1}^{T} \left( \frac{\bar{y}^*}{\lambda_t + \bar{y}^*} \right)^2} + 2 t \max_{t=1, \dots, T} \frac{\bar{y}^*}{\lambda_t + \bar{y}^*} \right) \leq e^{-t} \quad \forall t > 0. \label{eq:upper_tail}
\end{align}
We upper bound:
\begin{align}
m\sum_{t=1}^{T} \left(\frac{\bar{y}^*}{\lambda_t + \bar{y}^*}\right)^2
&\leq m \sum_{t=1}^{T} \left(\frac{\bar{y}^*}{c_0 t^2/T + \bar{y}^*}\right)^2 &&\text{using } \eqref{eq:lambda_bounds_c_0_c_1} \nonumber \\
&\leq m \int_0^T \left(\frac{\bar{y}^*}{c_0 x^2/T + \bar{y}^*}\right)^2 \,\mathrm{d} x \nonumber \\
&= \frac{m (\bar{y}^*)^2 T}{2 c_0 T \bar{y}^* + 2 (\bar{y}^*)^2} + \frac{\sqrt{T \bar{y}^*}}{2\sqrt{c_0}} \tan^{-1}\left(\sqrt{\frac{c_0 T}{\bar{y}^*}}\right) \nonumber \\
&\leq \frac{m \bar{y}^*}{2c_0} + \frac{\pi \sqrt{T\bar{y}^*}}{4\sqrt{c_0}} \nonumber \\
&\leq \frac{c_3}{2c_0} \frac{n_1^2}{mT} + \frac{\pi}{4}\sqrt{\frac{c_3}{c_0}} \frac{n_1}{m} &&\text{using } \eqref{eq:bar_y_star_bounds_c_2_c3} \nonumber \\
&= \left[ \frac{c_3}{128 c_0} + \frac{\pi}{4}\sqrt{\frac{c_3}{c_0}} \right] n_1 &&\text{since } mT \geq n \text{ and } m \geq 1 \nonumber \\
&=: c_4 n_1. \label{eq:first_bound}
\end{align}
Next, we immediately have:
\begin{align}
\max_{t=1, \dots, T} \frac{\bar{y}^*}{\lambda_t + \bar{y}^*} \leq 1. \label{eq:second_bound}
\end{align}
Thus, combining \eqref{eq:upper_tail},
\eqref{eq:first_bound}, and \eqref{eq:second_bound}, we have:
\begin{align}
\Pr\left( p(\bar{y}^*) > 2 \sqrt{t_u} \sqrt{c_4 n_1} + 2 t_u \right) \leq e^{-t_u} \:\forall t_u > 0. \label{eq:p_i_upper_bound}
\end{align}
\paragraph{Lower bound on $p'(\bar{y}^*)$.}
Differentiating $p(y)$ yields:
\begin{align*}
p'(y) = \sum_{j=1}^{m}\sum_{t=1}^{T} \frac{\lambda_t}{(\lambda_t + y)^2} g_{j,t}^2.
\end{align*}
Applying \Cref{lemma:chi_squared_tail_bounds} yields,
\begin{align}
\Pr\left( p'(\bar{y}^*) < m\sum_{t=1}^{T} \frac{\lambda_t}{(\lambda_t + \bar{y}^*)^2} - 2 \sqrt{t} \sqrt{ m\sum_{t=1}^{T} \frac{\lambda_t^2}{(\lambda_t + \bar{y}^*)^4} } \right) \leq e^{-t} \quad \forall t > 0. \label{eq:lower_tail}
\end{align}
Our first goal is to lower bound $m\sum_{t=1}^{T} \frac{\lambda_t}{(\lambda_t + \bar{y}^*)^2}$.
The function $x \mapsto x/(x+\bar{y}^*)^2$ is
increasing when $x \in [0, \bar{y}^*]$
and decreasing when $x \in (\bar{y}^*, \infty)$.
Let $t^* \in \{0, \dots, T\}$ be such that $c_1 t^2/T \leq \bar{y}^*$ for $t \in \{1, \dots, t^*\}$
and $c_1 t^2/T > \bar{y}^*$ for $t \in \{t^* + 1, \dots, T\}$
($t^* = 0$ if $c_1/T > \bar{y}^*$).
We write:
\begin{align*}
m \sum_{t=1}^{T} \frac{\lambda_t}{(\lambda_t + \bar{y}^*)^2}
&\geq \frac{c_0}{c_1}m \sum_{t=1}^{T} \frac{c_1 t^2/T}{(c_1t^2/T + \bar{y}^*)^2} &&\text{using } \eqref{eq:lambda_bounds_c_0_c_1} \\
&= \frac{c_0}{c_1} m \left[ \sum_{t=1}^{t^*} \frac{c_1t^2/T}{(c_1t^2/T + \bar{y}^*)^2} + \sum_{t=t^*+1}^{T} \frac{c_1t^2/T}{(c_1t^2/T + \bar{y}^*)^2} \right] \\
&\geq \frac{c_0}{c_1} m \left[ \int_0^{t^*} \frac{c_1 x^2/T}{(c_1x^2/T + \bar{y}^*)^2} \,\mathrm{d} x + \int_{t^*+1}^{T+1} \frac{c_1 x^2/T}{(c_1x^2/T + \bar{y}^*)^2} \,\mathrm{d} x \right] \\
&= \frac{c_0}{c_1} m \left[ \int_0^{T+1} \frac{c_1 x^2/T}{(c_1x^2/T + \bar{y}^*)^2} \,\mathrm{d} x - \int_{t^*}^{t^*+1}\frac{c_1 x^2/T}{(c_1x^2/T + \bar{y}^*)^2} \,\mathrm{d} x \right].
\end{align*}
The function $z \mapsto \frac{z}{(z+\bar{y}^*)^2}$ is upper bounded by $\frac{1}{4\bar{y}^*}$.
Therefore,
\begin{align*}
\int_{t^*}^{t^*+1}\frac{c_1 x^2/T}{(c_1x^2/T + \bar{y}^*)^2} \,\mathrm{d} x \leq \frac{1}{4 \bar{y}^*} \leq \frac{1}{4c_2} \frac{m^2 T}{n_1^2}.
\end{align*}
Next,
\begin{align*}
\int_0^{T+1} \frac{c_1 x^2/T}{(c_1x^2/T + \bar{y}^*)^2} \,\mathrm{d} x &= c_1 T \left[ \frac{1}{2c_1^{3/2} \sqrt{T\bar{y}^*}} \tan^{-1}\left( \frac{(T+1) \sqrt{c_1}}{ \sqrt{T\bar{y}^*}} \right) - \frac{T+1}{2 c_1^2 (T+1)^2 + 2 c_1 T \bar{y}^*} \right] \\
&\geq c_1 T \left[ \frac{m}{2c_1^{3/2} \sqrt{c_3} n_1} \tan^{-1}\left( \frac{(T+1) \sqrt{c_1}}{ \sqrt{T\bar{y}^*}} \right) - \frac{1}{2c_1^2T} \right] \\
&\geq c_1 T \left[ \frac{m}{2c_1^{3/2} \sqrt{c_3} n_1} \tan^{-1}\left(64\sqrt{\frac{c_1}{c_3}} \right) - \frac{1}{2c_1^2T} \right].
\end{align*}
The last inequality holds because:
\begin{align*}
\frac{(T+1)\sqrt{c_1}}{ \sqrt{T\bar{y}^*}} \geq (T+1) \sqrt{\frac{c_1}{c_3}} \frac{m}{n_1} \geq \sqrt{\frac{c_1}{c_3}} \frac{mT}{n_1} \geq 64 \sqrt{\frac{c_1}{c_3}}.
\end{align*}
Above, the first inequality holds using \eqref{eq:bar_y_star_bounds_c_2_c3} and the last inequality holds
since $mT \geq n$.
Therefore,
assuming that $mT \geq 2 \sqrt{\frac{c_3}{c_1}} \frac{1}{\tan^{-1}(64\sqrt{c_1/c_3})} n_1$,
\begin{align*}
\int_0^{T+1} \frac{c_1 x^2/T}{(c_1x^2/T + \bar{y}^*)^2} \,\mathrm{d} x &\geq
\frac{\tan^{-1}(64\sqrt{c_1/c_3})}{4\sqrt{c_1 c_3}} \frac{m T}{n_1}.
\end{align*}
Combining these inequalities,
assuming that $m \leq \frac{c_2}{2\sqrt{c_1c_3}} \tan^{-1}(64\sqrt{c_1/c_3}) n_1$, we have:
\begin{align}
m\sum_{t=1}^{T} \frac{\lambda_t}{(\lambda_t + \bar{y}^*)^2} \geq \frac{c_0}{c_1} m \left[ \frac{\tan^{-1}(64\sqrt{c_1/c_3})}{4\sqrt{c_1 c_3}} \frac{m T}{n_1} - \frac{m^2 T}{4c_2 n_1^2} \right] \geq \frac{c_0\tan^{-1}(64\sqrt{c_1/c_3})}{8 c_1^{3/2} \sqrt{c_3} } =: c_5 \frac{m^2 T}{n_1}. \label{eq:third_bound}
\end{align}
Next, we turn to upper bounding
$m\sum_{t=1}^{T} \frac{\lambda_t^2}{(\lambda_t + \bar{y}^*)^4}$.
Again the function $x \mapsto x^2/(x+\bar{y}^*)^4$
is increasing when $x \in [0, \bar{y}^*]$
and decreasing when $x \in (\bar{y}^*, \infty)$,
and therefore $x^2/(x + \bar{y}^*)^4 \leq \frac{1}{16 (\bar{y}^*)^2}$
for all $x \geq 0$.
Let $t^* \in \{0, \dots, T\}$ be such that
$c_0 t^2/T \leq \bar{y}^*$ for $t \in \{1, \dots, t^*\}$
and $c_0 t^2/T > \bar{y}^*$ for $t \in \{t^* + 1, \dots, T\}$.
In the case when $c_0/T > \bar{y}^*$, we set $t^* = 0$.
We have:
\begin{align*}
&~~~~m \sum_{t=1}^{T} \frac{\lambda_t^2}{(\lambda_t + \bar{y}^*)^4} \\
&\leq \frac{c_1^2}{c_0^2} m \sum_{t=1}^{T} \frac{ (c_0 t^2/T)^2 }{ ( c_0t^2/T + \bar{y}^* )^4 } \qquad\qquad\qquad\qquad\qquad~~~\text{using } \eqref{eq:lambda_bounds_c_0_c_1} \\
&= \frac{c_1^2}{c_0^2} m \left[ \sum_{t=1}^{t^*-1} \frac{ (c_0 t^2/T)^2 }{ ( c_0t^2/T + \bar{y}^* )^4 } + \sum_{t=t^*+2}^{T} \frac{ (c_0 t^2/T)^2 }{ ( c_0t^2/T + \bar{y}^* )^4 } + \frac{ (c_0 (t^*)^2/T)^2 }{ ( c_0(t^*)^2/T + \bar{y}^* )^4 } + \frac{ (c_0 (t^*+1)^2/T)^2 }{ ( c_0(t^*+1)^2/T + \bar{y}^* )^4 } \right] \\
&\leq \frac{c_1^2}{c_0^2} m \left[ \int_{1}^{t^*} \frac{ (c_0 x^2/T)^2 }{ ( c_0x^2/T + \bar{y}^* )^4 } \,\mathrm{d} x + \int_{t^*+1}^{T} \frac{ (c_0 x^2/T)^2 }{ ( c_0x^2/T + \bar{y}^* )^4 } \,\mathrm{d} x + \frac{ (c_0 (t^*)^2/T)^2 }{ ( c_0(t^*)^2/T + \bar{y}^* )^4 } + \frac{ (c_0 (t^*+1)^2/T)^2 }{ ( c_0(t^*+1)^2/T + \bar{y}^* )^4 } \right] \\
&\leq \frac{c_1^2}{c_0^2} m \left[ \int_{0}^{T}\frac{ (c_0 x^2/T)^2 }{ ( c_0x^2/T + \bar{y}^* )^4 } \,\mathrm{d} x + \frac{ (c_0 (t^*)^2/T)^2 }{ ( c_0(t^*)^2/T + \bar{y}^* )^4 } + \frac{ (c_0 (t^*+1)^2/T)^2 }{ ( c_0(t^*+1)^2/T + \bar{y}^* )^4 } \right] \\
&\leq \frac{c_1^2}{c_0^2} m \left[ \int_{0}^{T}\frac{ (c_0 x^2/T)^2 }{ ( c_0x^2/T + \bar{y}^* )^4 } \,\mathrm{d} x + \frac{1}{8 (\bar{y}^*)^2} \right] \qquad\qquad \text{since } \max_{x > 0} \frac{x^2}{(x + \bar{y}^*)^4} \leq \frac{1}{16(\bar{y}^*)^2} \\
&\leq \frac{c_1^2}{c_0^2} m \left[ \int_{0}^{T}\frac{ (c_0 x^2/T)^2 }{ ( c_0x^2/T + \bar{y}^* )^4 } \,\mathrm{d} x + \frac{1}{8c_2^2} \frac{m^4 T^2}{n_1^4} \right] .
\end{align*}
We now bound:
\begin{align*}
\int_{0}^{T}\frac{ (c_0 x^2/T)^2 }{ ( c_0x^2/T + \bar{y}^* )^4 } \,\mathrm{d} x &= c_0^2 T^2 \left[ \frac{(3c_0T + \bar{y}^*)(c_0T - 3\bar{y}^*)}{48 c_0^2 T \bar{y}_* (c_0 T + \bar{y}^*)^3} + \frac{\tan^{-1}\left(\sqrt{\frac{c_0 T}{\bar{y}^*}}\right)}{16 c_0^{5/2} T^{3/2} (\bar{y}^*)^{3/2}} \right] \\
&\leq c_0^2 T^2 \left[ \frac{1}{16c_0 \bar{y}^*(c_0 T + \bar{y}^*)^2} + \frac{\pi}{32 c_0^{5/2} T^{3/2} (\bar{y}^*)^{3/2}} \right] \\
&\leq c_0^2 T^2 \left[ \frac{1}{16c_0^3 \bar{y}^* T^2} + \frac{\pi}{32 c_0^{5/2} T^{3/2} (\bar{y}^*)^{3/2}} \right] \\
&\leq c_0^2 T^2 \left[ \frac{1}{16c_0^3 c_2} \frac{m^2}{n_1^2 T} + \frac{\pi}{32 c_0^{5/2} c_2^{3/2}} \frac{m^3}{n_1^3} \right] &&\text{using } \eqref{eq:bar_y_star_bounds_c_2_c3} \\
&\leq \left[ \frac{1}{1024 c_0 c_2} + \frac{\pi}{32 c_0^{1/2} c_2^{3/2}} \right] \frac{m^3 T^2}{n_1^3} &&\text{since } mT \geq n.
\end{align*}
Combining these inequalities, assuming that $m \leq n_1$:
\begin{align}
m\sum_{t=1}^{T} \frac{\lambda_t^2}{(\lambda_t + \bar{y}^*)^4} &\leq \frac{c_1^2}{c_0^2} \left[ \left[ \frac{1}{1024 c_0 c_2} + \frac{\pi}{32 c_0^{1/2} c_2^{3/2}} \right] \frac{m^4 T^2}{n_1^3} + \frac{1}{8c_2^2} \frac{m^5 T^2}{n_1^4} \right] \nonumber \\
&\leq \frac{c_1^2}{c_0^2} \left[\frac{1}{1024 c_0 c_2} + \frac{\pi}{32 c_0^{1/2} c_2^{3/2}} + \frac{1}{8c_2^2} \right] \frac{m^4 T^2}{n_1^3} &&\text{since } m \leq n_1 \nonumber \\
&=: c_6 \frac{m^4 T^2}{n_1^3}. \label{eq:fourth_bound}
\end{align}
Combining \eqref{eq:lower_tail},
\eqref{eq:third_bound}, and \eqref{eq:fourth_bound}
yields
\begin{align}
\Pr\left( p'(\bar{y}^*) < c_5 \frac{m^2 T}{n_1} - 2 \sqrt{t_\ell} \sqrt{c_6} \frac{m^2 T}{n_1^{3/2}} \right) \leq e^{-t_\ell} \:\forall t_\ell > 0. \label{eq:p_i_prime_lower_bound}
\end{align}
\paragraph{Lower bounds on $y^*$.}
We now combine \eqref{eq:p_i_upper_bound}
with \eqref{eq:p_i_prime_lower_bound} to established a lower
bound on $y^*$.
Equations \eqref{eq:y_i_lower_bound} and \eqref{eq:bar_y_star_bounds_c_2_c3} imply that:
\begin{align*}
y^* \geq \bar{y}^* - \frac{p(\bar{y}^*)}{p'(\bar{y}^*)} \geq \frac{c_2 n_1^2}{m^2 T} - \frac{p(\bar{y}^*)}{p'(\bar{y}^*)}.
\end{align*}
We first set $t_\ell = \frac{c_5^2}{16c_6} n_1$,
so that by \eqref{eq:p_i_prime_lower_bound},
\begin{align*}
\Pr\left( p'(\bar{y}^*) < \frac{c_5}{2} \frac{m^2 T}{n_1} \right) \leq e^{-\frac{c_5^2}{16c_6} n_1}.
\end{align*}
We next set $t_u = \beta n_1$ for a $\beta > 0$ to be specified.
By \eqref{eq:p_i_upper_bound},
\begin{align*}
\Pr\left( p(\bar{y}^*) > 2 (\sqrt{c_4 \beta} + \beta) n_1 \right) \leq e^{-\beta n_1}.
\end{align*}
Let $\mathcal{E}_2$ denote the event:
\begin{align*}
\mathcal{E}_2 := \left\{ p'(\bar{y}^*) \geq \frac{c_5}{2} \frac{m^2 T}{n_1} \right\} \cap \left\{ p(\bar{y}^*) \leq 2(\sqrt{c_4 \beta} + \beta)n_1 \right\}.
\end{align*}
By a union bound, $\Pr(\mathcal{E}_2^c) \leq e^{-\frac{c_5^2}{16c_6}n_1} + e^{-\beta n_1}$.
Furthermore,
\begin{align*}
\mathbf{1}\{ \mathcal{E}_2 \} \left[ \frac{c_2 n_1^2}{m^2 T} - \frac{p(\bar{y}^*)}{p'(\bar{y}^*)} \right] \geq \mathbf{1}\{\mathcal{E}_2\}\left[ c_2 - \frac{4(\sqrt{c_4 \beta} + \beta) }{c_5} \right] \frac{n_1^2}{m^2 T}.
\end{align*}
Setting $\beta = c_7 := \min\left\{ \frac{c_2c_5}{16}, \frac{c_2^2 c_5^2}{16^2 c_4} \right\}$, we have that
$c_2 - \frac{4(\sqrt{c_4 \beta} + \beta) }{c_5} \geq c_2/2$,
and therefore from \eqref{eq:y_i_lower_bound},
\begin{align}
\mathbf{1}\{ \mathcal{E}_{1} \} y^* \geq \mathbf{1}\{ \mathcal{E}_{1} \cap \mathcal{E}_2 \} \left[ \frac{c_2 n_1^2}{m^2 T} - \frac{p(\bar{y}^*)}{p'(\bar{y}^*)} \right] \geq \mathbf{1}\{ \mathcal{E}_{1} \cap \mathcal{E}_2\} \frac{c_2}{2} \frac{n_1^2}{m^2 T}. \label{eq:final_bar_y_star_lower_bound}
\end{align}
\paragraph{Finishing the proof.}
Define $\mathcal{E} := \mathcal{E}_1 \cap \mathcal{E}_2$ and
define $\underline{y}^* := \frac{c_2}{2} \frac{n_1^2}{m^2 T}$.
By a union bound,
\begin{align}
\Pr(\mathcal{E}) &\leq e^{-n/128} + e^{-mT/16} + e^{-\frac{c_5^2}{16c_6}n_1} + e^{-c_7 n_1} \nonumber \\
&\leq e^{-n/128} + e^{-n/16} + e^{-\frac{c_5^2}{1024 c_6} n} + e^{-\frac{c_7}{64} n} &&\text{since } mT \geq n \nonumber \\
&\leq 4 \exp\left(- \min\left\{ \frac{1}{128}, \frac{1}{16}, \frac{c_5^2}{1024 c_6} , \frac{c_7}{64}\right\} n \right) =: 4 e^{-c_8 n}. \label{eq:final_union_bound}
\end{align}
From \eqref{eq:bound_on_Z_i_good_v2},
since $y^* \geq \underline{y}^*$ on $\mathcal{E}$ by \eqref{eq:final_bar_y_star_lower_bound},
\begin{align}
\mathbf{1}\{ \mathcal{E} \} Z_i \leq \mathbf{1}\{ \mathcal{E} \} \frac{1}{\lambda_i + y^*}
\leq \frac{1}{\lambda_i + \underline{y}^*}. \label{eq:final_Z_i_E_i_bound}
\end{align}
Next, by \Cref{prop:invert_log_t_over_t},
if $n \geq 2 \max\{1, c_8^{-1}\} \log(4 \max\{1, c_8^{-1}\})$,
then we have
\begin{align*}
n \geq c_8^{-1} \log{n} \Longleftrightarrow n e^{-c_8 n} \leq 1.
\end{align*}
We now bound,
\begin{align*}
\sum_{i=1}^{T} \mathbb{E}[Z_i] &= \sum_{i=1}^{T} \left[\mathbb{E}[\mathbf{1}\{\mathcal{E}\} Z_i] + \mathbb{E}[\mathbf{1}\{\mathcal{E}^c\} Z_i]\right] \\
&\leq \sum_{t=1}^{T} \left[ \frac{1}{\lambda_t + \underline{y}^*} + \Pr(\mathcal{E}^c) (\Theta_{1,T,T})_{tt} \right] &&\text{using } \eqref{eq:final_Z_i_E_i_bound} \text{ and } Z_i \leq (\Theta_{1,T,T})_{ii} \\
&= \sum_{t=1}^{T} \frac{1}{\lambda_t + \underline{y}^*} + \Pr(\mathcal{E}^c) T &&\text{since } \Tr(\Theta_{1,T,T}) = T \\
&\leq \sum_{t=1}^{T} \frac{1}{c_0 t^2/T + \underline{y}^*} + 4 T e^{-c_8 n} &&\text{using } \eqref{eq:lambda_bounds_c_0_c_1} \text{ and } \eqref{eq:final_union_bound} \\
&\leq \int_0^T \frac{1}{c_0 x^2/T + \underline{y}^*} \,\mathrm{d} x + 4 T e^{-c_8 n} \\
&\leq \frac{\pi}{2}\sqrt{\frac{T}{c_0 \underline{y}^*}} + 4 T e^{-c_8 n}
= \frac{\sqrt{2}\pi}{2\sqrt{c_0c_2}} \frac{m T}{n_1} + 4 T e^{-c_8 n} \\
&\leq \left[\frac{\sqrt{2}\pi}{2\sqrt{c_0c_2}} + \frac{1 }{16} \right] \frac{m T}{n_1} =: c_8 \frac{mT}{n_1} &&\text{since } n e^{-c_8 n} \leq 1 \text{ and } m \geq 1.
\end{align*}
Plugging this upper bound into \eqref{eq:lower_bound_with_sum_Zis_inverse}:
\begin{align*}
\mathsf{R}(m,T,{T'}; \{\PxA{I_n}\}) \geq \sigma_\xi^2 p \cdot \frac{{T'}}{2T} \cdot \frac{n}{2m} \cdot \frac{1}{c_8} \frac{n_1}{mT} = \frac{1}{256c_8} \sigma_\xi^2 \cdot \frac{pn^2}{m^2 T} \cdot \frac{{T'}}{T}.
\end{align*}
The claim now follows.
\end{proof}
\subsection{Proof of \Cref{stmt:lower_bound_main}}
\lowerboundmain*
\begin{proof}
Let $\mathcal{P}_x := \{\PxA{0_{n \times n}}, \PxA{I_n}\}$.
We let $c'_0$, $c'_1$, $c'_2$, and $c'_3$ denote the universal positive
constants in the statement of \Cref{thm:lower_bound_r_equals_1}.
We first invoke \Cref{stmt:minimax_jensen_rate} to conclude that:
\begin{align}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) \geq \frac{\sigma_\xi^2}{2}
\cdot \frac{pn}{mT}
\cdot \max \left\{ \frac{{T'}}{T}, 1 \right\}. \label{eq:baseline_risk}
\end{align}
The proof now proceeds in three cases:
\paragraph{Case $n{T'}/(mT) \leq 1$.}
In this case, we trivially have
$\max\left\{ \frac{{T'}}{T}, 1 \right\} = \max\left\{ \frac{n{T'}}{mT}, \frac{{T'}}{T}, 1 \right\}$.
Therefore, \eqref{eq:baseline_risk} yields:
\begin{align*}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) \geq \frac{\sigma_\xi^2}{2}
\cdot \frac{pn}{mT}
\cdot \max\left\{ \frac{n{T'}}{mT}, \frac{{T'}}{T}, 1 \right\}.
\end{align*}
\paragraph{Case $n{T'}/(mT) > 1$ and $m \leq c_2' n$.}
In this case, we can invoke \Cref{thm:lower_bound_r_equals_1} to conclude that:
\begin{align}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) \geq c_3' \sigma_\xi^2 \cdot \frac{pn}{mT} \cdot \frac{n{T'}}{mT} = c_3' \sigma_\xi^2 \cdot \frac{pn}{mT} \cdot \max\left\{ \frac{n{T'}}{mT}, 1 \right\}. \label{eq:second_case}
\end{align}
Since $n/m \geq 1/c_2'$, we have that
$n{T'}/(mT) \geq {T'}/(c_2' T)$.
Therefore:
\begin{align*}
\max\left\{ \frac{n{T'}}{mT}, 1 \right\} = \max\left\{ \frac{n{T'}}{mT}, \frac{{T'}}{c_2' T}, 1 \right\} \geq \min\{1,1/c_2'\} \max\left\{ \frac{n{T'}}{mT}, \frac{{T'}}{T}, 1 \right\}.
\end{align*}
Hence, from \eqref{eq:second_case},
\begin{align*}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) \geq \min\{c_3',c_3'/c_2'\} \sigma_\xi^2 \cdot \frac{pn}{mT} \cdot \max\left\{ \frac{n{T'}}{mT}, \frac{{T'}}{T}, 1 \right\}.
\end{align*}
\paragraph{Case $n{T'}/(mT) > 1$ and $m > c_2' n$.}
In this case, we have
${T'}/T > c_2' n {T'}/(mT)$.
Therefore, we have:
\begin{align*}
\max\left\{ \frac{{T'}}{T}, 1 \right\} = \max\left\{ c_2' \frac{n{T'}}{mT}, \frac{{T'}}{T}, 1 \right\} \geq \min\{1, c_2'\} \max\left\{ \frac{n{T'}}{mT}, \frac{{T'}}{T}, 1 \right\}.
\end{align*}
Hence, from \eqref{eq:baseline_risk},
\begin{align*}
\mathsf{R}(m,T,{T'};\mathcal{P}_x) \geq \min\{1/2, c_2'/2\} \sigma_\xi^2
\cdot \frac{pn}{mT}
\cdot \max\left\{ \frac{n{T'}}{mT}, \frac{{T'}}{T}, 1 \right\}.
\end{align*}
The claim now follows taking
$c_0 = c_0'$, $c_1 = c_1'$, and
$c_2 = \min\{1/2, c_3', c_3'/c_2', c_2'/2\}$.
\end{proof}
\section*{Acknowledgements}
We thank Vikas Sindhwani for organizing a lecture series on
learning and control, during which this work was prompted,
and for encouraging us to pursue the ensuing questions.
We also thank Sumeet Singh for pointing out
that the claimed first-order optimality conditions
in \eqref{eq:critical_point_one} and \eqref{eq:critical_point_two}
require further conditions to hold for nonconvex/nonconcave games, e.g.,\ strictly concave stationary points of the inner maximization problem.
M.\ Soltanolkotabi is supported by the Packard Fellowship in Science and Engineering, a Sloan Research Fellowship in Mathematics, an NSF-CAREER under award \#1846369, DARPA Learning with Less Labels (LwLL) and FastNICS programs, and NSF-CIF awards \#1813877 and \#2008443.
\bibliographystyle{alpha}
\section{Problem formulation}
\label{sec:problem}
\paragraph{Notation.}
The real eigenvalues of a Hermitian matrix $M \in \ensuremath{\mathbb{C}}^{k \times k}$
are
$\lambda_{\max}(M) = \lambda_1(M) \ge \dots \ge \lambda_k(M) = \lambda_{\min}(M)$.
For a square matrix $M \in \ensuremath{\mathbb{C}}^{k \times k}$,
$M^*$ denotes its conjugate transpose, and
$\rho(M)$ denotes its spectral radius: $\rho(M) = \max\{ \abs{\lambda} \mid \lambda \text{ is an eigenvalue of } M \}$.
The space of $n \times n$ real-valued symmetric positive semidefinite
(resp.\ positive definite) matrices is denoted $\mathsf{Sym}^n_{\geq 0}$
(resp.\ $\mathsf{Sym}^n_{> 0}$).
The non-negative (resp.\ positive) orthant in $\ensuremath{\mathbb{R}}^n$ is denoted as
$\ensuremath{\mathbb{R}}^n_{\geq 0}$ (resp.\ $\ensuremath{\mathbb{R}}^n_{> 0}$), and
$\mathbb{S}^{n-1}$ denotes the unit sphere in $\ensuremath{\mathbb{R}}^n$.
Finally, the set of positive integers is denoted
by $\ensuremath{\mathbb{N}}_{+}$.
\subsection{Linear regression from sequences}
\label{sec:problem:regression}
\paragraph{Regression model.}
A \emph{covariate sequence} is an indexed set $\{x_t\}_{t \ge 1} \subset \ensuremath{\mathbb{R}}^n$.
Any distribution $\mathsf{P}_x$ over covariate sequences is assumed to have bounded second moments,
i.e., that $\mathbb{E}[x_t x_t^\mathsf{T}]$ exists and is finite for all $t \ge 1$.
Also for such a distribution $\mathsf{P}_x$, let
$\mathsf{P}_{\xi}[\mathsf{P}_x]$ be a distribution over
\emph{observation noise} sequences $\{\xi_t\}_{t \ge 1} \subset \ensuremath{\mathbb{R}}^p$.
Denoting by $\{\mathcal{F}_t\}_{t \geq 0}$ the filtration
with $\mathcal{F}_t = \sigma(\{x_k\}_{k=1}^{t+1}, \{\xi_k\}_{k=1}^{t})$,
we assume that
$\{\xi_t\}_{t \geq 1}$ is a $\sigma_\xi$-sub-Gaussian martingale difference sequence (MDS), i.e.,
for $t \geq 1$:
\begin{align*}
\mathbb{E}[ \ip{v}{\xi_t} \mid \mathcal{F}_{t-1}] = 0, \quad
\mathbb{E}[ \exp(\lambda \ip{v}{\xi_t}) \mid \mathcal{F}_{t-1} ] \leq \exp(\lambda^2 \norm{v}_2^2\sigma_\xi^2/2) \:\:\text{a.s.}\:\: \forall \lambda \in \ensuremath{\mathbb{R}}, v \in \ensuremath{\mathbb{R}}^{p}.
\end{align*}
Given a \emph{ground truth model} $W_\star \in \ensuremath{\mathbb{R}}^{p \times n}$,
define the \emph{observations} (a.k.a.\ ``responses'' or ``labels''):
\begin{align}
y_t = W_\star x_t + \xi_t, \quad t \geq 1. \label{eq:y_observation_model}
\end{align}
Denote by $\mathsf{P}_{x,y}^{W_\star}[\mathsf{P}_x,\mathsf{P}_{\xi}]$ the joint distribution
over covariates and their observations $\{(x_t, y_t)\}_{t \ge 1}$.
\paragraph{Regression task.}
Fix a ground truth model $W_\star \in \ensuremath{\mathbb{R}}^{p \times n}$,
a covariate distribution $\mathsf{P}_x$,
an observation noise model $\mathsf{P}_{\xi}$,
a training horizon $T$, and a test horizon ${T'}$.
Draw $m$ independent sequences $\{(x_t^{(i)}, y_t^{(i)})\}_{i \in [m], t \ge 1}$
from $\mathsf{P}_{x,y}^{W_\star}[\mathsf{P}_x,\mathsf{P}_{\xi}]$,
and call their length-$T$ prefixes
$\{(x_t^{(i)}, y_t^{(i)})\}_{i=1,t=1}^{m,T}$ the training \emph{examples}.
From these examples, the regression task is to find a hypothesis
$\hat{f}_{m,T} : \ensuremath{\mathbb{R}}^{n} \rightarrow \ensuremath{\mathbb{R}}^{p}$
that matches ground truth predictions $f_{W_\star}(x) := W_\star x$
in expectation over unseen trajectories of length ${T'}{}$.
Specifically, the excess \emph{risk} of a hypothesis $\hat{f}$ is:
\begin{align}
L(\hat{f}; {T'}, \mathsf{P}_x) :=
\mathbb{E}_{\mathsf{P}_x} \left[
\frac{1}{{T'}} \sum_{t=1}^{{T'}} \norm{ \hat{f}(x_t) - f_{W_\star}(x_t) }^2_2 \right]. \label{eq:risk_def}
\end{align}
We say that the evaluation horizon ${T'}$ is \emph{strict} if ${T'} \le T$
and \emph{extended} if ${T'} > T$.
When the hypothesis class is linear, meaning the hypotheses $\hat{f}$ are of the
form $\hat{f}(x) = \hat{W} x$ with $\hat{W} \in \ensuremath{\mathbb{R}}^{p \times n}$,
the risk expression \eqref{eq:risk_def} simplifies as follows.
For a positive definite matrix $\Sigma \in \ensuremath{\mathbb{R}}^{n \times n}$,
define the weighted square norm
$\norm{M}_\Sigma^2 := \Tr( M \Sigma M^\mathsf{T} )$ for $M \in \ensuremath{\mathbb{R}}^{p \times n}$.
Denoting, for $t \ge 1$:
\begin{align}
\Sigma_t(\mathsf{P}_x) := \mathbb{E}_{\mathsf{P}_x}[x_t x_t^\mathsf{T}], \quad\quad
\Gamma_{t}(\mathsf{P}_x) := \frac{1}{t} \sum_{k=1}^{t} \Sigma_k(\mathsf{P}_x),
\label{eq:covariance-def}
\end{align}
we overload notation and write:
\begin{align}
L(\hat{W}; {T'}, \mathsf{P}_x) = \norm{\hat{W} - W_\star}_{\Gamma_{{T'}}(\mathsf{P}_x)}^2.
\label{eq:risk_linear}
\end{align}
The risk \eqref{eq:risk_def}, being a notion of error averaged over time steps,
relates to that of \cite{ziemann2022single} in the study of learning dynamics
(the difference lies in whether the error norm is squared).
By allowing unequal training and test horizons $T \ne {T'}$, we cover two
related scenarios at once:
system identification in linear dynamical systems (when ${T'} = 1$) and
predicting past the end of a sequence (when ${T'} > T$).
For the latter, the risk definition \eqref{eq:risk_def}
is closely related to a commonly studied notion of ``final step''
generalization (see e.g.\ \cite[Eq.~5]{kuznetsov2017mixing}, \cite[Def.~10]{mcdonald2017timeseries})
that measures the performance of a hypothesis at ${T'} - T$ time steps beyond the training horizon:
$L_{\mathrm{end}}(\hat{f}; {T'}, \mathsf{P}_x) := \mathbb{E}_{\mathsf{P}_x}[ \norm{\hat{f}(x_{{T'}}) - f_{W_\star}(x_{{T'}})}_2^2 ]$.
Linear hypotheses enjoy the identity $L_{\mathrm{end}}(\hat{W}; {T'}, \mathsf{P}_x) = \norm{\hat{W}-W_\star}_{\Sigma_{{T'}}(\mathsf{P}_x)}^2$.
In turn:
\begin{align*}
L_{\mathrm{end}}(\hat{W};{T'},\mathsf{P}_x) \geq L(\hat{W};{T'},\mathsf{P}_x) \gtrsim L_{\mathrm{end}}(\hat{W};\floor{{T'}/2},\mathsf{P}_x).
\end{align*}
In other words,
provided the scale of the covariances $\Sigma_t(\mathsf{P}_x)$ does not grow
substantially over time $t$,
our risk definition $L$ is comparable to the final-step risk $L_{\mathrm{end}}$.
\paragraph{Minimax risk.}
To compare the hardness of learning across problem classes
(i.e., families of covariate distributions $\mathsf{P}_x$),
we measure the \emph{minimax rate} of the risk $L$---i.e.,
the behavior of the best estimator's worst-case risk
over valid problem instances---as a function of
the amount of training data $m, T$ and
other problem parameters such as $n$, $p$, $\sigma_\xi$, and ${T'}$.
Recall that $\mathsf{P}_{x,y}^{W_\star}$ %
denotes the distribution over labeled trajectories $\{(x_t, y_t)\}_{t \geq 1}$.
For a collection of covariate sequence distributions $\calP_x$,
the minimax risk over problem instances consistent with $\calP_x$ is:
\begin{align}
\mathsf{R}(m, T, {T'}; \calP_x) :=
\inf_{\mathsf{Alg}}
\sup_{\mathsf{P}_x \in \calP_x}
\sup_{W_\star, \mathsf{P}_{\xi}}
\mathbb{E}_{\otimes_{i=1}^{m} \mathsf{P}_{x,y}^{W_\star}[\mathsf{P}_x,\mathsf{P}_{\xi}]}
\left[ L \left(
\mathsf{Alg}(\{ (x_t^{(i)}, y_t^{(i)} ) \}_{i=1,t=1}^{m,T}); {T'}, \mathsf{P}_x \right) \right],
\label{eq:minimax_risk_def}
\end{align}
where the infimum ranges over estimators $\mathsf{Alg} : (\ensuremath{\mathbb{R}}^{n} \times \ensuremath{\mathbb{R}}^{p})^{mT} \to (\ensuremath{\mathbb{R}}^n \to \ensuremath{\mathbb{R}}^p)$
that map training samples to hypotheses,
the supremum
over $W_\star$ is over all $p \times n$ ground truth models, and
the supremum
over $\mathsf{P}_\xi$ is over all $\sigma_\xi$-sub-Gaussian MDS
processes determining the observation noise.
\paragraph{The ordinary least-squares estimator.}
Much like its classical role in iid{} learning, the
\emph{ordinary least-squares} (OLS) estimator will
be key to bounding the minimax risk \eqref{eq:minimax_risk_def}
from above.
We define the OLS estimator to be the linear hypothesis
$\hat{W}_{m,T} \in \ensuremath{\mathbb{R}}^{p \times n}$ that satisfies:
\begin{align}
\hat{W}_{m,T} \in \argmin_{W \in \ensuremath{\mathbb{R}}^{p \times n}} \sum_{i=1}^{m} \sum_{t=1}^{T} \norm{ W x_t^{(i)} - y_t^{(i)} }_2^2. \label{eq:ols_definition}
\end{align}
For $i=1, \dots, m$, let $X^{(i)}_{m,T} \in \ensuremath{\mathbb{R}}^{T \times n}$ be the data matrix for the $i$-th trajectory
(i.e., the $t$-th row of $X^{(i)}_{m,T}$ is $x_t^{(i)}$ for $t = 1, \dots, T$).
Define $Y^{(i)}_{m,T} \in \ensuremath{\mathbb{R}}^{T \times p}$ and $\Xi^{(i)}_{m,T} \in \ensuremath{\mathbb{R}}^{T \times p}$
analogously.
Put $X_{m,T} \in \ensuremath{\mathbb{R}}^{ mT \times n}$ as the vertical concatenation of $X_{m,T}^{(1)}, \dots, X_{m,T}^{(m)}$,
and similarly for $Y_{m,T} \in \ensuremath{\mathbb{R}}^{mT \times p}$ and $\Xi_{m,T} \in \ensuremath{\mathbb{R}}^{mT \times p}$.
Whenever $X_{m,T}$ has full column rank,
then we can write $\hat{W}_{m,T}$ as:
\begin{align}
\hat{W}_{m,T} = Y_{m,T}^\mathsf{T} X_{m,T} (X_{m,T}^\mathsf{T} X_{m,T})^{-1}. \label{eq:W_hat_expr}
\end{align}
\subsection{Problem classes}
\label{sec:prob:general-problems}
\input{problem-figure}
We formalize linear regression from sequential data generally as follows:%
\begin{myprob}[\problemName{Seq\smallDash{}LS}{}]
\label{prob:dsr}
Assume a covariate sequence distribution $\mathsf{P}_x$
in the linear regression model \eqref{eq:y_observation_model}.
Fix an evaluation horizon ${T'}$.
On input $m$ labeled trajectories of length $T$
drawn from this model,
in the form of examples $\{(x_t^{(i)}, y_t^{(i)})\}_{i=1,t=1}^{m,T}$,
output a hypothesis $\hat{f}_{m,T}$
that minimizes excess risk $L(\hat{f}_{m,T}; {T'}, \mathsf{P}_x)$.
\end{myprob}
\noindent
Our topmost goal is to study the effect of learning from sequentially
dependent covariates \emph{in comparison with} learning in the classical iid{} setup.
Linear regression is well understood in the latter setting.
Focusing on \emph{well-specified} linear regression further simplifies our presentation,
allowing us to isolate the effects of what interests us most---dependent covariates.
Generalizing the supervision aspect of \problemName{Seq\smallDash{}LS}{}
(say, to unrealizable and non-parametric regression, or to classification)
is left to future work. We return to discuss this in~\Cref{sec:conclusion}.
To study how dependent data affects learning,
we need to establish an ``independent data'' baseline.
The natural comparison point for \problemName{Seq\smallDash{}LS}{} is to remove all correlations across
time. Namely, instead of drawing covariates sequentially from the distribution $\mathsf{P}_x$,
consider learning separately from the marginals of $\mathsf{P}_x$ at each time step.
The resulting decorrelated distribution generates \emph{independent} examples,
but typically not iid{} ones.
We formalize linear regression from independent data generally as follows:
\begin{myprob}[\problemName{Ind\smallDash{}Seq\smallDash{}LS}{}]
\label{prob:psr}
Fix a sequence of distributions $\{\Pxt{t}\}_{t \ge 1}$.
Consider their product over time $\otimes_{t \ge 1} \Pxt{t}$
as the covariate sequence distribution in the linear regression model~\eqref{eq:y_observation_model}.
Fix an evaluation horizon ${T'}$.
On input $m$ labeled trajectories of length $T$
drawn from this model,
in the form of examples $\{(x_t^{(i)}, y_t^{(i)})\}_{i=1,t=1}^{m,T}$,
output a hypothesis $\hat{f}_{m,T}$
that minimizes $L(\hat{f}_{m,T}; {T'}, \otimes_{t \ge 1} \Pxt{t})$.
\end{myprob}
\noindent
This \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} problem generalizes the canonical iid{} learning setup slightly.
Existing theory can still characterize its minimax risk, provided the
covariances of the distributions $\{\Pxt{t}\}$ are roughly equal in scale across time $t$.
However, this equal-scale requirement rules out the marginals of interesting applications,
such as dynamical systems that are not stable or ergodic.
We therefore extend, in later sections, characterizations of the regression risk to
handle covariances that can scale \emph{polynomially} across time instead.
\subsection{Problem separations}
\label{sec:prob:separations}
To set up a baseline for a \problemName{Seq\smallDash{}LS}{} problem, we will specifically instantiate \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} over its marginals.
Namely, for a sequence distribution $\mathsf{P}_x$ over $\{x_t\}_{t \ge 1}$,
let $\marginal{\mathsf{P}_x}{t}$ be the marginal distribution of $x_t$ at time $t \ge 1$,
and consider \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} with covariates drawn from the sequence $\{ \marginal{\mathsf{P}_x}{t} \}_{t \ge 1}$.
\Cref{fig:problem-schematic} illustrates such a \problemName{Seq\smallDash{}LS}{} problem
and the corresponding \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} instance over its marginals.
This decorrelated baseline is a hypothetical benchmark:
in a practical context, collecting independent marginal data,
when nature only supplies its dependent form, can be expensive or infeasible.
However, we can expect that having such data on hand would make learning easier,
with risk rates that resemble iid{} learning.
In what follows, we outline scenarios where a sequential learning problem
and its decorrelated baseline coincide in difficulty, and others in which they diverge.
We then outline the possible assumptions that would allow us to always relate the two.
\paragraph{The iid{} special case.}
When $T = {T'} = 1$,
the example trajectories $\{x_1^{(i)}\}_{i=1}^m$ are trivially a set of iid{} covariates.
The problems \problemName{Seq\smallDash{}LS}{} and \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} thus coincide, and reduce to the well-specified
random design linear regression problem over $m$ iid{}
covariates.
It is well-known that under iid{} data, and mild regularity conditions, the
minimax risk scales as
$\sigma_\xi^2 pn/m$,
and is achieved by the OLS
estimator~\citep{hsu14randomdesign,mourtada19exactminimax,wainwright2019book}.
\paragraph{Extending the horizon.}
Considering nontrivial horizons $T = {T'} > 1$, both
\problemName{Seq\smallDash{}LS}{} and its corresponding \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} baseline become more involved,
but for different reasons.
The \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} problem, as we show in \Cref{sec:results:lower_bounds},
is not generally learnable with polynomially many examples. Specifically,
the minimax rate scales exponentially in the dimension $n$
provided the trajectory count $m$ is constant.
To address this, we will require that the covariances
of its constituent distributions $\{\Pxt{t}\}$ grow at most polynomially with time $t$.
Under this constraint,
the problem's minimax risk again scales as the iid{}-like rate $\sigma_\xi^2 pn/(mT)$
times, at most, a factor determined exponentially by the covariance growth.
The \problemName{Seq\smallDash{}LS}{} problem inherits the same growth limitation.
Even then, it is still not generally learnable without further assumptions
on the dependence structure of covariates:
the minimax risk %
is otherwise bounded away from zero as the horizon $T$ tends to infinity,
provided the trajectory count $m$ is constant.
To realize this, consider $x_1 \sim N(0, I_n)$ and $x_{t} = x_{t-1}$ for $t \ge 2$,
a sequence of identical covariates whose marginals are all independent Gaussians.
The resulting dataset presents an underdetermined regression problem if $m < n$.
In essence, its covariates lack sufficient ``excitation'' across time.
To rein \problemName{Seq\smallDash{}LS}{} back in to the realm of learnability, one must:
\begin{enumerate}[label=(\alph*)]
\item make further modeling assumptions about covariates, or
\item introduce excitation via independent resets.
\end{enumerate}
For (a), as detailed in~\Cref{sec:related},
the most common modeling assumption considers sequences that mix rapidly to a stationary distribution.
Another avenue---recently active in the literature, and sometimes overlapping with the
mixing approach---considers sequences generated by linear dynamical systems.
Among these two, %
mixing implies risk bounds that tend to zero with $T$,
but only hold in the worst case after a burn-in time that scales proportionally to the mixing time
\citep{bresler2020leastsquaresmarkov}.
This prevents a characterization of minimax risk uniformly across the full range of
problem instances $\mathsf{P}_x$ that mix, %
unless one caps the mixing time to a fixed constant.
Narrowing instead to LDS models in the sequel,
we manage to succinctly carve out a basic problem
family, with \emph{unbounded} mixing time, and to characterize
its minimax risk uniformly.
One still pays a price for sequential dependency,
as this minimax risk turns out to be larger than its \problemName{Ind\smallDash{}Seq\smallDash{}LS}{}
counterpart by a factor of the dimension $n$.
Turning in addition to (b), by introducing (sufficiently many) resets,
we can expand our data model substantially:
we manage to lift most of our LDS assumptions and extend to other dynamical systems.
Remarkably, we even show that for any controllable LDS---including ones
that are unstable and hence grow exponentially in time---having sufficiently many resets guarantees
that the risk exhibits, once again, the iid{}-like behavior of
$\sigma_\xi^2 pn/(mT)$, up to mere constants.
\subsection{Linear dynamical trajectories}
\label{sec:problem:lds-trajectories}
Fix a \emph{dynamics matrix} $A \in \ensuremath{\mathbb{R}}^{n \times n}$
and a \emph{control matrix} $B \in \ensuremath{\mathbb{R}}^{n \times d}$.
Consider the $n$-dimensional trajectory $\{x_t\}_{t \ge 1}$
defined by the linear dynamical system:
\begin{align}
x_t= A x_{t-1} + B w_t,
\:\: \text{where } w_t \sim N(0, I_d),
\:\: \text{for } t \ge 1,
\label{eq:lds_definition}
\end{align}
taking $x_0 = 0$ by convention.
We assume that the noise process $\{w_t\}_{t \ge 1}$ is independent across time, i.e.,
that $w_t \perp w_{t'}$ whenever $t \neq t'$.
Overloading notation,
let the matrix $\Sigma_t(A, B) := \sum_{k=0}^{t-1} A^k BB^\mathsf{T} (A^k)^\mathsf{T}$
denote the covariance of $x_t$,
and let the matrix $\Gamma_t(A, B) := \frac 1 t \sum_{k=1}^t \Sigma_k(A, B)$
denote the average covariance.
Denote by $\PxA{A,B}$ the distribution over the trajectory $\{x_t\}_{t \ge 1}$,
and let $\{x_t^{(i)}\}_{t \ge 1}$ for $i \ge 1$ denote independent draws from $\PxA{A,B}$.
When $B=I_n$, we use the respective shorthand notation
$\Sigma_t(A)$, $\Gamma_t(A)$, and $\mathsf{P}_x^A$.
Modeling regression covariates as linear dynamical trajectories gives us the \problemName{LDS\smallDash{}LS}{} problem,
a specialization of \problemName{Seq\smallDash{}LS}{} (\Cref{prob:dsr}):
\begin{myprob}[\problemName{LDS\smallDash{}LS}{}]
Assume a dynamics matrix $A \in \ensuremath{\mathbb{R}}^{n \times n}$,
a control matrix $B \in \ensuremath{\mathbb{R}}^{n \times d}$,
and a corresponding linear dynamical covariate distribution $\PxA{A,B}$
in the linear regression model \eqref{eq:y_observation_model}.
Fix an evaluation horizon ${T'}$.
On input $m$ labeled trajectories of length $T$,
drawn from this model,
in the form of examples $\{(x_t^{(i)}, y_t^{(i)})\}_{i=1,t=1}^{m,T}$,
output a hypothesis $\hat{f}_{m,T}$
that minimizes $L(\hat{f}_{m,T}; {T'}, \PxA{A,B})$.
\end{myprob}
Let $\PxAt{A,B}{t}$ be the marginal distribution of $x_t$ under $\PxA{A,B}$ at each $t \ge 1$.
The natural decorrelated baseline for \problemName{LDS\smallDash{}LS}{}
is a corresponding specialization of \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} (\Cref{prob:psr}) to LDS trajectories:%
\begin{myprob}[\problemName{Ind\smallDash{}LDS\smallDash{}LS}{}]
Assume a dynamics matrix $A \in \ensuremath{\mathbb{R}}^{n \times n}$,
a control matrix $B \in \ensuremath{\mathbb{R}}^{n \times d}$,
and a corresponding trajectory distribution $\PxA{A,B}$.
Consider covariates drawn independently from its marginals,
i.e., assume the linear regression model \eqref{eq:y_observation_model}
under the covariate sequence distribution $\otimes_{t \geq 1} \PxAt{A,B}{t}$.
Fix an evaluation horizon ${T'}$.
On input $m$ labeled trajectories of length $T$,
drawn from this model,
in the form of examples $\{(x_t^{(i)}, y_t^{(i)})\}_{i=1,t=1}^{m,T}$,
output a hypothesis $\hat{f}_{m,T}$
that minimizes $L(\hat{f}_{m,T}; {T'}, \otimes_{t \geq 1} \PxAt{A,B}{t})$.
\end{myprob}
\paragraph{Learning dynamical systems.}
\DLR{} generalizes \emph{linear system identification},
the problem of recovering the dynamics $A$ from data.
The reduction follows by setting $W_\star = A$
and $\xi_t^{(i)} = B w_{t+1}^{(i)}$,
so that $y_t^{(i)} = x_{t+1}^{(i)}$.
Note that when $B$ has full row rank,
the squared parameter error in the weighted $BB^\mathsf{T}$ norm $\norm{\cdot}_{BB^\mathsf{T}}$
is simply the risk $L(\hat{A};{T'}, \PxA{A,B})$ when ${T'} = 1$.
Recent related work typically assumes that $B$ indeed has full row rank,
but in later sections we touch on the more general case where
this is not required, so long as the pair $(A, B)$ is controllable.
Bounds in operator norm are also easily obtainable
from our proof techniques.
However, our lower bounds will not inform the system identification
problem specifically;
our hardness results rely on decoupling $W_\star$ from $A$ and
$\xi_t^{(i)}$ from $w_{t+1}^{(i)}$, whereas this reduction naturally ties them.
\section{Key proof ideas}
\label{sec:proof_ideas}
In this section, we highlight some of the key ideas
behind our results. Proofs of the upper bounds
are in \Cref{sec:appendix:upper_bound_proofs},
and proofs of the lower bounds are in \Cref{sec:appendix:lower_bounds}.
\paragraph{Additional notation.}
For $r \in \mathbb{N}_+$ and $M \in \ensuremath{\mathbb{R}}^{n \times n}$,
let $J_r \in \ensuremath{\mathbb{R}}^{r \times r}$ denote the
Jordan block of size $r$ with ones along its diagonal,
let $\mathsf{BDiag}(M, r) \in \ensuremath{\mathbb{R}}^{nr \times nr}$ denote the block
diagonal matrix with diagonal blocks $M$,
and let
$\mathsf{BToep}(M, r) \in \ensuremath{\mathbb{R}}^{nr \times nr}$ denote the block Toeplitz matrix with first column
$(I_n, M^\mathsf{T}, \dots, (M^{r-1})^\mathsf{T})^\mathsf{T}$.
\subsection{Upper bounds}
The proof of
\Cref{stmt:upper_bound_general}
decomposes the risk using a standard basic inequality,
which we now describe.
While \Cref{stmt:upper_bound_general} is stated quite generally, for simplicity of exposition we
restrict ourselves in this section to the case when
the matrix parameters $\{ \Psi_j\}_{j=1}^{S}$ in \Cref{def:trajectory_small_ball} are all set to $\underline{\Gamma}$.
Under this simplification, we have that
$\underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma}) = 1$.
Equation~\eqref{eq:y_observation_model} yields the identity
$Y_{m,T} = X_{m,T} W_\star^\mathsf{T} + \Xi_{m,T}$.
Plugging this relationship into the formula~\eqref{eq:ols_definition} for $\hat{W}_{m,T}$
gives $\hat{W}_{m,T} - W_\star = \Xi_{m,T}^\mathsf{T} X_{m,T} (X_{m,T}^\mathsf{T} X_{m,T})^{-1}$.
Define the whitened version of $X_{m,T}$ as
$\tilde{X}_{m,T} := X_{m,T} \underline{\Gamma}^{-1/2}$.
From these definitions and after some basic manipulations,
for any $\Gamma' \in \mathsf{Sym}^n_{> 0}$:
\begin{align}
\norm{\hat{W}_{m,T} - W_\star}^2_{\Gamma'}
\leq \min\{ n, p \} \frac{ \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 }{ \lambda_{\min}( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} ) \cdot \underline{\lambda}(\underline{\Gamma}, \Gamma') }. \label{eq:basic_inequality}
\end{align}
This decomposes the analysis into
two parts:
(a) upper-bounding
$\opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2$, which is a self-normalized
martingale term, and (b)
lower-bounding the term
$\lambda_{\min}( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} )$.
The analysis for the martingale term is fairly standard
\citep[cf.][Corollary~1]{abbasiyadkori2011selfnormalized},
so for the remainder of this section we focus on the
minimum eigenvalue bound, which contains much of what is novel in our analysis.
We first demonstrate
how the trajectory small-ball definition (\Cref{def:trajectory_small_ball})
can be used to establish \emph{pointwise} convergence of the quadratic form
$\chi(v) := \sum_{i=1}^{m} \sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2$
for $v \in \mathbb{S}^{n-1}$, where $\tilde{x}_t^{(i)} := \underline{\Gamma}^{-1/2} x_t^{(i)}$
is a whitened state vector.
Specifically, we show that
for a fixed $v \in\mathbb{S}^{n-1}$,
the probability of the event
$\{\chi(v) \leq \psi \cdot \varepsilon \}$
is small, for $\psi,\varepsilon$ to be specified.
The key idea is that for any non-negative random variable $X$
satisfying $\Pr\{ X \leq \varepsilon \} \leq (c \varepsilon)^\alpha$ for all $\varepsilon > 0$,
the moment generating function satisfies
$\mathbb{E}[ \exp(-\eta X) ] \leq (c/\eta)^\alpha$
for all $\eta > 0$ (cf.~\Cref{stmt:small_ball_to_mgf}).
Hence, by condition \eqref{eq:trajectory_small_ball}
from \Cref{def:trajectory_small_ball}, for any $\eta > 0$:
\begin{align*}
\mathbb{E}\left[\exp\left( -\frac{\eta}{k}\sum_{t=(j-1)k+1}^{jk} \ip{v}{\tilde{x}_t^{(i)}}^2 \right) ~\Bigg|~ \mathcal{F}_{(j-1)k} \right] \leq \left(\frac{c_{\mathsf{sb}}}{\eta}\right)^\alpha \:\: \textrm{a.s.}, \:\: i = 1, \dots m, \:\: j = 1, \dots S.
\end{align*}
By a Chernoff bound, the tower property of conditional expectation,
and the independence of the trajectories $\{\tilde{x}_t^{(i)}\}_{t \geq 1}$
and $\{\tilde{x}_t^{(i')}\}_{t \geq 1}$ when $i \neq i'$,
for any $\psi > 0$:
\begin{align*}
\Pr\left( \frac{1}{k}\sum_{i=1}^{m}\sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2 \leq \zeta \right) &\leq \inf_{\eta \geq 0} e^{\eta\zeta} \mathbb{E} \exp\left( -\frac{\eta}{k}\sum_{i=1}^{m}\sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2 \right) \\
&\leq \inf_{\eta \geq 0} e^{\eta\zeta} \left(\frac{c_{\mathsf{sb}}}{\eta}\right)^{mS\alpha} \\
&= \exp\left( -mS\alpha\left( \log\left(\frac{mS\alpha}{c_{\mathsf{sb}} \zeta}\right) - 1 \right)\right).
\end{align*}
Now with a change of variables $t := \log\left(\frac{mS\alpha}{c_{\mathsf{sb}} \zeta}\right) - 1$,
we obtain:
\begin{align}
\Pr\left( \sum_{i=1}^{m}\sum_{t=1}^{T} \ip{v}{\tilde{x}_t^{(i)}}^2 \leq \frac{mT \alpha}{2c_{\mathsf{sb}}} e^{-(t+1)} \right) \leq \exp(-mS\alpha t) \quad \forall t > 0. \label{eq:pointwise_lower_tail}
\end{align}
The key upshot of \eqref{eq:pointwise_lower_tail} is that
it controls tail probability at all scales.
This control is needed in order to bound the expected value of \eqref{eq:basic_inequality} by integration.
At this point, it remains to upgrade \eqref{eq:pointwise_lower_tail}
from pointwise to uniform
over $\mathbb{S}^{n-1}$.
A natural approach is to
use standard covering and union bound arguments,
as is done in \cite{simchowitz18learning}.
However, straightforward covering argument yields
un-necessary logarithmic factors in the covariate dimension $n$.
In order to circumvent this issue,
we utilize the PAC-Bayes argument from \cite{mourtada19exactminimax} (which itself
is an extension of \cite{oliveria2016lowertail}) to establish uniform
concentration.
The details are given in \Cref{sec:appendix:general_OLS_proof}.
\input{lower_bound_proof_sketch}
\section{Related work}
\label{sec:related}
Linear regression is a basic and well-studied problem.
The two treatments most closely related to our work are \cite{hsu14randomdesign} and \cite{mourtada19exactminimax},
who develop sharp finite-sample characterizations of the risk of random design linear regression
(i.e., from iid{} examples).
Discussion and references therein cover the broader problem over its long history.
A common approach to studying dependent covariates
is to assume that the data-generating process is
ergodic
(see e.g.\ \cite{yu1994mixing,meir2000timeseries,mohri2008rademachermixing,steinwart2009fastlearningmixing,mohri2010stability,duchi2012ergodicmd,kuznetsov2017mixing,mcdonald2017timeseries,shalizi2021book} and references therein).
The key phenomenon at play is that $N$ correlated examples
are statistically similar to $N/\tau_{\mathsf{mix}}$ independent examples,
where $\tau_{\mathsf{mix}}$ is the process \emph{mixing-time}.
Relying on this idea,
generalization bounds informing independent data can typically be
ported to the ergodic setting, where the effective sample size is simply
``deflated'' by a factor of $\tau_{\mathsf{mix}}$.
Since mixing-based bounds become vacuous
as $\tau_{\mathsf{mix}} \rightarrow \infty$,
they do not present an effective strategy for studying
dynamics that do not mix.
A critical instance of this arises in linear dynamical systems:
in LDS, the ergodicity condition amounts
to \emph{stability} of the dynamics matrix $A$ (i.e.,~$\rho(A) < 1$),
where $\tau_{\mathsf{mix}} \to \infty$ as $\rho(A) \to 1$~\citep[e.g.][Thm.~17.6.2]{meynandtweedie1993}.
Marginally unstable systems, in which $\rho(A) = 1$,
are thus not captured.
A recent line of work uncovers ways to sharpen generalization bounds based on the
specific structure of \emph{realizable} least-squares
regression problems over an ergodic trajectory.
For realizable linear regression with stationary covariates,
results from \cite{bresler2020leastsquaresmarkov}
imply that,
after the trajectory length exceeds an initial burn-in time
scaling as $\tau_{\mathsf{mix}} n$, the minimax (excess) risk coincides
with the classic iid rates.
Additionally, \cite{ziemann2022littlemixing}
show that the empirical risk minimizer exhibits similar behavior
in realizable nonparametric regression problems,
provided certain small-ball assumptions of the underlying process hold.
While these results sharpen our understanding of how the mixing time $\tau_{\mathsf{mix}}$
affects regression risk bounds, they ultimately rely on ergodicity.
Since learning from a single trajectory is generally impossible without ergodicity,
we are led to study other sequential learning configurations.
The two, however, are not mutually exclusive:
our results actually apply when mixing, and in fact show that
the empirical risk minimizer is minimax optimal (after a burn-in time scaling with the mixing time).
This eschews the need for algorithmic modifications to learning
from mixing trajectory data~\cite{bresler2020leastsquaresmarkov}.
We give details on this in \Cref{sec:app:high_prob_upper_bounds}.
\paragraph{Non-temporal dependency structures.}
Covariates and responses can be inter-dependent in many ways, not only via temporal structure.
A recent resurgence of work investigates learning under an Ising model structure over
covariates~\cite{bresler2015ising,dagan2019weaklydependent,ghosal2020ising,dagan2021multipleising},
as well as over responses ~\cite{daskalakis2019regression,dagan2021dependent} (conditioned on the covariates).
At a conceptual level, the extension from a single temporally dependent trajectory
to multiple trajectories is analogous to the extension from single observations
to Ising models with multiple independent observations.
Incidentally, in this area, investigations \emph{began} by studying learning under \emph{multiple} independent observations,
and progressed towards guarantees on learning from a single one.
Relating these two data models---trajectories and Ising grids---under intercompatible assumptions
may reveal interesting connections between these results.
\paragraph{System identification.}
A special case of our LDS-specific data model captures
\emph{linear system identification}
with full state observation:
the task of recovering the dynamical system parameters $A$
from observations of trajectories.
While classic results
are asymptotic in nature~(see e.g.~\cite{lai1982leastsquares,lai1983autoregressive,ljung1998sysidbook}),
recent work gives finite-sample guarantees for recovery of linear systems with fully observed states
\citep{simchowitz18learning,dean2020lqr,yassir2020sysid,faradonbeh2018unstable,sarkar2019sysid,jedra2019lowerbounds,tsiamis2021lowerbounds},
and also partially observed states
\citep{oymak2019lti,simchowitz2019semiparametric,tsiamis2019ssi,sarkar2021sysid,zheng2021ltimultiple}.
The proof of our upper bounds builds on
the ``small-ball'' arguments from \cite{simchowitz18learning}
(that, in turn, extend \cite{mendelson2015learningwithoutconc,koltchinskii2015smin}),
which do not require ergodicity.
To the best of our knowledge, our results are the first
to quantify the trade-offs between
few long trajectories and many short trajectories.
Nearly all finite-sample guarantees for linear system identification
consider a \emph{single} trajectory,
with a few notable exceptions. First, \cite{dean2020lqr}
allow for $m \ge 1$ trajectories with fully observed states
and make no assumptions on the dynamics matrix $A$.
However, their analysis discards all but the last state transition
within a trajectory, reducing to iid{} learning over only $m$ examples.
Second, \cite{zheng2021ltimultiple,xin2022multitraj} study the recovery of Markov parameters from
partially observed states over many trajectories.
However, their error bounds
do not decrease with longer training horizons $T$, since the
number of Markov parameters one must recover
scales with the trajectory length.
Third, \cite{xing2021multiplicative} consider multiple trajectories
where the noise enters \emph{multiplicatively} instead of additively.
Their main finite-sample parameter recovery result (Theorem 2) states that
the operator norm of the parameter error scales as $\sqrt{T/m}$,
with the additional restriction that $T \gtrsim n^2$. To achieve consistency, this result fixes
the trajectory length $T$ and takes the trajectory count $m \to \infty$.
By contrast, our analysis varies the two quantities $T$ and $m$ independently.
Finally, a line of work concurrent to ours investigates learning from multiple
sources of linear dynamical systems~\citep{chen2022learning,modi2022learning}.
This is a latent variable model, where the underlying index
of the LDS must be disambiguated from data.
This model is more general than the one studied
in this paper, and specializing the corresponding results to our
setup yields sub-optimal bounds and unnecessary requirements.
We discuss this in \Cref{sec:results:upper:comparison}, after presenting upper bounds in detail.
Furthermore, our LDS setup (\Cref{sec:problem:lds-trajectories})
decouples the covariate dynamics model $A$
from the observation model $W_\star$,
and our risk definition additionally allows for an arbitrary evaluation horizon ${T'}$.
The risk over an arbitrary evaluation horizon is harder to control than parameter
error, which corresponds to an evaluation length of one. This is because
the larger signal-to-noise ratio accrued by a less stable system
magnifies the prediction error over the entire evaluation horizon.
Although the observation model that we consider is mentioned in \cite{simchowitz18learning},
the general setup with matching upper and lower bounds are all,
to the best of our knowledge, new contributions.
A complementary line of work studies the
problem of online sequence prediction in a no-regret framework, where the
baseline expert class comprises of trajectories generated by a linear dynamical system~\citep{hazan2017spectralfiltering,hazan2018spectralfiltering,ghai2020noregret}.
These results also allow for marginally unstable dynamics but are otherwise not directly comparable.
Other efforts look beyond linear systems to identifying various non-linear classes, such as
exponentially stable non-linear systems~\citep{sattar2020nonlinear,foster2020nonlinear}
and marginally unstable non-linear systems~\citep{jain2021nonlinear}. These results again learn
from a single trajectory.
We believe that elements of our analysis can be ported over to
offer many-trajectory bounds for these
particular classes of non-linear systems.
\section{Risk lower bounds}
\label{sec:results:lower_bounds}
Our lower bounds rely on the following statement, that
the expected trace inverse covariance---a classic quantity in asymptotic
statistics---bounds the minimax risk from below:
\begin{restatable}[Expected trace of inverse covariance bounds risk from below]{mylemma}{traceinvlowerboundsminimaxrisk}
\label{thm:trace_inv_lower_bounds_minimax_risk}
Fix $m, T \in \ensuremath{\mathbb{N}}_{+}$ and a set of covariate distributions $\mathcal{P}_x$.
Suppose that for every $\mathsf{P}_x \in \mathcal{P}_x$,
the data matrix $X_{m,T} \in \ensuremath{\mathbb{R}}^{mT \times n}$ drawn from $\otimes_{i=1}^{m} \mathsf{P}_x$
has full column rank almost surely.
The minimax risk $\mathsf{R}(m, T, {T'}; \mathcal{P}_x)$ satisfies:
\begin{align*}
\mathsf{R}(m, T, {T'}; \mathcal{P}_x) \geq \sigma_\xi^2 p \cdot \sup_{\mathsf{P}_x \in \mathcal{P}_x} \mathbb{E}_{\otimes_{i=1}^{m} \mathsf{P}_x}\left[ \Tr\left( \Gamma_{{T'}}^{1/2}(\mathsf{P}_x) (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \Gamma_{{T'}}^{1/2}(\mathsf{P}_x) \right) \right].
\end{align*}
\end{restatable}
\noindent
\Cref{thm:trace_inv_lower_bounds_minimax_risk} is well known, possibly considered folklore;
we state and prove it for completeness.
Our proof is inspired by a recent argument from \cite{mourtada19exactminimax}.
It smooths over problem instances according to a Gaussian prior,
and analytically characterizes the posterior distribution
of the parameter $W_\star$ under a simple Gaussian observation model
detailed in \Cref{sec:lower_bound_proof_sketch}.
Our first lower bound underscores the need to make variance growth
assumptions \eqref{eq:variance_growth_condition},
in \Cref{stmt:upper_bound_ind_seq_ls},
for \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} in the few trajectories ($m \lesssim n$) regime:
\begin{restatable}[Need for growth assumptions in \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} when $m \lesssim n$]{mythm}{indseqlslowerbound}
\label{stmt:ind_seq_ls_lower_bound}
There exists universal constant
$c_0$, $c_1$, and $c_2$ such that the following holds.
Suppose that
$\mathsf{P}_x = \otimes_{t \geq 1} N(0, 2^t \cdot I_n)$,
$n \geq 6$,
$mT \geq n$,
and $m \leq c_0 n$.
Then:
\begin{align*}
\mathsf{R}(m,T,T;\{\mathsf{P}_x\}) \geq c_1 \sigma_\xi^2 \cdot \frac{p \cdot 2^{c_2 n/m}}{T} .
\end{align*}
\end{restatable}
\Cref{stmt:ind_seq_ls_lower_bound} states that if the variances $\Sigma_t$
are allowed to grow exponentially in $t$,
then the minimax risk of \problemName{Ind\smallDash{}Seq\smallDash{}LS}{} scales exponentially in $n/m$ when $m \lesssim n$.
Thus, some sub-exponential growth assumption is necessary
in order to have the risk scale polynomially in $n/m$.
We now turn to a lower bound for \problemName{LDS\smallDash{}LS}{}.
We consider
two particular hard instances for \problemName{LDS\smallDash{}LS}{} dynamics matrices $(A, B)$,
where we set $B = I_n$ and vary $A$.
The first instance corresponds to iid\ covariates,
i.e.,\ $A = 0_{n \times n}$.
The second instance corresponds
to an isotropic Gaussian random walk, i.e.,\ $A = I_n$.
These two hard instances satisfy
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability}, and
\Cref{assume:one_step_controllability}.
Together they show that our upper bounds are sharp up to logarithmic factors,
treating the condition number $\gamma(A, B)$
from \Cref{def:condition_number} as a constant:
\begin{restatable}[Risk lower bound]{mythm}{lowerboundmain}
\label{stmt:lower_bound_main}
There are universal positive constants $c_0$, $c_1$, and $c_2$ such that the following holds.
Recall that $\PxA{I_n}$ (resp.\ $\PxA{0_{n \times n}}$)
denotes the covariate distribution for a linear dynamical system
with $A=I_n$ and $B=I_n$ (resp.\ $A=0_{n \times n}$ and $B=I_n$).
If $T \geq c_0$, $n \geq c_1$, and $mT \geq n$, then:
\begin{align*}
\mathsf{R}(m, T, {T'}; \{\PxA{0_{n \times n}}, \PxA{I_n}\}) \geq c_2 \sigma_\xi^2 \cdot \frac{pn}{mT} \cdot \max\left\{ \frac{n {T'}}{m T}, \frac{{T'}}{T}, 1 \right\}.
\end{align*}
\end{restatable}
We can interpret this lower bound by a breakdown of the quantity
$\varphi := \max\{n{T'}/(mT), {T'}/T, 1\}$ across various regimes.
When trajectories are limited ($m \lesssim n$),
$\varphi \asymp \max\{ n{T'}/(mT), 1\}$, and therefore
the minimax risk is bounded below by
$\sigma_\xi^2 \cdot pn/(mT) \cdot \max\{ n{T'}/(mT), 1\}$.
This is the same rate prescribed by the OLS upper bound of
\Cref{thm:sublinear_trajectories_bound}, up to
the condition number $\gamma(A, B)$ and logarithmic factors
in $n/m$. We have thus justified the
summary statement \Cref{thm:simple-rate-few-traj}.
On the other hand, under many trajectories ($m \gtrsim n$),
$\varphi \asymp \max\{{T'}/T,1\}$
and the minimax risk is bounded below by
$\sigma_\xi^2 \cdot pn/(mT) \cdot \max\{{T'}/T,1\}$.
By \Cref{stmt:upper_bound_many_trajectories}, the OLS risk is
bounded above by the same quantity times $\gamma(A, B)$,
justifying the summary statement \Cref{thm:simple-rate-many-traj}.
\section{Risk upper bounds}
\label{sec:results:upper}
\label{sec:results:upper_bounds}
The trajectory small-ball definition
allows us to carve out conditions for learnability.
A key quantity for what follows is the minimum eigenvalue of
the ratio of two positive definite matrices:
\begin{align}
\underline{\lambda}(A, B) := \lambda_{\min}(B^{-1/2} A B^{-1/2}), \quad A,B \in \mathsf{Sym}^n_{> 0}. \label{eq:ulam_defn}
\end{align}
Our various upper bounds statements build on the following general lemma:
\begin{restatable}[General OLS upper bound]{mylemma}{upperboundgeneral}
\label{stmt:upper_bound_general}
There are universal positive constants $c_0$ and $c_1$ such that the following holds.
Suppose that $\mathsf{P}_x$ satisfies
the $(T,k,\{\Psi_j\}_{j=1}^{\floor{T/k}},c_{\mathsf{sb}},\alpha)$-TrajSB{} condition
(\Cref{def:trajectory_small_ball}).
Put $S := \floor{T/k}$ and $\Gamma_T := \Gamma_T(\mathsf{P}_x)$.
Fix any $\underline{\Gamma} \in \mathsf{Sym}^{n}_{> 0}$ satisfying
$\frac{1}{S} \sum_{j=1}^{S} \Psi_j \preccurlyeq \underline{\Gamma} \preccurlyeq \Gamma_T$,
and let $\underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma})$
denote the geometric mean of the minimum eigenvalues $\{\underline{\lambda}(\Psi_j, \underline{\Gamma})\}_{j=1}^{S}$, i.e.,
\begin{align}
\underline{\mu}( \{\Psi_j\}_{j=1}^{S}, \underline{\Gamma}) := \left[\prod_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma}) \right]^{1/S}. \label{eq:uMu_defn}
\end{align}
Suppose that:
\begin{align}
n \geq 2, \quad \frac{mT}{kn} \geq \frac{c_0}{\alpha} \log\left(\frac{\max\{e,c_{\mathsf{sb}}\}}{ \alpha \underline{\lambda}(\underline{\Gamma}, \Gamma_T) \underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma})}\right). \label{eq:mTkn_requirements}
\end{align}
Then, for any $\Gamma' \in \mathsf{Sym}^n_{> 0}$:
\begin{align}
\mathbb{E}[ \norm{ \hat{W}_{m,T} - W_\star }^2_{\Gamma'} ]
\leq c_1 c_{\mathsf{sb}} \sigma_\xi^2
\cdot \frac{pn}{ mT \alpha \underline{\lambda}(\underline{\Gamma},\Gamma') \underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma}) }
\cdot \log\left(\frac{ \max\{e,c_{\mathsf{sb}}\}}{\alpha \underline{\lambda}(\underline{\Gamma}, \Gamma_T) \underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma})}\right)
.
\label{eq:general_risk_bound}
\end{align}
\end{restatable}
The proof of \Cref{stmt:upper_bound_general}
blends ideas from the
analysis of random design linear regression
\citep{hsu14randomdesign,oliveria2016lowertail,mourtada19exactminimax}
with techniques from linear system identification with full state observation~\citep{simchowitz18learning,sarkar2019sysid,faradonbeh2018unstable,dean2020lqr}.
Note that \Cref{stmt:upper_bound_general} makes no explicit
assumptions on the ergodicity of the process $\mathsf{P}_x$.
The role of $\mathsf{P}_x$ is instead succinctly captured
by the trajectory small-ball condition,
together with the minimum eigenvalue quantities that appear in the bound.
The proof of \Cref{stmt:upper_bound_general} also yields, with some
straightforward modifications, bounds on
the risk that hold with high probability; we only present
bounds in expectation for simplicity.
Finally, if the square norm $\norm{X}_{M}^2$ is defined to be
$\lambda_{\max}(X M X^\mathsf{T})$ instead of $\Tr(X M X^\mathsf{T})$, then
\eqref{eq:general_risk_bound} holds with the expression $p + n$ replacing $pn$ in the numerator.
\input{results-table}
As long as the process $\mathsf{P}_x$ satisfies the trajectory small-ball condition
with excitation window $k=T$,
\Cref{stmt:upper_bound_general} (with $\Psi_1 = \underline{\Gamma} = \Gamma_T(\mathsf{P}_x)$)
immediately yields the following
result for learning from many trajectories in the \problemName{Seq\smallDash{}LS}\ problem:
\begin{restatable}[Upper bound for \problemName{Seq\smallDash{}LS}{}, many trajectories]{mythm}{upperboundseqlsmanytraj}
\label{stmt:upper_bound_seq_ls_many_traj}
There are univeral positive constants $c_0$ and $c_1$
such that the following holds.
Suppose that $\mathsf{P}_x$ satisfies
the trajectory small-ball condition (\Cref{def:trajectory_small_ball})
with parameters $(T, T, \Gamma_T(\mathsf{P}_x), c_{\mathsf{sb}}, \alpha)$.
If:
\begin{align*}
n \geq 2, \quad m \geq \frac{c_0 n}{\alpha}\log\left(\frac{\max\{e,c_{\mathsf{sb}}\}}{\alpha}\right),
\end{align*}
then, for any $\Gamma' \in \mathsf{Sym}^n_{> 0}$:
\begin{align}
\mathbb{E}[\norm{\hat{W}_{m,T} - W_\star}^2_{\Gamma'}] \leq c_1 c_{\mathsf{sb}} \sigma_\xi^2 \cdot \frac{pn}{mT\alpha \underline{\lambda}(\Gamma_T(\mathsf{P}_x), \Gamma')} \cdot \log\left(\frac{\max\{e, c_{\mathsf{sb}}\}}{\alpha}\right). \label{eq:T_T_traj_small_ball}
\end{align}
\end{restatable}
This result provides the upper bound for the summary statement \Cref{thm:simple-rate-many-traj-in-window}.
To interpret the bound \eqref{eq:T_T_traj_small_ball},
suppose that $c_{\mathsf{sb}}$ and $\alpha$ are universal constants.
Then, the requirement on $m$ simplifies to $m \gtrsim n$.
Under any strict evaluation horizon ${T'} \leq T$, taking $\Gamma' = \Gamma_{{T'}}(\mathsf{P}_x)$,
the risk $\mathbb{E}[L(\hat{W}_{m,T};{T'},\mathsf{P}_x)]$ scales as $\sigma_\xi^2 pn / (mT)$.
The lower bound for \Cref{thm:simple-rate-many-traj-in-window} follows from the fact that iid\ linear regression is a special case of \problemName{Seq\smallDash{}LS}{}.
Meanwhile, to obtain guarantees for parameter recovery,
consider taking $\Gamma' = I_n$. Then \Cref{stmt:upper_bound_seq_ls_many_traj}
implies that the parameter error
$\mathbb{E}[\norm{\hat{W}_{m,T}-W_\star}_F^2]$
scales as $\sigma_\xi^2 pn / [mT \cdot \lambda_{\min}(\Gamma_T(\mathsf{P}_x))]$.
Note that operator norm bounds on parameters also hold,
with the expression $p+n$ replacing $pn$ in the bound.
\Cref{stmt:upper_bound_general} also
yields a bound for \problemName{Ind\smallDash{}Seq\smallDash{}LS}, assuming polynomial growth
of the time-$t$ covariances $\Sigma_t$ \eqref{eq:covariance-def}.
To state the result, let $\phi : [1, \infty) \times [0, \infty) \rightarrow [1, \infty)$ be defined as:
\begin{align}
\phi(a, x) := \begin{cases} 1 &\text{if } x \leq 1, \\
ax &\text{otherwise.}
\end{cases}
\end{align}
Note that $1 \leq \phi(a, x) \leq \max\{ax, 1\}$.
\begin{restatable}[Upper bound for \problemName{Ind\smallDash{}Seq\smallDash{}LS}{}]{mythm}{upperboundindseqls}
\label{stmt:upper_bound_ind_seq_ls}
There are universal positive constants $c_0$ and $c_1$
such that the following holds.
Fix any sequence of distributions $\{\Pxt{t}\}_{t \geq 1}$,
and let $\Sigma_t := \mathbb{E}_{x_t \sim \Pxt{t}}[x_tx_t^\mathsf{T}]$ for $t \in \ensuremath{\mathbb{N}}_{+}$.
Suppose there exists $c_{\mathsf{sb}} > 0$ and $\alpha \in (0, 1]$ such
that for all $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$, $\varepsilon > 0$ and $t \in \ensuremath{\mathbb{N}}_{+}$:
\begin{align}
\Pr_{x_t \sim \Pxt{t}}\left\{ \ip{v}{x_t}^2 \leq \varepsilon \cdot v^\mathsf{T} \Sigma_t v \right\} \leq (c_{\mathsf{sb}} \varepsilon)^{\alpha}. \label{eq:small_ball_independent}
\end{align}
Furthermore, suppose there exists a
$c_\beta \geq 1$ and $\beta \geq 0$ such that
for all $s, t \in \ensuremath{\mathbb{N}}_{+}$ satisfying $s \leq t$:
\begin{align}
\frac{1}{\underline{\lambda}(\Sigma_s, \Sigma_t)} \leq c_\beta (t/s)^\beta. \label{eq:variance_growth_condition}
\end{align}
If:
\begin{align*}
n \geq 2, \quad mT \geq \frac{c_0 n}{\alpha} \left( \beta + \log\left(\frac{\max\{e,c_{\mathsf{sb}}\} c_\beta}{\alpha}\right) \right),
\end{align*}
then, for $\mathsf{P}_x = \otimes_{t \geq 1} \Pxt{t}$:
\begin{align}
\mathbb{E}[L(\hat{W}_{m,T};{T'}, \mathsf{P}_x)] \leq c_1 c_{\mathsf{sb}} \sigma_\xi^2 c_\beta e^\beta \cdot \frac{pn}{mT\alpha} \cdot \phi\left(c_\beta(\beta+1),
({T'}/T)^\beta
\right) \left[ \beta + \log\left(\frac{\max\{e,c_{\mathsf{sb}}\}c_\beta}{\alpha}\right)\right]. \label{eq:risk:ind_seq_ls}
\end{align}
\end{restatable}
Consider specializing \Cref{stmt:upper_bound_ind_seq_ls}
to the case when $\Sigma_t = \Sigma$ for all $t \in \ensuremath{\mathbb{N}}_{+}$.
Doing so yields random design linear regression from $mT$
covariates drawn iid\ from $\Pxt{1}$.
The growth condition \eqref{eq:variance_growth_condition} is trivially
satisfied with $c_\beta = 1$ and $\beta = 0$.
The small-ball assumption \eqref{eq:small_ball_independent}
simplifies to $\Pr_{x_1 \sim \Pxt{1}}\{ \abs{\ip{v}{x_1}} \leq \varepsilon \norm{v}_{\Sigma} \} \leq (\sqrt{c_{\mathsf{sb}}} \varepsilon)^{2\alpha}$ for all $v \neq 0$ and $\varepsilon > 0$,
which matches \cite[Assumption~1]{mourtada19exactminimax}
up to a minor redefinition of the constants $c_{\mathsf{sb}}, \alpha$.
Treating $c_{\mathsf{sb}}$ and $\alpha$ as constants,
the conclusion of \Cref{stmt:upper_bound_ind_seq_ls}
in this setting is that
$\mathbb{E}[\norm{\hat{W}_{m,T} - W_\star}_{\Sigma}^2] \lesssim \sigma_\xi^2 pn/(mT)$ as long as $n \geq 2$ and $mT \gtrsim n$,
which recovers \cite[Proposition~2]{mourtada19exactminimax}.
On the other hand, \Cref{stmt:upper_bound_ind_seq_ls}
does not require that the covariates are drawn iid\ from the
same distribution,
allowing the time-$t$ covariances $\Sigma_t$ to grow polynomially.
As an example, suppose that $\Sigma_t = t^\beta \cdot I_n$ for some $\beta > 0$.
In this case, $1/\underline{\lambda}(\Sigma_s, \Sigma_t) = (t/s)^\beta$, so we can take
$c_\beta=1$ in \eqref{eq:variance_growth_condition}.
Again treating $c_{\mathsf{sb}}$ and $\alpha$ as constants and taking ${T'} \leq T$, we have
$\mathbb{E}[ L(\hat{W}_{m,T}; {T'}, \mathsf{P}_x) ] \lesssim \sigma_\xi^2 \beta e^\beta \cdot pn/(mT)$
as long as $mT \gtrsim \beta n$.
If $\beta$ is also considered a constant, then we further
have the risk bound
$\mathbb{E}[ L(\hat{W}_{m,T}; {T'}, \mathsf{P}_x) ] \lesssim \sigma_\xi^2 pn/(mT)$.
This matches the minimax rate for iid\ linear regression.
It is natural to ask
if the covariance growth condition \eqref{eq:variance_growth_condition}
is needed under strict evaluation horizons ${T'} \le T$.\footnote{Some regularity
is needed when under extended evaluations ${T'} > T$, otherwise the risk could be arbitrarily large.}
In \Cref{sec:results:lower_bounds}, we show that
if the covariances are set to $\Sigma_t = 2^t \cdot I_n$
and $\Pxt{t} = N(0, \Sigma_t)$ (satisfying \eqref{eq:small_ball_independent}),
then the minimax risk $\mathsf{R}(m, T, T; \{\otimes_{t \geq 1} \Pxt{t}\})$
must scale at least $2^{cn/m}/T$ whenever $m \lesssim n$, for some positive constant $c$.
Sub-exponential
growth rates are therefore necessary for polynomial sample complexity.
Determining the optimal dependence of $\beta$
in \eqref{eq:risk:ind_seq_ls} is left to future work.
Note that \Cref{stmt:upper_bound_ind_seq_ls} is most interesting either when
trajectories are few ($m \lesssim n$)
or evaluations are extended (${T'} > T$).
When
$m \gtrsim n$ and ${T'} \leq T$, one can usually apply
\Cref{stmt:upper_bound_seq_ls_many_traj}
with $\Gamma' = \Gamma_{{T'}}(\mathsf{P}_x)$ instead,
and avoid placing any requirements on the growth of covariances.
Considering any of the small-ball examples in \Cref{sec:results:upper:small_ball_examples},
recall that when the excitation window $k$ and the horizon $T$ are equal,
\Cref{stmt:upper_bound_seq_ls_many_traj}
provides an upper bound on the risk of OLS estimation
for the corresponding \problemName{Seq\smallDash{}LS}\ problem.
Specifically, for \Cref{example:simple}
and \Cref{stmt:lds_traj_small_ball} with $k=T$,
if ${T'} = T$ and trajectories are abundant ($m \gtrsim n$), then
the OLS estimator's rate $\sigma_\xi^2 pn/(mT)$
matches its behavior in iid\ linear regression.
Meanwhile, for the degree-$D$ Volterra series (\Cref{example:volterra}),
we require that $m \gtrsim c_D \cdot n $,
and the OLS risk bound scales as $\sigma_\xi^2 c'_D \cdot pn/(mT)$,
for constants $c_D$ and $c'_D$ that only depend on $D$.
In order to cover scenarios in which trajectories may be relatively scarce,
namely $m \lesssim n$, we need additional structure.
More technically, when the small-ball condition is satisfied with
$k < T$, one needs to further control the various eigenvalues
that appear in \Cref{stmt:upper_bound_general} in order to bound the risk of
OLS.
Specifically for \problemName{Ind\smallDash{}Seq\smallDash{}LS}{}, a covariate growth assumption suffices: \Cref{example:independent_gaussians}
combined with \Cref{stmt:upper_bound_ind_seq_ls} yields an OLS risk bound.
Furthermore, both \Cref{example:partition} and \Cref{example:mixing_chain}
can be immediately combined with \Cref{stmt:upper_bound_general},
since the matrices $\Psi_j$ in these examples are bounded above and
below by $\Gamma_T(\mathsf{P}_x)$ up to universal constant factors.
But arbitrarily large risk can still be realized in the general \problemName{Seq\smallDash{}LS}{} problem, even
when the trajectory small-ball condition is satisfied.
To study the behavior of OLS across all regimes of trajectory count $m$, example dimensions $p$ and $n$,
and trajectory lengths $T$ and ${T'}$,
we focus specifically on linear dynamical systems and the \problemName{LDS\smallDash{}LS}{} problem
for our remaining upper bounds.
\subsection{Upper bounds for linear dynamical system}
\label{sec:upper_bounds:lds}
In this section, we focus exclusively on dynamics $\PxA{A,B}$ described
by a linear dynamical system \eqref{eq:lds_definition}.
As discussed previously, in order to apply
\Cref{stmt:upper_bound_general} in the few trajectories regime
when $m \lesssim n$ (or when $m \gtrsim n$ and ${T'} > T$), we must
(a) show that the process $\PxA{A,B}$ satisfies the
trajectory small-ball condition, and (b) bound the various
eigenvalues which appear in \Cref{stmt:upper_bound_general}.
\Cref{stmt:lds_traj_small_ball} establishes that $\PxA{A,B}$
satisfies the $(T, k, \Gamma_k(A, B), e, \frac 1 2)$-TrajSB{}
condition, as long as $(A, B)$ is $k_{\mathsf{c}}$-step controllable
and $k \geq k_{\mathsf{c}}$, thus taking care of (a). To handle (b),
we introduce additional assumptions on the dynamics matrices $(A, B)$:
\begin{myassump}[Marginal instability]
\label{assume:marginal_stability}
The dynamics matrix $A$ in \DLR{} is \emph{marginally unstable}.
That is, $\rho(A) \leq 1$, where $\rho(A)$ denotes the spectral radius of $A$.
\end{myassump}
\begin{myassump}[Diagonalizability]
\label{assume:diagonalizability}
The dynamics matrix $A$ in \DLR{} is \emph{complex diagonalizable} as $A = S D S^{-1}$,
where $S \in \ensuremath{\mathbb{C}}^{n \times n}$ is invertible
and $D \in \ensuremath{\mathbb{C}}^{n \times n}$ is a diagonal matrix comprising the eigenvalues of $A$.
\end{myassump}
\begin{myassump}[One-step controllability]
\label{assume:one_step_controllability}
The control matrix $B$ in \DLR{} has full row rank,
i.e., $\rank(B) = n$. Equivalently, the pair $(A, B)$ is
one-step controllable (\Cref{def:controllability}).
\end{myassump}
Assumption~\ref{assume:marginal_stability} is fairly standard
in the literature. Going beyond the regime
$\rho(A) = 1 + \varepsilon$, where $\varepsilon \lesssim 1/T$,
requires additional technical assumptions
on the dynamics matrix $A$ that we choose to avoid in the interest
of simplicity; the OLS estimator is in general not a consistent estimator when $\rho(A) > 1$ and $m=1$ (cf.~\cite{phillips2013inconsistent,sarkar2019sysid}).
The condition $\rho(A) \leq 1$ is often referred to as
\emph{marginal stability} in other work. We choose to
call it marginally \emph{unstable} instead, to emphasize the fact that
such systems, namely at $\rho(A)=1$, may not be ergodic and that the state can grow
unbounded (e.g.\ have magnitude roughly $t^n$ at time $t$).
Diagonalizability (\Cref{assume:diagonalizability})
is less standard in the literature.
We use it together with \Cref{assume:marginal_stability}
and \Cref{assume:one_step_controllability} to establish that
$\underline{\lambda}(k, t; A, B) := \underline{\lambda}(\Gamma_k(A, B), \Gamma_t(A, B)) \gtrsim c \cdot k/t$ whenever $k \leq t$, where $c$ is a constant that depends only on $A$ and $B$
(and not $k$ and $t$).
In previous work on linear system identification,
the term $\underline{\lambda}(k, t; A, B)$ only appears under a logarithm, and so
coarser analyses in the general case can
still establish polynomial rates
(cf.\ \cite[Proposition~A.1]{simchowitz18learning} and
\cite[Proposition~7.6]{sarkar2019sysid}).\footnote{Note, however,
that without diagonalizability,
\cite[Corollary~A.2]{simchowitz18learning} can only guarantee a $\sqrt{n^2/T}$ rate
for the operator norm of the parameter error in general, and this is likely not optimal.}
However, by allowing
for evaluation lengths ${T'} > 1$,
the dependence on $\underline{\lambda}(k, t; A, B)$ is no longer entirely confined under a logarithm
(cf.~\Cref{stmt:upper_bound_general}).
A sharp characterization is hence critical for deriving optimal rates.
In \Cref{sec:beyond-diag-results}, we conjecture the correct scaling of
$\underline{\lambda}(k, t; A, B)$ as a function of the ratio $k/t$ and the largest Jordan block size of $A$,
based on numerical simulation.
One-step controllability (\Cref{assume:one_step_controllability})
is also an assumption commonly made in linear system identification.
It is clear that some form of controllability is needed, otherwise
learning may be impossible (e.g.~consider the extreme case of $B=0$).
General multi-step controllability does not suffice either:
\cite[Theorem~2]{tsiamis2021lowerbounds} show that under a single trajectory ($m=1$),
$n$-step controllability (where $n$ remains the state dimension) does
not ensure finite risk, and even a more robust controllability definition \citep[Definition~3]{tsiamis2021lowerbounds}
cannot ensure risk bounds better than exponential in the dimension $n$.
Considering these barriers, we simply choose to rely on one-step
controllability in the few-trajectory setting ($m \lesssim n$).
Finally, we introduce a condition number quantity that will feature commonly in our bounds:
\begin{mydef}
\label{def:condition_number}
For dynamics matrices $(A, B)$ in \DLR{}
satisfying \Cref{assume:diagonalizability}
and \Cref{assume:one_step_controllability},
the \emph{condition number} $\gamma(A, B)$ is defined as:
$\gamma(A, B) := \frac{\lambda_{\max}(S^{-1} BB^\mathsf{T} S^{-*})}{\lambda_{\min}(S^{-1} BB^\mathsf{T} S^{-*})}$.
\end{mydef}
\subsubsection{Many trajectories}
Our first result instantiates \Cref{stmt:upper_bound_seq_ls_many_traj}
in the special case of $\Gamma' = I_n$, which yields a sharp bound for
parameter recovery without requiring stability of
the dynamics matrix $A$:
\begin{restatable}[Parameter recovery upper bound for \DLR{}, many trajectories]{mythm}{upperboundparameterrecovery}
\label{stmt:upper_bound_parameter_recovery}
There are universal positive constants $c_0$ and $c_1$
such that the following holds for any instance of \DLR{}.
Suppose that $(A, B)$ is $k_{\mathsf{c}}$-step controllable,
If $n \geq 2$, $m \geq c_0 n$, and $T \ge k_{\mathsf{c}}$, then:
\begin{align*}
\mathbb{E}[\norm{\hat{W}_{m,T} - W_\star}_F^2]
\leq c_1 \sigma_\xi^2
\cdot \frac{pn}{m T \cdot \lambda_{\min}(\Gamma_T(A, B))}.
\end{align*}
\end{restatable}
\Cref{stmt:upper_bound_parameter_recovery}
improves on existing linear system identification results in the following way:
it replaces stability assumptions on the dynamics matrix $A$
with a simpler assumption of relatively many trajectories ($m \gtrsim n$),
and it guarantees a rate that is inversely proportional
to the \emph{total} number of examples $mT$ instead of
only one example per trajectory.
In other words, our analysis does not need to ``discard'' the data within a trajectory,
which is the case in~\cite[Proposition~1.1]{dean2020lqr}.
Additionally, although OLS is generally not a consistent estimator
from one trajectory ($m=1$) if the dynamics $A$ are unstable,
the results of \cite{dean2020lqr} imply consistency as $m \to \infty$,
i.e., that $\hat{W}_{m,T}$ converges in probability to $W_\star$ as $m \to \infty$.
\Cref{stmt:upper_bound_parameter_recovery}
adds that, provided $m \gtrsim n$,
OLS is consistent under
unstable systems as $T \to \infty$ as well,
even if the trajectory count $m$ remains finite.
We will return to parameter recovery from relatively few trajectories $(m \lesssim n)$ by this section's end.
We now look beyond an evaluation horizon of length one, and
consider the setting with many trajectories ($m \gtrsim n$).
As noted previously, in order to handle an
arbitrary evaluation horizon ${T'}$ (in particular those that extend past the training horizon $T$),
some constraint on the admissible dynamics matrices
is needed to ensure that the minimax risk remains finite.
Without assumptions, the quantity $\underline{\lambda}(\Gamma_T(A, B), \Gamma_{{T'}}(A, B))$,
whose inverse inevitably bounds the risk \eqref{eq:risk_def} from below,
can be arbitrarily small
whenever ${T'} > T$, resulting in arbitrarily large risk.
We will use our stated assumptions from the beginning of this
section.
The following specializes \Cref{stmt:upper_bound_seq_ls_many_traj}
to \problemName{LDS\smallDash{}LS}{}:
\begin{restatable}[Risk upper bound for \DLR{}, many trajectories]{mythm}{upperboundmanytrajectories}
\label{stmt:upper_bound_many_trajectories}
There are universal positive constants $c_0$ and $c_1$ such that the following holds for any instance of \DLR{}.
Suppose that $(A, B)$ is $k_{\mathsf{c}}$-step controllable.
If $n \geq 2$, $m \geq c_0 n$, $T \geq k_{\mathsf{c}}$, and the evaluation horizon is strict (${T'} \leq T$), then:
\begin{align*}
\mathbb{E}[ L(\hat{W}_{m,T};{T'}, \PxA{A,B})]
\leq c_1 \sigma_\xi^2
\cdot \frac{p n}{m T}.
\end{align*}
On the other hand, suppose that $(A, B)$ satisfies
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability}, and
\Cref{assume:one_step_controllability},
with $\gamma := \gamma(A, B)$ (\Cref{def:condition_number}).
If $n \geq 2$,
$m \geq c_0 n$, and the evaluation horizon is extended
(${T'} > T$), then:
\begin{align*}
\mathbb{E}[ L(\hat{W}_{m,T};{T'}, \PxA{A,B})]
\leq c_1 \sigma_\xi^2
\cdot \frac{p n}{m T}
\cdot \gamma\frac{{T'}}{T}.
\end{align*}
\end{restatable}
Setting ${T'} = T$,
\Cref{stmt:upper_bound_many_trajectories} states that the risk of
\problemName{LDS\smallDash{}LS}{} in the many trajectories regime satisfies
$\mathbb{E}[ L(\hat{W}_{m,T};T,\PxA{A,B}) ] \lesssim \sigma_\xi^2 pn/(mT)$.
This rate matches the corresponding
independent baseline \problemName{Ind\smallDash{}LDS\smallDash{}LS}{} in the many trajectories regime.
To see this, first observe that the marginal distribution
$\PxAt{A,B}{t}$ at time $t \in \ensuremath{\mathbb{N}}_{+}$ is $N(0, \Sigma_t(A, B))$.
Hence, the covariate distribution for \problemName{Ind\smallDash{}LDS\smallDash{}LS}{} corresponds
to the product distribution $\otimes_{t \geq 1} N(0, \Sigma_t(A, B))$,
which is an instance of a Gaussian process.
Therefore, \Cref{example:gaussian_processes} combined with
\Cref{stmt:upper_bound_seq_ls_many_traj}
yields that
the \problemName{Ind\smallDash{}LDS\smallDash{}LS}{} problem also has a risk bound
that scales as $\sigma_\xi^2 pn/(mT)$ whenever $m \gtrsim n$.
Put differently, the dependent structure of the covariate distribution $\PxA{A,B}$
in \problemName{LDS\smallDash{}LS}{}
does not add any statistical overhead to the learning problem
(compared to the independent learning problem \problemName{Ind\smallDash{}LDS\smallDash{}LS}{}), as long as
$m \gtrsim n$.
\subsubsection{Few trajectories}
\label{sec:upper_bounds:lds:few_trajectories}
We now cover the regime in which relatively few training trajectories are available ($m \lesssim n$).
Our first result bounds the OLS risk for the \problemName{LDS\smallDash{}LS}{} problem:
\begin{restatable}[Risk upper bound for \DLR{}, few trajectories]{mythm}{fewtrajrate}
\label{thm:sublinear_trajectories_bound}
There are universal positive constants $c_0$, $c_1$, and $c_2$ such that the following holds for any instance of \DLR{}.
Suppose that $(A, B)$
satisfies
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability}, and
\Cref{assume:one_step_controllability},
with $\gamma := \gamma(A, B)$ (\Cref{def:condition_number}).
If $n \geq 2$, $m \leq c_0 n$, and $mT \geq c_1 n\log(\max\{\gamma n/m, e\})$, then:
\begin{align*}
\mathbb{E}[L(\hat{W}_{m,T}; {T'}, \PxA{A,B})]
\leq c_2 \sigma_\xi^2
\cdot \frac{p n \log(\max\{\gamma n/m, e\})}{mT}
\cdot \phi\left( \gamma, \frac{c_1 n \log(\max\{\gamma n/m, e\})}{m} \cdot \frac{{T'}}{T} \right).
\end{align*}
\end{restatable}
To interpret \Cref{thm:sublinear_trajectories_bound},
consider $\gamma$ a constant and suppose that ${T'} = T$.
Then \Cref{thm:sublinear_trajectories_bound} states that
$\mathbb{E}[L(\hat{W}_{m,T};T,\PxA{A,B})] \lesssim \sigma_\xi^2 \cdot pn/(mT) \cdot n\log^2(n/m) / m$.
We now see that this \problemName{LDS\smallDash{}LS}{} risk is an extra $n \log^2(n/m)/m$ factor larger than the
risk of the baseline problem \problemName{Ind\smallDash{}LDS\smallDash{}LS}{}:
\begin{restatable}[Risk upper bound for \problemName{Ind\smallDash{}LDS\smallDash{}LS}{}]{mythm}{upperboundindldsls}
\label{stmt:upper_bound_ind_lds_ls}
There are universal positive constants $c_0$ and $c_1$ such that the following holds for any instance of \problemName{Ind\smallDash{}LDS\smallDash{}LS}{}.
Suppose that $(A, B)$
satisfies
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability}, and
\Cref{assume:one_step_controllability},
with $\gamma := \gamma(A, B)$ (\Cref{def:condition_number}).
If $n \geq 2$ and $mT \geq c_0 n \log(\max\{\gamma, e\})$,
then:
\begin{align*}
\mathbb{E}[ L(\hat{W}_{m,T}; {T'}, \otimes_{t \geq 1} \PxAt{A,B}{t}) ]
\leq c_1 \sigma_\xi^2 \cdot \frac{pn \gamma \log(\max\{\gamma, e\}) }{mT}
\cdot \phi\left(\gamma, \frac{{T'}}{T}\right).
\end{align*}
\end{restatable}
\noindent
Treating $\gamma$ as a constant and setting ${T'} = T$,
\Cref{stmt:upper_bound_ind_lds_ls} states that
$\mathbb{E}[L(\hat{W}_{m,T};T, \otimes_{t \geq 1} \PxAt{A,B}{t})]$
scales as $\sigma_\xi^2 pn/(mT)$, matching the risk of
iid\ linear regression up to constant factors.
In \Cref{sec:results:lower_bounds}, we will see that the result of
\Cref{thm:sublinear_trajectories_bound} is sharp up to constants, and therefore the \problemName{LDS\smallDash{}LS}{} problem is fundamentally more difficult than its corresponding
baseline problem \problemName{Ind\smallDash{}LDS\smallDash{}LS}{} when trajectories are relatively scarce.
We conclude with our final upper bound,
using our assumptions to generalize \cite[Theorem~2.1]{simchowitz18learning} to the
few-trajectory setting:
\begin{restatable}[Parameter recovery upper bound for \DLR{}, few trajectories]{mythm}{paramrecoveryfewtrajrate}
\label{stmt:upper_bound_parameter_recovery_few_trajs}
There are universal positive constants $c_0$, $c_1$, and $c_2$ such that the following holds for any instance of \DLR{}.
Suppose that $(A, B)$
satisfies
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability}, and
\Cref{assume:one_step_controllability},
with $\gamma := \gamma(A, B)$ (\Cref{def:condition_number}).
If $n \geq 2$, and $mT \geq c_0 n \log(\max\{\gamma n/m, e\})$,
then:
\begin{align*}
\mathbb{E}[ \norm{\hat{W}_{m,T} - W_\star}_F^2 ]
\leq c_1 \sigma_\xi^2
\cdot \frac{ pn \log(\max\{\gamma n/m, e\})}{mT \cdot \lambda_{\min}(\Gamma_{k_\star}(A, B))},
\quad k_\star := \bigfloor{\frac{c_2 T}{n/m \cdot \log(\max\{\gamma n/m, e\})}}.
\end{align*}
\end{restatable}
\noindent
\Cref{stmt:upper_bound_parameter_recovery_few_trajs} complements \Cref{stmt:upper_bound_parameter_recovery};
together they cover parameter recovery across all problem regimes.
Again, operator norm bounds also hold with $p + n$ in place of $pn$.
\subsection{Comparison to learning from trajectories of multiple unknown systems}
\label{sec:results:upper:comparison}
As mentioned in \Cref{sec:related}, \cite{chen2022learning,modi2022learning} both study the
setup where a learner observes multiple independent
trajectories from $K$ different unknown
linear dynamical systems. The task is to identify the
parameters of the $K$ underlying systems. This is more general than the setting we consider, which is recovered by fixing $K=1$.
However, specializing these rates to our setting
yield either unnecessary requirements, suboptimal bounds, or both.
To see this, first, if we specialize
\cite[Theorem 1]{chen2022learning} to our setup, we generate
unnecessary assumptions.
Specifically, Theorem 1 requires strict stability,
one-step controllability, and
$mT \gtrsim \max\{ n^3, 1/(1-\rho) \}$, where $\rho$ is the spectral radius of $A$.
In comparison, \Cref{stmt:upper_bound_parameter_recovery}
only requires
$k_{\mathsf{c}}$-step controllability, $T \geq k_{\mathsf{c}}$, and $m \gtrsim n$.
However, note that Theorem 1, like \Cref{stmt:upper_bound_parameter_recovery}, does
have the property that the parameter error (in operator norm)
scales as $\sqrt{n/(mT)}$, reflecting that all collected datapoints contribute to reducing error.
Next, we specialize \cite[Theorem 2]{modi2022learning}.
Theorem 2 bounds the error of an estimation procedure which outputs $m$ different
estimates $\{\hat{A}_i\}_{i=1}^{m}$, one for each observed trajectory (cf.~Eq.~(3)).
Specifically, it gives an upper bound on the quantity
$\frac{1}{m}\sum_{i=1}^{m} \norm{\hat{A}_i - A_i}_F^2$, where $A_i$ is the dynamics matrix
associated with the $i$-th trajectory.
To specialize this to our setting, we average the estimates and apply Jensen’s inequality followed by Theorem 2.
This yields the bound $\| \hat{A} - A \|_F^2 \lesssim 1/T + n^2 / (mT)$, where $\hat{A} := \frac{1}{m}\sum_{i=1}^{m} \hat{A}_i$ is the
averaged estimate.
We see that, for a fixed $T$,
as $m \rightarrow \infty$, the rate tends to $1/T$ instead of zero
(compared with the $n^2/(mT)$ bound from \Cref{stmt:upper_bound_parameter_recovery}).
Additionally,
Theorem 2 requires both that the dynamics are one-step controllable and that
the spectral radius of $A$ is bounded by $1 + O(1/T)$.
\section{Trajectory small-ball definition and examples}
\label{sec:traj_sb}
We establish risk upper bounds by studying the behavior
of the ordinary least-squares estimator.
The key technical definition that drives the analysis is a ``small-ball'' condition
on covariate sequences:
\begin{mydef}[Trajectory small-ball (TrajSB{})]
\label{def:trajectory_small_ball}
Fix a trajectory length $T \in \ensuremath{\mathbb{N}}_+$,
a parameter $k \in \{1, \dots, T\}$,
positive definite matrices $\{\Psi_j\}_{j=1}^{\floor{T/k}} \subset \mathsf{Sym}^{n}_{> 0}$,
and constants $c_{\mathsf{sb}} \geq 1$, $\alpha \in (0, 1]$.
The distribution $\mathsf{P}_x$ satisfies the
$(T, k, \{\Psi_j\}_{j=1}^{\floor{T/k}}, c_{\mathsf{sb}}, \alpha)$-\emph{trajectory-small-ball (TrajSB{})}
condition if:
\begin{enumerate}
\item $\frac{1}{\floor{T/k}} \sum_{j=1}^{\floor{T/k}} \Psi_j \preccurlyeq \Gamma_T(\mathsf{P}_x)$,
\item $\{x_t\}_{t \geq 1}$ is adapted to a filtration $\{\mathcal{F}_t\}_{t \geq 1}$, and
\item for all $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$, $j \in \{1, \dots, \floor{T/k}\}$ and $\varepsilon > 0$:
\begin{align}
\Pr_{\{x_t\} \sim \mathsf{P}_x}\left\{
\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \ip{v}{ x_t}^2 \leq \varepsilon \cdot v^\mathsf{T} \Psi_j v
~\Bigg|~ \mathcal{F}_{(j-1)k} \right\} \leq (c_{\mathsf{sb}}\varepsilon)^\alpha \:\:
\textrm{a.s.}\label{eq:trajectory_small_ball}
\end{align}
\end{enumerate}
Above, $\mathcal{F}_0$ is understood to be the minimal $\sigma$-algebra.
Additionally, the distribution $\mathsf{P}_x$ satisfies the
$(T, k, \Psi, c_{\mathsf{sb}}, \alpha)$-TrajSB{} condition
if it satisfies $(T, k, \{\Psi_j\}_{j=1}^{\floor{T/k}}, c_{\mathsf{sb}}, \alpha)$-TrajSB{} with $\Psi_j = \Psi$.
Finally, we call the parameter $k$ the \emph{excitation window}.
\end{mydef}
In \Cref{def:trajectory_small_ball}, we typically
consider the matrices $\Psi_j$ to be the sharpest almost-sure lower bound
that we can specify (in the Loewner order) on the quantity
$\mathbb{E}[ \frac{1}{k} \sum_{t=(j-1)k+1}^{jk} x_tx_t^\mathsf{T} \mid \mathcal{F}_{(j-1)k} ]$.
\Cref{sec:results:upper:small_ball_examples} lists examples
of covariate sequence distributions $\mathsf{P}_x$ that satisfy the
TrajSB{} condition.
\Cref{def:trajectory_small_ball} draws inspiration from
the block martingale small-ball condition from
\cite{simchowitz18learning}.
There are two main differences:
(a) we consider the small-ball probability
of the \emph{entire} block $\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \ip{v}{x_t}^2$
at once, instead of
the \emph{average} of small-ball probabilities:
\begin{align*}
\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \Pr \left\{ \ip{v}{x_t}^2 \leq \varepsilon \cdot v^\mathsf{T} \Psi_j v \mid \mathcal{F}_{(j-1)k} \right\},
\end{align*}
and
(b) equation \eqref{eq:trajectory_small_ball} is required to hold
at all scales $\varepsilon > 0$, instead of at a single resolution.
We need the first modification (a) to prove optimal rates
under many trajectories without assuming stability or ergodicity.
Furthermore, condition \eqref{eq:trajectory_small_ball} is implied by
a bound on the average of small-ball probabilities
(\Cref{stmt:avg_small_ball_implies_block}).
We need the second modification (b) in order to
bound the expected value of the OLS risk.
Under weaker conditions, we would only have risk bounds
that hold with high probability, as detailed in the
following remark:
\begin{myremark}
\label{rem:mixing-epsilon}
\normalfont
In \Cref{sec:app:high_prob_upper_bounds}, we consider the following
modification to \Cref{def:trajectory_small_ball},
where we instead suppose that
\eqref{eq:trajectory_small_ball} holds for \emph{some} fixed $\varepsilon$
(such that the inequality's right-hand side is strictly less than one), rather than for all $\varepsilon$;
we refer to this modification as \emph{weak trajectory small-ball}
(\Cref{def:weak_trajectory_small_ball}).
As described above, a consequence of the weak trajectory small-ball
condition is that the main OLS risk bounds now
hold with high probability (i.e., polylogarithmic in $1/\delta$)
rather than in expectation (\Cref{stmt:upper_bound_main_high_prob}).\footnote{
Such a high probability bound does not, in turn,
imply a bound on the expected risk via integration over the tail.
The reason is that the high probability bound (\Cref{stmt:upper_bound_main_high_prob}) requires that the number
of data points $mT$ grow as $\log(1/\delta)$, where
$\delta$ is the failure probability.}
A key upshot (\Cref{stmt:phi_mixing_implies_weak_small_ball}), however,
is that this change allows
for an ergodic covariate sequence with $\phi$-mixing time bounded by $\tau_{\mathsf{mix}}$
to be considered (weak) trajectory small-ball (with excitation window $k \asymp \tau_{\mathsf{mix}}$),
provided the stationary distribution $\mu$ satisfies
a standard (weak) small-ball condition~\cite{mendelson2015learningwithoutconc,koltchinskii2015smin,oliveria2016lowertail}:
\begin{align*}
\sup_{v \in \mathbb{S}^{n-1}} \Pr_\mu\{ \ip{v}{x}^2 \leq \varepsilon \cdot \mathbb{E}_\mu[\ip{v}{x}^2] \} < 1 \text{ for some } \varepsilon > 0.
\end{align*}
This in turn yields upper bounds for \problemName{Seq\smallDash{}LS}\ in the few trajectories
($m \lesssim n$) regime of the following form: if $mT \geq \tilde{\Omega}(\tau_{\mathsf{mix}} n)$,
then
\begin{align*}
L(\hat{W}_{m,T}; T, \mathsf{P}_x) \leq \tilde{O}\left( \sigma_\xi^2 \frac{pn}{mT} \right),
\end{align*}
with high probability.
This statement generalizes the risk bound for a single ergodic trajectory from
\cite[Theorem 1]{bresler2020leastsquaresmarkov}
to the ordinary least-squares estimator~\eqref{eq:ols_definition}.
We can interpret the condition on $mT$ as a ``burn-in time'' requirement.
Meanwhile, at least in the single-trajectory ($m=1$) setting, \cite[Theorem 9]{bresler2020leastsquaresmarkov}
tells us that that such a burn-in assumption ($T \gtrsim \tau_{\mathsf{mix}} n$) is necessary for a non-trivial
risk guarantee.
\end{myremark}
\subsection{Examples of trajectory small-ball distributions}
\label{sec:results:upper:small_ball_examples}
We now turn to specific examples of distributions $\mathsf{P}_x$ which satisfy
the trajectory small-ball condition.
First is the example introduced in
\Cref{sec:prob:separations}, where $x_1$ is drawn from a multivariate Gaussian
and subsequently copied as $x_{t} = x_{t-1}$ for all $t \geq 2$:
\begin{restatable}[Copies of a Gaussian draw]{myexample}{examplesimple}
\label{example:simple}
Let $\Sigma \in \mathsf{Sym}^n_{> 0}$, and
let $\mathsf{P}_x$ denote the process
$x_1 \sim N(0, \Sigma)$ and $x_{t} = x_{t-1}$ for $t \geq 2$.
Fix any $T \in \ensuremath{\mathbb{N}}_{+}$.
Then $\mathsf{P}_x$ satisfies the $(T, T, \Sigma, e, \frac 1 2)$-TrajSB{} condition.
\end{restatable}
Note that this process only satisfies the trajectory small-ball condition
with excitation window $k=T$.
In other words, the conditional distribution $x_{t+k} \mid x_t$ for $k \geq 1$
(a Dirac distribution on $x_t$)
contains no excitation as needed for learning.
This example can actually be generalized to arbitrary
Gaussian processes indexed by time:
\begin{restatable}[Gaussian processes]{myexample}{examplegaussianprocess}
\label{example:gaussian_processes}
Let $\mathsf{P}_x$ be a Gaussian process indexed by time, i.e.,
for every finite index set $I \subset \ensuremath{\mathbb{N}}_{+}$, the
collection of random variables $(x_t)_{t \in I}$
is jointly Gaussian. Let $T_{\mathsf{nd}} := \inf\{ t \in \ensuremath{\mathbb{N}}_{+} \mid \det(\mathbb{E}[x_tx_t^\mathsf{T}]) \neq 0 \}$,
and suppose $T_{\mathsf{nd}}$ is finite.
Fix a $T \in \ensuremath{\mathbb{N}}_{+}$ satisfying $T \geq T_{\mathsf{nd}}$.
Then $\mathsf{P}_x$ satisfies the $(T, T, \Gamma_T(\mathsf{P}_x), 2e, \frac{1}{2})$-TrajSB{} condition.
\end{restatable}
Our next example involves independent, but not identically distributed, covariates:
\begin{restatable}[Independent Gaussians]{myexample}{exampleindependentgaussians}
\label{example:independent_gaussians}
Let $\{\Sigma_t\}_{t \geq 1} \subset \mathsf{Sym}^n_{>0}$, and
let $\mathsf{P}_x = \otimes_{t \geq 1} N(0, \Sigma_t)$.
Fix a $T \in \ensuremath{\mathbb{N}}_{+}$. Then
$\mathsf{P}_x$ satisfies the $(T, 1, \{\Sigma_t\}_{t=1}^{T}, e, \frac 1 2)$-TrajSB{}
condition.
\end{restatable}
\noindent
\Cref{example:independent_gaussians}
allows us to select $k=1$, reflecting the independence
of the covariates across time.
We can also craft an example around a process that does not mix,
but that still exhibits an excitation window of $k=2$:
\begin{restatable}[Alternating halfspaces]{myexample}{examplepartition}
\label{example:partition}
Suppose that $n \geq 4$ is even, and let
$u_1, \dots, u_n$ be a fixed orthonormal basis of $\ensuremath{\mathbb{R}}^n$.
Put $U_0 = \Span(u_1, \dots, u_{n/2})$ and $U_1 = \Span(u_{n/2+1}, \dots, u_n)$.
Let $i_1 \sim \mathrm{Bern}(\frac 1 2)$, $i_{t+1} = i_t \mod 2$ for $t \in \ensuremath{\mathbb{N}}_{+}$,
and let $\mathsf{P}_x$ denote the process with
conditional distribution $x_t \mid i_t$
uniform over the spherical measure on
$U_{i_t} \cap \mathbb{S}^{n-1}$.
For any $T \geq 2$, the process $\mathsf{P}_x$ satisfies the
$(T, 2, I_n / (2n), e, \frac 1 2)$-TrajSB{} condition.
\end{restatable}
\noindent
To see that the covariate distribution $\{x_t\}$ does not mix, observe that
the marginal distribution for all $t$ is uniform on $\mathbb{S}^{n-1}$,
whereas the conditional distribution $x_{t+k} \mid x_t$
for any $k \in \ensuremath{\mathbb{N}}_{+}$ is either uniform on
$U_0 \cap \mathbb{S}^{n-1}$ or uniform on $U_1 \cap \mathbb{S}^{n-1}$.
Although it does not mix at all, the trajectory supplies ample excitation for learning
in any mere two steps.
Even for a process that does mix, it may exhibit an excitation window
far smaller than its mixing time. The following sets up such an example,
where again where sufficient excitation is provided with $k=2$ steps:
\begin{restatable}[Normal subspaces]{myexample}{mixingchain}
\label{example:mixing_chain}
Suppose that $n \geq 3$.
Let $u_1, \dots, u_n$ be a fixed orthonormal basis in $\ensuremath{\mathbb{R}}^n$,
and let $U_{\neg i} := \Span( \{u_j\}_{j \neq i} )$
for $i \in \{1, \dots, n\}$.
Consider the Markov chain $\{i_t\}_{t \geq 1}$ defined by
$i_1 \sim \mathrm{Unif}(\{1, \dots, n\})$, and
$i_{t+1} \mid i_t \sim \mathrm{Unif}(\{1,\dots,n\}\setminus\{i_t\})$.
Let $\mathsf{P}_x$ denote the process with
conditional distribution $x_t \mid i_t$ uniform
over the spherical measure on $U_{\neg i_t} \cap \mathbb{S}^{n-1}$.
For any $T \geq 2$, the process $\mathsf{P}_x$ satisfies the
$(T, 2, I_n / (4n-4), e, \frac 1 2)$-TrajSB{} condition.
\end{restatable}
\noindent
In this example, a straightforward computation (detailed in
\Cref{stmt:mixing_time_simple_chain}) shows that
the mixing time $\tau_{\mathsf{mix}}(\varepsilon)$
of the Markov chain $\{i_t\}_{t \geq 1}$ scales
as $\log_{n}(1/\varepsilon)$.\footnote{
For concreteness, given a discrete-time Markov chain
over a finite state-space $S$ with transition matrix $P$
and stationary distribution $\pi$, we define the mixing time as:
$\tau_{\mathsf{mix}}(\varepsilon) := \inf\{ k \in \ensuremath{\mathbb{N}} \mid \sup_{\mu \in \mathcal{P}(S)} \tvnorm{ \mu P^k - \pi} \leq \varepsilon \}$.
Here, $\mathcal{P}(S)$ denotes the set of all probability distributions over $S$,
and $\tvnorm{\cdot}$ denotes the total variation norm over distributions.}
In most analyses which rely on mixing time arguments,
one requires that the mixing time resolution $\varepsilon$ tends to zero
as either the amount of data
and/or probability of success increases;
as a concrete example, \cite[Eq.~3.2]{duchi2012ergodicmd}
suggests to set $\varepsilon = 1/\sqrt{T}$, where $T$ is the number of
samples drawn from the underlying distribution.
On the other hand, the trajectory small-ball
condition in \Cref{example:mixing_chain}
holds with a short excitation window of length $k=2$,
independently of $T$.
Next we consider linear dynamical systems.
As setup, we first define the notion of controllability for a pair
of dynamics matrices $(A, B)$:
\begin{mydef}[Controllability]
\label{def:controllability}
Let $(A, B)$ be a pair of matrices with $A \in \ensuremath{\mathbb{R}}^{n \times n}$ and $B \in \ensuremath{\mathbb{R}}^{n \times d}$. For $k \in \{1, \dots, n\}$, we say that
$(A, B)$ is \emph{$k$-step controllable} if the matrix:
\begin{align*}
\begin{bmatrix} B & AB & A^2 B & \cdots & A^{k-1} B \end{bmatrix} \in \ensuremath{\mathbb{R}}^{n \times kd}
\end{align*}
has full row rank.
\end{mydef}
The classical definition of controllability in linear
systems \citep[cf.][Chapter~25]{rugh1996linear}
is equivalent to $n$-step controllability. \Cref{def:controllability}
allows the system to be controllable in fewer than $n$ steps.
Also note that $k$ is restricted to $\{1, \dots, n\}$, since
if a system is not $n$-step controllable, it will not be
$n'$-step controllable for any $n' > n$ (by the Cayley-Hamilton theorem).
A few special cases of interest to note are as follows.
If $B$ has rank $n$, then $(A, B)$ is trivially one-step controllable
for any $A$.
On the other hand, if $(A, B)$ are in canonical controllable form
(i.e., $A$ is the companion matrix associated
with the polynomial $p(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1} + z^n$
and $B$ is the $n$-th standard basis vector),
then $(A, B)$ is $n$-step controllable. The latter corresponds directly to
the state-space representation of
autoregressive processes of order $n$, e.g.\ $\mathsf{AR}(n)$.
\begin{restatable}[Linear dynamical systems]{myexample}{examplelds}
\label{stmt:lds_traj_small_ball}
Let $(A, B)$ with $A \in \ensuremath{\mathbb{R}}^{n \times n}$ and $B \in \ensuremath{\mathbb{R}}^{n \times d}$ be
$k_{\mathsf{c}}$-step-controllable (\Cref{def:controllability}).
Let $\PxA{A,B}$ be the linear dynamical system defined in \eqref{eq:lds_definition}.
Fix any $T,k \in \ensuremath{\mathbb{N}}_+$ satisfying $T \geq k \geq k_{\mathsf{c}}$. Then, $\PxA{A,B}$ satisfies the
$(T, k, \Gamma_k(A, B), e, \frac 1 2)$-TrajSB{} condition.
\end{restatable}
In all of the examples so far,
the time-$t$ marginal distribution of covariates $x_t$ has either been
a multivariate Gaussian or a spherical measure.
To underscore the generality of the small-ball method, we can create
additional examples where this is not the case.
In what follows, we consider Volterra series~\citep{mathews2000polynomialSP},
which generalize the classical Taylor series to causal sequences.
Analogous to how polynomials can approximate continuous functions arbitrarily
well on a compact set, Volterra series can approximate
signals that depend continuously (and solely) on their history
over a bounded set of inputs~\citep[cf.][Section~1.5]{rugh1981nonlinear}.
\begin{restatable}[Degree-$D$ Volterra series]{myexample}{examplegeneralvolterra}
\label{example:volterra}
Fix a $D \in \ensuremath{\mathbb{N}}_{+}$.
Let $\{c_{i_1, \dots, i_d}^{(d, \ell)}\}_{i_1,\dots,i_d \in \ensuremath{\mathbb{N}}}$
for $d \in \{1, \dots, D\}$ and $\ell \in \{1, \dots, n\}$ denote
arbitrary rank-$d$ arrays.
Let $\{w_t^{(\ell)}\}_{t \geq 0}$ be iid\ $N(0, 1)$ random variables
for $\ell \in \{1, \dots, n\}$.
Consider the process $\mathsf{P}_x$ where for $t \geq 1$, the $\ell$-th coordinate of $x_t$,
denoted $(x_t)_{\ell}$, is:
\begin{align}
(x_t)_{\ell} = \sum_{d=1}^{D} \sum_{i_1,\dots,i_d=0}^{t-1} c_{i_1,\dots,i_d}^{(d,\ell)} \prod_{d'=1}^{d} w_{t-i_{d'}-1}^{(\ell)}. \label{eq:volterra_series}
\end{align}
Let $T_{\mathsf{nd}} := \inf\{ t \in \ensuremath{\mathbb{N}}_{+} \mid \det(\Gamma_t(\mathsf{P}_x)) \neq 0 \}$,
and suppose $T_{\mathsf{nd}}$ is finite.
There is a constant $c_D > 0$, depending only on $D$, such that
for any $T \geq T_{\mathsf{nd}}$, $\mathsf{P}_x$ satisfies the
$(T,T,\Gamma_T(\mathsf{P}_x),c_D,1/(2D))$-TrajSB{} condition.
\end{restatable}
The main idea behind \Cref{example:volterra} is that,
while $x_t$ is certainly not Gaussian,
the quadratic form
$\sum_{t=1}^{T} \ip{v}{x_t}^2$
is a degree at most $2D$ polynomial in $\{w_t^{(\ell)}\}_{t=0}^{T-1}$.
It will hence exhibit anti-concentration, according to a landmark result from
\cite{carbery2001anticonc}.
The same result
actually provides an immediate extension of this
example---as well as the previous examples---to noise distributions
with log-concave densities, such as Laplace or uniform noise.
We next present a special case of the Volterra series, where
we can choose the excitation window $k$ in the small-ball definition strictly
between the endpoints $1$ and $T$.
To set up, a few more definitions are needed:
\begin{mydef}
\label{def:rank_d_array}
Fix an integer $d \in \ensuremath{\mathbb{N}}_{+}$.
A rank-$d$ array
of coefficients $\{c_{i_1,\dots,i_d}\}_{i_1,\dots,i_d \in \ensuremath{\mathbb{N}}}$ is called:
\begin{enumerate}[label=(\alph*)]
\item \emph{symmetric} if $c_{i_1, \dots, i_d} = c_{\pi(i_1, \dots, i_d)}$
for any permutation $\pi$ of indices $i_1, \dots, i_d \in \ensuremath{\mathbb{N}}$,
\item \emph{traceless} if $c_{i,\dots,i} = 0$ for all $i \in \ensuremath{\mathbb{N}}$, and
\item \emph{non-degenerate} if there exists an $k_{\mathsf{nd}} \in \ensuremath{\mathbb{N}}_{+}$ such that
the following set is non-empty:
\begin{align*}
\{ (i_1, \dots, i_d) \mid c_{i_1, \dots, i_d} \neq 0, i_1, \dots, i_d \in \{0, \dots, k_{\mathsf{nd}} - 1\} \}.
\end{align*}
\end{enumerate}
The smallest $k_{\mathsf{nd}}$
such that $\{c_{i_1,\dots,i_d}\}$ is the \emph{non-degeneracy index}.
\end{mydef}
\begin{restatable}[Degree-$2$ Volterra series]{myexample}{exampletwovolterra}
\label{example:volterra_degree_two}
Consider the following process $\mathsf{P}_x$.
Let $\{c_{i,j}^{(\ell)}\}_{i,j \geq 0}$ for $\ell \in \{1, \dots, n\}$ be
symmetric, traceless, non-degenerate arrays (\Cref{def:rank_d_array}).
Let $\{w_t^{(\ell)}\}_{t \geq 0}$ be iid\ $N(0, 1)$ random variables
for $\ell \in \{1, \dots, n\}$.
For $t \geq 1$, the $\ell$-th coordinate of $x_t$, denoted $(x_t)_{\ell}$, is:
\begin{align}
(x_t)_{\ell} = \sum_{i=0}^{t-1} \sum_{j=i}^{t-1} c_{i,j}^{(\ell)} w_{t-i-1}^{(\ell)} w_{t-j-1}^{(\ell)}. \label{eq:toy_vector_quadratic}
\end{align}
Let $k_{\mathsf{nd}} \in \ensuremath{\mathbb{N}}_{+}$ denote the smallest non-degeneracy index for all
$n$ arrays.
There is a universal positive constant $c$ such that
for any $T$ and $k$ satisfying $T \geq k \geq k_{\mathsf{nd}}$,
$\mathsf{P}_x$ satisfies the $(T,k,\Gamma_k(\mathsf{P}_x),c,\frac{1}{4})$-TrajSB{} condition.
\end{restatable}
The assumptions pulled in from \Cref{def:rank_d_array} help simplify the
construction of an almost sure lower bound for conditional covariances
$\mathbb{E}[ \frac{1}{k} \sum_{t=(j-1)k+1}^{jk} x_tx_t^\mathsf{T} \mid \mathcal{F}_{(j-1)k}]$,
to establish that \Cref{example:volterra_degree_two} satisfies
the trajectory small-ball condition.
We believe that generalizations to higher degree Volterra series with $k$
strictly between $1$ and $T$
are possible by more involved calculations.
Of course, many other examples are possible. To help in recognizing them,
the following statement
shows that condition \eqref{eq:trajectory_small_ball}
in the trajectory small-ball definition can be verified by
separately establishing small-ball probabilities
for the conditional distributions:
\begin{restatable}[Average small-ball implies trajectory small-ball]{myprop}{avgsmallballimpliesblock}
\label{stmt:avg_small_ball_implies_block}
Fix $T \in \ensuremath{\mathbb{N}}_+$, $k \in \{1, \dots, T\}$, $\{\Psi_j\}_{j=1}^{\floor{T/k}} \subset \mathsf{Sym}^{n}_{> 0}$, and
$\alpha,\beta \in (0, 1)$.
Let $\mathsf{P}_x$ be a covariate distribution, with $\{x_t\}_{t \geq 1}$ adapted
to a filtration $\{\mathcal{F}_t\}_{t \geq 1}$. Suppose
for all $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$ and
$j \in \{1, \dots, \floor{T/k}\}$:
\begin{align}
\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \Pr_{x_t \sim \mathsf{P}_x}\left\{ \ip{v}{x_t}^2 \leq \alpha \cdot v^\mathsf{T} \Psi_j v ~\Big|~ \mathcal{F}_{(j-1)k} \right\} \leq \beta \:\: \textrm{a.s.}, \label{eq:avg_weak_small_ball_inequality}
\end{align}
where $\mathcal{F}_0$ is the minimal $\sigma$-algebra.
Then,
for all $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$,
$j \in \{1, \dots, \floor{T/k}\}$, and $\varepsilon \in (0, \alpha)$
\begin{align}
\Pr_{\{x_t\} \sim \mathsf{P}_x}\left\{
\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \ip{v}{ x_t}^2 \leq \varepsilon \cdot v^\mathsf{T} \Psi_j v
~\Bigg|~ \mathcal{F}_{(j-1)k} \right\} \leq \frac{\beta}{1-\varepsilon/\alpha}\:\:
\textrm{a.s.} \label{eq:avg_weak_small_ball_implication}
\end{align}
\end{restatable}
\noindent
An immediate corollary of \Cref{stmt:avg_small_ball_implies_block} is the following:
suppose that for all
$v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$,
$j \in \{1, \dots, \floor{T/k}\}$,
and $\varepsilon > 0$,
\begin{align}
\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \Pr_{x_t \sim \mathsf{P}_x}\left\{ \ip{v}{x_t}^2 \leq \varepsilon \cdot v^\mathsf{T} \Psi_j v ~\Big|~ \mathcal{F}_{(j-1)k} \right\} \leq (c_{\mathsf{sb}} \varepsilon)^\alpha \:\: \textrm{a.s.} \label{eq:avg_small_ball_inequality}
\end{align}
Then,
the $(T, k, \{\Psi_j\}_{j=1}^{\floor{T/k}}, 2^{1+1/\alpha} c_{\mathsf{sb}}, \alpha)$-TrajSB\ condition holds.
Equation~\eqref{eq:avg_small_ball_inequality} can be easier to verify
than \eqref{eq:trajectory_small_ball}, since the former allows
one to reason about each conditional distribution individually,
whereas the latter requires reasoning about the entire excitation
window altogether.
The following two sections present upper and lower bounds for
learning from trajectories, involving various instances of the
trajectory small-ball assumption where applicable.
All main results are summarized in \Cref{tab:results}.
\section{Analysis for upper bounds}
\label{sec:appendix:upper_bound_proofs}
\subsection{Preliminaries}
We collect various technical results
which we will use in the proof of the upper bounds.
The first result gives us a bound
on the functional inverse of $T \mapsto T/\log{T}$.
\begin{myprop}[{\cite[Lemma~A.4]{simchowitz18learning}}]
\label{prop:invert_log_t_over_t}
For $\alpha \geq 1$, $T \geq 2\alpha \log(4\alpha)$ implies that $T \geq \alpha \log{T}$.
\end{myprop}
The next two results study various properties of functions
involving $\underline{\lambda}$.
\begin{myprop}
\label{stmt:ulam_concave_first_arg}
For $A \in \mathsf{Sym}^{n}_{> 0}$, the map $X \mapsto \underline{\lambda}(X, A)$ is concave over symmetric matrices.
\end{myprop}
\begin{proof}
Observe that $\underline{\lambda}(X, A) = \lambda_{\min}(A^{-1/2} X A^{-1/2}) = \inf\{ \ip{X}{A^{-1/2} vv^\mathsf{T} A^{-1/2}} \mid v \in \mathbb{S}^{n-1}\}$
is the pointwise infimum over a set of linear functions in $X$, and is therefore concave.
\end{proof}
\begin{myprop}
\label{stmt:Psi_leq_one}
Fix $T \in \ensuremath{\mathbb{N}}_{+}$,
$\{ \Psi_t\}_{t=1}^{T} \subset \mathsf{Sym}^n_{> 0}$, and $\Gamma \in \mathsf{Sym}^n_{> 0}$.
Suppose that $\frac{1}{T} \sum_{t=1}^{T} \Psi_t \preccurlyeq \Gamma$.
Then $\left[ \prod_{t=1}^{T} \underline{\lambda}(\Psi_t, \Gamma) \right]^{1/T} \leq 1$.
\end{myprop}
\begin{proof}
We have that:
\begin{align*}
\left[ \prod_{t=1}^{T} \underline{\lambda}(\Psi_t, \Gamma) \right]^{1/T} &\leq \frac{1}{T} \sum_{t=1}^{T} \underline{\lambda}(\Psi_t, \Gamma) &&\text{using the AM-GM inequality} \\
&\leq \underline{\lambda}\left(\frac{1}{T} \sum_{t=1}^{T} \Psi_t, \Gamma \right) &&\text{using \Cref{stmt:ulam_concave_first_arg} and Jensen's inequality} \\
&\leq \underline{\lambda}(\Gamma, \Gamma) &&\text{since } \frac{1}{T} \sum_{t=1}^{T} \Psi_t \preccurlyeq \Gamma \\
&= 1.
\end{align*}
\end{proof}
The next result relates the anti-concentration properties of a non-negative
random variable to its moment generating function on
$(-\infty, 0)$.
\begin{myprop}[{\cite[Lemma~7]{mourtada19exactminimax}}]
\label{stmt:small_ball_to_mgf}
Let $X$ be a non-negative random variable. Suppose
there exists an $\alpha \in (0, 1]$ and positive constant
$c$ such that:
\begin{align*}
\Pr(X \leq t) \leq (ct)^{\alpha} \quad \forall t > 0.
\end{align*}
Then:
\begin{align*}
\mathbb{E}[\exp(-\eta X)] \leq (c/\eta)^\alpha \quad \forall \eta > 0.
\end{align*}
\end{myprop}
The next few results involve various properties of
Gaussian and spherical distributions.
\begin{myprop}[{\cite[Lemma~6.2]{magnus1978gaussianforms}}]
\label{stmt:gaussian_fourth_moment}
For $w \sim N(0, I)$ and symmetric matrices $A,B$:
\begin{align*}
\mathbb{E}[ w^\mathsf{T} A w w^\mathsf{T} B w ] = 2\ip{A}{B} + \Tr(A)\Tr(B).
\end{align*}
\end{myprop}
\begin{myprop}[{\cite[Lemma~2.2]{dasgupta2003jl}}]
\label{stmt:small_ball_unit_sphere}
Let $n \geq 2$ and $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$ be fixed.
Suppose that $\psi$ is drawn uniformly at random from the uniform measure
over $\mathbb{S}^{n-1}$. We have that for all $\varepsilon > 0$:
\begin{align*}
\Pr\left\{ \ip{v}{\psi}^2 \leq \frac{\varepsilon}{n} \norm{v}_2^2 \right\} \leq (e \varepsilon)^{1/2}.
\end{align*}
\end{myprop}
Next, we state a classic result which gives
us anti-concentration of arbitrary Gaussian (more generally any log-concave distribution) polynomials of bounded degree.
\begin{mythm}[{\cite[Theorem~8]{carbery2001anticonc}}]
\label{thm:carbery_and_wright}
Fix an integer $d \in \ensuremath{\mathbb{N}}_{+}$. There exists a universal positive constant $c$
such that the following is true.
Let $p : \ensuremath{\mathbb{R}}^n \rightarrow \ensuremath{\mathbb{R}}$ be a degree $d$ polynomial,
and let $\varepsilon > 0$.
We have:
\begin{align*}
\Pr\{ \abs{p(x)} \leq \varepsilon \cdot \mathbb{E}\abs{p(x)} \} \leq c \cdot d \varepsilon^{1/d}, \quad x \sim N(0, I_n).
\end{align*}
For the case when $d=2$ and $p$ is non-negative, we can take
$c=\sqrt{e/2}$.
\end{mythm}
\Cref{thm:carbery_and_wright} can be further specialized as follows. Suppose $w \sim N(0, I)$,
$x$ is fixed, and $\bmattwo{Q_{11}}{Q_{12}}{Q_{12}^\mathsf{T}}{Q_{22}}$
is positive semidefinite. Then:
\begin{align}
\Pr\left\{ \cvectwo{x}{w}^\mathsf{T} \bmattwo{Q_{11}}{Q_{12}}{Q_{12}^\mathsf{T}}{Q_{22}} \cvectwo{x}{w} \leq \varepsilon \cdot \Tr(Q_{22}) \right\} \leq (e \varepsilon)^{1/2} \quad \forall \varepsilon > 0. \label{eq:cw_quadratic_explicit_constant}
\end{align}
Both \eqref{eq:cw_quadratic_explicit_constant} and
the explicit constant in \Cref{thm:carbery_and_wright} for $d=2$ and $p$ non-negative
can be derived by bounding the MGF of various Gaussian quadratic forms; see e.g.~\cite{tu2023gaussian}.
Next, we state a well-known result from \cite{abbasiyadkori2011selfnormalized},
which yields an anytime bound for the size of a
self-normalized martingale difference sequence (MDS).
\begin{mylemma}[{\cite[Theorem~3]{abbasiyadkori2011selfnormalized}}]
\label{lemma:yasin}
Fix a $\delta \in (0, 1)$
and positive definite matrix $V \in \ensuremath{\mathbb{R}}^{d \times d}$.
Let $\{x_t\}_{t \geq 1} \subset \ensuremath{\mathbb{R}}^d$ be a stochastic process
adapted to the filtration $\{\mathcal{F}_t\}_{t \geq 1}$.
Let $\{\eta_t\}_{t \geq 1} \subset \ensuremath{\mathbb{R}}$ be a martingale difference
sequence adapted to $\{\mathcal{F}_t\}_{t \geq 2}$.
Suppose there exists $R > 0 $ such that $\mathbb{E}[\exp(\lambda \eta_t) \mid \mathcal{F}_t] \leq \exp(\lambda^2 R^2/2)$ a.s. for all $\lambda \in \ensuremath{\mathbb{R}}$ and $t \geq 1$.
Define $V_t := \sum_{k=1}^{t} x_kx_k^\mathsf{T}$ for $t \geq 1$.
With probability at least $1-\delta$,
\begin{align*}
\bignorm{\sum_{k=1}^{t} \eta_k x_k }_{(V_t + V)^{-1}} \leq \sqrt{2 R^2 \log\left( \frac{\det(V_t + V)^{1/2} \det(V)^{-1/2} }{\delta} \right)} \quad \forall t \geq 1.
\end{align*}
\end{mylemma}
\Cref{lemma:yasin} is generalized to vector-valued self-normalized MDS via a covering argument:
\begin{myprop}[{\cite[Proposition~8.2]{sarkar2019sysid}}]
\label{prop:yasin_vector}
Fix a $\delta \in (0, 1)$
and positive definite matrix $V \in \ensuremath{\mathbb{R}}^{d \times d}$.
Let $\{x_t\}_{t \geq 1} \subset \ensuremath{\mathbb{R}}^d$ be a stochastic process
adapted to the filtration $\{\mathcal{F}_t\}_{t \geq 1}$.
Let $\{\eta_t\}_{t \geq 1} \subset \ensuremath{\mathbb{R}}^p$ be a stochastic process adapted to $\{\mathcal{F}_t\}_{t \geq 2}$.
Suppose that for every fixed $v \in \mathbb{S}^{p-1}$, for every $t \geq 1$:
\begin{enumerate}[label=(\alph*)]
\item $\mathbb{E}[ \ip{v}{\eta_t} \mid \mathcal{F}_t ] = 0$ a.s.
\item $\mathbb{E}[ \exp(\lambda \ip{v}{\eta_t}) \mid \mathcal{F}_t ] \leq \exp(\lambda^2 R^2/2)$ a.s. for every $\lambda \in \ensuremath{\mathbb{R}}$.
\end{enumerate}
Define $V_t := \sum_{k=1}^{t} x_kx_k^\mathsf{T}$ for $t \geq 1$.
With probability at least $1-\delta$,
for all $t \geq 1$:
\begin{align*}
\bignorm{ \sum_{k=1}^{t} \eta_k x_k^\mathsf{T} (V_t+V)^{-1/2}}_{\mathrm{op}} \leq 2 \sqrt{2 R^2 \log\left( \frac{5^p\det(V_t + V)^{1/2} \det(V)^{-1/2} }{\delta} \right)}.
\end{align*}
\end{myprop}
The next result assumes $V_t$ is invertible in order to simplify
\Cref{prop:yasin_vector}.
\begin{myprop}
\label{prop:yasin_vector_easier}
Under the same hypothesis of \Cref{prop:yasin_vector}, we have
with probability at least $1-\delta$, for all $t \geq 1$:
\begin{align*}
\mathbf{1}\{V_t \succcurlyeq V\} \bignorm{ \sum_{k=1}^{t} \eta_k x_k^\mathsf{T} V_t^{-1/2}}_{\mathrm{op}} \leq 4 \cdot \mathbf{1}\{V_t \succcurlyeq V\} \sqrt{R^2 \log\left( \frac{5^p\det(V_t + V)^{1/2} \det(V)^{-1/2} }{\delta} \right)}.
\end{align*}
\end{myprop}
\begin{proof}
Observe that when $V_t \succcurlyeq V$, we have:
\begin{align*}
2 V_t \succcurlyeq V_t + V \Longrightarrow V_t^{-1} \preccurlyeq 2 (V_t + V)^{-1}.
\end{align*}
For two positive definite matrices $M_1$ and $M_2$ satisfying
$M_1 \preccurlyeq M_2$,
and any matrix $N$,
\begin{align*}
\opnorm{N M_1^{1/2}} = \sqrt{\lambda_{\max}(N M_1 N^\mathsf{T})} \leq \sqrt{\lambda_{\max}(N M_2 N^\mathsf{T})} = \opnorm{N M_2^{1/2}}.
\end{align*}
Therefore,
\begin{align*}
\mathbf{1}\{V_t \succcurlyeq V\} \bignorm{ \sum_{k=1}^{t} \eta_k x_k^\mathsf{T} V_t^{-1/2}}_{\mathrm{op}} &\leq 2 \cdot \mathbf{1}\{V_t \succcurlyeq V\} \bignorm{ \sum_{k=1}^{t} \eta_k x_k^\mathsf{T} (V_t+V)^{-1/2}}_{\mathrm{op}} \\
&\leq 4 \cdot \mathbf{1}\{V_t \succcurlyeq V\} \sqrt{R^2 \log\left( \frac{5^p\det(V_t + V)^{1/2} \det(V)^{-1/2} }{\delta} \right)},
\end{align*}
where the last inequality holds for every $t$ with probability at least $1-\delta$ by \Cref{prop:yasin_vector}.
\end{proof}
\subsection{Examples of trajectory small-ball}
In this section, we prove that the examples listed in
\Cref{sec:results:upper:small_ball_examples} satisfying the
trajectory small-ball condition (\Cref{def:trajectory_small_ball}).
\examplesimple*
\begin{proof}
When $k=T$ and $\underline{\Gamma} = I_n$, the condition
\eqref{eq:trajectory_small_ball} simplifies to:
\begin{align*}
\sup_{v \in \mathbb{S}^{n-1}} \Pr\left\{ \frac{1}{T} \sum_{t=1}^{T} \ip{v}{ x_t}^2 \leq \varepsilon \right\} \leq (c_{\mathsf{sb}} \varepsilon)^{\alpha} \quad \forall \varepsilon > 0.
\end{align*}
Since $x_1 = x_2 = \dots = x_T$, this further simplifies to:
\begin{align*}
\sup_{v \in \mathbb{S}^{n-1}} \Pr\left\{ \ip{v}{x_1}^2 \leq \varepsilon \right\} \leq (c_{\mathsf{sb}} \varepsilon)^{\alpha} \quad \forall \varepsilon > 0.
\end{align*}
Since $\ip{v}{x_1} \sim N(0, 1)$,
\Cref{eq:cw_quadratic_explicit_constant} yields that
$\Pr_{X \sim N(0, 1)}\{ X^2 \leq \varepsilon \} \leq (e \varepsilon)^{1/2}$,
so we can take $c_{\mathsf{sb}} = e$ and $\alpha = 1/2$.
\end{proof}
\examplegaussianprocess*
\begin{proof}
Since the covariates $(x_1, \dots, x_T)$ are jointly Gaussian,
we can write,
\begin{align*}
\begin{bmatrix} x_1 \\ \vdots \\ x_T \end{bmatrix} = \begin{bmatrix} \mu_1 \\ \vdots \\ \mu_T \end{bmatrix} + \begin{bmatrix} M_1 \\ \vdots \\ M_T \end{bmatrix} w,
\end{align*}
where $\mu_1, \dots, \mu_T \in \ensuremath{\mathbb{R}}^n$
and $M_1, \dots, M_T \in \ensuremath{\mathbb{R}}^{n \times nT}$ are fixed, and
$w \sim N(0, I_{nT})$.
For any $v \in \ensuremath{\mathbb{R}}^n$,
\begin{align*}
\frac{1}{T}\sum_{t=1}^{T} \ip{v}{x_t}^2 = \frac{1}{T}\sum_{t=1}^{T} \ip{v}{\mu_t + M_t w}^2.
\end{align*}
This is a degree 2 non-negative polynomial in $w$, and therefore
by \Cref{thm:carbery_and_wright}, for all $\varepsilon > 0$:
\begin{align*}
\Pr\left\{ \frac{1}{T}\sum_{t=1}^{T} \ip{v}{x_t}^2 \leq \varepsilon \mathbb{E}\left[ \frac{1}{T} \sum_{t=1}^{T} \ip{v}{x_t}^2 \right] \right\} \leq (2e \varepsilon)^{1/2}.
\end{align*}
\end{proof}
\examplepartition*
\begin{proof}
For $i \in \{0, 1\}$, let
$\psi_i$ be uniform on the uniform measure over $U_i \cap \mathbb{S}^{n-1}$,
let $P_{U_i}$ denote the orthogonal projector onto $U_i$,
and let $v_i = P_{U_i} v$.
Fix any $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$. We observe that for any $t \in \ensuremath{\mathbb{N}}_{+}$,
$\ip{v}{x_{t+1}}^2 + \ip{v}{x_{t+2}}^2 \mid i_t$ is equal
in distribution to $\ip{v}{\psi_0}^2 + \ip{v}{\psi_1}^2$, which itself is equal in distribution to $\ip{v_0}{\psi_0}^2 + \ip{v_1}{\psi_1}^2$.
Suppose first that $\norm{v_0}_2 \geq \norm{v_1}_2$. Then,
since $\norm{v}_2^2 = \norm{v_0}_2^2 + \norm{v_1}_2^2 \leq 2 \norm{v_0}_2^2$:
\begin{align*}
\left\{ \ip{v_0}{\psi_0}^2 + \ip{v_1}{\psi_1}^2 \leq \frac{\varepsilon}{n} \norm{v}_2^2 \right\} &\subseteq \left\{ \ip{v_0}{\psi_0}^2 + \ip{v_1}{\psi_1}^2 \leq \frac{2\varepsilon}{n} \norm{v_0}_2^2 \right\} \\
&\subseteq \left\{ \ip{v_0}{\psi_0}^2 \leq \frac{2\varepsilon}{n} \norm{v_0}_2^2 \right\}.
\end{align*}
Writing $\alpha_0 = (\ip{u_1}{v}, \dots, \ip{u_{n/2}}{v}) \in \ensuremath{\mathbb{R}}^{n/2}$,
by a change of coordinates
we have that $\norm{\alpha_0}_2^2 = \norm{v_0}_2^2$, and that
$\ip{v_0}{\psi_0}$ is equal in distribution to
$\ip{\alpha_0}{\zeta_0}$, where $\zeta_0$ is uniform on $\mathbb{S}^{n/2-1}$.
Since we assumed $\norm{v_0}_2 \geq \norm{v_1}_2$, we must have that $\alpha_0 \neq 0$. Hence by \Cref{stmt:small_ball_unit_sphere},
\begin{align*}
\Pr\left\{\ip{v_0}{\psi_0}^2 + \ip{v_1}{\psi_1}^2 \leq \frac{\varepsilon}{n} \norm{v}_2^2\right\} \leq \Pr\left\{\ip{\alpha_0}{\zeta_0}^2 \leq \frac{2\varepsilon}{n} \norm{\alpha_0}_2^2 \right\} \leq (e \varepsilon)^{1/2}.
\end{align*}
Note that if $\norm{v_1}_2 > \norm{v_0}_2$, an identical argument
yields the same bound.
Hence, letting $\mathcal{F}_t = \sigma(x_1, \dots, x_t)$,
we have shown that for all $t \geq 0$:
\begin{align*}
\Pr\left\{ \frac{1}{2} \sum_{\ell=1}^{2} \ip{v}{x_{t+\ell}}^2 \leq \varepsilon \cdot v^\mathsf{T} \left(\frac{1}{2n} I_n\right) v \,\bigg|\, \mathcal{F}_t\right\} \leq (e \varepsilon)^{1/2},
\end{align*}
from which the claim follows.
\end{proof}
\mixingchain*
\begin{proof}
Fix any $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$,
and for $i \in \{1, \dots, n\}$,
let $v_i = P_{U_{\neg i}} v$, where $P_{U_{\neg i}}$
is the orthogonal projector onto $U_{\neg i}$
Let $\{\psi_i\}_{i=1}^{n}$ be independent random variables, where
each $\psi_i$ is uniform on the
uniform measure over $U_{\neg i} \cap \mathbb{S}^{n-1}$.
Let indices $j,k \in \{1, \dots, n\}$ with $j \neq k$.
We first observe that
since $j \neq k$, we have that $U_{\neg j}^\perp = \Span(u_j) \subset U_{\neg k}$.
Therefore:
\begin{align*}
\norm{v}_2^2 = \norm{v_j}_2^2 + \norm{P^\perp_{U_{\neg j}} v_j}_2^2 \leq \norm{v_j}_2^2 + \norm{v_k}_2^2.
\end{align*}
Hence, assuming that
$\norm{v_j}_2 \geq \norm{v_k}_2$, we have:
\begin{align*}
\left\{ \ip{v_j}{\psi_j}^2 + \ip{v_k}{\psi_k}^2 \leq \frac{\varepsilon}{2(n-1)} \norm{v}_2^2 \right\} &\subseteq \left\{ \ip{v_j}{\psi_j}^2 + \ip{v_k}{\psi_k}^2 \leq \frac{\varepsilon}{n-1} \norm{v_j}_2^2 \right\} \\
&\subseteq \left\{ \ip{v_j}{\psi_j}^2 \leq \frac{\varepsilon}{n-1} \norm{v_j}_2^2 \right\}.
\end{align*}
Writing $\alpha_j = ( \ip{u_i}{v} )_{i \neq j} \in \ensuremath{\mathbb{R}}^{n-1}$,
by a change of coordinates we have that $\norm{\alpha_j}_2^2 = \norm{v_j}_2^2$,
and that $\ip{v_j}{\psi_j}$ is equal in distribution to $\ip{\alpha_j}{\zeta_j}$,
where $\zeta_j$ is uniform on $\mathbb{S}^{n-2}$.
Since we assumed $\norm{v_j}_2 \geq \norm{v_k}_2$, we must have that $\alpha_j \neq 0$. Hence by \Cref{stmt:small_ball_unit_sphere},
\begin{align*}
\Pr\left\{ \ip{v_j}{\psi_j}^2 + \ip{v_k}{\psi_k}^2 \leq \frac{\varepsilon}{2(n-1)} \norm{v}_2^2 \right\} \leq \Pr\left\{ \ip{\alpha_j}{\zeta_j}^2 \leq \frac{\varepsilon}{n-1} \norm{\alpha_j}_2^2 \right\} \leq (e \varepsilon)^{1/2}.
\end{align*}
On the other hand if $\norm{v_k}_2 > \norm{v_j}_2$, an identical argument yields the same bound.
Now, for any $i \in \{1, \dots, n\}$ and $t \in \ensuremath{\mathbb{N}}_{+}$:
\begin{align*}
&~~~~\Pr\left\{ \ip{v}{x_{t+1}}^2 + \ip{v}{x_{t+2}}^2 \leq \frac{\varepsilon}{2(n-1)} \norm{v}_2^2 \,\bigg|\, i_t=i \right\} \\
&= \sum_{j \neq i, k \neq j} \Pr\left\{ \ip{v}{x_{t+1}}^2 + \ip{v}{x_{t+2}}^2 \leq \frac{\varepsilon}{2(n-1)} \norm{v}_2^2 \,\bigg|\, i_t=i, i_{t+1}=j, i_{t+2}=k \right\} \Pr\{i_{t+1}=j,i_{t+2}=k \mid i_t=i\} \\
&= \sum_{j \neq i, k \neq j} \Pr\left\{ \ip{v_j}{\psi_j}^2 + \ip{v_k}{\psi_k}^2 \leq \frac{\varepsilon}{2(n-1)} \norm{v}_2^2 \right\} \Pr\{i_{t+1}=j,i_{t+2}=k \mid i_t=i\} \\
&\leq (e \varepsilon)^{1/2} \sum_{j \neq i, k \neq j} \Pr\{i_{t+1}=j,i_{t+2}=k \mid i_t=i\} \\
&= (e \varepsilon)^{1/2}.
\end{align*}
Note we also have that
$\Pr\left\{ \ip{v}{x_{1}}^2 + \ip{v}{x_{2}}^2 \leq \frac{\varepsilon}{2(n-1)} \norm{v}_2^2 \right\} \leq (e \varepsilon)^{1/2}$ by a nearly identical argument.
Hence, letting $\mathcal{F}_t = \sigma(x_1, \dots, x_t)$, we have shown that
for all $t \geq 0$:
\begin{align*}
\Pr\left\{ \frac{1}{2} \sum_{\ell=1}^{2} \ip{v}{x_{t+\ell}}^2 \leq \varepsilon \cdot v^\mathsf{T} \left( \frac{1}{4(n-1)} I_n \right) v \,\bigg|\, \mathcal{F}_t \right\} \leq (e \varepsilon)^{1/2},
\end{align*}
from which the claim follows.
\end{proof}
For the next claim, recall that the mixing time of a Markov
chain over state-space $S$ with transition matrix $P$ and stationary
distribution $\pi$ is defined as:
\begin{align*}
\tau_{\mathsf{mix}}(\varepsilon) = \inf\left\{ k \in \ensuremath{\mathbb{N}} \,\bigg|\, \sup_{\mu \in \mathcal{P}(S)} \tvnorm{ \mu P^k - \pi} \leq \varepsilon \right\}.
\end{align*}
Here, $\mathcal{P}(S)$ denotes the set of distributions over $S$, and $\tvnorm{\cdot}$ is the total-variation norm
over distributions.
\begin{myprop}
\label{stmt:mixing_time_simple_chain}
Let $n \geq 2$.
Consider the Markov chain $\{i_t\}_{t \geq 1}$
where $i_1 \sim \mathrm{Unif}(\{1, \dots, n\})$ and
$i_{t+1} \mid i_t \sim \mathrm{Unif}(\{1, \dots, n\} \setminus \{i_t\})$.
We have that:
\begin{align*}
\tau_{\mathsf{mix}}(\varepsilon) = \inf\left\{ k \in \ensuremath{\mathbb{N}} \,\bigg|\, (n-1)^{-k} \leq \frac{2\varepsilon}{1-1/n} \right\}.
\end{align*}
\end{myprop}
\begin{proof}
Let $\mathbf{1} \in \ensuremath{\mathbb{R}}^n$ denote the all ones vector.
The transition matrix for this Markov chain is:
\begin{align*}
P = \frac{1}{n-1} (\mathbf{1} \mathbf{1}^\mathsf{T} - I_n),
\end{align*}
and its stationary distribution is uniform over $\{1, \dots, n\}$.
Note that for $j \geq 1$,
$(\mathbf{1}\ind^\mathsf{T})^j = n^{j-1} \mathbf{1}\ind^\mathsf{T}$.
Since $\mathbf{1}\ind^\mathsf{T}$ and $I_n$ commute, by the binomial theorem
we have that:
\begin{align*}
P^k &= \frac{1}{(n-1)^k} \sum_{j=0}^{k} \binom{k}{j} (\mathbf{1} \mathbf{1}^\mathsf{T})^{k-j} (-1)^j \\
&= \frac{1}{(n-1)^k} \left[ \sum_{j=0}^{k-1} \binom{k}{j} n^{k-j-1} (-1)^j \mathbf{1}\ind^\mathsf{T} + (-1)^k I_n \right] \\
&= \frac{1}{(n-1)^k} \left[ \frac{1}{n} \left( (n-1)^k - (-1)^k \right) \mathbf{1}\ind^\mathsf{T} + (-1)^k I_n \right] \\
&= \frac{1}{n} \mathbf{1}\ind^\mathsf{T} + \frac{(-1)^k}{(n-1)^k} \left[ I_n - \frac{1}{n} \mathbf{1}\ind^\mathsf{T} \right].
\end{align*}
Now, let $\mu \in \ensuremath{\mathbb{R}}^n_{\geq 0}$ satisfy $\mu^\mathsf{T} \mathbf{1} = 1$.
We have:
\begin{align*}
\bignorm{\mu^\mathsf{T} P^k - \frac{1}{n} \mathbf{1}^\mathsf{T}}_1 &= \frac{1}{(n-1)^k} \bignorm{\mu - \frac{1}{n} \mathbf{1}}_1.
\end{align*}
It is straightforward to check that
$\sup_{\mu \in \ensuremath{\mathbb{R}}^n_{\geq 0}, \mu^\mathsf{T} \mathbf{1} = 1} \bignorm{\mu - \frac{1}{n} \mathbf{1}}_1 = 1-\frac{1}{n}$,
from which the claim follows,
since the TV distance between two distributions
$\mu$, $\nu$ is $\tvnorm{\mu-\nu} = \frac{1}{2} \norm{\mu-\nu}_1$.
\end{proof}
\examplelds*
\begin{proof}
Let $\Gamma_k$ be shorthand for $\Gamma_k(\mathsf{P}_x)$
and $\Sigma_k$ be shorthand for $\Sigma_k(\mathsf{P}_x)$.
Let $w = (w_1, \dots, w_{k}) \in \ensuremath{\mathbb{R}}^{nk}$ denote the vertical
concatenation of the process noise variables.
Let $M_t := \rvectwo{A^t}{\Phi_t} \in \ensuremath{\mathbb{R}}^{n \times n(k+1)}$ denote the matrix
such that $x_t = M_t \cvectwo{x}{w}$.
With this notation, for any $v \in \ensuremath{\mathbb{R}}^n$:
\begin{align*}
\frac{1}{k} \sum_{t=1}^{k} \ip{v}{x_t}^2 = \cvectwo{x}{w}^\mathsf{T} \left( \frac{1}{k} \sum_{t=1}^{k} M_t^\mathsf{T} vv^\mathsf{T} M_t \right) \cvectwo{x}{w}.
\end{align*}
By \Cref{eq:cw_quadratic_explicit_constant},
for any $\varepsilon > 0$,
\begin{align*}
\Pr\left\{ \cvectwo{x}{w}^\mathsf{T} \left( \frac{1}{k} \sum_{t=1}^{k} M_t^\mathsf{T} vv^\mathsf{T} M_t \right) \cvectwo{x}{w} \geq \varepsilon \cdot \Tr\left( \frac{1}{k} \sum_{t=1}^{k} \Phi_t^\mathsf{T} vv^\mathsf{T} \Phi_t \right) \right\} \leq (e \varepsilon)^{1/2}.
\end{align*}
On the other hand:
\begin{align*}
\Tr\left( \frac{1}{k} \sum_{t=1}^{k} \Phi_t^\mathsf{T} vv^\mathsf{T} \Phi_t \right) = v^\mathsf{T} \left( \frac{1}{k} \sum_{t=1}^{k} \Phi_t \Phi_t^\mathsf{T} \right) v = v^\mathsf{T} \left( \frac{1}{k} \sum_{t=1}^{k} \Sigma_t \right) v = v^\mathsf{T} \Gamma_k v.
\end{align*}
Because we assumed that $k \geq k_c$, then $\Gamma_k$ is invertible.
Thus, we can take $c_{\mathsf{sb}} = e$ and $\alpha = 1/2$.
\end{proof}
\begin{myprop}
\label{stmt:toy_quadratic_filter}
Consider the scalar stochastic process $\{x_t\}_{t \geq 1}$ defined by:
\begin{align*}
x_t = \sum_{i=0}^{t-1} \sum_{j=0}^{t-1} c_{i,j} w_{t-i-1} w_{t-j-1},
\end{align*}
where $\{c_{i,j}\}_{i,j \geq 0}$ are the coefficients which describe the dynamics,
and
$\{w_t\}_{t \geq 0}$ are iid\ $N(0, 1)$ random variables.
Let $\{\mathcal{F}_t\}_{t \geq 1}$ denote the filtration
defined as $\mathcal{F}_t := \sigma(w_0, \dots, w_{t-1})$, so that $x_t$ is
$\mathcal{F}_t$-measurable.
Suppose that $\{c_{i,j}\}_{i,j \geq 0}$ is
symmetric and traceless.
For every $t \geq 1$ and $k \geq 0$, almost surely we have:
\begin{align*}
\mathbb{E}[ x_{t + k}^2 \mid \mathcal{F}_k ] \geq \mathbb{E}[ x_t^2 ] + (\mathbb{E}[ x_{t+k} \mid \mathcal{F}_k])^2.
\end{align*}
\end{myprop}
\begin{proof}
For $t \in \ensuremath{\mathbb{N}}_+$, define the
symmetric matrices $M_{t} \in \ensuremath{\mathbb{R}}^{t \times t}$ with $(M_{t})_{ii} = 0$
and $(M_{t})_{ij} = c_{(i-1),(j-1)}$.
With this notation and with $\bar{w}_t \sim N(0, I_t)$, we can write $x_t$ as:
\begin{align*}
x_t = \bar{w}_t^\mathsf{T} M_t \bar{w}_t.
\end{align*}
Therefore, by \Cref{stmt:gaussian_fourth_moment} and the
assumption that $\Tr(M_t) = 0$:
\begin{align*}
\mathbb{E}[x_t^2] = \mathbb{E}(\bar{w}_t^\mathsf{T} M_t \bar{w}_t)^2
= 2\norm{M_t}_F^2 + \Tr(M_t)^2 = 2\norm{M_t}_F^2.
\end{align*}
Now, partition $M_{t+k}$ as:
\begin{align*}
M_{t+k} = \bmattwo{M_t}{D_{t,k}}{D^\mathsf{T}_{t,k}}{E_{t,k}}.
\end{align*}
Let $\bar{v}_k = (w_{k-1}, \dots, w_0)$. Given $\mathcal{F}_k$, we can write $x_{t+k}$ as:
\begin{align*}
x_{t+k} = \cvectwo{\bar{w}_t}{\bar{v}_k}^\mathsf{T} \bmattwo{M_t}{D_{t,k}}{D^\mathsf{T}_{t,k}}{E_{t,k}}\cvectwo{\bar{w}_t}{\bar{v}_k}.
\end{align*}
With this notation:
\begin{align*}
\mathbb{E}[x_{t+k} \mid \mathcal{F}_k] = \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k, \quad \mathbb{E}[x_{t+k}^2 \mid \mathcal{F}_k] &= \mathbb{E}_{\bar{w}_t}\left(\cvectwo{\bar{w}_t}{\bar{v}_k}^\mathsf{T} \bmattwo{M_t}{D_{t,k}}{D^\mathsf{T}_{t,k}}{E_{t,k}}\cvectwo{\bar{w}_t}{\bar{v}_k}\right)^2.
\end{align*}
Expanding the square:
\begin{align*}
\left(\cvectwo{\bar{w}_t}{\bar{v}_k}^\mathsf{T} \bmattwo{M_t}{D_{t,k}}{D^\mathsf{T}_{t,k}}{E_{t,k}}\cvectwo{\bar{w}_t}{\bar{v}_k}\right)^2 &= ( \bar{w}_t^\mathsf{T} M_t \bar{w}_t + 2 \bar{w}_t^\mathsf{T} D_{t,k}\bar{v}_k + \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k)^2 \\
&= ( \bar{w}_t^\mathsf{T} M_t \bar{w}_t)^2 + 4 \bar{w}_t^\mathsf{T} M_t \bar{w}_t \bar{w}_t^\mathsf{T} D_{t,k}\bar{v}_k +
2 \bar{w}_t^\mathsf{T} M_t \bar{w}_t \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k \\
&\qquad+ 4 (\bar{w}_t^\mathsf{T} D_{t,k} \bar{v}_k)^2 + 4 \bar{w}_t^\mathsf{T} D_{t,k} \bar{v}_k \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k + ( \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k )^2.
\end{align*}
Using \Cref{stmt:gaussian_fourth_moment} again:
\begin{align*}
\mathbb{E}[x_{t+k}^2 \mid \mathcal{F}_k ] &= \mathbb{E}_{\bar{w}_t}\left(\cvectwo{\bar{w}_t}{\bar{v}_k}^\mathsf{T} \bmattwo{M_t}{D_{t,k}}{D^\mathsf{T}_{t,k}}{E_{t,k}}\cvectwo{\bar{w}_t}{\bar{v}_k}\right)^2 \\
&= \mathbb{E}_{\bar{w}_t}( \bar{w}_t^\mathsf{T} M_t \bar{w}_t)^2 + 2 \Tr(M_t) \bar{v}_k^\mathsf{T} E_{t,k}\bar{v}_k + 4 \norm{D_{t,k}\bar{v}_k}_2^2 + ( \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k )^2 \\
&= 2\norm{M_t}_F^2 + 4 \norm{D_{t,k}\bar{v}_k}_2^2 + ( \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k )^2 \geq 2\norm{M_t}_F^2 + ( \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k )^2 .
\end{align*}
To complete the proof, we recall that $\mathbb{E}[x_t^2] = 2\norm{M_t}_F^2$ and
$\mathbb{E}[x_{t+k} \mid \mathcal{F}_k] = \bar{v}_k^\mathsf{T} E_{t,k} \bar{v}_k$.
\end{proof}
\exampletwovolterra*
\begin{proof}
Fix a $v \in \ensuremath{\mathbb{R}}^n$.
The relation \eqref{eq:toy_vector_quadratic} shows that
$\ip{v}{x_t}^2$ is a degree four polynomial in $\{w_i^{(\ell)}\}_{i=0,\ell=1}^{t-1,n}$.
Let $\mathcal{F}_t = \sigma(\{w_i^{(\ell)}\}_{i=0,\ell=1}^{t-1,n})$,
so that $x_t$ is $\mathcal{F}_t$-measurable.
By \Cref{thm:carbery_and_wright},
there exists a univeral positive constant $c > 0$ such that
for any $s \geq 0$,
\begin{align*}
\Pr\left\{ \frac{1}{k} \sum_{t=1}^{k} \ip{v}{x_{t+s}}^2 \leq \varepsilon \mathbb{E}\left[ \frac{1}{k} \sum_{t=1}^{k} \ip{v}{x_{t+s}}^2 \,\bigg|\, \mathcal{F}_s \right] \,\bigg|\, \mathcal{F}_s \right\} \leq (c\varepsilon)^{1/4} \quad \textrm{a.s.}
\end{align*}
To conclude the proof, we need to lower bound
$ \mathbb{E}\left[ \frac{1}{k} \sum_{t=1}^{k} \ip{v}{x_{t+s}}^2 \,\bigg|\, \mathcal{F}_s \right]$.
For any $t \geq 1$,
\begin{align*}
\mathbb{E}\left[ \ip{v}{x_{t+s}}^2 \,\bigg|\, \mathcal{F}_s \right] &= \mathbb{E}\left[ \left( \sum_{\ell=1}^{n} v_\ell \cdot (x_{t+s})_{\ell} \right)^2 \,\bigg|\, \mathcal{F}_s \right] \\
&\stackrel{(a)}{=} \sum_{\ell=1}^{n} v_\ell^2 \cdot \mathbb{E}[ (x_{t+s})_{\ell}^2 \mid \mathcal{F}_s ] + \sum_{\ell_1 \neq \ell_2}^{n} v_{\ell_1} v_{\ell_2} \cdot \mathbb{E}[ (x_{t+s})_{\ell_1} \mid \mathcal{F}_s ] \cdot \mathbb{E}[ (x_{t+s})_{\ell_2}] \mid \mathcal{F}_s ] \\
&\stackrel{(b)}{\geq} \sum_{\ell=1}^{n} v_\ell^2 \cdot \mathbb{E}[ (x_t)_\ell^2 ] + \sum_{\ell=1}^{n} v_\ell^2 \cdot (\mathbb{E}[ (x_{t+s})_\ell \mid \mathcal{F}_s ])^2 \\
&\qquad + \sum_{\ell_1 \neq \ell_2}^{n} v_{\ell_1} v_{\ell_2} \cdot \mathbb{E}[ (x_{t+s})_{\ell_1} \mid \mathcal{F}_s ] \cdot \mathbb{E}[ (x_{t+s})_{\ell_2}] \mid \mathcal{F}_s ] \\
&= \sum_{\ell=1}^{n} v_\ell^2 \cdot \mathbb{E}[ (x_t)_\ell^2 ] + \left( \sum_{\ell=1}^{n} v_\ell \cdot \mathbb{E}[ (x_{t+s})_\ell \mid \mathcal{F}_s ]\right)^2 \\
&\geq \sum_{\ell=1}^{n} v_\ell^2 \cdot \mathbb{E}[ (x_t)_\ell^2 ] \stackrel{(c)}{=} v^\mathsf{T} \Sigma_t(\mathsf{P}_x) v.
\end{align*}
Above, (a) follows
since each coordinate of $x_t$ is independent by definition,
(b) follows from \Cref{stmt:toy_quadratic_filter},
and (c) follows since $\mathbb{E}[x_t] = 0$ and each coordinate is independent,
so $\mathbb{E}[ (x_t)_{\ell_1} (x_t)_{\ell_2} ] = \mathbb{E}[(x_t)_{\ell_1}] \mathbb{E}[(x_t)_{\ell_2}] = 0$
for $\ell_1 \neq \ell_2$.
Hence, we have shown:
\begin{align*}
\mathbb{E}\left[ \frac{1}{k} \sum_{t=1}^{k} \ip{v}{x_{t+s}}^2 \,\bigg|\, \mathcal{F}_s \right] \geq v^\mathsf{T} \left( \frac{1}{k} \sum_{t=1}^{k} \Sigma_t(\mathsf{P}_x) \right) v = v^\mathsf{T} \Gamma_k(\mathsf{P}_x) v.
\end{align*}
Note that because we assume that $k \geq k_{\mathsf{nd}}$,
the covariances $\Sigma_t(\mathsf{P}_x)$ are all invertible (and hence so is $\Gamma_k(\mathsf{P}_x)$).
The claim now follows.
\end{proof}
\examplegeneralvolterra*
\begin{proof}
Fix a $v \in \ensuremath{\mathbb{R}}^n$.
The definition \eqref{eq:volterra_series}
expresses $\ip{v}{x_t}$ as a degree at most $D$ polynomial in the noise
variables $\{w_t^{(\ell)}\}$.
By \Cref{thm:carbery_and_wright}, there exists a
positive constant $c_D$, only depending on $D$,
such that:
\begin{align*}
\Pr\left\{ \frac{1}{T} \sum_{t=1}^{T} \ip{v}{x_t}^2 \leq \varepsilon \mathbb{E}\left[ \frac{1}{T} \sum_{t=1}^{T} \ip{v}{x_t}^2 \right] \right\} \leq (c_D \varepsilon)^{1/(2D)}.
\end{align*}
Since $T \geq T_{\mathsf{nd}}$, the matrix $\Gamma_T(\mathsf{P}_x)$ is invertible.
The claim now follows.
\end{proof}
\subsection{Proof of \Cref{stmt:avg_small_ball_implies_block}}
\avgsmallballimpliesblock*
\begin{proof}
The following proof builds on the argument given
in \cite[Section~E.1]{simchowitz18learning}.
We note that a similar style of proof is used in
\cite[Lemma~15]{bartlett2020benign}.
Define the shorthand notation $\Pr_{t}\{\cdot\} := \Pr\{\,\cdot \mid \mathcal{F}_t \}$, and similarly $\mathbb{E}_{t}[\cdot] := \mathbb{E}[\,\cdot \mid \mathcal{F}_t]$.
Now fix a $v \in \ensuremath{\mathbb{R}}^n \setminus \{0\}$, $j \in \{1, \dots, \floor{T/k}\}$.
Markov's inequality yields that:
\begin{align*}
\frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \ip{v}{x_t}^2 \geq \alpha v^\mathsf{T} \Psi_j v \cdot \frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \mathbf{1}\{ \ip{v}{x_t}^2 > \alpha v^\mathsf{T} \Psi_j v \},
\end{align*}
and therefore for all $\varepsilon > 0$:
\begin{align*}
\Pr_{(j-1)k}\left\{ \frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \ip{v}{x_t}^2 \leq \varepsilon \cdot v^\mathsf{T} \Psi_j v \right\} \leq \Pr_{(j-1)k}\left\{ \frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \mathbf{1}\{ \ip{v}{x_t}^2 > \alpha v^\mathsf{T} \Psi_j v \} \leq \varepsilon/\alpha \right\}.
\end{align*}
Define $Z_j := \frac{1}{k} \sum_{t=(j-1)k+1}^{jk} \mathbf{1}\{ \ip{v}{x_t}^2 > \alpha v^\mathsf{T} \Psi_j v \}$, and observe that $Z_j \in [0, 1]$.
By \eqref{eq:avg_weak_small_ball_inequality}, we have:
\begin{align*}
\mathbb{E}_{(j-1)k}[Z_j] \geq 1 - \beta.
\end{align*}
On the other hand:
\begin{align*}
\mathbb{E}_{(j-1)k}[Z_j] &= \mathbb{E}_{(j-1)k}[ Z_j \mathbf{1}\{ Z_j > \varepsilon/\alpha \}] + \mathbb{E}_{(j-1)k}[ Z_j \mathbf{1} \{Z_j \leq \varepsilon/\alpha \}] \\
&\leq \Pr_{(j-1)k}\{ Z_j > \varepsilon/\alpha \} + \varepsilon/\alpha \cdot \Pr_{(j-1)k}\{Z_j \leq \varepsilon/\alpha \} && \text{since } Z_j \leq 1 \\
&= 1 - (1-\varepsilon/\alpha) \Pr_{(j-1)k}\{Z_j \leq \varepsilon/\alpha \}.
\end{align*}
Combining both these inequalities, and further
restricting $\varepsilon \in (0, \alpha)$, we obtain,
\begin{align*}
\Pr_{(j-1)k}\{Z_j \leq \varepsilon/\alpha\} \leq \frac{\beta}{1-\varepsilon/\alpha},
\end{align*}
which implies \eqref{eq:avg_weak_small_ball_implication}.
\end{proof}
\subsection{General ordinary least-squares estimator upper bound}
\label{sec:appendix:general_OLS_proof}
In this section, we supply the proof of
\Cref{stmt:upper_bound_general}.
We first start with a result which bounds the
minimum eigenvalue of the empirical covariance matrix.
\begin{mylemma}[Minimum eigenvalue bound via trajectory small-ball]
\label{stmt:small_ball_to_min_eval}
Suppose that $\mathsf{P}_x$ satisfies
the $(T,k,\{\Psi_j\}_{j=1}^{\floor{T/k}},c_{\mathsf{sb}},\alpha)$-trajectory-small-ball condition
(\Cref{def:trajectory_small_ball}).
Put $S := \floor{T/k}$, and $\Gamma_T := \Gamma_T(\mathsf{P}_x)$.
Fix any $\underline{\Gamma} \in \mathsf{Sym}^{n}_{> 0}$ satisfying
$\frac{1}{S} \sum_{j=1}^{S} \Psi_j \preccurlyeq \underline{\Gamma} \preccurlyeq \Gamma_T$.
Define
$\tilde{X}_{m,T} := X_{m,T} \underline{\Gamma}^{-1/2}$, and:
\begin{align}
\underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma}) := \left[\prod_{j=1}^{S} \underline{\lambda}(\Psi_j, \underline{\Gamma}) \right]^{1/S}. \label{eq:geo_mean_evals}
\end{align}
Suppose that:
\begin{align}
n \geq 2, \quad \frac{mT}{kn} \geq \frac{32}{\alpha} \log\left( \frac{320 c_{\mathsf{sb}}}{ \alpha \underline{\lambda}(\underline{\Gamma}, \Gamma_T) \underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma}) } \right). \label{eq:mS_assump1}
\end{align}
For any $t \geq 0$,
with probability at least $1 - 2e^{-t}$, the following statements
simultaneously hold:
\begin{align}
\Tr(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) &\leq \frac{mTn e^t}{\underline{\lambda}(\underline{\Gamma}, \Gamma_T)}, \quad
\lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) \geq \frac{mT\alpha \underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma})}{8ec_{\mathsf{sb}}} \exp\left(-\frac{16kn}{mT\alpha} t\right). \label{eq:traj_sb_min_eval_bound}
\end{align}
\end{mylemma}
\begin{proof}
The proof uses the PAC-Bayes argument for uniform convergence
from \cite{mourtada19exactminimax}.
The first step is to construct a family of random variables,
indexed by both $v \in \mathbb{S}^{n-1}$ and a scale parameter
$\eta > 0$, such that its moment generating function is pointwise
bounded by one.
For notational brevity, let:
\begin{align*}
\underline{\lambda} := \underline{\lambda}(\underline{\Gamma}, \Gamma_T), \quad \underline{\mu} := \underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma}).
\end{align*}
Since $\underline{\Gamma} \preccurlyeq \Gamma_T$ by assumption, we have
$\underline{\lambda} \in (0, 1]$.
Similarly, since
$\frac{1}{S} \sum_{j=1}^{S} \Psi_j \preccurlyeq \underline{\Gamma}$,
we also have $\underline{\mu} \in (0, 1]$ by \Cref{stmt:Psi_leq_one}.
The trajectory small-ball condition \eqref{eq:trajectory_small_ball}
implies for any $v \in \mathbb{S}^{n-1}$, $j \in \{1, \dots, S\}$, and
$\varepsilon > 0$,
\begin{align*}
\Pr\left\{ \frac{1}{k}\sum_{t=(j-1)k+1}^{jk} \ip{v}{\underline{\Gamma}^{-1/2} x_t}^2 \leq \varepsilon \underline{\lambda}(\Psi_j, \underline{\Gamma}) \,\Bigg|\, \mathcal{F}_{(j-1)k} \right\} \leq (c_{\mathsf{sb}} \varepsilon)^\alpha.
\end{align*}
Using a change of variables $\varepsilon \gets \varepsilon/\underline{\lambda}(\Psi_j,\underline{\Gamma})$,
\begin{align*}
\Pr\left\{ \frac{1}{k}\sum_{t=(j-1)k+1}^{jk} \ip{v}{\underline{\Gamma}^{-1/2} x_t}^2 \leq \varepsilon \,\Bigg|\, \mathcal{F}_{(j-1)k} \right\} \leq \left(c_{\mathsf{sb}}/\underline{\lambda}(\Psi_j,\underline{\Gamma}) \cdot \varepsilon \right)^\alpha.
\end{align*}
By \Cref{stmt:small_ball_to_mgf}, for any $\eta > 0$,
\begin{align}
\mathbb{E}\left[ \exp\left( -\frac{\eta}{k}\sum_{t=(j-1)k+1}^{jk} \ip{v}{\underline{\Gamma}^{-1/2} x_t}^2 + \alpha \log\left(\frac{\eta \underline{\lambda}(\Psi_j, \underline{\Gamma}) }{c_{\mathsf{sb}}} \right) \right) \,\Bigg|\, \mathcal{F}_{(j-1)k} \right] \leq 1 \quad \textrm{a.s.} \label{eq:laplace_transform_single}
\end{align}
For $i \in \{1, \dots, m\}$
and $j \in \{1, \dots, S\}$,
define the random variables
$Z_j^{(i)}(v; \eta)$, $Z^{(i)}(v; \eta)$,
and $Z(v;\eta)$:
\begin{align*}
Z_j^{(i)}(v;\eta) &:= -\frac{\eta}{k} \sum_{t=(j-1)k+1}^{jk} \ip{v}{\underline{\Gamma}^{-1/2} x_t^{(i)}}^2 + \alpha \log\left(\frac{\eta \underline{\lambda}(\Psi_j, \underline{\Gamma})}{c_{\mathsf{sb}}} \right), \\
Z^{(i)}(v;\eta) &:= \sum_{j=1}^{S} Z_j^{(i)}(v; \eta), \\
Z(v, \eta) &:= \sum_{i=1}^{m} Z^{(i)}(v;\eta).
\end{align*}
We first claim that $\mathbb{E}[\exp(Z(v;\eta))] \leq 1$ for every $v \in \mathbb{S}^{n-1}$
and $\eta > 0$.
Since $Z^{(i)}(v;\eta)$ is independent of $Z^{(i')}(v;\eta)$
whenever $i \neq i'$,
we have that:
\begin{align*}
\mathbb{E}[\exp(Z(v;\eta))] = \mathbb{E}\left[ \exp\left(\sum_{i=1}^{m} Z^{(i)}(v;\eta)\right)\right] = \prod_{i=1}^{m} \mathbb{E}[ \exp(Z^{(i)}(v;\eta)) ].
\end{align*}
Furthermore, by repeated applications of the tower property and \eqref{eq:laplace_transform_single},
for every $i \in \{1, \dots, m\}$,
\begin{align*}
\mathbb{E}[\exp(Z^{(i)}(v; \eta))] &= \mathbb{E}\left[ \exp\left( \sum_{j=1}^{S} Z_j^{(i)}(v;\eta) \right) \right] \\
&=\mathbb{E}\left[ \exp\left(\sum_{j=1}^{S-1} Z^{(i)}_j(v;\eta)\right) \mathbb{E}[ \exp(Z^{(i)}_{S}(v;\eta)) \mid \mathcal{F}_{(S-1)k}]\right] \\
&\leq \mathbb{E}\left[ \exp\left(\sum_{j=1}^{S-1} Z^{(i)}_j(v;\eta)\right) \right] \\
&~\vdots \\
&\leq 1.
\end{align*}
Hence $\mathbb{E}[\exp(Z(v;\eta))] \leq 1$ for every $v \in \mathbb{S}^{n-1}$
and $\eta > 0$.
Let us now import some notation from \cite{mourtada19exactminimax}.
First, let $\pi$ denote the spherical measure on $\mathbb{S}^{n-1}$,
and let $\rho_{v,\gamma}$ denote the uniform measure over the
spherical cap
\begin{align*}
\mathcal{C}(v,\gamma) := \{ w \in \mathbb{S}^{n-1} \mid \norm{v-w}_2 \leq \gamma \}.
\end{align*}
Next, let $F_{v,\gamma}(\Sigma) := \int_{\mathcal{C}(v,\gamma)} \ip{w}{\Sigma w} \,\mathrm{d}\rho_{v,\gamma}$ for any symmetric matrix $\Sigma$.
Fix any positive $t, \eta$.
For two measures $\mu$ and $\nu$ with
$\mu$ absolutely continuous w.r.t.\ $\nu$,
let $\mathsf{KL}(\mu, \nu) := \mathbb{E}_{\mu} \log\left(\frac{\mathrm{d}\mu}{\mathrm{d}\nu}\right)$ denote the KL-divergence
between $\mu$ and $\nu$.
By the PAC-Bayes deviation bound (cf.~\cite{catoni2007pacbayes}),
there exists an event $\mathcal{E}_{t,1}$
with probability at least $1-e^{-t}$,
such that on $\mathcal{E}_{t,1}$, we have
for every $v \in \mathbb{S}^{n-1}$ and $\gamma > 0$,
\begin{align}
&-\frac{\eta}{k} F_{v,\gamma}\left(\underline{\Gamma}^{-1/2} \sum_{i=1}^{m} \sum_{t=1}^{kS} x_t^{(i)} (x_t^{(i)})^\mathsf{T} \underline{\Gamma}^{-1/2}\right) + mS \alpha \log\left(\frac{\eta \underline{\mu}}{c_{\mathsf{sb}}} \right) \leq \mathsf{KL}(\rho_{v,\gamma},\pi) + t. \label{eq:pac_bayes_bound}
\end{align}
Next, by \cite[Sections~5.3 and 5.4]{mourtada19exactminimax}, we can write $F_{v,\gamma}$
in terms of a scalar function $\phi$ such that:
\begin{align}
F_{v,\gamma}(\Sigma) = (1-\phi(\gamma))\ip{v}{\Sigma v} + \phi(\gamma) \frac{1}{n} \Tr(\Sigma), \quad \phi(\gamma) \in \left[0, \frac{n}{n-1} \gamma^2\right]. \label{eq:F_v_gamma}
\end{align}
Furthermore,
for every $v \in \mathbb{S}^{n-1}$ and $\gamma > 0$,
the KL-divergence term can be upper bounded by:
\begin{align}
\mathsf{KL}(\rho_{v,\gamma},\pi) \leq n\log\left(1 + \frac{2}{\gamma}\right). \label{eq:KL_upper_bound}
\end{align}
Therefore on $\mathcal{E}_{t,1}$,
plugging \eqref{eq:F_v_gamma} and \eqref{eq:KL_upper_bound} into \eqref{eq:pac_bayes_bound},
\begin{align*}
\lambda_{\min}\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right) &\geq \frac{k}{\eta (1-\phi(\gamma))}\left[ mS\alpha \log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) - n \log\left(1 + \frac{2}{\gamma}\right) - t \right] \\
&\qquad- \frac{\phi(\gamma)}{1-\phi(\gamma)} \frac{1}{n} \Tr\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right).
\end{align*}
Restricting $\gamma \in [0, 1/2]$, we have
from \eqref{eq:F_v_gamma} that
$0 \leq \phi(\gamma) \leq \frac{n}{n-1} \gamma^2 \leq 2 \gamma^2 \leq 1/2$.
Hence, $1-\phi(\gamma) \in [1/2, 1]$.
Furthermore, $1+2/\gamma \leq 5/(4\gamma^2)$.
Therefore,
\begin{align*}
\lambda_{\min}\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right) \geq \frac{k }{\eta}\left[ mS\alpha\log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) - n \log\left(\frac{5}{4\gamma^2}\right) - t \right] - \frac{4\gamma^2}{n} \Tr\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right).
\end{align*}
Define the non-negative random variables $\psi_i := \sum_{t=1}^{T} \norm{\underline{\Gamma}^{-1/2} x_t^{(i)}}_2^2$, for $i=1, \dots, m$.
It is straightforward to verify that
$\Tr( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} ) = \sum_{i=1}^{m} \psi_i$.
By Markov's inequality, for any $\beta > 0$:
\begin{align*}
\Pr\left( \Tr(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) > \beta \right) &= \Pr\left( \sum_{i=1}^{m} \psi_i > \beta \right) \leq \frac{\mathbb{E}\left[ \sum_{i=1}^{m} \psi_i \right]}{\beta} \\
&=
\frac{mT \Tr(\underline{\Gamma}^{-1} \Gamma_T)}{\beta} \leq \frac{mTn \lambda_{\max}(\Gamma_T^{1/2} \underline{\Gamma}^{-1} \Gamma_T^{1/2})}{\beta} = \frac{mTn}{\underline{\lambda} \beta}.
\end{align*}
Therefore, setting $\beta = \frac{e^t mTn}{\underline{\lambda}}$, there exists an event $\mathcal{E}_{t,2}$ such that $\Pr(\mathcal{E}_{t,2}^c) \leq e^{-t}$ and on $\mathcal{E}_{t,2}$,
\begin{align*}
\Tr(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) \leq \frac{e^{t} mTn}{\underline{\lambda}}.
\end{align*}
Therefore on $\mathcal{E}_{t,1} \cap \mathcal{E}_{t,2}$, which we assume holds for the
remainder of the proof, we have:
\begin{align*}
\lambda_{\min}\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right) \geq \frac{k}{\eta}\left[ mS\alpha \log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) - n \log\left(\frac{5}{4\gamma^2}\right) - t \right] - \frac{4 mTe^{t}}{\underline{\lambda}} \gamma^2
\end{align*}
Next, we further restrict $\frac{\eta \underline{\mu}}{c_{\mathsf{sb}}} \geq e$ so that $\log(\eta\underline{\mu}/c_{\mathsf{sb}}) \geq 1$.
Now consider, for positive constants $A, B, C$, the function
$x \mapsto A \log(B/x) + Cx$
on the domain $(0, \infty)$.
The derivative vanishes at $x = A/C$, and the function attains
a minimum value of $A (1 + \log(BC/A))$ with this choice of $x$.
Let us set:
\begin{align*}
A \gets \frac{k n}{\eta}, \quad B \gets \frac{5}{4}, \quad C \gets \frac{4 mT e^{t}}{\underline{\lambda}}, \quad
x \gets \gamma^2.
\end{align*}
Then by choosing $\gamma^2 = \frac{k n \underline{\lambda}}{4 \eta mT e^{t}}$, we have that:
\begin{align*}
\frac{k n}{\eta} \log\left(\frac{5}{4\gamma^2}\right) + \frac{4 mT e^{t}}{\underline{\lambda}} \gamma^2 = \frac{k n}{\eta} \left[ 1 + \log\left( \frac{5 m T e^{t} \eta }{ kn \underline{\lambda} } \right)\right].
\end{align*}
Note that this choice of $\gamma$ satisfies $\gamma \in [0, 1/2]$, since:
\begin{align*}
\frac{k n \underline{\lambda}}{4 \eta mT e^{t}} \leq \frac{1}{4} &\Longleftarrow \frac{k n}{\eta mT} \leq 1 &&\text{since } t \geq 0 \text{ and } \underline{\lambda} \leq 1 \\
&\Longleftarrow \frac{k n \underline{\mu}}{e c_{\mathsf{sb}} mT} \leq 1 &&\text{since } \eta \geq ec_{\mathsf{sb}}/\underline{\mu} \\
&\Longleftrightarrow \frac{n \underline{\mu}}{e c_{\mathsf{sb}} } \leq \frac{mT}{k} \\
&\Longleftarrow \frac{n}{ec_{\mathsf{sb}}} \leq \frac{mT}{k} &&\text{since } \underline{\mu} \leq 1,
\end{align*}
and the last condition holds by \eqref{eq:mS_assump1}.
With this choice of $\gamma$,
we have:
\begin{align}
&~~~~\lambda_{\min}\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right) \nonumber \\
&\geq \frac{k}{\eta} \left[ mS\alpha \log\left(\frac{\eta \underline{\mu}}{c_{\mathsf{sb}}}\right) - t - n\left( 1 + \log\left( \frac{5 m T e^{t} \eta}{k n \underline{\lambda}} \right)\right) \right] \nonumber \\
&= \frac{k}{\eta} \left[ \left(mS \alpha - n \right) \log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) -t - n\left(1 + \log\left( \frac{5 c_{\mathsf{sb}} mT e^{t}}{kn \underline{\lambda} \underline{\mu}} \right)\right) \right] \nonumber \\
&\geq \frac{k}{\eta} \left[ \frac{mS\alpha}{2} \log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) -t - n\left(1 + \log\left( \frac{5 c_{\mathsf{sb}} mT e^{t}}{kn \underline{\lambda} \underline{\mu}} \right)\right) \right] &&\text{since } mS\geq 2n/\alpha \nonumber \\
&=\frac{k}{\eta} \left[ \frac{mS\alpha}{2} \log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) - (1 + n)t - n\left(1 + \log\left( \frac{5 c_{\mathsf{sb}} mT}{kn \underline{\lambda} \underline{\mu}} \right)\right) \right] \nonumber \\
&\geq \frac{k}{\eta} \left[ \frac{mS\alpha}{2} \log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) - 2nt - n\left(1 + \log\left( \frac{5 c_{\mathsf{sb}} mT}{kn \underline{\lambda} \underline{\mu}} \right)\right) \right] \nonumber \\
&\geq \frac{k}{\eta} \left[ \frac{mS\alpha}{4} \log\left(\frac{\eta\underline{\mu}}{c_{\mathsf{sb}}}\right) - 2n t \right] &&\text{since } \frac{mS\alpha}{4n} \geq 1 + \log\left( \frac{5 c_{\mathsf{sb}} mT}{kn \underline{\lambda} \underline{\mu}} \right) \nonumber \\
&= \frac{kmS\alpha}{4c_{\mathsf{sb}}/\underline{\mu}} \left[\frac{\log(\eta\underline{\mu}/c_{\mathsf{sb}}) - \frac{8nt}{mS\alpha}}{\eta\underline{\mu}/c_{\mathsf{sb}}}\right].
\label{eq:before_eta_optimization}
\end{align}
It remains to optimize over $\eta \in [e c_{\mathsf{sb}}/\underline{\mu}, \infty)$.
For any $G \in \ensuremath{\mathbb{R}}$, the function $\eta' \mapsto \frac{\log{\eta'} - G}{\eta'}$
on $(0, \infty)$
attains a maximum of $\exp(-1-G)$ at $\eta' = \exp(1 + G)$.
Hence, setting $\eta = \frac{c_{\mathsf{sb}}}{\underline{\mu}} \exp( 1 + 8nt/(mS\alpha) )$, which satisfies
$\eta \geq e c_{\mathsf{sb}}/\underline{\mu}$,
we have:
\begin{align*}
\lambda_{\min}\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right) &\geq \frac{kmS\alpha \underline{\mu}}{4ec_{\mathsf{sb}}} \exp\left( - \frac{8nt}{mS\alpha} \right) \\
&\geq \frac{mT\alpha \underline{\mu}}{8ec_{\mathsf{sb}}} \exp\left(-\frac{16kn}{mT\alpha} t \right) &&\text{since } S \geq T/(2k).
\end{align*}
The claim now follows by gathering the requirements on
the quantity $\frac{mT}{kn}$ and simplifying as in \eqref{eq:mS_assump1}.
\end{proof}
\begin{mycorollary}
\label{stmt:invertible_almost_surely}
Assume the hypothesis of \Cref{stmt:small_ball_to_min_eval} hold.
Then, $\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}$ is invertible almost surely.
\end{mycorollary}
\begin{proof}
For any $t \geq 0$, define the event $\mathcal{E}_t$ as:
\begin{align*}
\mathcal{E}_t := \left\{ \lambda_{\min}\left( \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \right) < \frac{mT\alpha\underline{\mu}}{8ec_{\mathsf{sb}}} \exp\left(-\frac{16kn}{mT\alpha} t\right) \right\}.
\end{align*}
The event $\{\lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) = 0\}$
is the intersection $\bigcap_{t = 1}^{\infty} \mathcal{E}_t$.
By \Cref{stmt:small_ball_to_min_eval}, we have that
$\Pr(\mathcal{E}_t) \leq 2 e^{-t}$.
Since the events $\mathcal{E}_{t'} \subseteq \mathcal{E}_t$ whenever $t' \geq t$,
by continuity of measure from above,
\begin{align*}
\Pr(\lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) = 0) = \Pr\left( \bigcap_{t=1}^{\infty} \mathcal{E}_t \right) = \lim_{t \rightarrow \infty} \Pr(\mathcal{E}_t) \leq \lim_{t \rightarrow \infty} 2 e^{-t} = 0.
\end{align*}
\end{proof}
We are now ready to restate and prove \Cref{stmt:upper_bound_general}.
\upperboundgeneral*
\begin{proof}
For notational brevity, let:
\begin{align*}
\underline{\lambda} := \underline{\lambda}(\underline{\Gamma}, \Gamma_T), \quad \underline{\mu} := \underline{\mu}(\{\Psi_j\}_{j=1}^{S}, \underline{\Gamma}).
\end{align*}
We choose $c_0 \geq 64$ such that \eqref{eq:mTkn_requirements}
implies \eqref{eq:mS_assump1},
and also so that $\frac{mT}{kn} \geq 64/\alpha$.
By \Cref{stmt:invertible_almost_surely},
$X_{m,T}$ has full column rank almost surely, hence:
\begin{align*}
\hat{W}_{m,T} - W_\star = \Xi_{m,T}^\mathsf{T} X_{m,T} (X_{m,T}^\mathsf{T} X_{m,T})^{-1}.
\end{align*}
Put $\tilde{X}_{m,T} := X_{m,T} \underline{\Gamma}^{-1/2}$.
With this decomposition, we have:
\begin{align*}
\norm{ \hat{W}_{m,T} - W_\star }_{\Gamma'}^2
&= \norm{\Xi_{m,T}^\mathsf{T} X_{m,T} (X_{m,T}^\mathsf{T} X_{m,T})^{-1}}^2_{\Gamma'} \\
&= \norm{\Xi_{m,T}^\mathsf{T} X_{m,T} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \underline{\Gamma}^{1/2}}_{\underline{\Gamma}^{-1/2} \Gamma' \underline{\Gamma}^{-1/2}}^2 \\
&\leq \lambda_{\max}(\underline{\Gamma}^{-1/2} \Gamma' \underline{\Gamma}^{-1/2}) \norm{\Xi_{m,T}^\mathsf{T} X_{m,T} (X_{m,T}^\mathsf{T} X_{m,T})^{-1} \underline{\Gamma}^{1/2}}_F^2 \\
&= \lambda_{\max}(\underline{\Gamma}^{-1/2} \Gamma' \underline{\Gamma}^{-1/2}) \norm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }_F^2 \\
&\leq \min\{ n, p \} \lambda_{\max}(\underline{\Gamma}^{-1/2} \Gamma' \underline{\Gamma}^{-1/2}) \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 \\
&\leq \min\{ n, p \} \lambda_{\max}(\underline{\Gamma}^{-1/2} \Gamma' \underline{\Gamma}^{-1/2}) \frac{ \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 }{\lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} )} \\
&= \min\{ n, p \} \frac{ \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 }{\underline{\lambda}(\underline{\Gamma}, \Gamma') \cdot \lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} )}.
\end{align*}
Fix any $t > 0$.
By \Cref{stmt:small_ball_to_min_eval},
there exists
an event $\mathcal{E}_{t,1}$ with probability at least $1 - 2e^{-t}$,
such that on $\mathcal{E}_{t,1}$ we have:
\begin{align*}
\Tr(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) \leq \frac{mnT e^t}{\underline{\lambda}}, \quad \lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) \geq \frac{m T \alpha \underline{\mu}}{8ec_{\mathsf{sb}}} \exp\left(-\frac{16kn}{mT\alpha} t\right).
\end{align*}
Recall by our cohice of $c_0$, we have $mT/k \geq 64 n/\alpha$.
Hence, on $\mathcal{E}_{t,1}$,
\begin{align*}
\lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}) \geq \zeta_t := \frac{m T \alpha \underline{\mu}}{8ec_{\mathsf{sb}}} \exp(-t/4).
\end{align*}
We now apply \Cref{prop:yasin_vector_easier}
with $V \gets M_t := \zeta_t I_n$ and:
\begin{align*}
&x_{1}, \dots, x_{T}, x_{T+1}, \dots, x_{2T}, \dots, x_{(m-1)T+1}, \dots, x_{mT} \gets \\
&\qquad \underline{\Gamma}^{-1/2} x_1^{(1)}, \dots, \underline{\Gamma}^{-1/2} x_T^{(1)}, \underline{\Gamma}^{-1/2}x_{1}^{(2)}, \dots, \underline{\Gamma}^{-1/2}x_{T}^{(2)}, \dots, \underline{\Gamma}^{-1/2} x_{1}^{(m)}, \dots, \underline{\Gamma}^{-1/2} x_{T}^{(m)},
\end{align*}
to conclude that there exists an event $\mathcal{E}_{t,2}$ with
probability at least $1-e^{-t}$ such that on $\mathcal{E}_{t,2}$:
\begin{align*}
&~~~~\mathbf{1}\{\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} \succcurlyeq M_t\} \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 \\
&\leq 16 \sigma_\xi^2 \left[ p \log{5} + \frac{1}{2}\log\det\left(I_n + \zeta_t^{-1} \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}\right) + t \right] \\
&\leq 32 \sigma_\xi^2 \left[ p + \log\det\left(I_n + \zeta_t^{-1} \tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T}\right) + t \right] \\
&\leq 32 \sigma_\xi^2 \left[ p + n \log(1 + \zeta_t^{-1} \Tr(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})/n) + t \right].
\end{align*}
Above, the last inequality holds since
$\log\det(X) \leq n \log(\Tr(X)/n)$ for any $X \in \mathsf{Sym}^n_{\geq 0}$
by the AM-GM inequality.
By \Cref{prop:invert_log_t_over_t}, whenever $t \geq 8 \log{16}$, we have
$t \leq e^{t/4}$.
Furthermore, for any $t \geq 0$ we have $1 \leq e^{t/4}$.
Therefore, for $t \geq 8\log{16}$,
on $\mathcal{E}_{t,1} \cap \mathcal{E}_{t,2}$:
\begin{align*}
&~~~~\frac{ \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 }{ \lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} )} \\
&\leq \frac{256ec_{\mathsf{sb}}}{mT\alpha} e^{t/4} \sigma_\xi^2 \left[ p + n \log\left( 1 + \frac{8ec_{\mathsf{sb}}}{\alpha \underline{\lambda} \underline{\mu}} e^{(1+1/4)t} \right) + t \right] \\
&\leq \frac{256ec_{\mathsf{sb}}}{mT\alpha} e^{t/4} \sigma_\xi^2 \left[ p + n \log\left( \frac{16 e c_{\mathsf{sb}}}{\alpha \underline{\lambda} \underline{\mu}}\right) + n(1+1/4)t + t \right] \\
&\leq \frac{256ec_{\mathsf{sb}}}{mT\alpha} e^{t/4} \sigma_\xi^2 \left[ p + n \log\left( \frac{16 e c_{\mathsf{sb}}}{\alpha \underline{\lambda} \underline{\mu}}\right) + 3nt \right] \\
&\leq \frac{256ec_{\mathsf{sb}}}{mT\alpha} \sigma_\xi^2 \left[ p + n \log\left( \frac{16 e c_{\mathsf{sb}}}{\alpha \underline{\lambda} \underline{\mu}}\right) + 3n \right] e^{t/2} .
\end{align*}
Define the random variable $Z$ as:
\begin{align*}
Z := \frac{ \opnorm{ (\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T})^{-1/2} \tilde{X}_{m,T}^\mathsf{T} \Xi_{m,T} }^2 }{ \lambda_{\min}(\tilde{X}_{m,T}^\mathsf{T} \tilde{X}_{m,T} )} \left( \frac{256ec_{\mathsf{sb}}}{mT\alpha} \sigma_\xi^2 \left[ p + n \log\left( \frac{16 e c_{\mathsf{sb}}}{\alpha \underline{\lambda} \underline{\mu}}\right) + 3n \right] \right)^{-1} .
\end{align*}
We have shown that:
\begin{align*}
\Pr(Z > e^{t/2}) \leq 3e^{-t} \:\: \forall t \geq 8 \log{16} \Longleftrightarrow
\Pr(Z > s) \leq 3 s^{-2} \:\: \forall s \geq 16^4.
\end{align*}
Hence,
\begin{align*}
\mathbb{E}[Z] = \int_0^\infty \Pr(Z > s) \,\mathrm{d} s \leq 16^4 + 3\int_{16^4}^{\infty} s^{-2} \,\mathrm{d} s = 16^4 + 3/16^4.
\end{align*}
That is, for some universal positive $c_1$,
\begin{align}
\mathbb{E}[\norm{\hat{W}_{m,T} - W_\star}^2_{\Gamma'}] \leq c_1 \sigma_\xi^2 \min\{n,p\} c_{\mathsf{sb}} \left[\frac{p + n\log\left(\frac{\max\{e,c_{\mathsf{sb}}\}}{\alpha \underline{\lambda} \underline{\mu}}\right)}{ mT \alpha \underline{\lambda}(\underline{\Gamma},\Gamma') \underline{\mu} }\right]. \label{eq:bound_with_min}
\end{align}
Now, if $p \leq n$, \eqref{eq:bound_with_min}
is upper bounded by:
\begin{align*}
c_1 \sigma_\xi^2 p c_{\mathsf{sb}} \left[\frac{p + n \log\left(\frac{ \max\{e,c_{\mathsf{sb}}\}}{\alpha \underline{\lambda} \underline{\mu}}\right) }{ mT \alpha \underline{\lambda}(\underline{\Gamma},\Gamma') \underline{\mu} }\right] \leq 2c_1 \sigma_\xi^2 p c_{\mathsf{sb}} \left[\frac{ n \log\left(\frac{\max\{e,c_{\mathsf{sb}}\}}{\alpha \underline{\lambda} \underline{\mu}}\right) }{ mT \alpha \underline{\lambda}(\underline{\Gamma},\Gamma') \underline{\mu} }\right].
\end{align*}
On the other hand, if $p > n$, \eqref{eq:bound_with_min}
is upper bounded by:
\begin{align*}
c_1 \sigma_\xi^2 n c_{\mathsf{sb}} \left[\frac{p + n \log\left(\frac{ \max\{e,c_{\mathsf{sb}}\}}{\alpha \underline{\lambda} \underline{\mu}}\right) }{ mT \alpha \underline{\lambda}(\underline{\Gamma},\Gamma') \underline{\mu} }\right] < 2 c_1 \sigma_\xi^2 n c_{\mathsf{sb}} \left[\frac{p \log\left(\frac{\max\{e,c_{\mathsf{sb}}\}}{\alpha \underline{\lambda} \underline{\mu}}\right) }{ mT \alpha \underline{\lambda}(\underline{\Gamma},\Gamma') \underline{\mu} }\right].
\end{align*}
\end{proof}
\subsection{Proof of \Cref{stmt:upper_bound_ind_seq_ls}}
\upperboundindseqls*
\begin{proof}
Equation~\eqref{eq:small_ball_independent}
shows that $\mathsf{P}_x$ satisfies the
$(T, 1, \{\Sigma_t\}_{t=1}^{T}, c_{\mathsf{sb}}, \alpha)$-TrajSB{} condition.
Let $\Gamma_t := \frac{1}{t} \sum_{k=1}^{t} \Sigma_k$ for $t \in \ensuremath{\mathbb{N}}_{+}$.
For any $s,t \in \ensuremath{\mathbb{N}}_{+}$ with $s \leq t$,
\begin{align*}
\underline{\lambda}(\Gamma_s, \Gamma_t) &\geq \underline{\lambda}(\Gamma_s, \Sigma_t) &&\text{since } \Gamma_t \preccurlyeq \Sigma_t \\
&\geq \frac{1}{s} \sum_{k=1}^{s} \underline{\lambda}(\Sigma_k, \Sigma_t) &&\text{using \Cref{stmt:ulam_concave_first_arg} and Jensen's inequality} \\
&\geq \frac{1}{c_\beta s} \sum_{k=1}^{s} (k/t)^\beta &&\text{using \eqref{eq:variance_growth_condition}} \\
&\geq \frac{1}{c_\beta (\beta+1)} (s/t)^\beta &&\text{since } x \mapsto x^\beta \text{ is increasing}.
\end{align*}
Next, the growth condition \eqref{eq:variance_growth_condition}
implies that:
\begin{align*}
\underline{\mu}(\{\Sigma_t\}_{t=1}^{T}, \Gamma_T) &= \left[ \prod_{t=1}^{T} \underline{\lambda}(\Sigma_t, \Gamma_T) \right]^{1/T} \\
&\geq \left[ \prod_{t=1}^{T} \underline{\lambda}(\Sigma_t, \Sigma_T) \right]^{1/T} &&\text{since } \Gamma_T \preccurlyeq \Sigma_T \\
&\geq \left[ \prod_{t=1}^{T} \frac{1}{c_\beta} (t/T)^\beta \right]^{1/T} &&\text{using \eqref{eq:variance_growth_condition}} \\
&= \frac{1}{c_\beta T^\beta} (T!)^{\beta/T} \\
&\geq \frac{1}{c_\beta e^\beta} &&\text{since } T! \geq (T/e)^T.
\end{align*}
We now apply \Cref{stmt:upper_bound_general}
with $\underline{\Gamma} = \Gamma_T$.
In doing so, the requirement \eqref{eq:mTkn_requirements}
simplifies to:
\begin{align*}
n \geq 2, \quad \frac{mT}{n} \geq \frac{c_0}{\alpha} \log\left( \frac{\max\{e, c_{\mathsf{sb}}\} c_\beta e^\beta}{\alpha} \right).
\end{align*}
We first assume that ${T'} \leq T$, in which case
\eqref{eq:general_risk_bound}
yields:
\begin{align*}
\mathbb{E}[L(\hat{W}_{m,T};{T'}, \mathsf{P}_x)] \leq c_1 c_{\mathsf{sb}} \sigma_\xi^2 \cdot \frac{pn}{mT\alpha} \cdot c_\beta e^\beta \cdot \log\left( \frac{\max\{e,c_{\mathsf{sb}}\} c_\beta e^\beta}{\alpha} \right).
\end{align*}
On the other hand, when ${T'} > T$, we have
$\underline{\lambda}(\Gamma_T, \Gamma_{{T'}}) \geq \frac{1}{c_b(\beta+1)} (T/{T'})^\beta$, and
\eqref{eq:general_risk_bound} yields:
\begin{align*}
\mathbb{E}[L(\hat{W}_{m,T};{T'}, \mathsf{P}_x)] \leq c_1 c_{\mathsf{sb}} \sigma_\xi^2 \cdot \frac{pn}{mT\alpha} \cdot c_\beta e^\beta \cdot c_\beta(\beta+1)\left(\frac{{T'}}{T}\right)^\beta \cdot \log\left( \frac{\max\{e,c_{\mathsf{sb}}\} c_\beta e^\beta}{\alpha} \right).
\end{align*}
\end{proof}
\subsection{Proofs for linear dynamical systems}
\subsubsection{Control of ratios of covariance matrices}
\begin{myprop}
\label{prop:ratio_covariances}
Let $(A, B)$ be the dynamics matrices for an \DLR{}
instance, and suppose $(A, B)$ satisfy
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability},
and \Cref{assume:one_step_controllability}.
Put $\Sigma_t := \Sigma_t(A, B)$ for $t \in \ensuremath{\mathbb{N}}_{+}$ and
$\gamma := \gamma(A, B)$.
For any integers $T_1, T_2$ satisfying $1 \leq T_2 \leq T_1$,
\begin{align*}
\lambda_{\min}(\Sigma_{T_1}^{-1/2} \Sigma_{T_2} \Sigma_{T_1}^{-1/2}) \geq \frac{1}{\gamma}\frac{T_2}{T_1}.
\end{align*}
\end{myprop}
\begin{proof}
Observe that for any $t \geq 1$,
\begin{align*}
\Sigma_t = \sum_{k=0}^{t-1} A^k BB^* (A^k)^* = \sum_{k=0}^{t-1} S D^k S^{-1} BB^* S^{-*} (D^k)^* S^{*}.
\end{align*}
By \Cref{assume:one_step_controllability}, we have that
$BB^*$ is invertible, and hence $S^{-1} BB^* S^{-*}$ is also invertible.
Therefore we have the following lower and upper bound on $\Sigma_t$:
\begin{align}
\lambda_{\min}(S^{-1}BB^*S^{-*}) \cdot S \left(\sum_{k=0}^{t-1} D^k (D^k)^* \right) S^* \preccurlyeq \Sigma_t \preccurlyeq \lambda_{\max}(S^{-1}BB^*S^{-*}) \cdot S \left(\sum_{k=0}^{t-1} D^k (D^k)^* \right) S^*.
\end{align}
Now recall that for two square matrices $X,Y$, the eigenvalues of $XY$ coincide with the eigenvalues of $YX$.
Letting $Q_t := \sum_{k=0}^{t-1} D^k (D^k)^*$, we have:
\begin{align*}
\lambda_{\min}( \Sigma_{T_1}^{-1/2} \Sigma_{T_2} \Sigma_{T_1}^{-1/2} ) &\geq \lambda_{\min}(S^{-1} BB^* S^{-*}) \lambda_{\min}(\Sigma_{T_1}^{-1/2} S Q_{T_2} S^{*} \Sigma_{T_1}^{-1/2}) \\
&= \lambda_{\min}(S^{-1} BB^* S^{-*}) \lambda_{\min}( (SQ_{T_2}S^*)^{1/2} \Sigma_{T_1}^{-1} (SQ_{T_2}S^*)^{1/2}) \\
&\geq \frac{\lambda_{\min}(S^{-1} BB^* S^{-*})}{\lambda_{\max}(S^{-1} BB^* S^{-*})} \lambda_{\min}( (SQ_{T_2}S^*)^{1/2} (S^{-*} Q_{T_1}^{-1} S^*) (S Q_{T_2} S^*)^{1/2}) \\
&= \frac{\lambda_{\min}(S^{-1} BB^* S^{-*})}{\lambda_{\max}(S^{-1} BB^* S^{-*})} \lambda_{\min}( Q_{T_2} Q_{T_1}^{-1} ).
\end{align*}
Let $\lambda \in \ensuremath{\mathbb{C}}$ be an eigenvalue of $A$.
We have
\begin{align*}
\sum_{k=0}^{t-1} \abs{\lambda}^{2k} = \begin{cases}
\frac{1 - \abs{\lambda}^{2t}}{1-\abs{\lambda}^2} &\text{if } \abs{\lambda} < 1, \\
t &\text{if } \abs{\lambda} = 1.
\end{cases}
\end{align*}
Therefore, $(Q_{T_2} Q_{T_1}^{-1})_{ii}$ is:
\begin{align*}
(Q_{T_2} Q_{T_1}^{-1})_{ii} = \begin{cases} \frac{1-\abs{\lambda_i}^{2T_2}}{1-\abs{\lambda_i}^{2T_1}} = \frac{1-(\abs{\lambda_i}^{2T_1})^{T_2/T_1}}{1-\abs{\lambda_i}^{2T_1}} &\text{if } \abs{\lambda_i} < 1, \\
T_2/T_1 &\text{if } \abs{\lambda_i} = 1.
\end{cases}
\end{align*}
Note that $\inf_{x \in (0, 1)} \frac{1-x^c}{1-x} = c$ for $c \in [0, 1]$.
Therefore, we can lower bound:
\begin{align*}
\lambda_{\min}(Q_{T_2} Q_{T_1}^{-1}) \geq \frac{T_2}{T_1}.
\end{align*}
The claim now follows.
\end{proof}
\begin{myprop}
\label{prop:lower_bound_ratio}
Let $(A, B)$ be the dynamics matrices for an \DLR{}
instance, and suppose $(A, B)$ satisfy
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability}, and
\Cref{assume:one_step_controllability}.
Put $\Gamma_t := \Gamma_t(A, B)$ for $t \in \ensuremath{\mathbb{N}}_{+}$ and
$\gamma := \gamma(A, B)$.
For any integers $k,t \in \ensuremath{\mathbb{N}}_{+}$ satisfying $k \leq t$, we have:
\begin{align*}
\underline{\lambda}(\Gamma_k, \Gamma_t) \geq \frac{1}{8\gamma} \frac{k}{t}.
\end{align*}
\end{myprop}
\begin{proof}
Let $\Sigma_t := \Sigma_t(A, B)$ for $t \in \ensuremath{\mathbb{N}}_{+}$.
We first consider the case when $k \geq 2$.
Observe that $\Gamma_t \preccurlyeq \Sigma_t$.
Furthermore, for any $k \geq 2$, we have:
\begin{align*}
\Gamma_k = \frac{1}{k} \sum_{k'=1}^{k} \Sigma_{k'} \succcurlyeq \frac{1}{k} \sum_{k'=\floor{k/2}}^{k} \Sigma_{\floor{k/2}} = \frac{k - \floor{k/2}+1}{k} \Sigma_{\floor{k/2}} \succcurlyeq \frac{1}{2} \Sigma_{\floor{k/2}}.
\end{align*}
Therefore,
\begin{align*}
\underline{\lambda}(\Gamma_k, \Gamma_t) = \lambda_{\min}(\Gamma_t^{-1/2} \Gamma_k \Gamma_t^{-1/2}) \geq \frac{1}{2} \lambda_{\min}( \Sigma_t^{-1/2} \Sigma_{\floor{k/2}} \Sigma_t^{-1/2} ) \stackrel{(a)}{\geq} \frac{1}{2\gamma} \frac{\floor{k/2}}{t} \geq \frac{1}{8\gamma} \frac{k}{t}.
\end{align*}
Above, (a) follows from \Cref{prop:ratio_covariances}.
When $k=1$, we have $\Gamma_1 = \Sigma_1$, and therefore by \Cref{prop:ratio_covariances}:
\begin{align*}
\underline{\lambda}(\Gamma_1, \Gamma_t) = \underline{\lambda}(\Sigma_1, \Gamma_t) \geq \underline{\lambda}(\Sigma_1, \Sigma_t) = \lambda_{\min}(\Sigma_t^{-1/2} \Sigma_1 \Sigma_t^{-1/2}) \geq \frac{1}{\gamma} \frac{1}{t}.
\end{align*}
The claim now follows.
\end{proof}
\begin{myfact}
\label{stmt:Gamma_t_is_monotonic}
Let $(A, B)$ be the dynamics matrices for an \DLR{} instance.
For any $s, t \in \ensuremath{\mathbb{N}}_{+}$ with $s \leq t$:
\begin{align*}
\Gamma_s(A, B) \preccurlyeq \Gamma_t(A, B).
\end{align*}
\end{myfact}
\begin{myprop}
\label{stmt:uMu_lower_bound_lds_marginals}
Let $(A, B)$ be the dynamics matrices for an \DLR{}
instance, and suppose $(A, B)$ satisfy
\Cref{assume:marginal_stability},
\Cref{assume:diagonalizability}, and
\Cref{assume:one_step_controllability}.
Put $\Gamma_t := \Gamma_t(A, B)$ for $t \in \ensuremath{\mathbb{N}}_{+}$,
$\Sigma_t := \Sigma_t(A, B)$ for $t \in \ensuremath{\mathbb{N}}_{+}$, and
$\gamma := \gamma(A, B)$.
For any $T$, we have:
\begin{align*}
\left[ \prod_{t=1}^{T} \underline{\lambda}(\Sigma_t, \Gamma_T) \right]^{1/T} \geq \frac{1}{8e\gamma}.
\end{align*}
\end{myprop}
\begin{proof}
By \Cref{prop:lower_bound_ratio},
we have that $\underline{\lambda}(\Gamma_t,\Gamma_T) \geq \frac{1}{8\gamma} \frac{t}{T}$
for all $t \in \{1, \dots, T\}$. Therefore,
since $\underline{\lambda}(\Sigma_t, \Gamma_T) \geq \underline{\lambda}(\Gamma_t, \Gamma_T)$,
and since $n! \geq (n/e)^n$ for all $n \in \ensuremath{\mathbb{N}}_{+}$,
\begin{align*}
\left[ \prod_{t=1}^{T} \underline{\lambda}(\Sigma_t, \Gamma_T) \right]^{1/T} \geq \frac{(T!)^{1/T}}{8\gamma T} \geq \frac{1}{8e \gamma}.
\end{align*}
\end{proof}
\subsubsection{Many trajectory results}
\begin{mylemma}
\label{stmt:upper_bound_many_trajs_helper}
There are universal positive constants $c_0$ and $c_1$ such that the
following holds for any instance of \DLR{}.
Suppose that $(A, B)$ is $k_{\mathsf{c}}$-step controllable.
If $n \geq 2$ and $m \geq c_0 n$, then
for any $\Gamma' \in \mathsf{Sym}^n_{> 0}$:
\begin{align}
\mathbb{E}[ \norm{\hat{W}_{m,T} - W_\star}_{\Gamma'}^2 ] \leq c_1 \sigma_\xi^2 \cdot \frac{pn}{mT \cdot \underline{\lambda}(\Gamma_T(A, B), \Gamma')}.
\label{eq:lds_ls_many_traj_general_rate}
\end{align}
\end{mylemma}
\begin{proof}
Let $\Gamma_T := \Gamma_T(A, B)$.
By \Cref{stmt:lds_traj_small_ball},
\DLR{} satisfies the $(T, T, \Gamma_T, e, 1/2)$-TrajSB{}
condition.
We therefore invoke \Cref{stmt:upper_bound_general} with $k = T$ and $\underline{\Gamma} = \Gamma_T$.
In this case,
$\underline{\mu}$ from \eqref{eq:uMu_defn} simplifies to
$\underline{\mu} = \underline{\lambda}(\Gamma_T, \Gamma_T) = 1$,
and the requirement \eqref{eq:mTkn_requirements}
simplifies to $n \geq 2$ and
$m \geq c_0 n$.
Finally, the rate \eqref{eq:general_risk_bound} simplifies to
\eqref{eq:lds_ls_many_traj_general_rate}.
\end{proof}
\upperboundparameterrecovery*
\begin{proof}
Follows by invoking \Cref{stmt:upper_bound_many_trajs_helper}
with $\Gamma' = I_n$.
\end{proof}
\upperboundmanytrajectories*
\begin{proof}
Let $\Gamma_t := \Gamma_t(A, B)$ for $t \in \ensuremath{\mathbb{N}}_{+}$.
Invoking \Cref{stmt:upper_bound_many_trajs_helper} with $\Gamma' = \Gamma_{{T'}}$
yields the bound:
\begin{align*}
\mathbb{E}[L(\hat{W}_{m,T};{T'}, \PxA{A,B})] \leq c_1 \sigma_\xi^2 \cdot \frac{pn}{mT \cdot \underline{\lambda}(\Gamma_T, \Gamma_{{T'}})}.
\end{align*}
If ${T'} \leq T$, then $\underline{\lambda}(\Gamma_T, \Gamma_{{T'}}) \geq 1$
since $\Gamma_{T} \succcurlyeq \Gamma_{{T'}}$ by
\Cref{stmt:Gamma_t_is_monotonic}.
On the other hand, if ${T'} > T$,
by \Cref{prop:lower_bound_ratio},
$\underline{\lambda}(\Gamma_T, \Gamma_{{T'}}) \geq \frac{1}{8\gamma} \frac{T}{{T'}}$.
The claim now follows.
\end{proof}
\subsubsection{Few trajectory results}
\fewtrajrate*
\begin{proof}
Let $\Gamma_t := \Gamma_t(A, B)$ for all $t \in \ensuremath{\mathbb{N}}_{+}$.
By \Cref{stmt:lds_traj_small_ball},
for any $k \in \{1, \dots, T\}$, \DLR{}
satisfies the $(T, k, \Gamma_k, e, 1/2)$-TrajSB{} condition.
We will apply \Cref{stmt:upper_bound_general} with $\underline{\Gamma} = \Gamma_k$.
The quantity $\underline{\mu}$ from \eqref{eq:uMu_defn}
simplifies to $\underline{\mu} = \underline{\lambda}(\Gamma_k, \Gamma_k) = 1$.
By \Cref{prop:lower_bound_ratio},
we have that $\underline{\lambda}(\Gamma_k, \Gamma_T) \geq \frac{1}{8\gamma} \frac{k}{T}$.
Hence the requirement \eqref{eq:mTkn_requirements}
simplifies to $n \geq 2$ and
\begin{align}
\frac{mT}{kn} \geq c \log\left(\gamma' \frac{T}{k}\right), \quad \gamma' := \max\{e, \gamma\} \label{eq:requirement_v1}
\end{align}
for some universal positive constant $c$.
Thus, for \eqref{eq:requirement_v1} to hold, it suffices to require:
\begin{align}
\frac{T}{k} \geq \max\left\{ \frac{2 c n}{m} \log{\gamma'}, \frac{2cn}{m} \log\left(\frac{T}{k}\right)\right\}. \label{eq:requirement_v2}
\end{align}
As long as $2c n/m \geq 1$, then by \Cref{prop:invert_log_t_over_t},
\begin{align*}
\frac{T}{k} \geq \frac{4cn}{m} \log\left(\frac{8 c n}{m}\right) \Longrightarrow \frac{T}{k} \geq \frac{2cn}{m} \log\left(\frac{T}{k}\right).
\end{align*}
Hence, for \eqref{eq:requirement_v2} to hold, it suffices to require
\begin{align}
\frac{T}{k} \geq \frac{4c n}{m} \log\left(\frac{8 c\gamma' n}{m}\right). \label{eq:requirement_v4}
\end{align}
Based on \eqref{eq:requirement_v4}, we choose $k$ as:
\begin{align}
k = \bigfloor{\frac{T}{4cn/m \cdot \log(8 c\gamma' n/m)}}. \label{eq:choice_of_k}
\end{align}
To ensure that $k \geq 1$, we need to ensure that:
\begin{align}
mT \geq 4c n \log(8 c\gamma' n/m).
\end{align}
On the other hand, since $2cn/m \geq 1$, we have that:
\begin{align*}
\frac{4c n}{m} \log\left(\frac{8c \gamma' n}{m}\right) \geq 1,
\end{align*}
which ensures that $k \leq T$.
Thus, our choice of $k$ from \eqref{eq:choice_of_k}
ensures that \eqref{eq:requirement_v1} holds.
We now ready to invoke \Cref{stmt:upper_bound_general}
with $\Gamma' = \Gamma_{{T'}}$,
and conclude for a universal $c'$:
\begin{align}
\mathbb{E}[ L(\hat{W}_{m,T};{T'}, \PxA{A,B}) ] \leq c' \sigma_\xi^2 \cdot \frac{p n \log(e/\underline{\lambda}(\Gamma_k, \Gamma_T)) }{mT \underline{\lambda}(\Gamma_k, \Gamma_{{T'}})}. \label{eq:bound_general_v1}
\end{align}
First, we assume that ${T'} \leq k$.
By \Cref{stmt:Gamma_t_is_monotonic} we have $\Gamma_k \succcurlyeq \Gamma_{{T'}}$, and therefore
$\underline{\lambda}(\Gamma_k, \Gamma_{{T'}}) \geq 1$.
Equation \eqref{eq:bound_general_v1} yields:
\begin{align}
\mathbb{E}[ L(\hat{W}_{m,T};{T'}, \PxA{A,B}) ] \leq c' \sigma_\xi^2 \cdot \frac{ pn \log(e/\underline{\lambda}(\Gamma_k, \Gamma_T)) }{mT }
\label{eq:bound_Tnew_leq_T}
\end{align}
By \Cref{prop:lower_bound_ratio},
\begin{align}
\underline{\lambda}(\Gamma_k, \Gamma_T) \geq \frac{1}{8\gamma} \frac{1}{T} \bigfloor{\frac{T}{4cn/m \cdot \log(8c\gamma' n/m)}} \geq \frac{m}{64c \gamma n \log(8 c \gamma' n/m)}. \label{eq:bound_v1}
\end{align}
Plugging \eqref{eq:bound_v1} into \eqref{eq:bound_Tnew_leq_T},
and using the inequalities
$\log{x} \leq x$ for $x > 0$ and $\phi(a, x) \geq 1$
for all $a \geq 1$ yields, for another universal $c''$:
\begin{align*}
\mathbb{E}[ L(\hat{W}_{m,T}; {T'}, \PxA{A,B}) ] &\leq c' \sigma_\xi^2 \cdot \frac{pn}{mT} \cdot \log(e \cdot 64c\gamma n/m \cdot \log(8c \gamma' n/m)) \\
&\leq c' \sigma_\xi^2 \cdot \frac{pn \log( 512e \cdot (c \gamma' n/m)^2 )}{mT} \\
&\leq c'' \sigma_\xi^2 \cdot \frac{pn \log(\max\{\gamma n/m, e\} )}{mT} \\
&\leq c'' \sigma_\xi^2 \cdot \frac{pn \log(\max\{\gamma n/m, e\} )}{mT} \cdot \phi\left( \gamma, c_1 \frac{n \log(\max\{\gamma n/m, e\})}{m} \frac{{T'}}{T}\right).
\end{align*}
On the other hand, if ${T'} > k$,
then by \Cref{prop:lower_bound_ratio},
\begin{align}
\underline{\lambda}(\Gamma_k, \Gamma_{{T'}}) \geq \frac{1}{8\gamma} \frac{1}{{T'}} \bigfloor{\frac{T}{4cn/m \cdot \log(8c\gamma' n/m)}} \geq \frac{m}{64c \gamma n \log(8c \gamma' n/m)} \cdot \frac{T}{{T'}}. \label{eq:bound_v2}
\end{align}
Plugging \eqref{eq:bound_v1} and \eqref{eq:bound_v2} into
\eqref{eq:bound_general_v1} and using again the inequality
$\log{x} \leq x$ for $x > 0$ yields,
for a universal $c'''$:
\begin{align*}
&~~~~\mathbb{E}[ L(\hat{W}_{m,T}; {T'}, \PxA{A,B}) ] \\
&\leq c' \sigma_\xi^2 \cdot \frac{pn}{mT} \cdot \log(e \cdot 64c\gamma n/m \cdot \log(8c \gamma' n/m)) \cdot 64c\gamma n / m \cdot \log(8c \gamma' n/m ) \cdot \frac{{T'}}{T} \\
&\leq c''' \sigma_\xi^2 \cdot \frac{pn\log(\max\{\gamma n/m, e\})}{mT} \cdot \gamma \frac{n\log(\max\{\gamma n/m, e\}) }{m} \cdot \frac{{T'}}{T}.
\end{align*}
Furthermore, when ${T'} > k$, by choosing $c_1$ sufficiently large:
\begin{align*}
8 c \frac{n \log(8c\gamma' n/m)}{m} \frac{{T'}}{T} > 1 %
&\Longrightarrow c_1 \frac{n \log(\max\{\gamma n/m, e\})}{m} \frac{{T'}}{T} > 1 \\
&\Longrightarrow \gamma c_1 \frac{n \log(\max\{\gamma n/m, e\})}{m} \frac{{T'}}{T} = \phi\left( \gamma, c_1 \frac{n \log(\max\{\gamma n/m, e\})}{m} \frac{{T'}}{T}\right).
\end{align*}
The claim now follows.
\end{proof}
\upperboundindldsls*
\begin{proof}
Let $\Gamma_t := \Gamma_t(A, B)$ and $\Sigma_t := \Sigma_t(A, B)$
for $t \in \ensuremath{\mathbb{N}}_{+}$.
From \Cref{example:independent_gaussians},
we have that
\problemName{Ind\smallDash{}LDS\smallDash{}LS}{} satisfies the $(T, 1, \{ \Sigma_t \}_{t=1}^{T}, e, 1/2)$-TrajSB{} condition.
We will apply \Cref{stmt:upper_bound_general} with
$\underline{\Gamma} = \Gamma_T$, $k=1$, and $\Gamma' = \Gamma_{{T'}}$.
By \Cref{stmt:uMu_lower_bound_lds_marginals}, we have that:
\begin{align*}
\underline{\mu}(\{\Sigma_t\}_{t=1}^{T}, \Gamma_T) \geq \frac{1}{8e \gamma}.
\end{align*}
The requirement \eqref{eq:mTkn_requirements} simplifies to
$n \geq 2$ and $\frac{mT}{n} \geq c \log(\max\{\gamma,e\})$
for a universal constant $c$.
By \Cref{stmt:upper_bound_general},
for a universal $c'$:
\begin{align*}
\mathbb{E}[L(\hat{W}_{m,T}; {T'}, \PxA{A,B})] \leq c' \sigma_\xi^2 \cdot \frac{pn \log(\max\{\gamma,e\})}{mT \cdot \underline{\lambda}(\Gamma_T, \Gamma_{{T'}}) } \cdot 8 e \gamma.
\end{align*}
If ${T'} \leq T$, then $\underline{\lambda}(\Gamma_T, \Gamma_{{T'}}) \geq 1$
since $\Gamma_T \succcurlyeq \Gamma_{{T'}}$ by \Cref{stmt:Gamma_t_is_monotonic}.
On the other hand, if ${T'} > T$, then
by \Cref{prop:lower_bound_ratio},
$\underline{\lambda}(\Gamma_T, \gamma_{{T'}}) \geq \frac{1}{8\gamma} \frac{T}{{T'}}$.
The claim now follows.
\end{proof}
\paramrecoveryfewtrajrate*
\begin{proof}
The proof is identical to that of \Cref{thm:sublinear_trajectories_bound}
until \eqref{eq:bound_general_v1}, after which we set ${T'} = 1$
from which the result follows.
\end{proof}
\input{high_probability_upper_bounds} |
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